Trust and Credit in Organizations and Institutions: As Viewed from the Evolution of Cooperation 9789811949791, 9811949794

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Theoretical Biology

Mayuko Nakamaru

Trust and Credit in Organizations and Institutions As Viewed from the Evolution of Cooperation

Theoretical Biology Series Editor Yoh Iwasa, Kyushu University, Fukuoka, Japan

The “Theoretical Biology” series publishes volumes on all aspects of life sciences research for which a mathematical or computational approach can offer the appropriate methods to deepen our knowledge and insight. Topics covered include: cell and molecular biology, genetics, developmental biology, evolutionary biology, behavior sciences, species diversity, population ecology, chronobiology, bioinformatics, immunology, neuroscience, agricultural science, and medicine. The main focus of the series is on the biological phenomena whereas mathematics or informatics contribute the adequate tools. Target audience is researchers and graduate students in biology and other related areas who are interested in using mathematical techniques or computer simulations to understand biological processes and mathematicians who want to learn what are the questions biologists like to know using diverse mathematical tools.

Mayuko Nakamaru

Trust and Credit in Organizations and Institutions As Viewed from the Evolution of Cooperation

Mayuko Nakamaru Department of Innovation Science, School of Environment and Society Tokyo Institute of Technology Minato-ku, Tokyo, Japan

ISSN 2522-0438 ISSN 2522-0446 (electronic) Theoretical Biology ISBN 978-981-19-4978-4 ISBN 978-981-19-4979-1 (eBook) https://doi.org/10.1007/978-981-19-4979-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022, corrected publication 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The earth’s wide variety of animals all interact with their species in various ways, in essence, forming societies that differ greatly from one species to another. Being highly social animals, humans live in groups for the entirety of their lives, as they generally do not enjoy living in solitude. To obtain such sociality, human beings have developed various capabilities required for social living through evolution, as did other animals that show strong social behaviors. In sum, these capabilities are directed to facilitate “cooperation among two or more individuals.”

Abilities Unique to Humans Since early times, humans have been considered distinct from other animals, as they are equipped with various capacities that no other animals have: language, culture and its inheritance, learning, teaching others, mathematical abilities, triadic relationships or representations, inferring others’ intentions, empathy, morality, and transitive inference (TI). However, recent experiential studies have noted that other animals can have similar capabilities. For instance, in one particular experiment on tufted capuchins, the monkeys were given a cucumber as a reward for completing an assignment. Interestingly, they stopped working on the assignment once they saw another monkey being given a grape as a reward for completing the same task, as they apparently considered grapes to be far more attractive. They also seemed particularly annoyed with the inequality, as if to say, “I can’t deal with this!” (Brosnan and de Waal 2003). Overall, this experiment indicated that animals are also adverse to inequality. In fact, inequality aversion is considered an origin of morality. TI refers to the ability to conclude that A is always more desirable than C (A > C), when A is more desirable than B (A > B) and B is more desirable than C (B > C). This ability has also been considered unique to adult humans, but experiential studies in the 1970s showed that children are also capable of TI. The v

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results prompted further research on other animals, and the findings suggested that some aspects of human capabilities are similar to those of other social animals. Subsequently, in the 2000s, the social complexity hypothesis emerged (Vasconcelos 2008), arguing that social animals, which form and live in large groups, evolved to have higher cognitive capacities to address the complexities of social interactions and relationships. The hypothesis concluded that they became capable of TI and other highly cognitively demanding reasoning for these reasons. Other researches that used evolutionary game theory to study TI’s evolution also obtained results that supported the social complexity hypothesis (Nakamaru and Sasaki 2003; Doi and Nakamaru 2018). Furthermore, triadic relationship or representation argues that if one can appropriately recognize the relationship between the self, other, and object, one can comprehend that the other is perceiving the same object as oneself. As a result, one can empathize with others. Some also argue that this cognitive ability is necessary for linguistic development (Kikusui 2019). Although it was originally considered unique to humans, research with chimpanzees has suggested their potential for having triadic relationships (Yamamoto et al. 2012). Similarly, the theory of mind (ToM), an ability to infer others’ intentions, was also initially considered unique to humans. Whether a person has ToM or not can be examined using the Sally–Anne task: Sally puts a doll in Box A and leaves, but once she is gone, Anne takes the doll out and puts it in an adjacent box (Box B), upon which Sally returns. In which box will Sally look for the doll? Children below the age of three 3 typically answer Box B and cannot correctly solve the test. However, the majority of 5-yearolds can answer Box A. Regarding animals, a previous experiential study revealed that chimpanzees and orangutans performed well in this test, albeit modified to suit the monkeys (Krupenye et al. 2016). This indicates the existence of ToM in some primates other than human beings. As such, future experiments on other animals, with suitable modifications, may reveal the prevalence of other abilities once considered unique to humans while clarifying where the distinction lies between human beings and other animals.

Why Animals Live in Groups As the aforementioned abilities are essential for forming and living in groups, what are the unique features of human groups? The major difference with other animals seems to lie in the group structures. Many animals form and live in groups, which greatly affects reproduction (Davis et al. 2012; Rubenstein and Alock 2019). For instance, lions form prides for hunting purposes and to ensure reproduction (e.g., the alpha male lions breed within the pride). In harem structure groups, such as those formed by gorillas, a single dominant male reproduces with all the females in the harem, while the other males cannot mate at all. Additionally, other animals show a variety of patterns (mating systems), including lek, scramble, polyandry, and polygyny (Table 9.1 in Davies et al. (2012)). For people, forming groups for reproduction,

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such as families, is also important, but groups consisting of unrelated members have a similar significance. This also applies to certain animals. For example, sardines and other fish form schools (e.g., Couzin et al. 2005, 2011). The school has a dilution effect against predators. In other words, whereas a single fish can be easily targeted, most of the fishes in a school are less likely to be preyed upon (dilution effect), although a few still become victims (Davis et al. 2012). There are other hypotheses to explain social defense in groups besides the dilution effect: the confusion effect and selfish herd (Rubenstein and Alock 2019). In the confusion effect hypothesis, predators cannot concentrate on one single prey if there are many preys moving around. In the selfish herd hypothesis, individuals can reduce the predation risk because other members in groups may be attacked by predators. Moreover, insects and birds, such as sparrows and starlings, also form groups, not for reproduction, but presumably to achieve this aforementioned social defense against predators (Davis et al. 2012; Rubenstein and Alock 2019). This purpose of forming groups may have also been applicable to humans at some point during the course of evolution. For instance, human beings may have found themselves in situations where being in a group was an advantageous tactic to protect themselves against predators. In modern society, however, predators rarely attack humans, making social defense an unlikely reason for group formation.

Groups of Human Beings Groups based on families bonded by kinship, which cultural anthropologists have extensively studied, are profoundly relevant for reproduction. As with groups of animals, the evolutionary ecology framework can explain these human kin groups, in which each individual considers how to maximize the amount of offspring. Collective behavior is another group mentality that humans reflect through their actions. Accordingly, many studies have discussed crowd behavior or fads involving hostile, destructive, and expressional actions. This applies to the panic that ensues when evacuating a building due to a fire, bank runs, acquisition madness (e.g., the housing bubble in Japan), riots, and hostile or unsystematic demonstrations (Coleman 1990). The sociologist Coleman (1990) listed the three sole characteristics which are common to collective behaviors: (1) a number of people carry out the same or similar simultaneous actions; (2) the exhibited behaviors are transient or continually changing (no equilibrium); and (3) there is some kind of dependency among the actions (they are not acting independently). Collective behaviors have been studied across various disciplines, including sociology and social psychology, and studies have recently been examining the relationship between the Internet, social networking services, and collective behaviors. In terms of other animals, these characteristics, which may be relevant to collective behaviors, are also observable in chickens and mice. While it is an interesting topic, as human beings may be exhibiting a specific type of collective behavior under the influence of a

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psychological mechanism unique to human beings, this topic is beyond the scope of our target research direction.

Organizations and Institutions Group organizations, rules, and institutions are another example of group behaviors which are specific to human beings. One difference between nonhuman animal groups and organizations in a human society seems to be that human members can understand an organization’s objectives and have hopes for the organization. The organizational sociologist Scott (1981) defined an organization as being “collectivity oriented to the pursuit of relatively specific goals and exhibiting a relatively highly formalized social structure” (p. 21), and other researchers also define an organization in the similar way (e.g., Yeager 1999). This definition also shows that the goals and objectives are critical for organizations in a human society. In addition, diverse types of organizations exist in human society, which are interconnected and have hierarchical and multilayered structures (e.g., Turchin 2016; Tverskoi et al. 2021). It is also unlikely that animals understand the bigger picture of aggregating various organizations (i.e., meta-concepts). Humans are seemingly the only beings that comprehend such concepts. Whether other animals also set collective objectives, have hopes for a group, or decide to join or leave an organization by judging its members’ objectives, expectations, or trustworthiness is yet to be discovered, to the best of my knowledge. The existence or absence of such objectives or expectations will eventually be clearer with the development of science. There are some definitions of institutions; Aoki (2001) defines them as “selfsustaining system of shared beliefs.” North (1990) defines them as “the rules of the game in a society or, more formally, the humanly devised constraints that shape human interaction” (p. 3). According to Yeager (1999), based on the definition of North (1990), institutions consist of three factors: formal rules, informal rules, and enforcement. Formal rules are codified ones. Informal rules are not codified and include culture, behavioral norms, moral code, and informal regulations. Enforcement makes people follow the rules. To the best of my knowledge, no animals, plants, and bacteria invent and develop institutions. The ability of triadic relationship or representation or ToM is required to share rules with others, and the cognitive ability to invent and understand the idea of codification as well as the language ability is also needed. However, these abilities have not developed enough in animals, plants, and bacteria, except humans. Punishment to promote cooperation is observed in animals, plants, and bacteria (e.g., Ratnieks and Wenseleers 2008), but it cannot be considered “enforcement” in this context because there is no scientific evidence that punishments carried out by animals, plants, and bacteria force their members to abide by codified or informal rules. I have no idea whether informal rules might exist in some primate societies or not, even though some have developed ToM to a primitive level.

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Trust in Organizations and Institutions For an organization or an institution, trust and confidence are highly significant, specifically with regard to members’ perceptions of the organization, as well as between members, between organizations, and between members and organizations. Organizations and institutions cannot be sustainable without trust and confidence. Fukuyama (1995) mentioned that trust is a more essential factor than rationality and efficiency in the economic life because all economic activities are based on groups rather than individuals. He considered that trust is a competence of people who collaborate together in organizations or groups. In Japanese, trust is often translated into either shinrai (信頼) or shiyo (信用). In the same manner, credit is often translated into shinyo (信用) and sometimes translated into shinrai (信頼) in Japanese. It depends on the context. They are used almost interchangeably in everyday conversations. You could say that a person is not trustworthy or that a person is not credible in the same context. Thus, there is virtually no difference in their everyday usage in Japanese. However, not shinrai but shinyo is often used in the financial industry. For example, credit union is translated into shinyo-kinko; credit association, shinyo-kumiai; and margin transaction or credit transaction, shinyo-torihiki. Furthermore, Toshio Yamagishi made a significant contribution to social psychology and conducted a variety of studies based on general trust. Yamagishi and Yamagishi (1994) distinguished confidence from trust and defined that trust is “expectation of goodwill and benign intent” and confidence is “expectation of competence” (p. 131). Furthermore, he defined general trust as “a brief in the benevolence of human nature in general and thus is not limited to particular objects,” and general trust is distinguished from knowledge-based trust which is defined as that “knowledge-based trust is limited to particular objects (people or organizations)” (Yamagishi and Yamagishi 1994, p. 139). When it comes to trust among members in an organization or trusting the organization, it does involve expectations towards not only another’s intentions, but also their competence. Regardless of intentions, competence is initially required to do anything as an organization. For instance, even a bank full of friendly employees will not be able to earn a reputation or any form of trust if they are incapable of appropriately conducting banking services and continue to make errors in interest calculations. At the same time, a bank’s reputation, stemming from customers, could also lower if most of its employees’ competences are highly prioritized over their intentions. Mayer et al. (1995) discussed organizational trust. Reviewing papers about trust, they defined trust as “the willingness of a party to be vulnerable to the actions of another party based on the expectation that the other will perform a particular action important to the trustor, irrespective of the ability to monitor or control that other party” (p. 712). They also pointed out that the three factors of trustworthiness often appear, ability, benevolence, and integrity, and taking risk in relationship is essential for trust. Therefore, trust in Mayer et al. (1995) rather than trust in Yamagishi and

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Yamagishi (1994) may be suitable for this book. However, Mayer et al. (1995) only focused on trust between two parties, a trustor and a trustee, in the organization. In my opinion, it is not enough to understand human organizations and institutions based on the dyadic trust. Moreover, various disciplines have various definitions of trust. Rousseau et al. (1998) reviewed various definitions of trust in various disciplines and found out that the common features of various definitions of trust are risk and interdependence. According to the sociologist Coleman (1990), systems of trust can be classified into a mutual trust, intermediaries in trust, and third-party trust (Chap. 8 in Coleman (1990)). Mutual trust applies to a situation where two parties trust each other, and the system of trust can, as a result, be established within a larger group of two mutually trusting parties. The second category, a situation involving intermediaries in trust, can be exemplified as follows: If person B trusts person A and person C trusts person B, when person A needs help from person C, C will help A with B acting as the intermediary. In a larger group of people, a system of trust is made possible by the presence of such intermediaries. Similar to this situation is the last classification, a third-party trust, but the intermediary still plays the most active role. In an attempt to amalgamate these definitions and perspectives, this book assumes the following: In a group of human beings with more than three parties, various rules are established to stabilize trust and confidence among members in groups and facilitate cooperation among the members towards a common objective. Then, those rules are turned into institutions and the group develops into an organization. They eventually form the complex institutions and organizations observed today. Overall, this book’s purpose is to enhance our understandings of institutions and organizations based on trust and confidence by analyzing the conditions of their early-stage establishment, especially focusing on the early-stage credit or insurance system. Coleman (1990) used “trust” to describe not only the system of mutual trust but also the system consisting of more than three parties, while “trust” implies “dyadic trust” in many papers, especially related to game theory and evolutionary game theory. Additionally, this book is concerned with the development of an organization’s or institution’s trust. As an organization is made up of three or more people, this book assumes that trust and confidence among members are a crucial cornerstone for an organization to earn credibility in its early stages. Once that becomes stable, it leads to building an organization’s credibility, which requires cooperation from members and a good exterior reputation. As I mentioned in the early part of this section, human beings especially have acquired the skill to understanding the triadic relationship or representation. As a result, I consider that trust and confidence among three or more individuals are possible, so that organizations can establish in our society. Therefore, “trust” is the most essential term to represent this book. As I mainly focus on the research of the “evolution” of the early-stage insurance or credit system as a derivation of mutual cooperation and trust in this book, building “credit” is also a key here. Therefore, not only “trust” but also “credit” is included in the title of this book.

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Chapters This book introduces many different studies, including one on trust and confidence employing evolutionary game theory, but not all studies presented across the chapters directly relate to building an organization’s trust and confidence, as some topics are still under investigation. On that note, in the upcoming chapters, we will elaborate on the following subjects. Chapter 1 presents the evolution of cooperation as the book’s theoretical background, highlighting this significant research topic’s ideas in the field of evolutionary ecology and some of its typical research examples. As underlying cooperative relationships between people support trust and confidence, we cannot ignore the importance of this aspect for enhancing understanding. It is also the foundation of a credit system. Chapters 2 and 3 present how the spatial structure and punishment to defectors promote the evolution of cooperation based on the author’s previous studies. These chapters do not deal with trust and confidence explicitly; however, they may be related to the evolution of cooperation and vice versa. In Chap. 3, we focus on punishment, which is a kind of enforcement, one of the factors institutions consist of. These theoretical studies would contribute to understanding the cultural or social evolution of institutions. Chapter 4 introduces studies on primordial insurance organizations and on those that operate under customary institutions, such as tanomoshi-ko (a rotation savings and credit association: ROSCA), a scheme that used to prevail all over the world and still exists in some countries. Tanomoshi-ko became the foundation of many banks and credit unions, including Grameen Bank in Bangladesh. For example, gojo-ko, founded by Kinjiro Ninomiya in 1814, is the first credit union in Japan and is considered to be based on ROSCAs. Two Germans, Friedrich Wilhelm Raiffeisen and Franz Hermann Schulze-Delitzsch, are well known as the founders of the modern credit unions, but they founded the credit unions around 50 years after the foundation of gojo-ko. Further information about Kinjiro Ninomiya appears in the Epilogue. Understanding the conditions under which tanomoshi-ko and other customary organizations can survive with stability will enable us to explore the evolution and development of credit systems. Chapter 5 presents the interviews we conducted regarding tanomoshi-ko in Sado Island located at the north part of Japan, and the subject experiments undertaken to verify the hypothesis posed in Chap. 4, and examines the validity of our simulations. Chapter 6 addresses the formation of a group, as an introduction to the study on the formation of organizations. People choose new members when forming a group or evaluate the group upon joining it. Thus, the presented study examines the type of decision criteria for choosing new members or evaluating groups that could promote the evolution of the group’s cooperation. This particular study can be further considered as research that explores the conditions for an organization to survive when adding new members or when members choose to join. It can also be applied to other researches on forming members in various sections inside an organization.

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In sum, it can be translated into a study on an organization’s credibility and trustworthiness. Chapter 7 draws on Sudgen’s (1986) study, which examined the primordial organization of insurance systems in a mathematical model and explained the latter’s evolutionary simulations. Some resemble the structure of tanomoshi-ko, but others do not. The analysis shows that a relationship of more than three parties is key, due to the limitations imposed by the rules. Most evolutionary game studies concerning cooperation only focused on two-party relationships, but this is not sufficient. Thus, we conclude that a relationship between more than three parties is essential for understanding human organizations and society. While we researched these things, we came to an idea of studying a credit system from the standpoint of mutual aid and tracing it back to the origin of financial systems. Both credit systems and “currency” are inseparable. The topic about the evolution of currency seems fairly complex, and you might think that it is widely different from the stance of this book. There is a book by Sehgal (2015) titled Coined: The Rich Life of Money and How Its History Has Shaped Us, which situates its topic on currency. The book was written by an investment banker, not a researcher. What is interesting about this book is that its first chapter introduces some studies from the field of evolutionary ecology and brain neuroscience to draw on his argument about the origin of exchange. It even cites Axelrod’s tournament of the iterated prisoner’s dilemma game. Furthermore, Chap. 4 touches on Japanese traditions of goshugi (wedding gift money) and koden (condolence money) paid at ceremonial occasions. Sehgal’s approach is extremely close to this book. Reading his book, I was reassured that my research approach was not so far off the mark. Chapter 8 introduces studies on an organization’s division of labor. Trust and confidence are important for dividing labor and proceeding with the job required for a particular objective. For example, this can refer to this division of production processes among three people instead of one: Person A first makes part of a product and then hands it to person B to complete other parts, who finally gives it to person C to complete the job. This can apply to the production of kimono, as a traditional craft, and car manufacturers. In this context, even if person A works hard on the product, its quality is compromised if person B cuts corners. Similarly, even if persons A and B make an excellent quality product, person C could jeopardize it with laziness. Therefore, person A will likely hire person B, who is a credible craftsman, and person B will hire a trustworthy person C. As illustrated here, the division of labor is made possible through underlying trust and confidence. Therefore, this book introduces the studies that explain the role of penalties in maintaining credibility and trustworthiness in organizations and how penalties promote credibility and trustworthiness and maintain the division of labor. Chapter 9 presents some fundamental studies concerning false or fake gossip and credibility, focusing on the effects of gossip exchanges between players. As a minimum, a triadic relationship is required to spread a gossip about others; player A tells gossip about player B to player C, and player A works as a mediator. Perhaps, thanks to the ability to understand the triadic relationship or representation as well as language ability, gossiping to others is possible.

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It is possible to portray the player as an organization; there is still a certain gap between this research and gossip research related to organizations. We wish to eventually expand the study presented in Chap. 9 to one on organizations. Chapters 10 and 11 present how cooperation among people promotes sustainability of biological resources such as rice and Chinese ingredients for bird’s nest soup. There are rules for sustainable use of ecological systems, which have been recognized by community members, and implicit institutions shared by members have been established and maintained there for a long time, even though the informal institution and membership have not been authorized officially by local governments. Therefore, Chaps. 10 and 11 are regarded as a study of institutions and organizations. These two studies do not seem to be directly related to trust and confidence in organizations. If farmers do not trust each other, no one would join informal organizations and then they would collapse; if harvesters do not trust each other, none of them would abide by informal institutions and they would vanish. In reality, there is a very complicated division of labor to harvest and sell the ingredients for bird’s nest soup, which influences the whole social and economic system for bird’s nest soup. Farming rice not only is an individual task but is also influenced by social, economic, and political institutions in a specific area. On this point, trust and confidence are inextricably connected with sustainability of biological resources. The last chapter presents a brief introduction to the ongoing research project related to the aim of this book and describes Kinji Ninomiya, who was a founder of the world’s first mutual-aid credit system during the Edo period in Japan. References Aoki M (2001) Towards a comparative institutional analysis. MIT Press, Cambridge, MA Brosnan SF, de Waal FBM (2003) Monkeys reject unequal pay. Nature 425(6955): 297–299. https://doi.org/10.1038/nature01963 Coleman JS (1990) Foundations of social theory. Belknap Press of Harvard University Press, Cambridge, MA Couzin ID, Krause J, Franks R, Levin SA (2005) Effective leadership and decisionmaking in animal groups on the move. Nature 433:513–516 Couzin ID, Ioannou CC, Demirel G, Gross T, Torney CJ, Hartnett A, Conradt L, Levin SA, Leonard NE (2011) Uninformed individuals promote democratic consensus in animal groups. Science 334:1578–1580 Davis NB, Krebs JR, West SA (2012) An introduction to behavioural ecology, 4th edn. Wiley-Blackwell, Hoboken, NJ Doi K, Nakamaru M (2018) The coevolution of transitive inference and memory capacity in the hawk-dove game. J Theor Biol 456:91–107. https://doi.org/10. 1016/j.jtbi.2018.08.002 Fukuyama F (1995) Trust: the social virtues and the creation of prosperity. Free Press, Florence, MA

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Kikusui K (2019) The origin of society: What does the animal group mean? Kyoritsu Shuppan Co., Ltd., Tokyo, Japan Krupenye C, Kano F, Hirata S, Call J, Tomasello M (2016) Great apes anticipate that other individuals will act according to false beliefs. Science 354(6308):110–114. https://doi.org/10.1126/science.aaf8110 Mayer RC, Davis JH, Schoorman FD (1995) An integrative model of organizational trust. Academy of Management Review 20:709–734. Nakamaru M, Sasaki A (2003) Can transitive inference evolve in animals playing the hawk-dove game? J Theor Biol 222(4):461–470. https://doi.org/10.1016/S00225193(03)00059-6 North D (1990) Institutions, institutional change and economic performance. Cambridge University Press, New York Ratnieks FLW, Wenseleers T (2008) Altruism in insect societies and beyond: voluntary or enforced? Trends in Ecology and Evolution 23(1):45–52. https:// doi.org/10.1016/j.tree.2007.09.013 Rousseau DM, Sitkin SB, Burt RS, Camerer C (1998) Not so different after all: a cross-discipline view of trust. Acad Manage Rev 23(3):393–404. https://doi.org/ 10.5465/Amr.1998.926617 Rubenstein DR, Alock J (2019) Animal behavior, 5th edn. Oxford University Press, New York Scott WR (1981) Organizations: rational, natural, and open systems. Prentice Hall, Upper Saddle River, NJ Sehgal K (2015) Coined: The rich life of money and how its history has shaped us. Grand Central Publishing, New York Sugden R (1986) The economics of rights, co-operation and welfare. Basil Blackwell, Oxford Turchin P (2016) Ultrasociety: how 10,000 years of war made humans the greatest cooperators on Earth. Beresta Books, Chaplin, CT Tverskoi D, Senthilnathan A, Gavrilets S (2021) The dynamics of cooperation, power, and inequality in a group-structured society. Sci Rep-Uk 11(1):18670. https://doi.org/10.1038/s41598-021-97863-7 Vasconcelos M (2008) Transitive inference in non-human animals: an empirical and theoretical analysis. Behav Process 78(3):313–334. https://doi.org/10.1016/j. beproc.2008.02.017 Yamagishi T, Yamagishi M (1994) Trust and commitment in the United-States and Japan. Motiv Emotion 18(2):129–166. https://doi.org/10.1007/Bf02249397 Yamamoto S, Humle T, Tanaka M (2012) Chimpanzees’ flexible targeted helping based on an understanding of conspecifics’ goals. P Natl Acad Sci USA 109(9): 3588–3592. https://doi.org/10.1073/pnas.1108517109 Yeager TJ (1999) Institutions, transition economies, and economic development. Westview Press, Boulder, CO Minato-ku, Tokyo, Japan

Mayuko Nakamaru

Acknowledgments

I would like to thank all collaborators in our studies presented in this book: Yoh Iwasa, Simon A. Levin, Akira Sasaki, Kazuto Doi, Takuya Sekiguchi, Hajime Shimao, Ulf Dieckmann, David G. Rand, Hisashi Ohtsuki, Shimpei Koike, Masahiro Tsujimoto, Tokinao Otaka, Ken-ichi Shimomura, Takehiko Yamato, Akira Yokoyama, Hayato Shimura, Yoko Kitakaji, Susumu Ohnuma, Masakado Kawata, Motohide Seki, Joung Hun Lee, Ryo Yamaguchi, Hiroyuki Yokomizo, Ayumi Onuma, Tomoki Takada, Souichiro Oishi, Shonan Matsumura, Kana Matsumura, Kazuo Ishizuka, Lim Chan Koon, Hiroo Sasaki, and Koichi Takase. I thank Masayuki Nakamura, the editor at Springer Nature, Japan, for his helpful support.

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Contents

Part I 1

2

Introduction to the Evolution of Cooperation

What Is “The Evolution of Cooperation“? . . . . . . . . . . . . . . . . . . . 1.1 Cooperation in Our Lives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What Is Evolutionary Game Theory? . . . . . . . . . . . . . . . . . . . . 1.3 Evolutionary Stable Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Games Between Two Players . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Prisoner’s Dilemma Game . . . . . . . . . . . . . . . . . . . . . 1.4.2 Chicken Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Snowdrift and Blizzard Game . . . . . . . . . . . . . . . . . . . 1.4.4 Which Is an ESS, Cooperation or Defection? . . . . . . . . 1.5 Why We Can Analyze Our Society by Means of Evolutionary Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 How to Make a Mathematical Model or an Agent-Based Model of Our Society by Means of Evolutionary Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The Mechanisms to Promote the Evolution of Cooperation Between Bilateral Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Kin Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Group Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 Direct Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.4 Indirect Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.5 Social Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.6 Punishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Games Among Three or More Players . . . . . . . . . . . . . . . . . . . 1.9 The Application of Evolutionary Game Theory to Our Society . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 13 13 15 17 19 20 21 23 25

The Evolution of Cooperation in a Lattice-Structured Population Under Two Different Updating Rules . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Completely Mixed Population . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 33

3 3 4 5 6 6 7 8 8 9

10

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2.3

One-Dimensional Lattice Under the Score-Dependent Viability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Computer Simulations on a One-Dimensional Lattice Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Dynamics of Density . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Invasion Success Probability in the One-Dimensional Lattice Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 The Pair-Edge Method . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Two-Dimensional Lattice Under the Viability Model . . . . . . . . . 2.4.1 Mean-Field Approximation in the Two-Dimensional Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Pair Approximation in the Two-Dimensional Lattice Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Score-Dependent Fertility Model . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Computer Simulations on a One-Dimensional Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Mathematical Analyses in a One-Dimensional Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Two-Dimensional Model . . . . . . . . . . . . . . . . . . . . . . 2.6 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 The Lattice-Structured Population vs. Completely Mixing Population . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Comparison Between the Viability Model and the Fertility Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Coexistence of Cooperators and Defectors . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

The Effect of Peer Punishment on the Evolution of Cooperation . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Two Updating Rules, Viability Model and Fertility Model . . . . . 3.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The Completely Mixing Model . . . . . . . . . . . . . . . . . . 3.2.3 The Lattice Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Expansion of This Study: Do Empty Sites Influence the Evolution of Cooperation and Punishment? . . . . . . 3.3 If Anti-Social Punishers Exist in the Population . . . . . . . . . . . . 3.3.1 Strategies and Payoffs . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Z-Mixed Population Model . . . . . . . . . . . . . . . . . . . . . 3.3.3 Lattice Model When Anti-Social Punishment Is Allowed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Which Promotes the Evolution of Cooperation, Strict or Graduated Punishment? . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 36 38 42 45 46 46 46 48 50 50 51 53 55 55 56 57 58 61 61 65 65 66 69 72 74 75 75 76 78 80

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3.4.1 What Is Graduated or Strict Punishment? . . . . . . . . . . . 3.4.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Spiteful Behavior in Social Sciences and Evolutionary Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Graduated or Strict Punishment in Society . . . . . . . . . . 3.5.3 Anti-Social Punishment in Society . . . . . . . . . . . . . . . . 3.5.4 Other Types of Punishment Beside Peer Punishment . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II 4

5

80 82 85 91 91 93 94 95 96

Cooperation, Trust, and Credit in the Early-Stage Mutual-Aid Systems

Rotation Savings and Credit Associations (ROSCAs) as Early-Stage Credit Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Models and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Model 1: Baseline Evolutionary Simulation Model of Rotating Indivisible Goods Game . . . . . . . . . 4.2.2 Results of Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Model 2: Rotating Indivisible Goods Game with the Peer Selection Rule . . . . . . . . . . . . . . . . . . . . 4.2.4 Results of Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Model 3: A Forfeiture Rule Is Introduced . . . . . . . . . . . 4.2.6 Results of Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.7 Effect of Labeling Rules . . . . . . . . . . . . . . . . . . . . . . . 4.2.8 Effect of Reputation Levels . . . . . . . . . . . . . . . . . . . . . 4.3 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 110 110 111 115 116 117 118

Tanomoshi-ko Field Study and Subjective Experiment . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Ko on Sado Island . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Tanomoshi-ko in Fukura-District . . . . . . . . . . . . . . . . . 5.2.2 Tanomoshi-ko in Ogi-District . . . . . . . . . . . . . . . . . . . 5.2.3 Nenbutsu-ko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Michibushin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Summary: Ko in Sado . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Experimental Study of ROSCA . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Theoretical Predictions . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Conclusions of the Experiment . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 123 123 126 130 133 134 134 135 141 142 151 156

103 103 107 107 108

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6

7

Contents

Who Does a Group Admit into Membership or Which Group Does a Player Want to Join? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Mutual-Aid Game as an Early-Stage Insurance System . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Eight Strategy Sets Are Categorized into Two . . . . . . . 7.3.2 Calculating the Expected Payoffs . . . . . . . . . . . . . . . . 7.3.3 What Happens When S-J Players Are the Majority? . . . 7.3.4 Can Rare S-J Players Invade the Population Occupied by AllD Players? . . . . . . . . . . . . . . . . . . . . . 7.3.5 The Effect of Large Group Size . . . . . . . . . . . . . . . . . . 7.3.6 Reputation vs. Experience . . . . . . . . . . . . . . . . . . . . . . 7.3.7 Why Cannot Conditional Cooperators Except S-J Players Invade the Population in n
4? . . . . . . . . . 7.3.8 Perception and Implementation Errors . . . . . . . . . . . . . 7.3.9 If Pure Cooperators Are Added in the Population . . . . . 7.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Comparison with the Previous Studies About the Mutual-Aid Game . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 The Mutual-Aid Game as an Institution . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part III 8

159 159 161 164 169 171 173 173 175 178 178 179 182 183 185 186 186 188 190 191 192 193 194

Cooperation, Trust, and Credibility in Society

Cooperation and Punishment in the Linear Division of Labor . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 The Linear Division of Labor . . . . . . . . . . . . . . . . . . . 8.1.2 The Industrial Waste Treatment Process in Japan . . . . . 8.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Baseline System in the Three-Role Model . . . . . . . . . . 8.2.2 Actor Responsibility System in the Three-Role Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Producer Responsibility System in the Three-Role Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Comparison with the Results of Kitakaji and Ohnuma (2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Comparison with Empirical Reality . . . . . . . . . . . . . . .

197 197 199 200 202 203 207 208 209 213 216

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8.3.3

Comparison Between the Three-Role and Two-Role Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 9

Can Cooperation Evolve When False Gossip Spreads? . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Comparison with Previous Studies About Indirect Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Comparison with Nakamaru and Kawata’s Gossip Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Unification of Direct and Indirect Reciprocity . . . . . . . 9.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Definition of the P-Score . . . . . . . . . . . . . . . . . . . . . . 9.2.3 The Giving-Game Session . . . . . . . . . . . . . . . . . . . . . 9.2.4 The Gossip Session . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.5 The Updating Rule Through Generations . . . . . . . . . . . 9.2.6 Parameter Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Competition Among Gossiping Reciprocators with Various Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Fair Gossipers with Various Criteria for Giving-Games (qG ¼ qB ¼ 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Biased Gossipers (qG ¼ qB ¼ k) . . . . . . . . . . . . . . . . . 9.4 Effects of Different Types of False Gossip on the Evolution of Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 ZDISCs Versus Non-gossiping ALLDs . . . . . . . . . . . . 9.4.2 ZDISCs Versus Fairly Gossiping ALLDs . . . . . . . . . . . 9.4.3 ZDISCs Versus Pure Self-Advertising ALLDs . . . . . . . 9.4.4 ZDISCs Versus ALLB-ALLDs . . . . . . . . . . . . . . . . . . 9.4.5 ZDISCs Versus ALLG-ALLDs . . . . . . . . . . . . . . . . . . 9.5 Effects of Selecting Gossip Based on the Trustworthiness of Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 ZDISCs Versus Fairly Gossiping ALLDs . . . . . . . . . . . 9.5.2 ZDISCs Versus Pure Self-Advertising ALLDs . . . . . . . 9.5.3 ZDISCs Versus ALLB-ALLDs . . . . . . . . . . . . . . . . . . 9.5.4 ZDISCs Versus ALLG-ALLDs . . . . . . . . . . . . . . . . . . 9.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part IV 10

223 223 224 226 227 227 228 229 229 230 231 232 232 233 235 236 239 244 245 249 250 251 253 253 254 254 255 258

Ecological Sustainability, Institutions, and Cooperation

Field Abandonment Problem in Rice Paddy Fields . . . . . . . . . . . . . 261 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 10.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

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10.2.1 Baseline Assumption in a Well-Mixed Population . . . . 10.2.2 Consideration of Spatial Structure . . . . . . . . . . . . . . . . 10.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Simulations for a Well-Mixed Population . . . . . . . . . . . 10.3.2 Simulations for a Spatially Structured Population . . . . . 10.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Method of Validating the Model Results . . . . . . . . . . . 10.4.3 Game Structure and Its Effect on the Dynamics . . . . . . 10.4.4 Application of the Current Model and Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.5 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

264 269 271 271 272 280 280 282 282 283 283 284

Ecological Features Benefiting Sustainable Harvesters in SocioEcological Systems: A Case Study of Swiftlets in Malaysia . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 An Appropriate Example: Swiftlets’ Nests in Sarawak, Malaysia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Comparison with the Previous Theoretical Studies About the Social and Ecological Dynamics . . . . . . . . . 11.1.3 The Findings of This Chapter . . . . . . . . . . . . . . . . . . . 11.2 Models and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Model 1: The Baseline Model . . . . . . . . . . . . . . . . . . . 11.2.2 Result 1: The Population Dynamics of Swiftlets in the Habitat Without Harvesters . . . . . . . . . . . . . . . . 11.2.3 Model 2: The Open-Access Model . . . . . . . . . . . . . . . 11.2.4 Result 2: The Results of the Open-Access Model . . . . . 11.2.5 Model 3: The Property Rights Model . . . . . . . . . . . . . . 11.2.6 Result 3: Results of the Property Rights Model . . . . . . 11.3 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Application of Our Model . . . . . . . . . . . . . . . . . . . . . . 11.3.2 One Direction of Our Future Work . . . . . . . . . . . . . . . 11.3.3 The Effect of the Price of a Nest . . . . . . . . . . . . . . . . . 11.3.4 Back to the Reality About Swiftlets . . . . . . . . . . . . . . . 11.3.5 Comparison with Theoretical Studies About SocioEcological Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

310 311

Correction to: Trust and Credit in Organizations and Institutions . . . . .

C1

Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Searching for tanomoshi-ko in Micronesia . . . . . . . . . . . . . . . . . . . . . The World’s First Credit System Was Established in Japan . . . . . . . . . Revival of Mutual-Aid Organizations in Japan? . . . . . . . . . . . . . . . . .

313 313 315 317

287 287 289 292 292 293 293 296 297 300 301 304 307 309 309 309 310

Part I

Introduction to the Evolution of Cooperation

Chapter 1

What Is “The Evolution of Cooperation“?

Abstract Our society is based on cooperation. In our daily lives, there are a lot of examples; the parent–teacher association (PTA), paying tax, participating in voting, and making an effort to maintain the public facilities such as irrigation systems and reservoirs in farmland. As a simple game such as the prisoner’s dilemma game showed, mutual cooperation is hard to achieved. In reality, even though free-riders exists, we cooperate together to some extent. Researchers in various fields have been investigating what mechanism promotes cooperation among people and can explain cooperation in our real world. This study field is named “the evolution of cooperation.” Chapter 1 summarizes the previous studies about mechanisms promoting mutual cooperation, providing readers with some basic knowledge to understand “the evolution of cooperation.”

1.1

Cooperation in Our Lives

Our society is based on cooperation. In our daily lives, there are a lot of examples; the parent–teacher association (PTA) is based on volunteer activities of parents and teachers to encourage their children’s school lives. Paying tax is cooperation with our local society because taxes are used for social welfare, to build and maintain public goods such as public facilities, medical care expenditure, and disaster prevention. Participating in voting is a cooperative behavior because people devote their time to go to the polling place. Making an effort to maintain the public facilities such as irrigation systems and reservoirs in farmland is also cooperation from farmers. Following rules in the institution is cooperation for the institution, which sustains the institution. However, some people do not make any effort to cooperate, and free-ride the contribution of cooperators. They are called defectors or free-riders. Even though defectors enjoy the benefit from the contribution from cooperators, some cooperators may change to defectors and other cooperators are still cooperators. Thanks to cooperators, our society can be sustainable. This raises questions about “cooperation”: (i) Why do cooperators exist in our society even though they are less rewarded

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Nakamaru, Trust and Credit in Organizations and Institutions, Theoretical Biology, https://doi.org/10.1007/978-981-19-4979-1_1

3

4

1 What Is “The Evolution of Cooperation“?

than defectors? (ii) What mechanism promotes cooperation? Evolutionary game theory is one of the research fields in which to challenge these questions.

1.2

What Is Evolutionary Game Theory?

Maynard Smith (1982) invented evolutionary game theory, which is the combination of game theory and the process of natural selection, to analyze the evolution of the social interaction between individuals. The big difference between evolutionary game theory and game theory is that evolutionary game theory assumes that players make a very simple decision-making and can describe how the frequencies of a player’s strategy in the population or the behavior of each player change with time. Therefore, evolutionary game theory can successfully describe the evolution of behavior of not only animals and plants but also microbes through natural selection. To enable readers to understand evolutionary game theory intuitively, I will explain the process of natural selection in the following. The three factors, mutation, selection, and inheritance, are required when natural selection occurs (Fig. 1.1). For simplicity, it is assumed that the population size is 6 in Fig. 1.1. The wild type in the population is X. Mutation produces a new evolutionary trait in reproduction, Y. So there are five individuals with X and one individual with Y. It is assumed that the individual with Y is matured and reproduces five offspring and the individual with X reproduces two offspring. The survivorships of individuals with X and Y are the same. In this case, the difference in the reproduction influences selection. There values, six offspring and two offspring, are fitness, which is a technical term in evolutionary biology and is simply defined as “the number of offspring which can survive and be mature enough to have offspring.” Fitness is often called the reproductive success of a biological entity (Futuyma 2009). If offspring inherit traits X and Y from parents, there are ten individuals with X and five individuals with Y in the population. The ratio of X is 2/3 and that of Y is 1/3 in the population if their parents die immediately after reproduction. If the carrying capacity of the population is six, offspring grow up and then the population has four individuals with X and two individuals with Y in the next generation. If this process is repeated and repeated over generations, individuals with Y occupy the population. We can say that Y is selected. In Fig. 1.1, the social interaction among individuals is not assumed. If individuals socially interact with each other and the interaction influences the reproduction and survivorship of each individual, this situation can be described by evolutionary game theory. In evolutionary game theory, the payoff or fitness resulting from the social interaction among players can be described by the game matrix, which can be used to calculate an evolutionary stable strategy (Maynard Smith and Price 1973; Maynard Smith 1982).

1.3

Evolutionary Stable Strategy

Fig. 1.1 The mechanics of natural selection

5 A mutation changes X in one offspring to Y . adults with X give birth when Y

X

frequency of X: 5/6 frequency of Y: 1/6

X X

X

X

They grow up to be adults Adults give birth One with X has 2 offspring One with Y has 5 offspring 10 offspring with X 5 offspring with Y

The offspring grow up to be adults: Assume that the probability of survivorship is 2/5 and the surivorship of individuals with X is the same as that with Y Y

X X X

1.3

X

Y

frequency of X: 2/3 frequency of Y: 1/3

Evolutionary Stable Strategy

To investigate which strategy or behavior the players adapt through an evolutionary process, an evolutionary stable strategy (ESS) is used (Maynard Smith and Price 1973; Maynard Smith 1982). I will explain the definition using Fig. 1.2. Assume that there are players which adapt either strategy X or Y. Players interact together and obtain a payoff. E(i, j) is the payoff of a player with strategy i interacting with a player with strategy j. If E(X, X) > E(Y, X), strategy X is an ESS against strategy Y. If E(X, X) ¼ E(Y, X), strategy X is an ESS against strategy Y when E(X, Y) > E(Y, Y). Figure 1.2 shows the image of the ESS. I will explain the meaning of the definition “E(X, X) > E(Y, X).” E(X, X) means the expected payoff of a player with strategy X in the population which only has players with strategy X, and E(Y, X) means the expected payoff of a player with strategy Y in the population where players with strategy X are the majority and there are few players with strategy Y. Players with strategy Y are mutants or invaders in the population occupied by players with strategy X; they cannot increase in number and then cannot

6

1

Fig. 1.2 The definition of strategy X being the evolutionary stable strategy (ESS) against strategy Y

What Is “The Evolution of Cooperation“?

X

X X XX X X X X X X

X

The expected payoff of X in the population occupied by X: E(X,X)

X

Y X XX X X X X X X

X

The expected payoff of Y in the population whose majority is X: E(Y,X)

Y

X Y YY Y Y Y Y Y Y

Y

The expected payoff of X in the population whose majority is Y: E(X,Y)

Y

Y Y

Y

Y

Y Y Y

YY Y Y

The expected payoff of Y in the population occupied by Y: E(Y,Y)

invade the population occupied by players with strategy X if E(X, X) > E(Y, X). Then, strategy X can be an ESS against strategy Y. If E(X, X) ¼ E(Y, X); we cannot judge if strategy X can be an ESS, and we have to see if rare players with strategy X can increase in number in the population where the majority consists of players with strategy Y. If E(X, Y) > E(Y, Y), rare players with X can increase in number and invade the population occupied by players with strategy Y. What determines the payoff of players interacting with others? The following shows the types of games which present social interactions between two players and among players.

1.4 1.4.1

Games Between Two Players Prisoner’s Dilemma Game

The Prisoner’s dilemma (PD) game clearly illustrates the difficulty of maintaining cooperation between players in spite of its advantage. The game is played by a pair

1.4

Games Between Two Players

7

Table 1.1 Prisoner’s dilemma game: The payoff matrix of the focal player when playing the Prisoner’s dilemma game with the opponent. Two conditions, T > R > P > S and 2R > T + S, are required

Focal player

Cooperation Defection

Opponent Cooperation R¼3 Reward T¼5 Temptation to defection

Defection S¼0 Sucker P¼1 Punishment

of individuals, who have two options: either to cooperate (C) or to defect (D). If both players cooperate, both get payoff R, standing for “reward.” If one defects while the other cooperates, the one who plays C gets a payoff S, standing for “sucker,” while the one who plays D gets a payoff of T, standing for “temptation to defect.” If both defect, both get a payoff P, standing for “punishment” (see Table 1.1). The order of the magnitude of payoffs is S < P < R < T. In addition, we assume that the payoffs satisfy 2R > T + S, which prevents a strategy of “Alternation of D and C” from invading in a cooperative population. Although both players receive a higher payoff if they cooperate than if they defect (R > P), cooperation is difficult to maintain because each player would have a higher payoff by defection than by cooperation irrespective of the partner’s action (T > R, and P > S). This causes a dilemma; each player’s attempt to increase their own payoff results in a smaller payoff (P) because of the failure to cooperate.

1.4.2

Chicken Game

The chicken game originates from a game where two cars are in line, facing each other, and then start to accelerate towards each other. If one driver steps on the brakes and stops, the driver is the loser (the “chicken”) and the other is the winner. The winner gets a reward, b (>0), and the loser does not obtain anything. If both do not step on the brakes, the two crash and receive serious damage, c (c > 0). If the two drivers step on the brakes simultaneously, they are in a draw and their payoff is b/2. Table 1.2 presents the payoff matrix of the chicken game. In the game, who steps on the brakes first can be interpreted as a cooperator and who does not do can be interpreted as a defector. The order of the payoff is Table 1.2 Chicken game: The payoff matrix of the focal player when playing the chicken game with the opponent. Assume b, c > 0 and T > R > S > P

Focal player

Cooperation Defection

Opponent Cooperation R ¼ b/2 T¼b

Defection S¼0 P ¼ c

8

1

What Is “The Evolution of Cooperation“?

P < S < R < T. The chicken game can describe the social situation where players obtain the lowest benefit if both players are defectors (S > P).

1.4.3

Snowdrift and Blizzard Game

This game was proposed by Sugden (1986: 132). We consider that two players in a car are stuck in snow. If the two players cooperate, get out of the car and push it, they can get back home. The cost of pushing a car is c (> 0) and the benefit of getting back home is b (> 0). The two cooperators share the cost of cooperation c and then each only pays for the cost, c/2. They obtain the benefit, b. Therefore, the payoff of a cooperator is b  c/2 (> 0). If both choose defection, they do not need to pay a cost but never get back home. So, their payoff is zero. If one player chooses cooperation, gets out of the car and pushes it, and the other chooses defection and does not do anything, what happens? Thanks to the effort of the cooperator, they can get back home and obtain the benefit, b. However, the defector does not pay for the cost and the cooperator pays for all of the cost. The payoff of a cooperator is b  c and that of a defector is b. Table 1.3 presents the matrix. If b > c, this game is called the snowdrift game, and if b < c, it is called the blizzard game, because players have to make more effort to push their car in the blizzard rather than in the snowdrift. The snowdrift game is equivalent to the chicken game and the blizzard game is equivalent to the one-shot PD game from the aspect of the order of the payoffs: T, R, P, S (Table 1.3). This game is often used because there are only two parameters and this game can describe both the one-shot PD game and the chicken game.

1.4.4

Which Is an ESS, Cooperation or Defection?

We would like to see if cooperation can be an ESS against defection. In the PD game (Table 1.1), E(C, C) ¼ R and E(D, C) ¼ T. According to the definition of the PD game, T is always higher than R. Therefore, cooperation is not an ESS against defection. Secondly, we will see if defection can be an ESS. E(D, D) ¼ P and E (C, D) ¼ S. According to the definition of the PD game, P is always higher than S. Therefore, defection is an ESS against cooperation. Table 1.3 Snowdrift game (b > c) and blizzard game (c < b): The payoff matrix of the focal player when playing the game with the opponent

Focal player

Cooperation Defection

Opponent Cooperation b  c/2 b

Defection bc 0

1.5

Why We Can Analyze Our Society by Means of Evolutionary Game Theory

9

In the chicken game (Table 1.2), cooperation is not an ESS because E(C, C) ¼ R < E(D, C) ¼ T. Defection is not an ESS because E(D, D) ¼ P < E(C, D) ¼ S. This means there are no pure ESSs, which only choose cooperation by probability of 100% or defection by probability of 100%, in the chicken game. A mixed ESS exists in the chicken game (Maynard Smith 1982). We can calculate the mixed ESS from the theorem in Bishop and Cannings (1978); this strategy chooses cooperation with probability of (S  P)/(T  R + S  P) and chooses defection otherwise (see Maynard Smith 1982). Returning to reality, cooperation is based on our society. We cooperate in some situations, and we do not cooperate in others. As the mixed ESS exists in the chicken game, the chicken game seems to present our society more than the PD game. However, the PD game describes some parts of our social interactions, the chicken game describes some parts of them, and other games describe other parts. Therefore, which is used to describe the social interaction between two players depends on the social situations. In Sect. 1.7, we will discuss what mechanism promote the evolution of cooperation when players play the PD game.

1.5

Why We Can Analyze Our Society by Means of Evolutionary Game Theory

Maynard Smith (1982) developed evolutionary game theory to investigate how the social interaction between animals, plants, and other creatures has evolved and been adapted through natural selection. Human beings are animals and evolutionary game theory can be used to describe how sociality, social behaviors and social skills highly developed in human beings have evolved: Evolution of parental care (e.g., Maynard Smith 1977; Houston et al. 2013); evolution of language (Nowak et al. 2000, 2002); evolution of transitive inference (Nakamaru and Sasaki 2003; Doi and Nakamaru 2018). In addition, evolutionary game theory is also used to describe the dynamics of social learning. In this case, we do not consider biological evolution. When we make a decision to change our behavior or way of thinking, we sometimes imitate the behavior of others. Whose behavior do we imitate? One answer is that we may imitate the behavior of successful persons. The definition of a successful player is simply a player who obtains a higher or highest payoff. This corresponds to the process of selection and named social learning. In addition, people may change their behavior on a whim or “randomly.” This process is equivalent to mutation in natural selection. In sum, if players imitate the behavior of the players obtaining a higher or the highest payoff and sometimes change their behavior randomly, the dynamics of the behavior can be described by evolutionary game theory (Fig. 1.3). In this book, we use evolutionary game theory as one of the methods to describe social learning. It is natural to consider that players do not always make a decision following the social learning and the random change. For example, Gavrilets and Richerson (2017)

10

1

Fig. 1.3 The process of social learning

What Is “The Evolution of Cooperation“?

. A player changes X to Y randomly

Y

X X X

X

X

frequency of X: 5/6 frequency of Y: 1/6

The payoff of a player with X is 2 The payoff of a player with Y is 5 The total payoff of X: 2*5 = 10 The total payoff of Y: 5 Each player changes their strategy proportionally to the total payoff

Y

X X X

X

Y

frequency of X: 2/3 frequency of Y: 1/3

considered that players, who have two strategies, cooperation for collective action and punishment, and one evolutionary trait which controls internalizing a social norm, change their strategies which maximize the payoff within each generation and then change one evolutionary trait by the social learning and mutation at the end of each generation. Macy and Flache (2002) did not assume the social learning as updating the strategy, but considered that players choose their strategy, cooperation or defection, based on the Bush-Mosteller stochastic learning model, where players increase the probability of cooperation when the player’s payoff is higher than the aspiration level which is changed with time. Arefin and Tanimoto (2020) considered the combination of two update mechanism; the imitation process and the aspiration process. The imitation process is the same as the social learning in this chapter. In the aspiration process, players change their strategy if the payoff is less than the aspiration level.

1.6

How to Make a Mathematical Model or an Agent-Based Model of Our Society by Means of Evolutionary Game Theory

Which strategy is an ESS can be obtained if the social interaction between players is described by the payoff matrix such as in Tables 1.1, 1.2, and 1.3, or the payoff functions which will be shown in the later chapters. It is just a static analysis.

1.6

How to Make a Mathematical Model or an Agent-Based Model of Our Society. . .

11

Evolutionary game dynamic changes with time. Even though the static analysis can predict the convergent state of the evolutionary game dynamics, sometimes it is not enough to understand the consequence of evolution. To know the consequence of the evolutionary dynamics, we can use differential or discrete equations. The replicator equation can describe the process of natural selection in the population when individuals interact with each other there. There are n types of strategies in the population. The value aij is the payoff of a player with strategy i interacting with a player with strategy j (i, j 2 n). The average payoff of player with strategy i, Ei, can n P be described as Ei ¼ aij xj in which xj is the frequency of a player with strategy j in j

the population, and the payoff of a player with strategy i can be xiEi. The average n n P P xi aij xj . Following the process of natural payoff of the population, Et, is i

j

selection (Sect. 1.2 and Fig. 1.1), the frequency of players with strategy i in the next time step, xi0 , can be modeled by the difference equation as xi 0 ¼ xi E i =Et:

ð1:1Þ

This is the replicator equation. In the differential equation, the replicator dynamics can be presented as dxi =dt ¼ xi ðE i  Et Þ:

ð1:2Þ

For more mathematical information, please see Hofbauer and Sigmund (1998), Nowak (2006), and Sigmund (2010). Equations (1.1) and (1.2) imply that all players stand on an equal footing. However, different players take different roles in our society. For example, males and females have different role for parental investment and the game matrix of males is different from that of females (e.g., Maynard Smith 1977). This situation can be described by the replicator equation for an asymmetric game (e.g., Hofbauer and Sigmund 1998). There are two populations; one population consists of players playing role A, and the other consists of players playing role B. A player gets a payoff when she interacts with the other playing different role from her (Fig. 1.4). This game assumes that players in the same role do not play the game together: as an example of parental investment, even though two females interact together, they

Fig. 1.4 The image of the replicator equation for an asymmetric game

Social interaction (game) role A

role B

1 What Is “The Evolution of Cooperation“?

12

cannot reproduce offspring. The payoff matrix of the players in role A is different from that in role B. The value aij is the payoff of a player with strategy i in role A interacting with a player j in role B. The value bji is the payoff of a player with strategy j in role B interacting with a player i in role A. The frequency of players with strategy i in role A is xi and the frequency of players with strategy j in role B is yi. The number of the strategy types in role A is na and that in role B is nb. If players in role A are selected for in the population of role A and players in role B are selected for in the population of role B, the replicator equation for an asymmetric game is as follows; dxi =dt ¼ xi

dyj =dt ¼ yj

nb X

aij yj 

na X

xi

nb X

j

i

j

na X

nb X

na X

i

bji xi 

j

yj

! aij yj ,

ð1:3aÞ

! bji xi :

ð1:3bÞ

i

The replicator equation for an asymmetric game can be applied to the division of labor when the number of roles is more than two. Please see the application in Chap. 8. If the model assumption cannot be described by the differential or discrete equations, agent-based simulations are used. An agent, which corresponds to a player in game theory, can be interpreted as one person, one group, one organization, one nation, and so on. The unit of an agent depends on the model assumptions in the agent-based simulations. The agent-based simulation is a very convenient tool to describe the evolutionary game dynamics. It is because it is not assumed that players are rational and infer other players’ decision-making in evolutionary game theory. The agent-based simulations are programmed in a language such as C, Python, and so on. If a player can infer the other players’ intention considering that the others infer the intention of the player, it is a difficult to make the agent-based model. The simple decision-making of each agent is appropriate to make the agent-based model. A relatively large population size can be assumed in the agent-based simulations, so that we can investigate what happens in the population after the simulation runs when agents interact socially and update their strategies. In the later chapters, we show how to use the agent-based simulations to describe the social learning process in the population.

1.7

The Mechanisms to Promote the Evolution of Cooperation Between Bilateral Players

The mechanisms to promote the evolution of cooperation between two players have been studied in recent decades and Nowak (2006) and Rand and Nowak (2013) categorized the mechanisms into five: kin selection, group selection, direct

1.7

The Mechanisms to Promote the Evolution of Cooperation Between. . .

13

reciprocity, indirect reciprocity, and network reciprocity. In addition to the five categories, “punishment” is added in this book. As a matter of course, these mechanisms can be applicable to games among three or more players in Sect. 1.8.

1.7.1

Kin Selection

Kin selection can explain the evolution of altruism among kin who share the same specific genes as their ancestors (Hamilton 1964a, 1964b). Good examples are eusocial insects such as ants and bees (Hymenoptera) and social amoebae, Dictyostelium discoideum (Kuzdzal-Fick et al. 2007). Most important for kin selection is the coefficient of relatedness, r, defined as “the probability that an allele present in one individual will be present in a close relative; the proportion of the total genotype of one individual present in the other as a result of shared ancestry” (G-2 in Rubenstein and Alcock 2019). The inclusive fitness theory can explain why relatives help each other, reducing their fitness; consider an individual who helps her relative and gives the relative a benefit, b (> 0), paying a cost (c > 0) when the genetic relatedness between two is r. She can get back a proportion of the benefit, rb, because they share the same genes by probability of r. If rb  c > 0, she can compensate for the cost of helping her relative, and then helping behavior can evolve. That rb  c > 0 is called Hamilton’s rule. Using evolutionary game theory as social learning, this book focuses on how cooperation among non-relatives is established and then sustain the social system and institutions. Readers interested in kin selection can see the original papers about kin selection (Hamilton 1964a, 1964b) or the textbook about animal behavior by Rubenstein and Alcock (2019). In our society, we often interact with others who have cultural or behavioral similarity with us and then may have a tendency to help them rather than others who do not have similarity. This is called assortative interaction. The mathematical model for kin selection can be applied to models for assortative interaction (e.g., CavalliSforza and Feldman 1981; Boyd and Richerson 1985).

1.7.2

Group Selection

Some animals, plants and other creatures live in a group and the sustainability of the group is essential to each individual; if enough individuals in the group die, the group may go extinct. In this case, individuals cooperate to sustain the group. However, if the group has some defectors or free-riders, they get a benefit from cooperators without paying any cost of cooperation. As the payoff of a defector is higher than that of a cooperator, cooperators would be selected against and then the group would go extinct.

14

1

What Is “The Evolution of Cooperation“?

If the productivity of the group which comes from cooperators lowers the extinction rate of the group (e.g., Levin and Kilmer 1974), or if the group with a large number of cooperators has an advantage over the group with a large number of defectors, the groups with a large number of cooperators can survive and increase in number. As a result, the number of cooperators increases. This idea can explain the coevolution of parochiality between groups and cooperation within a group in our ancient society (e.g., Choi and Bowles 2007; Turchin 2016; Rusch and Gavrilets 2020). This can be applied to our modern society: There is high competition among organizations in the same industry and cooperation within an organization is key to winning the competition, even though there are free-riders in an organization who get a higher benefit than cooperators in the short term. Evolutionary game theory can be used as social learning in the theme. Good examples of application of group selection to our society is team sports (Turchin 2016). Buchanan (2000) discussed group selection in the professional basketball team. For players, shooting a goal successfully is directly related to the reputation and outcome of the player. While, passing a ball to other teammates is important for the team to win the game. Pass is regarded as cooperation; shooting is as defection. If there are many players, called hot dogs, who want to shoot a goal as much as possible in a team, the team cannot win the game. However, the hot dogs are often in good standing and may earn a high payroll. Wiseman and Chaterjee (2003) analyzed the date about the payroll of the professional baseball players in major league baseball teams from 1985 to 2002 and showed that the teams where the disparity in payroll among players was low won the games more than those where the disparity was high. They also pointed out that the teams where the average payroll was high won the game more than those where it was low. This suggested that low disparity and high average payroll leaded to players’ willing to cooperate together, and as a result, they won the game more. This work also implies that the great disparity in which the superstars’ payroll is extremely high does not encourage players to cooperate together for the team and to win the games. Cultural traits in the cultural groups are also inherited in the alternation of generations. Here, cultural traits include social norms, beliefs such as religious beliefs, ways of thinking, and technological development. The process that cultural traits are transmitted from generation to generation is similar to that of inheriting genes (Fig. 1.3), but the process of transmitting cultural traits is not completely equivalent to the genes. Players imitate: (i) the cultural traits for which the player has a higher payoff (direct bias); (ii) the cultural traits for which players are prestigious (prestige bias) or similar to them (similarity bias), both of which are called indirect bias; (iii) the cultural traits which are shared by the majority or minority of the population which are termed conformity bias or anti-conformity bias (e.g., CavalliSforza and Feldman 1981; Boyd and Richerson 1985). Considering the cultural effect on the group selection, group selection is called cultural group selection (Richerson et al. 2016).

1.7

The Mechanisms to Promote the Evolution of Cooperation Between. . .

1.7.3

15

Direct Reciprocity

Direct reciprocity is the same as reciprocal altruism in evolutionary ecology (Trivers 1985). If two players play the PD game once, cooperation cannot be an ESS. In the repeated PD game, even though a cooperator gives a benefit to the opponent with paying a cost of cooperation, the cooperator receives a benefit from the partner who is a cooperator in the next round and can compensate for the cost. Then, direct reciprocity and reciprocal altruism can evolve if players can interact repeatedly. This has been known as the folk theorem in economics; however, no one had proved which strategy can be an ESS or a Nash equilibrium experimentally before Axelrod and Hamilton (1981). Axelrod and Hamilton (1981) did two experiments to investigate which strategy can be an ESS. In the first experiment, they recruited 14 strategies from eminent professors. Besides 14 strategies, a random strategy which chooses either cooperation or defection randomly was added. Fifteen strategies matched up in a round-robin and two played the repeated PD game 200 times. Among 15 strategies, Tit-for-Tat (TFT), which first cooperates, then adopts the same action as what the partner did previously, obtains the highest payoff. TFT was proposed by the eminent psychology professor, Anatol Rapoport. In the second experiment, TFT was also a winner among 62 strategies from all over the world. Here we will show the condition where TFT can be an ESS against All-Defect (AD), which always defects. Let w be the probability that the same two players interact in the following step as well, and wn1(1  w) be the probability that they interact exactly n times (n ¼ 1, 2, 3,. . .). The expected number of times the two players interact is 1/(1  w). Figure 1.5 shows the TFT strategy. When a TFT plays with an AD, the TFT gets payoff S in the first round. In the second round TFT gets P with the probability that they interact again, w. In the third round TFT gets P with the probability that they interact at least three times, w2 and so on. Then the expected score of TFT whose partner is AD is V(T/D) ¼ S + Pw + Pw2 + Pw3 + . . . ¼ S + wP/(1  w). Let V(i/j) be the expected score of a player with strategy i obtained from the interaction with a neighbor adopting strategy j (i, j ¼ T or D, indicating TFT and AD, respectively). A consecutive study of the iterated PD game reveals that if players make “errors” with a small probability, then strategies other than TFT can be better, such as the Win-Stay Lose-Shift strategy (WSLS), which is also called Pavlov, and Generous Tit-for-Tat (GTFT). WSLS cooperates in the first round and then changes behavior if the payoff in the previous move is low, such as S and P in the PD game; otherwise it does not change (Nowak and Sigmund 1993; see Fig. 1.6). GTFT always cooperates when its partner cooperated in the previous move and cooperates with probability defined as the minimum of 1  (T  R)/(R  S) and (R  P)/(T  P) when its partner defected in the previous move (Nowak and Sigmund 1992). A “strong” strategy in the repeated PD game was not discovered for 20 years from the early 1990s. Press and Dyson (2012) mathematically proved that the zerodeterminant strategy (ZD strategy) is a winner in the repeated PD game. Regardless

16 Fig. 1.5 Tit-for-Tat (TFT)

1

What Is “The Evolution of Cooperation“?

a Cooperate in the beginning

Imitate the behavior of the opponent in the previous round

TFT

C

C

D

C

Opponent

C

D

C

C

TFT

C

C

C

C

.....

TFT

C

C

C

C

.....

C

b

Make a mistake and choose D

c

Fig. 1.6 Pavlov or Win-Stay Lose-Shift (WSLS)

TFT

C

D

C

D

C

...

TFT

C

C

D

C

D

...

R

R

S

P

R

Pavlov

C

C

C

D

C

...

Pavlov

C

C

D T

D

C

...

Make a mistake and choose D

P Mutual cooperation, again

of the opponent’s strategy, the ZD strategy forces the payoff between the two to be linear. One of the ZD strategies is Equalizer, which forces the payoff of the opponent to be a fixed value. Another strategy is Extortion. If it is assumed that the payoff of the Extortion strategy is Pe, that of the opponent is Po and the value is p, the Extortion strategy forces the relationship of the payoff between two players to be Pe  p ¼ x(Po  p), where x > 1. The agent-based simulation studies which assume two players chosen randomly from the population play the repeated PD game showed that Extortion can evolve in a small population size, and that WSLS can evolve in a large population size (Hilbe et al. 2013). Sugden (1986) had already discovered before Nowak and Sigmund (1993) that mutual cooperation is not established if a player with TFT makes a mistake and chooses defection, even though the rule of TFT forces the player to choose cooperation. He considered that cooperation can be established again if the player with TFT chooses cooperation two rounds after the player makes a mistake. He assumed the modified version of TFT called T1 strategy. A player with T1 strategy chooses either

1.7

The Mechanisms to Promote the Evolution of Cooperation Between. . .

17

cooperation or defection depending on (i) the previous behaviors of both the focal player and the opponent and (ii) the reputation based on the previous behaviors. Sugden’s study of the repeated PD game has been applied to indirect reciprocity studies, and the T1 strategy in Sugden (1986) corresponds to the S-stand in the recent indirect reciprocal studies. The T1 strategy and S-stand will be explained in Chap. 7.

1.7.4

Indirect Reciprocity

In the repeated PD game, players can know the behavior of the opponent, and then choose cooperation if the focal player knows that the opponent has chosen cooperation. If a player has not met the opponent before, the player can choose either cooperation or defection when the player can obtain the opponent’s reputation. How do players get the reputation of others? One answer is to observe the social interaction between others; player A observes player B’s helping player C, and judges player B to be in good standing, and player A decides to help player B even though player B has not helped player A yet. Boyd and Richerson (1989) considered a strategy called Downstream TFT (DTFT). A player with DTFT decides to cooperate with a neighbor if the neighbor has cooperated with his/her neighbor. It is indirect reciprocity. Another answer is to obtain the gossip about the reputation of others. Nowak and Sigmund (1998) considered image score as the reputation of the players. The image score presents the past history of each player and all players know the image scores of all players publicly. Therefore, image score can be interpreted as either observing all players’ behavior or obtaining the gossip which can be shared among all players. If the image score of player B is high, player B is regarded as a cooperator since player B has cooperated with others often. Who is good or bad depends on the decision-making of each player; the threshold, k, is assumed. If the image score of player B (sB) is higher or equal to the threshold of player A (kA), player A thinks that player B is a cooperator. Then, player A cooperates with player B (Fig. 1.7). Then, the image score of player A increases by one unit, and then player C helps player A Fig. 1.7 Image score kA = -1 sA = 1

help

A

sB >= kA

B

sB = 1

kC = 1 sC = 1

sA = 2

help

C

sA t kC

A

18

1 What Is “The Evolution of Cooperation“?

because the image score of A (sA) is higher than the threshold of player C (kC). As the threshold of the player is low (high), the player does (does not) think that the opponent is a cooperator. Therefore, the high threshold means defectors and the low threshold means cooperators. In Chaps. 4 and 6, the assumption of the image score will be explained in more detail. In the real world, the idea of image score is close to credit score in Zhima Credit developed by the Ant group, related to the Alibaba group in China (e.g., Botsman 2017). The credit score presents the integration of five elements: the credit history of a credit card; the capability to fulfill the obligations of contract; private information such as the home address; the behavior and preferences deduced from shopping habits; academic background; and the social influence of friends and neighbors. If the credit score of person A is high, person A can get a special benefit; if it is low, person A has limited use of public transportation, which seems like a penalty person A pays for a low credit score. The company called Tala in Kenya and the Philippines loans money to people who have no credit to borrow money from banks (e.g., Botsman 2017). Tala collects the data from the smartphone of one who asks for a loan to estimate the range of the social network of the person and decides to lend money to the person based on the estimation. If player A helps player C in bad standing, is player A still in good standing? Some players may regard player A as good, others may regard player A as bad. However, in the assumption of the image score, helping a player in bad standing is not distinguished from helping a player in good standing. To distinguish these two cases, the strategies such as the T1 strategy in Sect. 1.7.4 or other strategies have been examined based on the framework of Sugden (1986). Ohtsuki and Iwasa (2004), following the framework of Sugden (1986), mathematically showed that eight strategies called the leading eight can be an ESS against players who only choose defection. The common feature of the leading eight will be explained in Chap. 7. These previous studies only considered a pair who only interact with other once and then their reputation spreads to all players immediately. In Chap. 7, the mutual-aid game, where more than two players play, is used when the reputation rule follows Sugden (1986). Chapter 9, which is an extension of Nakamaru and Kawata (2004), assumes that gossip spreads from one player to another, and players cannot only spread gossip from others but also start new gossip. They investigated the effect of false gossip on the evolution of cooperation and how the conditional cooperators can avoid being deceived by false gossip. These simulation models can deal with both indirect and direct reciprocity, and provide the unified framework of these two reciprocities; they showed that more gossip promotes the evolution of cooperation in less social interaction, more social interaction promotes the evolution of cooperation, and false gossip influences the evolutionary outcomes. Schmid et al. (2021) took the same approach using simple and beautiful mathematical models, in which they did not consider false gossip but errors. Their results are very close to Nakamaru and Kawata (2004) and Chap. 9 in a sense that the effect of false gossip is a bit similar to the effect of errors.

1.7

The Mechanisms to Promote the Evolution of Cooperation Between. . .

1.7.5

19

Social Network

In the simplification of the mathematical model, players are chosen randomly and play the game. This assumption works well in some social cases, but sometimes it does not work well in other cases. People do not interact with others chosen randomly from the population. Rather, people often interact with neighbors next to their house or to their desk in the workplace. Or, people have their own social networks and interact with each other on their networks both in reality and via the social network service (SNS). The lattice-structured population can present the abstract model of the interaction with neighbors in real space (Fig. 1.8a). The lattice is like chess, shogi, and go boards, and players or agents located at the lattice points interact with their neighboring points. The lattice model is often used because it is not so difficult to analyze the dynamics of agents and players in the lattice by computer simulations and by some approximation methods (see Chaps. 2 and 3), even though it is hard to find out real lattice-structured populations in human society. Chapters 2 and 3 show the regular lattice-structured population promotes the evolution of cooperation and the coevolution of cooperation and punishment. Barabási and Albert (1999), collecting mass data or “big data” and analyzing them, discovered that the scale-free network can be observed not only in the protein– protein interaction but also in our society such as the coauthor networks, co-starring Fig. 1.8 a Regular lattice model and b scale-free network when the number of nodes is 30 and the initial links is 2

a

b

1 What Is “The Evolution of Cooperation“?

20

of actors and actresses, and the World Wide Web. Briefly, a few vertices have a lot of links and a majority of vertices have a few links in the scale-free network (Fig. 1.8b). This founding triggers the development of complex network studies (e.g., Newman 2003; Barabási 2016). The effect of the scale-free network on the evolution of cooperation has been studied (e.g., Santos et al. 2006b) showed that cooperation evolves in the scale-free network more than the well-mixed population where players interact with each other randomly when the accumulated payoff, which is a sum of the payoffs of each player interacting with all of neighbors, is used in updating the strategy. While, cooperation does not evolve even in the scale-free network when the average payoff, which is calculated as the accumulated payoff over the number of neighbors, is used in updating the strategy (e.g., Tomassini et al. 2007). The above-mentioned studies implicitly assume that one individual or one agent is located at each vertex on the particular network, whereas in reality there are various networks among groups or populations (e.g., Hanski and Gaggiotti 2004). In this case, each group or population can be located at each vertex on a particular network and individuals may interact either with any individual randomly or with the neighbors in the group or population (e.g., Pecora and Carroll 1998; Brechtel et al. 2018). Beyond groups, some players in different groups also interact.

1.7.6

Punishment

Punishment or sanctions exist in our society and prevent defectors, deviants, and rule-breakers. Punishing them is costly: it takes time and money, and the player being punished by us may seek revenge. In current society, the legitimate police system is given a right to punish rulebreakers and to force them to be fined. On a daily basis, we have no right to force the rule-breakers to pay a fine even though they steal something from us. Instead of punishing the rule-breakers directly, we can go to the police or sue them in a court. It is also costly. From the viewpoint of evolutionary game theory, costly punishment is disadvantageous. What can promote the evolution of punishment? Frank (1989) explained that emotion, a non-rational behavior, triggers punishing of defectors, and, as a result, punishment hinders defectors. If cooperators are also punishers, cooperation with others can compensate for the cost of punishment. The coevolution of cooperation and punishment are investigated (see Chap. 3 for details). This type of punishment is called peer punishment. From the viewpoint of anthropology studies, peer punishment is rare in our society and the policing systems are often observed (e.g., Guala 2012), and Sigmund et al. (2010) investigated the evolution of the policing system. However, on condition that we have an institution of policing, people may have a psychological tendency to punish defectors. Chapter 3 shows the evolutionary conditions for which people have this tendency.

1.8

Games Among Three or More Players

21

Gossip about defectors can be interpreted as indirect and costless punishment and we use gossip instead of direct punishment in our daily lives. Chapter 9 also shows the effect of gossip as punishment. Or, deleting the social networks of the defectors is also indirect punishment (e.g., Santos et al. 2006a; Pathak et al. 2020). Choosing defection from the defector in a repeated interaction is also a kind of punishment (e.g., Axelrod 1984; Dreber et al. 2008). Social ostracism is also a punishment to rule-breakers and may promote cooperation in a group. Chapter 6 shows the effect of exclusion on the evolution of group cooperation. However, referring to our history, social ostracism has pros and cons. What I have explained concerns the theoretical and experimental studies of the evolution of punishment. In the laboratory experiments, anonymous subjects are given an opportunity to punish other anonymous subjects. In the theoretical studies, it is assumed that players have a strategy that can punish defectors in the mathematical models or the agent-based simulations. However, there are few studies about punishment in the “real” world. Di Stefano et al. (2015) did a field survey about punishment in the gourmet cuisine industry in Italy, where there are norms regulating social exchange of a new and original recipe but no legal system to punish the violators. There are debates whether punishment is either strong reciprocity or weak reciprocity (Guala 2012). In strong reciprocity, people punish others because they desire retribution; in weak reciprocity, people punish other with rational calculations about the cost–benefit of punishment. Di Stefano et al. (2015) examined whether punishment is strong reciprocity or weak reciprocity in the gourmet cuisine industry. They found that punishment is based on both. More field surveys about punishment in the real world should be done to examine what makes us punish norm violators.

1.8

Games Among Three or More Players

Not only bilateral players but also three and more players in a group interact socially. The multilateral interaction happens on a daily basis. The public goods game is often used to describe the multilateral interaction. In the public goods game, n players are in the group (n  3). Each player can choose either cooperation or defection. The number of players who choose cooperation (cooperators) is nc and the number of players who choose defection (defectors) is nd (nc + nd ¼ n). The cooperators invest their effort or money, b (> 0), to produce public goods such as public parks or public facilities and the effort or money can be interpreted as a cost of cooperation, c (> 0). It is assumed that b > c > b/n > 0. The sum of the efforts or money from all cooperators is bnc. Then, all players can get a benefit from the public goods because it is non-rivalry or non-exclusive. The benefit is bnc/n. Therefore, the payoff of a cooperator (Ec(nc)) is bnc/n  c and that of a defector (Ed(nc)) is bnc/n. The payoff of a defector in the group consisting of defectors is Ed(0) ¼ 0 and the payoff of a cooperator in the group consisting of cooperators is Ec(n)) ¼ b  c (> 0). It is obvious that Ed(nc) > Ec(nc). If one

22

1

What Is “The Evolution of Cooperation“?

Table 1.4 Four types of social interaction in groups Type Donor

(a) All-for-all

(b) All-for-one

(c) One-for-all

All

All

One or a few

All Garbage problem, global environmental problem

One or a few Roscas, credit union, labor union, mutual aid association, cooperatives

Public goods game

Mutual-aid game, Rotating indivisible goods game

All Garbage duty in local communities, annual meeting of the academic society Volunteer’s dilemma

(d) One-for-one

One

Recipient

Examples

Public goods game

One Cooperation between two parties in groups

Prisoner’s dilemma game giving game

cooperator changes into a defector, his/her payoff is Ed(nc  1) ¼ b(nc  1)/n. As Ed(nc  1) > Ec(nc) in c > b/n, the player has an incentive to change cooperation into defection. As a result, all cooperators change into defectors even though Ec(n) > Ed(0). This can correspond to “the tragedy of the commons” (Hardin 1968). A public goods game can be applied to environmental problems such as CO2 emission, the overexploitation of fisheries, and the use of ecological resources. In the case of the CO2 emission problem, regardless of the amount of each player’s contribution to the CO2 expenses, all players receive the same benefits from the improvement of the atmosphere, thus free-riders may proliferate. In the public goods game, all players have a chance to contribute to the pool and then all players receive the benefit from the pool. Here I call this system the all-for-all type (Table 1.4). Besides the all-for-all type, there are three possible types: the all-for-one type, and the one-for-all type, and the one-for-one type (Table 1.4). In the all-for-one type, all players have a chance to contribute to the pool and then only one player receives the benefit from the pool. This type can be observed in the mutual-aid system on a daily basis: the rotating savings and credit association (ROSCA) which is modeled as the rotating indivisible goods game (Koike et al. 2010; see also Chap. 4), and the earlystage insurance institution which is modeled as the mutual-aid game (Sugden 1986; see also Chap. 7). Other examples are customs in funerals and wedding parties in Japan such as kouden and shugi. This type will be discussed in Chaps. 4, 5 and 7. In the one-for-all type, only one player contributes to the pool and then all players receive the benefit from the pool. The annual meeting of an academic society follows this type; members belonging to one university organize the annual meeting, and hold the meeting on the campus of the university. Other members participate in it and

1.9

The Application of Evolutionary Game Theory to Our Society

23

talk about their research. The organizing committee incurs a large cost and other members obtain the benefit from participation. This type is modeled as the Volunteer’s Dilemma (Diekmann 1985). In the one-for-one type, in the population, pairs are formed in the population and each pair interacts using a bilateral game such as the PD game. Therefore, the one-for-one type is equivalent to the model assumption of the bilateral game in the replicator dynamics in the population.

1.9

The Application of Evolutionary Game Theory to Our Society

Evolutionary game theory can describe some social phenomena if the phenomenon has a process which is very close to the process of evolutionary game theory: players imitate the behavior of others whose payoff or utility is high and players sometimes change their behavior randomly. For example, if we consider that players may imitate the behavior of others who obtain more money in the mutual-aid systems, the mutual-aid system can be analyzed by evolutionary game theory. Chapters 4 and 7 show how to deal with this. Chapter 8 focuses on the division of labor, one of the topics in sociology and economics, which can be represented using evolutionary game theory. It is because members in the division of labor in Chap. 8 consist of dealers and they try to maximize their net benefit and may imitate the behavior of other dealers with high benefit. Chapter 11 shows that the ecological system can be used in a sustainable way even when harvesters only take into account the short-term benefit from ecological systems. This result implies that, even though players may imitate the behavior of others with high payoffs, harvesters may choose a sustainable method. In Chap. 10, field abandonment, one of the agricultural problems in many countries, can be formulated using evolutionary game theory if it is assumed that farmers may imitate the behavior of their neighbors with a high benefit. Besides this book, researchers are challenging the application of evolutionary game theory to social sciences, psychology, anthropology, linguistics, and so on. Here I would like to introduce three papers on: the application of cultural evolution to organization theory; explaining the economic and political inequality by means of evolutionary game theory; and the effect of intuition on deciding to either cooperate and defect. Cultural evolution, whose process is similar to the process of natural selection, can be applied to organization theory: cultural evolution of productive organizations (Brahm and Poblete 2021). Two types of players are needed for cultural evolution at least: individual learners, who individually invent a new culture including technology inventions, and social learners, who only imitate or copy the culture in the group or the population. As creating a new idea or product is more expensive than imitating an existing one, the cost of invention by individual learners is higher than the cost of imitation by social learners. So, social learners have an advantage over individual learners. However, if the social or natural environment changes, a cultural trait which

24

1 What Is “The Evolution of Cooperation“?

fits a new environment the most becomes different from the cultural trait which fitted the previous environment. For example, a cashmere sweater makes people warm in the cold weather. When it is warmer even in winter, cashmere sweaters are not bought more than before. If there are only social learners, producers only make cashmere sweaters, and then they cannot gradually make ends meet. If individual learners invent a new product made from cashmere which fits the warmer weather, social learners imitate them and the new product spreads in the market. Therefore, individual learners as well as social learners are needed when the environment changes. This is the famous “Rogers’ paradox”, in which environmental changes disadvantage social learners, make the mean payoff of the group with social and individual learners lower than the payoff of individual learners (Rogers 1988). As a result, individual learners are needed to help social learners update their knowledge even though social learning is cheaper than individual learning. Rogers (1988) showed that the environmental changes make both social learners and individual learners coexist. Brahm and Poblete (2021), applying Rogers (1988), studied the cultural evolution of productive organizations (POs), defined as organizations producing and delivering the goods and services which make the human population increase. They assumed that POs are exclusive, and social learners outside the POs incur more cost to learn technology in the POs than social learners inside the POs. This assumption is supported by empirical evidence (see the management and economics literature cited in Brahm and Poblete (2021)). They mathematically proved that, as the consequence of cultural evolution, the number of individual learners, who invent technology, outside the POs is high relative to that inside the POs; the number of social learners outside the POs is low relative to that inside the POs. This outcome can imply the situation that there are many startups that are innovative and there are many social learners but fewer innovators in the big firms. Inequality in our society arises not only for economic reasons but also political reasons. Tverskoi et al. (2021) investigated the evolutionary condition that economic and political inequality appears when cooperators contribute to making products for their group and defectors free-ride it and groups compete for the shares of resources in the group-structured population. They considered the political power and the incumbency effect, which plays a key role, and the two influence each other in their model. If the incumbency effect is zero, the political power is only dependent on the payoff produced by a group; if there is one, the political power enhances a new political power and then promotes the inequality in resource distribution. The incumbency effect is presented as the strength of democratic checks-and-balances. They showed that a higher incumbency effect increases the difference in the political power between groups and decreases the number of cooperating groups. We intuitively or deliberately decide either to cooperate or defect. Intuitive decision-making does not as take much time as deliberate decision-making which as a result, incurs a cost. Bear and Rand (2016) investigated which strategies evolve among intuitive players and dual-process players who use both intuitive and deliberate decision-making. Their analysis by using the evolutionary game theory showed

References

25

that selection favors two types of players: (i) intuitive players who always choose defection, and (ii) dual-process players who intuitively choose cooperation and deliberately choose cooperation in repeated games and defection in the one-shot PD game. They also discussed that their results can explain the reason why some people choose cooperation in the one-shot PD game in the experiment. How our cognition makes a decision in the game is essential and Bear and Rand (2016) is one of the challenging studies.

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Chapter 2

The Evolution of Cooperation in a Lattice-Structured Population Under Two Different Updating Rules

Abstract The evolution of cooperation is studied in a lattice-structured habitat under two different updating rules: the score-dependent viability model (viability model) and the score-dependent fertility model (fertility model). Each individual in each lattice site is assigned to Tit-for-Tat (TFT) or All Defect (AD). Each individual plays the iterated prisoner’s dilemma game with its nearest neighbors, and obtains the total payoff. In the viability model, its total payoff determines its mortality. After the death of an individual, the site is replaced with a copy of a randomly chosen neighbor. In the fertility model, an individual dies randomly, one of the neighbors is chosen proportionally to its total payoff, and the vacant site is replaced with a copy of the neighbor. The model is analyzed by invasion probability analysis, pair-edge method, mean-field approximation, pair approximation, and computer simulations. Results are: (1) In both updating rules, TFT players come to form tight clusters, and then rare TFT can invade and spread in a population dominated by AD, unlike in the completely mixing model where AD is always an ESS. The dynamics in a one-dimensional lattice are predicted by all the techniques except mean-field approximation. (2) The fertility model is more favorable for TFT than the viability model because spatial structure not only facilitates cooperation but also inhibits cooperation in the viability model because of the advantage of being spiteful by killing neighbors. Keywords Population structure · Pair approximation · Pair-edge method · Scoredependent viability model · Score-dependent fertility model

The original version of this chapter was revised: Figures 2.3 (b), 2.5 (b), 2.6 (d,e) and 2.8 (a,b,c,d) have been updated. The correction to this chapter can be found at https://doi.org/10.1007/978-981-19-4979-1_12 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022, corrected publication 2023 M. Nakamaru, Trust and Credit in Organizations and Institutions, Theoretical Biology, https://doi.org/10.1007/978-981-19-4979-1_2

29

30

2.1

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

Introduction

The prisoner’s dilemma game clearly illustrates the difficulty of maintaining cooperation between players in spite of its advantage. One way to resolve the dilemma is direct reciprocity (see Chap. 1 for more details). When two players are randomly chosen and play the repeated prisoner’s dilemma game, Tit-for-Tat (TFT) can be an evolutionary stable strategy (ESS) against All-Defect (AD, a strategy which defects on every move), whereas AD can be an ESS against TFT; this means that rare TFT can neither increase in number nor invade the population occupied by AD players. To solve this problem, a network structure such as the lattice model has been introduced into the model. There are pioneering works about network reciprocity; Axelrod (1984) studied the effect of the spatial structure in interaction and reproduction (he called it “territoriality”) on the evolution of cooperation. If interactions among individuals occur within an area much smaller than the whole population, and if the individuals of the same strategy tend to form tight clusters, then a cooperative strategy would be more likely to spread and be maintained in a spatially structured population than in a perfectly mixed one. Neighbors tend to become occupied by players of the same strategies if migration is limited in spatial range or if players tend to imitate their neighbors. To study the effect of spatial structure, Axelrod (1984) used a lattice model in which each lattice point is occupied by a single individual which plays the game only with its nearest neighbors. He carried out a computer simulation of the spatial patterns when one TFT (a cooperative strategy) mutated to an AD in a two-dimensional spatial lattice occupied by TFT only. Each individual repeatedly played the iterated prisoner’s dilemma game with its nearest neighbors. The average payoffs gained by this interaction were calculated, and then each individual player changed its strategy to the one adopted by the most successful nearest neighbor who had achieved the highest score. Nowak and May (1992) studied a prisoner’s dilemma game in a similar latticestructured population. To concentrate on the effect of spatial structure, rather than the effect of iteration in enhancing the evolution of cooperation, they simplified the game’s structure. For example, they used the strategy of All Cooperate (AC), instead of TFT, which implies that the game is played once only in each time we evaluate the score. They also assumed specific values for the payoffs: R ¼ 1, T ¼ b > 1, S ¼ P ¼ 0. They observed that, for a limited range of b, AC and AD can coexist and produce complex and constantly changing spatial patterns. There are pioneering theoretical studies of the evolution of altruistic social behavior which do not mention the prisoner’s dilemma game. For example, Matsuda (1987) and Matsuda et al. (1987) studied the evolution of cooperation in a lattice-structured model, in which two types differing in their social interaction compete with each other. Assuming that some sites are vacant and that reproduction occurs only to nearest neighbor vacant sites, each lattice site takes one of three alternative states. The model is a continuous time Markov chain, in which the state transition is not synchronized as

2.1

Introduction

31

in Axelrod (1984) or in Nowak and May (1992). Matsuda (1987) and Matsuda et al. (1987) discovered that the effect of the spatial clumping to the relative advantage of cooperation changes with the density. A similar conclusion was derived by Taylor (1992) and Wilson et al. (1992), who concluded that population viscosity (low mobility and a locally limited interaction) is not very effective in promoting the evolution of altruism. The major reason for this result is that the advantage of an enhanced probability for altruists to be surrounded by other altruists in a viscous population is canceled out by the disadvantage of the lack of surrounding vacant sites needed for reproduction. Interestingly, at the same time, not only in mathematical biology but also in theoretical economics, the effect of neighbors in playing a particular game was examined from the point of evolutionary game theory (Ellison 1993). This theory can be applied to the lattice-structured population. Since the 1990s, lattice-structured models have been used in describing ecological processes of sessile organisms, such as terrestrial plants or marine benthic invertebrates. The models have been studied mostly by computer simulation of the spatial stochastic processes. Sometimes, the results are compared with dynamics derived by the neglect of spatial correlation, or by assuming perfect mixing (meanfield approximation) (Caswell and Etter 1992; Durrett and Levin 1994a, 1994b). However, direct computer simulation of a stochastic lattice model is very costly in computation time, and in general it is often difficult to gain insight into a model’s behavior only from computer simulations. In addition, the effect of spatial structure sometimes produces predictions qualitatively different from nonstructured populations (e.g., Sato et al. 1994; Harada and Iwasa 1994). Matsuda et al. (1992) developed a method to construct a closed dynamical system of overall densities and correlation between nearest neighbors, by adopting pair approximation for latticestructured population dynamics. This approach constructs a system of ordinary differential equations for the average population densities and the local densities, the latter giving the nearest neighbor correlation of states. Nakamaru et al. (1997) studied both one-dimensional and two-dimensional lattice models where individuals using either TFT or AD play the iterated prisoner’s dilemma game with their neighbors. Two updating rules are examined. The accumulated payoffs then determine the mortality. After the death of an individual, the site is occupied by a copy of a randomly chosen neighbor (the nearest-neighbor migration model of Matsuda (1987)). This updating rule is called the scoredependent viability model (or viability model for short). The model is a continuous-time Markov chain. In contrast, in other updating rules of the evolution of cooperation in a lattice population (e.g., Axelrod 1984; Nowak and May 1992, 1993; Wilson et al. 1992), the game’s score affects the likelihood of being copied (i.e., reproductive rate), rather than the longevity. Therefore, we also make another assumption; an individual dies randomly independent of its score, and the probability that the open site is colonized

32

2 The Evolution of Cooperation in a Lattice-Structured Population Under. . .

by a particular neighbor is proportional to its total score of the game with its own neighbors. This updating rule is called the score-dependent fertility model or fertility model as abbreviation (Nakamaru et al. 1998). A biological example of score-dependent viability may be “allelopathy,” the production of toxic chemicals by plants and bacteria that harm neighboring competitors (Durrett and Levin 1997; Iwasa et al. 1998), or possibly food sharing by vampire bats to improve survivorship of other individuals (Wilkinson 1984), whilst an example of score-dependent fertility may be the enhancement of seed production of neighboring plants by attracting pollinators in self-incompatible flowers. In general, many modes of social interaction, such as competition for light among plants, would affect both mortality and fertility. Differences in the updating rules may cause qualitative differences in the model’s behavior to evolve. For example, Nowak and May (1992, 1993) observed the perpetual coexistence of cooperative and defective players for a range of parameters, forming constantly changing spatial patterns. In contrast, coexistence of the two different strategies is not observed in models studied by Nakamaru et al. (1997, 1998). Although we call the transition of state the “death” of an individual followed by “reproduction” throughout the chapter, the viability model may in fact represent an individual changing its strategy to adopt a new one randomly sampled from its neighbors, and the fertility model may also represent an individual occasionally changing its strategy by adopting the strategy of the neighbors with a higher score, and which would be more appropriate for human society. The interpretation of statechange as either the imitation or the change of one’s belief would be more appropriate for cultural evolution in human societies (Boyd and Richerson 1985) and economic games (Kandori et al. 1993). In the viability model, players tend to abandon the current strategy more quickly if its performance is not very good than if it performed well. It imitates one of its neighbors without considering their performance. In contrast, in the fertility model, players randomly change their strategy but attempt to adopt the strategy of the neighbor that achieves the high performance in terms of the game’s score. We do not ask why they change their strategy as described, but we simply assume these state transition rules as given. Note that the fertility model is more complicated than the viability model. In the latter, the state-change of a site depends on the states of itself and the nearest neighbors. In contrast, the transition in the fertility model is dependent upon the scores of the nearest neighbors which in turn are affected by their neighbors (two-step neighbors of the original site). Thus, in the viability model the transition rate at a site depends on three individuals in a one-dimensional lattice and nine individuals in a two-dimensional lattice (we adopt the Moore neighborhood). But in the fertility model the transition rate depends on five individuals in a one-dimensional lattice and 25 individuals in a two-dimensional lattice.

2.2

Completely Mixed Population

33

We observe a large difference in the condition for the evolution of cooperation between the two updating rules with the corresponding parameters. This is because less advantage is given to spiteful behavior of killing neighbors in the fertility model than in the viability model. Since the mid-2000s, there have appeared many theoretical studies about how different updating rules influence the evolution of cooperation when different types of game, such as the prisoner’s dilemma game and the chicken game, are played on various types of networks. The famous updating rules among them are birth–death (BD), death–birth (DB), imitation (IM), and pairwise-comparison (PC) (e.g., Ohtsuki and Nowak 2006). The DB rule is identical to the score-dependent fertility model (Nakamaru et al. 1998). In addition to direct computer simulations, we analyze the viability and fertility models by invasion success probability analysis, the pair-edge method (Ellner et al. 1998), mean-field approximation, and pair approximation for the one-dimensional model. For the two-dimensional model, we discuss the difference in behavior between the viability and fertility models and the difference between our fertility model and models similar to Nowak and May’s (1992, 1993) model, especially on the possibility of perpetual coexistence of cooperative and defective strategies by means of computer simulations.

2.2

Completely Mixed Population

Before examining the game in a lattice, we first study the completely mixing model, in which each player plays the iterated prisoner’s dilemma game with another player randomly chosen from the whole population (Nakamaru et al. 1997, 1998). We used the following parameter values for the payoffs in a single interaction: R ¼ 3, T ¼ 5, S ¼ 0, and P ¼ 1 (see Table 2.1A). We consider two strategies for each player to adopt: Tit-for-Tat (TFT) and All-Defect (AD) (see Chap. 1 for details). Let w be the probability that the same two players interact in the following step as well. Let V(i/j) be the expected score of a player with strategy i obtained from the interaction with a neighbor adopting strategy j (i, j ¼ T or D, indicating TFT and AD, respectively). Table 2.1B shows the expected total payoffs V(i/j). The reproductive rate and and survivorship are dependent on the score. If the total population size is sufficiently large, the fitnesses of TFT and AD players, denoted by FT and FD, are expressed by using ρT and ρD, the fractions of TFT and AD players, respectively in the population (note ρT + ρD ¼ 1, as there is no vacancy):
 R wP ρT þ S þ ρ , 1w 1w D
 wP P ρ þ ρ , FD ¼ F0 þ T þ 1w T 1w D FT ¼ F0 þ

ð2:1aÞ ð2:1bÞ

where F0 is the baseline fitness that a player enjoys without social interaction. Other terms are the average payoffs in Table 2.1 with the weight of the fraction of two

34

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

Table 2.1 Payoff matrix for games: (A) The scores of the prisoner’s dilemma game to player A interacting with player B. The prisoner’s dilemma game satisfies the relations, T > R > P > S and 2R > T + S. Here, T ¼ 5, R ¼ 3, P ¼ 1 and S ¼ 0; (B) The expected total score obtained by a player A through interacting with a player B (A)

Player A

Cooperate Defect

Player B Cooperate R T

Defect S P

(B)

Player A

TFT AD

Player B TFT V(T/T ) ¼ R/(1  w) V(D/T ) ¼ T + Pw/(1  w)

AD V(T/D) ¼ S + Pw/(1  w) V(D/D) ¼ P/(1  w)

Fig. 2.1 The phase plane of the completely mixed model, the evolutionary game with fitness given by Eqs. (2.1a) and (2.1b). Horizontal axis is the probability of reiteration w, vertical axis is the density of TFT ρT. The region in which ρT increases is indicated by shade, given by Eq. (2.2). For w less than 0.5, TFT should always decrease with time and AD is the only evolutionarily stable strategy. In Contrast, for w > 0.5, TFT can be stable as it refuses the invasion of rare AD. However, AD is always evolutionarily stable because rare TFT cannot invade the population dominated by AD

strategies in the population. Then the density of TFT increases if FT > FD. This inequality is rewritten as: ρT >

ð1  wÞðP  SÞ 1w ¼ : ðS þ T  2PÞw þ R  S  T þ P 3w  1

ð2:2Þ

The shaded area of Fig. 2.1 illustrates Eq. (2.2). The horizontal axis is for w. The range of w for TFT to be evolutionarily stable is given by Eq. (2.2) with

2.3

One-Dimensional Lattice Under the Score-Dependent Viability Model

35

ρT ¼ 1. It is rewritten as w > wa ¼ (T  R)/(T  P) ¼ 0.5. In contrast, Eq. (2.2) always fails if ρT ¼ 0 and w < 1, implying that TFT if rare cannot invade a population dominated by AD. Hence AD is evolutionarily stable when w < 1 (see Axelrod and Hamilton 1981).

2.3

One-Dimensional Lattice Under the Score-Dependent Viability Model

Now, we study the iterated prisoner’s dilemma in a lattice-structured habitat under the viability model. In the initial population, each lattice site is filled either by TFT or AD randomly with a given probability. Then each player engages in the iterated prisoner’s dilemma game with their neighbors. Let z be the number of neighbors with whom a player interacts. In a one-dimensional lattice, z ¼ 2. In a two-dimensional lattice, each player has eight neighboring sites (z ¼ 8). The mortality of the player is determined by its total score, which is the sum of the scores obtained by interacting with the z neighbors. After the death of an individual, the empty lattice site is filled immediately by a copy of a neighbor randomly chosen among z possible sites. This is called the nearest neighbor migration model by Matsuda (1987). Hereafter, we call it the score-dependent viability model or the viability model. The model is a continuous time Markov chain, and in a sufficiently short time interval only a single event of state transition occurs. The total score B of an individual depends on the strategy adopted (TFT or AD) and on the number of TFT neighbors, denoted by n (0  n  z). Let BT,n and BD,n be the scores of a TFT player and an AD player, respectively, if surrounded by n TFT neighbors and (z  n) AD players. They are: BT,n ¼ nV ðT=T Þ þ ðz  nÞV ðT=DÞ

ð2:3aÞ

BD,n ¼ nV ðD=T Þ þ ðz  nÞV ðD=DÞ:

ð2:3bÞ

and

In the viability model, the mortality of a player is a decreasing function of the total score. Specifically, we assume that the instantaneous mortality of a TFT player surrounded by n TFT and (z  n) AD decreases exponentially with B: M T,n ¼ exp ðαBT,n Þ,

ð2:4aÞ

36

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

and that of an AD player is: M D,n ¼ exp ðαBD,n Þ:

ð2:4bÞ

In a short time interval of length Δt, a player dies with probability MT, nΔt or MD, nΔt. Constant α was chosen so that the mortality varies between sites of different states and different combinations of neighbors. For a reiteration probability w close to one, the total score becomes very large, because the expected number of iterations is proportional to 1/(1  w), and the mortality of different types in Eq. (2.4b) are all very small. To prevent this situation, we chose α as a decreasing function of w, specifically α ¼ 0.1(1  w) in this chapter.

2.3.1

Computer Simulations on a One-Dimensional Lattice Model

We carried out computer simulation of the model, in a one-dimensional lattice (z ¼ 2). To remove the effect of edges, we used a periodic boundary condition: i.e., the lattice is circular (the rightmost site is the nearest neighbor of the leftmost site), with a lattice size of 500. In the initial pattern, sites were filled independently either by TFT or AD with a given probability. We computed the model until a time in which either one of the two types occupied the whole population. We computed runs with the initial population with different fractions of TFT: ρT with 0.1, 0.3, 0.5, 0.7, and 0.9. Examples of spatial patterns of the model are shown in Fig. 2.2a. Starting from a random spatial pattern, clusters composed of the same type are quickly formed. Let ρT be the total fraction of TFT in the whole lattice, called the global density of TFT, and ρD be the global density of AD. Figure 2.2b illustrates the final outcome of the simulation for different initial density ρT and parameter w. When w is less than 0.6, AD is finally fixed in the population for any initial global density ρT. When the parameter w is larger than 0.6, TFT became fixed for any initial density ρT. No evolutionary bistability was observed. These results are quite different from the behavior of the completely mixing model (Fig. 2.1). Next, we apply analytical calculations to confirm the computer simulation in Fig. 2.2b.

2.3

One-Dimensional Lattice Under the Score-Dependent Viability Model

37

Fig. 2.2 Computer simulations of one-dimensional lattice model. a The spatial patterns generated by the model when w ¼ 0.8 and initial ρT is 0.5. There are 500 sites, and black and white points are for those occupied by TFT and those occupied by AD, respectively. Vertical axis is for time. The initial population (t ¼ 0) is random (as shown by the top row), but a strongly clumped spatial distribution is quickly formed, in which the lattice is composed of long runs of TFT and those of AD. Finally TFT won in this simulation. b The outcome of simulation for different iteration probability w and initial density ρT. For each pair of parameters, 100 runs were computed. Open circles indicate sets of parameters for which all 100 runs end up with the extinction of TFT and the fixation of AD, while solid circles are parameter values for which all the runs show the fixation of TFT. Shaded circles are those for which some runs end up with the fixation of TFT (the fractions are indicated by numerals), but others with the fixation of AD. When w is less than 0.6, AD is an evolutionarily stable strategy. When 0.6 is larger than 0.6, TFT is evolutionarily stable

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2.3.2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

The Dynamics of Density

Here we present the time-differential equations for the score-dependent viability model. The time change of ρT, the global density of TFT, is given by: dρT ¼ M T,1 ½density of DTT  M T,0 ½density of DTD dt þ M D,1 ½density of DDT þ M D,2 ½density of TDT:

ð2:5Þ

Here we call a site occupied by TFT player a T-site, and a site occupied by an AD player as a D-site for short. The first term of the RHS of Eq. (2.5) indicates the rate at which a T-site located between a D-site and a T-site changes to a D-site. It is a product of the mortality of T, denoted by MT, 1, and the density of the triplet “DTT” in the lattice, i.e., the probability of a randomly chosen triplet is “DTT.” We need to consider a factor 2 because there is another triplet “TTD” having the same contribution as “DTT.” However, this factor is canceled out by another factor 1/2, the probability for the middle T to be replaced by D instead of T. Similarly, the second term in the RHS is for the transition of T in a triplet “DTD” to D. The third and the fourth terms are for the transition of D to T in the middle of “DDT” and in the middle of “TDT,” respectively. The frequency of a triplet “DTT” cannot be expressed only using global densities, such as ρT and ρD. We need to introduce a conditional probability for a site to be T if it is chosen next to a T site. This is expressed as qT/T and called conditional density or local density of T sites (Matsuda et al. 1992; Harada and Iwasa 1994). In a similar way, qa/bc {a, b, c ¼ T or D} indicates a conditional density of a higher order. For example, qT/TD is the probability that TFT is in the neighborhood of TFT whose neighbor is AD. Using these notations, the density of triplet DTT is ρTqT/TqD/TT. We can rewrite the frequencies of the various triplets in Eq. (2.5) as: dρT ¼ M T,1 ρT qT=T qD=TT  M T,0 ρT qD=T qD=TD þ M D,1 ρD qT=D qD=DT dt þ M D,2 ρD qT=D qT=DT:

ð2:6Þ

Hence, to calculate the dynamics of the average density of TFT, we need to know conditional densities, such as qT/T and qT/TT, which include information concerning the correlation of states between close neighbors. In the following sections, two methods of constructing a closed dynamical system, based on “mean-field approximation” and “pair approximation,” are developed.

2.3

One-Dimensional Lattice Under the Score-Dependent Viability Model

2.3.2.1

39

Mean-Field Approximation

Mean-field approximation means to neglect spatial structure or to assume a random spatial configuration. Under this assumption, the local density and other probability on the interaction between more than two players are the same as the global density: qT/ab ¼ qT/c ¼ ρT {a, b, c ¼ T or D}. Then the density of triplet DTT is simply ρ2T ρD. If we adopt this simplification and ρD ¼ 1  ρT, the dynamics of global density given by Eq. (2.6) become: dρT ¼ ρT ð1  ρT ÞfM T,1 ρT  M T,0 ð1  ρT Þ þ M D,1 ð1  ρT Þ þ M D,2 ρT g: ð2:7Þ dt These dynamics of a single variable have two trivial equilibria: ρT¼ 0 and ρT¼1. In addition, there may be an intermediate equilibrium with 0 < ρT < 1, such as: ρT ¼

M T,0  M D,1 : M T,0  M D,1  M T,1 þ M D,2

ð2:8Þ

The intermediate equilibrium (2.8) is feasible only when w > wa ¼ (2 T  S  R)/ (2 T  S  P), which is wa ¼ 7/9 for payoffs in Table 2.1. By examining the sign of Eq. (2.7), we know that, if 0  w  wa, ρT ¼ 1 is unstable and ρT ¼ 0 is globally stable. The extinction of TFT is inevitable, and AD is the unique evolutionarily stable strategy. In contrast, if wa < w < 1, both ρT ¼ 1 and ρT ¼ 0 are locally stable, and the intermediate equilibrium given by Eq. (2.8) is unstable. The system is bistable and the evolutionary end point depends on the initial density of TFT. If the density of TFT players is higher than the unstable point, TFT players tend to increase and become fixed. If not, AD players occupy the whole population instead. Hence TFT is evolutionarily stable when reiteration probability w is larger than wa, and AD is always evolutionarily stable. However, both Eqs. (2.7) and (2.8) are unable to explain the results of computer simulation (Fig. 2.2b). First, the threshold value of w for a TFT population to be locally stable from computer simulation is about 0.6, clearly smaller than 7/9 (¼ 0.7777) predicted by the mean-field approximation. Second, the system does not show bistability for any value of w – either a TFT population or an AD population is globally stable depending on w. Especially notable is that the area in which TFT is predicted to increase is much smaller than the shaded area in Fig. 2.1. This implies that cooperation is much more unlikely to evolve in the dynamics of a mean-field approximation of the model than in the completely mixing model, in spite of the fact that the spatial configuration of individuals is neglected in both models. We will discuss the reason for this difference later.

40

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2.3.2.2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

Pair Approximation

Pair approximation is a method of constructing a system of ordinary differential equations for the global and local densities. In some cases, it predicts the population dynamics of lattice-structured models accurately, while the equations based on the mean-field assumption, such as Eq. (2.7), fail to do so (Sato et al. 1994; Harada et al. 1995). Pair approximation keeps the distinction between the average density and the local density, but does not consider the correlation beyond nearest neighbors. Note that pair approximation does not assume perfect independence between non-nearest neighbor sites. It assumes that the correlation between non-nearest neighbor sites can be approximated by the product of the nearest neighbor correlation. For an example, consider a triplet, ABC, in which A and B are correlated in state and B and C are correlated because they are neighbors. Then A and C are also correlated. Pair approximation assumes that the correlation between A and C is caused only by the indirect effect via their common neighbor B. In reality, A and C can be more or less strongly correlated than the simple product of correlation coefficients between the two nearest neighbors, and hence the pair approximation may not be exact. Let ρTT be the doublet density, i.e., the probability that a randomly chosen pair of nearest neighbors are both TFT. Then the local density of TFT is the ratio, qT/ T ¼ ρTT/ρT. Hence we have: dqT=T 1 dρTT qT=T dρT ¼  , dt ρT dt ρT dt which implies that the dynamics of doublet density are needed to compute the dynamics of local density. The time change is derived as follows: dρTT ¼ M T,1 ρT qT=T qD=TT þ M D,1 ρD qT=D qD=DT þ 2M D,2 ρD qT=D qT=DT : dt

ð2:9Þ

The first term of the RHS of Eq. (2.9) indicates the rate of transition from a triplet TTD to another triplet TDD. The second term is the rate of transition from DDT to DTT, which produces a new TT pair. Then the last term is the rate of transition from TDT to TTT, which creates two new “TT” pairs expressed by factor 2. To construct a closed dynamical system of global density ρT and local density qT/ T, we neglect the correlation of state beyond nearest neighbor pairs: qa=bc ¼ qa=b fa, b, c ¼ T or Dg,

ð2:10Þ

2.3

One-Dimensional Lattice Under the Score-Dependent Viability Model

41

which is called pair approximation, or doublet decoupling approximation (Matsuda et al. 1992; Sato et al. 1994). Note also ρTD ¼ ρDT. In addition, we have the following relations, coming from the definition of conditional probabilities: ρ D ¼ 1  ρT ,

ð2:11aÞ

qD=T ¼ 1  qT=T ,

ð2:11bÞ

1  2ρT þ ρT qT=T , 1  ρT
 ρT 1  qT=T ¼ : 1  ρT

qD=D ¼

ð2:11cÞ

qT=D

ð2:11dÞ

Then a pair of ordinary differential equations for global density and local density are derived:  
 M ρ M ρ dρT ¼ ρT 1  qT=T qT=T M T,1 þ M T,0 þ D,1 T  D,2 T dt 1  ρT 1  ρT  M 1  2ρT þ D,2 ,  M T,0 þ M D,1 1  ρT 1  ρT   2  dqT=T
M D,1 ρT M D,2 ρT qT=T M T,1 þ M T,0 þ  ¼ 1  qT=T dt 1  ρT 1  ρT  1  2ρT 2M D,2 :  þM D,1 þ 1  ρT 1  ρT

ð2:12aÞ

ð2:12bÞ

This system of autonomous equations can be analyzed by the standard technique for nonlinear dynamics. All the points on the line qT/T ¼ 1 are equilibria. The asymptotic behavior of the system is different depending on whether the reiteration probability w is larger than a critical value wb ¼ (T  S  R + P)/(T  S) ¼ 0.6. The trajectories converge to (ρT, qT/T) ¼ (0, 1) for w < wb; they converge to (ρT, qT/T) ¼ (1, 1) for w > wb. When w ¼ wb, different trajectories converge to different point on the line of equilibria qT/T ¼ 1, which are neutrally stable. Figure 2.3a shows the trajectories of pair-approximation dynamics Eqs. (2.12a) and (2.12b) when w ¼ 0.8. Figure 2.3b is the phase plane for pair-approximation dynamics Eqs. (2.12a) and (2.12b). Then when w is smaller than wb, which is 0.6 for payoffs in Table 2.1, AD is evolutionarily stable, and when w is larger than wb. TFT is evolutionarily stable. No bistability is predicted. These are consistent with the results of computer simulation (Fig. 2.2b) and the invasion success probability in Sect. 2.3.3.

42

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The Evolution of Cooperation in a Lattice-Structured Population Under. . .

Fig. 2.3 Analyses of one-dimensional lattice model: a Curves are the trajectories of the dynamics of pair approximation when w is 0.8. Dots are from computer simulation where the time interval between points is 10. A broken line is for qT/T ¼ ρT implying a random spatial pattern. The horizontal axis is global density ρT and the vertical axis is local density qT/T. Arrows indicate the direction of movement along trajectories. qT/T keeps increasing and hence TFT clusters become larger and larger. b The phase plane of the dynamics based on pair approximation, Eqs. (2.12a) and (2.12b). Horizontal axis is the probability of reiteration w, vertical axis is the density of TFT ρT. The region in which ρT increases is indicated by shade. When w < wb, AD is an evolutionarily stable strategy. When w > wb, TFT is an evolutionarily stable strategy and the AD is no longer evolutionarily stable. The predictions by the dynamics based on pair approximation are consistent with the results of computer simulation of Fig. 2.2b

2.3.3

Invasion Success Probability in the One-Dimensional Lattice Model

Although a cooperative behavior is evolutionarily stable in the completely mixing model, it is quite difficult for the cooperation to establish itself in a population initially dominated by noncooperative behavior, because AD is always evolutionarily stable and repels the invasion of rare TFT. However, in the one-dimensional lattice model, both computer simulation and pair-approximation dynamics show that TFT can invade successfully a population dominated by AD if the probability of iteration w is sufficiently high. Computer simulation shows that TFT and AD quickly form clusters of the same strategy and that the movement of their boundaries determines the fate of the system (Fig. 2.2a). Changes of spatial patterns occur if TFT on the edge of a TFT cluster becomes AD or if AD on the edge of the AD cluster changes to TFT. The location of the boundary between two clusters follows a random walk. Then the condition for the successful invasion of TFT in the population occupied by AD players can be derived by the standard techniques for birth-and-death processes (Goel and RichterDyn 1974). Starting from a single TFT, it may increase or decrease its number, and

2.3

One-Dimensional Lattice Under the Score-Dependent Viability Model

43

with a large probability, TFT goes extinct, leaving the population with AD only. However, there can be a positive probability for TFT to increase in number, and TFT has a chance to occupy all the sites in the population. If this probability is positive, however small, TFT will eventually replace the AD population after many trials of repeated invasions or recurrent mutations. We note that the descendants of the initial single invader TFT always form a continuous sequence of T. We call this a “cluster,” and the number of TFTs the “cluster size.” Let Qn be the probability that a cluster containing n TFTs goes extinct eventually. Let λnΔt be the transition probability in a short time interval of Δt from the cluster containing n TFTs to a cluster with (n + 1) TFTs. Let μnΔt be the transition probability from n TFT’s to (n  1) TFTs. Thus we have the basic equation: Qn ¼ Qn ð1  λn Δt  μn Δt Þ þ Qnþ1 λn Δt þ Qn1 μn Δt:

ð2:13Þ

We first consider the case in which the cluster will not go extinct once the cluster reaches a sufficiently large size k + 1. Later, we make k infinitely large. Then the boundary conditions are: Qn ¼ 0 ðn  k  1Þ, Q0 ¼ 1: The transition rates are:  λn ¼  μn ¼

0 M D,1

n¼0 , n ¼ 1, 2, . . .

M T,0

n¼1

M T,1

n ¼ 2, 3, . . .

,

where n ¼ 0 is an absorbing point. We can calculate Q1 !

1 1þ

Q1 ! 1

eαðPþT2SÞ

 eαðPSþ1wÞ RP

if w > wb , if w  wb :

when k !1. We derive the probability QT ¼ 1  Q1 that TFT survives ultimately, starting from a single TFT player (see Nakamaru et al. (1997) for more details). It is positive for w > wb, where wb ¼ (T  S  R + P)/(T  S) is the critical value, which is wb ¼ 0.6 if payoffs in Table 2.1 are used. In contrast, QT ¼ 0 for w wb.

44

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

A single AD invades TFT population

A single TFT invades AD population

Fig. 2.4 The probability of invasion success. A solid and broken lines are the predictions by QT and RT derived by the birth-and-death processes. Solid and open circles show the results of the computer simulations, which are consistent with the predictions. A single TFT invader in the population occupied by AD can increase in number and becomes fixed with a positive probability if w > 0.6 but must go extinct if w < 0.6. In contrast, a single AD invader in the population occupied by TFT can increase and becomes fixed with a positive probability if w < 0.6 but cannot increase if w > 0.6. These results are consistent with the predictions by pair-approximation (Fig. 2.3c)

Similarly, the probability for a single AD player to invade a population composed of TFT can be derived by the equations based on birth-and-death processes. The probability that AD survives in the lattice where only one AD existed in the initial population is denoted by RD ¼ 1  R1 (see Nakamaru et al. (1997), where R1 !

1 1

R1 ! 1

eαðTPÞ

þ eαð2RPS1wÞ RP

if w > wb , if w  wb :

when k !1. RD is positive for w < wb, but zero for w  wb. The two lines in Fig. 2.4 illustrate the probabilities of successful invasion for different w. We see that TFT is evolutionarily stable for w larger than wb and that AD is the evolutionarily stable strategy for w less than wb. Computer simulation results using a finite sized lattice (500 sites) are indicated by solid and open circles, and are consistent with the curves given by the calculations in Nakamaru et al. (1997). If the invasion of a single mutant is not a unique event but occurs recurrently, then a small but positive probability of establishment is sufficient for the ultimate success of invasion. Figure 2.3 therefore implies that TFT beats AD for w > wb and AD beats TFT for w < wb. This result is consistent with the results of pair approximation derived from Eqs. (2.12a) and (2.12b) (Fig. 2.3b). In the following, we will show that this is also consistent with the results of pair-edge method.

2.3

One-Dimensional Lattice Under the Score-Dependent Viability Model

2.3.4

45

The Pair-Edge Method

The computer simulation shows that players form clusters of the same type, and tend to play assortatively, i.e., more often with the same type than randomly. Hence, the dynamics of the model depend on the movement of the boundary between a cluster of TFT and a cluster of AD that are located side by side. By considering the speed of the movement of the boundary between two large clusters, we can estimate the condition in which either TFT or AD can spread, which is a simple version of “pairedge method” that is very useful in analyzing the dynamics of lattice-structured models (Ellner et al. 1998). We consider the case in which the whole system is occupied only by a large cluster of TFT and a second cluster of AD, and examine the movement of their boundary. Now we calculate the speed and the direction of the boundary movement between clusters. In terms of the pair-edge method please see Ellner et al. (1998) for more details. We assume that the clusters of TFT and AD are arranged in a one-dimensional lattice such as TTTTDDDD, and that right is the positive direction of the velocity, v, and left is negative. For example, in order for the boundary to move to the right in one site, it is necessary that AD next to TFT dies and TFT colonizes the vacant site, which is expressed by the first term of the following: 1 1 1 v ¼ 1  d  M D,1  þ 0  d  M D,1   1  d  M T,1  þ 0  d 2 2 2 1  M T,1  , ð2:14Þ 2 where d indicates the velocity that the boundary moves one site during a unit time. We assume that d is positive. The second and fourth terms indicate that if an AD (or TFT) next to a TFT (or AD) dies and an AD (or a TFT) colonizes the vacant site then the boundary does not move. The third term shows that if TFT next to AD dies and that AD colonizes the open site, then the boundary moves to the left by one. If v > 0, the cluster of TFT expands at the expense of AD, whilst, if v < 0, the cluster of AD expands. Eq. (2.14) is equivalent to BT,1 > BD,1. Then we can get wb at which the boundary between TFT and AD stops moving as follows: wb ¼ ðT  S  R þ PÞ=ðT  SÞ: If w > wb, TFT can spread, whilst, if w < wb, AD can spread.

ð2:15Þ

46

2.4

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

Two-Dimensional Lattice Under the Viability Model

We also examined the model in a two-dimensional square lattice. The number of nearest neighbors with whom a single player interacts is z ¼ 8 (i.e., we assume the Moore neighborhood instead of the Neumann neighborhood of z ¼ 4). Just as for Eq. (2.6) for the one-dimensional model, the time change of global density can be expressed as: z1   z X z dρT zn X M T,n ½density of fT, ng þ ¼ z dt n n¼0 n¼1   z n  M D,n ½density of fD, ng , z n

ð2:16aÞ

where {T, n} indicates a T-site surrounded by n T-sites and (z  n) D-sites. With pair approximation, it can be expressed as: ½densityoffT,ng  ρT ðqT=T Þn ðqD=T Þzn :

ð2:16bÞ

The first term in Eq. (2.16a) indicates the transition of a TFT to an AD given that it interacts with n TFT and (z  n) AD. The second term comes from the transition of an AD to a TFT if it interacts with n TFT and (z  n) TFT.

2.4.1

Mean-Field Approximation in the Two-Dimensional Lattice

The equations derived from the mean-field approximation (see Nakamaru et al. 1997) have two trivial equilibria: ρT ¼ 0 and ρT ¼ 1. In addition, there may be an intermediate equilibrium (0 < ρT < 1) if w > wd ¼ (40  S  7 T)/(32  S + P), which is wd ¼ 19/33 for the payoffs in Table 2.1. This result suggests that AD is always the evolutionarily stable strategy (when w < 1). When w > wd, TFT is also evolutionarily stable, and the system is bistable.

2.4.2

Pair Approximation in the Two-Dimensional Lattice Model

By adopting pair approximation, we can construct a closed dynamical system of ρT and qT/T, as in Eq. (B4) in Nakamaru et al. (1997). A system of autonomous equations such as Eq. (B4) can be analyzed by the standard techniques of nonlinear

2.4

Two-Dimensional Lattice Under the Viability Model

47

Fig. 2.5 Analyses of the two-dimensional lattice model. a Curves are the trajectories of the dynamics of pair approximation when w is 0.6. Dots are from computer simulation where the time interval between points is 10. The horizontal axis is global density ρT and the vertical axis is local density qT/T. The broken line is for qT/T ¼ ρT implying a random spatial pattern. Arrows indicate the direction of movement along trajectories. The system is bistable. b The phase plane of the dynamics based on pair approximation, Eq. (B.4) in Nakamaru et al. (1997). Horizontal axis is the probability of reiteration w, vertical axis is the density of TFT ρT. The region in which ρT increases is indicated by shade, given by Eq. (B.4). When w is less than 0.77, AD is an evolutionarily stable strategy. When w is larger than about 0.53, TFT is an evolutionarily stable strategy. When the range of w is between 0.53 and 0.77, whether TFT or AD will be fixed in the lattice depends on ρT. The prediction is intermediate between the complete mixing case (Fig. 2.1) and one-dimensional case (Fig. 2.3b). Note that the rare TFT can invade the population composed of AD if probability of iteration w is sufficiently high

dynamics. All the points on the line qT/T ¼ 1 are equilibria. The asymptotic behavior of the system depends on the value of w. According to numerical analysis, all the trajectories starting from internal points converge to ρT ¼ 0 for 0  w  0.49, and all the trajectories converge to (ρT, qT/ T) ¼ (1, 1) for w  0.77. For w between 0.77 and 0.49, whether the trajectories converge to ρT ¼ 0 or to (ρT, qT/T) ¼ (1, 1) depends on the initial density of TFT. The system is bistable. For example, for w ¼ 0.6, two trajectories of pair-approximation dynamics given by Eq. (B4) in Nakamaru et al. (1997) are illustrated in Fig. 2.5a. Figure 2.5b shows the change in ρT for different w. This indicates that AD is evolutionarily stable when w is less than 0.77 and that TFT is evolutionarily stable when w is larger than 0.49. The prediction of pair-approximation dynamics for the two-dimensional model in Fig. 2.5b is intermediate between the completely mixing case (Fig. 2.1) and the one-dimensional case (Fig. 2.3b). This can be understandable as the number of neighbors is 2 for the one-dimensional lattice model, 8 for the two-dimensional lattice, and infinitely large for the completely mixing model.

48

2.4.3

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

Computer Simulations

The method of computer simulation of the two-dimensional square lattice model (z ¼ 8, with the Moore neighborhood) was the same as for the one-dimensional model explained before. We assumed a periodic boundary condition: i.e., the lattice was a torus (the rightmost column is the nearest neighbor of the leftmost column, and the top row is the nearest neighbor of the bottom row). We have carried out simulations both on a lattice of size 2020 and on a lattice of size 100100. We normally computed the model until a time at which one of the two types is fixed in the population. If the system is bistable, the outcome depends on the initial density. The initial population is of a random spatial pattern with different fractions of TFT: We computed ρT with 0.1, 0.3, 0.5, 0.7, and 0.9. Examples of spatial patterns of the model on the lattice of 2020 are illustrated in Fig. 2.6a–c. The phase plane on w-ρT for the lattice of size 2020 is given in

Fig. 2.6 The spatial patterns generated by the two-dimensional lattice model with w ¼ 0.8 and initial density ρT ¼ 0.5 when lattice size is 20  20. a Initial random distribution (t ¼ 0), b patchy distribution is quickly formed (t ¼ 100), and c TFT is eventually fixed (t ¼ 700). d This illustrates change in initial ρT with time for different w when lattice size is 20  20. Open circles, shaded circles and solid circles in the graph show the computer simulations of AD fixed, either AD or TFT fixed and TFT fixed in the lattice, respectively. The numbers below shaded circles are the fractions of runs in which TFT become fixed. When w is less than about 0.8, AD is an evolutionarily stable strategy. When w is larger than about 0.4, TFT is an evolutionarily stable strategy. For w between 0.5 and 0.8, the system shows bistability, i.e., whether or not TFT is fixed depends on the initial global density of TFT. The computer simulation results are explained well by the dynamics based on pair approximation (Fig. 2.5b). e This illustrates change in initial ρT with time for different w when lattice size is 100  100

2.4

Two-Dimensional Lattice Under the Viability Model

49

Fig. 2.6d. TFT is the evolutionarily stable strategy for w larger than a critical value which lies between 0.5 and 0.6. AD is the evolutionarily stable strategy for w less than another threshold between 0.6 and 0.8. For w between these two critical values, this system shows bistability and the evolutionary outcome depends on the initial global density of TFT. For several sets of parameters indicated by gray circles, some runs end up with fixation of TFT but other runs end up with fixation of AD. This is caused by stochasticity due to the finiteness of the lattice size. The phase plane on w-ρT for a large lattice of 100100 is in Fig. 2.6e. The parameter region showing fixation of TFT and fixation of AD are quite similar, but gray circles are absent. Depending on the parameter w and initial condition ρT, the evolutionary outcome is either fixation of TFT (solid circles) or fixation of AD (open circles). Bistability was observed for the case of w ¼ 0.6; TFT is fixed in all 50 runs if the initial ρT is 0.7 and 0.9, but AD is fixed if the initial ρT is 0.5 or smaller. The results of the computer simulation, in Fig. 2.6d, e, are again consistent with the pair-approximation dynamics (as shown in Fig. 2.5c), but not with the mean-field approximation. The mean-field dynamics predict that AD always repels the invasion by rare TFT. However, according to the computer simulation, rare TFT can invade an AD population if w is larger than 0.7, which is similar to the 0.77 predicted by the pair-approximation dynamics. We must note, however, that our results of computer simulation on lattices of finite size (2020 and 100100) are different from behavior predicted for an infinitely large lattice, as there is a general mathematical argument that no bistability is possible on an infinitely large lattice irrespective of the dimensionality (Durrett 1980; Gray 1982; Liggett 1978). The arguments can be explained intuitively as follows: For bistability to occur in the system, we must consider the situation where one type is dominating all the local areas in the system. However, because the model is stochastic, and because the lattice size is infinitely large, there will always appear an area in which one of the two types dominates, and another area in which the other type dominates. Then, whether one type or the other wins is determined simply by the movement of the boundary between two areas, in each of which one of the two types dominates. Hence, if we analyze an infinitely large lattice, the phase plane should show no bistability, and the prediction by pair approximation, such as in Fig. 2.5b, is not valid. It will look more like Fig. 2.3b. Pair approximation assumes that the correlation between sites can be approximated considering correlation between neighbors. It is quite plausible that pair approximation fails if the whole system is segregated into large subareas in which the relative density of TFT and AD greatly differ and long-range spatial correlation of states is large. The computer simulation of the model in a finite-size lattice showed bistability, which should disappear if we examine an infinitely large lattice.

50

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

On the other hand, if in biological or social situations there are a finite number of lattice sites, then the model in a lattice of reasonably large (finite) size may be more suitable as a modeling tool. The behavior of this lattice was closely described by pair-approximation dynamics.

2.5

Score-Dependent Fertility Model

Here, we present the score-dependent fertility model (or the fertility model) in a lattice-structured habitat (Nakamaru et al. 1998). In the initial population, each lattice site is filled either by TFT or AD chosen randomly with a given probability. Then players engage in the iterated prisoner’s dilemma game with each of their neighbors, and the results of the iterated game affect the fitness of the players. Individuals die randomly, independent of the game’s score. We chose the time unit so that any one site is killed on average once in a unit time. After an individual dies randomly, the created open lattice site is filled immediately by a copy of a neighbor, and the probability for a particular neighbor to be chosen is proportional to the game’s score. The model is a continuous time Markov chain, and in a sufficiently short time interval only a single event of state transition occurs. Consider the way to colonize the open site in the one-dimensional lattice model. For short, T stands for TFT and D stands for AD in the following. Suppose that there is a sequence TDDTD in the one-dimensional lattice, and that D in the middle of this sequence dies by chance. Then either D or T next to the open site colonizes to it. The score of D next to the middle D is BD,1, and the score of T next to the middle D is BT,0. We assume that the probabilities that D and T can colonize to the vacancy is proportional to these scores, namely BD,1/(BD,1 + BT,0) and BT,0/(BD,1 + BT,0), respectively. Since BD,1 is larger than BT,0, AD is more likely to fill the vacant site than TFT.

2.5.1

Computer Simulations on a One-Dimensional Lattice

We carried out computer simulation of the model in a one-dimensional and a two-dimensional lattice, but we here focus our attention to the one-dimensional model, and we will discuss the two-dimensional model later. According to computer simulations, TFT did not coexist with AD in our system in one- and two-dimensional lattices, except for versions with modified state-transition rules. In such cases, we computed the model until one of the two types occupied the whole population. We computed runs with the initial population having different initial fractions of TFT.

2.5

Score-Dependent Fertility Model

51

Fig. 2.7 The results of the computer simulation on the one-dimensional lattice (the lattice size is 1000). The horizontal axis is the initial density of TFT and the vertical axis is w. Solid circles indicate the parameters for which all the 50 runs converged to ρT ¼ 1. The gray one indicates the parameter in which some runs converged to ρT ¼ 1 and other runs converged to ρT ¼0. The number over the gray circle indicates the ratio of 50 runs converges to ρT ¼ 1. The payoffs are: T ¼ 5, R ¼ 3, P ¼ 1 and S ¼ 0

We used a one-dimensional lattice of 1000 sites in the following. We set the time unit so that each site is on average examined once for the state-change possibility in a unit time. An example of the time-change in spatial patterns of the one-dimensional model is close to Fig. 2.2a, in which the vertical axis is the time. Figure 2.7 shows the final outcome of the computer simulation for different initial densities of TFT (initial ρT) and probabilities of iteration w. From this result of the one-dimensional lattice model, we can conclude that TFT is always ESS and that AD cannot invade a TFT population even if w is very small. In contrast, TFT can always invade an AD population. These results are quite different from those of the completely mixing model (Fig. 2.1). We conclude that the local interaction and explicit spatial structure in the lattice model give a great advantage to TFT in an AD population. The model’s behavior near the threshold depends on the lattice size, in a similar way to the scoredependent viability model (Nakamaru et al. 1997).

2.5.2

Mathematical Analyses in a One-Dimensional Lattice

We analyze the score-dependent fertility model in a one-dimensional lattice using the same methods as the viability model.

52

2.5.2.1

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

Mean-Field Dynamics

According to the same method in the viability model, mean-field approximation can be applied to the fertility model (see Appendix D in Nakamaru et al. (1998)). The steady states are ρT ¼ 0 and ρT ¼ 1, which are both stable solutions. If (T  R)/ (T  P)  w < 1, there is an unstable solution ρT with 0 < ρT < 1. When (T  R)/ (T  P) < w < 1, TFT is an ESS, and the system is bistable. If the initial frequency is larger than ρT , TFT increases and eventually occupies the population at equilibrium. Otherwise, AD eventually occupies the whole population at equilibrium. The prediction of the mean-field dynamics is similar to the completely mixing model (Fig. 2.1), but slightly less favorable for TFT; ρT > (1  w)/(3w  1). However, it is very different from the computer simulations of the lattice models shown in Fig. 2.7. The mean-field dynamics do not distinguish between global and local densities, thus neglect the spatial structure, and fail to predict the behavior of the lattice model.

2.5.2.2

Pair Approximation

Pair approximation can be applied to the fertility model. We construct a pair of ordinary differential equations for global and local densities of TFT (see Appendix D in Nakamaru et al. 1998). These pair-approximation dynamics in a one-dimensional lattice show that qT/T increases with time and that all points on a line qT/T ¼ 1 are equilibria. Starting from any density of TFT with random spatial pattern, many small clusters of TFT are formed quickly, and then these clusters merge or disappear. We can show that, if w > wc, qT/T ! 1 and ρT ! 1 with time, in which wc ¼ (P(P + T )  R(R + S))/ (R(P  S) + P(T  P)). Whilst if w > wc, qT/T ! 1 and ρT ! 0. The predictions of pair approximation are the same as those of invasion success probability, the pair-edge method, and the computer simulations appearing in the following sections, but are quite different from the predictions of the mean-field approximation.

2.5.2.3

Invasion Success Probability

In the one-dimensional model, we can calculate the probability of successful invasion by a single TFT to a population occupied by AD using the same technique of birth-and-death processes (Goel and Richter-Dyn 1974) as in the viability model (see Appendix B in Nakamaru et al. (1998)). Then we can obtain the invasion success probability by a single TFT, QT. One TFT cannot invade the population occupied by AD (QT ¼ 0) if w  wc and one TFT can invade the population occupied by AD if

2.5

Score-Dependent Fertility Model

53

w > wc, in which wc ¼ (P(P + T )  R(R + S))/(R(P  S) + P(T  P)). When the payoffs are as in Table 2.1A, wc is negative, and TFT can invade for all w. In a similar way, we derive the probability RD for a single AD to survive and ultimately dominate the population as follows (see Nakamaru et al. (1998)): RD ¼ 0 if w wc, and RD is positive if w < wc.

2.5.2.4

The Pair-Edge Method

Here we also apply the pair-edge method to the score-dependent fertility model (Ellner et al. 1998). We show that wc ¼ (P(P + T )  R(R + S))/(R(P  S) + P(T  P)) is the threshold of w (see Nakamaru et al. (1998) for more details). The results are: if w > wc, there is a positive probability for a cluster of TFT to spread and occupy the whole population; whilst clusters of AD will become smaller and go extinct ultimately. In contrast, if w < wc, the cluster of AD spreads, and TFT will go extinct.

2.5.3

Two-Dimensional Model

The methods of analyses applicable to the one-dimensional model are either not applicable (e.g., invasion success probability) or too complicated (pair approximation, pair-edge approximation) for the two-dimensional model. We have done the computer simulations of the two-dimensional model. The lattice was of size 100100, and of torus shape (the rightmost sites were the nearest neighbor of the leftmost sites, and the top sites were next to the bottom sites). Examples of spatial patterns of the two-dimensional model are shown in Fig. 2.8a–d, in which the payoffs are: T ¼ 5, R ¼ 3, P ¼ 1 and S ¼ 0. The simulations were started from a random spatial pattern, and clusters of the same strategies were formed quickly. In the two-dimensional lattice model, AD is an ESS when w is less than 0.6, and TFT is an ESS when w is more than 0.5 (Fig. 2.9). The condition for TFT to increase for the fertility model in the two-dimensional lattice is broader than those for the viability model studied (Fig. 2.6e), again supporting that the fertility model favors the evolution of cooperation more than the corresponding viability model. However, their difference is much smaller than the corresponding difference of the models in the one-dimensional lattice. In the two-dimensional model, unlike in one-dimensional model, neighbors have some probability to be the direct competitor, because a neighbor’s neighbor can also be the neighbor, which gives an advantage to spiteful behavior of reducing the payoffs of the opponents in the game.

54

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

Fig. 2.8 a–d Illustrate the patterns of two-dimensional lattice with time. Black is TFT and white is AD. The lattice size was 100  100, w was 0.6 and the initial density of TFT was 0.1. We set the time unit so that the possible state change of each site was examined on average once in a unit time; a is the initial pattern; b–d are the patterns when time is 10, 50 and 200, respectively. When t ¼ 292, TFT occupied the whole lattice

Fig. 2.9 The results of the computer simulation on the two-dimensional lattice (100  100). Payoffs are: T ¼ 5, R ¼ 3, P ¼ 1 and S ¼ 0. Other parameters are the same as in Fig. 2.7

2.6

Discussion and Conclusion

2.6 2.6.1

55

Discussion and Conclusion The Lattice-Structured Population vs. Completely Mixing Population

The problem of the difficulty in invasion and establishment of the cooperative strategy in a population dominated by a noncooperative one can be resolved in the lattice-structured population. In both the one-dimensional and in the two-dimensional lattices, initially rare TFT can increase in an AD-dominated population and replace AD if the iteration probability w is sufficiently high in the viability model and w is very small in the fertility model. In the lattice population, cooperation can evolve from an initially noncooperative society. Then we conclude that the spatial structure of the population enables rare cooperative strategies to spread when the individual interacts with the neighbor repeatedly, as proposed by Axelrod (1984). The lattice model, especially the one-dimensional lattice model, does not show bistability. In a lattice population the movement of the boundary between a TFT cluster and an AD cluster determines the fate of evolution. The total fraction of TFT in the whole population does not affect the relative advantage of two strategies. Rare cooperative individuals can spread not only if there is local interaction (e.g., Killingback and Doebeli 1996; Matsuda 1987; Matsuda et al. 1987; Nowak and May 1992, 1993; Taylor 1992; Wilson et al. 1992; Hutson and Vickers 1995; Ferriere and Michod 1996) but also if there is assortative interaction (e.g., Boyd and Richerson 1988; Wilson and Dugatkin 1997). Both the spatial structure and assortative interaction facilitate the evolution of cooperative strategy. One mechanism which causes assortative interaction is nepotism, the tendency to be more cooperative with relatives than nonrelatives (Alexander 1979). The spatial structure population such as the lattice model makes rare cooperators increase in number because cooperators increase in the neighborhood of cooperators in the spatial structured population. However, if the chicken game is used, the latticestructured population does not promote the evolution of cooperation more than the completely mixing population where cooperators and defectors coexist (e.g., Hauert and Doebeli 2004). In this chapter, we assumed that players only interact with nearest neighbors in the lattice and replace their strategy with the strategy of one of nearest neighbors who has a higher score; the range of social interaction is one and the range of replacement is one. These ranges also influence the evolution of cooperation; if both or either are larger, cooperation evolves less (e.g., Ohtsuki et al. 2007).

56

2.6.2

2

The Evolution of Cooperation in a Lattice-Structured Population Under. . .

Comparison Between the Viability Model and the Fertility Model

Now we compare the viability model with the fertility model, both for the one-dimensional lattice. From the birth-and-death process and the pair approximation, Nakamaru et al. (1997) also calculated the threshold in the viability model, wb ¼ (T  S  R + P)/(T  S), and the threshold in the fertility model, wc ¼ (P(P + T )  R(R + S))/(R(P  S) + P(T  P)). The difference in the threshold probability of iteration between two models, wb  wc, is always positive because of the assumptions for the prisoner’s dilemma game. This implies that the fertility model is more favorable to TFT than the viability model. Then we can conclude that TFT is easier to be ESS in the fertility model, and that TFT can invade the AD population for a wider range of parameters than in the viability model of the same payoffs. Why is the fertility model more favorable to TFT than the viability model? In the viability model it is profitable for an individual to decrease the payoff of the neighbors. Then because neighbors of a lower score are more likely to die and to give a higher chance of colonizing the vacant site. Then a selfish or spiteful strategy such as the one which decreases the score of the neighbor has an advantage. In contrast, in the fertility model in the one-dimensional lattice, mortality of two neighbors is independent of the score, and reducing the neighbor’s score gives no advantage, which explains why a cooperative strategy such as TFT is easier to evolve in the fertility model than in the viability model. However, TFT is evolutionarily stable only when w is larger than wb in the one-dimensional lattice of viability model, when w is larger than wa in the completely mixed model, and when w is larger than wc in the one-dimensional lattice of fertility model. The order is wb > wa > wc. The lattice structure facilitates the evolution of cooperation when TFT, a cooperative strategy, is rare, but inhibits it when TFT is common in the viability model. The lattice models in the viability model are different from the completely mixed model on two points: the possible advantage of being spiteful makes cooperation less likely to evolve and the correlation between neighbors makes it more likely to evolve. In a lattice population how the boundary between a TFT cluster and an AD cluster moves determines the dynamics of evolution. The fertility model is more favorable for TFT than the viability model. This interpretation is supported by the results of the fertility model and the viability model in the two-dimensional lattice. But their difference in the two-dimensional lattice is much smaller than in the one-dimensional lattice. In the two-dimensional lattice, unlike in the one-dimensional lattice, the neighbor’s neighbor of a player can also be a direct neighbor, and hence can be a potential competitor for a vacancy to reproduce when their common neighbors have died. Thus this gives an advantage to the spiteful behavior of reducing the score of neighbors by defects. A biological example of spiteful behavior is toxin production by bacteria. Some strains of the bacteria E. coli produce a toxin, called colicin, but they themselves are

2.6

Discussion and Conclusion

57

immune to it. However, colicin-producing strains have a slower rate of population growth than colicin-sensitive strains. Durrett and Levin (1997) studied the coexistence of colicin-producing and colicin-sensitive strains in a lattice, as they cannot coexist in a perfectly mixed population. The model is similar to the nearestneighbor-migration model for the evolution of altruism by Matsuda (1987). Matsuda (1987) and Matsuda et al. (1987) studied the extinction–invasion lattice model in which the dynamics of three states (sites occupied by the cooperative type, those occupied by selfish type and vacant sites) are examined. In the absence of kin recognition, cooperation with neighbors can evolve if the migration range is small and the habitat is not saturated (i.e., many sites are vacant). However, this effect was not very strong if most sites are occupied. The reason is that a high score for cooperative individuals in the middle of tight clusters would not contribute to their spread. Similarly, in our model TFT forms tight clusters and vacant sites within a cluster of AD would be filled by AD even if they have low scores. What matters is the movement of the boundary between a tight cluster of TFT and a neighboring cluster of AD. However, the model studied in this chapter is simpler than the one studied by Matsuda, because we assumed only two states (sites occupied by TFT and those occupied by AD) neglecting the possibility that the site may be vacant. A biological example of score-dependent fertility may be the enhancement of seed production of neighboring plants by attracting pollinators in self-incompatible flowers. In general, many modes of social interaction, such as competition for light among plants, would affect both mortality and fertility.

2.6.3

Coexistence of Cooperators and Defectors

The perpetual coexistence of cooperative players and defective players in a single population is a very interesting result of the lattice models with deterministic stage change. The result is even more interesting if the coexistence of the two occur by showing constantly changing spatial patterns as in Nowak and May (1992, 1993), rather than a coexistence forming a fixed spatial pattern. This may possibly provide an explanation of the variability among people with respect to the degree of cooperation in societies. However, the study of many different versions of the lattice game in Appendix E in Nakamaru et al. (1998) concluded that the coexistence of the two strategies is less likely for the models with stochastic state transition. Since the stochastic state transition is probably more realistic than the deterministic state transition, the two strategies less likely to coexist in the spatial population than suggested by deterministic models. Durrett and Levin (1997) have shown that more than three types can coexist in the spatially structured population with stochastic transition results, if the competitive advantage is circular, so that the first type is replaced by the second type, and the second is replaced by the third, which is then replaced by the first type: the rock–scissors–paper relationship. This circular competition may give a more robust mechanism allowing the coexistence of multiple types in the spatially structured population (e.g., Kerr et al. 2002).

58

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The Evolution of Cooperation in a Lattice-Structured Population Under. . .

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Matsuda H, Ogita N, Sasaki A, Sato K (1992) Statistical-mechanics of population—the lattice Lotka-Volterra model. Prog Theor Phys 88(6):1035–1049. https://doi.org/10.1143/Ptp.88.1035 Matsuda H, Tamachi N, Ogita N, Sasaki A (1987) A lattice model for population biology. In: Teramoto E, Yamaguti M (eds) Mathematical topics in biology: lecture notes in biomathematics, vol 71. Springer, New York, pp 154–161 Nakamaru M, Matsuda H, Iwasa Y (1997) The evolution of cooperation in a lattice-structured population. J Theor Biol 184(1):65–81. https://doi.org/10.1006/jtbi.1996.0243 Nakamaru M, Nogami H, Iwasa Y (1998) Score-dependent fertility model for the evolution of cooperation in a lattice. J Theor Biol 194(1):101–124. https://doi.org/10.1006/jtbi.1998.0750 Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature 359(6398):826–829. https://doi.org/10.1038/359826a0 Nowak MA, May RM (1993) The spatial dilemmas of evolution. Int J Bifurc Chaos 3(1):35–78. https://doi.org/10.1142/S0218127493000040 Ohtsuki H, Nowak MA (2006) The replicator equation on graphs. J Theor Biol 243(1):86–97. https://doi.org/10.1016/j.jtbi.2006.06.004 Ohtsuki H, Pacheco JM, Nowak MA (2007) Evolutionary graph theory: Breaking the symmetry between interaction and replacement. J Theor Biol 246(4):681–694. https://doi.org/10.1016/j. jtbi.2007.01.024 Sato K, Matsuda H, Sasaki A (1994) Pathogen invasion and host extinction in lattice structured populations. J Math Biol 32(3):251–268. https://doi.org/10.1007/Bf00163881 Taylor PD (1992) Altruism in viscous populations—an inclusive fitness model. Evol Ecol 6(4): 352–356. https://doi.org/10.1007/Bf02270971 Wilkinson GS (1984) Reciprocal food sharing in the vampire bat. Nature 308(5955):181–184. https://doi.org/10.1038/308181a0 Wilson DS, Dugatkin LA (1997) Group selection and assortative interactions. Am Nat 149(2): 336–351. https://doi.org/10.1086/285993 Wilson DS, Pollock GB, Dugatkin LA (1992) Can altruism evolve in purely viscous populations. Evol Ecol 6(4):331–341. https://doi.org/10.1007/Bf02270969

Chapter 3

The Effect of Peer Punishment on the Evolution of Cooperation

Abstract Punishment is an important mechanism promoting the evolution of cooperation among nonrelatives. We investigate the coevolution of cooperation and punitive behavior from the perspective of spiteful behavior. Firstly, we analyze the effect of selfish punishers, which are defectors that punish other defectors in the coevolution of cooperation and punishment. We show that the updating rule and the spatial structure influence the role of selfish punishers in the coevolution of cooperation and punishment. In particular, the score-dependent viability model presented in chapter 2 promotes the evolution of spiteful behavior, and selfish punishers promote the evolution of cooperation and punishment. Then, in light of empirical evidence of punishment targeted at cooperators, we also study the effect of so-called “anti-social punishment.” Secondly, assuming the cooperation level and punishment level, we investigate whether graduated or strict punishment can promote a higher cooperation level in the score-dependent viability model. Our evolutionary simulation outcomes demonstrate that stricter punishment promotes increased cooperation in a spatially structured population, whereas graduated punishment increases cooperation when players interact with randomly chosen opponents from the population. Keywords Punishment · Population structure · Score-dependent viability model · Score-dependent fertility model · Graduated punishment · Strict punishment

3.1

Introduction

Punishment for selfish behaviors is an important mechanism for promoting the evolution of cooperation. In humans, punishment improves the level of cooperation in a group (Yamagishi 1986; Fehr and Gächter 2002), sometimes via promoting social

The original version of this chapter was revised: Figure 3.6 have been updated. The correction to this chapter can be found at https://doi.org/10.1007/978-981-19-4979-1_12 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022, corrected publication 2023 M. Nakamaru, Trust and Credit in Organizations and Institutions, Theoretical Biology, https://doi.org/10.1007/978-981-19-4979-1_3

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3 The Effect of Peer Punishment on the Evolution of Cooperation

norms (e.g., Ostrom 1990). Punishment is also observed in animals besides humans, plants and even bacteria (e.g., Ratnieks and Wenseleers 2008). The male in a harem of red deer punishes females who attempt to escape (Clutton-Brock and Parker 1995). Punishment has also been reported in soybeans (Kiers et al. 2003). In paper wasps, subordinates that cheat by signaling an inflated status of dominance receive more aggression from other wasps (Tibbetts and Dale 2004). Punishment is also known as “policing” in studies of social insects, where queens or workers sometimes attack other nestmate workers that attempt to produce offspring by themselves (Monnin and Ratnieks 2001; Ratnieks et al. 2006; Wenseleers and Ratnieks 2006). Punishment seems difficult to evolve because the punisher must pay a cost to reduce the fitness of others. However, theoretical studies have shown that the evolution of cooperation is encouraged by punishment in a completely mixed population (e.g., Axelrod 1986; Sigmund et al. 2001; Fowler 2005; Sigmund 2007), in a metapopulation (e.g., Boyd et al. 2003; Bowles and Gintis 2004), and in a lattice-structured population (e.g., Brandt et al. 2003; Nakamaru and Iwasa 2005, 2006), if the cooperation-punisher, who cooperates with any opponent but also punishes defectors, can obtain a benefit from cooperation with other cooperators, and if this benefit is large enough to compensate for the fitness loss due to the punishment of defectors. Sigmund et al. (2001), for example, analyzed the replicator dynamics of four strategies: a cooperator who punishes defectors (denoted by G1), a defector who punishes defectors (G2), a pure defector (G3) and a pure cooperator (G4). They concluded that either G1 or G3 can be evolutionarily stable strategies, and that G1 cannot invade a population dominated by defectors. Recent models of cultural evolution suggest that weak conformist transmission can stabilize punishment and hence promotes the evolution of cooperation (Henrich and Boyd 2001; Henrich 2004). The option to abstain from the game also favors the co-evolution of cooperation and punishment (Fowler 2005; Hauert et al. 2007; Traulsen et al. 2009). Punishment is similar to a spiteful behavior in a sense that the players who are punished by punishers have to pay a cost such as a fine and the punishers have to pay a cost for punishment. In a spatially structured population, neighbors tend to compete for space with each other for the opportunity to reproduce, fostering the evolutionary tendency toward spiteful behavior that pays a cost to damage others (e.g., Nakamaru et al. 1997; Nakamaru et al. 1998; Durrett and Levin 1997; Iwasa et al. 1998; Nakamaru and Iwasa 2000; Frean and Abraham 2001; Czaran et al. 2002; Kerr et al. 2002; Czaran and Hoekstra 2003; see Chap. 2). Therefore, we can guess that the coevolution of cooperation and punishment is favored in the viability model more than in the fertility model from the viewpoint of the evolution of spiteful behavior, even though cooperation is favored in the fertility model more than in the viability model. Therefore, we will make it clear how the updating rules, such as the viability model and the fertility model, influence the coevolution of cooperation and punishment in the spatially populations in Sects. 3.2 and 3.3. In particular, we focus on the role of defectors who punish other defectors in Sect. 3.2, which is based on Nakamaru and Iwasa (2006), because the effect of the defector who punishes other defectors on the evolution of cooperation has not been discussed in detail in previous theoretical studies of punishment (e.g., Sigmund

3.1

Introduction

63

et al. 2001; Brandt et al. 2003). For example, Boyd et al. (2003) and Bowles and Gintis (2004) studied games with three strategies such as a cooperator who punishes defectors, a pure cooperator, and a pure defector, but did not discuss the effect of defectors who punish other defectors. After Nakamaru and Iwasa (2006), Eldakar and Wilson (2008) investigated the effect of selfish punishers on the evolution of cooperation when players play the public goods game, and basically their results support Nakamaru and Iwasa (2006). In human societies, however, not only cooperators but also defectors punish other defectors, as shown by an experiment (Falk et al. 2001). In many situations, cooperators punish defectors, but defectors also punish other defectors and cooperators, especially when the damage that results from being punished is high. Shinada et al. (2004) performed a series of experiments to study the effects of in-group favoritism. They found that cooperators punish defectors in the same group more often than the defectors in a different group, and that defectors tend to punish other defectors who belong to a different group. The questionnaire study by Price et al. (2002) showed that punishment is not necessarily linked with cooperation, but many theoretical studies of the altruist-punisher indicated that punishment is linked to the evolution of cooperation. A study in brain physiology demonstrated that a player who punishes defectors is likely to be rewarded with positive emotion from observers (de Quervain et al. 2004). The theoretical models have studied the effect of allowing players to punish non-cooperators. In addition to such pro-social punishment, however, numerous behavioral experiments have found that a significant fraction of non-cooperators will pay to punish cooperators (Shinada et al. 2004; Denant-Boemont et al. 2007; Dreber et al. 2008; Herrmann et al. 2008; Nikiforakis 2008; Gächter and Herrmann 2009; Wu et al. 2009). For example, a series of cross-cultural public goods game experiments found a great deal of cross-cultural variation in the extent to which cooperators versus non-cooperators were targeted with punishment (Herrmann et al. 2008). In the most extreme cases, participants in countries such as Greece and Oman were as likely to punish those who contributed more than them as those who contributed less. The high level of punishment directed at cooperators in this and other experiments indicates that this behavior is not merely the result of errors or lack of comprehension, but is instead a surprising aspect of human behavior requiring explanation. Throughout Sect. 3.3, we refer to this punishment targeted at cooperators as “anti-social punishment,” in order to distinguish it from the (usual) punishment that is targeted at non-cooperators. Antisocial punishment runs counter to the common assumptions about why people choose to punish (Johnson et al. 2009; Fehr and Gächter 2000, 2002). These strategies that pay a cost for cooperators to incur a cost for punishment are often excluded from evolutionary models of cooperation and punishment. In addition to the empirical evidence for anti-social punishment, it seems most appropriate for evolutionary models to allow the full set of possible strategies (of a given complexity) and to ask what strategies emerge via natural selection, rather than restricting the strategy set to only include particularly attractive strategies. We ask how including anti-social punishment affects the evolution of cooperation. Is this “dark side” of punishment in behavioral experiments just irrational, erroneous behavior, or can it in fact be favored by natural selection? Does punishment still promote cooperation when anti-social punishment is possible? In Sect. 3.3,

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we adopt the framework used in Sect. 3.2 and study the consequences of including anti-social punishment (Rand et al. 2010). Although numerous models of punishment have been developed, most assume discrete or binary strategies for both cooperation (cooperator or defector) and punishment (punisher and non-punisher). Section 3.2, for example, assumes four strategies: AP, AN, SN and SP (see also Sect. 3.2). These simple settings are convenient for mathematical analysis and are thus helpful for the discussion of basic mechanisms. However, in reality, the cooperation level between two players or the amount of contribution to the public goods in the public goods game is continuous (Chen et al. 2012; Killingback et al. 1999). The cooperation level in certain collective actions is continuous, and there may be several discrete cooperation levels, e.g., levels 1, 2,. . .,10. We may consider the example of house cleaning by several housemates. Some housemates help clean the house completely, some do so almost completely, some do a bit, whereas others do not help at all. Thus, a continuous cooperation level can encompass a situation in which there are more than two choices. The punishment level is also continuous depending on the level of continuous cooperation. If we adopt these realistic assumptions, we can investigate the punishment level in response to the opponent who contributes a certain level of cooperation to the public goods. Section 3.4 showed the effect of strictness of punishment on the evolution of the continuous cooperation level. Strict punishment means a policy of “zero tolerance”: a punisher severely punishes an opponent whose cooperation level is below the threshold and weakly punishes an opponent whose cooperation level is above the threshold. If punishment is graduated, a punisher gradually changes the severity, adjusting to the cooperation level. This principle is followed by the criminal laws of most western countries (Iglesias et al. 2012). Other examples include the following. Ostrom found that graduated punishment is one of seven design principles for long-enduring common-pool resource institutions and graduated punishments for violators are likely to be assessed depending on the seriousness and context of the violation (Ostrom 1990). Cox found that graduated punishments progress on the basis of either the severity or repetition of violations to deter participants from excessive violations of community rules (Cox et al. 2010). In many legal systems, repeat offenders are punished more severely than first-time offenders, and theoretical studies of criminal sanctions have shown that the erroneous conviction of innocent offenders and learning contribute toward making this sanction system optimal (Mungan 2010; Emons 2003; Cyrus et al. 2000). Therefore, in Sect. 3.4, we investigate whether graduated or strict punishment depending on the cooperation level increases cooperation. We also investigate how spatial structure affects the results; this outcome depends on an updating rule that prescribes how the game score affects fitness and generational changes. Updating rules can dramatically change evolutionary dynamics (see Chap. 2 and Sect. 3.2). Therefore, we expect that the viability model can foster the coevolution of cooperation and punishment even in a completely mixed population.

3.2

Two Updating Rules, Viability Model and Fertility Model

3.2 3.2.1

65

Two Updating Rules, Viability Model and Fertility Model Model

We consider the coevolution of altruism and punishment (Nakamaru and Iwasa 2006). Players engage in a two-person game. Each player is either altruistic (A) or selfish (S) and is also either a punisher (P) or a nonpunisher (N). There are four possible combinations of these two traits in the population: altruist-punisher (denoted by AP), altruist-nonpunisher (AN), selfish-punisher (SP) and selfishnonpunisher (SN). An altruist pays a cost c to increase the fitness of its opponent by benefit b (b, c > 0). If two altruists interact with each other, both increase their fitness by b
c. In contrast, a selfish player does not improve the fitness of its opponent, but it also pays no cost. When two selfish players interact, their fitnesses remain unchanged and are lower than in the case where both are altruists, because b
c > 0. However, when a selfish player engages in a game with an altruistic player, the selfish one increases its fitness by b and the altruist decreases its fitness by c. Hence selfish players can invade a population of altruists. To overcome this difficulty of maintaining altruism, we consider punishers who punish selfish players. In the presence of a punisher, the selfish individual has to pay a “fine,” which decreases the benefit of being selfish and promotes the spread of altruists. One the other hand, the punisher also has to pay a cost of execution. If a punisher (P) interacts with a selfish player (S), S suffers the cost of being punished (
p) and then P also has to pay the cost of imposing punishment (
q). We assume that p  0 and that q  0. Table 3.1 illustrates the payoffs of the four strategies shown in the left column, when each type of player plays with an opponent specified in the top row. If the population is composed of only two strategists, AP and AN, both can get a positive payoff, b
c. When a selfish individual (SN or SP) is introduced into this population, AN is exploited by SN or SP, but AP can prevent the selfish invader (SN or SP) from exploiting others. Hence, in the absence of AP, AN cannot be maintained in the population with SN or SP, whereas AP does not need AN. AP suffers the cost of imposing punishment (
q), which AN does not pay. To study the effect of spatial structure on the coevolution of altruism and punishment, we can compare a completely mixed population with a latticestructured population. For a lattice model we assume a regular square lattice, in

Table 3.1 Payoff matrix of a focal player interacting with an opponent. We assumed b
c  0, q  0 and, p  0

Focal AP AN SP SN

Opponent AP b
c b
c b
p b
p

AN b
c b
c b b

SP
c
q
c
q
p
p

SN
c
q
c
q 0

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3 The Effect of Peer Punishment on the Evolution of Cooperation

which each player interacts with the four nearest neighbors (Neumann neighborhood). In a completely mixing population, each player interacts with four players chosen randomly from the whole population. Each individual obtains a game score. After the interaction with the four players, the score for the player is the sum of the payoffs specified in Table 3.1. For example, if an AP player interacts with an AP, an SN, and two ANs, the score of the focal AP players is 3(b
c) + (
c
q). In the score-dependent viability model (also see Chap. 2), the death probability of each individual decreases with its score. After the death of an individual, the site becomes empty, and one of the four nearest neighbors colonizes the vacant site randomly. In contrast, in the score-dependent fertility model (also see Chap. 2), an individual dies randomly, independent of its score, and the vacant site is colonized by one of the four nearest neighbors in proportion to its score. For completely mixed populations, we can write ordinary differential equations for both the viability and the fertility models (see Appendixes A and B in Nakamaru and Iwasa (2006)), respectively. We did a Monte Carlo simulation to examine the accuracy of the analysis. We also performed a Monte Carlo simulation of both the viability and the fertility models in a two-dimensional lattice population. In the Monte Carlo simulation, each site is chosen once in one unit of time. We used a 5050 lattice and a 100100 lattice (population size (N) ¼ 2500 or 10,000, respectively). Initially, the sites were filled with four strategies in a random spatial configuration. We assumed a periodic boundary condition in order to remove the effect of the edge of the lattice.

3.2.2

The Completely Mixing Model

3.2.2.1

Score-Dependent Viability Model

Figure 3.1 shows the results of the dynamics of the strategies analyzed in Appendix A in Nakamaru and Iwasa (2006). Let Y1 ¼ b + 4q + 4c, Y2 ¼ (b + q)/ 4 + c, Y3 ¼ (b + 4c)/3
q, and Y4 ¼ 4q. Three lines, p ¼ Y2, p ¼ Y3, and p ¼ Y4, intersect at b ¼ 15q
4c. We set G ¼ 15q
4c. The four lines Y1 to Y4 divide the p
b graph into seven areas, (i)–(vii) in Fig. 3.1a: (i) satisfies p > Y1; (ii) Y3 < p < Y1 when b > G and Y4 < p < Y1 when b < G; (iii) p < Y4 when b > G and p < Y3 when b < G; (iv) Y2 < p < Y4 when b < G; (v) Y3 < p < Y2 when b < G; (vi) Y2 < p < Y3 when b > G; and (vii) Y4 < p < Y2 when b > G. The following are the predictions from both the analysis (see Appendix A in Nakamaru and Iwasa (2006)) (Table 3.2) and the numerical calculations (Fig. 3.1b, c). In (i), (ii), and (vi) AP and AN win against others; in (iii) and (v) SN wins against others; in (iv) the dynamics converge in favor of either SN or AP + AN; in (vi) AP wins against others if SP and SN successfully drive AN to extinction; and in (vii) SP wins against others (Fig. 3.1c). In (i), (ii), and (vi), the final population includes AP and AN, which coexist. First, SP drives SN to extinction, and then AP is stronger than SP, which becomes extinct;

3.2

Two Updating Rules, Viability Model and Fertility Model

67

Fig. 3.1 a Results of a mathematical analysis of the score-dependent viability model in a completely mixing population, as derived from Appendix A of Nakamaru and Iwasa (2006). Horizontal axis is p, vertical axis is b, and c ¼ q ¼ 1. The four lines are p ¼ Y1 (rightmost solid line), p ¼ Y2 (leftmost solid line), p ¼ Y3 (dashed line), and p ¼ Y4 (vertical solid line). The black dot is located at (4q, 15q
4c), where three lines (Y2, Y3, Y4) intersect. We labeled the seven areas divided by these four lines as (i), (ii) . . . (vii). b Outcomes of numerical analysis of the scoredependent viability model when the initial densities of AP, AN, SN, SP are 0.03, 0.9, 0.03, 0.04, c ¼ q ¼ 1. Horizontal axis is p, vertical axis is b. The numerical calculation for each b and p is done for 200,000 time steps. Black dots mean that SN wins against others; gray, AP and AN win against others; “s,” SN and SP are neutral and exist together; “p,” SP wins against others. c Density changes with time when the initial densities of AP, AN, SN, SP are 0.3, 0.3, 0.3, 0.1, b ¼ 15, p ¼ 4.5, and c ¼ q ¼ 1 in area (vii) of a. The thin dashed line is AP; the thick solid line, AN; the thin solid line, SN; the thick dashed line, SP (as labeled on the graph)

and finally AP and AN coexist, as they are neutral (Fig. 3.1b). Because AN is naive, it should be exploited by SN; hence we can conjecture that AN would indirectly hinder the evolution of AP. However, we did not observe a negative effect of AN on the evolution of AP (Fig. 3.1b). This is because SP wins against SN first. Because SP beats SN, our results differ from the case with the initial population consisting of AP and SN only. In the parameter regions (ii) and (vi), the system has bistability in the population consisting only of SN and AP. In contrast, in the

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The Effect of Peer Punishment on the Evolution of Cooperation

Table 3.2 Results derived from the analysis in Appendix A of Nakamaru and Iwasa (2006)

(i) (ii) (iii) (iv) (v) (vi) (vii)

p > Y1 Y3 < p < Y1 in b > G Y4 < p < Y1, in b < G p < Y3 in b < Gp < Y4 in b>G Y2 < p Y4 in b < G Y3 < p < Y2 in b < G Y2 < p < Y3 in b > G Y4 < p < Y2 in b > G

The population consisting of two strategies AP/SN AP/SP SN/SP AN/SP AP > SN AP > SP SN < SP Coexist Bistable AP > SP SN < SP Coexist

AN/SN SN > AN SN > AN

AP < SN

AP < SP

SN > SP

AN AN

Bistable AP < SN Bistable AP < SN

AP > SP AP < SP AP > SP AP < SP

SN > SP SN > SP SN < SP SN < SP

Coexist Coexist AN AN SN > AN SN > AN

(i)–(vii) in the leftmost column correspond to numbers in Fig. 3.1a. Each column indicates the result when the population has two strategies, as listed above the columns to the right. For example, “AP > SN” means that AP wins against SN in the population consisting of AP and SN (labeled as AP/SN)

population of four strategists, no bistability is observed, and AP and AN combined always win against others. It is because the relative strength of three strategies is AP > SP > SN in both p > Y4 and p > Y2, and then SP always prevents SN from increasing. Figure 3.1b also shows that the total area where SN finally wins ((iv), (v), and (iii), including the bistability region) is narrower than that in the initial population consisting of SN and AP, which corresponds to (ii)–(vii), including the bistability region. To investigate whether an interior stable equilibrium existed, we performed 10,000 runs of a numerical simulation analysis of the differential equations of the four strategies, in which the initial densities of the four strategists were chosen randomly and independently for different runs. We did not observe interior points in any of the runs. Therefore, we concluded that the values Y1, Y2, Y3, and especially Y4, controlled the evolutionary dynamics of the four strategies (Fig. 3.1), that SP promoted the evolution of AP, and that AN did not have a negative effect on the evolution of AP ((i), (ii), and (vi) in Fig. 3.1b).

3.2.2.2

Score-Dependent Fertility Model

The mathematical analyses (see Appendix B in Nakamaru and Iwasa (2006)) and the outcomes of the numerical analysis (Fig. 3.2a) predicted the following: (1) When c < p, the density dynamics of the four strategies converge either to SN or to a mixture of AP and AN (Fig. 3.2a). (2) When p < c, the population converges to SN (Fig. 3.2a). In addition, if c
q < p, AN and SP coexist, but a mixture of AN and SP can be invaded by SN. In the fertility model, altruistic punishment is unable to invade the population occupied by selfish individuals for any choice of parameters; this is in sharp contrast to the viability model, in which altruistic punishment can invade a completely mixed

3.2

Two Updating Rules, Viability Model and Fertility Model

69

Fig. 3.2 Result of a mathematical analysis of the score-dependent viability model in a completely mixing population, as derived from Appendix B of Nakamaru and Iwasa (2006) (line), and the outcomes of a numerical analysis (dots). Horizontal axis is p, vertical is b, and c ¼ q ¼ 1. Line is p ¼ c. Black dots mean SN wins against others; gray, AP and AN win against others. a The initial densities of AP, AN, SN, SP are 0.9, 0.03, 0.03, 0.04. b Density changes with time when the initial densities of AP, AN, SN, SP are 0.03, 0.03, 0.03, 0.91, b ¼ 3, p ¼ 3, and c¼q¼1

population occupied by SN, given that the punishment fine is sufficiently large. Figure 3.2b also shows that SP has a negative effect on the evolution of AP. This is because the presence of many SPs in the initial population increases AN when c
q < p, leading to an indirect increment in SN; and as a result, SN finally wins, even when c < p. AN fails to make AP evolve, because AN is merely exploited by SN, an evolutionary dynamics different from that of the viability model.

3.2.3

The Lattice Model

3.2.3.1

Score-Dependent Viability Model

In our model, the same strategies made clusters in the lattice (Fig. 3.3a). Brandt et al. (2003) showed that SP (denoted as G2 in their paper) quickly vanishes in the public

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The Effect of Peer Punishment on the Evolution of Cooperation

Fig. 3.3 Example of a simulation run of the scoredependent viability model on the lattice. The parameters are: b ¼ 1, p ¼ 4, and c ¼ q ¼ 1. The initial densities of AP, AN, SP, and SN are 0.3, 0.3, 0.3, and 0.1, respectively. a The lattice pattern in 300 units of time. White sites are those occupied by AP; gray sites by AN; black by SN; light gray sites by SP. b Dynamics of the densities of the four strategists over time. AP (thin dashed line) and AN (thick solid line) fluctuate together. The thin solid line is SN, and the thick dashed line is SP

goods game played in a lattice population. We observed a similar behavior (Fig. 3.3b). Hence SP was expected to be unable to help the evolution of AP. Figure 3.4 shows that, in the two-dimensional-lattice structured population, a larger benefit from cooperation does not affect the evolution of AP, but a larger fine as punishment ( p) encourages AP. AP can win against SN when p is larger than a threshold value (which is between about 2 and 3 in c ¼ q ¼ 1). To determine the effect of the SP and the pure altruist (AN) on the evolution of AP, we fixed the initial total density of the altruist (AP + AN) and that of the selfish player (SP + SN) and then changed the ratio of SN to SP and that of AN to AP (Fig. 3.4). Figure 3.4 shows that the parameter region on a p–b plane where SN wins against others (black dots) is small when the SN/SP ratio is high, and that the region becomes larger when the AN/AP ratio is higher. Hence a higher initial density of AN hinders the evolution of altruism. This result can be explained as follows: even if AN gives a benefit to AP, AN is always exploited by selfish players, and as a result, AN helps SN to win against AP. To understand why SP promotes the evolution of AP even if SP quickly vanishes in the lattice population, we did a pair-edge approximation (see Appendix C in

3.2

Two Updating Rules, Viability Model and Fertility Model

71

a a

0.5/0.0

0.4/0.1

0.1/0.4

0.0/0.5

SN/SP

0.0/0.5

0.2/0.3

0.4/0.1

0.5/0.0

AN/AP Fig. 3.4 Results of application of the viability model in a regular lattice population: outcomes of a Monte Carlo simulation when c ¼ q ¼ 1. Vertical axis of each graph is p, and horizontal axis is b. The dark gray portion of each circle indicates the fraction of runs (for 100 simulations) in which SN won; the light gray portion, the fraction of runs in which AP won or both AP and AN coexist; the open portion, the fraction of runs in which the dynamic did not converge; the lightest gray portion labeled “a,” the fraction of runs in which SP won; the lighter gray portion labeled “b,” the fraction of runs in which AN won. The initial densities of altruist and selfish populations are both 0.5. The ratios of AN/AP and SN/SP were changed to determine the effects of AN and SP on the evolution of AP. Each simulation was run for 50,000 units of time in a 50  50 lattice

Nakamaru and Iwasa (2006)). Starting from a random initial pattern, players using the same strategy make clusters in the lattice (Fig. 3.3a), and then the movement of the boundary between different clusters determines the outcome of competition. The

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3 The Effect of Peer Punishment on the Evolution of Cooperation

pair-edge approximation is a convenient tool for analyzing the direction and speed of the boundary between two clusters (see Chap. 2). Figure 3.4 is the outcome for a two-dimensional lattice. Because the pair-edge approximation for a two-dimensional lattice is complicated, we could analyze only that for a one-dimensional lattice. Although the dimension of the population is not the same, the result of the pair-edge approximation for a one-dimensional model helps us to understand the dynamics of lattice populations in higher-dimensional lattices (e.g., Nakamaru and Iwasa 2005). To determine why SP helps the evolution of AP, we focused on the dynamics of three strategists, AP, SN, and SP, in the one-dimensional lattice. The pair-edge approximation in Appendix C in Nakamaru and Iwasa (2006) shows that AP can win against SP and SN when p > 2c + q; AP wins against SP, but SN wins against both SP and AP when c < p < 2c + q; and SP wins against AP, but SN wins against both SP and AP when p < c. As Nakamaru and Iwasa (2005) indicated, the two-dimensional lattice model whose lattice size is finite has a parameter region where the evolutionary outcome depends critically on the initial density. In this parameter region, which is very close to the parameter region of c < p < 2c + q, if SP is replaced by AP, which can increase in number, AP has an advantage in the competition with SN (Fig. 3.3b). Since Liggett (1978) proved that a very large lattice size reduces the parameter region of bistability, we examined whether or not lattice size would affect the role of SP, but we did not observe a dramatic difference between the two lattice sizes, 50  50 and 100  100 (see Fig. 4b of Nakamaru and Iwasa (2006)). Hence we tentatively conclude that AN generally hinders, and SP promotes, the evolution of AP, regardless of the population size.

3.2.3.2

Score-Dependent Fertility Model

In the two-dimensional lattice-structured population, not only the fine of punishment but also the benefit from altruism affects the evolution of altruistic punishment. When b is small, a high p can encourage AP to win against SN; but when b is large, AP always wins against SN for any value of p. This result holds regardless of the initial frequency of occurrence of AP. Figure 3.5 illustrates this result. Just as in Fig. 3.4, AN hinders the evolution of AP, but SP promotes it.

3.2.4

Summary

We investigated if the effect of punishment on the evolution of altruism depends on the population structure and updating rules affected by the game’s score. In the completely mixing population, punishment promotes the evolution of altruism in both the fertility and viability models. Although punishment does not enable altruist players to invade the population occupied by selfish players in the fertility model, punishment can do this in the viability model. In the lattice population, punishment promotes the evolution of altruism when both the benefit from altruism and the fine

3.2

Two Updating Rules, Viability Model and Fertility Model

73

0.5/0.0

0.4/0.1

0.1/0.4

0.0/0.5

SN/SP

0.0/0.5

0.2/0.3

0.4/0.1

0.5/0.0

AN/AP Fig. 3.5 Fertility model in a regular lattice population. The dark gray portion of each circle indicates the fraction of runs (for 100 simulations) in which SN won; the light gray portion, the fraction of runs in which AP or AN won. See the figure for additional information

of punishment is high in the fertility model and when the fine of punishment is high regardless of the benefit from altruism in the viability model. One of our main conclusions is that the presence of selfish punishers (SPs) tends to encourage the evolution of altruism. However, SPs do not coexist with APs in our model.

74

3.2.5

3

The Effect of Peer Punishment on the Evolution of Cooperation

Expansion of This Study: Do Empty Sites Influence the Evolution of Cooperation and Punishment?

The above-mentioned studies have assumed that players occupy all the lattice sites. However, if some sites are empty or inactive, the number of neighboring opponents interacting with a focal player decreases. Concerning the effect of empty sites, we may have the following two scenarios: the first is that spiteful behavior does not promote the coevolution of cooperation and punishment in the viability model, especially in a lattice-structured population. This is because a punisher’s effectiveness of punishing defectors is reduced as their neighboring sites might be empty. The second is that it would be difficult for selfish players to evolve in the viability model, because selfish players have a low chance of interacting with cooperators owing to the existence of empty sites and therefore cannot benefit by exploiting cooperators. In this section, we consider empty sites and study their effect on the evolution of cooperation by punishment, not only in a lattice-structured population but also in a completely mixing population where players interact with opponents chosen randomly (Sekiguchi and Nakamaru 2009).

3.2.5.1

Model and Results

We also considered four types of players: the altruist punisher (AP), the altruist non-punisher (AN), the selfish punisher (SP), and the selfish non-punisher (SN). We use the game payoff in Table 3.1. We introduced empty sites in the lattice-structured population of the scoredependent viability model (see Sects. 3.2.1–3.2.4); we call this modified model the “E-viability model,” in which “E” signifies “Empty.” The score-dependent viability model is simply called the viability model. In the E-viability model of the lattice-structured population, it is assumed that players are initially distributed in the lattice sites randomly in accordance with the initial densities of AP, SN, AN, SP, and empty sites. One site can be occupied by only one player. Players are updated asynchronously in each time step (Δt), and the overlapping of generations is taken into account. At each time step (Δt), one player plays the game (Table 3.1) once with each of the players in z nearest sites. Our main results come from the Neumann neighborhood (z ¼ 4). We also used the Moore neighborhood (z ¼ 8) and conclude that the outcomes are almost the same as with the Neumann neighborhood (z ¼ 4). For example, if two of the four nearest neighbor sites are empty and players occupy the other two sites, the focal player interacts only with each of two players and gets a game score that is the sum of the payoffs. In each time step (Δt), one site is chosen randomly from all the sites. (i) If a player is occupying the chosen site, the focal player (i) dies, with the death probability (di) decreasing with her game score (si); di ¼ α exp (
β  si), and a new empty site is created after the death of that player. (ii) If the chosen site is empty, one of the nearest neighbor sites, called site X, is chosen randomly. If site X is occupied by a player, the

3.3

If Anti-Social Punishers Exist in the Population

75

player reproduces to occupy the focal empty site. If site X is also empty, nothing happens. In the following, we study the effect of α on the evolutionary dynamics with β kept constant (β ¼ 0.1). Our individual-based simulations and mathematical analysis showed that the empty sites change the evolutionary dynamics in the viability model, which are close to the evolutionary dynamics in the fertility model without empty sites in the population. In particular, in the completely mixed population, punishment does not promote the evolutionary invasion of cooperation in the population with empty sites (see Sekiguchi and Nakamaru (2009) for a more detailed explanation).

3.3

If Anti-Social Punishers Exist in the Population

Here we introduce anti-social punishers to four strategies to see the effect of antisocial punishment on the coevolution of cooperation and punishment (Rand et al. 2010).

3.3.1

Strategies and Payoffs

We consider a two-stage game with cooperation followed by punishment. In the first stage, each player has two choices, defection (D) or cooperation (C). Defection means doing nothing, such that all players receive zero payoffs. Cooperation means paying a cost c for another to get a benefit b (b > c > 0). In the second stage, each player can choose to punish each other player or not, conditioned on the other person’s action in the first stage. By punishing, a player pays a cost α for the other person to incur a loss β (α, β > 0). The values of α and β correspond to q and p respectively in Sect. 3.2. Withholding punishment results in both players receiving zero payoffs in the second stage. We do not consider mixed strategies in the following analysis, and restrict our attention to pure reactive strategies. Since the action in the second stage is conditioned on the other players’ action in the first stage, there are four possible strategies in the second stage. A non-punisher (N) punishes no one. A pro-social punisher (P) punishes defectors only. An antisocial punisher (A) punishes cooperators only. A spiteful punisher (S) punishes both cooperators and defectors. A combination of the action in the first stage (D, C) and the strategy in the second stage (N, P, A, S) defines one’s strategy in the game. Thus we have eight possible strategies: DN, DP, DA, DS, CN, CP, CA and CS. For example, a CP- strategist cooperates in the first stage and harms defectors in the second stage (therefore deemed as a “strong reciprocator” (Fehr and Fischbacher 2003; Bowles and Gintis 2004)). CP and CN correspond to AP and AN in Sects. 3.2.1–3.2.3, respectively. Table 3.3 summarizes these eight strategies. We can classify these eight strategies into two categories: strategies which do not punish other players using the same strategy (hereafter “self-consistent”), and

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Table 3.3 The eight strategies considered in Sect. 3.3 Strategy DN DP DA DS CN CP CA CS

Cooperate? No No No No Yes Yes Yes Yes

Harm cooperators? No No Yes Yes No No Yes Yes

Harm defectors? No Yes No Yes No Yes No Yes

strategies which do punish other players using the same strategy (hereafter “self in-consistent”). The strategies DN, DA, CN and CP are self-consistent. Conversely, the strategies DP, DS, CA and CS are self-inconsistent. The 8  8 payoff matrix of the pairwise game is given in Table 3.4. As can be seen, the strategies DN and DA are Nash equilibria for all payoff values, and CP is a Nash equilibrium when β > c.

3.3.2

Z-Mixed Population Model

First we study the viability model’s dynamics in populations with no spatial structure. Groups of size z + 1 are randomly selected from the population, and each player interacts with her z other group members to obtain a game payoff, f. In each generation a random player is given a chance to update her strategy (i.e., has some chance of death). With probability d( f) ¼ γ exp[
θf], she abandons her current strategy (i.e., dies) and randomly adopts the strategy of one of the z players she just interacted with (i.e., one of the z players is randomly chosen to reproduce). This model is called the “z-mixed population model.” With the introduction of execution errors, in which players mistakenly choose D instead of C and vice versa, the evolutionary stability of each strategy in the game is summarized in Fig. 3.6. Once anti-social punishment is available, cooperators no longer have an advantage, and spiteful defection, DS, fares best. The β > zα condition reveals that the success of spiteful punishers relies on agents interacting and competing with a limited number of others. In the z ! 1 limit of a well-mixed infinite population, DN is the unique evolutionary stable strategy (ESS) for all payoff values.

Focal DN DP DA DS CN CP CA CS

Opponent DN 0
α 0
α
c
c
α
c
c
α

DP
β
α
β
β
α
β
c
c
α
c
c
α

DA 0
α 0
α
c
β
c
α
β
c
β
c
α
β

DS
β
α
β
β
α
β
c – β
c
α
β
c
β
c
α
β

CN b b b
α b
α
c + b
c + b
c + b
α
c + b
α

CP b
β b
β b
α
β b
α
β
c + b
c + b
c + b
α
c + b
α

Table 3.4 Payoff matrix of a focal player interacting with an opponent when there are eight strategies CA b b b
α b
α
c + b
β
c + b
β
c + b
α
β
c + b
α
β

CS b
β b
β b
α
β b
α
β
c + b
β
c + b
β
c + b
α
β
c + b
α
β

3.3 If Anti-Social Punishers Exist in the Population 77

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The Effect of Peer Punishment on the Evolution of Cooperation

Fig. 3.6 Evolutionary stability under the z-mixed population model. The strategies that are ESS are indicated in each region of the (b, β) parameter space. We set c ¼ 1 and α ¼ 1. We consider small errors. When anti-social punishment is allowed, CP is never ESS. Either DN or DS is the unique ESS

3.3.3

Lattice Model When Anti-Social Punishment Is Allowed

We now consider a structured population in which players are arranged on a square lattice. Each player interacts with the four players in the von Neumann neighborhood and obtains a game payoff. Thus, this lattice model is the structured counterpart to the z-mixed population model with z ¼ 4, and is equivalent to a series of overlapping linear public goods games (e.g., Santos et al. 2008), where each player’s cooperative actions affect her four neighbors. Other features of the model are the same as the completely mixing model as described in the previous section. Agent-based simulations are used to explore the evolutionary dynamics in this structured population. We explore the dynamics on the lattice when strategies which punish cooperators are included (Table 3.4). Simulation results are shown in Fig. 3.7. Results are dependent on initial strategy frequencies. When DN or CP is most abundant initially, we see a similar pattern: the majority of the time DN wins when β is small, DA wins when β is intermediate, and CP wins when β is large. When DS is most abundant initially, DN again wins when β is small, DA wins when β is large, and further simulations find that CP wins when β > 10. When DA is most abundant initially, however, the outcome is very different. Regardless of b or β, DA wins in the majority of cases. DN wins occasionally. Cooperation never invades a resident population of anti-social defectors. These results are qualitatively different from what we saw in the z-mixed population model. First, DS never wins on the lattice, whereas DS was the sole ESS in the z-mixed population model when β was large. A possible explanation for this difference involves spatial correlations. Since offspring disperse locally on the lattice, DS players are very likely to interact with other DS players (i.e., their “relatives”). Because DS is self-inconsistent in the sense that one DS player harms other DS players, the target of the DS players’ spiteful punishment tends to be other

3.3

If Anti-Social Punishers Exist in the Population

79

Fig. 3.7 Evolutionary outcomes of the full strategy set on a regular lattice. Cooperation cannot invade a population of antisocial defectors, DA. Starting from the specified initial condition, 50 agent-based simulations are run and the winning strategy recorded. The blue portion of each circle indicates the fraction of runs where CP wins or CP, CN, CA and/or CS coexist; the red portion, the fraction of runs where DN wins; the yellow portion the fraction of runs where DA wins; the white portion, the fraction of runs in which there was no convergence after 125,000,000 generations. DP and DS never win. We considered small errors, ε ¼ 0.01, and a 5050 lattice for a total population size N ¼ 2500. We used viability updating with parameter values γ ¼ 0.1 and θ ¼ 0.1. We explored the (b, β) parameter space, setting c ¼ 1 and α ¼ 1. Additional simulations found qualitatively similar results for α ¼ 0.5 and α ¼ 1.5. Initial population density a DN ¼ 0.79, all others 0.03; b CP ¼ 0.79, all others 0.03; c DS ¼ 0.79, all others 0.03; and d DA ¼ 0.79, all others 0.03. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of Rand et al. (2010))

DS players. Therefore DS, as well as the other self-inconsistent strategies DP, CA and CS, cannot propagate on the lattice (see also Appendix B in Rand et al. (2010)). A pairwise invasion analysis based on computer simulations (Table 2 of Rand et al. (2010)) shows that across β values, each self-inconsistent strategy is invaded by the first-round equivalent self-consistent strategies (i.e., DP and DS invaded by DN and DA, CA and CS invaded by CN and CP).

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The second major difference we observe is that on the lattice, anti-social defectors (DA) win in a large region of the parameter space, whereas DA is never an ESS in the the z-mixed population model. This is because DP and DS, which are potential invaders of DA in the z-mixed population model, are self-inconsistent and thus do not perform well on the lattice as discussed above. In contrast, DA is self-consistent and harms only cooperators, which protects DA from invasion by CP. The pairwise invasion analysis in Table 2 in Rand et al. (2010) suggests that strategies other than DN cannot invade DA on the lattice. DN can occasionally invade DA, but only due to a small advantage from the tiny execution error probability. Hence DN and DA are almost neutral. In the z-mixed population model DA can invade DN when β is large. In contrast, DA never invades DN on the lattice. Again this is because in the presence of errors, DA engages in self-inconsistent punishment of erroneous cooperation. Overall, the lattice simulation results using the full strategy set are quite different from what occurs in the restricted strategy set where anti-social punishment is not possible (Sect. 3.2). When anti-social punishment is excluded, there is a wide parameter region where cooperation wins regardless of initial conditions (see Sect. 3.2). However, the introduction of anti-social punishment eliminates this region, hinders cooperation, and anti-social defectors prevail (Fig. 3.7).

3.4

Which Promotes the Evolution of Cooperation, Strict or Graduated Punishment?

In the previous sections, players only chose either cooperation or defection and either non-punishment or punishment; binary choice was assumed. However, in our society, binary choice does not always occur. People can sometimes choose cooperation level and punishment level. If they are continuous traits, we can investigate the effect of graduated punishment or strict punishment on the coevolution of cooperation and punishment.

3.4.1

What Is Graduated or Strict Punishment?

In graduated punishment, punishment level corresponds to the opponent’s cooperation level (Fig. 3.8a); if player A’s cooperation level is low, the punishment level player A incurs is high. If it is high, the punishment level is low. Graduated punishment is a design principle for long-enduring common-pool resource institutions (Ostrom 1990). Strict punishment is the all-or-none type, where player A is punished severely if A’s cooperation level exceeds the threshold, and is not punished if it is below the threshold (Fig. 3.8b). Either strict punishment or graduated

Which Promotes the Evolution of Cooperation, Strict or Graduated Punishment?

Fig. 3.8 Punishment severity. a, b show the relationship between the cooperation level of the opponent (horizontal axis) and the damage from punishment (vertical axis) when punishment is graduated (a ¼ 2) and strict (a ¼ 1000), respectively. The other parameters are f ¼ 1.0, u ¼ 0.5, and β ¼ 1.0

81

a 1.0

damage from punishment

3.4

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

cooperation level

b damage from punishment

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

cooperation level

punishment is executed in institutions such as the police and the judicial system. Which promotes the evolution of cooperation in our society? Nakamaru and Dieckmann (2009) mathematically proved that if each individual interacts with a randomly chosen individual from the population, then neither cooperation nor punishment can evolve, and showed that the rule “the stricter, the better” had to be applied to punishment if cooperation were to evolve in a population with a lattice structure. As Chap. 2 and Sect. 3.2 showed, the updating rule would influence the coevolution of cooperation and punishment in the completely mixing model and in the lattice model. Whether graduated or strict punishment promotes the evolution of cooperation may depend on both the updating rule and the spatial structure. The updating rule in Nakamaru and Dieckmann (2009) is that a player gives a birth or dies according to the birth rate and the death rate, and a new offspring can stay at an

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3 The Effect of Peer Punishment on the Evolution of Cooperation

empty site. The birth rate of each player is the benefit from cooperators and the death rate consists of the cost of cooperation, the cost of punishment and the cost of being punished. Spiteful behavior such as punishment does not work because, even though the spiteful players increase the death rate of the opponent, the spiteful players cannot colonize the place of the opponent immediately after the opponent dies in the model of Nakamaru and Dieckmann (2009). Therefore, we will investigate the effect of the score-dependent viability model on the evolution of cooperation and the strictness of the punishment in Sect. 3.4 (Shimao and Nakamaru 2013). Although Sect. 3.3 considered the effect of anti-social punishment, the following only considers punishment against the defectors. We will discuss anti-social punishment from the viewpoint of the continuous trait in the last section.

3.4.2

Models

It is assumed that a population consists of N  N individuals. Each individual i has four adaptive traits: propensity for altruism (xi), severity of punishment ( fi), threshold of punishment or tolerance level (ui), and strictness of punishment (ai). Higher values of x and f represent greater cooperation and more severe punishment, respectively. The values of u and a determine the punishment function. The value of u is the threshold of the cooperation level, below which the focal individual punishes the opponent, and the value of a determines the level of punishment in response to the opponent’s cooperation level. The former three parameters are continuous and limited to the range 0–1. Strictness is also continuous but ranges from 0 to infinity. During each turn, an individual i is randomly chosen from the population and plays the two-stage game in each group of four members, which includes individual i. The two-stage game consists of the cooperation and punishment stage. In the cooperation stage, the public goods game is played; each member contributes his/her resource to the public goods. The amount of contributions is determined by their propensity for altruism x. The pool of contributions from all the members is multiplied by r, defined as the efficiency of public goods, and equally divided among the four members so that the score for an individual i with a cooperation level xi after the cooperation stage ends is rX x
xi : 4 j j In the punishment stage, the four members punish each other according to the others’ cooperation levels and pay the cost of punishment. The cost that an individual i pays to punish a member j whose cooperation level is xj is defined as

3.4

Which Promotes the Evolution of Cooperation, Strict or Graduated Punishment?

83

  ai  xj : f i exp
ui Figure 3.8a, b present the examples of punishment when a ¼ 2 and a ¼ 1000, respectively. The damage suffered by a punished opponent is   ai  xj , βf i exp
ui where β (the efficiency of punishment) is above 1. These functions represent the relationship between the strength of punishment and the opponents’ cooperation level. If a is small, the strength of punishment gradually decreases as the cooperation level increases, whereas it decreases steeply around the threshold u if a is high. The total score of an individual i (si) in a game is the sum of (i) the benefit and cost of the public goods game, (ii) the cost of punishing the other three opponents in the group, and (iii) the damage from the other members’ punishment. That is, si ¼

rX x
xi 4 j j

!


X i6¼j

  ai  X   aj  xj x
: f i exp
βf j exp
i ui uj

ð3:1Þ

i6¼j

After the two-stage game, the death and birth events occur. In this study, we used an updating rule called the score-dependent viability model or viability model, in which death probability is inversely proportional to one’s score. Here, the death probability of individual i is c exp ð
dsi Þ,

ð3:2Þ

where c and d are constant and positive and si is the individual’s score. If an individual i dies as predicted by his/her death probability, a new empty site exists. One of the other individuals reproduces and that individual’s offspring immediately fills the empty site. The offspring inherits the four adaptive traits of the parent, and mutation occurs with probability m for x, u, and f and ma for a. If a mutation occurs, the offspring’s trait is normally distributed around the parental trait, with standard deviation σ for x, u, and f and σ a for a. In all the results presented in this chapter, we set these parameters as m ¼ 0.01, ma ¼ 0.1, σ ¼ 0.01, σ a ¼ 1.0, and N ¼ 50. At this point, one turn ends. We did not run simulations in a lattice larger than N ¼ 50. If a larger lattice was used, we would have expected the simulation outcome to be almost the same as that presented in this chapter. Our assumptions differ from those of Nakamaru and Dieckmann (2009). Unlike Nakamaru and Dieckmann (2009), we did not allow the existence of empty sites in the spatial structure. Sekiguchi and Nakamaru (2009) demonstrated that in the scoredependent viability model, the existence of empty sites promotes the evolution of

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altruistic punishers more than the non-existence of empty sites. Thus, it is more difficult for punishment and cooperation to evolve in our setting than in that of Nakamaru and Dieckmann (2006). If evolutionary dynamics is interpreted as social learning, the death probability in Eq. (3.2) can be considered the probability of a change in behavior. The individual decides to change behavior when the behavior’s score is low. After deciding to change behavior, the individual imitates and exhibits another behavior randomly chosen from the neighbors or the population. In each generation, N  N turns are completed; thus, every individual experiences this birth and death process once on average. We compared two conditions for spatial structure: the spatially structured condition and the random-matching condition. These two conditions differ in whom each individual interacts with and who produces offspring in the empty site created by a death. By comparing the results under these two conditions, we investigated how spatial structure affects the evolution of cooperation and punishment. First, let us describe the spatially structured condition. We assumed a population consisting of N  N individuals set in an N  N square lattice. Under these conditions, each individual plays the public goods game with their nearest neighbors. We assume that all the payoffs for the focal player are totaled for all the games played by the focal player. Let us explain our detailed assumptions. We assume that one group has four members who play the public goods game and decide to punish other members in the same group. In the Moore neighborhood, each individual has eight nearest neighbors. If we assume that four nearest neighbors make one group, each individual belongs to four groups. For example, the focal player is known as X, and X’s eight neighbors are known as A, B, C, D, E, F, G, and H, who are positioned clockwise from the top. X belongs to four groups. The first group comprises A, B, C, and X; the second, C, D, E, and X; the third, E, F, G, and X; and the fourth, G, H, A, and X. Therefore, the total payoff for X is derived from the payoff accumulated in all four groups. After the focal individual dies according to the total score, one of the eight neighbors is randomly chosen to reproduce and fill the empty site with an offspring. Second, we explain the random-matching condition, corresponding to the completely mixed population in Chap. 2 and Sects. 3.2 and 3.3. One individual and eight opponents are randomly chosen from the entire population during each turn. The selected eight opponents are put into the Moore neighborhood, and the interaction process and the reproduction process are the same as those in the spatially structured condition. After one turn, a new focal individual and eight new opponents are again randomly chosen and the game and reproduction process start anew. This process repeats until the end of the simulations (1000,000 generations). Thus, each individual plays the public goods game with almost different opponents in each generation. We made the complicated assumption about the interaction partners to use the same assumption as the spatially structured condition and compare the results of the two conditions.

3.4

Which Promotes the Evolution of Cooperation, Strict or Graduated Punishment?

3.4.3

Results

3.4.3.1

Agent-Based Simulations

85

The results of the simulations indicated that determining whether punishment should be strict or graduated for the evolution of cooperation depends heavily on how constantly the individuals interact with their neighbors. Note that all the values below are the average of 50 simulation trials. First, to distinguish the effect of the lattice from that of punishment on the evolution of the cooperation level, we examined whether the lattice-structured population can increase the cooperation level without punishment. Figure 3.9 depicts the outcomes of the evolutionary simulations and illustrates the cooperation level’s slight increase in r ¼ 4, but not in r < 4 in the spatially structured condition (Fig. 3.9a). Figure 3.9b illustrates the cooperation level’s complete failure to increase in the random-matching condition. These results indicate that the cooperation level does not increase without punishment regardless of the spatial structure. Next, we examined how spatial condition and strictness of punishment (a) affect the evolution of each trait when the strictness of the punishment was fixed for all individuals (Fig. 3.10). In the initial state, all traits were zero, that is, at first there was no punishment or cooperation. The evolved level of cooperation with punishment was much higher than that without punishment (Fig. 3.10a) and we concluded that punishment could increase cooperation. Cooperation and punishment evolved to greater levels with strict punishment than with graduated punishment in the spatially structured condition (Fig. 3.10a). These results were compatible with those of Nakamaru and Dieckmann (2009) despite the differences in assumptions and procedures. In contrast, in the random-matching condition, the simulation outcomes demonstrated that cooperation and punishment evolved with graduated punishment

b

c

benefit-to-cost ratio r

a benefit-to-cost ratio r

Fig. 3.9 Effectiveness of public goods (r) and evolution of cooperation level without punishment. a, b present the cooperation level after 1000,000 generations in the spatially structured condition and the random-matching condition, respectively. Other conditions are the same as noted in the main text. Each data block is the average of 50 trials. c presents the relationship between a value and color in both a and b

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b

cooperation level x

cooperation level x

a

strictness of punishment a

strictness of punishment a

Fig. 3.10 Evolution of cooperation level, punishment severity, and punishment threshold in the spatially structured condition. Values are averages of 50 trials at 1000,000 generations. The horizontal and vertical axes represent log a and the value of each trait, respectively. The cooperation level (x) is denoted by a solid red line and points, the punishment severity ( f ) is denoted by an orange dotted line and triangles, and the punishment threshold (u) by a solid blue line and squares. The final level of cooperation with no punishment is 0.06272092 in the spatially structured condition and 0.03539616 in the random-matching condition. a presents the spatially structured condition and b the random-matching condition. The value of a used in this figure is 0.01, 0.07, 0.1, 1, 2, 5, 20, 100, and 1000. When a ¼ 0.0, (x, f, u) ¼ (0.0747, 0.0327, 0.3223). The parameters are r ¼ 3, β ¼ 10

(low a) rather than severe punishment (Fig. 3.10b). If a is too low (roughly 0.0), where the punishment level is nearly the same regardless of the opponent’s cooperation level, the cooperation level will not increase. We also examined the outcome when strictness (a) was not fixed but assumed to be an adaptive trait. We analyzed how two parameters, r and β (the effectiveness of the public goods and that of the punishment, respectively), affected the evolution of each trait in the spatially structured and random-matching conditions, respectively (Figs. 3.11 and 3.12). In the random-matching condition, the value of β had to be sufficiently high and higher than that in the spatially structured condition for the coevolution of punishment and cooperation. Strictness (a) was higher with a higher cooperation level in the spatially structured conditions (Fig. 3.11) but lower with a higher cooperation level in the random-matching condition (Fig. 3.12). By contrast, higher levels of cooperation and punishment occurred in both conditions when the scale factor of public goods, r, was smaller. The efficiency of punishment β had to be sufficiently high for cooperation to increase in the spatially structured and random-matching conditions because the indirect advantage of fitness of punishment must outweigh the direct cost of punishment. This result is common in both modeling and experimental studies (Egas and Riedl 2008). In contrast, the efficiency of the public goods r had to be sufficiently low in the spatially structured condition. Cooperation has two negative effects on the evolution of cooperation in the viability model: (i) it is costly and (ii) it reduces opponents’ death probabilities because the focal individual gives a benefit, the function of r, to

3.4

Which Promotes the Evolution of Cooperation, Strict or Graduated Punishment?

87

b benefit-to-cost ratio r

benefit-to-cost ratio r

a

efficiency of punishment

efficiency of punishment

c benefit-to-cost ratio r

benefit-to-cost ratio r

d

efficiency of punishment

efficiency of punishment

Fig. 3.11 Effects of parameters on evolution in the spatially structured condition. The horizontal and vertical axes represent the effectiveness of public goods (r) and punishment (β), respectively. Each data block is the average of 50 trials. a–d represent the average values of a (strictness), x (cooperation level), f (punishment level), and u (punishment threshold) at 1000,000 generations, respectively. Deeper color means that the trait evolved to a higher value. Each bar below each graph presents the relationship between a value and color

88

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a

b

c

d

Fig. 3.12 Effects of parameters on evolution in the random-matching condition. The horizontal and vertical axes represent the effectiveness of public goods (r) and punishment (β). a–d represent the average values of a (strictness), x (cooperation level), f (punishment level), and u (punishment threshold) at 1000,000 generations, respectively. Deeper color means the trait evolved to a higher value. Each data block is the average of 50 trials. Each bar below each graph presents the relationship between a value and color

3.4

Which Promotes the Evolution of Cooperation, Strict or Graduated Punishment?

89

opponents and then helps to increase the opponent’s score in the viability model. A higher r facilitates the second negative effect but restrains the first effect. As a result, the evolved cooperation level is not high in the high r condition in both Figs. 3.11 and 3.12. This result is counterintuitive from the viewpoint of reality. If the viability model can be introduced into the experiment, we can compare our results in the spatially structured condition with the experiment and discuss whether the viability model actually has a low cooperation level with higher r. It may seem strange that strictness evolves at a higher level in the randommatching condition when β is low (Fig. 3.12a), but this is a trivial result. When this parameter is low in the random-matching condition, neither cooperation nor punishment evolves (Fig. 3.12b–d), and so punishment is only a cost. Strict punishment can avoid the cost of punishment more than graduated punishment can, even when the severity of punishment is low; an individual is never punished if his/her propensity of altruism (x) is higher than the threshold in strict punishment, whereas an individual is punished even though his/her x is higher than the threshold in graduated punishment.

3.4.3.2

The Mathematical Model of the Random-Matching Condition

To determine whether graduated punishment can evolve in the random-matching condition, we devised a mathematical equation to describe the random-matching condition. The assumption in the evolutionary simulations is so complicated that we simply reassume that a focal individual and z opponents are randomly chosen from the entire population. Then z + 1 members play a public goods game and the score of the focal player in the game gives the death rate (Eq. 3.2). If z ¼ 8 in this model is considered as eight opponents in the random-matching condition, this analytical result would predict the evolutionary simulation outcomes. To investigate the evolution of continuous traits, the cooperation level x, severity of punishment f, and threshold of punishment u, we can obtain the invasion fitness or the mutant’s growth rate following adaptive dynamics (Dieckmann and Law 1996; Geritz et al. 1997). Adaptive dynamics is a mathematical framework that deals with eco-evolutionary dynamics based on simple assumptions such as rare mutations or small mutational effects (Hastings and Gross 2012). For simplicity, we assume that strictness of punishment a is a constant. Let ρ and ρm, respectively, be the frequency of the wild type whose traits are x, f, and u, and that of the mutant type whose traits are xm, fm, and um. The time differential equation of ρm is obtained as

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The Effect of Peer Punishment on the Evolution of Cooperation

X dρm z
i z! c exp ½
dsw,i  ρi ρm z
i ¼ρ z dt i! ð z
i Þ! i¼0 z
1


ρm

z X i¼1

ρi ρm z
i

i z! c exp ½
dsm,i  , z i!ðz
iÞ!

ð3:3Þ

in which sw,i is the total score of a focal player who adopts the wild type and plays the public goods game with i wild types and (z
i) mutants. Let sm,i be the total score of a focal player who is a mutant and plays the public goods game with i wild types and (z
i) mutants. The first term in the right side of Eq. (3.3) means the increase of the mutants: the focal individual with the wild type’s traits plays the game with i wild types and (z
i) mutants and then dies with the death probability defined by Eq. (3.2). After the individual’s death, the site previously occupied by the focal individual becomes a new empty site. Next, one mutant is randomly chosen from z opponents, who then colonizes the empty site. The second term means the decrease of the mutants: the focal individual with the mutant type’s traits plays the game with i wild types and (z
i) mutants and then dies with the death probability defined by Eq. (3.2). After the individual’s death, the site previously occupied by the focal individual becomes a new empty site. Next, one wild type is randomly chosen from z opponents, who then colonizes the empty site. It is assumed that the frequency of mutants is rare in the population and ρm2 is assumed to be zero. Therefore, Eq. (3.3) can be simplified as follows: dρm  cρz ρm ð exp ½
dsw,z
1 
exp ½
dsm,z Þ  cρz ρm I, dt

ð3:4Þ

where I is defined as the invasion fitness or the mutant’s growth rate (Dieckmann and Law 1996). The gradients of each trait are followed as if the traits are independent:  dx dI  / dt dxm xm ¼x,f ¼f ,um ¼u m   a
1    a  x x exp ð
dGÞ, exp
¼ d
1 þ ðβz
1Þaf ua u    a  df dI  x ¼ ðβ
zÞd exp
exp ð
dGÞ, /  dt df m xm ¼x,f ¼f ,um ¼u u m    a  du dI  adf xa x ¼ ð β
z Þ exp
exp ð
dGÞ, / dt dum xm ¼x,f ¼f ,um ¼u u uaþ1

ð3:5aÞ ð3:5bÞ ð3:5cÞ

m

in which G ¼ x(r
1)
(1 + β)zf exp.(
(x/u)a). Eqs. (3.5b) and (3.5c) indicate that traits f and u always increase through the evolutionary process when β > z, and other parameters do not affect the sign of

3.5

Discussion and Conclusions

91

Eq. (3.5b and 3.5c) if u is not zero in Eq. (3.5c). If z ¼ 8, in this model considered as eight opponents in the random-matching condition, this analytical result can predict the evolutionary simulation outcomes f and u in the random-matching condition (Fig. 3.12c, d); the evolved values of f and u is high in β > 8. In a ¼ 0, the sign of Eq. (3.5a) is always negative. This indicates that the cooperation level converges to zero (only for defectors) if the punishment level is constant, regardless of the opponent’s cooperation level. If a is less than zero, in which case cooperators are punished, the sign of Eq. (3.5a) is negative. If a is higher than zero, the sign of Eq. (3.5a) is dependent on other parameters. Figure 3.13 (r ¼ 3, β ¼ 10, z ¼ 8) shows the pairwise invasibility plots of x generated by Eq. (3.4) when values close to the evolved values of f and u in Fig. 3.10b are substituted into f and u in Eq. (3.5a). Figure 3.13a shows that when the punishment level remains the same regardless of the opponent’s cooperation level (a ¼ 0), the population is eventually dominated by defectors (x ¼ 0). When a ¼ 0.01, the value of x converges to a low level, which is an ESS. When punishment becomes graduated (a is 0.07, 1, and 2 in Fig. 3.13), the cooperation level converges to 1 (Fig. 3.13c–e). When punishment becomes strict, there are at least two singular points in Fig. 3.13f–i, and these figures imply that the cooperation level may converge to the higher value of the singular point when x starts from a high value. The stricter the punishment, the lower the converged value of x (around 0.5). Even though f and u are not evolutionary traits in Fig. 3.13, these theoretical results can roughly predict the simulation outcomes of x in Fig. 3.10b.

3.5 3.5.1

Discussion and Conclusions Spiteful Behavior in Social Sciences and Evolutionary Evolution

Punishment reduces the player’s fitness in order to damage the opponent’s fitness, and hence it is a form of spite. A good example of spite is toxin production by Escherichia coli (e.g., Chao and Levin 1981). The colicin-sensitive strain is killed by colicin. The colicin-producing strain is immune to it but must pay a cost. The colicinproducing strain can increase in numbers when rare in a spatially structured habitat, but not in a perfectly mixed population, as shown by theoretical and experimental studies (Durrett and Levin 1997; Iwasa et al. 1998; Nakamaru and Iwasa 2000; Frean and Abraham 2001; Czaran et al. 2002; Kerr et al. 2002; Czaran and Hoekstra 2003). Experimental economics also observes spiteful behavior in humans (Levine 1998; Pillutla and Murnighan 1996; Saijo and Nakamura 1995; Cason et al. 2004). We need more theoretical and experimental studies on the evolution and maintenance of spite to understand cooperation in societies of human and nonhuman species.

92

3

b

c

e

f

h

i

mutant type

a

The Effect of Peer Punishment on the Evolution of Cooperation

d

g

wild type

Fig. 3.13 Pairwise invasibility plots (PIPs) of cooperation level when other traits are not adaptive. The horizontal and vertical axes represent the wild and mutant type, respectively. The black region means the mutant type can invade the population occupied by the wild type (I > 0 in Eq. 3.4), and the white region means the mutant type cannot invade (I < 0). The black diagonal means I ¼ 0. The small graph in each figure presents the punishment cost assumed in each of a–i. In each small graph, the horizontal and vertical axes represent the opponent’s cooperation level and the punishment cost of the focal player. Parameters are z ¼ 8, r ¼ 3, β ¼ 10, d ¼ 0.1, and u ¼ 0.5; a a ¼ 0.0 and f ¼ 0.9; b a ¼ 0.01 and f ¼ 0.9; c a ¼ 0.07 and f ¼ 0.9; d a ¼ 1 and f ¼ 0.5; e a ¼ 2 and f ¼ 0.9; f a ¼ 5 and f ¼ 0.9; g a ¼ 20 and f ¼ 0.9; h a ¼ 100 and f ¼ 0.9; and i a ¼ 1000 and f ¼ 0.9

The noteworthy definition of spite is proposed by West and Gardner (2010) from the viewpoint of evolutionary ecology. A spiteful player incurs a cost, c, and an opponent which is damaged by the spiteful behavior obtains the negative benefit (b < 0). If br
c > 0 holds, where r is the relatedness between the spiteful player and the opponent, the value of r becomes negative (Hamilton 1970). The negative relatedness means that the relatedness between two players is lower than the average

3.5

Discussion and Conclusions

93

relatedness among the population. After Hamilton (1970), there have been studies about spite from the viewpoint of evolutionary ecology. West and Gardner (2010) defines the condition for the evolution of spiteful behavior, focusing on the triad relationship: players A, B and C, for example. Player A spites player B. Not only is player B damaged by player A but also player A incurs a cost of spite. When the damage of player B gives player C a benefit and player C’s benefit gives player A the benefit, spiteful behavior by player A can evolve if three conditions hold: (i) the difference between player A and B in relatedness is high, (ii) each player can distinguish their relatives from the non-relatives, and (iii) the local competition among players is high. A good example of spite from the perspective of West and Gardner (2010) is toxin production by Escherichia coli (e.g., Chao and Levin 1981; Durrett and Levin 1997; Iwasa et al. 1998; Nakamaru and Iwasa 2000). However, the spiteful behavior in experimental economics and altruistic punishment is not spiteful from the viewpoint of West and Gardner (2010), because it will promote cooperation and then receive a higher payoff in the long-run. Therefore, we have used the term “spiteful behavior” to describe the spiteful behavior in the dyad relationship in Chaps. 2 and 3 and it is not the same as the spiteful behavior in West and Gardner (2010).

3.5.2

Graduated or Strict Punishment in Society

The major finding in Sect. 3.4 is that strict punishment is more effective in increasing cooperation in a spatially structured population and graduated punishment is more effective without spatial structure. From an evolutionary psychology viewpoint, if a function of punishment becomes experimentally clear, we can speculate the type of situation that would cause the evolution of punishment behavior in humans. Strict punishment would evolve in a small, fixed society, such as a band, whereas graduated punishment would more likely evolve in a fluid society. Another possible hypothesis is that people change their punishing function depending on the relationship with their opponents. Experimental studies have shown that membership of the opponent in the in-group or the out-group determines the punishment strategy (Shinada et al. 2004; Bernhard et al. 2006; Goette et al. 2006). Based on our study, we can infer that the strategy will be adaptive if people employ strict punishment to in-group members and graduated punishment to out-group members. This is because members of the same group often encounter each other, which is similar to the spatially structured condition. Individuals will encounter others in different groups only occasionally, which corresponds to the random-matching condition. These hypotheses should be tested in psychological experiments. Although people have a tendency to punish defectors, the function of this type of punishment has not, to our knowledge, been discussed. The experiments of Egas and Riedl have some relevance to our model in that they mention the importance of a punishment

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3 The Effect of Peer Punishment on the Evolution of Cooperation

threshold for maintenance of cooperative behaviors (Egas and Riedl 2008). Although strictness of punishment was not a factor, and they used linear regressions to represent punishment increases in relation to cooperation decreases, their experimental settings would be useful in studies of people’s punishment functions with appropriate modifications. Numerous studies have examined cooperation and punishment, but the number that have addressed continuous strategies of cooperation and punishment is far from adequate. The strictness of punishment or of any other functional strategy is an important consideration in the punishment’s success, as this study has proven. Our study provides further information for application in various experiments, simulations, and field work. Our approach can be applied to adaptive punishment or reward, which would change the effort of punishment or reward, depending on the success of cooperation (Szolnoki and Perc 2012; Perc and Szolnoki 2012), whereas many theoretical studies on punishment assume that the cost of punishing and being punished is fixed for simplicity. In future research, we plan to examine specifically the evolution of severely punishing repeat offenders, a subject that applies to our real society in the punishment of criminals and certain long-standing common-pool resource institutions (Cox et al. 2010; Mungan 2010; Cyrus et al. 2000).

3.5.3

Anti-Social Punishment in Society

Experimental studies have shown the existence of anti-social punishers (Saijo and Nakamura 1995; Shinada et al. 2004; Denant-Boemont et al. 2007; Dreber et al. 2008; Herrmann et al. 2008; Nikiforakis 2008; Gächter and Herrmann 2009; Wu et al. 2009) and should not be ignored. The theoretical studies show that anti-social punishment collapses the evolution of cooperation and punishment (Rand et al. 2010; Rand and Nowak 2011; Hilbe and Traulsen 2012). However, it is highly possible that the social or innate motivation to punish cooperators is completely different from the motivation to punish defectors from the viewpoint of reality and social context. For example, envy or malicious intent might make people punish cooperators, while the anger provoked by social justice might make people punish defectors. Herrmann et al. (2008) demonstrated that the weak norms of civic cooperation and the weakness of the rule of law in a country are predictors of antisocial punishment. This implies that social structure influences punishment behavior. Therefore, anti-social punishers and punishers who punish cooperators belong to socially or innately different categories and should not be dealt with together. In future, we will formulate appropriate models that will test assumptions regarding what causes people to punish cooperators in order to investigate the emergence of anti-social punishment in our society. From this viewpoint, it would be useful to study the coevolution of functional forms of punishment and cooperation without assuming anti-social punishment. In our society, graduated sanctions are used for resource management and in the sanction system used by the police (Ostrom 1990; Cox et al. 2010). Peoples’

3.5

Discussion and Conclusions

95

minds are equipped with graduated punishments, so that they may establish a graduated sanction system for resource management and for sanctioning law violators. If this is true, our study can prove why we have a graduated sanction system in our society. However, future studies should examine what happens when anti-social punishment is introduced when cooperation and punishment are continuous traits. The mathematical analysis (see Appendix S1 in Shimao and Nakamaru (2013)) demonstrated that cooperation and punishment did not evolve when players behave as antisocial punishers (the value a is fixed and less than zero). When we assume that the evolutionary trait a is between
1 and +1, it is difficult to predict whether a new result would follow the previous theoretical studies (Rand et al. 2010; Rand and Nowak 2011; Hilbe and Traulsen 2012) because these studies, on the evolution of anti-social punishment, assumed discrete strategies, and then mutation changed non-punishers to punishers and cooperators to defectors. However, our model does not exhibit this behavior because we assume that, when mutation occurs, the offspring’s trait is normally distributed around the parental trait with standard deviation. For example, mutation changes non-punishers to players who punish little if the standard deviation is small. Therefore, the difference in the strategy space between the previous theoretical studies (Rand et al. 2010; Rand and Nowak 2011; Hilbe and Traulsen 2012) and our model may cause different evolutionary dynamics, which may depend on the initial condition and the magnitude of standard deviation. When the initial population is monomorphic, the initial value of a is positive and high, and the standard deviation is small, the result may be the same as that observed in this study. However, when the initial value of a is equal to or less than zero and the standard deviation is very high, anti-social punishment may collapse the evolution of cooperation and punishment in both the random-matching condition and the spatially structured condition. However, when the initial value of a is zero and the standard deviation is small, we require further simulations to investigate whether cooperation and punishment evolve.

3.5.4

Other Types of Punishment Beside Peer Punishment

In this section, we focused on costly punishment or peer punishment where a punisher directly punishes an opponent, incurring the cost of punishment. In reality, instead of costly punishment, we use other punishment ways to avoid the punishment cost; we spread the reputation or gossip about defectors, exclude the defectors from the groups, and retaliate against the defectors. Boehm (1999) showed in anthropological field work that people who want to be a leader are receive spite from other members and that rule breakers are excluded from the group in a small egalitarian society. In Chaps. 4 and 6, we show the effect of group exclusion on the evolution of cooperation when the group is formed. The gossip works as costless punishment and we investigate the effect of gossip in Chap. 9. Anthropological field work has also shown that not costly punishment but spreading negative gossip about

96

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free-riders and excluding free-riders from a group in a small society works as punishment (Boehm 1999; Guala 2012). However, it is very difficult to do experiments to investigate the effect of bad gossip and exclusion as costless punishment on cooperation in a laboratory from the viewpoint of morals and ethics. Therefore, the computer simulation and model studies will help us understand the role of exclusion and bad gossip as costless punishment (see Chaps. 6 and 9). In a large society, it is not efficient for each individual to make efforts to punish others in either costly or costless ways. Institutions such as the police system or judicial system have been invented in our society. Even in a small society, punishment coordinated by institutions exists rather than costly punishment (Guala 2012). One direction for the evolution of institutional punishment has been studied by Sigmund et al. (2010), who compared the effect of peer punishment with that of pool punishment, in which players invest a cost to maintain the policing institution and the strength of an institutional sanction is proportional to the number of players who invest in the policing institution. Their study showed that pool-punishers do better than peer-punishers if not only the first-order punishment, in which defectors are punished, but also the second-order punishment, in which cooperators who do not perform punishment are punished, is allowed. This may suggest that all members in a community should pay for the policing institution as tax, and so on. In a small society, monitoring defectors is cheap because all members know everyone’s behavior, whereas in a large society, it is expensive because it is hard to know all members’ behavior. In Chap. 8, we will consider in detail the effect of two types of punishment by the administration or local government on the evolution of cooperation in the division of labor. The first type requires monitoring defectors; however, this is very difficult because the local government cannot pursue their behavior in a large society. The other type does not require monitoring defectors, and here the punishment works better than the first type.

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Part II

Cooperation, Trust, and Credit in the Early-Stage Mutual-Aid Systems

Chapter 4

Rotation Savings and Credit Associations (ROSCAs) as Early-Stage Credit Systems

Abstract Collective behavior is theoretically and experimentally studied through an “all-for-all type” game such as a public goods game. However, if goods are indivisible, only one player can receive the goods. In this case, the problem is how to distribute indivisible goods, and here therefore we propose a new game with an “allfor-one” type, namely the “rotating indivisible goods game.” In this game, the goods are not divided but distributed by regular rotation. An example is rotating savings and credit associations (ROSCAs), which exist all over the world and serve as efficient and informal institutions for collecting savings for small investments. In a ROSCA, members regularly contribute money to produce goods and to distribute them to each member on a regular rotation. It has been pointed out that ROSCA members are selected based on their reliability or reputation, and that defectors who stop contributing are excluded. We elucidate mechanisms that sustain cooperation in rotating indivisible goods games by means of evolutionary simulations. First, we investigate the effect of the peer selection rule by which the group chooses members based on the players’ reputation, also by which players choose groups based on their reputation. Regardless of the peer selection rule, cooperation is not sustainable in a rotating indivisible goods game. Second, we introduce the forfeiture rule that forbids a member who has not contributed earlier from receiving goods. These analyses show that employing these two rules can sustain cooperation in the rotating indivisible goods game, although employing either of the two cannot. Finally, we prove that evolutionary simulation can be a tool for investigating institutional designs that promote cooperation. Keywords Peer selection · Institution · ROSCA · Rotating indivisible goods game

4.1

Introduction

Cooperation among more than three individuals, such as collective behavior, has often been studied using a public goods game (e.g., Ostrom 2000), which is categorized as the all-for-all type (Table 1.4). In human society, the goods are not always divided, or are not enjoyed by all players concurrently. In some social and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Nakamaru, Trust and Credit in Organizations and Institutions, Theoretical Biology, https://doi.org/10.1007/978-981-19-4979-1_4

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4 Rotation Savings and Credit Associations (ROSCAs) as Early-Stage. . .

economic situations, goods such as private land or internal organs for patients should not be divided. In game theory, the problem of fairly allocating indivisible goods has been investigated when, for example, the number of people who need them is higher than the number of indivisible goods (e.g., Alkan et al. 1991; Young 1995). Following Table 1.4, this situation is categorized as the all-for-one type. Some types of indivisible goods are distributed in rotation. For example, in rural villages, farmers cooperate in the busy farming season: on 1 day all the villagers work on farmer A’s land, the next day they all work on farmer B’s land, and so on. In this case, the goods (aggregated farmers’ labor) are not divided, instead given to each farmer in rotation. Ekeh (1974) called this type of cooperation “individual-focused net generalized exchange.” The form of individual-focused net generalized exchange is the normal usage of the common or natural resources in some local areas. In an irrigation system without storage, water is strictly tied to the land, and some form of rotation is used (Ostrom 2000). Even though money itself is divisible, a small amount of money is useless for the purpose of investing in, for example, business and building a new house. Therefore the cooperative exchange of money also takes the form of individual-focused net generalized exchange. The outstanding example of this is the “rotating savings and credit association (ROSCA),” which exists all over the world (Geertz 1962; Ardener 1964). ROSCAs are informal financial institutions formed by groups of friends and neighbors who typically meet regularly. Ardener (1964) points out that small-scale capital formation is a function of a ROSCA. The ROSCA is one of the most common finance institutes in the developing countries such as Africa and some Asian countries, supporting socioeconomic development and improving human welfare for those people (Bouman 1995; Morduch et al. 2009). Ambec and Treich (2007) defined the benefit of ROSCAs: facilitating an early purchase of a durable good, as a substitute to insurance, coping with self-control problems. One important characteristic of ROSCAs is that contributions are voluntary. ROSCA participants typically use their returns payouts to buy durable goods (Besley and Levenson 1996), invest in a business (Light 1972), or make precautionary savings to meet unplanned expenses (Handa and Kirton 1999). Some microfinance organizations, such as the Grameen Bank and credit cooperatives, have originated from ROSCAs (Armendáris and Morduch 2010). The advantage of joining a ROSCA is that, with the exception of the last recipient, participants obtain access to money earlier than would have been possible through independent saving. Generally, ROSCAs can be classified into three types according to the means used to determine the order of payment of returns: a random ROSCA, a fixed ROSCA, and a bidding ROSCA. With the random ROSCA, at each meeting a recipient is selected by lottery from among those participants who have not yet received their returns. On average, participants in a random ROSCA can expect to receive their fund at the (n + 1)/2-th meeting. In the case of the fixed ROSCA, the order of payment is determined before the first meeting. Finally, in the bidding ROSCA, participants bid for the available returns payout at each meeting, and payment goes to the highest bidder.

4.1

Introduction

105

Although there are various schemes for ROSCAs, in this chapter we demonstrate a rather simple one. A ROSCA holds meetings regularly; if the number of members is n, the meetings are held n times. At every meeting, each member contributes the same amount of money to a central fund, and the fund is given to one of the members (through bidding or by lot). When all n regular meetings have been completed, each of the n members has contributed n times and has received the fund once. In the case of a ROSCA, the free-rider problem is that some members may fail to contribute, or abscond with the money after receiving the fund. Henceforth we call such non-contributors defaulters. Defaulters receive more benefits than members who always contribute, because the defaulters do not contribute after receiving the fund. Although defaulters receive large benefits, ROSCAs have been conducted actively all over the world. Therefore, it is important to investigate the mechanism that suppresses defaulters and sustains the ROSCAs. It may be thought that a public goods game can describe the system of ROSCAs, because n meetings of a ROSCA which consists of n members seem to be the same as n repetitions of a public goods game. However, a ROSCA is fundamentally different from a public goods game. ROSCAs are sustainable by suppressing defaulters who contribute before receiving the fund and stop thereafter; they also indicate that cooperation (or contribution) before receiving the fund should be distinguished from cooperation after receiving the fund. In contrast, in a public goods game there is no need to distinguish these two kinds of cooperation. Investigating the conditions that sustain cooperation under ROSCAs is not included in studies of public goods games from the perspective of evolutionary game theory. We propose a new model called the “rotating indivisible goods game,” which can be applied not only to ROSCAs but also to various social and economic situations. There have been theoretical studies of ROSCAs by economists (e.g., Besley et al. 1993; van den Brink and Chavas 1997; Anderson et al. 2009). Besley et al. (1993) conducted a mathematical analysis based on the lifetime utility function, in which individuals join only one ROSCA throughout their lives, invest the money obtained from the ROSCA in durable goods, and receive the use of the durable good until the end of their lives. They investigated the condition under which players joining a ROSCA receive more benefits than players who do not join a ROSCA but save money on their own, and they assumed that all individuals always contribute money to the pot. They showed that ROSCAs can be sustainable when the benefit to a defaulter who receives the pot in the first meeting does not exceed the cost that a defaulter has to pay, such as a sanction. Anderson et al. (2009) assumed a group whose members never change and repeats ROSCAs infinitely. One of their conclusions was that introducing social sanctions prevents ROSCAs from collapsing, as defaulters’ benefits are high without social sanctions even though they are excluded from the ROSCA. These earlier theoretical studies calculated the lifetime utility, and did not consider the effect of the reliability of each individual and peer selection in organizing a ROSCA group. There have been studies that argue the importance of peer selection in sustaining a ROSCA (e.g., Hechter 1987; Tsujimoto et al. 2007). Based on field research, Tsujimoto et al. (2007) argued that the peer selection of a ROSCA is based on

106

4 Rotation Savings and Credit Associations (ROSCAs) as Early-Stage. . .

face-to-face relationships, which means that the members know each other, as a way of solving the free-rider problem. Coleman (1990) and Putnam (1993) referred to ROSCAs, and argue the importance of social capital such as trust or networks. As Putnam (1993) wrote, “The risk of default is well recognized by participants, and organizers select members with some care. Thus, a reputation for honesty and reliability is an important asset for any would-be participant” (p. 168). Coleman (1990) wrote, “One could not imagine such a rotating credit association operating successfully in urban areas marked by high degree of social disorganization—or, in other words, by a lack of social capital” (pp. 306–307). These studies imply that peer selection based on reputation is essential to sustain a ROSCA. In this chapter, we conduct evolutionary game simulations of the rotating indivisible goods game with reference to ROSCAs (Koike et al. 2010), because ROSCAs have been investigated thoroughly. First, based on the image score (Nowak and Sigmund 1998), we investigate whether reputation can promote cooperation in a rotating indivisible goods game. In contrast to studies of public goods games with punishment (e.g., Fehr and Gächter 2002; Gürerk et al. 2006), we do not investigate the effect of punishment in sustaining cooperation in a rotating indivisible goods game, because punishment cannot be introduced into such informal institutions as ROSCAs. Instead of punishment, peer selection based on reputation may work as a cost-free sanction in informal institutions. We also formulate the decision-making processes in peer selection as a “peer selection rule.” It is not obvious what kind of decision players make when they join a particular group, or when they allow others to join their group. Simulation can clarify a decision-making process concretely, whereas in previous studies that reported the importance of peer selection only verbally, these details were not clarified. Second, we investigate the effectiveness of the forfeiture rule, which prohibits any member who has not contributed in the preceding meetings from receiving the goods. Our simulation outcomes will demonstrate that peer selection based on reputation is not enough to suppress defaulters. To sustain cooperation, not only peer selection based on reputation, but also a forfeiture rule that is shared among the players is required. Finally, we will show that evolutionary simulation can clarify which mechanism or rule of the institution can sustain cooperation, and we can then propose that evolutionary simulation is one research tool for an institutional design that promotes cooperation.

4.2

Models and Results

4.2 4.2.1

107

Models and Results Model 1: Baseline Evolutionary Simulation Model of Rotating Indivisible Goods Game

Here we explain the baseline evolutionary simulation model of a rotating indivisible goods game (Koike et al. 2010). The population has N players. Each generation time consists of r rotating times. During one rotating time, there are n regular meetings when there are n members in the group. At the beginning of one rotating time, N players are divided into m groups randomly. Therefore each group has N/m (¼ nave) candidates on average. In the baseline model, we assume that candidates become members who play the rotating indivisible goods game in each group. Then the n regular meetings of the rotating indivisible goods game start. At the t-th meeting (1  t  n), each member decides to cooperate or not. If she cooperates, she has to pay a cost, x, to contribute to and to produce indivisible goods. If she does not, she does not have to pay any cost. The indivisible goods are given to one of the members by lot. Members who receive the goods earlier can obtain more profits than those who receive the goods later. This is because the earlier a member receives the fund or service from the indivisible goods, the earlier the member can, for example, invest in a business. Therefore we assume the future benefit factor, w (1). The parameter w > 1 can be interpreted as a future benefit, and w ¼ 1 means no future benefit. If all the members always contribute x to the goods during one rotating time, the total payoff of the member who receives the indivisible goods at the t-th meeting is x(n  1)wn tþ 1  (n  1)x. We assume that, if the member who receives the goods in the t-th meeting contributes x to the goods, she loses x, and her contribution increases the indivisible goods by x. These cancel out. If the member does not contribute x to the goods, she loses any cost and does not increase the goods. The players’ strategies are as follows. Each strategy consists of three heritable traits, q1, q2, and k. Let q1 be the cooperation probability before receiving the goods, and let q2 be the cooperation probability after receiving the goods. A player cooperates according to the probability of q1 (or q2) before (or after) receiving the goods. The player with q1 ¼ 1 and q2 ¼ 1, defined as the cooperation strategy, always cooperates. The member with q1 ¼ 1 and q2 ¼ 0, defined as the default strategy, does not cooperate after receiving the goods. For simplicity, we assume q1, q2 ¼ 0 or 1. Let k be the threshold of the reputation level, between 6 and 6, which will be introduced in Models 2 and 3. Therefore there are four strategies in Model 1, and 4  13 strategies in Models 2 and 3; each member adopts one of these strategies. The total payoff of the member who receives the goods at the t-th meeting in the u-th rotating time (Iu,t) is

108

4

I u,t ¼

t1 X j¼1

Rotation Savings and Credit Associations (ROSCAs) as Early-Stage. . .

q2j þ

n X

! q1j xwnþ1t  ½ðt  1Þq1t þ ðn  t Þq2t x,

ð4:1Þ

j¼tþ1

where q1t and q2t are the q1 and q2 of the member who receives the goods at the t-th meeting. The first term of the right-hand side represents the goods multiplied by the future benefit. The second term represents the contribution of the member in one rotating time. We assume that if the member who receives the goods in the t-th meeting contributes x to the goods, that member suffers a cost, x, the member’s contribution increases the goods by x, and these therefore cancel out. If the member does not contribute x to the goods, she has no cost and does not increase the goods. Therefore, whether or not a member contributes x in the t-th meeting does not affect the total payoff of the member, Eq. (4.1). When the n-th meeting is ended, the members are dispersed. Then m new groups are formed at the beginning of the next rotating time. At the beginning of each generation, all players are given the baseline payoff xnaver, and the reputation level (s) of each player is zero. During each generation, players accumulate the payoffs. Let us explain which strategy players adopt at the beginning of the next generation. At the end of each generation, the total payoffs of players who adopt the same strategy are accumulated, and this is called the total payoff of the strategy. We assume the mutation probability, μ, which can be interpreted as the probability that players adopt one of strategies randomly, if the process of evolution is applied to describe the decision-making model. Then, at the beginning of the next generation, each player adopts one of the strategies in proportion to the total payoff of each strategy in the previous generation with probability 1  μ, and each player adopts one of the strategies uniformed randomly with probability μ.

4.2.2

Results of Model 1

At the beginning of one simulation run, the heritable traits are uniformed randomly (q1, q2 ¼ 0 or 1). Here we do not consider k. The baseline parameters are N ¼ 100, m ¼ 5, r ¼ 20, and μ ¼ 0.005. Figure 4.1a is the average of q1* and q2* in 50 simulation runs, in which q1* and q2* are the average through 10,000 generation times in each simulation run. Both q1 and q2 become almost zero through the evolution (Fig. 4.1a), and cooperation collapses under the baseline model.

4.2

Models and Results

109

b

q1 or q2

q1 or q2

a

q2 q1

w

w

Fig. 4.1 Relationship between average cooperation probability and the future benefit factor. The horizontal axis represents the future benefit factor, w, and the vertical represents average cooperation probability (q1 or q2). See the main text for the method of calculating average probability. The black line represents average cooperation probability before receiving the goods (q1), and the gray line represents average probability after receiving the goods (q2). The parameters are N ¼ 100, m ¼ 5, r ¼ 20, μ ¼ 0.005. a Neither the peer selection rule based on labeling rule 1 nor the forfeiture rule is assumed. b Not the forfeiture rule, but the peer selection rule based on labeling rule 1, is assumed

4.2.3

Model 2: Rotating Indivisible Goods Game with the Peer Selection Rule

Here we will introduce the peer selection rule into the baseline model. We assume that members are selected according to the peer selection rule at the beginning of each rotating time. Before specifying the rule, we will explain the other assumptions related to the rule. At the beginning of each rotating time, N players are divided into m groups randomly. Therefore, each group has N/m (¼ nave) candidates on average. The group chooses members, and members also decide whether to join the group or not, according to the peer selection rule. After n members have been selected, n regular meetings of the rotating indivisible goods game start. If candidates cannot be members, they do not do anything during the rotating time, and their payoffs remain unchanged. Each player has a reputation level, which is similar to the image score of Nowak and Sigmund (1998), which is a non-heritable trait and can be observed correctly by every player. The reputation level, s, is between 5 and 5, and a high s means that the player has a high reputation. We will explain how to label the reputation level (this is called labeling rule 1). When a member cooperates and contributes x, “the number of contributions” increases by one. Whenever n meetings have been completed, the reputation level is changed by the following rule: calculate pb and pa, followed by pb ¼ (the number of contributions before receiving the goods)/(the number of meetings before receiving the goods), and pa ¼ (the number of contributions after receiving the goods)/(the number of meetings after receiving the goods). If a member does not contribute x before receiving the goods, the reputation level of

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4 Rotation Savings and Credit Associations (ROSCAs) as Early-Stage. . .

this member can decrease by 1 with a probability of 1  pb. Otherwise, the reputation level increases by 1. If a member does not contribute x after receiving the goods, the reputation level of this member can decrease by 1 with a probability of 1  pa. Otherwise, the reputation level of this member increases by 1. At the beginning of each generation, the reputation level is reset: s ¼ 0. Now we specify the peer selection rule, according to which candidates can be members of the rotating indivisible goods game. Here we assume that the group chooses the members, and that the members decide to join the group. Therefore the peer selection rule has two conditions: (i) if the reputation level of the candidate j in the group i (si,j) is higher than or equal to the average threshold ki,ave ¼ (1/nc,i) P nc,i j¼1 k i,j among all candidates in the group i where ki,j is the threshold of a candidate j in the group i, ki,ave is average threshold among all candidates in the group i, and nc,i is the number of candidates in the group i, the group permits the candidate j to join the group i (si,j  ki,ave). If ki,ave is high, the group i permits only a candidate with a high reputation to join the group i. (ii) If the average reputation level (si, Pnlevel c,i s ) among all candidates in the group i is higher than or equal i,j ave ¼ (1/nc,i) j¼1 to the threshold ki,j of the candidate j (si,ave  ki,j), the candidate j wants to join the group i. If ki,j is low, the candidate j wants to join the low-reputation group. If these two conditions hold, a candidate can become a member.

4.2.4

Results of Model 2

At the beginning of one simulation run, the heritable traits are uniformed randomly (6  k  6, q1, q2 ¼ 0 or 1). The parameters are N ¼ 100, m ¼ 5, r ¼ 20, and μ ¼ 0.005. Figure 4.1b shows the results when we introduce the peer selection rule based on reputation, and it shows that the peer selection rule does not help to increase the number of contributions. Later we will discuss the reason why cooperation is not established.

4.2.5

Model 3: A Forfeiture Rule Is Introduced

Here we introduce a “forfeiture rule,” which forbids a member who has not contributed in the preceding meetings from receiving the goods. Some empirical investigations indicate that this rule is supposed to be employed in ROSCAs. In addition, we introduce labeling rule 2: if a member cooperates and contributes x after receiving the goods, the number of contributions of this member increases by one. If a member does not cooperate after receiving the goods, the number of contributions of this member decreases by one. The reason why we use labeling rule 2 and not labeling rule 1 is that counting the number of contributions before receiving the goods does not provide any information about the reliability of members under the forfeiture

4.2

Models and Results

111

rule, because all members come to contribute x to the goods before receiving the goods under this rule (see Sect. 4.2.7). At the end of each rotating time, the reputation level of each member increases by one with a probability of pa. Otherwise, it decreases by one.

4.2.6

Results of Model 3

Figure 4.2a shows the simulation outcome when we introduce the peer selection rule based on labeling rule 2 later in the rotating indivisible goods game. The values of q1 and q2 are higher than those without the peer selection rule (Fig. 4.1a). However, under this institution cooperation collapses. Figure 4.2b shows the results when we introduce the forfeiture rule into the rotating indivisible goods game without the peer selection rule, and it is natural that q1 may be almost one, due to the forfeiture rule. This figure indicates that the default strategy can evolve. When both the peer selection rule and the forfeiture rule are introduced, the cooperation strategy can evolve for high w (see Fig. 4.2c). Figure 4.3 shows the change in three heritable traits with generation time in one simulation run, when

a

b q1 or q2

q1 or q2

q1

q2

q2

q1

w

w

c q1 or q2

q1

q2

w Fig. 4.2 Relationship between average cooperation probability and the future benefit factor. a Not the forfeiture rule, but the peer selection rule based on labeling rule 2, is assumed. b Not the peer selection rule based on labeling rule 2, but the forfeiture rule, is assumed. c Both the forfeiture rule and the peer selection rule based on labeling rule 2 are assumed. See Fig. 4.1 for further details

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k or reputation level

a 6 4 2 0 -2 -4 -6 0

s k 5000

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15000

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q1

q1 or q 2

0.8 0.6 0.4

q2

0.2 0.0 0

5000

10000 generation time

number of members

c 20 15 10 5 0 0

5000

10000 generation time

Fig. 4.3 The evolutionary dynamics of average cooperation probabilities per generation (q1, q2), the threshold (k) and the average number of members per group in one simulation run, when both the forfeiture rule and the peer selection rule based on labeling rule 2, are assumed. The horizontal axis represents generation times, and the vertical represents the average of k in each generation in a, d. The black line represents the average threshold (k) in the population per generation, and the gray line represents the average reputation level (s) in the population per generation. In b, e, the vertical represents average q1 and q2 in each generation, the black line represents the average of q1, and the gray line represents q2. In c, f, the vertical represents the average number of members per group per generation. The value of w is 1.06 in a–c and 1.37 in d–f. Other parameters are N ¼ 100, m ¼ 5, r ¼ 20 and μ ¼ 0.005

4.2

Models and Results

113

k or reputation level

d 6 4 2 0 -2 -4 -6

s k

0

5000

e

10000 generation time

15000

20000

15000

20000

15000

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0.8 0.6 0.4

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0.2 0.0 0

5000

number of members

f

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20 15 10 5 0

0

5000

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Fig. 4.3 (continued)

w ¼ 1.06 and 1.37 when both are assumed. For w ¼ 1.06, the value of q2 goes down to zero after about 2000 generation times, and the value of k also reaches the minimum (Fig. 4.3a–c). For w ¼ 1.37, the oscillations generate the q2-generation time graph: the value of q2 becomes one, and then becomes zero by change. It then goes back to one again (Fig. 4.3e). Figure 4.3a,d show that, when the reputation level is higher than the threshold (k), the number of members is almost nave on average. This means that all the candidates become members. Let us explain what happens when w ¼ 1.37 (Fig. 4.3d). When q2 is one during some periods, the reputation level is high and the value of k is also high. When all the players adopt the cooperation strategy and the reputation level (s) is high, the threshold k does not affect the decision-making for choosing members, and any

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Rotation Savings and Credit Associations (ROSCAs) as Early-Stage. . .

Fig. 4.4 Parameter dependence on the evolutionary traits, q1, q2, and k, when both the forfeiture rule and the peer selection rule based on labeling rule 2 are assumed. The black line represents r ¼ 5, the gray line represents r ¼ 20, and the light gray line represents r ¼ 40. Other parameters are N ¼ 100, m ¼ 5, μ ¼ 0.005. a Average cooperation probability before receiving the goods (q1). b Average cooperation probability after receiving the goods (q2). c Average threshold (k). See the main text for how to calculate the average values of the three traits

value of k is acceptable. As a result, k goes down (see around 3000–5000 generation times in Fig. 4.3d). Players with low k allow the players who do not contribute anything to become members. As a result, the value of q2 becomes low and the reputation level also becomes low. However, if players with high threshold (k) are reproduced by mutation, the threshold can be high, and the peer selection rule can exclude from the group those players who do not contribute anything. Therefore in Fig. 4.3 the generations oscillate. When w ¼ 1.06, the threshold k remains at its lowest throughout the generations. Even though mutation produces players with high k, the players with high k do not increase in number because the average reputation level (s) does not increase as average k increases: s becomes lower than k and the number of members in each group is less than nave (Fig. 4.3c). When the number of members in the group is small, the low future benefit (w ¼ 1.06) does not provide high benefit to players who contribute x to the goods after receiving the goods. Therefore players with high k are not selected for, and q2 never reverts to one. Figure 4.4 shows how the number of repetitions, r, affects the evolutionary outcomes. When r is high, the contribution increases in number (high q2) and the threshold k can be high.

4.2

Models and Results

4.2.7

115

Effect of Labeling Rules

Labeling rules other than those we have assumed in the previous sessions are possible. Let D ¼ (d1, d2, d3, d4) be defined as the labeling rule, where di ¼ 1, 0, or + 1 (i ¼ 1, 2, 3, 4). If a member does not contribute x before receiving the fund, the reputation level of this member can increase by d1 with a probability of 1  pb. Otherwise, the reputation level increases by d2. If a member does not contribute x after receiving the goods, the reputation level of this member can increase by d3 with a probability of 1  pa. Otherwise, the reputation level of this member can increase by d4. Therefore there are 34 (¼ 81) possible labeling rules. Labeling rule 1 in Model 2 is D ¼ (1, +1, 1, +1) and labeling rule 2 in Model 3 is D ¼ (0, 0, 1, +1). Table 1 in Koike et al. (2010) shows the labeling rules that make the average of q2 for w ¼ 1.0–1.5 high when the forfeiture rule and the peer selection rule are assumed. The other labeling rules in Table 1b in Koike et al. (2010) do not promote cooperation (average q2 is less than 0.3). Table 1 in Koike et al. (2010) indicates that the value of q2 can be high only when d3 < d4, and d2 is not one. This is for the following reasons: (i) It is obvious that d3 < d4 is required to distinguish cooperation strategies from default strategies after receiving the goods. (ii) Under the forfeiture rule, members who cooperate and contribute x before receiving the goods are selected for through evolution. As generations go, the whole population has cooperation strategies (q1 ¼ 1 and q2 ¼ 1) and default strategies (q1 ¼ 1 and q2 ¼ 0). The labeling rule has to distinguish cooperation strategies from default strategies. If d2 ¼ +1, not only members with cooperation strategies, but also those with default strategies, can obtain a high reputation. Therefore labeling rule 1 in Model 2, D ¼ (1, +1, 1, +1), cannot sustain cooperation when the forfeiture rule is introduced. Our simulation outcomes indicate that labeling rule 1 promotes cooperation slightly more than labeling rule 2 (see Figs. 4.1b and 4.2a), because members who cooperate before receiving the goods can be distinguished from those who do not cooperate before receiving the goods under labeling rule 1. However, Table 2 in Koike et al. (2010) shows that any labeling rule cannot promote cooperation when the forfeiture rule is not assumed, but the peer selection rule is average q1 and q2 for w ¼ 1.0–1.5 are less than around 0.5 in any labeling rule. This means that the labeling rule itself cannot exclude default strategies from the population. The reason why any labeling rule cannot promote cooperation in the rotating indivisible goods game is as follows. Even though some labeling rules can distinguish cooperators from defectors (players with q1 ¼ 0 and/or q2 ¼ 0), mutation always creates a small number of defectors. If defectors have more benefits than cooperators by chance, and players adopt the behavior of defectors, then defectors can increase in number and dominate the population. Therefore the peer selection rule cannot exclude defectors from the population. However, when the forfeiture rule is introduced, this rule enables players with q1 ¼ 1 to evolve. Even though mutation provides players with q1 ¼ 0, the forfeiture rule can resist the invasion of such players. Then, if the proper labeling rule can distinguish players with q2 ¼ 1 from those with q2 ¼ 0,

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Rotation Savings and Credit Associations (ROSCAs) as Early-Stage. . .

cooperation can be selected for. Therefore both the peer selection rule and the forfeiture rule are required to sustain cooperation in the rotating indivisible goods game.

4.2.8

Effect of Reputation Levels

We assume that the reputation label is zero at the beginning of each generation. Offspring can inherit a reputation level from their parents, especially in human society. Or, if a generation is interpreted not as a biological generation, but as a time step, it is not necessary to reset the reputation label at the beginning of each generation. Here we assume that the reputation is not reset, but is accumulated until one simulation run is over. This assumption means that all players remember past behaviors, whereas resetting the reputation label is interpreted as deleting the past memory of players. Figure 4.5a–c shows the cooperation probability and the threshold when the peer selection rule based on labeling rule 2 and the forfeiture rule are assumed. Comparing Figs. 4.4 and 4.5, the cooperation probability is lower in Fig. 4.5 than in Fig. 4.4. When reputation levels are not reset, cooperators who used to be defaulters have lower reputation levels, and defectors who used to be

Fig. 4.5 Parameter dependence on the evolutionary traits, q1, q2 and k when both the forfeiture rule and the peer selection rule based on labeling rule 2 are assumed. Here the reputation levels are not reset in each generation. See Fig. 4.4 for details

4.3

Discussion and Conclusions

117

cooperators have higher reputation levels. Therefore the peer selection rule based on reputation cannot choose proper players when the reputation is not reset.

4.3

Discussion and Conclusions

We proposed evolutionary simulation models to analyze the condition that cooperation is established in the rotating indivisible goods game, with reference to an extensively investigated informal institution, the rotating savings and credit association (ROSCA). Our evolutionary simulation outcomes show that peer selection based on reputation cannot by itself promote cooperation. In the rotating indivisible goods game, only a peer selection rule based on reputation in combination with a forfeiture rule shared by the group members promotes cooperation under this informal institution. In our simulation model we assumed the peer selection rule based on reputation. It is not obvious how players make decisions when the group chooses players, and when players choose a group. The peer selection rule in our model is one of the candidates for this decision-making. Our assumption of the peer selection rule suggests that a few players whose decision-making is extreme can affect the decision-making of the group. In our daily life, extreme opinions sometimes control a group’s decision-making, and our assumption describes one of our collective actions. There are other possible decision-makings besides this rule, and Chap. 6 shows which decision-makings can promote the evolution of cooperation in groups. We found that the labeling rule for reputation also affects the evolutionary dynamics. With the forfeiture rule, if players who contribute x to the goods before receiving the goods are labeled as high reputation (d2 ¼ +1), the labeling rule cannot distinguish players who adopt default strategies from those who adopt cooperation strategies, and thus it cannot promote the evolution of cooperation. This is because players with q1 ¼ 0 are excluded by the forfeiture rule (Fig. 4.3). Without the forfeiture rule, no labeling rule promotes the evolution of cooperation. Therefore the proper labeling rules combined with the forfeiture rule enable cooperation to evolve under the institution or rules. Our results also show the mechanism that sustains cooperation under an informal institution in which sanctions cannot be implemented. The peer selection rule based on reputation can be interpreted as a cost-free sanction, because the peer selection rule can exclude players who do not contribute the necessary effort, x, to the goods from the group. However, we also showed that this did not work effectively by itself. The combination of the forfeiture rule and the peer selection rule works as a cost-free sanction, and sustains cooperation under such informal institutions as ROSCAs. Some empirical investigations support the view that members share rules, such as the forfeiture rule, implicitly or tacitly. However, in empirical studies, it is difficult to obtain convincing evidence that members share implicit or tacit rules, and to clarify whether the implicit rule works as an informal sanction, and thus promotes cooperation. However, simulation studies can predict that the rules that members share

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Rotation Savings and Credit Associations (ROSCAs) as Early-Stage. . .

tacitly help to sustain cooperation under such informal institutions. Therefore collaboration between empirical and simulation studies enables us to understand human society more deeply. We focused on some of the rules and behaviors from much information on the empirical studies, and then made the simple assumption. To verify if people behave in the way that our simulations concluded, we performed the economic experiments, which will be shown in Chap. 5. The analyses in this study show that evolutionary simulation is a convenient tool by which to investigate which kinds of rules are required to establish cooperation (Koike et al. 2010). Although our model of the rotating indivisible goods game corresponds to a random ROSCA, there are more complicated forms of ROSCA, such as the bidding type. We could create more complicated models based on our simple model, compare the evolutionary simulation outcomes under different institutions, and investigate which rules or characteristics of each model are essential to sustain cooperation. There used to exist a lot of ROSCAs in Japan, and many ROSCAs have vanished except on Okinawa Island, Yamanashi prefecture, which is famous for Mt. Fuji, and some rural areas. To make simulations and mathematical models more appropriately, detailed information about rules is required. Upon hearing that some people on Sado Island located in the northern part of Japan were still enjoying ROSCAs, we decided to visit the Island to learn about the rules, which are shown in Chap. 5. In this study, we do not investigate how players invent the rules, or how all players come to share the rules. We simply assume that players obey the given rules. A human is only a creature that has the ability to create a rule or an institution. In the future, we will investigate the evolution of social rules, institutions, rules, and social norms.

References Alkan A, Demange G, Gale D (1991) Fair allocation of indivisible goods and criteria of justice. Econometrica 59:1023–1039 Ambec S, Treich N (2007) Roscas as financial agreements to cope with self-control problems. J Dev Econ 82:120–137 Anderson S, Baland JM, Moene OK (2009) Enforcement in informal saving groups. J Dev Econ 90: 14–23 Ardener S (1964) The comparative study of rotating credit associations. J R Anthropol Inst G B Irel 94:201–229 Armendáris B, Morduch J (2010) The economics of microfinance. MIT Press, Cambridge, MA Besley T, Coate S, Loury G (1993) The economics of rotating savings and credit associations. Am Econ Rev 83:792–810 Besley T, Levenson A (1996) The role of informal finance in household capital accumulation: evidence from Taiwa. Econ J 106:39–59 Bouman F (1995) Rotating and accumulating savings and credit associations: a development perspective. World Dev 23:371–384

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Coleman JS (1990) Foundations of social theory. Belknap/Harvard University Press, Cambridge, MA Ekeh PP (1974) Social exchange theory: the two traditions. Harvard University Press, Cambridge, MA Fehr E, Gächter S (2002) Altruistic punishment in humans. Nature 415(6868):137–140. https://doi. org/10.1038/415137a Geertz C (1962) The rotating credit association: a “middle rung” in development. Econ Dev Cult Chang 10:241–263 Gürerk Ö, Irlenbusch B, Rockenbach B (2006) The competitive advantage of sanctioning institutions. Science 312:108–111 Handa S, Kirton C (1999) The economics of rotating savings and credit associations: evidence from the Jamaican ‘partner’. J Dev Econ 60:173–194 Hechter M (1987) Principles of group solidarity. University of California Press, Berkeley, CA Koike S, Nakamaru M, Tsujimoto M (2010) Evolution of cooperation in rotating indivisible goods game. J Theor Biol 264(1):143–153. https://doi.org/10.1016/j.jtbi.2009.12.030 Light IH (1972) Ethnic enterprise in America: business and welfare among Chinese, Japanese, and Blacks. University of California Press, Berkeley, CA Morduch JRS, Collins D, Ruthven O (2009) Portfolios of the poor. Princeton University Press, Princeton, NJ Nowak MA, Sigmund K (1998) Evolution of indirect reciprocity by image scoring. Nature 393(6685):573–577 Ostrom E (2000) Collective action and the evolution of social norms. J Econ Perspect 14:137–158 Putnam RD (1993) Making democracy work: civic traditions in modern Italy. Princeton University Press, Princeton, NJ Tsujimoto M, Kuniyoshi M, Yokuda I (2007) The genesis of re-source exchange: a study of rotating savings and credit associations in the Okinawa Islands. Jpn J Soc Psychol 23:162–172 van den Brink R, Chavas JP (1997) The microeconomics of an indigenous African institution: the rotating savings and credit association. Econ Dev Cult Chang 45:745–772 Young HP (1995) Equity: in theory and practice. Princeton University Press, Princeton, NJ

Chapter 5

Tanomoshi-ko Field Study and Subjective Experiment

Abstract Chapter 4 showed the conditions that rotating savings and credit associations (ROSCAs) can be sustainable by means of agent-based simulations and evolutionary game theory. To verify that our simulation assumption and results match the real ROSCAs and to know the existence rules of various ROSCAs, we visited Sado Island, located at the northern part of Japan, where some people are still enjoying ROSCAs. We interviewed Sado people about several types of ROSCAs, some of which had already vanished, and about the rules regarding their ROSCAs. We also conducted a laboratory experiment to investigate the mechanisms that can prevent default in a fixed ROSCA, in which the order of receipt of returns is determined before starting and is also known to members. The findings are as follows: (i) Low contributors were excluded from ROSCA groups by voting for increased contribution rates both before and after the receipt of returns; (ii) ROSCA members exhibited reciprocity and a sense of revenge, that is, members contributed to the returns payments of other members who had contributed to them, and did not contribute to the returns payments of non-contributors. Voluntary behaviors thus sustained ROSCAs. Meanwhile, an exogenous punishment whereby subjects were prevented from receiving returns payments unless they had themselves contributed previously did not increase contribution rates. Keywords Tanomoshi-ko · Sado · Random-ROSCA · Experiments

5.1

Introduction

Rotating savings and credit associations (ROSCAs) exist all over the world (see Chap. 4 for the definition). In Japan, tanomoshi-ko, which is a Japanese word equivalent to ROSCAs, used to be common in daily life until the end of World War II. People at that time started a tanomoshi-ko if a round sum was needed urgently. Currently it is hard to find ROSCAs in daily life and many people have not heard of tanomoshi-ko. However, ROSCAs are still popular on Okinawa Island, which is located in the southern part of Japan, and Yamanashi, which is famous for Mt. Fuji. ROSCAs are called moai in Okinawa. I heard that some ROSCAs still

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Nakamaru, Trust and Credit in Organizations and Institutions, Theoretical Biology, https://doi.org/10.1007/978-981-19-4979-1_5

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Fig. 5.1 The map of Sado Island, Niigata, in Japan

Sado Island

Fukura

Ryotsu Port

Kamo Lake Iwakubi

Ogi

Japan

existed on Sado Island (Fig. 5.1) from the researchers who often visited Sado Island for their project on revitalization of local communities. In the first half of this chapter, we present the results from the interviews conducted on Sado Island to determine the realities of tanomoshi-ko, also called Sado-ko on this particular island (Nakamaru and Koike 2015a, 2015b). The research took place in Fukura-district, an area nearby Ryotsu Port and facing Kamo Lake, and in Ogi-district, which used to be a prosperous gold shipping port and a central port for Kitamae ships, with flourishing mercantile culture mainly centered around wholesalers, especially in the Edo era. We also establish a comparison with the tanomoshi-ko simulation introduced in Chap. 4 (Koike et al. 2010), touch upon the other ko on Sado Island, and discuss their relationship with the evolution of cooperation. In sum, we conducted a subjective experiment using an experimental economics framework, based on the simulation framework described in Chap. 4, to examine how people behave when they engage with tanomoshi-ko (Koike et al. 2018). In the latter half of this chapter, we provide descriptions of the subjective experiment, the results, and a comparison of the findings with those shown in Chap. 4.

5.2 Ko on Sado Island

5.2 5.2.1

123

Ko on Sado Island Tanomoshi-ko in Fukura-District

We visited a household in Fukura-district on January 25 and 26, 2013 to conduct an interview, as it came to our knowledge that the wife of this household participated in tanomoshi-ko, whose primary purpose was ladies’ luncheons, and that her entire family also used to participate in nenbutsu-ko (described in Sect. 5.2.3). Several years after our interview, the tanomoshi-ko was dissolved. In the past, the rules were different from those in 2013, and at the time, bidding appeared to decide the deposit recipients. The deposits here have two meanings: the amount of money a recipient receives and the amount of money each member invests in ko. Suppose the tanomoshi-ko consists of n members and holds t meeting times (1
t
n). A member, who has never won a bid and wishes to place one, submits a note with an amount of written dividends (kaisen) to participate. The member with the highest amount of kaisen wins and receives the deposit. Kaisen, in other words, is an interest or dividend that a member who has not won before can receive. The winning bidder pays out the dividend to all non-winning members, but those who have won before are not eligible to receive the kaisen. This particular ko also has a special dividend, called hanadai, that the members with the second- and thirdhighest kaisen bid could receive. The third-place bidder receives 50% of the hanadai amount given to the second-place bidder. The winning bidder also pays hanadai, meaning that the amount of deposit the winning bidder can receive is the total amount of deposits collected from the members subtracted by the distributed kaisen and hanadai. Considering that the last winning bidder does not have to pay kaisen or hanadai, he/she can receive the full deposit amount. Each member’s deposit and hanadai amount are determined before the ko begins. The winning bidder becomes in charge of hosting the next tanomoshi-ko, and as the primary purpose of this ko was to have luncheons, the deposits are paid immediately. Using a more tangible example (Fig. 5.2), suppose each member’s deposit contribution is 10,000 yen, the hanadai is 500 yen, and the ko consists of seven members. Ms. A won the first ko bid, Ms. B won the second, and the five remaining members, Ms. C to Ms. G, can then bid on the kaisen this time, unless intending otherwise. Suppose all five members bid. Ms. C had the highest kaisen bid with 300 yen, Ms. D had the second with 200 yen, and the third was Ms. E with 100 yen. Ms. A and Ms. B respectively pay 10,000 yen, and Ms. D pays 9200 yen, which is 10,000 yen minus the 300-yen kaisen and 500 yen hanadai. Ms. E pays 9450 yen, which is 10,000 yen minus the 300-yen kaisen and 250-yen hanadai. Ms. F and Ms. G pay 9700 yen, which is 10,000 yen minus the 300-yen kaisen. Therefore, Ms. C, the winning bidder, receives 58,350 yen as the deposit (Ms. C’s 10,000-yen deposit is not included here.) This shows that the earlier one wins the bid, the smaller the deposit amount one receives. In contrast, if one wins in the later ko, one receives more and the last winning bidder receives the full amount. While the deposit one receives is smaller in

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each member’s deposit contribution

dividend “kaisen”

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The winning bidder, C, receives the deposit Fig. 5.2 The example of tanomoshi-ko in Fukura-district presents that Ms. C is a bid-winner at the third meeting

the earlier ko, one can still gain a large sum of money earlier, which can be used to invest in buying a car or other expensive purchases, or for unexpected expenses, such as ceremonial occasions for relatives. This could be the advantage of receiving the deposit early, but it is undoubtedly a better deal to win in the later ko, unless one needs a large sum of money in advance. Provided the ko continues until the very last bid, if the ko is defaulted and terminated halfway, it will be a substantial loss for the members who are yet to win the bidding. In such riskier ko, members attempt to win early and the kaisen soars as a result, increasing the dividend for non-winning members. Overall, it reflects the same mechanism as riskier, but higher-paying trust investments. Interestingly, they purposely set hanadai, considered to represent bidders” mindfulness of those with higher kaisen who need the funding early. Furthermore, distributing hanadai to second- and third-place bidders is thought to strengthen the bidding’s bargaining aspect. Tsujimoto (2000) termed this sage-moai to refer to a moai in which the receiving amount is lower than the deposit’s full amount (¼ deposit  number of participants). For instance, in the example in Fig. 5.1, while the

5.2 Ko on Sado Island

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full amount is 60,000 yen (excluding the dividends), Ms. C only received a net amount of 58,050 yen, as she had to pay dividends to the participants. In other words, this ko in Fukura-district is also a sage-moai. “Sage” is a Japanese word equivalent to “down” or “low” in English; the opposite of “sage” is “age,” which is equivalent to “up” or “high” in English. There also exists age-moai in Okinawa. In age-moai, which is also a bidding ROSCA, participants bid for the repayment interest rate (Tsujimoto 2000). The participant whose repayment interest rate is highest among the participants can get the right to make a successful bid. The successful bidder can obtain not only the deposit’s full amount but also the sum of the repayment interest from the previous successful bidders. Therefore, the amount the successful bidder receives is higher as the meeting is proceeds. This is the reason this ROSCA is called age-moai. As of 2013, the tanomoshi-ko no longer use the previous bidding system. Instead, they pre-define the order, which is modifiable, to receive the deposit before starting a ko. The interview respondent explained that they mainly continue to practice tanomoshi-ko to have an opportunity to dine together and exchange information, and that the ko’s deposit recipients are responsible for finding a venue for the next gathering. Furthermore, this tanomoshi-ko does not have a “parent,” as explained in Sect. 5.2.2 on Ogi-district’s tanomoshi-ko, and is considered to have similarities with the evolutionary simulation presented in Chap. 4. A new member may join the ko if an existing member leaves the group, due to relocation or other reasons, but requires a recommendation from the participants. Finally, with unanimous approval from the participants, the candidate becomes an official member. This corresponds to the “exclusion condition” in the evolutionary simulation’s peer selection rule presented in Chap. 6. However, the simulation in Chap. 4 did not assume the necessity for unanimous approval to become a ko member. Thus, we discuss the modification of the assumed exclusion condition in Chap. 4. When withdrawing from a ko, a member either has to wait until all turns are completed or must continue paying the deposit until all turns are completed. These rules are in place to prevent defaults, and as tanomoshi-ko is a gathering of neighborhood wives, the members are unlikely to run away, as leaving the group will profoundly affect future relationships. Instead of applying the forfeiture rule, as explained in Chap. 4, it is implied that a member who refuses to pay is socially ostracized. Thus, we wondered if just having a function to remove an uncooperative member, such as the peer selection rule, will be enough for the ko to function as a system. However, the evolutionary simulation results indicate that both peer selection and forfeiture rules are required to maintain a ko. The challenge lies in interpreting the gaps between the results and the ko rules in reality. At first, we thought it would be possible for non-professional surveyors to obtain information about the rules through an interview, but we came to realize that it may be harder than we thought to question the rules associated with punishments, such as the forfeiture rule, as such negative information could be emotionally difficult to discuss. We also wanted to understand their criteria for choosing new members, but verbalizing and explaining one’s decision-making processes is often difficult. In

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this sense, a model analysis is an effective way to complement the missing information.

5.2.2

Tanomoshi-ko in Ogi-District

Although Ogi-district is the island’s southern gateway, it is now out-shone by Ryotsu Port, the main entrance to Sado Island, and has lost its previous vibrancy. Even today, businesses in Ogi-district seek funding in tanomoshi-ko, instead of lending from banks or credit unions, suggesting that it assumes the role of a mutual financing system. On July 6 and October 27, 2013, we conducted interviews on a type of tanomoshi-ko that had a unique and complex structure. We imagine that the scheme is more efficient for larger-scale funding. Figure 5.3 offers an overview of this particular tanomoshi-ko. Specifically, when a person needs a sizeable amount of funds, he/she becomes an initiator of a tanomoshi-ko, called a “parent” (oya). The parent then recruits and borrows money from “followers” (ko) who are willing to invest, and gradually repays the debt. Here, ko means offspring in Japanese and is different from the ko in tanomoshi-ko. Thus far, it seems that the scheme is merely an interpersonal money-lending process. Tanomoshi-ko comes into the picture when the parent repays all the followers: as it is not possible to pay them all back at once, the parent instead does so one by one in order. During this process, the followers hold a tanomoshi-ko in which the bidding decides the deposit recipient. The winning bidder receives not only the deposit, but also the parent’s repayment. In other words, the repayment order is essentially decided during the followers’ tanomoshi-ko. To elaborate on the details, Box 5.1 indicates the terms of a tanomoshi-ko and we use the hypothetical numbers shown below to explain (Fig. 5.3). When a person needs a large sum of money to start a new business, remodel a house, or for other necessities, he/she becomes a parent and invites followers by talking to past tanomoshi-ko participants. As shown in Box 5.1 and Fig. 5.3, the parent gathered 45 followers and each funded 250,000 yen, meaning that the parent received a total of 11,250,000 yen. This is considered the first ko. The parent is supposed to repay 250,000 yen to a follower every time a ko is held. As the primary purpose of tanomoshi-ko is mutual aid, the debt is interest free, for the followers offer assistance to the parent. This notion seems to apply to other ko as well (Tsujimoto 2000). The ko is held 46 times. From the second ko, the parent’s repayments and the followers’ tanomoshi-ko both begin. As the followers’ tanomoshi-ko consists of 45 members, with a 50,000 yen deposit each, once everyone has contributed, the deposit adds up to 2,250,000 yen (50,000 yen  45 people). With the addition of the parent’s 250,000 yen repayment, the total becomes 2,500,000 yen. As one of the followers receives this amount, the recipient is determined similarly as in the tanomoshi-ko of Fukura wives: the followers bid kaisen and the highest bidder wins the 2,500,000 yen. The kaisen amount will be the dividend, which the winning bidder pays to all non-winning followers. Furthermore, the winning bidder is also responsible for refreshment expenses during the tanomoshi-ko. In sum, the sooner

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250,000 yen㽢45 members =11,250,000 yen The 1st ko: mutual aid for the parent

parent 250,000 yen

250,000 yen

follower 1

......

2

repayment 250,000yen

total 2,500,000 yen The follower’s ko starts from the 2nd ko This is the example of the 2nd ko The 46th ko is the last

45

parent

50,000 yen㽢45 members = 2,250,000 yen 50,000 yen follower 1

50,000 yen ......

2

45

decide kaisen bidding price not bid follower 1

kaisen bidding price is 10,000 yen

......

2

45

5,000 yen

Follower wins 250 million yen for the bid deposit: 2,500,000 yen follower 1

2

dividend 10,000 yen

......

45 dividend 10,000 yen

Fig. 5.3 Tanomoshi-ko in Ogi-district

one wins the bid, the lower the deposit amount becomes, as there are more followers to pay the dividend to. The follower who receives the money in the 46th ko receives the full amount of 2,500,000 yen. Unless one needs funding immediately, the better deal is to receive the deposit in later ko, with more dividends to receive and fewer dividends to pay upon winning. However, the 46th ko is held in the 12th year from the first ko, similar to investing in a super 12-year fixed deposit. As 12 years is a long time, it may default halfway through and one may end up not receiving the deposit. Thus, as with the tanomoshi-ko of the Fukkura wives, the kaisen bidding price becomes higher in riskier ko, as the participants try to win early. We used the specific numbers above to better illustrate the scheme, but we also offer more general formulae as follows: A winning bidder who wins the i-th ko with the kaisen of qi yen will pay qi(n + 1  i) yen to non-winning bidders. Therefore, the net deposit the winner receives will be z  qi(n + 1  i) yen, provided z ¼ nw + y,

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with w being the followers’ paid deposits and y being the parent’s regular repayment. Furthermore, n + 1  i will be the number of non-winning bidders (2
i
n). As bidding will not take place on the last ko (n + 1-th session), the recipient earns the full amount (z yen). The amount a person who won the i-th ko can receive from the followers’ ko is z  qi(n + 1  i) + Σ2
j < i qj. Those who receive the money at a later held ko during the parent’s repayment period continue to receive dividends (kaisen) throughout that time. Thus, the total amount of earned dividends may become higher than the dividend one pays as the winner, as in qi(n + 1  i) < Σ2
j < i qj. For this case, ko offers an opportunity for investment management, as the participant receives a larger amount of money than the deposit (z yen). Still, it has the risk of potentially becoming an uncollectible loan, due to the timing of participants’ defaults or changes in socio-economic situations. When asked about kaisen, our interview respondents noted that many used to bid higher kaisen to win. In fact, the kaisen often went up to about 10–20% of the deposit, and in a particular case, the dividend for a deposit of 50,000 yen was 10,000 yen. However, the kaisen bidding price is seemingly falling to 1–5% in more recent sessions, with a corresponding dividend of around 100 yen. Therefore, the winning bidders receive almost the full amount (z yen), as they only have to pay a marginal amount of dividends. On the contrary, for non-winning bidders, the dividends are no longer fruitful. The respondents explained that this is because present-day tanomoshi-ko is no longer practiced for procuring funds, but more for the sake of good neighborhood relationships. Regarding the tanomoshi-ko’s default-prevention mechanisms, there are many rules, as shown in Box 5.1. The terms state that the winning bidder shall repay w yen from the next ko date until the last. As the winning bidder is indebted to the ko, the terms obligate the participant to repay to prevent defaults. Only after submitting an IOU to the treasurer (zenisho), the winning bidder can receive the money he/she won. Within the district, two joint sureties for the parent and one surety for a follower must be assigned, and they only guarantee the deposit amount. Appointing a follower’s surety is a measure to prevent his/her default after winning the bid. The extent of a surety’s liabilities is within those of a general guarantor. They are called joint sureties, only because there are multiple guarantors. Sakurai (1988) indicated a few similar systems in his study. While joint sureties are responsible for assuming a parent or a follower’s debt in case they fail to repay, it is apparently a very rare event and only happens once every several decades at most. When such a situation occurs, the delinquent is no longer allowed to become a parent of another ko or may even get ostracized. We can see from Box 5.1 that they appoint facilitators (sewanin), who are trusted district members, and whose duty is to arbitrate any issues and handle unexpected events, such as followers becoming insolvent or participants dying during the ko period. Apparently, such troubles that require a facilitator’s intervention rarely happen, and no cases of a dispute developing into a suit were reported, as all participants, under the facilitator’s leadership, cooperate to solve issues. The treasurer (zenisho) mentioned in Box 5.1 oversees the management of the deposits. Within all the aforementioned parties involved, the treasurer is virtually the only one

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who actually engages in the tanomoshi-ko operations. In exchange for lending interest-free loans, the parent is responsible for carrying out the necessary tasks for operating the ko. Some examples include collecting deposits, hosting the ko at home, and distributing kaisen. The treasurer manages the parent’s collected money and secures credibility. The facilitator will also address any shortcomings that the treasurer discovers regarding the collected money. The only penalty mentioned in Box 5.1 is as follows: A participant who fails to pay the deposit for tanomoshi-ko by the bidding time shall donate Japanese liquor, sake. When we inquired about this penalty’s background during an interview, the respondent explained: “Usually, the parent collects the deposits from the followers ahead of the ko, but sometimes they bring the money on the day of the session. If a participant arrives late to the session, perhaps because he forgot about it, he brings the deposit along with sake as a fine.” On that note, not all followers participate in every ko session. Some members who previously won the bidding often simply pay the deposit and skip the session. Joint sureties, facilitators, and treasurers all seem to play an essential role in preventing the participants from defaulting. The rule corresponding to the forfeiture rule, as assumed in the simulation in Chap. 4, seems to be the following description: A bidder who does not arrive on time will lose the right to bid (Box 5.1). However, according to the interviews, unlike the forfeiture rule, this rule only applies to the very session for which the participant arrived late. Thus, the member does not lose the right to bid in the upcoming ko, because if a follower loses the right to bid altogether, by simply not being able to pay the deposit once, he/she would lose the chance to obtain the parent’s refund (250,000 yen). Upon inspecting Table 1.4 in Chap. 1 one can understand that the tanomoshi-ko of Ogi-district is a complex system of an all-for-one type (financing for a parent and followers” tanomoshi-ko) connected with a one-for-one type (repayment by the parent). From another perspective, it can also be understood as a single all-for-one type, with the tanomoshi-ko simply having different deposit amounts for the first ko and the rest. Either way, the forfeiture rule cannot exist. We also conducted interviews in Iwakubi-district, located in the Maehama area in the southern part of Sado. It is a village with 60 households that stretches along the coastline, surrounding the former Iwakubi Primary School. The majority of residents are farmers who work on rice terraces along the mountain (Fig. 5.4), and the village’s main industries are agriculture and marine products. The scenery of rice terraces facing the coastline was registered as Japan’s first Globally Important Agricultural Heritage System in 2011. While this area also had the tanomoshi-ko system in the past, under the name of tanomushi, details of its rules remain unknown, as it has not been held since approximately 1972. For the sake of simplicity, the evolutionary simulation in Chap. 4 assumed a rule for group formation under which a participant decides which group to join by judging the group’s reputation (other participants’ average evaluation), whereas the group approves the participant based on his/her reputation. We did not assume the presence of a leader, such as an initiator (zamoto or hokkinin). The examples of moai in Okinawa (Tsujimoto 2000) and tanomoshi-ko in Ogi-district operate with a

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Fig. 5.4 Iwakubi Shoryu terraced rice fields, registered as Globally Important Agricultural Heritage System. The photo was taken by the author on July 6th, 2013

framework in which a parent invites a follower to participate based on friendships and the follower’s reputation, and the follower approves participation based on the parent’s reputation and his/her own financial situation. For such cases, a group’s reputation and criteria are dependent on the parent. This structure originated from the tanomoshi-ko’s primary objective: to offer financial assistance to the parent. Our intention is to use the simulations and eventually further examine the conditions that allow the system’s stable operation in which a parent and other participants form a hierarchical structure, as observed in Ogi’s tanomoshi-ko.

5.2.3

Nenbutsu-ko

Many mutual-aid mechanisms other than the tanomoshi-ko have been passed down on Sado Island, but we were able to learn a very limited amount about all the ko practiced on the island. The yaban, eban, and shinmei-ko of Iwakubi-district, and the kobara-ko of Fukura-district all had a one-for-all type (see Table 1.4 in Chap. 1), while the Nenbutsu-ko of Fukura-district was of both the one-for-all and the all-forone type.

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131

Regarding this one-for-all type, we introduce four examples of ko that follow this construct, starting with the yaban of Iwakubi-district. Mutual aid systems remained present throughout the village when we conducted the interviews in 2013. The yaban, a patrolling activity aimed at preventing crimes and fires, involves two neighboring households providing one person each to patrol the entire village every night and to handle any unexpected events that cannot be managed by a single person. In addition, the patrolling also offers the villagers an opportunity to have a chat, and exchange information about themselves and the entire village’s situations. Due to its objective being crime and fire prevention, the night patrol cannot expediently exclude non-members. Thus, aside from the elderly, all households are assigned to take the duty in turn. If one wants to avoid the duty, because they are away from the village, for instance, they can switch with other villagers who are assigned for patrol during any of the following days. Every evening, someone must offer their time for everyone’s benefit. Therefore, yaban is a perfect example of a one-for-all type. As far as I remember, the same practice existed in my home town until the 1970s–80s. As another example, Eban of Iwakubi-district was a duty related to the irrigation of rice fields. As the water temperature would be too cold if spring water from the mountains was irrigated directly into the fields, to adjust the temperature, the villagers first filled the egoda with spring water, warmed it up, and then directed the water into the rice fields. Once a member was appointed to be on duty, he/she also did maintenance work on the water routes and egoda. As these duties died out after the implementation of agricultural infrastructures, we could not learn more details on the eban or its rules, but the Shinmei-ko still remains extant in Iwakubidistrict. The on-duty member was determined by drawing a lottery and between approximately 1955 and 1965, they began by conducting religious rituals at the on-duty member’s residence, followed by a supper, previously held three times a year. As much as members looked forward to participating in a ko and holding a feast, the burden was becoming heavier for the on-duty member to prepare for the gathering and sometimes a member who was struggling to make a living left the group for this reason. Thus, the participants began holding the feast at Japanese inns or restaurants, instead of a member’s house. Nowadays, they hold gatherings on November 11th and February 11th, at a temple with a fixed fee for participants. We discovered the next one-for-all type ko example during the interview process. In Fukura-district, 10 shops used to hold a ko called kobara-ko with the leadership of a Japanese confectionery shop. It represented a gathering of businesses that use fire for cooking and the duty was passed on by rotation. The members gathered once a year at the on-duty member’s shop for fire prevention activities and to exchange information. They conducted religious rituals for prayer, and held a party where they wined and dined. In other words, the on-duty member prepared a kakejiku, with candles and sacred sake (Fig. 5.5), which are all associated with the Furumine Shrine. Out of the 3000 yen participation fee, 1000 yen was saved for a reserve fund. Additionally, every year on the first Sunday of June, some members voluntarily visited the Furumine Shrine in Tochigi-prefecture to offer prayers to Yamato

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Fig. 5.5 Kakejiku with candles and sacred sake, associated with the Furumine Shrine. The photo was taken by the author on January 26th, 2013

Takeru no Mikoto, known as a fire prevention deity. However, as time passed, these practices all died out. Finally, a ko in Fukura-district, called nenbutsu-ko, is of both the one-for-all and the all-for-one type. We interviewed a family in Fukura-district and they said, “During the early Showa era, 10 households in Fukura-district gathered to initiate nenbutsu-ko for the purpose of mutual aid, which involved ceremonial occasions and promoting social bonding.” The nenbutsu-ko has been passed down to subsequent generations until today. Regarding its one-for-all type, this ko, held mid-winter every year, determines the household in charge that year through counter-clockwise rotation, as it appears on the map. The members gather at the on-duty household, line up rice, salt, and sacred sake; they beat a drum, and strike a bell while chanting nenbutsu. If any member is experiencing some difficulties or a tragedy (yakudoshi), the members beat the drum and chant the nenbutsu to drive off the ill-luck. Then, they all have supper. The duty rotation is flexible and the members can decide the next household in charge during a nenbutsu-ko session. For instance, a household could nominate themselves if they wished to hold a ritual to drive away the evil. Another nenbutsu-ko purpose is ceremonial occasions held at the member’s house, especially funerals, usually managed by participants that have shown

5.2 Ko on Sado Island

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leadership. The group of nenbutsu-ko participants is called gonin-gumi (five members), which consist of dogyo (10 households). Therefore, this ko is an all-for-one type, as the members help one member who is in trouble. Contrary to tanomoshi-ko, which decides the fund-winning member by rotation or bidding, rotation cannot dictate this process as funerals occur unexpectedly. In fact, the same household could hold funerals consecutively if luck is not on their side. This has a similar structure to the mutual-aid game (Sugden 1986), the details of which are illustrated in Chap. 7. Thus, whereas the regular nenbutsu-ko sessions have a one-for-all type, and aim to foster social bonding and a sense of fellowship, a nenbutsu-ko also is of the all-for-one type to help fellow members during unprecedented times. Although several households have joined and left since the beginning, the approximate size of 10 households has been maintained, some comprising samesect believers. In addition, one cannot simply join a group without first obtaining consent from the existing members. Being a nenbutsu-ko member leads to community trust and there was even a time when a specific nenbutsu-ko membership was considered a social status symbol. More recently, the rules have been modified to follow current trends. For instance, a member can withdraw from the group voluntarily, but they must inform that year’s on-duty household in advance. Funeral or nenbutsu-ko assistance has also been simplified to fit the social conditions of the moment.

5.2.4

Michibushin

The michibushin of Iwakubi-district, which is of an all-for-all type, involves members undertaking maintenance works and mowing the farm roads that lead to the rice terraces twice a year, once in early April after the snow has dissipated and once in late August before harvesting. All households in the village are meant to participate and work simultaneously. As everyone goes out of their way to care for the roads and everyone benefits from more accessible roads, it is of an all-for-all type. When a member cannot participate in this work, their absence is recorded in a book and cleared every 6 months under the district chief’s supervision. If a member has fewer workdays than the average of other members, due to absences, a male member would pay 4500 yen to the village, as a daily allowance, and a female member would pay 4000 yen. The members who did the work for them would receive the allowance. Through such a practice, the village mediation balances the workload. However, this is not a punishment for not participating. Rather, it is a system to substitute an effort with daily allowances, based on the principle of accommodating each other’s workloads in the long run. Those who received the daily allowance for a year may also pay the allowance in the following year. To limit the costs, the daily allowance is designed to be lower than a general hourly wage. Their system for adjusting the excess workload once every 6 months is quite interesting from the standpoint of preserving order.

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Only the michibushin of Iwakubi-district among ko in Sato Island was of the allfor-all type. While the public goods game, which is of the all-for-all type, is frequently employed in theoretical and empirical studies on the evolution of group cooperation, such structures are rarely seen in ko.

5.2.5

Summary: Ko in Sado

While many forms of mutual aid exist, one of their problems is determining the legitimacy of a member’s absence. Specifically, it is difficult to distinguish whether a member genuinely has a scheduling conflict or is simply not participating (freeriding). To some extent, the daily allowance may function as a signal to show that one is not being cooperative and must bear the cost. However, if the system is abused and members start to substitute efforts with money, the entire work will eventually fall on just a few members. The worst-case scenario would be if a member neither pays, nor helps with the work. Perhaps these kinds of people, who try to buy-off or do nothing at all, are very rare, but regardless, our questions on how they handle such cases remained unanswered during the interviews. As mentioned earlier, the respondents were reluctant to disclose negative information, such as penalties, during the few survey days. It was difficult to obtain the answers, even when we were simply asking about their rules, but this may also be because they actually do not use penalties. For example, Guala’s (2012) cultural anthropological research review concluded a surprisingly low prevalence of penalties. In recent times, the ageing of the local population is a more serious issue than free-riders. For example, the michibushin and yaban were already struggling with maintaining the mutual-aid system, as the workload per member increases with fewer people sharing the work due to ageing. When the co-researcher, Shimpei Koike, revisited in March 2015, he learned that the yaban had already been terminated in December 2014. In addition, many residents of Iwakubi-district are increasingly becoming unable to continue farming on rice terraces, due to ageing. As rice terraces, which are one of the main industries of Iwakubi-district, require extensive water route and slope maintenance, the remaining farmers’ burden increases whenever someone quits.

5.3

Experimental Study of ROSCA

From our field work in the first half of this chapter, we cannot know how people make decisions when they decide to join tanomoshi-ko or to contribute money in tanomoshi-ko. To investigate whether the results in the computer simulations really support human decision-making, we conducted a laboratory experiment to examine whether these two factors, the peer selection rule and the forfeiture rule (see Chap. 4), prevented subjects from becoming defaulters and sustained fixed

5.3

Experimental Study of ROSCA

135

ROSCAs, where the order of payment is determined before each cycle starts (Koike et al. 2018); during each cycle, participants decide to contribute the fund or not, and the fund is given to the predetermined receiver in each meeting. To the best of our knowledge, no experimental study of ROSCAs has been undertaken, even though several theoretical studies have investigated the conditions for solving the default problem and sustaining ROSCAs (see Chap. 3). Using public goods experiments, several studies have found that exclusion based on reputation can promote contribution level by expelling low contributors from the game (Cinyabuguma et al. 2005; Maier-Rigaud et al. 2010; Feinberg et al. 2014; Croson et al. 2015). This exclusion can be interpreted as a costless punishment. In contrast, a number of experimental studies have investigated costly punishments. Costly punishment of free-riders has been found to increase contribution levels in public goods experiments (Yamagishi and Cook 1993; Ostrom et al. 1992; Fehr and Gächter 2000, 2002; Carpenter 2007; Page et al. 2005; Bochet et al. 2006). However, Herrmann et al. (2008) suggested that antisocial punishment, in which free-riders pay a cost to punish contributors, is observed in societies with weak civic cooperation and rule of law. Under some social conditions, costly punishment is ineffective in increasing contribution level (Nikiforakis 2008; Gächter and Herrmann 2009, 2011). A ROSCA resembles both the simultaneous and sequential public goods games, although each is structurally unique. In joining a ROSCA, all players decide whether to make a series of simultaneous contributions of a fixed sum to a fund while each taking turns to receive a return. In the public goods games, players decide their contribution level to a public good, either simultaneously or sequentially, and all benefit from the public good. Additionally, the payoff function of a player in our ROSCA game depends on the contributions of others rather than themselves. Conversely, the payoff function of a player in the public goods games is determined by the contributions of all players, including themselves.

5.3.1

Experiment

5.3.1.1

Design

The Tokyo Institute of Technology Ethics Committee approved our study on November 19, 2012, and October 10, 2014 (approval numbers 2012030 and 2014041, respectively). All subjects provided their written informed consent to participate in the experiment. The experiment consisted of four types of sessions: Treatment B (Base); Treatment P (Punishment); Treatment V (Voting); and Treatment VP (Voting with Punishment). In Treatment P, subjects played the fixed ROSCA on condition that non-contributing members could not receive return payouts (punishment rule). In Treatment V, subjects mutually select group members by voting. In Treatment VP, both the voting and punishment rules were employed. In Treatment B, neither the voting rule nor punishment rule was used.

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Treatment B (Base)

For each treatment, each session consisted of 10 rounds, each comprising four periods. At the beginning of a session, 20 subjects were each assigned to a laboratory computer and given instructions (see S1–S4 Instructions in Koike et al. (2018). After listening to the experimenter read aloud the instructions, the subjects were randomly divided into five groups, each comprising four members. We informed the subjects up-front that they would play the four periods at each round. At the beginning of each round, an order of payment was randomly determined and displayed on subjects’ computer screens, such that each group member knew the order in which all members of their group would receive their payouts. Notice that the number of periods is equal to the number of members and that each member could receive her payout exactly one time. The member scheduled to receive their payout in the n-th period was termed the n-th recipient. In each period, the experimenter gave 100 points to all group members, and all members other than the scheduled recipient had to choose whether to contribute these points to the communal fund or retain them. Members that decided to contribute lost 100 points, while members that decided not to contribute saved 100 points. We assumed that members who received their returns payouts earlier would profit more than those who received them later. This was because the earlier members received their payouts, the earlier they could purchase durable goods, invest in their business and make a profit. If mC is the number of contributing members and n is the number of periods, the real payoff to each recipient is π mc ,n

! mc ¼ 100 þ1 , ð1 þ δÞn4

where δ is the discount factor. We used δ ¼ 0.3 in this experiment. We informed the subjects up-front that their payoffs are determined by the above equation and that the discount factor δ is equal to 0.3. We also provided each subject a table showing how his/her payoff in one round consisting of four periods is determined by both “the order of payout receipt” and “the number of other members of his/her group who give 100 points to the fund” with this discount factor (Table 5.1).

Table 5.1 Payoffs for receiving returns payouts in each period in δ ¼ 0.3

Period First Second Third Fourth

Number of contributors to the fund 0 1 2 100 320 539 100 269 438 100 230 360 100 200 300

3 759 607 490 400

5.3

Experimental Study of ROSCA

137

Fig. 5.6 Example of a confirmation screen. ○: choosing to give; : choosing not to give; –: scheduled recipient in that period

After the receipt of each returns payout, all members were informed which members had contributed to the fund and which had received a returns payout (Fig. 5.6). Figure 5.7a summarizes the experimental procedure for Treatment B. Let us describe how the experiment proceeds and payoffs are determined in Treatment B by using the following example illustrated in Fig. 5.6. This example was explained to the subjects in the experimental instructions. Example 5.1 Consider the group of four members {A1, B1, C1, D1}. Now suppose that the order of payout receipt is “A1 ! B1 ! C1 ! D1.” In the first period, A1 receives the payout. Each of the three members, B1, C1, and D1, chooses either to “Give” 100 points to A1 or to “Not Give” any point (0 point) to A1. Suppose that all three members except A1 choose “Give” (“○” in Fig. 5.6). Then A1 receives 300 points and his or her payoff will be 759 at the end of the fourth period: 300  1.3(4  1) + 100 ≒ 759. The payoffs of the other three members are 0. In the second period, B1 receives the payout. Suppose that A1 and C1 choose “Give,” and D1 chooses “Not Give” (“” in Fig. 5.6). Then B1 receives 200 points and his/her payoff will be 438 at the end of the fourth period: 200  1.3(4  2) + 100 ¼ 438. The payoffs of A1 and C1 are 0 and the payoff of D1 is 100. In the third period, C1 receives the payout. Suppose that all three members except C1 choose “Not Give.” The payoffs of all four members are 100. In the fourth period, D1 receives the payout. Suppose that all three members except D1 choose “Give.” Then D1 receives 300 points and his or her payoff is 400 at the end of the fourth

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Fig. 5.7 Experimental procedures for treatments B and P

period: 300  1.3(4  4) + 100 ¼ 400. The payoffs of the other three members are 0. The total payoff of B1 (you) in this round is 0 + 438 + 100 + 0 ¼ 538, as appearing at the bottom of the screen in Fig. 5.6. After the fourth period, each member was shown his/her total payoff in that round.

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Experimental Study of ROSCA

139

At the beginning of the next round, new groups were randomly formed, with members being unaware of whether their new group still contained any members from their group in the previous round. After the end of the final round, the subjects answered a questionnaire and were paid a reward proportional their total payoff for the session.

5.3.1.3

Treatment P (Punishment)

We introduced a rule called the punishment rule: this rule holds that no member who had not contributed in preceding periods could receive a returns payout. In Example 5.1 for Treatment B, because the scheduled fourth recipient, D1, did not contribute in the second period, D1 cannot receive the payout in the fourth period, but saves 100 points in Treatment P. In this situation the other members also save 100 points because they do not need to contribute 100 points. Koike et al. (2010) introduce this rule and they term it the forfeiture rule. Notice that this punishment rule is applied to a member who has not contributed before receiving his/her payout. There is no penalty for a member who has stopped contributing after receiving his/her payout, such as A1 in Example 5.1. Figure 5.7b summarizes the experimental procedure for Treatment P.

5.3.1.4

Treatments V (Voting) and VP (Voting With Punishment)

We introduced a voting system. In this system, 20 subjects were randomly divided into five groups at the beginning of each round. All group members were aware of both the number of times each member had contributed in previous rounds and the number of previous rounds. Based on this information, starting in the second round, each member voted on who to exclude from the group. A group member was expelled if at least two members voted to exclude them. Excluded members were paid 100 points in each period and saved a total of 400 points during one round but could not participate in the round in which they were expelled. At the end of each round, the 20 subjects were randomly divided into five groups for the next round. Therefore, subjects excluded from their group in a previous round had the chance to join a new group in the next round. Treatment VP was the same as Treatment P with the addition of the voting system. Figure 5.8 summarizes the experimental procedures for Treatments V and VP. We conducted two sessions for each treatment at the Tokyo Institute of Technology from December 2011 to June 2013 using the program z-Tree (Fischbacher 2007). We recruited 160 (20  4  2) subjects from a Tokyo Institute of Technology subject pool. Subjects consisted mainly of undergraduate students who had never participated in similar experiments and were not familiar with ROSCAs. The average payment made to the subjects was 3506 yen (approximately 35.06 US dollars at a rate of 1 US dollar ¼ 100 yen).

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Fig. 5.8 Experimental procedures for treatments V and VP

5.3

Experimental Study of ROSCA

5.3.2

141

Theoretical Predictions

We analyzed four-person four-period ROSCA games in the four treatments, and obtained the subgame-perfect equilibrium, whereby no player pays a contribution during any period in any of the treatments (see the proof appearing in S1 Appendix in Koike et al. 2018). Applying backward induction to the ROSCA game with Treatment B, the fourth scheduled recipient gets nothing in the fourth period, because the other players have no incentive to contribute at the end of the round. The fourth recipient has no incentive to contribute in the preceding periods and saves 100 points in each period on his own. Therefore, the third scheduled recipient also gets nothing in the third period because he has not received a contribution from the first and second recipients, who have already received their returns payouts, or from the fourth scheduled recipient. Similarly, the other recipients get nothing in the period when they receive the returns payout and have no incentive to contribute before or after receiving that payout. We also considered what happens in the ROSCA game with Treatment P. In this treatment the other players also have no incentive to contribute to the fund in the fourth period and then the fourth scheduled recipient receives zero. If the fourth recipient cannot receive their payout because of the punishment rule for non-contribution before receiving the fund, he chooses no-contribution in the preceding periods and so independently saves 100 points in each period. As with the game in Treatment B, the third scheduled recipient also gets nothing in the third period because he has received no contribution from the other recipients. Hence, in Treatment P, no player pays any contribution during rounds 1–10. The theoretical prediction in Treatment V is as follows. In round 10, the players allowed to participate in a group pay no contribution to save 100 points each, regardless of history. Meanwhile, the excluded members also save 100 points per period. At the beginning of the round, each player knows that all participants will receive the same points and his vote does not affect his payoff. Applying backward induction, this argument is true not only for round 10 but also for rounds 2–9; hence, in rounds 2–10, all players elected as members of groups pay no contribution in any period. As the argument for round 1 is identical to that for Treatment B, in round 1, no player pays any contribution in any period. Treatment VP is a combination of Treatments P and V. Therefore, no player pays any contribution at any period in rounds 1–10, and no player cares about voting in rounds 2–10.

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5.3.3

Results

5.3.3.1

Average Contribution Rates Over Time and Distribution of Total Profit

We categorize the contribution rates into two types: the contribution rate before receiving the payout (β0) and the contribution rate after receiving the payout (α0). We consider Table 5.2 to understand β0 and α0. Because A1 is the first recipient, β0 cannot be calculated, but α0 is 1/3 because he has made a contribution once out of a possible three times after receiving the funds. Likewise, the β0 and α0 of B1 are 1/1 (¼ 1) and 2/2 (¼ 1), and those of C1 are 2/2 and 0/1, respectively. The β0 of D1 is 2/3, but α0 cannot be calculated because he is the last recipient. Result 1 The average contribution rate before receiving the payout in Treatment V was significantly higher than in Treatments B, P, or VP. The average contribution rate before receiving the payout in Treatment B was not significantly different from that in Treatment P. Support Fig. 5.9a shows the average β0 per subject in each round. We conducted a Steel–Dwass rank sum test to compare the four treatments. The average β0 per subject in Treatment V (red line) was significantly higher than the rates in Treatment B (yellow line) except in the final round, and was significantly higher than the rates in Treatment P (blue line) in 8 of 10 rounds and in Treatment VP (green line) in 5 of 10 rounds ( p < 0.05). However, except in round 3, no significant difference existed between the average β0 per subject in Treatments B and P. The average values of β0 in Treatment VP did not significantly differ from those in Treatment B in 9 of 10 rounds and did not differ from those in Treatment P in 8 of 10 rounds. In the final round, the end-game effect caused the average values of β0 in Treatments VP and V to decrease. This tendency is the same as in results observed in public goods experiments with majority voting (Cinyabuguma et al. 2005; Page et al. 2005; Maier-Rigaud et al. 2010; Croson et al. 2015).

Table 5.2 Example of calculating β0, α0, α1, α2 and α3 Period

β0 α0 α1 α2 α3

First Second Third Fourth

Subject A1 – ○   – 1/3 ½ – 0/1

B1 ○ – ○ ○ 1/1 2/2 1/1 1/1 –

C1 ○ ○ –  2/2 0/1 – 0/1 –

D1  ○ ○ – 2/3 – – – –

○: choosing to give; : choosing not to give. –: scheduled recipient in that period

5.3

Experimental Study of ROSCA

143

Fig. 5.9 Average contribution rates in each round. The vertical axis is for the average contribution rate and the horizotal is for rounds. a shows the average contribution rate before receiving (β0); b, the average contribution rate after receiving (α0); c, the average contribution rate to contributed donors (α1)

Result 2 The average contribution rate after receiving the payout in Treatment V was significantly higher than in Treatments B, P, or VP. The average contribution rates after receiving the payout do not differ significantly among Treatments B, P, and VP. Support Fig. 5.9b shows the average α0 per subject in each round. The average α0 per subject in Treatment V was significantly higher than in Treatment B, except in rounds 9 and 10 ( p < 0.05). To canvass the results of the treatments with and without the punishment rule, we categorized the contribution rate after receiving the returns payout (α0) into three types: the contribution rate of a subject to a recipient who has contributed to both the subject and all other members (α1); the contribution

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rate of a subject to a recipient who has contributed to the subject but not to some other members (α2); and the contribution rate of a subject to a recipient who has not contributed to the subject (α3). We reconsider Table 5.2 to confirm these values. To put is differently, α0 means the contribution rate to anyone; α1 indicates both direct reciprocity and indirect reciprocity; α2 only means direct reciprocity; α3 means a player helps others even though they have never helped the player. In Table 5.2, α1 of subject A1 is 1/2 because subjects B1 and C1 contribute to A1 in period 1, while A1 contributes to B2 in period 2 and does not contribute to C1 in period 3. Moreover, the α2 of B1 is 1/1 because B1 contributes to D1, who contributes to both B1 and C1 but does not contribute to A1 in period 1. Moreover, the α3 of A1 is 0/1, because D1 does not contribute to A1 in period 1 and A1 does not contribute to D1 in period 4. The punishment rule prohibited subjects from contributing to any scheduled recipient who had previously not contributed themselves, and so α0 in Treatments P and VP was represented by α1. Moreover, α0 in Treatments B and V was represented by the contribution rate of a subject to the recipient regardless of that recipient’s past behavior. Hereafter, we use α1 rather than α0 in comparing the contribution rate after receipt of the funds in the four treatments under the same criteria. Figure 5.9c shows the average value of α1 per subject in each round. In Treatment V, the average α1 per subject was significantly higher than in Treatment P in 6 of 10 rounds, while in Treatment VP it was higher in 7 of 10 rounds ( p < 0.05). However, the average α1 per subject did not differ significantly among Treatments B, P, and VP in any round ( p > 0.05). Result 3 The mean payoff in Treatment V significantly exceeded that in Treatments B, P, and VP. The mean payoffs in Treatments B, P, and VP were not significantly different. Support We now consider the distribution of the total payoff at the end of the treatments (Fig. 5.10). Figure 5.10b shows that the payoffs for all subjects in Treatment V exceeded 4000 points, which equaled the amount that all subjects received during one treatment (100 points  4 times  10 rounds). The payoffs for seven, eight, and five subjects were below 4000 points in Treatments B, P, and VP, respectively. The mean payoff in Treatment V was 4980 points, significantly higher than the mean payoffs in Treatments B, P, and VP, which were 4681, 4566, and 4494, respectively ( p < 0.01 in Wilcoxon rank-sum test). The mean payoffs thus were not significantly different ( p > 0.10).

5.3.3.2

Why Did the Voting System Work Well?

To show why the voting system boosted contribution rates, we investigated both the number of participants in each group and the number of subjects excluded from each group. In Fig. 5.11, the subjects are categorized into three types based on cumulative contribution rate: low contributors, with a cumulative contribution rate of less than 0.40; medium contributors, with a cumulative contribution rate of 0.40–0.80; and high contributors, with a cumulative contribution rate equal to or greater than 0.80.

Experimental Study of ROSCA

a

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Mean : 4,556 Max : 6,442 Min : 3,538 Std : 662

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0 3000 3500 4000 4500 5000 5500 6000 6500

0 3000 3500 4000 4500 5000 5500 6000 6500

Total Payoff

Total Payoff

d

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15

Mean : 4,980 Max : 6,142 Min : 4,159 Std : 548

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Number of Subjects

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Mean : 4,681 Max : 6,120 Min : 3,484 Std : 677

10

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10

5

0 3000 3500 4000 4500 5000 5500 6000 6500

0 3000 3500 4000 4500 5000 5500 6000 6500

Total Payoff

Total Payoff

Fig. 5.10 Total profits after the treatments. a is for Treatment B; b, Treatment V; c, Treatment P; d, Treatment VP

The upper sections in each part of Fig. 5.11 show the number of participants who were never excluded from the groups (non-excluded participants are represented by white bars) and the number who were excluded in previous rounds (excluded participants are represented by gray bars). The lower section in each part of Fig. 5.11 indicates the number of subjects who were excluded from the groups (excluded subjects are represented by black bars). The horizontal axis represents the number of rounds; the upper and lower vertical axes represent the numbers of participants and subjects, respectively. Result 4 The voting system excluded low contributors while letting medium and high contributors participate in Treatments V and VP. Support In Treatment V, low contributors were frequently excluded from the groups (Fig. 5.11a); medium and high contributors were less frequently excluded (Fig. 5.11b, c). Over the 10 rounds in Treatment V, the voting system excluded 72% of low contributors and 6% of medium and high contributors. The difference was significant ( p < 0.01 by Fisher’s exact test). Therefore, because the voting system excluded low contributors from the groups while favoring participation by medium and high contributors, the contribution rates in Treatment V exceeded those in Treatments B and P, which lacked the voting system. In Treatment VP, the number of excluded subjects did not always exceed the number of participants in low contributors (Fig. 5.11d); however, among medium

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Tanomoshi-ko Field Study and Subjective Experiment

2

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Fig. 5.11 Total numbers of participants and excluded subjects in each round over two sessions. a presents the results of low contributors in Treatment V; b presents the results of medium contributors in Treatment V; c presents the results of high contributors in Treatment V; d presents the results of low contributors in Treatment VP; e presents the results of medium contributors in Treatment VP; f presents the results of high contributors in Treatment VP

and high contributors, excluded subjects were outnumbered by actual participants in each round (Fig. 5.11e, f). Because the average contribution rates (β0, α1) continued declining with number of rounds after this number reached the halfway point (Fig. 5.9), the number of high contributors decreased while that of medium contributors increased. Over the 10 rounds in Treatment VP, the voting system excluded 51% of low contributors versus only 8% of medium and high contributors. The exclusion rates thus differed significantly among the three groups ( p < 0.01).

5.3

Experimental Study of ROSCA

147

Fig. 5.12 Average number of participants in each round. The red line is for Treatment V; the green one, Treatment VP

Result 5 The average exclusion rate of low contributors in Treatment V significantly exceeded that in Treatment VP. Support The average exclusion rate of low contributors in Treatment V (72%) was significantly higher than in Treatment VP (51%, p ¼ 0.043). However, in a comparison of Treatments V and VP, the exclusion rates of medium and high contributors were 6% and 8%, respectively, and thus were not significantly different ( p ¼ 0.405). In sum, the contribution rates in Treatment V exceeded those in Treatment VP. Result 6 The voting system gave excluded low contributors the chance to participate in future groups in Treatments V and VP. Support The upper part of Fig 5.11a shows that all the low contributors available to participate were excluded. In Treatment VP, similarly, 39 out of 50 low contributors available to participate were excluded (Fig. 5.11b). As low contributors who had been excluded had a chance to participate in a group again, the number of participants included in each round in Treatment V averaged at least three out of every four participants available for selection (Fig. 5.12).

5.3.3.3

Why Did the Punishment Rule Not Work Well?

The previous sections showed that the contribution rates did not differ significantly between Treatments B and P, and that the contribution rates in Treatment V exceeded those in Treatment VP. These results suggest that punishment did not work well. To examine the reason, we focus on the differences among the average values of α1, α2, and α3 per subject over 10 rounds. Result 7 The average contribution rate to non-contributors was significantly lower than that to contributors in Treatments B and V without the punishment rule. Support Table 5.3 shows that the average values of α1, α2, and α3 in Treatment B were 0.40, 0.33, and 0.04, respectively. Thus the average values of α1 and α2 were not significantly different ( p ¼ 0.59 by the Steel–Dwass rank sum test), while that of

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Table 5.3 Average contribution rates in the four treatments over 10 rounds

Treatment B N Treatment P N Treatment V N Treatment VP N

Before receiving β0 0.44 (0.05) 40 0.40 (0.05) 40 0.81 (0.03) 40 0.56 (0.04) 40

After receiving α0 α1 0.18 0.40 (0.03) (0.05) 40 39 0.29 0.29 (0.06) (0.06) 38 38 0.62 0.71 (0.03) (0.04) 40 40 0.38 0.38 (0.05) (0.05) 39 39

α2 0.33 (0.08) 30 –

α3 0.04 (0.02) 40 –

0.60 (0.09) 28 –

0.15 (0.04) 37 –

Averages with standard errors in parentheses

α3 was significantly smaller than the other two ( p < 0.01). This suggests that the subjects in Treatment B discriminated between recipients based on whether those recipients had contributed to them, and rejected making contributions themselves to non-contributors. This behavior can be interpreted as a kind of costless punishment of non-contributors. Hence, subjects tended to punish recipients who had not contributed to them in Treatment B. As a result, even though Treatment P combines Treatment B with the punishment rule that prohibited subjects from contributing to a recipient who had not previously contributed, the contribution rates in Treatments B and P did not differ significantly. Similarly, in Treatment V, the average values of α1, α2, and α3 were 0.71, 0.60, and 0.15, respectively (Table 5.3). The average values of α1 and α2 did not differ significantly ( p ¼ 0.99), while the average value of α3 was significantly smaller than the others ( p < 0.01). This suggests that the subjects in Treatment V also retaliated against those who had not contributed to them.

5.3.3.4

Order Effect for the Receipt of Payouts

In our experiment, each subject was informed of the order in which they would receive the returns payout in each round. Here, we examine how this order influenced contribution rates. Result 8 The average contribution rates before recipients received their returns payouts gradually decreased from the second recipient, to the third, and finally the fourth in all four treatments. Support In Table 5.4, the average value of β0 of the second recipient significantly exceeded that of the third recipient in all treatments ( p < 0.05 by the Steel–Dwass

5.3

Experimental Study of ROSCA

Table 5.4 Order effect of contribution rates

Order of payout receipt First Second Contribution rate before payout receipt (β0) Treatment B – 0.77 (0.04) N 100 Treatment P – 0.69 (0.05) N 100 Treatment V – 0.95 (0.02) N 98 Treatment VP – 0.78 (0.04) N 97 Contribution rate after payout receipt (α1) Treatment B 0.39 0.44 (0.05) (0.07) N 85 43 Treatment P 0.32 0.38 (0.05) (0.07) N 74 43 Treatment V 0.71 0.71 (0.04) (0.05) N 96 75 Treatment VP 0.38 0.38 (0.05) (0.07) N 78 48

149

Third

Fourth

0.52 (0.04) 100 0.38 (0.05) 100 0.85 (0.03) 94 0.56 (0.05) 81

0.25 (0.04) 100 0.24 (0.04) 100 0.73 (0.05) 66 0.33 (0.06) 51

0.38 (0.14) 13 0.45 (0.11) 22 0.74 (0.07) 38 0.36 (0.13) 14

–

–

–

–

Averages with standard errors in parentheses

rank sum test). The average value of β0 of the third recipient significantly exceeded that of the fourth recipient for all treatments ( p < 0.05, but p ¼ 0.066 in Treatment P). Focusing on the discount factor (δ ¼ 0.3), we explain this phenomenon as follows. Suppose that the second recipient contributes to the first recipient. If the first recipient is a directly reciprocal player, he contributes to the returns payout in the second period. The payoff lost by the second recipient in the first period is less than the payoff for receiving the returns payout because the total contribution of the returns payout is multiplied by (1 + δ)2. Hence, the second recipient has an incentive to make a contribution before receiving the returns payout. However, if the other players behave reciprocally, the payoff that the fourth recipient lost before receiving their own returns payout, which is 400 points, equals the payoff from receiving that payout given that the total contribution of the payout is multiplied by (1 + δ)0 ¼ 1. The last recipient has no incentive to contribute to the other players, even where direct reciprocity is in effect.

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Result 9 The average contribution rates after receiving the returns payouts among the first, second, and third recipients did not differ significantly among all four treatments. Support The average values of α1 for the first, second, and third recipients did not differ significantly among the four treatments (Table 5.4). We expected that the average value of α1 of the earlier recipients would be lower than that of the later recipients in Treatments B and P because earlier recipients would stop making contributions after receiving their payouts. However, because the subjects tended to behave reciprocally, the order of payout receipt did not change the average values of α1 among the first, second, and third recipients.

5.3.3.5

Panel Regression of Average Contribution Rates

We conducted panel regression analysis to confirm the results in the previous sections. Table 5.5 shows the results of random and fixed effects panel regressions using β0 and α1 as dependent variables (using Breusch–Pagan Lagrange multiplier tests, p ¼ 0.000). The independent variables are as follows: “Voting” and “Punishment” are dummy variables for the voting system (1 ¼ if the subject is in Treatment Table 5.5 Linear panel regressions of average contribution rates

Voting Punishment Voting  Punishment No. contributors No. members Excluded Order Intercept Observations F-test Adj. R2

Before payout receipt (β0) Random effects 0.360*** (0.058) 0.078 (0.056) 0.171** (0.081) – 0.084*** (0.031) 0.116** (0.048) 0.213*** (0.014) 0.702*** (0.140) 1087 50.594*** 0.219

After payout receipt (α1) Random effects Fixed effects 0.229*** – (0.074) – 0.060 (0.071) – 0.155 (0.099) 0.155*** 0.146*** (0.023) (0.024) 0.083* 0.108** (0.044) (0.047) 0.109* 0.083 (0.064) (0.087) 0.004 0.016 (0.022) (0.024) – 0.300 (0.187) 629 629 12.817*** 14.214*** 0.127 0.108

Robust standard errors in parentheses * Significant at 10%; ** significant at 5%; *** significant at 1%

5.3

Experimental Study of ROSCA

151

V and VP) and the punishment rule (1 ¼ if the subject is in Treatment P and VP). “Voting  Punishment” is the interaction term between “Voting” and “Punishment.” “Order” is the order of payout receipt, and subjects who receive a payout in the first period are assigned the value 1. “No. Contributor” is the number of members who contributed to the returns payout received by a subject. “No. Member” is the number of members participating in the same group as a subject. “Excluded” is a dummy for a subject having been excluded (¼1). A panel data analysis on β0 using the Hausman test supports the random effects model ( p ¼ 0.401). The result of the model in the second column of Table 5.5 shows that the voting rule had a positive and highly significant effect on the contribution rate before receipt of payout (β0) and that the punishment rule had a negative but insignificant effect on β0. However, “Voting  Punishment” significantly and negatively affected β0. These results support Result 1. A weak positive correlation was observed between “No. Member” and β0. “Excluded” and β0 had a positive and significant correlation, which suggests that the experience of exclusion from the previous group promoted the contribution rate of excluded subjects. As mentioned in Result 8, the order of payout receipt negatively and significantly affected β0. The Hausman test rejects the null hypotheses of zero correlation between the error term and the independent variables in the random effects model regression on α1 only at the 10% significance level but not at the 5% level ( p ¼ 0.069). Therefore, we conduct the random and fixed effects models. The result of the random effects model in the third column of Table 5.5 shows that the voting rule positively and significantly affected the contribution rate after receipt of payout (α1) and that the punishment rule and “Voting  Punishment” had a negative but insignificant effect on α1. These results support Result 2. The results of the random and fixed effects models show that “No. Contributors” positively and significantly affected α1. This suggests that subjects reciprocally contribute to the returns payouts of those who contribute to them, which partially supports Result 7. In the fixed effects model, “No. Member” positively affects α1 at the 5% significance level. Both models suggest that the order of payout receipt does not significantly affect α1, which is consistent with Result 9.

5.3.4

Conclusions of the Experiment

We conducted a laboratory experiment to show a mechanism for solving the default problem in a fixed ROSCA whose participants are shuffled randomly for each ROSCA cycle. We observed that the exclusion of defaulters through voting increased contribution rates before and after receiving returns payouts. We also found that a punishment rule, corresponding to that of Sugden (1986) and the forfeiture rule of Koike et al. (2010), did not improve contributions: without the punishment rule, the subject voluntarily took revenge on others who had not contributed to the subject (see Table 5.3). It is because all subjects know who receives the fund when in the fixed ROSCA. On the other hand, the punishment rule hinders voluntary revenge (see Table 5.3) and the external punishment works

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Table 5.6 Comparison with the public goods experiments with exclusion Rosca experiment

Game stricture

Exclusion system

Result What promotes the contribution rate or level? Expelling low contributors? Reciprocity and retaliation? Threatening?

Treatment V Players decide whether to contribute a fixed sum to a fund or not Each player takes turns to receive the return Majority voting Excluded players can join the group if they are not excluded in the next round Contribution rate increases before and after each player receives the fund

Public goods experiments with exclusion MaierCinyabuguma Rigaud et al. (2010) et al. (2005) Players choose their contribution level to a public good Players share the total benefit from the public good Majority voting Excluded players cannot join the group once they are excluded Contribution levels increase

Yes

Yes

Yes

No

Yes

Yes

instead. As a result, there is no difference in contribution between Treatment P and Treatment B. Consequently, contrary to our theoretical predictions (see Chap. 4), the contribution rates and mean payoff of the subjects in Treatment V were significantly higher than in the other treatments. These results support those of field studies of ROSCAs, in which social sanctions through exclusion suppressed the problem of defaults even without external enforcement (Ardener 1964; Armendáris and Morduch 2010). Previous studies using the public goods game have shown that exclusion (Cinyabuguma et al. 2005; Maier-Rigaud et al. 2010; Feinberg et al. 2014) and other endogenous group-formation (Page et al. 2005; Ehrhart and Keser 1999; Charness et al. 2014; Charness and Yang 2014) promoted contribution level. The present investigation also supported those results even though our game structure differed from the public goods game (Table 5.6 and Fig. 5.13). However, an interesting finding in our ROSCA game was the observed voluntary direct reciprocity and voluntary retaliation (Treatment B and V). These two phenomena are not observed in the public goods game, where all the players contribute to the fund, which is distributed equally among all players. Another interesting finding of the present study was that, although we introduced an explicit costless punishment rule, it did not improve the contribution rate (Treatments P and VP). In our experiment, we could not clarify the relationship between “voluntary direct reciprocity and retaliation” and “explicit punishment.”

5.3

Experimental Study of ROSCA

Rotating indivisible goods game (Rosca game)

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deposit

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deposit

deposit

Fig. 5.13 The comparison between two games: the rotating individual goods game (Rosca game) and the public goods game

Voluntary direct reciprocity and retaliation are reminiscent of the repeated game between two players, in which a player can behave either reciprocally or revengefully toward their opponent. However, the ROSCA game differs from the repeated game: the order of payout receipt influences reciprocity and retaliation in the ROSCA game. Further investigation of the ROSCA game should offer us new perspectives on the repeated game between two players. Hechter (1988) examined ROSCAs and argued that members can easily suppress non-contribution before receiving their payouts. This is because if a member fails to contribute to the fund before receiving their payout, the other members can prevent them from obtaining any funds. However, in our treatments, the subjects often failed to make a contribution before receiving their payout despite the punishment rule prohibiting the receipt of payouts by such non-contributors. This result reinforces the importance of exclusion based on reputation.

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Exclusion based on majority voting has two characteristics: (1) low contributors are excluded from groups; and (2) low contributors who have been excluded sometimes have a chance to participate in a group again. Item (1) accords with the results of public goods experiments (Cinyabuguma et al. 2005; Maier-Rigaud et al. 2010), but no experimental studies on exclusion have examined item (2). Sugden (1986) proposed a mutual-aid game based on informal health insurance in England, with three elements: insurance members pay a subscription, one randomly chosen member receives the insurance money in each period, and a member previously awarded a payout remains eligible for future payouts. The mutual-aid game differs from ROSCAs in that every member has an equal chance of receiving the returns payout in each period. Sugden’s theoretical analysis showed that a kind of tit-for-tat strategy, which has the three features of “brave reciprocity,” “punishment,” and “reparation,” produces a stable equilibrium. Our results correspond to Sugden’s titfor-tat strategy: in our experiment, the subjects behaved reciprocally and punished defectors by excluding them from the group. Thus, we consider that allowing excluded members to rejoin a group gives them a chance to make reparations for past defaults. In the latter part of Chap. 5, we investigated how social exclusion based on peer selection can prevent the problem of default in the fixed ROSCA. Besides fixed ROSCAs, random and bidding ROSCAs are generally observed in the rural areas of developing countries. Future research is needed to examine the effectiveness of such social exclusion in the random and bidding ROSCAs and so verify the robustness of our results. In the random ROSCA, the subjects cannot use a retaliation strategy because the payout recipient is chosen randomly after subjects decide whether to contribute. Therefore the punishment rule, which did not work in the fixed ROSCA in our experiment, may be needed to sustain the random ROSCA. Because our study did not assume the existence of a costly punishment, it would be interesting to investigate how a costly punishment such as confiscating the property of the defaulter influences the sustainability of ROSCAs, even in a laboratory setting. Further investigations in the fixed ROSCA are also needed to verify the robustness of our results. For instance, we will investigate how the knowledge that the subjects know in advance the game will be over at the tenth round affect the outcomes and how what points players start off with will change their behaviors. We will also examine if we can obtain the same results as our experiment when subjects are chosen from local people who often join real-world ROSCAs. Box 5.1 Tanomoshi-ko terms Name: This ko shall be named the Tanomoshi-ko of Mr. A. Ko contribution: This ko shall consist of 45 lots of 250,000 yen per lot, with a total of 11,250,000 yen. Loan: The contribution for this ko shall be collected on (Month)/(Day)/ (Year) and shall be loaned to the initiator with no interest. (continued)

5.3

Experimental Study of ROSCA

155

Box 5.1 (continued) Deposit: Starting with the second ko, the deposit shall be 50,000 yen per lot, with a total of 2,500,000 yen. Repayment: The initiator shall repay 250,000 yen 45 times starting from the ko on (Month)/(Day)/(Year) until the last ko. Starting from the second ko, the winning bidder shall repay 50,000 yen between the following ko and the last ko. Should the initiator or the winning bidder fail to repay on the day of ko, the sureties will be obligated to take over. Date: This ko shall be held four times a year, on the 23rd of February, May, August, and November. Location: At Mr. A’s residence. Time: Bidding shall start at XX:XX (a bidder who arrives late shall lose the right to bid). If a date rearrangement is needed, the members shall be notified 3 days in advance. Notification: The initiator shall provide notifications 3 days in advance. Bid: The bidding shall be done on the day of ko. The highest bidder shall win the bid. • The bid increment shall be 100 yen. Any fraction of less than 100 yen shall be rounded down. • Should the highest bid be a tie, the concerned parties shall rebid. The rebid shall not be lower than the original bid. Dividend: Starting from the second ko, kaisen shall be distributed to the followers who have not previously won the bid. Confectionery expenses: The confectionery fee shall be 1000 yen per session, which the winning bidder shall pay. Joint: The initiator’s joint sureties Sureties: Mr. XXXXX, Mr. XXXXX • • • •

The joint sureties cannot win the bid before the 23rd ko. A winning bidder’s surety must be a resident of XX district. An IOU submission shall be required upon providing the money. The treasurer must be able to verify the winning bidder’s surety by telephone.

Officers: Treasurer Mr. XXX Facilitators Mr. XXX, Mr. XXX, Mr. XXX, and Mr. XXX Penalty: A member shall provide sake (1.8 L) should he/she fail to pay the deposit by the given time.

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References Ardener S (1964) The comparative study of rotating credit associations. J R Anthropol Inst G B Irel 94:201–229 Armendáris B, Morduch J (2010) The economics of microfinance. MIT Press, Cambridge, MA Bochet O, Page T, Putterman L (2006) Communication and punishment in voluntary contribution experiments. J Econ Behav Organ 60:11–26 Carpenter JP (2007) Punishing free-riders: how group size affects mutual monitoring and the provision of public goods. Games Econ Behav 60:31–51 Charness G, Cobo-Reyes R, Jiménez N (2014) Identities, selection, and contributions in a publicgoods game. Games Econ Behav 87:322–338 Charness G, Yang CL (2014) Starting small toward voluntary formation of efficient large groups in public goods provision. J Econ Behav Organ 102:119–132 Cinyabuguma M, Page T, Putterman L (2005) Cooperation under the threat of exculsion in a public goods experiment. J Public Econ 89:1421–1435 Croson R, Fatas E, Neugebauer T, Morales AJ (2015) Excludability: a laboratory study on forced ranking in team production. J Econ Behav Organ 114:13–26 Ehrhart KM, Keser C (1999) Mobility and cooperation: on the run. CIRANO Scientific Series 99s, Montréal Fehr E, Gächter S (2000) Cooperation and punishment in public goods experiments. Am Econ Rev 90:980–994 Fehr E, Gächter S (2002) Altruistic punishment in humans. Nature 415(6868):137–140. https://doi. org/10.1038/415137a Feinberg M, Willer R, Schultz M (2014) Gossip and ostracism promote cooperation in groups. Psychol Sci 25:656–664 Fischbacher U (2007) Z-Tree: Zurich toolbox for ready-made economic experiments. Exp Econ 10:171–178 Gächter S, Herrmann B (2009) Reciprocity, culture and human cooperation: previous Insights and a new cross-cultural experiment. Philos Trans R Soc B Biol Sci 364:791–806 Gächter S, Herrmann B (2011) The limits of self-governance when cooperators get punished: experimental evidence from urban and rural Russia. Eur Econ Rev 55:193–210 Guala F (2012) Reciprocity: weak or strong? What punishment experiments do (and do not) demonstrate. Behav Brain Sci 35(1):1–15. https://doi.org/10.1017/S0140525X11000069 Hechter M (1988) Principles of group solidarity. University of California Press, Berkeley, CA Herrmann B, Thoni C, Gächter S (2008) Antisocial punishment across societies. Science 319:1362– 1367 Koike S, Nakamaru M, Otaka T, Shimao H, Shimomura KI, Yamato T (2018) Reciprocity and exclusion in informal financial institutions: An experimental study of rotating savings and credit associations. PLoS One 13(8):e0202878. https://doi.org/10.1371/journal.pone.0202878 Koike S, Nakamaru M, Tsujimoto M (2010) Evolution of cooperation in rotating indivisible goods game. J Theor Biol 264(1):143–153. https://doi.org/10.1016/j.jtbi.2009.12.030 Maier-Rigaud FP, Martinsson P, Staffiero G (2010) Ostracism and the provision of a public good: experimental evidence. J Econ Behav Organ 73:387–395 Nakamaru M, Koike S (2015a) The institutional structure in which non-cooperative behavior promotes non-banding among people. In: Takagi O, Takemura K (eds) Isolated society: why do people delete ties among them? Seisin Shobo, Tokyo, pp 150–171 Nakamaru M, Koike S (2015b) The structure of cooperation in a group and rules sustaining cooperation: evolutionary game simulations and interview survey. In: Kameda T, Saijo T (eds) How to determine “the rule of our society”. Keiso Shobo, Tokyo, pp 49–83 Nikiforakis N (2008) Punishment and counter-punishment in public good games: can we really govern ourselves? J Public Econ 92:91–112 Ostrom E, Walker J, Gardner R (1992) Covenants with and without a sword: self-governance is possible. Am Polit Sci Rev 86:404–417

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Chapter 6

Who Does a Group Admit into Membership or Which Group Does a Player Want to Join?

Abstract The evolution of cooperation in a large group is an evolutionary puzzle, because defectors always obtain a higher benefit than cooperators. When people participate in a group, they evaluate group members’ reputations and then decide whether to participate in it. In some groups, membership is open to all who are willing to participate in the group. In other groups, a candidate is excluded from membership if group members regard the candidate’s reputation as bad. We developed an evolutionary game model and investigated how participation in groups and ostracism influence the evolution of cooperation in groups when group members play the voluntary public goods game, by means of computer simulation. When group membership is open to all candidates and those candidates can decide whether to participate in a group, cooperation cannot be sustainable. However, cooperation is sustainable when a candidate cannot be a member unless all group members admit them to membership. Therefore, it is not participation in a group but rather ostracism, which functions as costless punishment on defectors, that is essential to sustain cooperation in the voluntary public goods game in a large group. Keywords Reputation · Sociality · Partner choice · Exclusion · Participation

6.1

Introduction

Chapters 4 and 5 showed that who can be chosen as a new group member or how a new member decides to participate in a group is important to maintain cooperation or friendship within the group, such as a club or informal institutions (Sugden 1986). Experimental study shows that when adolescent school students form groups, members can come to resemble each other after they have become a group, rather than by choosing members assortatively (Chang et al. 2013). This indicates that the premise that groups are formed assortatively is not always a proper assumption. Another example is the screening by a secret organization, such as the Freemasons, of those who apply for membership before deciding whether to admit them as members. How group members are selected and whether an individual participates in a group are essential for collective action. Theoretical studies of animal behavior have © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Nakamaru, Trust and Credit in Organizations and Institutions, Theoretical Biology, https://doi.org/10.1007/978-981-19-4979-1_6

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addressed the evolutionary advantage of group forming or joining a group in resource allocation and food exploitation (Vehrencamp 1983; Giraldeau and Beauchamp 1999). However, these studies did not answer the question. Previous studies about evolutionary game theory have shown that nonparticipation in the public goods game promotes the evolution of cooperation (Hauert et al. 2002a, 2002b; Semmann et al. 2003) and coevolution of cooperation and punishment (Hauert et al. 2007). However, these studies assumed that nonparticipants never participate in the public goods game, regardless of group members. Theoretical studies have investigated whether defector exclusion or ostracism from groups promotes the evolution of cooperation (Bowles and Gintis 2004; Tavoni et al. 2012; Lade et al. 2013; Sasaki and Uchida 2013). If defectors suffer the cost of exclusion from the group because that exclusion causes them some damage in common pool resource use, cooperators increase in number and common-resource management can be sustainable (Tavoni et al. 2012; Lade et al. 2013). Players called excluders who cooperate and exclude defectors from the group can solve the problem of second-order free-riders, in which punishing cooperators diminishes after the population is dominated by pure cooperators and punishing cooperators, when excluders find and exclude defectors by paying a cost; excluded defectors cannot get benefits from cooperators (Sasaki and Uchida 2013). Reputation can be used to choose members or groups. The following are evolutionary game studies about reputation. Theoretical studies of the evolution of indirect reciprocity investigate the evolution of cooperation in the two-player donation game in which a player cooperates with an opponent whose reputation is good; otherwise, the player does not cooperate (Nowak and Sigmund 1998). Recent studies have focused on the effect of the assessment rule, which determines who is good or bad, on the evolution of cooperation. Other works investigate if group favoritism can be proven by indirect reciprocity in the two-player donation game (Nakamura and Masuda 2012; Ohtsuki and Iwasa 2004; Uchida and Sigmund 2010). Suzuki and Akiyama (2005) applied the framework of Nowak and Sigmund (1998) to the nperson prisoner’s dilemma game in which more than two players participate, and each player can choose cooperation when the average reputation of the other members is equal to or higher than the threshold of the focal player (Suzuki and Akiyama 2005). There are studies in which players choose a partner without using reputation as a cue. For example, Aktipis (2004) used the repeated prisoner’s dilemma game and showed that cooperation can be established when cooperators can run away from defectors in a spatially structured population (Aktipis 2004). Chiang (2008) showed that partner choice based on payoff reinforcements from past interaction promotes the evolution of fairness between two players in the ultimatum game (Chiang 2008). Here, we focus on how individuals decide to participate in a group and/or how the group members decide to exclude some individuals from membership when they contribute to collective action, using the voluntary public goods game (Nakamaru and Yokoyama 2014). The decision-making of a player who chooses a partner in a dyadic interaction is different from that of a player who participates in a group consisting of more than two players, or who excludes individuals from membership.

6.2

Models

161

The reasons for this are as follows. When we want to participate in a group, we may evaluate all members or representative members of the group. We decide to (or not to) participate in the group after we observe the best (or worst) member in the group and then regard the group as good (or bad). Similarly, when an individual wants to participate in the group, group members evaluate the individual. In some groups, when all members agree that a candidate is good, that individual is allowed to participate in the group. In other groups, when at least one member decides that a candidate is good, that individual is allowed to join. Therefore, we investigate what types of decision-making of groups or individuals influence the evolution of cooperation in the voluntary public goods game, based on the framework of Nowak and Sigmund (1998). When members of a group exclude some individuals from membership, this results in ostracism and is interpreted as costless punishment. Therefore we investigate the effects of ostracism as costless punishment, and the effects of participation in the group, on the evolution of cooperation in the voluntary public goods game. Chapter 4 presents that the peer selection rule is one of the necessary factors to promote the evolution of group cooperation when players play the rotating indivisible goods game based on Koike et al. (2010). Koike et al. (2010) only considered one type of decision-making as the peer selection. That is, if the following two conditions, (i) and (ii), are satisfied, the player can be a member of the group; (i) if the average reputation level over the group members is higher or equal to the player’ criteria, the player decides to join the group, and (ii) if the reputation of the candidate is higher than or equal to the average criteria over the group members’ criterion, the group permits the player to join the group. There are other possible types of the decision-making when a player’s choosing a group or a group’s choosing a new member, and we investigate which type promotes the evolution of group cooperation more in voluntary public goods game in this chapter (Nakamaru and Yokoyama 2014).

6.2

Models

The population has N players. Each player has three evolutionary traits, one concerning the contribution to the public goods game (PGG) and two concerning the threshold of decision-making (kpa and kex). There are two types of traits regarding the contribution to the PGG, a cooperator (C) and defector (D). Trait kpa is the threshold of decision-making when participating in a group. Trait kex is the threshold of decision-making when choosing a new member. In addition to these three traits, each player has his own reputation, called image score (s) (Nowak and Sigmund 1998). If a player cooperates, his image score increases by one unit; if the player does not cooperate, his image score decreases one unit. Therefore, a high image score means that the player invests substantially in the PGG, and a low image score means that the player does not invest much. It is assumed that every player knows the reputations of all players. Initially, the s of each player is zero.

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The following outlines the flow during one unit of time. (i) N players in a population are randomly divided into N/m groups. Each group has m players on average, who are called candidates for group members. (ii) The candidates can be members of the group, according to decision-making called the peer selection rule, which is dependent on the traits (kpa, kex) and image score, s. The peer selection rule consists of two conditions: the participation condition and the exclusion condition. In the participation condition, each candidate decides to participate in the group if the group reputation, Sg, is equal to or higher than kpa (Sg
kpa). We assume that the group reputation is based on its members’ image scores in a group. We define four types of group reputation: Average Sg, Median Sg, Maximum Sg and Minimum Sg. Average Sg is defined as the average value of all candidate reputations in the group; Median Sg, the median value of all candidate reputations in the group; Maximum Sg, the maximum value for all candidate reputations in the group; Minimum Sg, the minimum value for all candidate reputations in the group. Maximum Sg can be interpreted as that a candidate decides to participate in the group if the best reputation in the group is equal to or higher than the candidate’s threshold (Maximum Sg
kpa). This assumption corresponds to the situation that people decide to join the group because the group has such a good player. Minimum Sg can be interpreted as that a candidate decides to remain in the group if the worst reputation in the group is equal to or higher than the candidate’s threshold (Minimum Sg
kpa): if all reputations in the group do not meet the threshold of a candidate, the candidate never joins the group. This assumption describes the situation in which people decide not to join a group because it has a very bad player. In the exclusion condition, each group excludes some candidates from membership if a candidate’s image score (s) is less than the group criterion, Kg (s < Kg). We define four types of group criteria: Average Kg, Median Kg, Maximum Kg and Minimum Kg. Average Kg is the average of kex in the group; Median Kg, the median for kex in the group; Maximum Kg, the maximum for kex in the group; Minimum Kg, the minimum for kex in the group. Average or Median Kg means that candidates are selected as group members according to the group average or median of kex. Maximum Kg can be interpreted as that a candidate can be a member if the most stringent player in the group accepts the candidate, or if all players in the group accept or vote in favor of the candidate. Minimum Kg can be interpreted as that a candidate can be a member if the most lax player in the group accepts or votes in favor of the candidate. There are four kinds of peer selection rules: participation selection, exclusion selection, participation–exclusion selection with the same threshold (or same PE selection), and participation–exclusion selection with different thresholds (or different PE selection). The participation selection consists of the participation condition; the exclusion selection, the exclusion condition; the same PE selection, both the participation and exclusion conditions with kpa ¼ kex; the different PE selection, both the participation and exclusion

6.2

Models

163

conditions with kpa 6¼ kex. In the same and different PE selections, the participation and exclusion conditions occur simultaneously: a candidate can be a member if the reputation level (s) of the candidate is higher than or equal to Kg (s
Kg) and Sg is higher than or equal to kpa of the candidate (Sg
kpa). Therefore, will investigate the effect of 16 types of peer selection rules on the evolution of cooperation in the voluntary PGG. That is, the participation selection with one of four group reputations, exclusion selection with one of four group criteria, the same PE selection with one of four criteria (Average Kg and Average Sg, Median Kg and Median Sg, Maximum Kg and Maximum Sg, Minimum Kg and Minimum Sg), and the different PE selection with one of four criteria (Average Kg and Average Sg, Median Kg and Median Sg, Maximum Kg and Maximum Sg, Minimum Kg and Minimum Sg). Later, we investigate three other possible types of peer selection rule. (iii) Members in each group play the PGG once in which a cooperator (C) contributes to public goods and a defector (D) does not contribute at all. The payoffs for C (π C) and D (π D) in the group are defined as π C(nc) ¼ bxnc/ n  x and π D(nc) ¼ bxnc/n, in which n (n
2) is the number of group members and nc (0  nc  n) is the number of Cs in the group; b is the benefit factor produced by cooperators and x is the contribution to the public goods or the cost of cooperation. Here, two conditions, π D(nc) > π C(nc + 1) and π D(n) > π C(0), meet the definition of social dilemma. We assume 1 < b < 2, because n ¼ 2 is the minimum group size after peer selection. The payoff for the players who are excluded from membership is zero, since they do not play the PGG. If the group only consists of one member, the payoff for that player is zero. Therefore, if defectors participate in groups, their payoff is higher than zero if the group has one or more cooperator. If cooperators participate in groups, their payoff is higher than zero if nc/n > 1/b. Because the value of b is between 1 and 2, nc/n should be higher than 0.5 at least, if cooperators within groups receive benefits from participating in them. Otherwise, it is better for cooperators not to participate in groups than to do so. (iv) The image score (s) of each player is given after playing the PGG. The image score of a player increases by one unit if the player cooperates or invests in the PGG, and decreases by one unit if the player is a defector. If a player is eliminated from the group according to the peer selection rule and does not play the PGG, his image score does not change. After steps (i)–(iv) are repeated h times, the total score of each player, defined as his accumulated payoff during one generation, is obtained. Each player updates his traits to those of a player called A, in proportion to the total score of player A over the sum of the total scores of all players. This is interpreted as social learning. Each player changes his traits randomly with probability μ, which corresponds to a probability of mutation in evolutionary game theory. This algorithm corresponds to natural selection from the standpoint of evolution. Then one generation, consisting of h units of time, ends.

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In the next generation, image score and the total score are reset to zero. We analyzed this model through 10,000 generations by computer simulation. Baselines for the three parameters are N ¼ 100, m ¼ 5, h ¼ 10, x ¼ 1, and μ ¼ 0.005. The values of kpa or kex are uniform random numbers from integers between 6 and + 6 at the beginning of the simulation run and when a mutation occurs. Initially, the image score (s) of each player is zero and is constrained between 5 and + 5 (Nowak and Sigmund 1998). The initial population consists of defectors, to investigate if cooperators can invade a population dominated by defectors.

6.3

Results

In the baseline model, in which all members of each group can participate in the PGG because the peer selection rule is not implemented, cooperation never evolves, regardless of N, h, m and b. This is because the payoff of D is always higher than that of C. We next introduced the peer selection rules to the baseline model to determine how each of the peer selection rules influenced the evolutionary outcomes. Figure 6.1 shows the evolutionary simulation outcomes for the four criteria and four peer selection rules (see also Figs. S1–S5 in Nakamaru and Yokoyama (2014)). A nonparticipating defector is in the majority in any selection with any criterion in which benefit factor b is low. As b increases, the number of participants increases; a higher b increases the payoff of the players when they join the group and play the PGG (see Fig. S1 in Nakamaru and Yokoyama (2014)). Naturally, the number of nonparticipating cooperators declines when b increases. As a result, the number of cooperators, which is the sum of participating and nonparticipating cooperators, decreases as b slightly increases when b is small and is less than ~1.1 (see Fig. S1 in Nakamaru and Yokoyama (2014)). In the participation selection, although b is high, cooperation never evolves regardless of criterion type (Fig. 6.1). This result indicates that the existence of nonparticipants does not promote the evolution of cooperation, if players can evaluate a group and decide whether to participate in it. This is contrary to previous studies, in which nonparticipants who never evaluated groups promoted such evolution. Cooperation especially evolves in the exclusion selection and the different PE selection, with average or maximum criteria (Fig. 6.1a, c). Figure 6.1d indicates that the minimum criterion in any peer selection rule does not increase the cooperation rate more than other criteria. Simulation outcomes in the average criterion are different from the median one, but similar to the maximum criterion. This implies that the distribution of image score or thresholds after selection is not a normal or uniform one, even though thresholds are randomly determined to be uniform and the image score of all players is zero at the beginning of simulation. In the following, we explain why bursts of defectors and nonparticipants occur (see Figs. S2–S5d, e, g, h, k, l in Nakamaru and Yokoyama (2014)). When mutation produces many defectors by chance, the image score of a player can be less than Kg (see Figs. S2–S5d–f in Nakamaru and Yokoyama (2014)). As a result, many players are excluded from membership and payoffs are low. If

6.3

Results

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Fig. 6.1 Simulation outcomes of baseline model with peer selections. Vertical axis is average percentage of cooperators in entire population over 100 runs, in each of which 10,000 generations were simulated. Horizontal axis represents benefit factor for cooperation b. a Is for Average criterion; b median criterion; c maximum criterion; d minimum criterion. Red line represents the participation selection, green line the exclusion selection, blue line the same PE selection, and orange line the different PE selection. In d, red and blue lines overlap. Other parameters are N ¼ 100, m ¼ 5, h ¼ 10, and μ ¼ 0.005

cooperators can join the group and they dominate the group, they can obtain high payoffs. Then, the frequency of cooperators increases again. The same mechanism is depicted in Figs. S2–S5g–n in Nakamaru and Yokoyama (2014). In conclusion, ostracism promotes the evolution of cooperation if the group excludes some individuals who are regarded as bad, either by all members or by the average. Now, we discuss why the exclusion selection promotes the evolution of cooperation but the participation selection does not. In the latter selection, players can decide to participate in a group. Participating defectors obtain more benefits than nonparticipating ones with higher b. Then, defectors want to join the PGG, and the number of nonparticipating defectors is high with low b and the number of participating ones is high with high b. Since the participation selection has no mechanism for excluding defectors from membership, defectors come to dominate groups. Consequently, cooperators do not obtain high payoffs although they participate in the group, so their numbers diminish. For example, when Maximum Sg is used in the

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participation selection, other members except the one with the maximum image score in the group do not have an incentive to choose cooperation, and then do not have a high image score. Even if the group has one member with a very good reputation and others with a bad reputation, a candidate decides to participate in the group. The members with bad reputation, many of whom are defectors, receive a benefit from a newcomer if he is a cooperator. As a result, cooperation never evolves. Since the exclusion selection excludes defectors from membership, cooperators have high payoffs when they join groups and play the PGG. As a result, cooperators dominate groups, especially with higher b (see Fig. S1 in Nakamaru and Yokoyama (2014)). The following shows why the maximum criterion promotes cooperation but the minimum one never does so in the exclusion selection. The maximum criterion in that selection means that a candidate cannot become a group member unless all players in the group accept him. This is very strict exclusion or ostracism and consequently, defectors are excluded from membership and cooperation can then evolve. The minimum criterion in the exclusion selection means that when one player in a group accepts a candidate, that candidate can obtain membership. This corresponds to the situation in which only one vote is needed to become a member. This means that defectors can obtain a membership more easily with the minimum criterion than with the maximum one (see Fig. S1g, h; the exclusion selection in Fig. S6c, d in Nakamaru and Yokoyama (2014)). As a result, the minimum criterion in the exclusion selection does not promote cooperation. Cooperation evolves in the different PE selection more than in the same PE selection. The participation and exclusion conditions are independent of each other in the different PE selection. Then, the exclusion condition in that selection works as efficiently as the exclusion selection does. If the threshold of the participation condition (kpa) is the same as that of the exclusion condition (kex) in the same PE selection, the exclusion condition in the latter selection does not work efficiently because the participation condition hinders the exclusion condition in the same PE selection. Figure 6.2 shows the effect of h, m and b on the evolution of cooperation with N ¼ 100 and N ¼ 1000 (see also Figs. S7 and S8 in Nakamaru and Yokoyama (2014)). When h and b are high, cooperation is favored. The result for N ¼ 1000 is basically the same as for N ¼ 100. However, the evolved cooperation level is higher with N ¼ 1000 than N ¼ 100 when b and h are high, and that level is lower with N ¼ 1000 than N ¼ 100 when b and h are low. This is because stochasticity influences evolutionary dynamics more with smaller N. Figure 6.2 also shows that smaller m and larger m promotes the evolution of cooperation. When m is higher than 20, the larger group size appears to promote cooperation. Even though the net group size after peer selection m0 is smaller than m (see Fig. S6 in Nakamaru and Yokoyama (2014)) and the ratio of nonparticipants increases, we can conclude that cooperation can evolve even in large group size under peer selection. If the group can properly exclude defectors from membership, members may obtain a high benefit despite the application of Minimum Sg in the participation condition. In the minimum criterion, the cooperation rate in the different PE

6.3

Results

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1.40

1.60

b

1.80 1.95

1.00

1.20

1.40

1.60

1.80 1.95

1.60

1.80 1.95

15

15

10

10 h

h 5

5

1

1

c

d

b 1.00

1.20

1.40

1.60

b

1.80 1.95

1.00

1.20

1.40

500 250 200 125 100 50 m 40 25 20 10 8 5 4 2

50 25 20 m 10 5 4 2

e 0

100%

average percentage of cooperators

Fig. 6.2 Parameter dependence when exclusion selection with Average Kg is applied. The average percentage of cooperators in the entire population is shown over 100 runs, in each of which 10,000 generations were simulated. a, b show the effect of parameters h and b on simulation outcomes when m ¼ 5. c, d show the effect of parameters m and b on simulation outcomes when h ¼ 10. a, c are for N ¼ 100, and b, d for N ¼ 1000. e presents the relationship between percentage and color in all graphs. The other parameter is μ ¼ 0.005

selection is much higher than that in the exclusion selection (Fig. 6.1d). However, for other criteria, cooperation rates in both the exclusion selection and different PE selection are nearly the same (Fig. 6.1a–c). This result implies that the combination of the exclusion condition with other criteria (Maximum, Average, or Medium Kg) and the participation condition with Minimum Sg promote the evolution of cooperation in the different PE selection. Figure 6.3 shows that cooperation is slightly more favored when Maximum, Average or Medium Kg is used in the exclusion condition and Minimum Sg is used in the participation condition, relative to when the same criterion is used in both the participation and exclusion conditions, especially when the benefit from the PGG (b) is high. However, the cooperation rate in the different PE selection with Maximum (or Average) Kg and Minimum Sg is nearly the same as

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6

Who Does a Group Admit into Membership or Which Group Does a Player. . .

b

a

100

80

cooperators (%)

cooperators (%)

100

60 40 20 0 1.0

80 60 40 20

1.2

1.4 1.6 benefit factorb

1.8

2.0

0 1.0

1.2

1.6

1.8

1.4 1.6 benefit factor b

1.8

2.0

c cooperators (%)

100 80 60 40 20 0 1.0

1.2

1.4

2.0

benefit factor b

Fig. 6.3 Comparison of different PE selections between the same and different criteria. Vertical axis is the average percentage of cooperators in the entire population over 100 runs, in each of which 10,000 generations were simulated. Horizontal axis represents benefit factor for cooperation b. a Black line represents simulation outcomes when the different PE selection with Average Kg and Minimum Sg is used, and red line when the different PE selection with Average Kg and Average Sg is used. b Black line represents simulation outcomes when the different PE selection with Median Kg and Minimum Sg is used, and red line when the different PE selection with Median Kg and Median Sg is used. c Black line represents simulation outcomes when the different PE selection with Maximum Kg and Minimum Sg is used, and red line when the different PE selection with Maximum Kg and Maximum Sg is used. Dashed lines indicate double standard deviation (95% confidence interval). The other parameters are N ¼ 100, m ¼ 5, h ¼ 10, and μ ¼ 0.005

in the exclusion selection with Maximum (or Average) Kg. Therefore, if both the exclusion and participation conditions are needed, cooperation is established when two decision-making principles are obeyed: (i) if a candidate wishes to participate in a group, they should see if the worst person or all persons can satisfy that candidate’s criterion and then decide to participate in it; and (ii) if all players in a group accept a candidate, that candidate can become a member.

6.4

6.4

Discussion and Conclusions

169

Discussion and Conclusions

We investigated the effect of optional participation and ostracism on the evolution of cooperation in contributing to collective action, which is described by the voluntary PGG based on the framework of Nowak and Sigmund (1998). We showed that the existence of nonparticipants does not promote cooperation when players evaluate others in a group that they are deciding whether to join. When players in a group evaluate others and decide whether to exclude them from membership, cooperation can evolve. This indicates that exclusion from groups, which can be interpreted as ostracism and functions as costless punishment, promotes the evolution of cooperation. This is especially so when a candidate for group member is excluded unanimously, or when a candidate is excluded because their reputation or image score is below the average of thresholds for the group; then, the cooperation rate can be high through evolution. Our results indicate that if group membership is open to all who are willing to participate in it, group cooperation is not sustainable. The exclusion condition, in which group members exclude some candidates from membership if their reputation does not satisfy the threshold, is essential from the standpoint of sustaining cooperation in a large group. However, in daily life, group members do not exclude individuals who are unwilling to participate in the group; thus, PE selection consisting of both the participation and exclusion conditions is more realistic than the exclusion selection. Examples of institutions that require PE selection are a rotating savings and credit association (ROSCA) in Chaps. 4 and 5 and microcredit (Armendáris and Morduch 2010; Koike et al. 2010; Geertz 1962). As mentioned in Chaps. 4 and 5, when forming a group, its members must properly choose new members or individuals have to choose a proper group in a ROSCA. Otherwise, people lose their investments. As Chap. 5 mentioned, my coworker and I interviewed a woman who joined a ROSCA or tanomoshi-ko with seven or eight other elderly women on January 25, 2013. She stated that a person who wanted to participate in her ROSCA could not be a member unless members admitted that person to membership unanimously. This corresponds to Maximum Kg in the exclusion condition of our model. A ROSCA candidate may use Minimum Sg because the person wants to join the ROSCA if all members’ reputations satisfy his threshold. Therefore, the different PE selection with Maximum (or Average) Kg and Minimum Sg may more accurately describe reality. To verify how to choose a new member and a group in reality, we must do more field research. Microcredit, which originated from a ROSCA, is another example with the same characteristics (Armendáris and Morduch 2010). A microfinance bank lends money to a group and then if some of its members do not repay this money, other members cannot make or receive any more loans. Our results indicate that cooperation can be better sustained even in a large group when individuals use different thresholds for participation and exclusion (kpa 6¼ kex), more so than when such individuals use the same threshold for participation and exclusion (kpa ¼ kex) in the PE selection. We also showed that “Minimum Sg and

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Who Does a Group Admit into Membership or Which Group Does a Player. . .

Maximum Kg” or “Minimum Sg and Average Kg” should be used to create a high cooperation rate. The experimental studies showed that exclusion promotes cooperation in the PPG when players can cast a vote for excluding others from the group based on their past behavior, gossip, or reputation (Maier-Rigaud et al. 2010; Feinberg et al. 2014), This is partially supported by our results concerning the exclusion selection. To further verify our conclusion, experimental work should be undertaken to help examine people’s decision-making with respect to participation in and exclusion from groups. Now, we compare our results of participant selection with Hauert et al. (2002a). They assumed that: (i) the population has three strategies, cooperators, defectors and nonparticipants; and (ii) payoff of the nonparticipants (σ) is between 0 and xb  x, in which x is the contribution to the pool. Hauert et al. showed that the population is dominated by nonparticipants in equilibrium when 1 < b < 2 and three strategies coexist when b > 2. We assumed that: (i) there were two strategies, cooperators and defectors, and whether to participate in a group was not determined by an evolutionary trait but by the decision-making of each player; and (ii) σ ¼ 0. Our result showed that nonparticipating and participating defectors exist in equilibrium when the participant selection is applied. The number of participating defectors increases and that of nonparticipating defectors decreases when b increases (see Fig. S1a–d in Nakamaru and Yokoyama (2014)). This is because the payoff of participating defectors is slightly higher than that of nonparticipating defectors, as the population has a small number of participating cooperators (see Figs. S2a, b, S3a, b, S4a, b, S5a, b in Nakamaru and Yokoyama (2014)). We did not perform simulations for b > 2, because social dilemma in a group consisting of n members (n
2) is solved if b > n. In a future study, we will investigate the effect of b > 2 on the evolution of cooperation, but we must check if b is less than the group size of all groups in the population during simulations. Even if we had assumed that σ was positive, our results would have been different from Hauert et al. (2002a). This is because players never evaluate a group when they decide whether to join it (Hauert et al. 2002a). We assumed that the reputation (or image score) of nonparticipants did not change. However, the reputation of nonparticipants may decrease by one unit if they are excluded from membership. Or, the image score of nonparticipants may increase by one unit if they decide not to join a group with bad reputation. Group members may pay a cost of exclusion, and then excluded players may suffer the cost of being excluded (Sasaki and Uchida 2013; Tavoni et al. 2012; Lade et al. 2013; Bowles and Gintis 2004). This suggests that there are other possible ways for defining reputation and costs. In the future, we will elucidate the effects of σ and of the costs of exclusion and being excluded on the evolution of group cooperation, and examine how to define the reputation of nonparticipants influences the evolution of group cooperation. We assumed that individuals screen others based on the history of their behaviors. A field study of an Andean community showed that reputation is related to the contribution to collective action (Lyle III and Smith 2014). In our model, reputations of individuals who have not cooperated with defectors are the same as those of individuals who have not cooperated with cooperators. However, some indirect

References

171

reciprocity studies of dyadic interaction have shown that cooperation is more sustainable when the reputation of an individual who has not cooperated with cooperators can be distinguished from that of one who has not cooperated with defectors than when the reputations of defectors are the same, regardless of whom defectors have not cooperated with (Sugden 1986; Ohtsuki and Iwasa 2004). We do not know whether to use second- and third-order information about the reputation of others when forming a group. We may be confused by complicated information when evaluating people or groups. Some people may not have a sufficiently high cognitive ability to manage such complex information, whereas others can do so. Our model may be appropriate for investigating the effect of participation and exclusion on the evolution of cooperation in groups, because people may not use complex information in forming a group. The effect of cognitive ability on managing reputation is suitable for future study. In our model, we did not deal with the following situations: (i) people willing to join a particular group and who can compare more than one group, then choose one group; and (ii) people excluded from membership who look for other groups and then attempt to participate in another group. We will tackle these cases in future research.

References Aktipis CA (2004) Know when to walk away: contingent movement and the evolution of cooperation. J Theor Biol 231(2):249–260. https://doi.org/10.1016/j.jtbi.2004.06.020 Armendáris B, Morduch J (2010) The economics of microfinance. MIT Press, Cambridge, MA Bowles S, Gintis H (2004) The evolution of strong reciprocity: cooperation in heterogeneous populations. Theor Popul Biol 65(1):17–28. https://doi.org/10.1016/j.tpb.2003.07.001 Chang C-Y, Ho H-C, Wu C-I (2013) Social influence or friendship selection: coevolution of dynamic network and adolescent attitudes and behavior. Paper presented at the the international network for social network analysis conference, Xi’an, China, July 12–15, 2013 Chiang YS (2008) A path toward fairness: preferential association and the evolution of strategies in the ultimatum game. Ration Soc 20(2):173–201. https://doi.org/10.1177/1043463108089544 Feinberg M, Willer R, Schultz M (2014) Gossip and ostracism promote cooperation in groups. Psychol Sci 25(3):656–664. https://doi.org/10.1177/0956797613510184 Geertz C (1962) The rotating credit association: a “middle rung” in development. Econ Dev Cult Change 10:241–263 Giraldeau L-A, Beauchamp G (1999) Food exploitation: searching for the optimal joining policy. Trends Ecol Evol 14:102–106 Hauert C, De Monte S, Hofbauer J, Sigmund K (2002a) Replicator dynamics for optional public good games. J Theor Biol 218:187–194. https://doi.org/10.1006/yjtbi.3067 Hauert C, De Monte S, Hofbauer J, Sigmund K (2002b) Volunteering as Red Queen mechanism for cooperation in public goods games. Science 296(5570):1129–1132. https://doi.org/10.1126/ science.1070582 Hauert C, Traulsen A, Brandt H, Nowak MA, Sigmund K (2007) Via freedom to coercion: the emergence of costly punishment. Science 316(5833):1905–1907. https://doi.org/10.1126/ science.1141588 Koike S, Nakamaru M, Tsujimoto M (2010) Evolution of cooperation in rotating indivisible goods game. J Theor Biol 264(1):143–153. https://doi.org/10.1016/j.jtbi.2009.12.030

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Lade SJ, Tavoni A, Levin SA, Schlüter M (2013) Regime shifts in a social-ecological system. Theor Ecol 6(3):359–372. https://doi.org/10.1007/s12080-013-0187-3 Lyle HF III, Smith EA (2014) The reputational and social network benefits of prosociality in an Andean community. Proc Natl Acad Sci U S A 111(13):4820–4825. https://doi.org/10.1073/ pnas.1318372111 Maier-Rigaud FP, Martinsson P, Staffiero G (2010) Ostracism and the provision of a public good: experimental evidence. J Econ Behav Organ 73(3):387–395. https://doi.org/10.1016/j.jebo. 2009.11.001 Nakamaru M, Yokoyama A (2014) The effect of ostracism and optional participation on the evolution of cooperation in the voluntary public goods game. PLoS One 9(9):e108423. https://doi.org/10.1371/journal.pone.0108423 Nakamura M, Masuda N (2012) Groupwise information sharing promotes ingroup favoritism in indirect reciprocity. BMC Evol Biol 12:213 Nowak MA, Sigmund K (1998) Evolution of indirect reciprocity by image scoring. Nature 393(6685):573–577 Ohtsuki H, Iwasa Y (2004) How should we define goodness? Reputation dynamics in indirect reciprocity. J Theor Biol 231(1):107–120. https://doi.org/10.1016/j.jtbi.2004.06.005 Sasaki T, Uchida S (2013) The evolution of cooperation by social exclusion. Proc R Soc B 280(1752):20122498. https://doi.org/10.1098/rspb.2012.2498 Semmann D, H-Jr K, Milinski M (2003) Volunteering leads to rock–paper– scissors dynamics in a public goods game. Nature 425:390–393 Sugden R (1986) The economics of rights, co-operation and welfare. Basil Blackwell, New York Suzuki S, Akiyama E (2005) Reputation and the evolution of cooperation in sizable groups. Proc R Soc B 272:1373–1377 Tavoni A, Schlüter M, Levin S (2012) The survival of the conformist: social pressure and renewable resource management. J Theor Biol 299:152–161. https://doi.org/10.1016/j.jtbi.2011.07.003 Uchida S, Sigmund K (2010) The competition of assessment rules for indirect reciprocity. J Theor Biol 263(1):13–19. https://doi.org/10.1016/j.jtbi.2009.11.013 Vehrencamp SL (1983) A model for the evolution of despotic versus egalitarian societies. Anim Behav 31:667–682

Chapter 7

The Mutual-Aid Game as an Early-Stage Insurance System

Abstract We consider another game with the all-for-one type besides the rotating indivisible goods game, the mutual-aid game, in which one member of the group is chosen randomly as the aid recipient, and other members decide whether to help the recipient. This game can describe the early stage of insurance provision in England, for example. With reference to existing indirect reciprocity studies, we investigate what promotes the evolution of cooperation in the mutual-aid game by means of replicator equations and agent-based simulations. Following are the findings. In a multilateral relationship in which members play the mutual-aid game in a group whose size is greater than and equal to four, once two or more defectors are in bad standing, they remain bad under the rule that a donor’s bad reputation remains bad, whether or not the donor helps a recipient with a bad reputation. Then, cooperators never help them, and mutation helps the invasion of rare cooperators more when the group size is larger in the finite population, even when implementation and perception errors occur. Meanwhile, this rule never helps the invasion of rare cooperators in the bilateral relationship, in which the group size is two. Our results suggest that large group cooperation can be possible when social institutions are equipped with systems such as those in the mutual-aid game. Keywords Reputation · Institution · Evolutionary invasion · Replicator dynamics · Evolutionary simulations

7.1

Introduction

In our society, there exist various mutual systems not only in the traditional community and also in the sharing economy on the internet. Table 1.4 showed that mutual systems can be categorized into four types: “one for one,” “all for all,” “one for all,” and “all for one.” The “all for one” type corresponds to some mutual-aid systems, such as rotating savings and credit associations (ROSCAs) shown in Chaps. 4 and 5 (Geertz 1962; Andener 1964; Besley et al. 1993; Anderson et al. 2009; Koike et al. 2010; see Chaps 4 and 5), an early stage insurance institution (Bell 1907), and customs in funerals and wedding parties in Japan such as kouden and shugi. Gojo-ko, which was established by Mr. Kinjiro Ninomiya and is the world’s © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Nakamaru, Trust and Credit in Organizations and Institutions, Theoretical Biology, https://doi.org/10.1007/978-981-19-4979-1_7

173

174

7 The Mutual-Aid Game as an Early-Stage Insurance System

first credit system, originates from these mutual-aid systems (see Epilogue for more information). ROSCAs can be modeled by the rotating indivisible goods game (Koike et al. 2010). There are other games with the all-for-one type in our society; Sugden proposed the mutual-aid game, which can describe the system based on Bell (1907), in which Lady Florence Bell observed the daily life of ironworkers in England and reported, “It often happens that if one of their number is struck down by accident or sudden illness, a ‘gathering’ is made at the works, the hat is passed round, and each one contributes what he can to tide over the time of illness, or, in case of death, to contribute toward funeral expenses” (p. 76, lines 17–23) (Sugden 1986). In the mutual-aid game, there are n members in a group, and one of the members is randomly chosen as a recipient (Sugden 1986). Accidents and sudden illnesses can be considered to occur randomly, making the game’s assumption appropriate. The others are assigned as donors. Then, the donors decide whether to give money to the recipient. In the next round, one of the members of the group, possibly including the previous recipient, is chosen as a recipient, and this step continues further on all the rounds. We can easily imagine that members have knowledge of others’ reputations. Sugden (1986) proposed the T1 strategy, which “prescribes that all donors should contribute if j is in good standing, but defect if he is not [Note: j is a recipient at the start of round i]. If j was in good standing at the start of round i, then at the start of round i + 1 the players who are in good standing are j and all those donors who co-operated in round i. If j was not in good standing at the start of round i, then at the start of round i + 1 the players who are in good standing are simply those who were in good standing at the start of round i” (p. 129, lines 24–30). Sugden (1986) then proved that T1 is a stable equilibrium strategy. This resulted in consecutive theoretical studies into the evolution of indirect reciprocity in the giving game (e.g. Leimar and Hammerstein 2001; Panchanathan and Boyd 2003; Brandt and Sigmund 2004; Ohtsuki and Iwasa 2004; Takahashi and Mashima 2006). Ohtsuki and Iwasa (2004) showed that conditional cooperators can resist the invasion of defectors under eight assessment rules called the leading eight, which have the following common features: (i) if a donor helps (does not help) a recipient with good standing, the donor’s reputation becomes good (bad), and (ii) if a donor with good standing does not help a recipient with bad standing, the donor’s reputation remains good. The leading eight include the T1 strategy. These previous studies have been applied to multilevel selection (Pacheco et al. 2006) and in-group favoritism (Nakamura and Masuda 2012; Matsuo et al. 2014). It is notable that the conditional cooperators never invade a population occupied by defectors under any assessment rules in these previous studies. To the best of our knowledge, there is a dearth of theoretical studies analyzing the mutual-aid game from the viewpoint of evolutionary dynamics, the exceptions being Panchanathan and Boyd (2004) and Inaba et al. (2016). These studies considered two stages: players play the one-shot collective action game (which is similar to the public goods game) in the first stage and then play the mutual-aid game repeatedly in the second stage (Panchanathan and Boyd 2004; Inaba et al. 2016). In the first stage,

7.2

Model

175

each player decides on his or her contribution to the collective action. In the second stage, each player decides whether to help the recipient in the mutual-aid game. Both studies assumed the linkage strategy, named Shunner, which contributes to the collective action and helps the recipient who has a good reputation in the mutualaid game. They showed the evolutionary condition that Shunners can resist the invasion of rare defectors who always choose defection in both stages. Their results derive from the mutual interactions between two stages and do not show how the mutual-aid game influenced evolutionary dynamics in and of itself. Empirically speaking, the collective action game is not always linked to the mutual-aid game. Here, we assume only the mutual-aid game. Focusing on the leading eight (Ohtsuki and Iwasa 2004), we investigate what kinds of features in the assessment rules can resist the invasion of defectors when conditional cooperators are common and what kinds of features in the assessment rules increase the number of rare conditional cooperators when the defectors are common (Shimura and Nakamaru 2018). Generally, a small group size promotes the evolution of group cooperation, especially when the public goods game is played (Boyd and Richerson 1988). Contrary to the general expectation, we show that a large group size promotes the evolution of cooperation in the mutual-aid game, and we examine the reason behind this effect.

7.2

Model

We assume N players in the population. These N players are randomly divided into N/n groups, and each group comprises n members at the beginning of each generation (2
n
N ). All members in a group play the mutual-aid game through m rounds per generation. The extant literature, especially experimental studies, suggests that the fixed number of repetitions causes an “end effect” (Axelrod and Hamilton 1981), and then the probability, w, by which the mutual-aid game is played in the next round, is often introduced. We do not assume that players decide their tactics on the basis of the number of rounds; thus, the end effect is negligible. Furthermore, the effect of the fixed number of repetitions on the evolutionary game is the same as that of the probability (Leimar and Hammerstein 2001). Therefore, m is equivalent to the expected number of rounds per generation 1/(1  w). In the mutual-aid game, one member chosen randomly from n members is assigned to be a recipient. The other n  1 members are assigned to be donors. Each donor decides whether to give a benefit (b; b  0) to the recipient, following his or her strategy. If the donor gives, he or she must pay a cost (c; b  c  0). In the next round, one member is again chosen randomly to be the recipient, and the other n  1 members are assigned to be donors. This process is repeated m times. Each player has one strategy as a heritable trait, and one assessment rule is adopted across the entire population. Four possible strategies are assumed: Always Defect (AllD), Always Cooperate (AllC), Co-strategy (Co) and Or-strategy (Or) (Ohtsuki and Iwasa 2004). Donors who adopt AllD never give a benefit to the recipient, while those who

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The Mutual-Aid Game as an Early-Stage Insurance System

Table 7.1 The eight assessment rules for the new reputation, R(X,Y,Z), in which X is the reputation of a donor (X ¼ G or B), Y is the reputation of a recipient (Y ¼ G or B), and Z is the behaviour of a donor (Z ¼ C or D) G

Reputation of donor (X)

B

G

Reputation of recipient (Y)

B

G

B

C

D

C

D

C

D

C

D

S-stand

G

B

G

G

G

B

B

B

Behaviour of donor (Z)

Judge

G

B

B

G

G

B

B

B

Sugden rule

G

B

G

G

G

B

G

G

Kandori rule

G

B

B

G

G

B

B

G

SugKan

G

B

G

G

G

B

B

G

KanSug

G

B

B

G

G

B

G

G

Stand

G

B

G

G

G

B

G

B

G-judge

G

B

B

G

G

B

G

B

Note: G and B denote good and bad reputation, respectively. C means that the donor gives a benefit to the recipient and D means that the donor does not give a benefit. Five of the assessment rules have been named S-Stand, Judge, Sugden rule, Kandori rule and Stand in previous studies. For convenience, we name the other three rules as SugKan, KanSug and G-Judge. For example, in R(G, G, C), the reputation of the donor becomes G, regardless of the assessment rules. The green columns highlight common features of the “leading eight”

adopt AllC always give a benefit to the recipient. Regardless of his or her own reputation, a donor who adopts the Co-strategy gives a benefit to a recipient whose reputation is good (G) but not to a recipient with a bad reputation (B). A donor who adopts the Or-strategy gives a benefit to a recipient whose reputation is G but not to a recipient who is B, only if the donor’s reputation is G. If his or her reputation is B, he or she gives the benefit to the recipient regardless of the recipient’s reputation. We assume that all donors whose strategy is either Co or Or observe the recipient’s reputation and then decide whether to give a benefit to the recipient following his or her own strategy and his or her own reputation. Then, the recipient receives the benefit nc,i  b if the number of donors who have given a benefit to him or her in the i-th round is nc,i. Before the first round starts, all players are labeled as G. Furthermore, according to the assessment rule, each donor’s reputation is reviewed based on the donors’ previous reputation (X), the recipient’s previous reputation (Y), and the donor’s behavior (Z) (see Table 7.1). The reputation of the recipient does not change because the recipient only receives donors’ help. The new reputation (G or B) is represented by the function R(X, Y, Z). In Table 7.1, the leading eight assessment rules are listed. Five of the assessment rules, S-stand, Judge, Sugden rule, Kandori rule, and Stand, have been applied in previous studies. For convenience, we named the remaining three rules: SugKan rule, KanSug rule,

7.2

Model

177

and Generous-judge (or G-judge). The common feature of the eight assessment rules is as follows: R(*, G, C) ¼ G, R(*, G, D) ¼ B, and R(G, B, D) ¼ G, in which * is either G or B (see the green columns in Table 7.1). Ohtsuki and Iwasa (2004) showed that the Co-strategy can be an evolutionarily stable strategy (ESS) against AllD players under implementation and perception errors when the assessment rule is of one of the following six types: S-stand, Judge, Sugden rule, Kandori rule, SugKan rule, or KanSug rule. The Or-strategy can be an ESS when the assessment rule is either Stand or G-judge. Hereafter, the eight strategy sets are described as (S-stand, Co), (Judge, Co), (Sugden rule, Co), (Kandori rule, Co), (SugKan, Co), (KanSug, Co), (Stand, Or) and (G-judge, Or). These eight strategy sets are termed the conditional cooperators or CCs. (S-stand, Co) is equivalent to the T1 strategy in Sugden (Sugden 1986). (Sugden rule, Co), also called simple-standing or the Sugden rule, appears in the repeated prisoner’s dilemma game (Chap. 6 of Sugden (1986); Note that different T1 strategies appear in different chapters of Sugden (1986) and correspond to RDICS (Panchanathan and Boyd 2003). (Kandori rule, Co) is called either stern-judging or the Kandori rule (Kandori 1992; Pacheco et al. 2006; Matsuo et al. 2014). CTFT in Panchanathan and Boyd (2003) and the standing strategy in Inaba et al. (2016) correspond to (Stand, Co) but not (Stand, Or). The standing strategy in Leimar and Hammerstein (2001) corresponds to Stand in this chapter. In this study, we only focus on the eight strategies out of 4096 possible strategies, as these eight strategies are ESSs against AllD players in the previous theoretical studies. The reason is as follows; our setting in the mutual-aid game is more complicated than that in the previous theoretical studies regarding the evolution of indirect reciprocity; the benefit from helping, b, and the cost of helping, c, are the only parameters controlling the dynamics in the previous theoretical studies (b > c > 0). In our study, not only b and c, but also the group size, n, and the number of repetitions, m, influence the dynamics. It takes much time to do simulation runs to investigate whether each of all possible strategies can be an ESS against AllD players and can invade the population occupied by AllD players, changing the parameters b, n, and m even in fixed c. If we had made general mathematical equations which can describe the dynamics in the mutual-aid game such as Ohtsuki and Iwasa (2004) for the one-shot game between two players, we could have saved time in analyzing the 4096 combinations. However, it is impossible to make general equations to describe the dynamics in the mutual-aid game, which is more complicated. After the m-round of the mutual-aid game is completed, the accumulated payoff of each player is calculated. When uA is the accumulated payoff of player A, uA is m m P P ΦA,i nc,i b  ΩA,i c þ u0, in which u0 is the baseline payoff. ΦA, i is one, if player i¼1

i¼1

A is chosen as a recipient in the i-th round or zero otherwise. ΩA, i is one, if player A is a donor and gives a benefit to the recipient in the i-th round or zero otherwise. nc,i is the number of cooperators in the i-th round. These definitions lead to

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The Mutual-Aid Game as an Early-Stage Insurance System

X

nc,i ¼

Ωk,i :

k2fthe donor set in the i‐th roundg

Then, we obtain the ratio of the payoff of strategy k, Uk, calculated as the total payoff of players who adopt strategy k over the total payoff of N players: Uk ¼

X A0 s strategy is k

uA =

N X

uj :

j

We introduce the mutation probability, e. If mutation does not occur with a probability of 1  e, then each player changes his or her strategy to strategy k proportional to Uk. If mutation occurs with a probability of e, then each player changes his or her strategy to one of the alternative strategies uniformly randomly. Then, one generation is completed, and the next generation begins. We also consider the effects of perception and implementation errors. Let δ be the probability of a perception error. In perception errors, when a new reputation is given to the donor after the donor’s action, all members take G (or B) for B (or G) when they observe the donor’s new reputation. It is assumed that the donor’s reputation is public information, and that all players share the same reputations of all players. Let α be the probability of an implementation error. In implementation errors, donors make a mistake and give a benefit to the recipient, even though they do not want to give a benefit, or they do not give a benefit, even though they want to. Panchanathan and Boyd (2004) and Inaba et al. (2016) pointed out that if the conditional cooperators make a mistake and do not help recipients with good standing, this can be approached mathematically. Our assumptions about errors do not converge on their assumptions. Therefore, we conducted agent-based simulations to investigate the effect of errors on evolutionary dynamics. The baseline parameters are N ¼ 1000, c ¼ 1.0, e ¼ 0.0001, δ ¼ 0, and α ¼ 0.

7.3 7.3.1

Results Eight Strategy Sets Are Categorized into Two

First, we considered the case in which players do not make any implementation or perception errors. We found that four strategy sets, (S-stand, Co), (Judge, Co), (Stand, Or), and (G-judge, Or), behave in the same manner, because their six features, R(G, G, C) ¼ G, R(G, G, D) ¼ B, R(G, B, D) ¼ G, R(B, G, C) ¼ G, R (B, G, D) ¼ B, and R(B, B, D) ¼ B, influence the outcomes. On the other hand, the common features of the leading eight do not include R(B, B, D) ¼ B. Here, S-J is used as a general term for (S-stand, Co), (Judge, Co), (Stand, Or), and (G-judge, Or); other four strategy sets, (Sugden rule, Co), (Kandori rule, Co), (SugKan, Co), and (KanSug, Co), are called non-S-J.

7.3

Results

179

Here, we explain why R(B, B, D) ¼ B is not one of the common features in the leading eight in the consecutive theoretical studies regarding the evolution of indirect reciprocity. It is assumed that two players are randomly chosen from the population, one assigned as a donor and the other as a recipient, and that these two players interact with each other only once. All players can know the reputation of all other players. At the beginning of the interaction, all players are assigned to G. We consider that S-J players are common in the population because the conditional cooperators can be an ESS but do not invade the population occupied by AllD players. Two players randomly chosen from the population are either (S-JG, S-JG) or (S-JG, AllDG), in which S-JG means an S-J player with the good standing, and AllDG means an AllD player with the good standing. The pair (AllDG, AllDG) is not chosen because there are few AllD players in the population. Then, the pair (AllDB, AllDB) is not chosen, in which AllDB means an AllD player with the bad standing. As a result R(B, B, D) does not matter. I will explain this in Sect. 7.3.4 in more detail.

7.3.2

Calculating the Expected Payoffs

When there are i players with AllD and (n  i) S-J players in a group comprising n players, and they play the mutual-aid game m times, the expected payoffs of S-J and AllD, fi(S-J) and fi(AllD), are calculated as follows: f i ðS‐JÞ ¼ m

m   ðn  i  1Þ c X i j1 i ð b  cÞ  c  , n n n j¼2 n

m  j2 X i 1 1 f i ðAllDÞ ¼ ðn  iÞb þ 2 ðn  iÞb : n n n j¼2

ð7:1aÞ ð7:1bÞ

The following provides the derivation of Eqs. (7.1a) and (7.1b). The S-J player, called player B, can get a benefit in the following ways: (i) Player B can get a benefit in the first round when player B is chosen as a recipient and S-J donors help him/her. Player B’s payoff in the first round is (n  i  1)b. (ii) Player B can get a benefit in the j-th round (m  j  2) when player B is chosen as a recipient in the j-th round and S-J donors help him/her. The payoff of player B in the j-th round is also (n  i  1)b. (iii) Player B pays a cost in the first round when not chosen as a recipient by a probability of (n  1)/n. (iv) Player B pays a cost in the j-th round (m  j  2) when (a) one of the S-J players except player B is chosen as a recipient by a probability of (n  i  1)/n, and (b) the AllD recipient has good standing in the j-th round and then player B helps the AllD recipient by probability of (1/n)(i/n)-j-1. The expected payoff of player B in the first round is (1/n)(n  i  1)b  ((n  1)/n)c, and that in the j-th round (m  j  2) is (1/n)(n  i  1)b  ((n  i  1)/n)c  (1/n)(i/n)j-1c. Therefore, the expected payoff of player B through m rounds is Eq. (7.1a).

180

7

The Mutual-Aid Game as an Early-Stage Insurance System

Table 7.2 Example of the mutual-aid game in a group which has two S-J players and three AllD players when n ¼ 5 and m ¼ 4 1st

2nd

G

S-J

G

− C

D

D

AllD

B

D

0 B

D B

D

b−c B

B

−

D

b−c G

−

D

B

D

G

B

B

D

G

AllD

D

−

Total payoff

C G

B

B

D

G

G

B

G

4th

D

G

G

AllD

G

D

G

S-J

3rd

0 B

D

0

Note: One S-J player is chosen as a recipient in the first round. C and D represent “help the recipient” and “not help the recipient,” respectively. G denotes good standing and B is bad standing. “” means the recipient

The AllD player, called player A, can get a benefit in the j-th round (m  j  2) only when player A keeps good standing even in the j-th round; player A is chosen as a recipient in the first round, one of the AllD players keeps being chosen as a recipient from the second to j  1-th round, and then player A is chosen as a recipient in the j-th round (see Table 7.2). In this case, player A is in good standing and S-J donors help player A if he/she is a recipient. Therefore, player A’s expected payoff in the first round is (1/n)(n  i)b, and that in the j-th round (m  j  2) is (1/n)2(i/n)j  2(n  i)b. If one of the S-J players is chosen as a recipient in the first round, player A changes his/her reputation to bad standing. Once AllD players are in bad standing, the reputation remains bad and S-J donors never help them. Therefore, the expected payoff of player A through m rounds is Eq. (7.1b). When we assume that n group members are randomly chosen from an infinite population, we can derive the approximate expected payoffs of S-J and AllD as follows: F SJ ¼

 n1  X n1 i

i¼0

F AllD ¼

 n1  X n1 i¼0

i

yi ð1  yÞn1i f i ðS‐JÞ,

ð7:2aÞ

yi ð1  yÞn1i f iþ1 ðAllDÞ,

ð7:2bÞ

7.3

Results

181

Fig. 7.1 S-J players vs. AllD players. a ESS condition of S-J players against AllD players. This graph shows the ratio of the number of runs in which S-J players occupy the population at the 10,000th generation relative to 100 runs, starting from an initial population occupied by S-J players. b Invasion of S-J players in the population initially occupied by AllD players. The invasion success of the S-J players is calculated as the ratio of the number of runs in which the frequency of S-J players is 1.0 for 10,000 simulation times relative to 100 runs, beginning from the initial population occupied by AllD players. In a and b, the horizontal and vertical axes represent the benefit, b, and the number of repetitions, m, respectively. The color scale bar shows the number of simulation runs in which S-J players make up the entire population divided by 100 simulation runs. Red indicates that S-J players always occupy it, and purple indicates that S-J players never occupy it. The black line in a corresponds to Eq. (7.4). In a, the initial frequency of S-J players is one; in b, this frequency is zero. The other parameters are N ¼ 1000, n ¼ 5, c ¼ 1, δ ¼ 0, α ¼ 0, and e ¼ 0.0001

where y is the frequency of players with AllD in the infinite population. Then, we obtain the replicator dynamics:

dy=dt ¼ yð1  yÞðF AllD  F SJ Þ:

ð7:3Þ

The evolutionary stable condition of S-J against AllD can be derived from Eq. (7.3) if it is assumed that y2  0 and 1 > > (1/n)m  1: m > ðn=ðn  1ÞÞ  ðb=ðb  cÞÞ:

ð7:4Þ

Inequality (7.4) shows that the region where S-J can be an ESS is larger when the group size, n, is larger (Fig. 7.1a). If n is infinite, S-J can be an ESS when m > b/ (b  c), which also predicts that the conditional cooperator can be an ESS after the group members play the mutual-aid game more than twice, when b is greater than 2c.

182

7.3.3

7

The Mutual-Aid Game as an Early-Stage Insurance System

What Happens When S-J Players Are the Majority?

Here, we examine what happens in a group when S-J players are common in the population. Each group has zero or one AllD player. When they play the mutual-aid game, S-J players never change their reputations into B without errors. In the first round, if one of the S-J players is chosen as a recipient, the S-J donors give a benefit to the S-J recipient and the AllD player does not (e.g., Table 7.2). As a result, the reputation of the AllD player is changed to B. Alternatively, in the first round, if the AllD player is chosen as a recipient, S-J donors help him or her, and the AllD player receives a benefit from them. If the AllD player is always chosen as a recipient, then his or her reputation remains G, and he or she receives a high payoff. The probability that the AllD player is always chosen as a recipient, (1/n)m, is smaller for larger n and m. Therefore, when m and n are large, the AllD players can no longer obtain a high payoff, and S-J players can resist the invasion of rare AllD players better. We also found that R(B, B, D) does not matter when S-J players are common in the population, because two AllD players are needed when R(B, B, D) works, but each group has zero or one AllD player. Even in non S-J players, R(B, B, D) does not influence the evolutionary dynamics when one of these four strategy sets is common, and so their ESS condition is the same as that of S-J.

Fig. 7.2 Comparison between the expected payoffs of S-J players and AllD players. The horizontal and vertical axes denote the benefit, b, and the number of S-J players in the entire population, respectively. The blue and red areas are calculated using Eqs. (7.2a) and (7.2b). The blue and red points are from the exact expected payoffs of S-J players and AllD players in the supplemental information in Shimura and Nakamaru (2018), respectively. The blue (red) area or points indicate that the expected payoff of S-J players is higher (lower) than that of AllD players. The other parameters are N ¼ 1000, n ¼ 5, m ¼ 10, δ ¼ 0, α ¼ 0, and c ¼ 1

7.3

Results

7.3.4

183

Can Rare S-J Players Invade the Population Occupied by AllD Players?

Equation (7.3) also shows that rare S-J cannot invade the infinite population occupied by AllD players, whereas the outcomes of the individual-based simulations show that mutation helps rare S-J increase in number and successfully occupy the finite population when AllD players are common at the beginning of the simulations (Fig. 7.1b). To understand why, we compare Eqs. (7.2a) and (7.2b) (Fig. 7.2). Figure 7.2 shows that even though the frequency of S-J is very low, the expected payoff of S-J is higher than that of AllD when b and m are large enough to promote the evolution of S-J. If mutation increases the number of S-J players, which can be higher than the threshold of the two expected payoffs being the same (see Fig. 7.2), rare S-J players can invade the population occupied by AllD players. The exact expected payoffs of S-J and AllD when n ¼ 5 can be calculated as described in Appendix B in Shimura and Nakamaru (2018), and Fig. 7.2 presents that the exact expected payoffs are the same as the approximate expected payoffs derived from Eqs. (7.2a) and (7.2b). Why is the expected payoff of S-J higher than that of AllD even with a low frequency of S-J in the population? We clarify what happens in the group when n  3. If one S-J player and n  1 AllD players are in the group, the expected payoff of S-J is always lower than that of AllD. If mutation increases the number of S-J players slightly, and there are more than two S-J players in a group (n  3), then the expected payoff of S-J is not always lower than that of AllD. For instance, imagine what happens when two S-J players and three AllD players are in the group whose size is five (Table 7.2). In the first round, one S-J player is chosen as a recipient and the other players are assigned as donors. Only one S-J donor gives a benefit to the S-J recipient. AllD donors never give benefit, and their reputation changes to B. In the second round, even though one AllD player is chosen as a recipient, two S-J donors never give a benefit to the AllD recipient, and then these two can save a cost for cooperation. Due to R(B, B, D) ¼ B, the AllD donors with B who do not help the AllD recipient with B remain as B (see red-colored columns in Table 7.2). As a result, the S-J donors can avoid helping the AllD recipient when the AllD player is chosen as a recipient again. In the third round, another AllD is chosen as a recipient. As the AllD recipient is in bad standing, two S-J donors with good standing never help the recipient and then avoid being exploited. The AllD donors never help the AllD recipient and they are still in bad standing. In the fourth round, if one S-J player is chosen as a recipient with a probability of 2/5, the S-J donor gives a benefit to the S-J recipient. Thus, two S-J players help each other, increasing their payoffs. This process indicates that R(B, B, D) ¼ B keeps the AllD players in bad standing once they are in bad standing. However, R(B, B, D) ¼ B never influences the decisionmaking of the conditional cooperators in n ¼ 2 without errors (Table 7.3). Table 7.3 gives two examples about the social interaction between one S-J player and one AllD player. Regardless of who is the first recipient, R(B, B, D) ¼ B does not appear in the

184

7

The Mutual-Aid Game as an Early-Stage Insurance System

Table 7.3 Example of the mutual-aid game in a group which has one conditional cooperator (CC), who is either a S-J player or a non S-J player, and one AllD player when n ¼ 2 and m ¼ 3. (A) The conditional cooperator is chosen as a recipient at the first round, and then the AllD player is chosen twice. (B) The AllD player is chosen in the first round, the conditional cooperators is chosen, and then the All D player is chosen again. Both (A) and (B) show that R(B, B, D) ¼ B does not appear in the game between two players without errors (A) 1st G

2nd G



CC G AllD (B)

D B

1st CC G AllD

B

3rd G

 G



B 

2nd G

C

G D



D

G

3rd G

G D

B D

B 

interaction. This indicates that at least two AllD and two S-J players in a group are needed when R(B, B, D) ¼ B can promote the evolution of cooperation. Second, we explain what happens when one AllD player is chosen as a recipient in the first round. The AllD recipient remains as G. Then, if one of the AllD players is always chosen as a recipient, the AllD player who used to be a recipient in the first round remains as G because of R(G, B, D) ¼ G. If the AllD player with G is chosen as a recipient again, S-J donors help him or her. The probability that the AllD player remains as G through m rounds, (1/n)(i/n)m  1, is lower for larger n and m, and the expected payoff of AllD players is likely to be lower. Once the AllD player with G is chosen as a donor, his or her reputation changes to B, and then S-J donors do not help him or her ever again. Therefore, the expected payoff of AllD players is smaller than the expected payoff of S-J players ( fi(S  J) > fi(AllD)) when n and m are larger. This mechanism shows that if mutation increases the number of S-J players when there are fewer S-J players initially, one group has two or more S-J players, and the S-J players can increase their payoff. If the payoff of the S-J player in the group with two S-J players and three AllD players is also higher than the expected payoff of the AllD player in the group with one S-J player and four AllD players, the S-J player begins to be selected for. Further, when mutation occurs, rare S-J players can invade the population occupied by AllD players. Therefore, R(B, B, D) ¼ B facilitates the evolution of cooperation, especially when the number of AllD players in a group is two or more.

7.3

Results

185

Fig. 7.3 Effect of group size on the invasion success of conditional cooperators. The horizontal axis denotes group size, n, with n ¼ 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, and 1000. The vertical axis corresponds to the invasion success of the S-J players. The invasion success of the S-J players is calculated as the ratio of the number of runs in which the frequency of S-J players is greater than 0.99 at the 10,000th generation relative to 100 runs, beginning from an initial population occupied by AllD players. The red solid line represents (b, m) ¼ (5, 5); the orange thick line, (10, 5); the blue dashed line, (5, 10); and the green dashed line, (10, 10). The other parameters are N ¼ 1000, c ¼ 1, δ ¼ 0, α ¼ 0, and e ¼ 0.0001

7.3.5

The Effect of Large Group Size

Now, we consider the effect of large group size on the evolution of S-J players. Figure 7.3 shows that (i) S-J can evolve even in a population initially occupied by AllD players, especially when the group sizes, b and m are larger and that (ii) the smaller group inhibits the evolution of S-J for smaller b and m. Here, we explain why the large group promotes the evolution of cooperation, which is counterintuitive from the viewpoint of previous studies on the evolution of cooperation in subpopulations in which players play the public goods game (Boyd and Richerson 1988). Basically, the same logic as in the previous paragraph can be used. To satisfy FS-J > FAllD, each group has two or more S-J players regardless of the group size. If oversimplification is allowed, two S-J players are required for N ¼ n ¼ 1000, and 400 S-J players are required for N ¼ 1000 and n ¼ 5 to increase the number in the AllD population. As a matter of course, this cannot be correct. The probability that one of two S-J players in a group whose size is 1000 is randomly chosen as a recipient (2/1000) is too small for S-J players to be chosen as recipients and to help mutually. Then, the minimum number of S-J players that need to increase in the AllD population is expected to be higher than the two players for n ¼ 1000. Even though some groups do not have S-J players in small groups, the S-J players can increase in number if the expected payoff of the S-J players is higher than that of the AllD players in the entire population. Therefore, some of the groups do not need to have at least two S-J players in small groups. Our analysis derived from the comparison between FS-J and FAllD in Eqs. (7.2a) and (7.2b) predicts that the minimum number

186

7

The Mutual-Aid Game as an Early-Stage Insurance System

Fig. 7.4 Invasion of S-J players in the experiencebased model. The horizontal and vertical axes denote the benefit, b, and the number of repetitions, m, respectively. See Fig. 7.1b for further details

of S-J players is smaller in the larger group; the minimum number of S-J is approximately 25 for n ¼ 5 and approximately 12.5 for n ¼ 1000 when b ¼ 10, m ¼ 10, and N ¼ 1000. Consequently, we can explain that larger groups promote the evolution of cooperation more.

7.3.6

Reputation vs. Experience

We show that a higher number of repetitions (m) promotes the evolution of cooperation in the mutual-aid game more (see Fig. 7.1a, b), which reminds us of the repeated game. To investigate whether the number of repetitions rather than reputation influences the results, we introduce a new model called the experience-based model, in which each player knows only his or her experiences and only the recipient can know the donors’ actions and reputations; moreover, none of the donors can observe other donors’ actions and reputation. Figure 7.4 shows that S-J players can increase in number and can occupy a population in which AllD players are initially common when b and m are very high in the experience-based model. Figs. 7.1b and 7.4 indicate that reputation rather than the number of repetitions (m) promotes the evolution of cooperation in Fig. 7.1b.

7.3.7

Why Cannot Conditional Cooperators Except S-J Players Invade the Population in n  4?

Mutation does not help the non S-J players increase in number when AllD players are common and non S-J players are rare, except when n ¼ 2 (Fig. 7.5). The reason for n  4 is as follows. When AllD donors with B do not help an AllD recipient with B, the AllD donors change their reputations into G again owing to R(B, B, D) ¼ G, which is the common feature of these four strategy sets (Table 7.4). Then,

7.3

Results

187

Fig. 7.5 The invasion of non S-J players, starting from the population occupied by AllD players. The horizontal axis denotes the group size, n, and n ¼ 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500 and 1000. The vertical axis denotes invasion success, calculated as the ratio of the number of runs, in which the frequency of the conditional co-operators is more than 0.99 at the 10,000th generation, starting from the initial population occupied by AllD players, relative to 50 runs. The graph has four overlapping lines corresponding to (Sugden rule, Co), (Kandori rule, Co), (SugKan, Co) or (KanSug, Co). The parameters are N ¼ 1000, c ¼ 1, b ¼ 10, m ¼ 10, δ ¼ 0, α ¼ 0 and e ¼ 0.0001

Table 7.4 Example of the mutual-aid game in a group which has two non S-J players and three AllD players when n ¼ 5 and m ¼ 4

1st G

non S-J

G

− G

non S-J

D

C

AllD AllD

D B

D

D

B

D G

− G

0 B

D B

D

b − 2c

−

D

D

G

B

G

b − 2c

C

C

−

Total payoff G

G

B

B

G

C

D

D

4th G

G

B

G

3rd G

G

G

AllD

2nd

2b B

D

0

Note: One non S-J player is chosen as a recipient in the first round. See Table 7.2 for more information

188

7 The Mutual-Aid Game as an Early-Stage Insurance System

donors who are conditional cooperators give a benefit to the AllD player with G and suffer from the cost of helping. Therefore, the assessment rules of these four strategy sets do not promote the evolution of cooperation more than those of S-J. However, R (B, B, D) ¼ G neither influences the decision-making of the conditional cooperators, nor then hinders the evolution of conditional cooperators for n ¼ 2 (see Table 7.3). Four or more players in a group are required when R(B, B, D) ¼ G can hinder the evolution of cooperation (Fig. 7.5).

7.3.8

Perception and Implementation Errors

Next, we examine the effects of perception and implementation errors. This investigation shows that a larger group size also promotes the evolution of conditional cooperation, especially (S-stand, Co) and (Judge, Co), even for these two types of errors. Here, we introduce perception and implementation errors to the framework. Each of the eight strategy sets can be an ESS against AllD, and the black parameter region, indicating that the strategy set is an ESS, is smaller than the region without errors (Fig. C.1 in Shimura and Nakamaru (2018)). Mutation also helps rare S-J players increase in number in the population occupied by AllD players even when errors occur (Fig. 7.6). These figures indicate that (S-stand, Co) and (Judge, Co) are advantageous over (Stand, Or) and (G-judge, Or), especially when the implementation error (α) is relatively high. The reason is as follows. We consider what happens when the perception error (δ) occurs by small probability when the majority of the group is AllD players. In these four strategies, AllD players’ reputations change into B and J-S players have good reputations without the perception error. We consider that the perception error occurs and the AllD donor’s new reputation is changed from B to G after his/her action. Then, the AllD donor is chosen as a recipient in the next round, donors regard the AllD recipient as G, and the S-J donors give a benefit to the AllD recipient. Or, we consider that the perception error changes the S-J donor’s new reputation from G to B and the S-J donor is chosen as a recipient in the next round. The donors regard the S-J recipient as B, and then, the S-J donors do not give a benefit to the S-J recipient. This occurs in four strategies and Fig. 7.6 shows that the effect of the perception error (δ) is qualitatively identical across the four strategies. Let us consider a small implementation error when the majority of the group is AllD players. In this situation, each group is dominated by AllD players and has zero or one S-J player. In (S-stand, Co) and (Judge, Co), even if an AllD donor commits a mistake and gives a benefit to an AllD recipient, the AllD donor keeps his/her bad reputation because R(B, B, C) ¼ B. In (Stand, Or) and (G-Judge, Or), if an AllD donor makes a mistake and gives a benefit to an AllD recipient, the AllD donor changes his/her reputation to G because R(B, B, C) ¼ G. In the next round, if the AllD player with G is chosen as a recipient, S-J donors give a benefit to the AllD recipient. Hence, AllD players obtain more benefit in (Stand, Or) and (G-Judge, Or)

7.3

Results

189

Fig. 7.6 The invasion success of each S-J player when perception and implementation errors occur. The horizontal and vertical axes denote the probabilities of the implementation error, α, and the perception error, δ, respectively. The graph shows the ratio of the number of runs where S-J players can occupy the population through 100,000 generations, starting from the initial population occupied by AllD players, relative to 100 runs. The gray scale presents the ratio; black means the ratio is one, and light gray means it is zero. The strategy set of S-J players is (S-stand, Co) in a, (Judge, Co) in b, (Stand, Or) in c and (G-judge, Or) in d. The parameter values are N ¼ 1000, n ¼ 5, m ¼ 8, c ¼ 1 and e ¼ 0.0001

than in (S-stand, Co) and (Judge, Co), if the implementation error occurs. Figure 7.6 supports this statement; higher implementation error inhibits the invasion of (Stand, Or) and (G-Judge, Or) but not of (S-stand, Co) and (Judge, Co). Figure 7.7 shows that the larger group size also promotes the evolution of conditional cooperation, especially (S-stand, Co) and (Judge, Co), even under the two types of errors.

190

7

The Mutual-Aid Game as an Early-Stage Insurance System

Fig. 7.7 The effect of group size on the invasion success of S-J players when perception and implementation errors occur. The horizontal axis denotes group size, n, and n ¼ 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500 and 1000. The vertical axis denotes the invasion success of the S-J players. The invasion success of the S-J players is calculated as the ratio of the number of runs in which the frequency of S-J players is over 0.99 at the 10,000th generation relative to 100 runs, starting from the initial population occupied by AllD players. The black solid line represents (S-stand, Co); the black dashed line represents (Judge, Co); the gray solid line represents (Stand, Or) and the gray dashed line represents (G-judge, Or). The other parameters are N ¼ 1000, m ¼ 10, c ¼ 1, b ¼ 10, δ ¼ α ¼ 0.025 and e ¼ 0.0001

7.3.9

If Pure Cooperators Are Added in the Population

We introduce the AllC strategy which always gives a benefit to the recipient in the population with AllD and one of the S-J strategies. First, we compare (S-stand, Co) with (Judge, Co). In the absence of errors, (S-stand, Co) and (Judge, Co) behave in a different manner when AllC exists because R(G, B, C), which is the only difference between the two, influences the dynamics. After an AllD player obtains a bad reputation, an AllC player with G gives a benefit to the AllD player with B and then the AllC player changes his/her reputation from G to B, if Judge is used. Then, the conditional cooperators do not give a benefit to the AllC player with B unless the AllC gives a benefit to the recipient with G and obtains G again. This scenario does not occur under S-stand because R(G, B, C) ¼ G. Therefore, Judge inhibits the increment of AllC players more than S-stand, and then AllD players exploit AllC players under Judge more than under S-stand. Further, conditional cooperators are advantageous over AllD players under Judge to a slightly greater extent than under S-stand. When errors are introduced, (Judge, Co) is slightly more advantageous over (S-stand, Co) (Fig. 7.8). The reason for this can be explained as follows. For example, if the implementation error occurs, an S-J donor commits a mistake and chooses defection instead of cooperation even if his/her reputation is G and the recipient’s reputation is G. According to R(G, G, D) ¼ B, the S-J player’s reputation is changed into B. We assume that the S-J player with B is chosen as a recipient in the next round. AllC donors with G help him/her. Their reputation remains G under S-stand, but changes to B under Judge, because of R(G, B, C). As S-J donors do not

7.4

Discussion and Conclusions

191

Fig. 7.8 Average frequencies of AllD players, AllC players and conditional co-operators through 100,000 generations. The horizontal and vertical axes denote generations and average frequencies over 100 runs, respectively. The thick black line represents the conditional co-operators; the thin black line represents AllC players and the dotted line represents the AllD players. The strategy set of conditional co-operators is (S-stand, Co) in a and (Judge, Co) in b. The parameters are N ¼ 1000, n ¼ 5, m ¼ 10, c ¼ 1, b ¼ 10, δ ¼ α ¼ 0.025 and e ¼ 0.0001

help the AllC recipient with B, the AllC’s expected payoff is lower under Judge than under S-stand. Consequently, the number of AllC players in the next generation decreases and then AllD players do not benefit from exploiting AllC players more than before. In sum, errors inhibit the increment of AllC players and then inhibit the increment of AllD players. If we also consider (Stand, Or) and (G-Judge, Or), the only difference between the two is R(G, B, C). Then, (G-Judge, Or) promotes the evolution of cooperation more than (Stand, Or).

7.4

Discussion and Conclusions

Group cooperation is the foundation of our society, but what kinds of systems or institutions are required to sustain cooperation in large groups? In this chapter, we investigated how the mutual-aid game, the game with the all-for-one type, influences

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7 The Mutual-Aid Game as an Early-Stage Insurance System

the evolution of group cooperation, especially in a large group (Sugden 1986). We focused on eight reputation assessment rules called the leading eight (Ohtsuki and Iwasa 2004), which have common features: R(*, G, C) ¼ G, R(*, G, D) ¼ B, and R (G, B, D) ¼ G. Even in the mutual-aid game, conditional cooperation can be an ESS against defectors under the leading eight, especially when the group size is large (inequality (7.4)). We also found that mutation helps rare conditional cooperators increase in number when defectors are initially common under four assessment rules, S-stand, Judge, Stand and G-judge, which includes R(B, B, D) ¼ B, especially when the group size is large. The reason is as follows: when the group size is four or more and each group has two defectors at least, R(B, B, D) ¼ B keeps defectors in bad standing and conditional cooperators can avoid helping defectors with B. If mutation increases the number of conditional cooperators and the group has two or more conditional cooperators by chance, they help each other and get more benefit than defectors with larger numbers of repetitions (m). As a result, rare cooperators can invade the population occupied by defectors. On the other hand, under the other four assessment rules, i.e., Sugden rule, Kandori rule, SugKan rule, and KanSug rule, R (B, B, D) ¼ G cannot keep defectors in bad standing and mutation never promotes the invasion of conditional cooperators when the group size is three or more. Sugden (1986) only considered the T1 strategy in the mutual-aid game, and here, we proved why the T1 strategy, which corresponds to (S-stand, Co), obtains a higher expected payoff than defectors. When the group size is two, R(B, B, D) does not influence the reputation dynamics, and the dynamics correspond to the repeated game between two players (see Table 7.3). Then, conditional cooperators adopting one of the eight assessment rules can increase in number when the benefit (b) and the number of repetitions (m) are high. When implementation and perception errors are introduced, two assessment rules, specifically, S-stand and Judge, including R(B, B, C) ¼ B, promote the invasion of conditional cooperators in a population occupied by defectors, especially in a large group. When AllC players are introduced to the population, the Judge assessment rule promoted the invasion of conditional cooperators more than S-stand because Judge’s feature, R(G, B, C) ¼ B, inhibits the increment of AllC players and then the increment of defectors exploiting AllC players. Therefore, we conclude that Judge promotes the evolution of conditional cooperators when errors occur and the population has AllD and AllC players.

7.4.1

Comparison with the Previous Studies About the Mutual-Aid Game

Panchanathan and Boyd (2004) presented an evolutionary analysis in a two-stage game context. They investigated whether Shunner, which chooses cooperation in the collective action game and behaves as (Stand, Co) in the mutual-aid game, can

7.4

Discussion and Conclusions

193

evolve when there are defectors who always choose defection in both games in the population. After the collective action game, the defectors’ reputations turn bad and then Shunners know the reputation of defectors. As a result, Shunners never help the recipient who is a defector in the first round of the mutual-aid game. Therefore, Shunners avoid being exploited by defectors in the two-stage game more than in our setting here. We compare inequality (7.4) in our chapter with Panchanathan and Boyd (2004). When errors do not occur, inequality (1) in Panchanathan and Boyd (2004) can be written as m > (n/(n  1))(C/(b  c)), where C is the cost of contribution in the one-shot collective action game. To compensate for the loss in the collective action game, Shunners help other Shunners mutually in the mutual-aid game. This inequality indicates how many repetitions of the mutual-aid game compensate for the Shunner’s loss in the collective action when the group size is n. This inequality is very close to inequality (7.4) in this chapter. For C > b, the parameter region in which conditional cooperators can be an ESS against defectors is wider in this chapter than in the study by Panchanathan and Boyd (2004). Otherwise, the region is narrower in this chapter than in their study. They simply assumed the implementation error whereby players make a mistake and do not help the recipient with G in the mutual-aid game. As a result, R(B, B, C) does not matter in their model. In our framework, defectors also make a mistake and choose cooperation. If this error occurs, R(B, B, C) matters; R(B, B, C) ¼ B inhibits defectors from exploiting conditional cooperators.

7.4.2

The Mutual-Aid Game as an Institution

Decision-making depends upon social systems and institutions, which tend to operate in large group contexts. Our results predict that if the structure of groups corresponds to “all-for-one type,” such as in the mutual-aid game, institutional cooperation is sustainable even in a large group. This result is expected to be applied to not only the mutual-aid system in the community but also to other systems such as common-pool resource management. Additionally, the mutual-aid game originated in the early stage of insurance provision and helps us understand how the credit system developed or evolved in a large group such as our society. Previous studies have shown that smaller groups promote the evolution of cooperation in the public goods game. This means that an institution whose structure is similar to “all-for-all type” or the public goods game cannot work for a large group. However, as Nakamaru and Yokoyama (2014) pointed out, cooperation in the public goods game is sustainable even in a large group if groups can choose members (see Chap. 6). Therefore, we shed light on the effect of both the structure of institutions and the process of choosing group members vis-à-vis the evolution of group cooperation in understanding our societal credit system, in which cooperation is sustainable in a large group.

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References Andener S (1964) The comparative study of rotating credit associations. J Roy Anthropol Inst 94: 201–229 Anderson S, Baland JM, Moene OK (2009) Enforcement in informal saving groups. J Dev Econ 90: 14–23 Axelrod R, Hamilton WD (1981) The evolution of cooperation. Science 211(4489):1390–1396 Bell F (1907) At the works. A study of a manufacturing town. Edward Arnold, London Besley T, Coate S, Loury G (1993) The economics of rotating savings and credit associations. Am Econ Rev 83:792–810 Boyd R, Richerson PJ (1988) The evolution of reciprocity in sizable groups. J Theor Biol 132:337– 356 Brandt H, Sigmund K (2004) The logic of reprobation: assessment and action rules for indirect reciprocation. J Theor Biol 231(4):475–486. https://doi.org/10.1016/j.jtbi.2004.06.032 Geertz C (1962) The rotating credit association: a “middle rung” in development. Econ Dev Cult Change 10:241–263 Inaba M, Takahashi N, Ohtsuki H (2016) Robustness of linkage strategy that leads to large-scale cooperation. J Theor Biol 409:97–107. https://doi.org/10.1016/j.jtbi.2016.08.035 Kandori M (1992) Social norms and community enforcement. Rev Econ Stud 59:63–80 Koike S, Nakamaru M, Tsujimoto M (2010) Evolution of cooperation in rotating indivisible goods game. J Theor Biol 264(1):143–153. https://doi.org/10.1016/j.jtbi.2009.12.030 Leimar O, Hammerstein P (2001) Evolution of cooperation through indirect reciprocity. Proc R Soc Lond B 268(1468):745–753. https://doi.org/10.1098/rspb.2000.1573 Matsuo T, Jusup M, Iwasa Y (2014) The conflict of social norms may cause the collapse of cooperation: indirect reciprocity with opposing attitudes towards in-group favoritism. J Theor Biol 346:34–46. https://doi.org/10.1016/j.jtbi.2013.12.018 Nakamaru M, Yokoyama A (2014) The effect of ostracism and optional participation on the evolution of cooperation in the voluntary public goods game. PLoS One 9(9):e108423. https://doi.org/10.1371/journal.pone.0108423 Nakamura M, Masuda N (2012) Groupwise information sharing promotes ingroup favoritism in indirect reciprocity. BMC Evol Biol 12:213 Ohtsuki H, Iwasa Y (2004) How should we define goodness? Reputation dynamics in indirect reciprocity. J Theor Biol 231(1):107–120. https://doi.org/10.1016/j.jtbi.2004.06.005 Pacheco JM, Santos FC, Chalub FA (2006) Stern-judging: a simple, successful norm which promotes cooperation under indirect reciprocity. PLoS Comput Biol 2(12):e178. https://doi.org/10. 1371/journal.pcbi.0020178 Panchanathan K, Boyd R (2003) A tale of two defectors: the importance of standing for evolution of indirect reciprocity. J Theor Biol 224(1):115–126. https://doi.org/10.1016/s0022-5193(03) 00154-1 Panchanathan K, Boyd R (2004) Indirect reciprocity can stabilize cooperation without the secondorder free rider problem. Nature 432:499–502 Shimura H, Nakamaru M (2018) Large group size promotes the evolution of cooperation in the mutual-aid game. J Theor Biol 451:46–56. https://doi.org/10.1016/j.jtbi.2018.04.019 Sugden R (1986) The economics of rights, co-operation and welfare. Basil Blackwell, New York Takahashi N, Mashima R (2006) The importance of subjectivity in perceptual errors on the emergence of indirect reciprocity. J Theor Biol 243(3):418–436. https://doi.org/10.1016/j.jtbi. 2006.05.014

Part III

Cooperation, Trust, and Credibility in Society

Chapter 8

Cooperation and Punishment in the Linear Division of Labor

Abstract We focus on the evolution of cooperation in linear division of labor, taking the industrial waste treatment process in Japan as an example. We consider three organizational roles and Bi is defined as the i-th role (i ¼ 1–3). The player of Bi can choose two strategies: legal treatment or illegal dumping, which can be interpreted as cooperation or defection. With legally required treatment, the player of Bj pays a cost to ask the player of Bj + 1 to treat the waste ( j ¼ 1, 2). Then, the cooperator of Bj + 1 pays a cost to treat the waste properly. With illegal dumping, the player of Bi dumps the waste and does not pay any cost. However, the waste dumped by the defector has negative environmental consequences, which all players in all roles suffer from. This situation is equivalent to a social dilemma. The administrative organ introduces two sanction systems to address the illegal dumping problem: the actor responsibility system and the producer responsibility system. We analyze this situation using the replicator equation for asymmetric games. We reveal that the producer responsibility system promotes the evolution of cooperation more than the system without sanctioning, and that the actor responsibility system does not promote the evolution of cooperation if monitoring defectors is unsuccessful. Keywords Illegal dumping · Social dilemma · Common-pool resource management · Monitoring · Replicator equation for asymmetric games

8.1

Introduction

The previous chapters in Part II, mainly focusing on the game with the all-for-one type, investigated what factors can make the informal mutual systems sustainable by the evolutionary game theory as well as the field survey and the experiment. The game with the all-for-one type is considered as one of human inventions and helps build trust and credibility among members. This chapter focuses on another human invention, the division of labor, which trust and credibility are required to maintain. In the division of labor, the different players play the different social roles. Besides the division of labor, in actual, empirical contexts we cooperate not only © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Nakamaru, Trust and Credit in Organizations and Institutions, Theoretical Biology, https://doi.org/10.1007/978-981-19-4979-1_8

197

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Cooperation and Punishment in the Linear Division of Labor

a role A1 role A2

goal

role An

role A3 role A4

b

role B1

role B2

role B3

....

role Bn

goal

Fig. 8.1 The division of labor. a shows general division of labor. Players in each role Ai (i ¼ 1, . . ., n) work to achieve their goal. b shows linear division of labor. Players in role Bi interact with those in role Bi + 1 (i < n). Then, players in the final role Bn achieve the goal

with peers, but also among players with different social roles, between a leader and a subordinate, within groups under hierarchy, or among groups which exhibit hierarchical relationships (Henrich and Boyd 2008; Powers and Lehmann 2014; Roithmayr et al. 2015; Tverskoi et al. 2021). However, many previous theoretical studies about the evolution of cooperation have assumed that players are peers. Therefore, investigating the social interaction among different players with different roles with considering the division of labor will give a new insight on “the evolution of cooperation” study. Of course, various animals, such as social insects and naked mole rats, developed division of labor and different individuals have different roles. In social insects, each individual plays various roles such as attending the mother queen, grooming larvae, guarding the nest entrance, and foraging, and which role s/he plays depends on his/her age; an individual attends the mother queen when s/he is younger than 10 days, rolls and carries mature larvae at the age of 10 days, then defends the nest when s/he is older than 14 days, which is termed temporal division of labor (Hölldobler and Wilson 1990). We consider the basic structure of the division of labor in animals, where individuals belonging to a specific role (A1, . . . or An) can work independently and cooperate together for the purposes of goal attainment (Fig. 8.1a). If some of them are not cooperators, all cannot achieve the goal. We humans have also developed division of labor (Kuhn and Stiner 2006; Henrich and Boyd 2008; Nakahashi and Feldman 2014). Humans differ from other animals in terms of their approach to the division of labor because they can innovate and create

8.1

Introduction

199

new styles of division of labor suitable for group, institutional or societal goals. Powers et al. (2016) noted that natural selection has shaped our cognitive ability in terms of language usage, a theory of mind, shared intentionality; these abilities then facilitate the development of institutions. The same logic can be applied to human division of labor since some institutions are equipped with the division of labor to run or manage institutions efficiently. We can consider that not the division of labor but the cognitive ability to innovate new styles of the division of labor is involved with natural selection. Cognitive anthropologists have discussed that episodic memory, one of the high cognitive abilities typically evolved and developed in humans, but not in Neanderthals, is required to innovate age and gender divisions thereof. It is because episodic memory not only stores and retrieves past events but also makes future planning or simulating future scenarios possible; to innovate or maintain the division of labor requires such ability (Coolidge and Wynn 2008). Cognitive sciences, some branches of anthropology, and natural selection fall under the domain of biology, and therefore, the division of labor in human society can be studied from the perspective of biology.

8.1.1

The Linear Division of Labor

There are various types of division of labor existing in our society and we focus on one such type, illustrated in Fig. 8.1b, the linear division of labor. Well-known examples thereof include the car assembling process and the manufacturing process of some traditional crafts such as kimonos and Buddhist altars in Japan (Ohnuma, personal communication). In Fig. 8.1b, a player in role B1 cooperates to achieve the goal and works towards that goal accordingly. After s/he completes his/her work, s/he brings the product to a player in role B2. Then, the player in B2 completes further goal-oriented work. After s/he finishes it, s/he brings it to a player in role B3. Role Bj is dependent on role Bi (n  j > i). This process is repeated and then a player in role Bn produces the final product. The final product is a goal of the division of labor and the quality influences all players in all roles. For example, if the quality is good, the price is expensive and the group members obtain a good reputation. However, if players are not cooperative and the final product is bad, they cannot accrue the desirable benefits. However, cooperation is costly; if a player in Bi is not a cooperator, a player in Bj is a cooperator and these two roles are similar, the player in Bj can compensate for the imperfection of the player in Bi ( j > i), and the final product may be good. As a result, the player in Bi does not need to pay a cost for doing a good job and get a high benefit from the good final product. However, if a cooperative player in Bj cannot compensate for the imperfection of a player in Bi because of the high specialty in each role, the final product may not be good. As a result, no players accrue desirable benefits from the final product. The division of labor has a special feature. If a player in Bi knows the reputation of players in Bi + 1, a player in Bi can choose a cooperator in Bi + 1. Then, players in both Bi and Bi + 1 would accrue desirable benefits. However, if a player in Bj

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8 Cooperation and Punishment in the Linear Division of Labor

( j > i + 1), who is in the far position from a player in Bi, does not choose a cooperator in Bj + 1, the quality of the final product becomes low and then all players in all roles do not accrue desirable benefits even though the player in Bi chose a cooperator in Bi + 1. This means that a player who the focal player has not known yet possibly hinder cooperation in the whole system. While, if a player can distinguish cooperators from defectors and only helps the cooperators, cooperation can evolve in the bilateral relationship. What kinds of system promote the evolution of cooperation in linear division of labor? We focus on the effect of sanctions and monitoring. Previous empirical and theoretical studies have shown that monitoring and/or sanctions inhibit the violators who break institutional rules in common-pool resource management contexts, such as with forest logging, in order to run the institution efficiently (Ostrom 1990; Rustagi et al. 2010; Chen et al. 2015; Lee et al. 2015). After successful monitoring detects violators, sanctions can be imposed on them. Monitoring is effective in small villages where people have knowledge of the behavior of others, through direct observation and gossip (see Chap. 9). However, detecting violators is sometimes hard work and it is very costly in larger societies because it is almost impossible to have knowledge of the behavior of all members in society. Then, sanctions cannot be imposed on the violators. Consequently, both monitoring and sanctions are not meaningful anymore. For example, detecting illegal logging far from human habitation deep in the mountains to which there is no access by roads is almost impossible and monitoring does not work. How do we deal with this situation?

8.1.2

The Industrial Waste Treatment Process in Japan

Here, we take the industrial waste treatment process in Japan as an example of the linear type of division of labor, and then investigate the effect of sanctions and monitoring on the evolution of cooperation, using the industrial waste illegal dumping game which Ohnuma and Kitakaji proposed based on their field survey and government publications (Ohnuma and Kitakaji 2007; Kitakaji and Ohnuma 2014; Kitakaji and Ohnuma 2016; Ohnuma 2021). In what follows, the industrial waste treatment process in Japan based on their experimental works (Ohnuma and Kitakaji 2007; Kitakaji and Ohnuma 2014; Kitakaji and Ohnuma 2016; Ohnuma 2021) is explained. The industrial waste treatment process consists of five roles: generators (B1), the first waste haulers (B2), the intermediate treatment facilities (B3), the second waste haulers (B4) and the landfill sites (B5). The generators produce industrial waste as a secondary product, and commit the waste to the first waste haulers. When the first waste haulers can commit the waste to the intermediate treatment facilities, the first waste haulers bring it to the intermediate treatment facilities. The intermediate treatment facilities crush the waste, treat it chemically, or incinerate it. Then, the intermediate treatment facilities decide to commit it to the second waste hauler. When the second waste haulers can commit it to the landfill sites, they bring it to the landfill sites. The landfill sites dispose of it in sanitary

8.1

Introduction

201

landfills. In the industrial waste treatment process, cooperation means that a player in Bi commits the waste to a player in Bi + 1, paying a commission cost, or a player in B3 also treats waste, paying a treatment cost. If all players in all roles cooperate, the volume of waste is reduced and its risk, such as its toxicity, is removed; then the waste is landfilled safely and does not deleteriously impact the natural environment. Therefore, the final product in the industrial waste treatment process is the safely landfilled waste. Defection means that a player in Bi does not commit the waste to a player in Bi + 1 but illegally dumps the waste far from human habitation deep in the mountains, which are hard to access (1  i  4). Defectors do not need to pay a commission cost or a treatment cost. Once defection occurs, illegal dumped waste damages the natural environment. Hence, the final product in this case is the environmental damage and all players suffer from the damage. Actually, if the local administrative organ detects the damage caused by the illegal dumped waste but cannot know which player dumped the waste, the organ forces all players in all roles to pay for restoration. Players in all roles reserve a fund in advance to pay for future restoration. This situation in the industrial waste treatment process is interpreted as a social dilemma (Ohnuma and Kitakaji 2007; Kitakaji and Ohnuma 2014; Kitakaji and Ohnuma 2016; Ohnuma 2021). We explain the reason as follows. If all players in all roles are cooperators, they pay a cost, such as a commission cost and a treatment cost, but do not need to pay for restoration. However, if a player in B1 chooses defection, the player does not need to pay a commission cost. Not only the defector in B1 but also other players in Bj ( j > 1) have to pay for restoration. If players in B1 and B2 are cooperators and a player in B3 is a defector, players in B1 and B2 have to pay a commission cost as well as for restoration. Players in B4 and B5 also have to pay for restoration even though they are cooperators. If the restoration cost is expensive, the payoff when all players choose cooperation can be higher than the payoff when a player in B1 chooses defection. However, if a player changes behavior from cooperation to defection in Bi (1  i  5), the defector gets a higher payoff than the cooperator because the defector does not need to pay for a cost such as a commission cost or a treatment cost. To inhibit illegal dumping, two sanction systems are put into effect in Japan (Ohnuma and Kitakaji 2007; Kitakaji and Ohnuma 2014; Kitakaji and Ohnuma 2016; Ohnuma 2021). Here, we term these two systems the actor responsibility system and the producer responsibility system. In the actor responsibility system, the local administrative organ attempts to detect the illegal dumping. As many firms illegally dump the industrial waste deep in the mountains or in rivers far from habitation, the organ hardly detects the waste. When the organ luckily detects the waste, s/he has to detect who illegally dumped it. To detect who dumped it is very hard work. If the organ detects who dumped it, the firm is fined; the maximum fine in Japan is 100 million yen, which is equivalent to one million US dollars. Previous theoretical studies concerning the evolution of pool-punishment also make the same assumption: pool-punishers detect the violators and punish them, and pool-punishers do not fail to detect these violators (Sigmund et al. 2010).

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Cooperation and Punishment in the Linear Division of Labor

However, as the local administrative organ had difficulty in monitoring and detecting the illegal dumping, a new system was introduced in 1990, which we call the producer responsibility system herein. In this system, a manifest is important. The local administrative organ prepares the manifest, which all players in all roles have to fill in when committing waste. Then, after the generator fills in the manifest, it is handed to the first waste hauler. After the first waste hauler fills in the manifest, it is handed to the intermediate treatment facility. Then, after the intermediate treatment facility hands it to the second waste hauler, the second waste hauler fills it in and hands it to the landfill site. Next, the landfill site fills it in and hands it back to the second waste hauler, who then hands it back to the intermediate treatment facility. This process continues and finally the generator hands it to the local administrative organ. If the generator fails to hand it back to the local administrative organ, the local administrative organ punishes the generator and the generator has to pay a fine, even though another player in another role does not fill in it and hand it back. Data from the Ministry of the Environment in Japan shows that the total number of illegal dumping activities detected annually increased directly after introducing the manifest system; the number has subsequently decreased since 2000 (see http:// www.env.go.jp/press/103219.html). Does the data show that the new system inhibits the number of illegal dumping activities? Or, does the data only show that monitoring fails to detect illegal dumping because illegal dumping is more secret than before? Consecutive experimental studies by Ohnuma and Kitakaji tackled this question (Ohnuma and Kitakaji 2007; Kitakaji and Ohnuma 2014; Kitakaji and Ohnuma 2016; Ohnuma 2021). They showed that either monitoring or sanctions in the actor responsibility system did not prevent illegal dumping under the producer responsibility system (Kitakaji and Ohnuma 2014). In this chapter, using the replicator equation in evolutionary game theory, we investigate the effect of either of two sanction systems on the evolution of cooperation in the industrial waste process in Japan as an example of linear division of labor (Nakamaru et al. 2018). Generally, firms pursue profits and change tactics or strategy based on profit considerations. They may imitate the strategy of others to accrue more profits. Therefore, replicator dynamics, which can be interpreted as a social learning model, is a useful tool to describe the behavior of firms.

8.2

Models

It is a complex task to construct a mathematical model assuming five roles in the linear division of labor. We consider three cases: (i) There are only generators which can treat the industrial waste after producing the product. We call this model the no-role model. See Appendix A in Nakamaru et al. (2018) for model assumptions and results. (ii) There are generators and landfill sites; this is the two-role model (see Sect. 8.3.3 in more details). (iii) There are generators, intermediate treatment facilities, and landfill sites; this is the three-role model.

8.2

Models

203

In the following, we delineate the assumptions of the three systems in the threerole model: the baseline system, the actor responsibility system, and the producer responsibility system. The reason that we consider the baseline system, which has no sanctions, is to facilitate examination and comparison of the effects of two sanction types on the evolution of cooperation in linear division of labor.

8.2.1

Baseline System in the Three-Role Model

We consider that there are three roles: the generator group, the intermediate treatment facility (ITF) group, and the landfill site (LS) group. The player is the firm. A player in the generator group, called a generator, plays a generator’s role. One in the ITF group, called an ITF, plays an ITF’s role. One in the LS group, called an LS, plays an LS’s role. Each group has an infinite number of players. The generator is either a cooperator or a defector. The cooperator commits industrial waste to the ITF, and the defectors dump the waste illegally. The generator’s benefit from production is b. If the generator is a cooperator, the commission cost is x1 (b > x1 > 0). Hereafter, the cooperators and defectors in the generator group are termed g-cooperators and g-defectors, respectively. The ITF obtains the commission cost of the g-cooperator, x1, as his/her benefit. Then, the ITF has to choose either cooperation or defection in each of two stages. S/he can choose either cooperation or defection in the first stage. Cooperation in the first stage (cooperation-1) indicates that s/he treats the industrial wastes brought from the generator by paying an intermediate treatment cost, cmid. An example of cooperation-1 is breaking waste into pieces, treating it chemically, and rendering it harmless. Defection in the first stage (defection-1) means that the ITF does not treat the waste. After the ITF makes the decision in the first stage, s/he chooses either cooperation or defection in the second stage. Cooperation in the second stage (cooperation-2) means committing the industrial waste to a landfill site, and defection in the second stage (defection-2) means illegal dumping. Let the C-C player, the C-D player, the D-C player and the D-D player be defined as the ITF choosing cooperation-1 and cooperation-2, one choosing cooperation-1 and defection-2, one choosing defection-1 and cooperation-2, and one choosing defection-1 and defection-2, respectively. The C-C player pays the commission cost, x2. The commission cost of the D-C player is x20 . We assume x20  x2, because the D-C player does not treat industrial waste at all, and the LS pays a greater cost for treating the waste than the waste from the C-C player. The C-D or D-D player does not pay any commission cost. The LS can harness the commission cost of ITF (x2 or x20 ) as his/her profit. The LS can be either a cooperator or a defector. Defectors dump the waste illegally and do not pay any cost for treatment. The cooperator buries the waste in the landfill that the LS possesses, paying a treatment cost (ct or ct0 ; x1 > ct, ct0 ). If the ITF has treated the waste, the treatment cost is ct. If not, the treatment cost is ct0 . The cooperator incurs a greater cost in burying non-treated waste than treated waste because

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8 Cooperation and Punishment in the Linear Division of Labor

non-treated waste is larger or more dangerous than treated waste. Therefore we assume that ct0  ct > 0. Hereafter, cooperators and defectors in the LS group are termed ls-cooperators and ls-defectors, respectively. The g-defectors, C-D and D-D players or ls-defectors illegally dump industrial waste in places such as rivers, seas, mountains, and forests, and they do damage to the environment in the form of water pollution, soil pollution, and other types of pollution. Consequently, the population size of animals and plants will decrease in the future. As a result, all players equally suffer from the damage. To prevent future environmental damage, the local administrative organ mandates all players to pay an environmental restoration fee. Illegal dumping by a g-defector damages the natural environment and g0 is defined as the amount of industrial waste dumped by a g-defector. Illegal dumping by a C-D player damages the natural environment and g1 is defined as the amount of industrial waste dumped by a C-D player. The amount of industrial waste dumped by a D-D player is g0 because the D-D player has not treated the industrial waste from the g-cooperator. We assume that g0  g1, because treatment by the ITF can reduce the amount of waste. If the industrial waste has not been treated at all, the amount of waste dumped by an ls-defector is still g0. If the industrial waste has been treated, the amount of waste dumped by an ls-defector is g1. Let r be defined as the environmental restoration fee or damage per unit of waste. Therefore, rgi is the restoration fee (i ¼ 0 or 1) to restore the damaged environment. We consider that the local administrative organ determines that the generator group is forced to pay a fee s times as high as the ITF and the LS group (s  1). If a player dumps industrial waste illegally, srgi is imposed on a generator and rgi is imposed on an ITF or an LS. As illegal dumping damages all players’ utility or health, the problem of illegal dumping can be interpreted as a social dilemma problem. The parameters in the system are listed in Table 8.1. The payoff matrix of generators when the LS is an ls-cooperator is:
A1 ¼

b  x1

b  x1  srg1

b  x1

b  x1  srg1

b  srg0

b  srg0

b  srg0

b  srg0

 ,

in which the element a1j in A1 is the payoff of g-cooperators committing the industrial waste to the ITF whose type is j ( j ¼ 1–4). The ITF’s strategies 1, 2, 3, and 4 correspond to a C-C player, a C-D player, a D-C player, and a D-D player, respectively. The element a2j is the payoff of g-defectors ( j ¼ 1–4). The payoff matrix of generators when the LS is an ls-defector is:
A2 ¼

b  x1  srg1

b  x1  srg1

b  x1  srg0

b  x1  srg0

b  srg0

b  srg0

b  srg0

b  srg0

 ,

in which the element a1j in A2 is the payoff of cooperators of the generator committing the industrial waste to the ITF whose type is j ( j ¼ 1–4). The element a2j in A2 is the payoff of defectors in the generator ( j ¼ 1–4).

8.2

Models

205

Table 8.1 Parameters in the three systems in the three-role model b x1 cmid x2 x20 ct ct0 g0 g1 r s f d f1 p2

p3

p4

q3 q4

Generator’s benefit from production Commission cost of the g-cooperator Intermediate treatment cost of the C-C or C-D player Commission cost paid by the C-C player Commission cost paid by the D-C player Treatment cost of an ls-cooperator when the ITF has treated the waste Treatment cost of an ls-cooperator when the ITF has not treated the waste Amount of industrial waste when a g-defector or a D-D player dumps the waste or a ls-defector dumps the waste from a D-C player Amount of industrial waste when a C-D player dumps the waste or when a ls-defector dumps the waste from a C-C player Environmental restoration expense or damage per unit of waste dumped illegally Generator group is forced to pay for the environmental restoration expense s times more than the ITF and LS group Fine in the actor responsibility system model Probability of being detected by the local government in the actor responsibility system model Fine in the producer responsibility system model

65 s 45 s 10 s 20 s 40 s 10 s 20 s 100 tons 50 tons 0.1 4 30 s 0.0065 100 s/ 50 s

Probability that the g-defector does not hand the manifest to the local government in the producer responsibility system model 1  p2, the probability that the g-defector hands back a fictitious manifest Probability that the C-D or D-D player does not hand the manifest back to the generator in the producer responsibility system model 1  p3, the probability that the C-D or D-D player hands back a fictitious manifest Probability that the ls-defector does not hand the manifest back to the ITF in the producer responsibility system model 1  p4, the probability that the ls-defector hands back a fictitious manifest Probability that the C-C or D-C player does not hand the manifest back to the generator in the producer responsibility system model (very low q3) Probability that the ls-cooperator does not hand the manifest back to the ITF in the producer responsibility system model (very low q4)

Note: The rightmost column indicates the estimated values based on Kitakaji and Ohnuma (2014) when the industrial waste is 100 tons (see Sect. 8.3 for more information)

The payoff matrix of the ITF interacting with the ls-cooperator is C1, and that of the ITF interacting with the ls-defector is C2. They present:

206

8

Cooperation and Punishment in the Linear Division of Labor

0

x1  cmid  x2 B x  c  rg mid B 1 1 C1 ¼ B @ x1  x2 0 0

x1  rg0

1 rg0 rg0 C C C, rg0 A rg0

x1  cmid  x2  rg1

rg0

x1  rg0

rg0

B x  c  rg mid B 1 1 C2 ¼ B @ x1  x2 0  rg0

1

rg0 C C C, rg0 A

in which the (i, 1) element in C1 and C2 is the payoff of ITF’s strategy i interacting with a g-cooperator and the (i, 2) element in C1 and C2 is the payoff of ITF’s strategy i interacting with a g-defector. The payoff matrices of landfill sites when the generator chooses cooperation (committing the waste to ITF) and defection (illegal dumping) are B1 and B2, respectively,


 rg1 x2 0  ct 0 rg0 x2  ct , B1 ¼ x2  rg1 rg1 x2 0  rg0 rg0
 rg0 rg0 rg0 rg0 , B2 ¼ rg0 rg0 rg0 rg0 in which the (1, j) element in B1 and B2 is the payoff of the ls-cooperator interacting with ITF’s strategy j and the (2, j) element in B1 and B2 is the payoff of the ls-defector interacting with ITF’s strategy j. Here we assume that each group size is infinite. One generator is chosen randomly from the generator group and interacts with one ITF chosen randomly from the ITF group, following the model assumptions. Then, the ITF interacts with one LS chosen randomly from the LS group, following the model assumptions. We assume an absence of intra-group interaction. Therefore we can apply the replicator equation of an asymmetric game to our model, and the time-differential equations are h i h i du1 ! ! ! ! ! ! ¼ v1 u1 A1 z 1  u A1 z þ ð1  v1 Þu1 A2 z 1  u A2 z , dt h  i dzi ! ! ! ¼ v1 z i C 1 u  z C 1 u dt i h  i ! ! ! þ ð1  v1 Þzi C 2 u  z C 2 u ði ¼ 1  3Þ,

ð8:1bÞ

h i h i dv1 ! ! ! ! ! ! ¼ u1 v1 B1 z 1  v B1 z þ ð1  u1 Þv1 B2 z 1  v B2 z , dt

ð8:1cÞ

i

ð8:1aÞ

8.2

Models

207

in which we let u1 be the frequencies of the g-cooperators in the generator group; u2, the frequencies of the g-defectors (u1 + u2 ¼ 1; u1, u2  0). Let zi be the frequency of players with the ITF’s strategy i (i ¼ 1–4; z1 + z2 + z3 + z4 ¼ 1; z1, z2, z3, z4  0). Let v1 be the frequencies of the ls-cooperators in the landfill site group; v2, the frequencies of the ls-defectors in the landfill site group (v1 + v2 ¼ 1; v1, v2  0). We assume ! ! ! that u ¼ (u1 u2)t, z ¼ (z1 z2 z3 z4)t, and v ¼ (v1 v2)t.

8.2.2

Actor Responsibility System in the Three-Role Model

If the local administrative organ successfully monitors and detects the illegal dumping and the illegal dumper, the local administrative organ punishes the dumper and then forces the dumper to pay a fine, f. Let d be the probability of being monitored and detected by the local administrative organ. We call this sanction the actor responsibility system. The parameters in the system are listed in Table 8.1. The local administrative organ has difficulty in monitoring perfectly and detecting the illegal dumping and the illegal dumper because the illegal dumper dumps the waste in inconspicuous places such as deep in the mountains, destroying evidence. As a result, perfect monitoring is very costly and d is very low. However, the fine f is very expensive. For example, f can be 100 million yen which is roughly equivalent to one million US dollars. The expected value of the fine which the dumper has to pay, df, is added to the baseline system, and then the payoff matrices in Eqs. (8.1a), (8.1b) and (8.1c) are replaced by:
A1 ¼
A2 ¼

b  x1

b  x1  srg1

b  x1

b  x1  srg1

b  srg0  df

b  srg0  df

b  srg0  df

b  srg0  df

b  x1  srg1 b  srg0  df

b  x1  srg1 b  x1  srg0 b  x1  srg0 b  srg0  df b  srg0  df b  srg0  df 0 1 x1  cmid  x2 rg0 B x  c  rg  df rg C mid B 1 1 0C C1 ¼ B C, @ x1  x2 0 rg0 A 0

x1  rg0  df

x1  cmid  x2  rg1 B x  c  rg  df mid B 1 1 C2 ¼ B @ x1  x2 0  rg0 x1  rg0  df

rg0

1 rg0 rg0 C C C, rg0 A rg0

 ,  ,

208

8


B1 ¼

8.2.3

Cooperation and Punishment in the Linear Division of Labor

x 2  ct x2  rg1  df
rg0 B2 ¼ rg0

rg1 rg1 rg0 rg0

 x2 0  ct 0 rg0 , x02  rg0  df rg0  rg0 rg0 : rg0 rg0

Producer Responsibility System in the Three-Role Model

Herein, the local administrative organ prepares the manifest, which all industries have to fill in. Then after the producer or the generator fills in the manifest, it is handed to the ITF. After the ITF fills in the manifest, it is hand to the LS. Then, the LS hands it back to the ITF, who then hands it back to the generator. Next, the generator hands it to the local administrative organ. If the generator fails to hand it back to the local administrative organ, the local administrative organ punishes the generator and s/he has to pay a fine, f1. Reasons why the generator cannot hand the manifest back to the local administrative organ are: (i) The generator is a g-defector, and then s/he cannot hand the manifest to the ITF because there is no transaction between him/her and the ITF. Let p2 be the probability that the g-defector does not hand the manifest to the local administrative organ. The probability 1  p2 means that the g-defector hands a fictitious manifest to the local administrative organ. (ii) The ITF is a C-D or D-D player and the generator does not receive the manifest from the ITF. Let p3 be the probability that the C-D or D-D player does not hand the manifest back to the generator. The probability 1  p3 means that the C-D or D-D player hands a fictitious manifest to the generator. (iii) The LS is an ls-defector and does not hand the manifest back to the ITF. As a result, the generator does not receive the manifest. Let p4 be the probability that the ls-defector does not hand the manifest back to the ITF. The probability 1  p4 means that the ls-defector hands a fictitious manifest to the ITF. For simplicity, we assume that the local administrative organ does not distinguish the genuine manifest from the fictitious one. We also consider that the g-cooperator always hands the manifest back to the local administrative organ, and that the cooperator in the ITF or LS group accidentally fails to hand back the manifest. (i) Let q3 (1  q3) be the probability that the C-C or D-C player does not (does) hand the manifest back to the generator. (ii) Let q4 (1  q4) be the probability that the ls-cooperator does not (does) hand the manifest back to the ITF. We can calculate the probabilities that the g-cooperator does not hand the manifest back to the local administrative organ. (i) q4 + (1  q4)q3 is the probability that the g-cooperator does not receive the manifest when there is a transaction with a C-C or D-C player and the ls-cooperator. (ii) p4 + (1  p4)q3 is the probability that the g-cooperator does not receive the manifest when there is a transaction with a C-C

8.3

Results

209

or D-C player and the ls-defector. (iii) p3 is the probability that the g-cooperator does not receive the manifest from the C-D or D-D player when there is a transaction between them. If (i)  (iii) occur, the local administrative organ punishes the g-cooperator, who has to pay a fine, f1. The parameters in the system are listed in Table 8.1. The expected value of the fine which the generator has to pay is added to the baseline system, and then the payoff matrices in Eq. (8.1) are replaced by


 bx1 srg1 f 1 p3 bx1 F 1 bx1 srg1 f 1 p3 bx1 F 1 , A1 ¼ bsrg0 f 1 p2 bsrg0 f 1 p2 bsrg0 f 1 p2 bsrg0 f 1 p2 in which F1 ¼ f1(q4 + (1  q4)q3),
A2 ¼

bx1 srg1 F 2 bx1 srg1 f 1 p3 bx1 srg0 F 2 bx1 srg0 f 1 p3 bsrg0 f 1 p2

bsrg0 f 1 p2

bsrg0 f 1 p2

bsrg0 f 1 p2

 ,

in which F2 ¼ f1( p4 + (1  p4)q3), 0

x1  cmid  x2

rg0

x1  rg0

rg0

B x  c  rg mid B 1 1 C1 ¼ B @ x1  x2 0 0

rg0 C C C, rg0 A

x1  cmid  x2  rg1

rg0

x1  rg0

rg0

B x  c  rg mid B 1 1 C2 ¼ B @ x1  x2 0  rg0
B1 ¼

1

x2  ct

rg1

x2 0  ct 0

1

rg0 C C C, rg0 A rg0

x2  rg1 rg1 x2 0  rg0 rg0
 rg0 rg0 rg0 rg0 : B2 ¼ rg0 rg0 rg0 rg0

 ,

Next, we calculate the equilibrium points in three systems and conduct the local stability analysis. We also did Monte Carlo simulations and we obtained almost the same results as our following analysis of the replicator dynamics.

8.3

Results

We analyze Eqs. (8.1a), (8.1b) and (8.1c) and obtain the equilibrium points in three systems, which are categorized into seven types:

210

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Cooperation and Punishment in the Linear Division of Labor

Table 8.2 The local stability of seven equilibrium points in each of the three systems of the threerole model Equilibrium points (u1, z1, z2, z3, v) ¼ (G, ITF, LS) (1, 1, 0, 0, 1) ¼ (C, C-C, C)

(1, 0, 0, 0, *) ¼ (C, D-D, *)

Baseline system Stable/ unstable Stable/ unstable Stable/ unstable Stable/ unstable Unstable

(1, 0, 0, 1, 0) ¼ (C, D-C, D)

Unstable

(1, 1, 0, 0, 0) ¼ (C, C-C, D)

Unstable

(1, 0, 0, 1, 1) ¼ (C, D-C, C) (0, *, *, *, *) ¼ (D, *, *) (1, 0, 1, 0, *) ¼ (C, C-D, *)

Actor responsibility system Stable/ unstable Stable/ unstable Stable/ unstable Stable/ unstable Stable/ unstable Stable/ unstable Stable/ unstable

Producer responsibility system Stable/unstable Stable/ unstable Stable/unstable Stable/unstable Stable/ unstable Unstable

Interpretation Cooperation in the division of labor Illegal dumping of G Illegal dumping of ITF

Illegal dumping of LS

Unstable

Note: C denotes cooperation; D, defection. “Stable/unstable” means that some parameters cause the equilibrium point to be locally stable and others make it unstable; “unstable” means that the equilibrium point is always locally unstable

ðu1 , z1 , z2 , z3 , vÞ ¼ ð1, 1, 0, 0, 1Þ, ð1, 0, 0, 1, 1Þ, ð0,   ,   Þ, ð1, 0, 1, 0 Þ, ð1, 0, 0, 0 Þ, ð1, 0, 0, 1, 0Þ, ð1, 1, 0, 0, 0Þ, in which the asterisk “*” means any value between 0 and 1. Equations (8.1a), (8.1b) and (8.1c) shows that there is one inner equilibrium point besides seven types of equilibrium points in either the baseline system or the producer responsibility system: (u1, z1, z2, z3, v) ¼ (1, (rg0  ct0)/(r(g0  g1) + ct  ct0), 0, (rg1 + ct)/(rr(g0  g1) + ct  ct0), 1  (cmid + x2  x20)/(r(g0  g1))). One inner equilibrium point in the actor responsibility system is: (u1, z1, z2, z3, v) ¼ (1, (df + rg0  ct0)/(r g 1) + ct  ct0), 0, (df  rg1 + ct)/(r(g0 0 r(g0  g1) + ct  ct ), 1  (cmid + x2  x20)/(r(g0  g1))). The numerical calculations show they are unstable. These seven equilibrium points indicate that all members adopt the same strategy in each industry in equilibrium, and then (u1, z1, z2, z3, v) can be represented by (G, ITF, LS) which is the strategy of the G, the ITF and the LS, respectively. For example, (1, 1, 0, 0, 1) is equivalent to (C, C-C, C), in which the first column denotes that the generator is a g-cooperator, the second that the ITF is a C-C player, and the third that the LS is an ls-cooperator. Table 8.2 shows the seven equilibrium points and the corresponding strategies. Out of seven equilibrium points, two points, (1, 1, 0, 0, 1) and (1, 0, 0, 1, 1) are interpreted as cooperation evolving in the division of labor (see Table 8.2). (0, *, *, *, *) is interpreted as illegal dumping by G; two

8.3

Results

211

Table 8.3 The local stability conditions of (1, 1, 0, 0, 1) ¼ (C, C-C, C) in each of the three systems Baseline model x1 < srg0 cmid + x2  x20 < 0 cmid + x2  rg0 < 0 x2  rg1 < 0 ct  rg1 < 0

Actor responsibility system x1  df < srg0 cmid + x2  x20 < 0 cmid + x2  rg0  df < 0 x2  rg1  df < 0 ct  rg1  df < 0

Producer responsibility system x1 + f1(q3  p2 + q4(1  q3)) < srg0 cmid + x2  x20 < 0 cmid + x2  rg0 < 0 x2  rg1 < 0 ct  rg1 < 0

Table 8.4 The local stability conditions of (1, 0, 0, 1, 1) ¼ (C, D-C, C) in each of the three systems Baseline model x1 < srg0 cmid + x2  x20 > 0 cmid  x20 + rg1 > 0 x20  rg0 < 0 ct0  rg0 < 0

Actor responsibility system x1  df < srg0 cmid + x2  x20 > 0 cmid  x20 + rg1+ df > 0 x20  rg0  df < 0 ct0  rg0  df < 0

Producer responsibility system x1 + f1(q3  p2 + q4(1  q3)) < srg0 cmid + x2  x20 > 0 cmid  x20  rg1 > 0 x20  rg0 < 0 ct0  rg0 < 0

Table 8.5 The local stability conditions of (0, *, *, *, *) ¼ (D, *, *) in each of the three systems Baseline model sr(g0  g1)(z1* + z2*) + v*sr(g1z1* + g0z3*)  x1 < 0

Actor responsibility system sr(g0  g1)(z1* + z2*) + v*sr(g1z1* + g0z3*)  x1 + df < 0

Producer responsibility system sr(g0  g1)(z1* + z2*) + v*sr(g1z1* + g0z3*)  x1  f1(z1* + z3*) {p4 + (1  p4)q3  p3 + v*(q4  p4) (1  q3)}  f1( p3  p2) < 0

Table 8.6 The local stability conditions of (1, 0, 1, 0, *) ¼ (C, C-D, *) in each of the three systems Baseline model x1  sr(g0  g1) < 0 r(g0  g1) > cmid cmid + r(g1  g0) + rg0v*  x20 < 0  x2 + rg1 v* < 0

Actor responsibility system x1  sr(g0  g1)  df < 0 r(g0  g1) > cmid cmid + r(g1  g0) + rg0v*  x20 + df < 0  x2 + rg1 v* + df < 0

Producer responsibility system x1  sr(g0  g1) + f1( p3  p2) < 0 r(g0  g1) > cmid cmid + r(g1  g0) + rg0v*  x20 < 0  x2 + rg1 v* < 0

points, (1, 0, 1, 0, *) and (1, 0, 0, 0, *), illegal dumping by ITF; two points, (1, 0, 0, 1, 0), (1, 1, 0, 0, 0), illegal dumping by LS. We conducted the local stability analysis of the seven points. The following shows the condition of the local stability of each point. The stability condition of seven equilibrium points in the baseline model, the actor responsibility system model, and the producer responsibility system model (see Table 8.2) are calculated using Eqs. (8.1a), (8.1b) and (8.1c); we calculate the eigenvalues for each equilibrium point, and they are real numbers. Therefore, when all eigenvalues for one equilibrium point are negative, the equilibrium is locally stable. Tables 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, and 8.9 show the condition that all eigenvalues are negative, which is the local stability condition of each equilibrium point.

212

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Cooperation and Punishment in the Linear Division of Labor

Table 8.7 The local stability conditions of (1, 0, 0, 0, *) ¼ (C, D-D, *) in each of the three systems Baseline model x1 < 0 r(g0  g1) < cmid cmid + x2 + r(g1  g0)  rg1v* > 0  x20 + rg0 v* < 0

Actor responsibility system x1  df < 0 r(g0  g1) < cmid cmid + x2 + r(g1  g0)  rg1v*  df > 0  x20 + rg0 v* + df < 0

Producer responsibility system x1 + f1( p3  p2) < 0 r(g0  g1) < cmid cmid + x2 + r(g1  g0)  rg1v* > 0  x20 + rg0 v* < 0

Table 8.8 The local stability conditions of (1, 0, 0, 1, 0) ¼ (C, D-C, D) in each of the three systems Baseline model x1 < 0

Actor responsibility system x1  df < 0

x20 < 0 x20  cmid + r(g0  g1)  x2 < 0 x20  cmid + r(g0  g1) < 0  ct0 + rg0 < 0

x20  df < 0 x20  cmid + r(g0  g1)  x2 < 0 x20  cmid + r(g0  g1)  df < 0  ct0 + rg0 + df < 0

Producer responsibility system x1  f1{p2 + p4 + (1  p4) q3} < 0 x20 < 0 x20  cmid + r(g0  g1)  x2 < 0 x20  cmid + r(g0  g1) < 0  ct0 + rg0 < 0

Table 8.9 The local stability conditions of (1, 1, 0, 0, 0) ¼ (C, C-C, D) in each of the three systems Baseline model x1  sr(g0  g1) < 0 cmid + x2  r(g0  g1) < 0 x2 < 0 cmid  r(g0  g1) + x2  x20 < 0  ct0 + rg1 < 0

Actor responsibility system x1 sr(g0  g1)  df < 0 cmid + x2  r(g0  g1)  df < 0 x2  df < 0 cmid  r(g0  g1) + x2  x20 < 0  ct0 + rg1 + df < 0

Producer responsibility system x1 sr(g0  g1) + f1{p2 + p4 + (1  p4)q3} < 0 cmid + x2  r(g0  g1) < 0 x2 < 0 cmid  r(g0  g1) + x2  x20 < 0  ct0 + rg1 < 0

In (1, 1, 0, 0, 1) ¼ (C, C-C, C) and (1, 0, 0, 1, 1) ¼ (C, D-C, C), the actor responsibility system promotes the local stability if df is large enough to influence the dynamics (see Tables 8.3 and 8.4). Tables 8.3 and 8.4 also show that, if (1, 1, 0, 0, 1) ¼ (C, C-C, C) is locally stable, (1, 0, 0, 1, 1) ¼ (C, D-C, C) is not locally stable, and vice versa. It is natural to assume that p2 is high, q3 is low, and q4 is low, and then q3  p2 + q4(1  q3) can be negative. This indicates that the producer responsibility system also favors the evolution of cooperation in the division of labor more than the baseline system (see Tables 8.3 and 8.4). In (0, *, *, *, *) ¼ (D, *, *), both sanction systems discourage the stability and then both disfavor illegal dumping by the generator more than the baseline system. In (1, 0, 1, 0, *) ¼ (C, C-D, *), the effect of the sanction systems on the stability of illegal dumping by ITF depends on the parameter values (see Table 8.6). As x1 < 0 does not hold due to our

8.3

Results

213

assumption that x1 > 0 (see Table 8.7), the equilibrium point, (1, 0, 0, 0, *) ¼ (C, D-D, *), is always unstable in the baseline system. However, both sanction systems promote the stability of this equilibrium point (see Table 8.7). Therefore, both sanction systems promote illegal dumping by ITF. In (1, 0, 0, 1, 0) ¼ (C, D-C, D) and (1, 1, 0, 0, 0) ¼ (C, C-C, D), as three inequalities, x1 < 0, x2 < 0 and x20 < 0, do not hold because of our assumption that x1 > 0, x2 > 0 and x20 > 0, this equilibrium point is always unstable in both the baseline system and the producer responsibility system. However, the actor responsibility system promotes illegal dumping by LS (see Tables 8.8 and 8.9). Table 8.2 summarizes the stability conditions of the seven equilibrium points. (i) Both sanction systems promote the evolution of cooperation in the division of labor, and inhibit illegal dumping by generators. (ii) There are two types of illegal ITF dumping: ITF is either a C-D player or a D-D player in equilibrium. Illegal dumping by a D-D player does not occur in the absence of a sanction system because the equilibrium point, (C, D-D, *), is not stable. However, both sanction systems promote illegal dumping by a D-D player. (iii) Illegal dumping by LS does not occur in the absence of a sanction system and in the producer responsibility system, while the actor responsibility system promotes illegal dumping by LS. In the producer responsibility system, when any defector never hands back the fake manifest ( pi ¼ 1, i ¼ 2, 3, 4) and any cooperator always hands back the manifest (qi ¼ 0, i ¼ 3, 4), the sanction promotes (C, C-C, C) and (C, D-C, C), and the sanction inhibits (D, *, *). However, the sanction does not influence the other four types of equilibrium point resulting in illegal dumping, as per the baseline system.

8.3.1

Comparison with the Results of Kitakaji and Ohnuma (2014)

Following Kitakaji and Ohnuma (2014) and their personal communication, model parameters can be estimated. “s” is used as the currency in their experiment and 1 s is equivalent to one million yen or 10,000 US dollars. In their experiment, when the generator produces 100 tons of industrial waste, b ¼ 65 s, x1 ¼ 45 s, cmid ¼ 10s, x2 ¼ 20s, x20 ¼ 40s, ct ¼ 10s, ct0 ¼ 20s, g0 ¼ 100 tons, g1 ¼ 50 tons, r ¼ 0.1 s/ton, s ¼ 4 in which “s” is not the same as the currency “s” here, f ¼ 30s, d ¼ 0.0065, and f1 ¼ 100 s or 50s (Table 8.1). For simplicity, p2 ¼ p3 ¼ p4 ¼ 1 and q3 ¼ q4 ¼ 0. Using these parameter values, we obtain that (D, *, *) is locally stable, and other equilibrium points are locally unstable not only in the baseline model but also in both sanction systems. Now we investigate which sanction system promotes the evolution of cooperation based on parameters used in Kitakaji and Ohnuma (2014). The local administrative organ has difficulty in monitoring perfectly and detecting the illegal dumping and the illegal dumper because the illegal dumper dumps waste in inconspicuous places such as deep in the mountains, destroying evidence. As a result, perfect monitoring is very

214

8

Cooperation and Punishment in the Linear Division of Labor

costly. The experimental results from Kitakaji and Ohnuma (2014) indicate that the probability of being monitored and detected by the local administrative organ (d ) is very low and they estimated d at 0.0065. In reality, d is considered to be much smaller than their experimental observation because monitoring and detecting illegal dumping is not as difficult in the experimental space as in real space. Even though the fine f is very expensive, the value df is very small relative to other parameters. Then, the actor responsibility system converges near the baseline model (see Tables 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, and 8.9); the actor responsibility system hardly promotes the evolution of cooperation. If monitoring and detecting illegal dumping were straightforward and df were not neglected, (C, C-C, C) would be stable when df > 20 and other parameters are as per Kitakaji and Ohnuma (2014). Here we consider the producer responsibility system. The value r can be controlled by the local administrative organ and Ishiwata (2002) suggests that x1 is much higher than x2. We examine for which value of r and x1 the producer responsibility system promotes the evolution of cooperation based on the local stability conditions in Tables 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, and 8.9. Our calculations show that, if both inequalities, r > 0.4 and 400r < x1 < 400r + f1p2, hold, (C, C-C, C) is locally unstable in the baseline model, and locally stable in the producer responsibility system (see Figs. 8.2 and 8.3). Figure 8.2 shows that the initial frequencies influence the dynamics when r ¼ 0.5, x1 ¼ 250, which satisfy r > 0.4 and 400r < x1 < 400r + f1p2, and other parameters are as per Kitakaji and Ohnuma (2014). Numerical simulations show that (u, z1, v) almost converges to (1, 1, 1) even in the low initial value of u when the initial values of z1 and v are high (Fig. 8.2a, b). When the initial value of u is higher, the area where (u, z1, v) almost converges to b

initial v

initial v

a

initial z1

initial z1

Fig. 8.2 The initial frequency dependency of the dynamics in the producer responsibility system. These graphs present the effect of the initial frequencies on the dynamics. The horizontal axis is the initial frequency of C-C players (z1) and the vertical is the initial frequency of ls-cooperators (v). The initial values of z2 and z3 are (1  (the initial value of z1))/3. The blue point signifies that the dynamics converge to (C, C-C, C); the pink point signifies that the dynamics converge to (D, *, *). a The initial frequency of g-cooperators (u) is 0.95, and we executed numerical simulations through 500 time steps (the interval between time steps is 0.01 time). b The initial frequency of g-cooperators is 0.05, and we executed numerical simulations through 1000 time steps. In both a and b, r ¼ 0.5, x1 ¼ 250, and other values are as per Kitakaji and Ohnuma (2014)

u, z1, z2, z3, v

a

Results

215

b

v z1

z1 v

u, z1, z2, z3, v

8.3

u

u

z2 z3

z2 z3

time 㽢0.01

u, z1, z2, z3, v

c

time 㽢0.01

z1 v

u z2 z3

time 㽢0.01

Fig. 8.3 The time-change of the frequencies in the producer responsibility system. These graphs present the numerical simulation outcomes. The horizontal axis is time, and the vertical is the frequencies of u, z1, z2, z3, and v. The thick red line is u; the thick blue, z1; the thick dotted blue line, z2; the thick purple gray line, z3; the thick green line, v. a the initial values of (u1, z1, z2, z3, v) are (0.5, 0.8, 0.066, 0.066, 0.95); b, (0.5, 0.8, 0.066, 0.066, 0.75); c, (0.5, 0.8, 0.066, 0.066, 0.7). In a–c, r ¼ 0.5, x1 ¼ 250, and other values are as per Kitakaji and Ohnuma (2014)

(1, 1, 1) is wider (see Fig. 8.2a, b). In Fig. 8.3a, the value of u increases and converges to one. When the initial value of v is 0.75, the value of u decreases and then increases until u becomes one (Fig. 8.3b). When the initial value of v is 0.7, the value of u immediately converges to zero. Figure 8.3b indicates that the frequency of g-cooperators (u) decreases at the beginning and then increases before the dynamics finally converge to (C, C-C, C) even though the producer responsibility system is effective. Kitakaji and Ohnuma (2014) showed that sanctioning increased the number of defectors, whereas our analysis only shows that (D, *, *) is stable in the baseline model and two sanction systems using their parameters, and does not show that sanctions increase the number of defectors. Instead, we can indicate that, if those authors use the parameters we estimated in the previous paragraph, the producer responsibility system may promote cooperation. However, our model assumptions do not perfectly match the experimental design in Kitakaji and Ohnuma (2014), and the behavior of examinees in their experiment was not identical to those of the players in our case: some examinees formed a coalition, the examinees monitored

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each other in the same role, or examinees divided the waste into pieces and committed them to some examinees. Therefore, our suggestion is tentative; it may not work well in their experimental design.

8.3.2

Comparison with Empirical Reality

We can compare these equilibrium points with Japanese field survey data, captured by our coauthors, Ohnuma and Katakaji. (1, 0, 0, 1, 1) ¼ (C, D-C, C) applies in Hokkaido, the second largest of Japan’s principal islands. The land which the LS has is abundant and real estate is very inexpensive relative to Tokyo. As a result, LS has huge and inexpensive land and does not mind landfilling the waste which ITF does not treat properly prior to it being landfilled by LS. By contrast, (1, 0, 1, 0, *) ¼ (C, C-D, *) applies in the Kanto area including the Tokyo metropolitan area. There is little room in this area and real estate is very expensive. Thus the land of the LS is very limited. As a result, LS cannot accept the offer from the ITF. ITF has no choice but to dump the waste. Before illegal dumping, ITF treats the waste properly following the reasoning: (i) if valuable materials such as precious metals are extracted from the waste successfully through the treatment process, ITF can sell them and obtain benefits in the metropolitan area in which there are big markets for reuse. It is easy for ITF to access the market and the transportation cost is low for ITF. (ii) The amount of treated waste is smaller than that of untreated waste, and the smaller amount of waste is not easily found if dumped. We denote this as the Kanto type. According to data from the Ministry of the Environment in Japan, in 1998 (2015), 60% (56.6%) of illegal dumping resulted from generators, 10% (2.1%) from unlicensed dealers, and 8% (4.9%) from licensed dealers. The data shows that the ratio of generators’ illegal dumping is much higher than others. If we consider that df is too small to influence the dynamics and p2 ¼ p3 ¼ p4 ¼ 1 and q3 ¼ q4 ¼ 0, illegal dumping by generators occurs, but illegal dumping by firms in other roles does not occur, except the Kanto type. Our theoretical model is supported by empirical data suggesting that generators exhibit a high propensity to engage in illegal dumping activities. Data from the Ministry of the Environment in Japan also shows that, directly following the introduction of the producer responsibility system in 1990, illegal dumping first increased and then decreased. The dynamics illustrated in Fig. 8.3b may explain this empirical phenomenon.

8.3

Results

8.3.3

217

Comparison Between the Three-Role and Two-Role Models

We presented results in the previous section whereby the number of industry types was reduced from five to three types. We also constructed a simpler model called the two-role model, constituted by the generator and the landfill site. In the two-role model, we consider that there are two types of industries: generators and landfill sites. The group size of each industry is infinite. Each generator is either a g-cooperator or a g-defector. The generator’s benefit from production is b. When a g-cooperator commits industrial waste to the landfill site, the commission cost is x1 (b > x1 > 0). The landfill site can capture the commission cost as his/her profit, x1. The landfill site is either an ls-cooperator or an ls-defector. The ls-cooperator pays a treatment cost, ct (x1 > ct > 0). The definition of the environmental load caused by g-defectors and ls-defectors is the same as per the three-role model. Let g0 be defined as the amount of industrial waste dumped by a g-defector. We consider that the local administrative organ determines that the generator group be forced to pay an expense s times as high as the LS group (s  1). Here let r be defined as the environmental restoration expense or damage per unit of waste. If a player dumps industrial waste illegally, srg0 is imposed on a generator and rg0 is imposed on an LS. The payoff of a generator and a landfill site are presented by the payoff matrices A and B as follows:
A¼

b  x1 b  g0 sr

b  x1  g0 sr b  g0 sr




and B ¼

x1  c t x1  g0 r

 g0 r : g0 r

ð8:2Þ

Let u1 and u2 be the frequencies of g-cooperators and g-defectors in the generator group (u1 + u2 ¼ 1), and v1 and v2 be the frequencies of ls-cooperators and ls-defectors in the landfill site group (v1 + v2 ¼ 1). We assume that one player is randomly chosen from the generator’s group and the other is chosen from the landfill site’s group, and then the two interact, following our assumptions. We also assume that there are no dealings between generators or between landfill sites. We can apply this model to replicator dynamics of asymmetric games (Hofbauer and Sigmund 1998):  dui ¼ ui ðAvÞi  uAv , dt  dvi ¼ vi ðBuÞi  vBu ði ¼ 1 or 2Þ, dt

ð8:3aÞ ð8:3bÞ

in which u ¼ (u1, u2)t and v ¼ (v1, v2)t. The equation below can be derived from Eqs. (8.3a) and (8.3b):

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du1 ¼ u1 ð1  u1 Þðg0 srv1  x1 Þ, dt dv1 ¼ u1 v1 ð1  v1 Þðg0 r  ct Þ: dt

ð8:4aÞ ð8:4bÞ

The equilibrium points are: (u1, v1) ¼ (1, 1), (1, 0), (0, v1*), in which v1* means any value between 0 and 1. When x1/(sr) < g0 and ct/r < g0, (1, 1) is locally stable. (1, 0) is locally stable when x1 < 0 and ct/r > g0. (0, *) is unstable if g0rsv1* < x1, as the other eigenvalue is zero. When the actor responsibility system is introduced, the payoff matrices are:
A¼

b  x1

b  x1  g0 sr



b  g0 sr  df b  g0 sr  df
 g0 r x1  ct , B¼ x1  g0 r  df g0 r

, ð8:5Þ

in which the local administrative organ punishes the dumper and then forces the dumper to pay a fine, f, if the local administrative organ successfully monitors and detects the illegal dumping and the illegal dumper. Let d be the probability of being detected by the local administrative organ. The matrices in Eqs. (8.3a) and (8.3b) are replaced with those in Eq. (8.5), thus: du1 ¼ u1 ð1  u1 Þðg0 srv1 þ df  x2 Þ, dt dv1 ¼ u1 v1 ð1  v1 Þðg0 r þ df  ct Þ: dt

ð8:6aÞ ð8:6bÞ

We obtain the equilibrium points (1, 0), (1, 1), (0, v1*). The local stability conditions of (1, 1) are (ct  df)/r < g0 and (x1  df)/(sr) < g0. Those of (1, 0) are (ct  df)/ r > g0 and x1  df < 0. (0, v1*) is unstable if g0rsv1* + df < x1 as the other eigenvalue is zero. When the producer responsibility system is introduced, the payoff matrices are
A¼

 b  x1  g0 sr  f 1 p1 b  x1  f 1 q1 , b  g0 sr  f 1 p2 b  g0 sr  f 1 p2
 g0 r x1  ct , B¼ x 1  g0 r  g 0 r

ð8:7Þ

in which we let p1 be the probability that the ls-defector does not hand the manifest back to the g-cooperator. 1  p1 is the probability that the ls-defector hands in a fictitious manifest. Let p2 be the probability that the g-defector does not hand the manifest back to the local administrative organ. 1  p2 is the probability that the

8.4

Discussion and Conclusions

219

g-defector hands in a fictitious manifest. Let q1 be the probability that the ls-cooperator does not hand the manifest back to the g-cooperator. The matrices in Eqs. (8.3a) and (8.3b) are replaced with those in Eq. (8.7) and then the equations are du1 ¼ u1 ð1  u1 Þ ½ðg0 sr þ f 1 p1  f 1 q1 Þv1  ðx1 þ f 1 ðp1  p2 ÞÞ, dt dv1 ¼ u1 v1 ð1  v1 Þðg0 r  ct Þ: dt

ð8:8aÞ ð8:8bÞ

We obtain the equilibrium points (u1, v1) ¼ (1, 0), (1, 1), (0, v1*). The local stability conditions of (1, 1) are ct/r < g0 and (x1  f1( p2  q1))/(sr) < g0. Those of (1, 0) are ct/r > g0 and x1 + f1( p1  p2) < 0. (0, v1*) is unstable if g0rsv1 + f1(p1 + p2) + f1v1(q1 + p1) < x1 as the other eigenvalue is zero. The two-role model shows that there are three equilibrium points in the three systems: (u1, v1) ¼ (1, 1), (1, 0), (0, *). (u1, v1) ¼ (1, 1) corresponds to (u1, z1, z2, z3, v) ¼ (1, 1, 0, 0, 1) and (1, 0, 0, 1, 1); (u1, v1) ¼ (1, 0), (u1, z1, z2, z3, v) ¼ (1, 0, 0, 1, 0) and (1, 1, 0, 0, 0); (u1, v1) ¼ (0, *) corresponds to (u1, z1, z2, z3, v) ¼ (0, *, *, *, *). However, other equilibrium points in the three-role model, such as (1, 0, 1, 0, *), which is interpreted as the Kanto type, cannot be described by the results of the two-role model. Therefore, we conclude that the three-role model has greater empirical credibility than the two-role model.

8.4

Discussion and Conclusions

We investigate the effect of sanctions on the evolution of cooperation in linear division of labor. As an example, we institute the replicator dynamics in the context of an industrial waste illegal dumping game proposed by Ohnuma and Kitakaji (2007). We introduce two sanction systems, the actor responsibility system and the producer responsibility system, and then compare each of these two systems with a baseline model devoid of sanctions. Our main conclusion is that both sanction systems seem to promote the evolution of cooperation and inhibit illegal dumping by generators. However, where fines do not influence evolutionary dynamics because monitoring is ineffective, the actor responsibility system no longer promotes the evolution of cooperation. Monitoring violators is arduous not only in the case of illegal industrial waste but in other contexts too, such as illegal logging and overfishing. The industrial waste treatment process in Japan embodies linear division of labor; and the sanction system which does not require monitoring violators can be put into practice rather than the sanction system with monitoring. Our analysis also shows that the producer responsibility system, which does not require monitoring, promotes the evolution of cooperation and inhibits illegal dumping more than the actor responsibility system. If logging or fishing in some areas is configured according to linear division of labor,

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sanction systems like the producer responsibility system may work to inhibit illegal logging or overfishing. In the producer responsibility system, the generator is sanctioned if the manifest is not handed to the local administrative organ. There is another possible sanction system derived from the producer responsibility system; not a generator but an intermediate treatment facility or a landfill site is punished when the manifest is not handed to the local administrative organ. However, generators are expected to choose illegal dumping more if intermediate treatment facilities or landfill sites are punished, and then the amount of illegal dumping is larger. It is because g-defectors do not need to pay not only the cost of cooperation but also the fine. To confirm our guess, we will analyze the model with the new sanction system as a future study. Sanctions are only one of a number of potential interventions. The following are other potential solutions which may promote the evolution of cooperation. If we can configure a group that chooses good players in all roles, the members maximize their efforts towards goal attainment, and produce a final good product. In this case, how to choose group members and/or to choose a group is crucial. Chapter 6 examined how choosing a new member, or choosing a good group, influences the evolution of group cooperation when players are peers or on an equal footing in a group consisting of more than three members (Nakamaru and Yokoyama 2014; see Chap. 6). We may apply the study in Chap. 6 to the linear division of labor context. However, if players in role Bi do not interact with players in role Bj and then cannot evaluate the quality of players in Bj correctly ( j > i + 1), the group fails to choose optimal players and the evolution of cooperation in the linear division of labor does not occur. Or, without choosing group members, we can consider that a player in role Bi chooses a player with good reputation in role Bi + 1 and then the player in role Bi + 1 can accept the player’s offer if s/he has a good reputation. In this case, we guess that the evolution of cooperation in linear division of labor is promoted and we will analyze the model in future work. Linear division of labor (Fig. 8.1b) has a similar structure to the centipede game (Rosenthal 1981). In the standard setting, two players play the centipede game alternately and repeatedly. The player can choose “Right” or “Down” on his/her turn. If the player chooses Down, the game is over and then both players can receive the payoff. If the player chooses Right, the game is continued and the other player can choose either of the two options. This is repeated. It is assumed that: (i) The payoff of player A who chooses Down in the i-th stage, sA,i, is higher than that of player A who chooses Right when the other player B chooses Down in the i + 1-th stage, sA,i + 1, and (ii) sA,i + 2 is higher than sA,i., and (iii) If two players always choose Right, they achieve the final stage. The payoff in the final stage is higher than in any previous stage. There is one study, Smead (2008), from the viewpoint of evolutionary game theory; this showed that a finite population promotes the evolution of cooperation in the quasi-centipede game, where two players choose their tactics simultaneously and the payoff is symmetric. It is often assumed that two players repeatedly and alternately play the centipede game, whereas, in our assumptions, players in different roles play the game in different stages and the same players do not play the game repeatedly. As a result, we can describe the model through the replicator dynamics for asymmetric games.

References

221

We only assumed players or organizations in three roles, and did not assume a local administrative organ player. If we introduce such a player, which also maximizes payoff in evolutionary game theory, the player may prefer the producer responsibility system to the actor responsibility system because the latter is very costly. Further exploration thereof remains for future research. We assumed the generators, the intermediate treatment facilities, and the landfill sites as the three-role model in the linear division of labor. We can consider another combination of the three roles, which are the generators, the haulers, and the landfill sites. The model assumption in the new three roles is different from our current model, and we will investigate which three-role model promotes the evolution of cooperation in linear division of labor in future work. Then, we will attempt to analyze the five-role model. It is complicated to analyze the equations and the analytical results may be complicated. To understand the complicated results and obtain the general conclusion, we will compare the results in the five-role model with our current results and results in the new three-role model. In empirical contexts, there are many types of division of labor besides the linear variant; the structure of the division of labor may influence the efficiency of the institution or the social goal. The division of labor has long been studied in sociology and organization theory, especially industrial ecology (Durkheim 1893; Frosch and Gallopoulos 1989; Milgrom and Roberts 1992; Axtell et al. 2001; Giddens 2006). We studied the division of labor from the viewpoint of evolutionary game theory and then showed that a special sanction such as the producer responsibility system can promote cooperation among organizations with linear division of labor. Network structure among roles is also key from the viewpoint of evolutionary game theory. In this chapter, we assumed the simplest network: the linear-type network or one-dimensional lattice structure. The network structures among organization or roles in other institutions are more complicated than what we dealt with here. We can proffer a new possibility that organizations can be studied from the viewpoint of the evolution of cooperation and complex networks by means of replicator dynamics.

References Axtell RL, Andrews CJ, Small MJ (2001) Agent-based modeling and industrial ecology. J Ind Ecol 5(4):10–13 Chen X, Sasaki T, Perc M (2015) Evolution of public cooperation in a monitored society with implicated punishment and within-group enforcement. Sci Rep 5:17050. https://doi.org/10. 1038/srep17050 Coolidge FL, Wynn T (2008) The role of episodic memory and autonoetic thought in upper Paleolithic life. PaleoAnthropology 2008:212–217 Durkheim É (1893) De la division du travail social. Bloomsbury Publishing, London Frosch RA, Gallopoulos NE (1989) Strategies for manufacturing. Sci Am 261:144–152 Giddens A (2006) Sociology, 5th edn. Polity Press, Cambridge Henrich J, Boyd R (2008) Division of labor, economic specialization, and the evolution of social stratification. Curr Anthropol 49:715–724. https://doi.org/10.1086/587889 Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge

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Hölldobler B, Wilson EO (1990) The ants. The Belknap Press of Havard University Press, Cambridge, MA Ishiwata M (2002) Industrial waste connections (In Japanese). WAVE Publisher, Tokyo Kitakaji Y, Ohnuma S (2014) Demonstrating that monitoring and punishing increase non-cooperative behavior in a social dilemma game. Jpn J Psychol 85(1):9–19 Kitakaji Y, Ohnuma S (2016) Even unreliable information disclosure makes people cooperate in a social dilemma: development of the industrial waste illegal dumping game. In: Kaneda T, Kanegae H, Toyoda Y, Rizzi P (eds) Simulation and gaming in the network society. Springer, Singapore, pp 369–385 Kuhn SL, Stiner MC (2006) What’s a mother to do? The division of labor among Neandertals and modern humans in Eurasia. Curr Anthropol 47:953–980 Lee JH, Sigmund K, Dieckmann U, Iwasa Y (2015) Games of corruption: how to suppress illegal logging. J Theor Biol 367:1–13. https://doi.org/10.1016/j.jtbi.2014.10.037 Milgrom P, Roberts J (1992) Economics, organization & management. Prentice Hall, Englewood Cliffs, NJ Nakahashi W, Feldman MW (2014) Evolution of division of labor: emergence of different activities among group members. J Theor Biol 348:65–79. https://doi.org/10.1016/j.jtbi.2014.01.027 Nakamaru M, Yokoyama A (2014) The effect of ostracism and optional participation on the evolution of cooperation in the voluntary public goods game. PLoS One 9(9):e108423. https://doi.org/10.1371/journal.pone.0108423 Nakamaru M, Shimura H, Kitakaji Y, Ohnuma S (2018) The effect of sanctions on the evolution of cooperation in linear division of labor. J Theor Biol 437:79–91. https://doi.org/10.1016/j.jtbi. 2017.10.007 Ohnuma S (2021) Consensus building: process design toward finding a shared recognition of common goal beyond conflicts. In: Metcalf GS, Kijima K, Deguchi H (eds) Handbook of systems sciences. Springer, Singapore, pp 645–662. https://doi.org/10.1007/978-981-13-03708_68-1 Ohnuma S, Kitakaji Y (2007) Development of the “industrial waste illegal dumping game” and a social dilemma approach—effects derived from the given structure of asymmetry of incentive and information. Simul Gaming 17(1):5–16 Ostrom E (1990) Governing the commons: the evolution and institutions for collective action. Cambridge University Press, Cambridge Powers ST, Lehmann L (2014) An evolutionary model explaining the Neolithic transition from egalitarianism to leadership and despotism. Proc Biol Sci 281(1791):20141349. https://doi.org/ 10.1098/rspb.2014.1349 Powers ST, van Schaik CP, Lehmann L (2016) How institutions shaped the last major evolutionary transition to large-scale human societies. Philos Trans R Soc Lond Ser B Biol Sci 371(1687): 20150098. https://doi.org/10.1098/rstb.2015.0098 Roithmayr D, Isakov A, Rand DG (2015) Should law keep pace with society? Relative update rates determine the co-evolution of institutional punishment and citizen contributions to public goods. Games 6:124–149. https://doi.org/10.3390/g6020124 Rosenthal RW (1981) Games of perfect information, predatory pricing and the chain-store paradox. J Econ Theory 25:92–100 Rustagi D, Engel S, Kosfeld M (2010) Conditional cooperation and costly monitoring explain success in forest commons management. Science 330(6006):961–965. https://doi.org/10.1126/ science.1193649 Sigmund K, De Silva H, Traulsen A, Hauert C (2010) Social learning promotes institutions for governing the commons. Nature 466(7308):861–863. https://doi.org/10.1038/nature09203 Smead R (2008) The evolution of cooperation in the centipede game with finite populations. Philos Sci 75:157–177 Tverskoi D, Senthilnathan A, Gavrilets S (2021) The dynamics of cooperation, power, and inequality in a group-structured society. Sci Rep 11(1):18670. https://doi.org/10.1038/s41598021-97863-7

Chapter 9

Can Cooperation Evolve When False Gossip Spreads?

Abstract Indirect reciprocity is considered to be important for explaining cooperation among humans, and has been modeled using several types of reputational scores, most of which were assumed to be updated immediately after each game session. In this study, we introduce gossip sessions held between game sessions to capture the spread of reputation and examine the effects of false information intentionally introduced by some players. Analytical and individual-based simulation results indicated that the frequent exchange of gossip favored the evolution of cooperation when no players started false information. In contrast, intermediate repetitions of gossip sessions were favored when the population included liars or biased gossipers. In addition, we found that a gossip listener’s strategy of incorporating any gossip regardless of speakers usually worked better than an alternative strategy of not believing gossip from untrustworthy players. Keywords Direct and indirect reciprocity · Discriminator · Image score · Lie · Reputation dynamics

9.1

Introduction

Thanks to our language skill and the cognitive abilities, we can tell a lie; we can spread false information to deceive others and obtain benefits from them. Even though false information spread everywhere in our world, we can still build trust and credibility in our society. Why can we do that? In this chapter, we will investigate what prevents cooperators being deceived by false information, such as false reputation, spread by defectors from the viewpoint of “the evolution of cooperation.” The effect of reputation on the evolution of cooperation has been theoretically studied as indirect reciprocity (Sect. 1.7.4), and many prior studies assumed that reputation is correct. Following shows the history of indirect reciprocity study and then the originality of our study by comparison with the previous studies. Alexander (1987) explicitly distinguished the case where a player’s current recipient will be his/her future potential donor (namely, direct reciprocity) from the other case, where a third party will be his/her donor (indirect reciprocity). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. Nakamaru, Trust and Credit in Organizations and Institutions, Theoretical Biology, https://doi.org/10.1007/978-981-19-4979-1_9

223

224

9 Can Cooperation Evolve When False Gossip Spreads?

Researchers have focused on the effect of reputation on the evolution of indirect reciprocity (see Chap. 1); if players can discriminate cooperators from defectors according to reputation, then a cooperative player can be helped by a third party who has not met the player. Following the pioneering studies of Pollock and Dugatkin (1992), Nowak and Sigmund (1998a, 1998b) proposed a reputational system called image scoring, where a player who chooses to give a benefit to his/her recipient in the last session is regarded as good, while one who chooses not to give a benefit is regarded as bad. The key strategy was named discriminating (DISC), in which players choose to give a benefit to good co-players and not to bad co-players. These researchers showed that the DISC strategy is evolutionarily stable against the all-defection (ALLD) strategy if the probability with which a player knows the past behavior of his/her current partner, namely the degree of social transparency, is higher than the cost–benefit ratio of one altruistic behavior. Multiple studies (e.g. Leimar and Hammerstein 2001; Panchanathan and Boyd 2003; Brandt and Sigmund 2004; Ohtsuki and Iwasa 2004; Takahashi and Mashima 2006) have indicated the importance of not only first-order information (who cooperated) but also second-order information (who cooperated with whom [and why]). In the simplest framework of image scoring used in the study of Nowak and Sigmund (1998b), the conditional cooperator, who does not cooperate with defectors, is regarded as bad due to his/her action of not giving a benefit. Consequently, cooperators such as DISC cannot sustain their mutually beneficial relation when players use only first-order information. In contrast, the collapse of cooperative relations in the whole population does not occur if players not giving a benefit to bad players are not regarded as bad (Brandt and Sigmund 2004; Ohtsuki and Iwasa 2004; Takahashi and Mashima 2006). Such a strategy has already been proposed in the study on the iterated prisoner’s dilemma game (Sugden 1986; see also Kandori 1992) and is called the standing norm. Considering both first- and second-order information, standing players can appropriately infer the reason for other players’ choices and maintain cooperative relations among them even under the presence of many ALLD players.

9.1.1

Comparison with Previous Studies About Indirect Reciprocity

Here, we investigate the effects of two factors on the evolution of cooperation: (i) the intentional manipulation of information and (ii) the process of information transmission. These two factors are worth examining when considering a relatively large human society in which a player cannot observe all events occurring in the population and must depend on gossip from other players to know who did what to whom and why. In the following, we show the reason why (i) is examined. Previous studies on indirect reciprocity (Leimar and Hammerstein 2001; Panchanathan and Boyd 2003; Brandt and Sigmund 2004; Ohtsuki and Iwasa 2004; Takahashi and Mashima

9.1

Introduction

225

2006) have investigated the types of assessment rules promoting the evolution of indirect reciprocity when perceptional errors occur. These studies do not focus on the effect of intentional manipulation of information or fake information on the evolution of indirect reciprocity. However, perceptional errors qualitatively differ from intentional manipulation of information or lies in that (1) errors occur basically at random, while a lying speaker can continue to fake information throughout, and (2) there are various types of intentional manipulation of information; some misrepresent their own behaviors to make themselves sound good, whereas others spread false bad news about targets in order to damage their reputation. The reason why (ii) is investigated is as follows. Previous studies on indirect reciprocity assumed that all players share the same reputation about any member in the population (public information; e.g., Nowak and Sigmund 1998a, 1998b; Ohtsuki and Iwasa 2004; Takahashi and Mashima 2006; Ghang and Nowak 2015) or that different players have different reputations (private information; e.g., Brandt and Sigmund 2004; Rauwolf et al. 2015). We consider an explicit process of information transmission: gossip spreads via the repetition of a one-by-one contact process (e.g., Mohtashemi and Mui 2003; Nakamaru and Kawata 2004). Consequently, gossip does not spread homogeneously in the population; that is, some players are subject to information delays and different players may evaluate a specific player differently (i.e., some regard a player as good and others regard him/her as bad). When gossip is spread, a triadic relationship is needed at least: a speaker, a listener, and a gossip target (e.g., Giardini and Wittek 2019; see also Preface). Introducing this process, we can investigate the effect of speed of spreading gossip in the population on the success of detecting fake information (Seki and Nakamaru 2016). We use a framework similar to the image scoring framework introduced by Nowak and Sigmund (1998a) to present the reputation of each player, simplifying and sophisticating the simulation model on gossip proposed by Nakamaru and Kawata (2004) for the following reasons. In the study of Nowak and Sigmund (1998a), the image score of player i is the accumulation of the past behavior of player i. More specifically, a player’s image score increases (respectively, decreases) by one unit every time he/she chooses to give (resp. not to give) a payoff benefit to the player chosen as his/her potential recipient at a cost to his/her own payoff in a one-way gift-giving game. In the subsequent round in which another player is assigned the role of potential donor of him/her, the donor player can evaluate him/her by referring to his/her image score. That means that the image score totally belongs to the public domain of knowledge, and thus we can classify the image score as a type of public score. As described above, the DISC strategy can be evolutionarily stable within this framework. Most of the following studies (e.g., Leimar and Hammerstein 2001) used similar types of reputational scores (i.e., public scores). In contrast, we assume that each player has a score (called P-score later) regarding each of other players, which is thus regarded as a kind of private score. Using this framework of private scores, we consider not only that (i) a player can choose either cooperation or defection based on the P-score, but also that (ii) a player can make and spread not only gossip about another player but also gossip about him/herself by

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9 Can Cooperation Evolve When False Gossip Spreads?

him/herself, and (iii) a player can decide whether to believe gossip that another player spreads based on his/her own P-score. Previous theoretical studies on indirect reciprocity using a type of private score (e.g., Brandt and Sigmund 2004) cannot accommodate scenarios (ii) and (iii) since they assumed that update of scores is basically based on private observation. In those models, a player had to behave altruistically to keep or recover a good reputation of him/her. However, it is possible for each player to try to manipulate the reputation of him/herself by spreading false and fake information regarding him/herself without an altruistic behavior. In addition, a player may also make and spread false and fake information regarding other players in order to reduce payoffs of them, which leads to the higher relative fitness of him/herself. Such types of information manipulation cannot be represented by simple probabilistic errors when some players always try to spread false and fake information while the others do not. Recent empirical studies have provided evidence of such intentional manipulation of information (Cohn et al. 2014; Toyokawa et al. 2014) and strong hate against that (Ohtsubo et al. 2010; Konishi and Ohtsubo 2015), weakly suggesting the evolutionary origin of lying strategies and counter strategies to the liars. The image score framework proposed by Nowak and Sigmund (1998a) is considered to be first-order information. The results of experimental studies suggest that people use only first-order information to make decisions, even though both firstand second-order information are presented (Milinski et al. 2001). Therefore, some factors that were not incorporated into the previous models may make a standing strategy disadvantageous. In the present model, even though the P-score is regarded as first-order information (see Sect. 9.2.4), a recipient of gossip can use his/her P-score against the speaker to logically judge the validity of the speaker’s evaluation of the other player when the number of the gossip events is quite large (see Sect. 9.6 for more details); then, he/she can decide whether to believe gossip from the speaker based on the P-score.

9.1.2

Comparison with Nakamaru and Kawata’s Gossip Model

As stated above, the present model in Seki and Nakamaru (2016) is a non-simple extension of Nakamaru and Kawata’s gossip model. Nakamaru and Kawata (2004) considered not a single but two types of scores; Players update the first type of score (called G-score) by direct observation of their opponents’ actions in the game sessions. On the other hand, they update the second type of score (called R-score) according to gossip that they receive in the gossip sessions. Nakamaru and Kawata (2004) then examined a limited number of strategies in a rather non-numeric way unlike the other previous studies, in which each strategy was represented as a point in a strategic space. Specifically, players in Nakamaru and Kawata (2004) choose to cooperate or to defect based on either G-scores or R-scores (not both), and which

9.2

Models

227

type of score is used in the decision-making depends on their strategies. They may start new gossip based on G-scores and spread gossip based on R-scores. Among the few strategies, Nakamaru and Kawata (2004) focused on the strategy called LIAR, with which a defector starts good gossip about him/herself, and investigated strategies which are stable against the LIAR strategy. The significant contributions of the present study are fourfold. First, we show that several results so far unique to Nakamaru and Kawata (2004) also hold in the widely used framework characterized by the use of a single type of score and a variety of strategies defined on a numerical space. Second, we introduce new strategies with which players spread false and fake information, e.g., a strategy with which a player does not believe any strangers and starts bad gossip about them. Third, we also introduce strategies weakly corresponding to the standing strategy, one of winning strategies in the observation models, into our gossip model and examine whether they can serve as counter strategies against the strategies starting false and fake information. Fourth, we have developed an approximation method to obtain analytical results that explain the outcomes of individual-based simulations.

9.1.3

Unification of Direct and Indirect Reciprocity

Both Nakamaru and Kawata (2004) and Seki and Nakamaru (2016) can consider the effect of both indirect reciprocity and direct reciprocity. In both studies, the speed of spreading gossip and how many times the same pair can be randomly chosen and interact each other are controlled. From the aspect of indirect reciprocity, these studies can investigate if cooperation can evolve not only in spreading correct gossip but also under the spread of false and fake gossip when the same pair does not interact again. These studies can examine the effect of fast or slow spread of gossip on indirect reciprocity. From the aspect of direct reciprocity, when the same pair, chosen randomly from the population, can interact often, the effect of fast or slow speed of spreading correct or false gossip on direct reciprocity can be examined. These studies provide the first model to unify direct and indirect reciprocity. Schmid et al. (2021) also studied the unified framework of direct and indirect reciprocity by using the simple mathematical model. In their model, gossip is not spread between players. Their study did not consider the effect of false and fake gossip, but the effect of errors. Interestingly, the main result in Schmid et al. (2021) is close to both Nakamaru and Kawata (2004) and Seki and Nakamaru (2016).

9.2

Models

We assume a finite population comprising n players and two types of events: givinggame and gossip sessions (see below for details), which occur at the rates of g and r times, respectively, per unit time step (g > 0 and r
0). The value of T denotes the number of time steps per generation (T > 0). In a giving-game session, one potential

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donor and one recipient are randomly chosen from n players. The expected number that the same donor–recipient pair is chosen in a unit time is denoted by wg, where wg ¼ g=ðnðn  1ÞÞ:

ð9:1Þ

The parameter range of interest in this study is T < 1/wg (T < 1980 for g ¼ 5 and n ¼ 100; T < 86,988 for g ¼ 5 and n ¼ 660), with which each pair is expected to meet less than once at game sessions. Similarly, in a gossip session, one speaker and one listener are chosen randomly from n players. Let wr be the expected number that an identical speaker–listener pair is chosen in a unit time, where wr ¼ r=ðnðn  1ÞÞ:

ð9:2Þ

In addition, we define λ as the ratio of gossip sessions per game (i.e., r∕g), which turned out to be an important index to predict if a certain type of lying strategy will fix in the population.

9.2.1

Strategies

Each player possesses a strategy set F, which comprises six numeric strategies: F ¼ (ρ, k, qG, qB, qR, a). Following his/her F, a player modifies his/her private images of the other players (P-scores; see Sect. 9.2.2). Using the P-scores and F, the player decides his/her action at each event. All parameters and strategies given above are summarized in Table 9.1. In addition, we almost fully replicate the model of Nakamaru and Kawata (2004) in the framework with P-scores by using nine strategies denoted as (γ C, γ D, ρG, ρB, k, qG, qB, qR, a) in total.

Table 9.1 Numeric strategies for each individual k qG qB qR γ C (¼ 1) γ D (¼ 1) ρG (¼ ρ) ρB (¼ ρ)

Donor’s criterion for giving to a recipient at a game stage Speaker’s criterion for delivering G gossip at a gossip stage Speaker’s criterion for delivering B gossip at a gossip stage Listener’s criterion for receiving a speaker’s gossip at a gossip stage Magnitude of positive effect of a donor’s giving on a recipient’s image score of the donor Magnitude of negative effect of a donor’s non-giving on a recipient’s image score Magnitude of positive effect of G gossip on a listener’s image score Magnitude of negative effect of B gossip on a listener’s image score

Note: In each interaction, the strategy of the underlined player is referred to. The four parameters (γ C, γ D, ρG, and ρB) were described in Sects. 9.2.1 and 9.2.2

9.2

Models

9.2.2

229

Definition of the P-Score

Each player has P-scores against the other n  1 players; thus, there exist n(n  1) P-scores. A player can always refer to and modify his/her own P-scores. In contrast, he/she cannot directly access the P-scores of the other players. Based on the image score proposed by Nowak and Sigmund (1998a, 1998b), a P-score at a certain time step is defined as sij ¼ Cði

jÞ  Dði

jÞ þ ρΣu6¼i ½Gj ði

uÞ  Bj ði

ð9:3aÞ

uÞ:

where a real number sij (i 6¼ j) represents player i’s private image for/against player j, the initial value of which is zero. In Eq. (9.3a), C(i j) and D(i j) denote the total numbers of game sessions wherein a potential donor j chooses to give and not to give u) and Bj(i a benefit to a recipient i through one generation, respectively. Gj(i u) denote the total numbers of gossip sessions wherein listener i receives good gossip (G gossip) and bad gossip (B gossip) about player j from speaker u, respectively. A higher P-score represents a better image of one player as perceived by another. The parameter ρ is defined as the magnitude of gossip relative to a direct experience on one’s P-scoring. A player with ρ 6¼ 0 is called a gossip user. In contrast, a player with ρ ¼ 0 is called a pure direct reciprocator who does not refer to any gossip-mediated information and starts gossip purely based on his/her own experience. In addition, we assume that sij is between +5.0 and 5.0, and do not incorporate a newly occurring event that will make sij larger than +5.0 or smaller u), or Bj(i u). than 5.0 into C(i j), D(i j), Gj(i We show that our full model with nine parameters can imitate a larger portion of strategies that were defined in Nakamaru and Kawata (2004). In the full model, the strategy set is denoted as (γ C, γ D, ρG, ρB, k, qG, qB, qR, a) and player i’s P-score against player j, sij, is defined as sij ¼ γ C Cði

jÞ  γ D Dði

jÞ þ ρG Σu6¼i Gj ði

uÞ  ρB Σu6¼i Bj ði

uÞ,

ð9:3bÞ

where strategic parameters γ C, γ D, ρG, and ρB represent the relative importance of giving, not giving, receiving good gossip, and receiving bad gossip, respectively, on the P-score. Note that Eq. (9.3a) is a special form of Eq. (9.3b) with γ C ¼ γ D ¼ 1 and ρG ¼ ρB ¼ ρ. Hereafter we use Eq. (9.3a) as a P-score and F ¼ (ρ, k, qG, qB, qR, a) as the strategy set.

9.2.3

The Giving-Game Session

Suppose that players i and j are chosen as the potential donor and recipient, respectively. Player i decides whether to give a benefit b (cooperation) or not

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Can Cooperation Evolve When False Gossip Spreads?

Intrinsic cooperator (k ≤ 0) Fair gossipers qG = q B = 0 Biased gossipers qG = q B = k

D B

C k

0

D B

D sij B

G C

k

0

Intrinsic defector (k > 0) C 0

k

D sij

G

B

sij G C

0

k

sij G

Fig. 9.1 Graphical images of unbiased and biased gossipers

(defection) to player j at a cost of c. We set b > c > 0 so that one can expect a higher accumulated payoff if the current recipient or the third party reciprocates directly or indirectly, respectively. Player i makes a decision to give a benefit to player j by comparing the value of his/her numeric strategy k with his/her P-score against j, sij. If sij
k, then he/she decides to give a benefit to player j. If sij < k, then he/she decides not to give a benefit to player j. As the initial value of sij is zero, players with k  0 choose to give a benefit to those who are unknown; that is, to those of whom they have not heard or met. In contrast, players with k > 0 choose not to give a benefit to such “strangers”. We call the former intrinsic cooperators and the latter intrinsic defectors (Fig. 9.1). Those with k  5.0 (i.e., the minimum value of sij) and those with k > +5.0 (i.e., the maximum value of sij) are unconditional cooperators and unconditional defectors (so-called ALLD players), respectively. Subsequently, j modifies his/her P-score against i, sji, following the rule already mentioned above. Specifically, sji is increased by one if i gives benefit to j, and it is decreased by one if i does not give benefit to j.

9.2.4

The Gossip Session

Let player i be chosen as the speaker and player j as the listener, where player i tells player j his/her evaluation of n players (including him/herself) as a series of gossip stories. We assume that all gossip can be qualitatively classified into three types: good (G), bad (B), and neutral or nothing-to-speak (N ). For simplicity, we assume that the magnitude of a good or bad evaluation (i.e., the size of the absolute value of a P-score) cannot be expressed in the G or B gossip. In other words, we consider a quite primitive language that consists of only nouns and basic adjectives, and the sentence “player l [is] G (respectively, B)” represents a speaker’s expectation that player l usually chooses to give (resp. not to give) to someone. The gossip in the present study comprises a subject and a complement implying behavior, whereas it does not contain an object required to extract the second-order information in the previous assessment rule studies (e.g., Ohtsuki and Iwasa 2004). In addition,

9.2

Models

231

listeners have no means to discriminate whether the gossip is based on the speaker’s actual experience or it is hearsay information. In this study, we regard gossip as false when it is started without any corresponding event. More specifically, G (resp. B) gossip about player l that is started before player l actually chooses to give (resp. not to give) to someone at least once is called false and fake gossip. The false and fake gossip may be a lie or may represent an expectation bias of a player who makes it. The P-score, sil, and two strategic values of player i, qG and qB (qG
qB), determine player i’s evaluation of and his/her gossip about l (l 6¼ i). Specifically, player i gossips about player l to player j, following the rule: if sil > qG, then i regards l as G and gossips that l is G, which we call G gossip about l. If sil < qB, then he/she regards player l as B, which we refer to as B gossip about player l. If qB  sil  qG, then player i regards player l as N. In this case, he/she does not gossip about player l. Note that it might be natural to assume qB  0  qG, since the case sil ¼ 0 includes the situation that player i does not yet know the name of player l. We call the strategy with qB  0  qG the unbiased gossiper. However, we discuss other possible gossiping strategies with qG < 0 (called G-biased) and qB > 0 (B-biased) later in this study. Player i’s gossip about him/herself is determined by his/her binary strategy a. If player i has the strategic value a ¼ 1, then player i tactically self-advertises to another player that he/she is a G player. If a ¼ 0, then player i does not gossip about him/herself, in which case gossip about him/herself is N. We do not consider the case that a speaker delivers B gossip about him/herself because it is obviously a maladaptive behavior in this model. The listener j first decides either to consider i’s gossip as valuable information (believe) or to dismiss all as a plain lie or unfair judgment (not believe), comparing sji with qR of listener j. When sji
qR, j believes all of the G and B gossip from speaker i. If speaker i engages in G gossip about player l, then sjl increases by ρ. Similarly, if speaker i engages in B gossip about player l, then sjl decreases by ρ. Note that G or B gossip about player j from any speaker (player i in this case) has no effect, since we do not consider sjj in this study. When sji < qR, listener j does not believe any gossip from speaker i. If this is the case, then listener j’s P-scores remain unchanged in that gossip session.

9.2.5

The Updating Rule Through Generations

At the end of each generation, players asexually reproduce offspring depending linearly on their accumulated payoffs obtained through the giving-games. Specifically, we assumed that player i is chosen as a parent of a newborn with the following probability:

232

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Can Cooperation Evolve When False Gossip Spreads?


 ð1 þ yi  y min Þ=Σj 1 þ yj  y min ,

ð9:4Þ

where yx and ymin stand for payoff gained by player x and the worst player at that generation, respectively. Under the above rule, every player has non-zero probability of being chosen as a parent. We repeated this sampling of a parent n times with replacement to determine all parents of n newborns. A newborn’s F0 is inherited from his/her parent without mutations while payoffs and P-scores are reset to zero. The population size is n in each generation. We expect that this process does not largely differ from imitation dynamics without actual death and birth.

9.2.6

Parameter Settings

Numerical and individual-based simulations were performed in addition to an algebraic approach to investigate our model. In those simulations, we assigned b ¼ 10, c ¼ 1 (following Nowak and Sigmund 1998a), g ¼ 5, and relatively small integers to r compared with n (0  r  25, thus 0  λ  5 when n ¼ 660 [Sect. 9.3] and 0  r  10, thus 0  λ  2 when n ¼ 100 [Sects. 9.4 and 9.5]). A large benefitto-cost ratio (b/c) is necessary for cooperation to be evolutionarily stable with not many repetitions of a game session (Axelrod and Hamilton 1981). Note that the effect of indirect reciprocity on the evolution of cooperation is less important for a larger number of games, in which case the same two players are involved in multiple games and direct reciprocity can work. Since our interest is in the indirect reciprocity, we use the relatively large b/c. The case that every gossip user unconditionally believes all gossip (qR ¼ 5.0 for every player gossip user with ρ 6¼ 0) is examined in Sects. 9.3 and 9.4, and the results of the other cases (qR > 5.0 for some players) are shown in Sect. 9.5.

9.3

Competition Among Gossiping Reciprocators with Various Criteria

Nowak and Sigmund (1998a) showed that when all possible strategic values are uniformly distributed at the first generation, “stern discriminators” (k ¼ 0) are more likely to evolve than more generous discriminators (k < 0). We performed similar computer simulations in our framework of P-scoring with gossip and subsequently investigated the types of discriminators who played an important role for the evolution of reciprocity. It was assumed that gossiping discriminators believe what the speaker says (qR ¼ 5.0). We consider the baseline zero-discriminating strategy (ZDISC) to be defined as k ¼ qG ¼ qB ¼ 0. A ZDISC player gives a benefit to a recipient whose P-score is equal to or higher than k ¼ 0 and engages in G gossip about that player. In contrast,

9.3

Competition Among Gossiping Reciprocators with Various Criteria

233

the ZDISC player does not give a benefit to a recipient whose P-score is lower than k ¼ 0 and engages in B gossip about that player. Every ZDISC player in this section and in Sect. 9.4 believes what any speaker says to him/her (qR ¼ 5.0). In the following, we will focus on two types of derivative strategies of ZDISC: fair gossipers (qG ¼ qB ¼ 0 with various k) and gossipers with special types of bias simply called biased gossipers in this chapter (qG ¼ qB ¼ k; Fig. 9.1). Actions of fair gossipers are not always consistent with their words (k 6¼ qG and k 6¼ qB), whereas actions of biased gossipers are consistent with their words (k ¼ qG ¼ qB). We will discuss the strategic values that can be advantageous when the population comprises one of the above-mentioned two types of gossipers.

9.3.1

Fair Gossipers with Various Criteria for Giving-Games (qG = qB = 0)

In the following, we will investigate what values of k among various “fair gossipers”, defined as players who have qG ¼ qB ¼ 0 and any value of k (Fig. 9.1), are favored under various environmental parameters λ and T when players believe what the speakers say (qR ¼ 5.0). ZDISC players have k ¼ 0. However, various players have various values of k (see Fig. 9.1). When player i has k(i)  sij < 0, player i gives a benefit to player j while player i engages in B gossip about player j to a third party as sij < qB ¼ 0. Meanwhile, when player i has 0 < sij < k(i), player i does not give a benefit to player j while player i engages in G gossip about player j to a third party as sij > qG ¼ 0. We performed a series of individual-based simulations in which population size was fixed to n ¼ 660 for any generation. At the first generation of every simulation, strategic values were evenly distributed. More specifically, 22 values (from 5.0 to +5.5 with intervals of 0.5) were chosen as the value for k of each player. Six values (1, 1∕2, 1∕4, 1∕8, 1∕1024, and 0) were chosen as the value for ρ. We do not assume selfadvertisement; all players are assigned to a ¼ 0. Accordingly, there are 22  6 sets of strategic values {k, ρ} in the population; every five players shared each set of strategic values at the beginning of individual-based simulations. Trials were ended once (i) the population was occupied by intrinsic cooperators (k  0), (ii) the population was occupied by intrinsic defectors (k > 0), or (iii) after 1000 generations. The number of trials at the end of which the population was occupied by the fair gossipers with k  0 was counted and is shown in Fig. 9.2a. Almost all trials ended within 100 generations, indicating that there are no sustainable coexisting states for intrinsic cooperators (k  0) and intrinsic defectors (k > 0). Figure 9.2a shows that intrinsic cooperators are more likely to fix for larger λ or T. This indicates that intrinsic cooperators perform better when they have more opportunities for gossip exchange. In addition, we found that the ZDISC strategy (k ¼ qG ¼ qB ¼ 0) plays an important role, especially at the marginal regions on the parameter planes of λ and

234

9

Can Cooperation Evolve When False Gossip Spreads?

a

b

 5

4

3

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0.985 0.966 0.957 0.949 0.93 0.899 0.85 0.813 0.747 0.611 0.456 0.334 0.169 0.071 0.012 0.001 0 0 0 0 0 0 0 0 0 0 660

1 1 1 1 1 1 1 1 1 1 1 0.999 0.999 0.996 0.992 0.974 0.946 0.902 0.758 0.387 0.049 0.001 0 0 0 0 1320

1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.999 0.998 1 0.999 0.998 0.985 0.931 0.651 0.044 0 0 0 1980

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.999 0.997 0.98 0.793 0.016 0 0 2640

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.996 0.974 0.515 0 0 3300

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.993 0.896 0 0 3960

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.998 0.975 0.028 0 4620

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.997 0.244 0 5280

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.996 0.637 0 5940

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.999 0.834 0 6600 T

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13200

0 0 0 0 0 0 0 0.001 0.001 0 0 0.001 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19800

 5 0

1

4

3

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0 0 0.001 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 660

0.004 0.008 0.01 0.015 0.012 0.022 0.028 0.033 0.049 0.056 0.066 0.076 0.058 0.062 0.082 0.079 0.061 0.056 0.031 0.008 0.003 0 0 0 0 0 1320

0.032 0.04 0.064 0.097 0.133 0.165 0.232 0.271 0.32 0.437 0.455 0.516 0.573 0.644 0.656 0.683 0.658 0.628 0.604 0.531 0.386 0.165 0.003 0 0 0 1980

0.088 0.111 0.159 0.201 0.282 0.335 0.402 0.511 0.636 0.672 0.741 0.791 0.855 0.871 0.896 0.916 0.953 0.946 0.935 0.908 0.856 0.744 0.423 0.008 0 0 2640

0.124 0.181 0.244 0.327 0.395 0.485 0.585 0.663 0.726 0.807 0.84 0.896 0.916 0.953 0.968 0.976 0.974 0.988 0.985 0.984 0.972 0.954 0.831 0.334 0 0 3300

0.194 0.27 0.315 0.368 0.482 0.578 0.662 0.749 0.83 0.861 0.903 0.935 0.957 0.973 0.983 0.992 0.992 0.992 0.993 0.996 0.995 0.99 0.959 0.767 0 0 3960

0.234 0.302 0.41 0.475 0.54 0.642 0.722 0.796 0.835 0.885 0.931 0.948 0.974 0.988 0.987 0.995 0.999 0.998 1 1 0.999 0.996 0.996 0.926 0.014 0 4620

0.289 0.344 0.404 0.513 0.575 0.697 0.74 0.832 0.857 0.91 0.946 0.966 0.978 0.99 0.995 0.999 0.998 0.999 1 0.997 1 0.999 0.999 0.971 0.162 0 5280

0.3 0.39 0.444 0.538 0.643 0.717 0.776 0.857 0.892 0.926 0.953 0.986 0.981 0.986 0.995 0.997 0.999 1 1 1 1 1 1 0.996 0.549 0 5940

0.324 0.39 0.485 0.568 0.645 0.706 0.797 0.856 0.907 0.949 0.97 0.98 0.989 0.99 0.998 1 0.997 1 1 1 1 1 1 0.999 0.802 0 6600 T

0

1

c  5

4

3

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6600

0 0 0.002 0 0 0.002 0.001 0 0 0 0.001 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26400

0 0.002 0.001 0 0.003 0 0.001 0.001 0 0.001 0 0.001 0 0.002 0.001 0 0.001 0.002 0 0 0 0 0 0.002 0 0 33000

0.003 0.002 0.002 0.001 0.003 0.002 0 0.002 0.003 0.003 0.001 0.003 0.002 0.002 0.001 0.001 0.001 0.001 0 0.002 0 0.001 0.001 0.001 0.006 0 39600

0.003 0.002 0.001 0.004 0 0 0.003 0.003 0.001 0.004 0.003 0.003 0.003 0.003 0.002 0.003 0.003 0 0.001 0.003 0.001 0 0.001 0.001 0.021 0.004 46200

0.001 0.001 0 0.001 0.002 0.001 0.005 0.005 0.002 0.002 0.005 0.003 0.003 0.004 0.002 0.002 0.002 0.001 0.002 0.002 0.002 0.001 0.005 0 0.031 0.024 52800

0.001 0 0.004 0.003 0.002 0.002 0.003 0.001 0.003 0.001 0.001 0.003 0.004 0.005 0.004 0.002 0.007 0.005 0.007 0.003 0.004 0.003 0.007 0.005 0.041 0.084 59400

0.004 0 0.001 0.002 0.006 0.002 0.002 0.004 0.004 0.003 0.003 0.002 0.008 0.009 0.01 0.003 0.008 0.006 0.007 0.004 0.008 0.006 0.005 0.007 0.051 0.16 66000 T

0

1

Fig. 9.2 Graphical tables for the proportion of trials in which the population is dominated by intrinsic cooperators (k  0) among the total 100 trials. Red, yellow, and blue background colors correspond to the numbers zero, 50, and 100, respectively. Each population comprises a fair gossipers with a ¼ 0, b fair gossipers, two-fifths of which were initially assigned a ¼ 1, and c biased gossipers with a ¼ 0. The trials were performed for various T and λ. Note that the scales of T differ between a, b and c. At the first generation, every five players share the same strategic values for k (a value chosen from 5.0 to +5.5 at intervals of 0.5) and ρ (1, 1∕2, 1∕4, 1∕8, 1∕1024, or 0). The other parameter values are n ¼ 660, b∕c ¼ 10, g ¼ 5, qR ¼ 5.0 for all players

T (yellow cells in Fig. 9.2a). More generous intrinsic cooperators than ZDISCs (i.e., intrinsic cooperators with k < 0 and any ρ) and pure direct reciprocating ZDISCs (ZDISCs with ρ ¼ 0) are rapidly wiped out and intrinsic defectors (k > 0) increase in the early stages of the simulations. Then, ZDISCs with positive ρ gradually increase and finally wipe out the fair gossipers with k > 0. Though such explicit dynamics were not observed within the parameter region that was more favorable for the evolution of intrinsic cooperators (blue cells in Fig. 9.2a), we also found that ZDISCs were almost always present at the end. Comparatively, we can say that intrinsic cooperators with k equal to a specific negative value were not a key strategy based on the fact that they were often absent in the final generations of trials where intrinsic cooperators won (e.g., the final generation only comprised DISCs with

9.3

Competition Among Gossiping Reciprocators with Various Criteria

235

k ¼ 2.5 and 0 in one trial, while it only comprised DISCs with k ¼ 4.0, 0.5, and 0 in another trial; note that only one strategy present in both populations was DISCs with k ¼ 0, i.e., ZDISCs). The result that ZDISC is the key strategy is consistent with that of previous studies (e.g., Nowak and Sigmund 1998a, in which the key strategy was those with k ¼ 0 called “stern discriminators”). Therefore, a framework of indirect reciprocity, regardless of whether information spreads via observation (Nowak and Sigmund 1998a) or via gossip (this study), would generally require intrinsic cooperators to withdraw as rapidly as possible from giving a benefit to intrinsic defectors, at which ZDISCs (k ¼ 0) are better when compared with other intrinsic cooperators (k < 0). Since there are some pieces of negative information regarding intrinsic cooperators (k  0) that are originally started by intrinsic defectors (k < 0), selfadvertisement (a ¼ 1) by intrinsic cooperators may have a certain effect on the maintenance of reciprocity among them. In this case, however, some intrinsic defectors would also engage in self-advertisement. Figure 9.2b shows that the selfadvertisement, which we assumed was assigned to two-fifths of the players in the initial population, makes reciprocity evolve under the upper limit of λ. This is in good accordance with the basic property of false self-advertisement examined using bistrategic populations (see Sect. 9.4).

9.3.2

Biased Gossipers (qG = qB = k)

For fair gossipers, qG and qB are equal to 0. Here, we assume that a player’s qG and his/her qB are not necessarily equal. It is also not necessary to assume that players speak nothing (i.e., N gossip) about those whom they have not met: some players may consider that strangers are good or bad. In this subsection, we assume qG ¼ qB ¼ k for all individuals, where everyone’s actions are in perfect accord with the gossip passed by him/her; that is, when sij
k, player i gives a benefit to player j, and when sij > qG, he or she engages in G gossip about player j. When sij < k, player i does not give a benefit to player j and engages in B gossip about player j. We call players with this property “biased gossipers”. As qG ¼ qB, the range of neutral evaluation is narrowest; that is, player i regards player j as N only when sij ¼ k in which case i chooses to give a benefit to j. Theoretically, ZDISCs are classified as a special type of biased gossipers who have k ¼ 0 thus no bias. It is also assumed that qR ¼ 5.0 and a ¼ 0. Now, all biased gossipers, except ZDISCs, initially presume either that everyone in the population is G (in case of k < 0) or that everyone is B (in case of k > 0) because all P-scores are zero at the beginning. We believe that, in reality, we often hear statements such as “Be kind to strangers” or “Do not trust strangers easily.” Intrinsic cooperators, except ZDISCs, tell their listeners to be kind to strangers, whereas intrinsic defectors tell them not to trust strangers easily until some of their P-scores are far from zero. Biased gossipers with qG ¼ qB ¼ k ¼ +5.5 do not always give a benefit to a recipient, regardless of the P-score (we call this behavior ALLD)

236

9 Can Cooperation Evolve When False Gossip Spreads?

and unconditionally tell their listeners not to trust strangers easily (we call this gossiper ALLB). Hence, biased gossipers with qG ¼ qB ¼ k ¼ +5.5 are named ALLB-ALLD. The values for ρ have no influence on their actions or gossip. Figure 9.2c indicates that biased gossipers with k < 0 can fix almost only when λ ¼ 0 and T is very large. Note that the scale of the horizontal axis in Fig. 9.2c is larger than that in Fig. 9.2a, b. Although the biased gossip surely contains appropriate and correct information, such as B gossip about intrinsic defectors and G gossip about intrinsic cooperators, the simulation results show that a conditional giving strategy (k  +5.0) hardly worked under a flood of false gossip defined as B gossip about a player spread before the player actually chooses not to give for the first time or G gossip about a player before he/she chooses to give. Therefore, there should be no gossip sessions through which non-factual information spreads when we assume the population of biased gossipers, for intrinsic cooperators (k < 0) to evolve. Figure 9.2c also shows that reciprocation using gossip did fix in a few trials. For example, the cell in T ¼ 46,200 and λ ¼ 0.2 (r ¼ 1) in Fig. 9.2c has the value 0.021, which is larger than the value of the lower cell (0.004; T ¼ 46,200; λ ¼ 0). In most of those trials, interestingly, all survivors were not ZDISCs with ρ ¼ 0 (pure direct reciprocating ZDISCs) but ZDISCs with ρ ¼ 1∕1024, which was the smallest non-zero value in our simulation setting. A ZDISC player with ρ ¼ 1∕1024 puts his/her own experience at game sessions above all else and does not change his/her good evaluation to those who have given a benefit to him/her even after repeatedly listening to bad gossip against them. This is a key property for this strategy to evolve (see Sect. 9.4 for more details).

9.4

Effects of Different Types of False Gossip on the Evolution of Reciprocity

As shown in Sect. 9.3, the ZDISC strategy, defined as the strategy set of k ¼ qG ¼ qB ¼ 0, plays an important role in the evolution of reciprocity when the population has various types of derivative strategies of ZDISC. We focus on ZDISC with a ¼ 0 and qR ¼ 5.0 (non-advertising and gossip-sensitive ZDISC strategy, called simply ZDISC hereafter in this section) and analyze a bistrategic population comprising x ZDISCs and n  x ALLDs. Any types of ALLDs are defined to have k > +5.0 so that they will choose not to give a benefit to their recipients, regardless of their P-scores of the recipients. The gossip strategies of ALLDs vary, depending on the values for qG, qB, and a. In the following, we assume five types of ALLDs: non-gossiping ALLDs, fairly gossiping ALLDs, pure self-advertising ALLDs, ALLB-ALLDs, and ALLG-ALLDs (Fig. 9.3). First, the non-gossiping ALLD strategy is defined as the strategy set of min{k, qG} > +5.0, qB  5.0, and a ¼ 0. A non-gossiping ALLD player always engages in N gossip about any players, including him/herself. Second, the fairly gossiping ALLD is, as in the previous section, defined as k > +5.0 (practically, we

9.4

Effects of Different Types of False Gossip on the Evolution of Reciprocity

237

D Non-gossiping ALLD

−5.0

0

5.0

k

sij

a=0

k

sij

a=0

N D Fairly-gossiping ALLD

−5.0

5.0

0

B

G D

Pure self-advertising ALLD

−5.0

0

k

5.0

sij

a=1

N D

ALLB-ALLD

0

5.0

k

sij

a=0

B D ALLG-ALLD

−5.0

5.0 k

0

sij

a=1

G Fig. 9.3 Graphical images of non-gossiping, fair-gossiping, pure-self-advertising, ALLB-, and ALLG-ALLDs

have assigned k ¼ +5.5) and qG ¼ qB ¼ a ¼ 0. A fairly gossiping ALLD player engages in G and B gossip of a player if his/her P-score of that player is positive and negative, respectively. Third, the pure self-advertising ALLD strategy is defined as min{k, qG} > +5.0, qB  5.0, and a ¼ 1. A pure self-advertising ALLD player only engages in false and fake gossip stating that he/she is cooperative. Fourth, the ALLB-ALLD strategy is, as in the previous section, defined as min{k, qG, qB} > +5.0 (k ¼ qG ¼ qB ¼ +5.5 here) and a ¼ 0. An ALLB-ALLD player engages in B gossip about all players. ALLB-ALLD can be considered as prejudiced toward everyone and gossips that everyone is bad. ALLB-ALLD can be interpreted as a false gossiper if he/she intensively gossips that every player is bad even in cases where his/her P-scores for some players are positive. Finally, the ALLG-ALLD strategy is defined as k > +5.0, max{qG, qB} < 5.0, and a ¼ 1. An ALLG-ALLD player engages in G gossip about all players, including him/herself. The same interpretation as in the case of the ALLB-ALLDs can be possible for ALLGALLDs. Here, these strategies believe what the speaker says (qR ¼ 5.0). Henceforth, in every simulation, ρZ is ρ of every ZDISC and ρA is ρ of every ALLD. Note that behaviors of ALLDs at any game session are completely fixed (choosing non-giving), regardless of the values of their P-scores, and the values of ρ and qR for ALLDs do not influence their decision-making at game sessions. In addition, the behaviors of the four types of ALLDs (the non-gossiping ALLD, the

238

Can Cooperation Evolve When False Gossip Spreads?

9

pure self-advertising ALLD, the ALLB-ALLD, and the ALLG-ALLD) at gossip sessions are also fixed. We will discuss the effects of different types of false gossip started by them on the evolution of reciprocity by comparing three strategies (the pure self-advertising ALLD, the ALLB-ALLD, and the ALLG-ALLD) with the non-gossiping ALLD and the fairly gossiping ALLD. The results are summarized, as shown in Fig. 9.4, in the estimated coordinates of unstable equilibria for the number of ZDISC players x*, above and below which the population is more likely to reach the states of x ¼ n ¼

a

 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

28.12 30.07 32.4 35.26 38.86 43.56 49.99 59.45 75.07 100 100

34 35 37 40 43 47 53 62 76 99 100

ρZ (A) (IBM) vs

21 22 24 27 30 35 41 51 68 99

21 22 24 27 30 35 41 51 68 99

1

1/2

21 22 24 27 30 35 41 51 68 99

21 22 24 27 30 35 41 51 68 99

57 56 56 55 56 59 62 69 82 99

52 52 51 52 53 55 60 66 80 99

1/8

1/1024

1

1/2

100

Non

T = 200 49 49 49 49 49 49 50 50 52 52 55 55 59 59 66 66 80 80 99 99

Fair

10 10 11 11 12 13 14 16 18 24 46

ρZ (A) (IBM) Vs

Non

6 6 6 6 6 6 7 7 9 13

6 6 6 6 6 6 7 7 9 13

1

1/2

1/1024

1/2

Self-advertising

5 5 5 6 6 6 7 7 9 13

5 5 5 5 6 6 7 7 9 13

38 34 31 28 26 24 22 21 22 26

31 28 25 23 21 19 19 19 20 25

1/8

1/1024

1

1/2

46 Fair

1

94 94 93 92 89 88 85 83 87 100

92 90 88 87 87 87 85 83 87 100

99 99 98 98 98 98 97 97 97 100

98 98 98 98 98 97 96 96 95 100

1/8

1/1024

1

1/2

100 1/8

b 7.77 8.16 8.63 9.2 9.91 10.83 12.06 13.86 16.78 22.64 45.3

96 95 95 94 94 92 91 88 88 100

100

 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

96 96 96 95 95 94 93 92 92 100

95 95 95 95 94 93 92 90 85 70

94 94 94 94 93 92 90 85 78 58

1/8

1

1/2

46 Self-advertising

1/8

1/1024

0

100

0

100

ALLG

94 93 93 90 87 84 73 62 49 30

31 31 31 31 30 29 28 27 27 27

98 98 98 98 98 97 97 96 95 91

98 98 98 98 97 97 97 96 94 86

1/8

1/1024

1

1/2

46 1/1024

97 97 97 97 97 96 96 95 95 100

100

ALLB

T = 1000 21 18 19 18 18 17 17 17 17 17 17 17 17 17 18 18 20 20 25 25

98 98 97 97 97 96 96 95 95 100

98 98 98 97 97 97 96 93 83 53

46 46 46 46 46 46 46 45 45 44

1/8

1/1024

46

ALLB

ALLG

Fig. 9.4 Graphical tables for the expected minimum number of ZDISC players required for the fixation of the ZDISC strategy (i.e., the coordinate of unstable equilibrium x*) in a bistrategic population of ZDISCs and one of the following five types of ALLDs: non-gossiping ALLDs (indicated by “Non” in the bottom row), fairly gossiping ALLDs (“Fair”), pure self-advertising ALLDs (“Self-advertising”), ALLB-ALLDs (“ALLB”), and ALLG-ALLDs (“ALLG”). Blue, yellow, and red backgrounds indicate zero, 50, and 100, respectively. Each x* was obtained using numerical simulations of the analytical model only if the column is indicated with “(A)” in the second-bottom row, and using individual-based simulations otherwise. The column indicated with “(IBM)” is also the outcomes of the individual-based simulations. We obtained x* using individualbased simulations as follows: for each parameter set of (T, ρZ, λ) corresponding to each cell, we examined 99 types of different populations comprising different numbers of the ZDISCs (x) from x ¼ 1 to x ¼ 99. For each x-value, we ran 999 repeated simulations (defined as “the same-x simulation set”) for one generation. In each trial, the average payoffs for ZDISCs and ALLDs at the end (i.e., at the T-th time step) were compared. We regarded that the ZDISC strategy is more likely to fix for the x-value if an average ZDISC player had more payoffs than an average ALLD in more than a half of runs (i.e., at least 500 runs) of the same-x simulation set, and explored the expected minimum number x*. The trials were performed for various ρZ and λ (r). Note that the results of the bistrategic population, including non-gossiping ALLDs, are independent of the value of ρZ. The other parameter values are n ¼ 100, b∕c ¼ 10, g ¼ 5, qR ¼ 5.0 for all players, ρA ¼ 0, and (a) T ¼ 200 or (b) T ¼ 1000

9.4

Effects of Different Types of False Gossip on the Evolution of Reciprocity

239

100 and x ¼ 0, respectively. Thus, the ZDISC strategy is more likely to fix with a lower coordinate when the bistrategic population contains ZDISC players more than x*.

9.4.1

ZDISCs Versus Non-gossiping ALLDs

First, we consider a bistrategic population of ZDISCs with ρ > 0 and non-gossiping ALLDs using an approximated deterministic mathematical model. ZDISCs only spread gossips, and it is not needed to consider how fair gossips spread by ALLDs influence P-scores because non-gossiping ALLDs do not spread fair gossips about others. As a result, the mathematical model can be made. Consider a population that comprises x ZDISC players and n  x non-gossiping ALLD players (1 < x < n  1). We neglect any types of stochastic effect to obtain differential equations, which would approximately hold for a sufficiently large population. We focus on a particular ALLD individual and categorize all ZDISC individuals at the same generation based on their image scores of ALLD. In this section, there are no factors that improve the image of ALLD. It follows that every ZDISC shall have either a zero (N ) or a negative (B) image score of ALLD. Hence, we consider only two classes, the xN- and the xB-classes, which comprise ZDISCs that have N and B images of ALLD, respectively. Every ZDISC belongs to the xN-class at the beginning of each generation (t ¼ 0) and the flow of individuals is one way from the xN- to the xB-class. Here, we subdivide the xB-class into two groups, the xg-class and the xr-class, according to whether a ZDISC has entered the xB-class directly (through his/her own experience with ALLD) or indirectly (by receiving B gossip from another ZDISC). Specifically, if an xN-ZDISC is chosen as a recipient at a giving-game where the potential donor is the focal ALLD, then the xN-ZDISC is changed into the xg-ZDISC. In contrast, if the xN-ZDISC is chosen as a listener of a speaker who belongs to the xB-class, he/she enters into the xr-class. The expected numbers of individuals of the above-mentioned classes at time t (0  t  T ) are denoted by xN(t), xB(t), xg(t), and xr(t), where xN(t) + xB(t) ¼ x, xB(t) ¼ xg(t) + xr(t), xN(0) ¼ x, and xB(0) ¼ xg(0) ¼ xr(0) ¼ 0. Note that, with regard to within-generation dynamics of these classes, x is referred to as a constant parameter. In a short time interval [t, t + Δt], giving-game sessions and gossip sessions occur at ratios of gΔt and rΔt, respectively. In that interval, the probability that a game where the donor is ALLD and the recipient is a ZDISC of the xN-class occurs is gΔt

1 xN ðt Þ ¼ wg xN ðt ÞΔt: n n1

ð9:5Þ

On the other hand, the probability that a gossip where the speaker is an xB-ZDISC and the listener is an xN-ZDISC occurs is

240

9

rΔt 

Can Cooperation Evolve When False Gossip Spreads?

xB ðt Þ xN ðt Þ  ¼ wr ½x  xN ðt ÞxN ðt ÞΔt, n n1

ð9:6Þ

Equations (9.5) and (9.6) correspond to the flows of ZDISC individuals from the xNclass into the xg- and xr-classes, respectively, within the time interval. Taking the limit of Δt ! 0, we have the following differential equations:
   dxN ðt Þ ¼  wg þ wr x  wr xN ðt Þ xN ðt Þ, dt

ð9:7aÞ

dxg ðt Þ ¼ wg xN ðt Þ, dt

ð9:7bÞ

dxr ðt Þ ¼ wr ½x  xN ðt ÞxN ðt Þ, dt

ð9:7cÞ

When r ¼ 0, xr-ZDISCs never appear and Eq. (9.7a) is reduced to a simple exponential growth model with a negative growth rate, where xN ðt Þ ¼ xewg t , x g ð t Þ ¼ x ð1  e

wg t

ð9:8aÞ Þ,

xr ðt Þ ¼ 0,

ð9:8bÞ ð9:8cÞ

When r > 0, Eq. (9.7a) can be considered a logistic growth equation with a negative growth rate. First, xg(t) grows linearly with xN(t)  x while the increase of xr(t) is suppressed by the term x  xN(t)( xr(t) for small t. The gap between them may be lessened as time advances and xN(t) decreases, and xr(t) may overtake xg(t) at some point. In fact, Eqs. (9.7a), (9.7b) and (9.7c) have general solutions as below: 1 þ λx x, λx þ eð1þλxÞwg t
 ln 1 þ λxeð1þλxÞwg t xg ð t Þ ¼  þ xg ð1Þ, λ xN ð t Þ ¼

xr ð t Þ ¼ x  x N ð t Þ  x g ð t Þ ,

ð9:9aÞ ð9:9bÞ ð9:9cÞ

where λ ¼ r∕g and xg(1) ¼ (1 ∕ λ) ln (1 + λx). xN(t), xg(t), and xr(t) monotonically approach 0, xg(1), and x  xg(1), respectively, with increasing t. At a certain time step t, the relative frequencies xN(t)∕x and xg(t)∕x decrease, whereas xr(t)∕x increases with increasing λx. Thus, the xr-ZDISCs occupy a relatively larger proportion with the larger gossip/game ratio or the larger total number of ZDISCs at any time step. Obviously, from the fact that Eqs. (9.9a), (9.9b) and (9.9c) is reduced to Eqs. (9.8a), (9.8b) and (9.8c) when we take the limit of λ ! 0, the monotonicity of these relative frequencies with respect to λ is valid for λ
0. In addition, according to our

9.4

Effects of Different Types of False Gossip on the Evolution of Reciprocity

241

numerical analysis, the population converges to the equilibrium state satisfying xg(t) ¼ xg(1) more quickly with increasing λ or decreasing n. When the same value is assigned to n, increasing x seems to hasten the convergence. This is because two ZDISCs meet at gossip sessions more frequently and the number of “idle talks,” where no information is transferred as a speaker and/or a listener is an ALLD, is reduced. Note that xN(t)∕x is the same as an average giving rate when a ZDISC is chosen as a donor and an ALLD is chosen as a recipient in a game session occurring at t. In this sense, the aforementioned within-generation reputation dynamics determine the direction of between-generation evolutionary dynamics of the bistrategic population. In both cases of r ¼ 0 and r > 0, the focal ALLD is expected to be given a payoff benefit from wgxN(t) ZDISCs at each interval [t, t + Δt]. His/her payoff at the end of the generation (t ¼ T ) is calculated as Z

T

bwg xN ðt Þdt ¼ bxg ðT Þ,

ð9:10Þ

t¼0

which can be considered an average accumulated payoff for ALLDs. In contrast, a ZDISC continues to give a benefit to and be given a benefit by each of the other x  1 ZDISCs, although he/she may give a benefit to some ALLDs in vain. To calculate the average value of the latter, the total payoff losses of x ZDISCs caused by n  x ALLDs are divided by x: Z

T t¼0

Z bwg ðx  1Þdt 

T

cwg ðx  1Þdt 

t¼0

¼ ðb  cÞwgðx  1ÞT 

nx x

nx cxgðT Þ: x

Z

T

cwg xn ðt Þdt

t¼0

ð9:11Þ

We define the function δ as the difference between Eqs. (9.10) and (9.11) (the latter minus the former), where   cn δ ¼ ðb  cÞwg ðx  1ÞT  b  c þ x ðT Þ: x g

ð9:12Þ

As long as the reproductive success of an individual is proportional to his/her total payoff, ZDISCs are likely to increase in frequency in the next generation if and only if δ > 0. The frequencies of the two strategies may not change if δ ¼ 0. Note that xg(T ) is bounded with respect to T and its value is at most x. Hence, if we choose a sufficiently large T, δ can be positive for any set of the other parameters. More importantly, it can be shown that dδ∕dx > 0 for all x(>1) and δ|x ! 1 < 0. It follows that there is at most one internal equilibrium, which is always unstable if it exists. The coordinate of the unstable equilibrium x ¼ x*(b, c, g, r, n, T) can be obtained implicitly by solving δ ¼ 0 for x numerically. If x* exists in the range of 1 < x* < n, then the population may asymptotically approach either of the two trivial equilibria,

242

9 Can Cooperation Evolve When False Gossip Spreads?

x ¼ 0 or x ¼ n, depending on the initial number of ZDISCs. Otherwise, the latter state is unstable and ZDISCs may monotonically decrease to disappear. The analytical results and Fig. 9.4 indicate that ZDISCs are more advantageous for larger T in the absence of any gossip sessions and that the gossip from ZDISCs helps the evolution of ZDISCs for large T and high λ in the presence of any gossip sessions. The reasons are as follows. When λ ¼ 0, i.e., in the absence of any gossip sessions, the number of ZDISCs that possess negative P-scores against each ALLD players grows negative exponentially (Eqs. (9.8a), (9.8b) and (9.8c)). In this case, every ZDISC behaves as a direct reciprocator. A ZDISC continues to give a benefit to an ALLD until the ZDISC is chosen as a recipient of the same ALLD at a givinggame session, where ALLD does not give a benefit to ZDISC; thereafter, ZDISC chooses not to give benefit to ALLD. As every player has the same chance to be a potential donor or a recipient, each ZDISC is expected to give a benefit to each ALLD 1∕2 times, regardless of the magnitude of T. Meanwhile, every pair of ZDISCs continues to give a benefit to one another until the end of the generation. Hence, ZDISCs are more advantageous for larger T due to their mutual cooperation. When λ > 0, i.e., in the presence of gossip sessions, some ZDISCs stop giving a benefit to an ALLD more quickly after receiving B gossip regarding that ALLD player than when r ¼ 0. In this case, the number of ZDISCs that have a negative P-score against ALLDs increases as a sigmoidal function of time (Eqs. (9.9a), (9.9b) and (9.9c)). Note that the value of ρZ (> 0) has no influence on the outcome in this section. This is because ZDISCs use the zero criterion and all pieces of information regarding each player that are available for a ZDISC have the same direction. More specifically, ZDISC is always given to (respectively, not given to) by ZDISCs (resp. by ALLDs), and thus ZDISC engages in only G (resp. B) gossip about ZDISCs (resp. ALLDs). Even though a ZDISC does not give a benefit to an ALLD, any B gossip about ZDISC does not spread, since ALLDs in this subsection do not gossip. Consequently, there are only two types of gossip: G gossip about ZDISCs and B gossip about ALLDs. After a ZDISC’s P-score against a player becomes either positive or negative at a giving-game or gossip session, its sign never changes throughout a generation. Further, G gossip about ZDISCs is meaningless because it spreads only among ZDISCs and never influences ZDISCs’ actions of giving a benefit to those with non-negative P-scores. What matters is only how fast a ZDISC player acquires the first piece of negative information about each ALLD. The spreading rate of the negative information is higher for larger λ. A positive relation exists between the spreading rate and the number of ZDISCs in the population, x, when λ > 0 because it is only ZDISCs that spread gossip. For both the cases λ ¼ 0 and λ > 0, we can numerically obtain the coordinate x ¼ x* above which an average ZDISC is expected to have a higher accumulated payoff than an average ALLD at the end of each generation (see Eqs. (9.5)–(9.12)). Note that x* is a function of parameters b, c, g, r, n, and T. If we neglect a stochastic effect due to the finite population size, then we can regard the threshold x ¼ x* as the unstable equilibrium of evolutionary dynamics above which a population is likely to reach x ¼ n and below which it is likely to reach x ¼ 0. The threshold curves x ¼

9.4

Effects of Different Types of False Gossip on the Evolution of Reciprocity

243

x∗ 100 90 80 70

=0

60

= 0.2

50

= 0.6

40

=1

30

= 1.4 = 1.8

20 10 0 0

100

200

300

400

500

600

700

800

900

1000

T

Fig. 9.5 Plots of the expected minimum number of ZDISC players required for the fixation of the ZDISC strategy in a bistrategic population of ZDISCs and non-gossiping ALLDs (i.e., the coordinate of unstable equilibrium x*) against the length of each generation (T ). The curves and points are obtained by numerical simulations of the analytical model and individual-based simulations, respectively. The calculations were performed for various r. The other parameter values are n ¼ 100, b∕c ¼ 10, g ¼ 5, qR ¼ 5.0 for all players, ρZ ¼ 1, and ρA ¼ 0

x*(T ) for various λ (i.e., various r) are plotted in Fig. 9.5. Figure 9.5 indicates that each threshold curve crosses the line x ¼ n  1 (x ¼ 99 in Fig. 9.5) at a point T ¼ T*, which means that the ZDISC strategy is evolutionarily stable against invasion by a single player following the ALLD strategy for T > T*. The range of T within which ZDISC can be evolutionarily stable against non-gossiping ALLD is wider when λ is larger. It is also shown that each curve approaches asymptotically to x ¼ 1 with increasing T. In short, the basin of attraction for x ¼ n is enlarged with increasing T or λ. This is in accord with the intuition that ZDISCs more easily beat ALLDs with larger iteration of the games or gossip, which increases the accumulated payoff of each ZDISC with additional game sessions in which ALLDs are not given to. The increased iteration of gossip decreases the payoff of each ALLD by accelerating the propagation of negative information about him/her. Among the curves plotted in Fig. 9.5, the curve x ¼ x*(T) for λ ¼ 0 needs much larger T than others to cross x ¼ n  1 and to converge to x ¼ 1. The same curve is obtained in the analysis of bistrategic populations of ZDISCs and any type of ALLDs when we assign either ρZ ¼ 0 or λ ¼ 0. We call it the curve for pure direct reciprocators and deviation from this curve indicates the magnitude of advantage or disadvantage of gossiping. When a part of a curve for λ > 0 lies above that for λ ¼ 0, ZDISCs need to stop gossiping and behave as pure direct reciprocators for such a parameter range. If each player can hardly control the frequency of gossip sessions, then each ZDISC may simply reduce the value of ρ to zero. Figure 9.5 shows that such a region does not appear when ALLDs do not spread any gossip.

244

9

Can Cooperation Evolve When False Gossip Spreads?

The effect of population size is discussed (see Fig. S2 in Seki and Nakamaru (2016)). In short, gossiping is more effective for larger population size, although it might not be realistic when n is large to assume that each player completely memorizes the n  1 P-scores.

9.4.2

ZDISCs Versus Fairly Gossiping ALLDs

ZDISCs had more difficulty beating fairly gossiping ALLDs when we assigned ρA ¼ 0 than when we assigned the other values for ρA, and the results for ρA ¼ 0 are shown in this subsection. Decline of cooperation rates within the initial 500 time steps in Fig. 9.6 indicates that ZDISCs can effectively exclude fairly gossiping ALLDs from their reciprocal relationship with B gossip about ALLDs at an earlier stage of each generation. B gossip, which is started not only by ZDISCs but also by fairly gossiping ALLDs, promotes ZDISCs’ not giving a benefit to ALLDs. Figure 9.6 also shows that reciprocity among ZDISCs begins to collapse at a certain point in time (around the 2000th time step), which occurs a bit earlier for larger λ. This is due to B gossip about ZDISCs that is started by fairly gossiping ALLDs when they choose not to give benefit to the fairly gossiping ALLDs. Though G gossip about a ZDISC can maintain the other ZDISCs’ positive P-scores of ZDISC for a while, repeatedly invading B gossip about ZDISC started by ALLDs may finally change

Cooperation rate 0.6 0.5 0.4

=0 = 0.6 =1 = 1.8 (

0.3 0.2

(

0.1 0

0

500

1000

1500

2000

2500

3000

Time step within a generation

Fig. 9.6 Changes in the rate of a donor’s giving a benefit in one generation of a bistrategic population of ZDISCs and fairly gossiping ALLDs (the vertical axis) against time steps within one generation (the horizontal axis). These are obtained using individual-based simulations. A simulation with a certain parameter set involved 999 repetitions and their average was plotted. The calculations were performed for various r. The other parameter values are n ¼ 100, x ¼ 50, b∕c ¼ 10, g ¼ 5, qR ¼ 5.0 for all players, ρZ ¼ 1, and ρA ¼ 0

9.4

Effects of Different Types of False Gossip on the Evolution of Reciprocity

245

ZDISCs’ minds. Note that Fig. 9.6 shows the results when half the population is ALLD who start B gossip about ZDISCs (n ¼ 100 and x ¼ 50). However, ZDISCs have an advantage over the fairly gossiping ALLDs for larger λ within the range of T that we investigated (Fig. 9.4). Similar to the previous subsection, the larger λ makes each ZDISC give no benefit to each fairly gossiping ALLD at an earlier stage (compare the initial declines of the giving rates in Fig. 9.6). Moreover, even for large λ, ZDISCs can maintain the reciprocal relation among them for a while. An average ZDISC accumulates a higher payoff than an average ALLD during the time period until the collapse of the reciprocal relation starts with our choice of b∕c ¼ 10. Consequently, the collapse hardly affects the evolutionary outcome summarized in Fig. 9.4. We confirmed that the collapse does not occur for smaller T (T < 2000 in Fig. 9.6) or larger x (not shown). In the case of larger x, meaning that there are more ZDISCs than fairly gossiping ALLDs, the inflow of B gossip about a ZDISC into the other ZDISCs’ P-scores is small because this B gossip is started by the minority (i.e., fairly gossiping ALLDs). This B gossip can be wiped out by more G gossip that the majority (i.e., ZDISCs) starts and spreads. In addition, we investigated whether the dependency of giving rates on ρZ. ZDISCs with small ρZ can keep, or at least prolong, the altruistic relations even for larger λ (not shown). However, our simulation results suggest that it hardly affects the form of the threshold curve for at least T  1000, again indicating that the collapse hardly affects the evolutionary outcome. It is also noteworthy that the value of ρZ has no positive or negative influence on the initial spreading of negative images against fairly gossiping ALLDs, which is the most important factor for the evolution of reciprocity in this bistrategic population.

9.4.3

ZDISCs Versus Pure Self-Advertising ALLDs

It is possible that some players deliberately introduce false and fake information to benefit themselves at future game sessions. As the simplest example of such lies, we examined a self-advertisement given by ALLD players. A pure self-advertising ALLD is defined as one who only chooses defection, only engages in false gossip that he/she is a good player, and never engages in gossip about the other players. We obtained the average giving rates at each time step similar to Fig. 9.5 (not shown), in which every curve of the giving rates monotonically decreases as time advances, and it flattens after certain time steps. The initial decline of the giving rate is faster for larger λ; however, the convergence level is higher for larger λ. It is only when λ ¼ 0 that the giving rate converges to x(x  1) ∕ [n(n  1)] corresponding to the state in which no ZDISCs give benefits to any self-advertising ALLDs. The estimated threshold curves x ¼ x* are also obtained using individual-based simulations. As extracted in Fig. 9.4 and detailed in Fig. 9.7, the dependency of the threshold on T and λ is not monotonic. When T was small (T ¼ 200 in Fig. 9.4; T < 250 in Fig. 9.7), larger λ was favored for the ZDISC strategy. The ZDISC

246

9

Can Cooperation Evolve When False Gossip Spreads?

x∗ 100 90 80 70 60 50 40 30

=0 = 0.6 =1 = 1.8

20 10 0

Fig. 9.7 Plots of the expected minimum number of ZDISC players required for the fixation of the ZDISC strategy in a bistrategic population of ZDISCs and pure self-advertising ALLDs (i.e., the coordinate of unstable equilibrium x*) Against the length of each generation (T ). These are obtained using individual-based simulations. To obtain the coordinate using individual-based simulations, we specifically tested 99 types of the initial number of ZDISCs (x) from x ¼ 1 to x ¼ 99 for every parameter set. For each initial number, simulation runs for one generation were repeated 999 times. At the end of each generation, average payoffs for ZDISCs and ALLDs were calculated. If an average ZDISC player had more payoff than an average ALLD in more than 500 runs, then we regarded that the ZDISC strategy is more likely to fix and vice versa. The coordinate is strictly defined as “n minus [the number of initial frequencies with which the ZDISC strategy is more likely to fix] minus 0.5.” The trials were performed for various λ (r). The other parameter values are n ¼ 100, x ¼ 50, b∕c ¼ 10, g ¼ 5, qR ¼ 5.0 for all players, ρZ ¼ 1, and ρA ¼ 0

strategy was stable against the invasion by a few ALLD players with smaller T, and initial decline of x*(T) was more rapid for larger λ. When T took an intermediate value (T ¼ 1000 in Fig. 9.4; T > 250 in Fig. 9.7), a moderate iteration of gossip sessions (0 < λ  1) was more favored than the plethora of gossip sessions (λ > 1) or the total withdrawal from gossiping (λ ¼ 0). When T was very large, λ ¼ 0 was most favored (not shown in Figs. 9.4 and 9.7). For such a small T that no one is chosen as a listener more than once, larger λ may make ZDISC more advantageous over pure self-advertising ALLD as well as in the previous subsections. This is because a ZDISC player i chooses to give benefit to an ALLD player j not only when sij > 0 but also when sij ¼ 0. In this sense, the positive but fake image for player j created by player j does not affect decisionmaking at game sessions by ZDISCs, and the larger λ may only promote the spread of negative images against each ALLD and non-giving to him/her by ZDISCs. If T is a bit larger, however, the dependency of evolutionary dynamics on λ is not so simple because a player that once received B gossip about an ALLD player can receive G gossip about the same ALLD player. To overview what occurs in that case, a simplified version of our model, where P-scores were not continuous but

9.4

Effects of Different Types of False Gossip on the Evolution of Reciprocity

247

trichotomous (1, 0, or 1) and ρZ was fixed to 1, was analyzed as follows. Here, as ZDICSs only spread fair gossips about other players and self-advertising ALLDs only start their own fake gossips, the mathematical analysis is possible; it is not needed to consider how fair gossips spread by ALLDs influence P-score because ALLDs here do not spread fair gossips about others. Consider a population that comprises x ZDISC individuals and n  x selfadvertising ALLD individuals (1 < x < n). For simplicity, we assume that all ZDISCs have the same numeric strategy ρ ¼ 1; that is, they value the evaluations exchanged in the dialogues as highly as their own experiences. In addition, we make a minor modification to our model: the maximum and minimum values of the image scores are +1 and  1, respectively, in this section. Consequently, every P-score can take one of three states: 0 (N ), 1 (B), or + 1 (G). Similarly to the analysis in Sect. 9.4.1, a particular ALLD individual is focused on and all ZDISCs are classified according to their image scores of ALLD. In this section, there are three classes of ZDISCs: the xN-, xB-, and xG-class. In contrast, xN-ZDISCs and xG-ZDISCs change into the xB-ZDISCs and the xN-ZDISCs, respectively, either when ALLD gives them no benefit or when they hear B gossip about ALLD from xB-ZDISCs. On the other hand, xN-ZDISCs and xB-ZDISCs change into xG-ZDISCs and xN-ZDISCs, respectively, when they receive false and fake gossip from ALLD or G gossip of ALLD from xG-ZDISCs. The expected numbers of individuals in the three classes at time t are denoted by xN(t), xB(t), and xG(t), where xN(t) + xB(t) + xG(t) ¼ x. Using this restriction, the dimension of this system can be reduced from third to second. The within-generation dynamics of ZDISCs on the phase plane {(xB, xG)| xB
0, xG
0, 0  xB + xG  x} is described as follows:  dxB ðt Þ
¼ wg þ wr xB ðt Þ xN ðt Þ  wr ½1 þ xG ðt ÞxB ðt Þ, dt
 dxG ðt Þ ¼ wr ½1 þ xG ðt ÞxN ðt Þ  wg þ wr xB ðt Þ xG ðt Þ, dt

ð9:13aÞ ð9:13bÞ

where (xB(0), xG(0)) ¼ S0 ¼ (0, 0). Note that xG(t) is constantly zero and Eqs. (9.13a) and (9.13b) is the same as Eqs. (9.7a), (9.7b) and (9.7c) when r ¼ 0. It can be analytically shown that there are at most three equilibria onto the phase plane, the coordinates of which depend only on λ (¼ r∕g) and x. When λ ¼ 1, Eqs. (9.13a) and (9.13b) are symmetric and an equilibrium SC ¼ (x∕3, x∕3) always exists. If x > 3 (i.e., in most cases), then two locally stable equilibria are additionally found, SB ¼ (α, 1∕α) and SG ¼ (1∕α, α), where α¼

x1þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð x þ 1Þ ð x  3Þ , 2

ð9:14aÞ

248

9

Can Cooperation Evolve When False Gossip Spreads?

1 x1 ¼ α

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð x þ 1 Þ ð x  3Þ : 2

ð9:14bÞ

Note that x – 2 < α < x  1, 1∕α < 1, and xN(1) ¼ x  (α + 1∕α) ¼ 1. Thus, almost all ZDISCs have negative images of ALLD at the state SB while most have positive images of him/her at the state SG. In any case, the trajectory starting from S0 approaches to SC straight along the diagonal line xG ¼ xB. However, SC is a saddle equilibrium that is stable only against perturbations along the line of symmetry (i.e., xG ¼ xB) whenever SB and SG exist. According to our numerical study, trajectories released from the diagonal quickly leave it and go closer to one of those two stable equilibria. Hence, in a finite population, each state of ZDISCs’ P-scores of each ALLD may eventually fall into either SB or SG due to a small stochastic effect. For any λ, we define, of the three possible equilibria, SC as a saddle equilibrium and SB and SG as the states that have more and less xB, respectively, than SC. When λ < 1, the numerical analysis suggests that the trajectory rooted in S0 asymptotically reaches SB. In this case, xB(t) and xG(t) are a monotonically increasing function and a one-humped function of time, respectively. This also suggests that the coordinates of SB asymptotically approach to (x, 0) with decreasing λ. Note that the coordinates of SB are exactly (x, 0) when λ ¼ 0. Conversely, when λ > 1 the orbit connects the initial state S0 and the state SG, coordinates of which asymptotically approach the boundary (0, x) with increasing λ. The one-humped function xB(t) converges to nearly zero, while xG(t) monotonically increases to nearly x. In any cases of λ 6¼ 0, however, a slight deviation from the expected trajectory at an earlier stage of the generation (i.e., near S0) can lead the population to an opposite steady state in our numerical simulations. As long as the stochastic drift is neglected, the difference of average payoffs for ALLDs and manner as in Sect. 9.4.1, by R R ZDISCs can be obtained in the same replacing xN(t)dt of Eqs. (9.10) and (9.11) by [xN(t) + xG(t)]dt: 

cn δ ¼ ðb  cÞwg ðx  1ÞT  b  c þ x

Z

T

½xN ðT Þ þ xG ðT Þdt:

ð9:15Þ

t¼0

When ZDISCs’ image scores of every ALLD converge to the state SB (i.e., in the case of λ < 1) the delta of the payoffs δ is positive for sufficiently large T, as in Sect. 9.4.1. In contrast, when they converge to SG (i.e., λ > 1), δ can be positive only for an intermediated range of T, or always negative. This is because the number of “suckers” (i.e., xN(t) + xG(t)) is larger than x  1 at the equilibrium state SG. If the P-scores of an ALLD converge to this, then ALLD is expected to gain more payoff than an average ZDISC. Hence, the condition λ  1 is highly important for the ZDISC strategy to beat the self-advertising ALLD strategy. Note that the optimal value of λ for the evolution of the ZDISC strategy is not always zero. In the numerical simulation, a moderate iteration of gossip (i.e., 0 1, respectively, in the deterministic model. This result indicates that reputation with the higher initial spreading rate (G gossip about ALLD when λ > 1; B gossip about ALLD when λ < 1) wipes out the other (B gossip about ALLD when λ > 1; G gossip about ALLD when λ < 1) eventually. We also found that, as far as λ > 0, the evaluation dynamics are sensitive to perturbations at an earlier stage of each successive generation. Hence, under the presence of a stochastic effect, it is possible that ZDISCs’ P-scores against some ALLDs reach SB and their P-scores against the other ALLDs reach SG. In the framework of the present model, a ZDISC is better off choosing a smaller but non-zero ρ. Though ZDISCs cannot avoid starting and spreading G gossip about ALLD, each ZDISC can make the ZDISC’s P-score to ALLD permanently negative if ALLD does not give to ZDISC at least once. Even under abundant false advertisement about ALLD, those ZDISCs continuously spread true B gossip about him/her without changing their minds, weakening the spread of the opposite G gossip about him/her. P-scores of every ALLD are more likely to approach the state SB when ρZ 0 and λ ¼ 0 in Fig. 9.4). Note that changing the value of a of some ZDISCs from 0 to 1 and allowing them to self-advertise do not influence the outcome because there is no negative information about them in this bistrategic population. It is also notable that for a ZDISC with ρG ¼ 0 and ρB > 0, the self-advertising ALLG-ALLDs do not differ from the non-gossiping ALLDs.

9.5

9.5

Effects of Selecting Gossip Based on the Trustworthiness of Speakers

251

Effects of Selecting Gossip Based on the Trustworthiness of Speakers

Cooperation may evolve in a wider parameter range compared with what was obtained in the previous section, in the case where ZDISCs do not believe bad players’ words. This is because a stream of gossip unfavorable to ZDISCs is always started by ALLDs, some of which are already categorized as bad by their listeners due to their own actions. Such a strategy for listeners is, to some degree, similar to a standing strategy in previous studies. Players with a standing strategy initially assign a “good standing” to all players in the population and discount an uncooperative action against the player with “bad standing,” or non-giving against the player that has chosen not to give benefit to a player with “good standing” (Sugden 1986). For this strategy to be realized, players need to observe not only “who defected” but also “who defected against whom,” which is called second-order information. In contrast, the strategy that uses only first-order information came to be called “scoring” after the original study of image scoring by Nowak and Sigmund (1998a, 1998b). In our framework of the gossiping model, gossip stories are regarded as alternatives of first-order information. However, players are not provided with second-order information. The only additional information is who speaks to them (i.e., the listener’s P-score against the speaker). Thus far, in the previous sections, every player was assumed to have qR ¼ 5, with which they indiscriminately (i.e., regardless of their P-scores against their speakers) incorporated all gossip into their P-score. In this section, we show the results of individual-based simulations in which we considered bistrategic populations comprising one of four types of the gossiping ALLDs that appeared in the previous section and one of four types of ZDISCs (Fig. S3 in Seki and Nakamaru (2016)): ZDISCs with qR ¼ 5, those with qR ¼ 1∕2 (more generally those with qR ¼ qgenerous, where 1 < qgenerous < 0) regarded as generous choosy listeners, those with qR ¼ 0 regarded as precise choosy listeners, and those with qR ¼ ρZ (more generally those with qR ¼ qstrict, where 0 < qstrict  min{1, ρZ}) regarded as strict choosy listeners. We assigned ρZ ¼ 1∕1024 for all of the following simulations. Consequently, the ZDISCs with qR ¼ 1∕2 and those with qR ¼ 0 initially believe what everyone gossips about and stop believing the players that have not given a benefit to them at least once. In addition, the ZDISCs with qR ¼ 0 stop believing those whom they have heard are B. In contrast, the ZDISCs with qR ¼ ρZ only believe those whom have given them a benefit in the past giving-game sessions or those whom they have heard (from the “trustworthy” players) are G in past gossip sessions. We analyzed 16 combinations of bistrategic populations. We chose those four strategies among many other ZDISC strategies with different qR because they effectively cover the characteristics of the whole set of strategies (see Appendix D in Seki and Nakamaru (2016)). Let us first focus on the estimated coordinates of unstable equilibria obtained using individual-based simulations of bistable populations of the ZDISCs with qR ¼ ρZ and one type of five gossiping ALLDs shown in Fig. 9.8, in which a higher

252

9

Can Cooperation Evolve When False Gossip Spreads?

Fig. 9.8 Graphical tables for the expected minimum number of ZDISC players required for the fixation of the ZDISC strategy (i.e., the coordinate of unstable equilibrium x*) in a bistrategic population of ZDISCs and one of the following five types of ALLDs: fairly gossiping ALLDs (indicated by “Fair” in the bottom row), pure self-advertising ALLDs (“Self-advertising”), ALLBALLDs (“ALLB”), and ALLG-ALLDs (“ALLG”). Blue, yellow, and red backgrounds indicate zero, 50, and 100, respectively. Each x* was obtained using individual-based simulations as follows: for each parameter set of (T, qR, λ) corresponding to each cell, we examined 99 types of different populations comprising different numbers of ZDISCs (x), from x ¼ 1 to x ¼ 99. For each x-value, we ran 999 repeated simulations (defined as “the same-x simulation set”) for one generation. In each trial, the average payoffs for ZDISCs and ALLDs at the end (i.e., at the T-th time step) were compared. We regarded that the ZDISC strategy is more likely to fix for the x-value if an average ZDISC player had more payoffs than an average ALLD in more than a half of runs (i.e., at least 500 runs) of the same-x simulation set, and vice versa, and explored the expected minimum number x*. The strict definition of x* obtained using individual-based simulations was “n minus [the number of same-x simulation sets with which the ZDISC strategy is more likely to fix]”. The trials were performed for various qR and λ (r). The other parameter values are n ¼ 100, b∕c ¼ 10, g ¼ 5, ρZ ¼ 1∕1024, ρA ¼ 0, and (a) T ¼ 200 or (b) T ¼ 1000

coordinate indicates more difficulty for the evolution of the ZDISC strategy. Under the settings of bistrategic populations, the ZDISCs with qR ¼ ρZ reject gossip from any ALLD, and thus ALLDs’ gossiping strategies are totally neutralized. Hence, the estimated coordinate of an unstable equilibrium is independent from the types of gossiping ALLDs. However, the coordinates obtained from this bistrategic population are always higher than that obtained from the bistrategic population of ZDISCs with qR ¼ 5 and non-gossiping ALLDs (Fig. 9.8) because each ZDISC with qR ¼ ρZ does not believe the ZDISC with qR ¼ ρZ whom he/she does not yet know. This means that against the non-gossiping ALLDs, the ZDISC strategy with qR ¼ 5 is

9.5

Effects of Selecting Gossip Based on the Trustworthiness of Speakers

253

stable in a wider range of parameters or initial frequencies than the ZDISC with qR ¼ ρZ strategy. However, the ZDISCs with qR ¼ ρZ are more favored than the other three types of ZDISCs when T is sufficiently large (Fig. 9.8b) and ALLDs adopt the gossiping strategy with which the gossiping system is largely confused, such as the ALLB or the ALLG strategy.

9.5.1

ZDISCs Versus Fairly Gossiping ALLDs

The simulation results suggest that each of three types of ZDISC strategies with qR
1∕2 is evolutionarily stable in almost the same or even narrower ranges of T against the fairly gossiping ALLDs compared with ZDISCs with qR ¼ 5 (not shown). This may be a counterintuitive result because it is expected that the decrease in the rate of cooperation among ZDISCs, which begins with an ALLD’s B gossip about a ZDISC having not given a benefit to him/her (as shown in Fig. 9.6), can be avoided if ZDISCs adopt qR
1∕2 instead of qR ¼ 5. Indeed, for the reason argued above, ZDISCs with qR ¼ ρZ never meet such a disruption of cooperative relations. The reason why ZDISCs with qR
1∕2 are sometimes less advantageous over fairly gossiping ALLDs than ZDISCs with qR ¼ 5 is that the fairly gossiping ALLDs engage in not only B gossip about ZDISCs but also B gossip about some of the other ALLDs, all of which can be rejected by the ZDISCs with qR
1∕2. The results suggest that, in terms of evolutionary stability, rapid withdrawal from giving a benefit to ALLDs is more important than the avoidance of corruption in the cooperative relationships among ZDISCs as long as we set a high benefit–cost ratio (i.e., b∕c ¼ 10). We should also note again that the corruption of cooperative relations among ZDISCs can be avoided if only they have a small ρZ, and ZDISCs with qR ¼ 5 and ρZ ¼ 1∕1024 do not meet the corruption, at least when T  1000. Unlike ZDISCs with qR
1∕2, we have not yet found any adverse effect of taking a small but positive ρZ.

9.5.2

ZDISCs Versus Pure Self-Advertising ALLDs

Self-advertisement by ALLDs loses its effectiveness after certain repetitions of giving-game sessions if only ZDISCs do not follow the the strategy with qR ¼ 0. According to our simulations within the range of qR  0, the giving rate was slightly lower for higher qR at any time step, which means that ZDISCs with non-positive higher qR can more rapidly identify the self-advertising ALLDs (not shown). This is because ZDISCs with 5 < qR  0, including ZDISCs with qR ¼ 1∕2 and ZDISCs with qR ¼ 0, gradually stop believing what pure self-advertising ALLDs gossip. We also found that those differences depending on qR are smaller for smaller ρZ. In the population containing ZDISCs with qR ¼ 5, self-advertisement by ALLDs keeps its effectiveness until the end of each generation, but it has small

254

9 Can Cooperation Evolve When False Gossip Spreads?

effect because there is abundant B gossip about each ALLD repeatedly passed by his/her victim ZDISCs (note that ρZ ¼ 1∕1024 in this section). We found quite a small difference among the coordinates of the unstable internal equilibria among various qR  0 (Fig. 9.8).

9.5.3

ZDISCs Versus ALLB-ALLDs

Against ALLB-ALLDs and for sufficiently large T, ZDISCs are evolutionarily favored in a wider range of x and the other parameter values when they share positive qR than when they share non-positive qR (see Figs. 9.4 and 9.8). In this simple bistrategic population, ZDISCs with qR ¼ ρZ are favored in the widest range of parameters. This suggests that under abundant gossip unfavorable to them, ZDISCs should initially give up incorporating all gossip, including the few pieces that are favorable to them. Along with the distinctive difference between ZDISCs with qR ¼ ρZ and the other three types of ZDISCs, we can see another distinctive difference; that is, the difference between ZDISCs with qR ¼ 0 and those with qR ¼ 1∕2 or qR ¼ 5. The ZDISC strategy with qR ¼ 0 has more difficulty in dealing with ALLB-ALLDs than the ZDISC strategy with qR ¼ 1∕2 or qR ¼ 5 (Fig. 9.8). Suppose that a ZDISC player (called player i) is chosen as a listener to an ALLB-ALLD player (called player u), a listener to another ALLB-ALLD player (called player v), and a potential donor to the player u in this order. If the player i has sufficiently small qR, then siu becomes negative due to the ALLB gossip from player v, and thus player i chooses not to give a benefit to player u in the giving-game session. On the other hand, when player i has qR ¼ 0, player i does not believe the ALLB gossip from player v because siv < 0 due to the ALLB gossip from player u. Consequently, siu ¼ 0 holds and player i chooses to give a benefit to player u. Although there is an opposite scenario that might make the ZDISC strategy with qR ¼ 0 resist ALLBALLDs more than the ZDISC strategy with qR ¼ 5, the simulation results indicate that the major factor influencing the evolutionary consequence is the one mentioned above.

9.5.4

ZDISCs Versus ALLG-ALLDs

Similar to Sect. 9.5.3, ZDISCs with qR ¼ ρZ performed the best (Figs. 9.4 and 9.8), whereas little difference was detected among the other three types of ZDISCs: those with qR ¼ 5, qR ¼ 1∕2, and qR ¼ 0. These ZDISCs, who believe what a stranger (i.e., player j for player i with sij ¼ 0) gossips, behave almost equally in the bistrategic population with ALLG-ALLDs. The reason is the following: ALLG gossip started by ALLG-ALLDs keeps sij, which is a P-score of ZDISC player i with qR ¼ 1∕2 or qR ¼ 0 against player j with the ALLG-ALLD strategy, positive for a long period, especially when the initial frequency of ALLG-ALLD is relatively

9.6

Discussion and Conclusions

255

high. During the period, player i believes the ALLG gossip of player j. Consequently, player i’s behavior does not differ from the qR ¼ 5.0 ZDISC player’s behavior, namely, believing what any player gossips.

9.6

Discussion and Conclusions

In this chapter, we developed a model for evolution of cooperation with gossip sessions and P-score, defined as a private image score. The P-score sij indicates player i’s image against player j and is formed by both player i’s direct experience with player j and gossip about player j that player i has received. In our study, the key strategy for the evolution of cooperation was the ZDISC strategy, with which player i chooses not to give a benefit to player j and spreads bad gossip about player j if sij is less than zero (the initial value of every P-score). The problem of how many reputational statuses are necessary has recently come under the spotlight within the framework of public image scoring (Tanabe et al. 2013). In the framework of gossip-mediated private image scoring, our tentative answer to this question is “at least more than three (good/bad plus neutral)” because the ZDISC strategy with ρ ¼ 1, with which a player classifies others into three categories (bad, sij  1; no information, sij ¼ 0; good, sij
+1), was easily invaded by defectors who engage in non-factual gossip (biased gossip and lies). On the other hand, a population comprising players adopting the ZDISC strategy placing emphasis more on their own experience than on gossip (i.e., ρZ