Delegate Apportionment in the US Presidential Primaries: A Mathematical Analysis 3031249534, 9783031249532

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Studies in Choice and Welfare

Michael A. Jones David McCune Jennifer M. Wilson

Delegate Apportionment in the US Presidential Primaries A Mathematical Analysis

Studies in Choice and Welfare Editors-in-Chief Marc Fleurbaey, Paris School of Economics, Paris, France Maurice Salles, University of Caen, Caen, France Series Editors Bhaskar Dutta, Department of Economics, University of Warwick, Coventry, UK Wulf Gaertner, FB Wirtschaftswissenschaften, Universität Osnabrück, Osnabrück, Niedersachsen, Germany Carmen Herrero Blanco, Faculty Economics and Business, University of Alicante, Alicante, Spain Bettina Klaus, Faculty of Business & Economics, University of Lausanne, Lausanne, Switzerland Prasanta K. Pattanaik, University of California, Riverside, CA, USA William Thomson, Department of Economics, University of Rochester, Rocchester, NY, USA

Studies in Choice and Welfare is a book series dedicated to the ethical and positive aspects of welfare economics and choice theory. Topics comprise individual choice and preference theory, social choice and voting theory (normative, positive and strategic sides) as well as all aspects of welfare theory (Pareto optimality; welfare criteria; fairness, justice and equity; externalities; public goods; optimal taxation; incentives in public decision making; cost-benefit analysis, etc.). All titles in the book series are peer-reviewed.

Michael A. Jones · David McCune · Jennifer M. Wilson

Delegate Apportionment in the US Presidential Primaries A Mathematical Analysis

Michael A. Jones Mathematical Reviews American Mathematical Society Ann Arbor, MI, USA

David McCune Department of Mathematics and Data Science William Jewell College Liberty, MO, USA

Jennifer M. Wilson Department of Natural Sciences and Mathematics Eugene Lang College, The New School New York, NY, USA

ISSN 1614-0311 ISSN 2197-8530 (electronic) Studies in Choice and Welfare ISBN 978-3-031-24953-2 ISBN 978-3-031-24954-9 (eBook) https://doi.org/10.1007/978-3-031-24954-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Caroline, for her patience. MAJ To my mentors Susan Hermiller, John Meakin, and Brian Raines. DM To Loie, for pulling me away just the right amount. JW

Preface

The purpose of this book is to examine the mathematics of the delegate allocation process in the US presidential primaries. The US presidential primaries are a series of state elections held every four years that determine the Democratic and Republican Parties’ nominees for president in the general election. Presidential candidates in each party are awarded delegates based on their shares of the vote in each state contest. At the end of the primary season, each party holds a national convention. The candidate with a majority of state delegates, after one or more rounds of voting, is selected to be the party’s nominee for president. From a mathematical standpoint, the question of how to allocate delegates is largely a problem of apportionment. Apportionment problems arise when a fixed quantity (such as a number of delegates) must be divided among several constituents (such as candidates) in proportion to some attribute (such as vote share), so that each portion is a whole number. Since delegates are individuals and cannot be divided, their allocation is a matter of apportionment. Apportionment or similar allocation problems occur at multiple stages in the delegate selection process from determining how many delegates each state receives, and how they should be divided between statewide and district delegates, to determining what happens after each primary when the delegates must be awarded to the candidates based on how well they do in the election. Apportionment has been well-studied in the context of allocating state representatives to the US House of Representatives in proportion to state populations. It has also been widely analyzed for its role in proportional representation systems where party seats are awarded to parties based on the vote distribution. It is less well-known in the case of apportioning delegates in the US primaries. Thus the examination of apportionment applied to delegate allocation involves both an identification and analysis of new apportionment methods as well as an evaluation of old and new criteria by which to evaluate their properties. The goal of this book is to do both. We have spent several years researching delegate allocation in presidential primaries and were inspired to write this volume because of the interesting aspects of apportionment arising in this context. The book’s

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origin can be traced to 2016, when we observed that the 2016 Georgia Republican party adopted an apportionment method for its state primary that was unique and perhaps the most interesting method we had encountered. This motivated us to investigate the methods used by other state parties, many of which are new, and to consider how the allocation of delegates is affected by elimination thresholds—the minimum levels of support required to receive delegates—and the overall structure of the state delegates selection plans in which delegates are apportioned based on vote distributions in each district and statewide. This led us also to consider how the goals of delegate apportionment differ from that of house or party apportionment— most noticeably in that the state primaries occur over time, and that the end result is not political representation but the declaration of a winner. This means that the usual criteria for evaluating apportionment need to be rethought in an effort to determine what features of traditional apportionment theory are most relevant to delegate apportionment and which features are not. This book is a work of mathematics, not of political science. We do not attempt to address questions of campaign strategy, conventional voting patterns, or the role of superdelegates at the national convention, etc. Instead, we study the mathematical rules by which delegates are allocated to presidential candidates, combining formal investigation with simulations and empirical research based on data from recent primaries. Our purpose is to provide a rigorous mathematical description and analysis of these rules, similar to the prior work that has been completed on house seat and party seat apportionment. Our hope is that the reader will find the delegate apportionment problem as interesting as we do, and ultimately will appreciate the complexity of the problem and the mathematical creativity of state and national party officials who grapple with how to fairly allocate delegates outside the academic traditions of apportionment theory. Since the completion of this book in November of 2022, several changes have been proposed for the 2024 Democratic Party presidential primary. On February 3, 2023, the Democratic National Committee approved a reordering of the 2024 state primaries in which South Carolina will hold the first primary on February 3, 2024, followed by New Hampshire and Nevada on February 7, 2024, Georgia on February 13 and Michigan on February 29. (In a separate decision in June 2021, the Nevada Democratic Party voted to switch their contest from a caucus to a primary.) This new ordering represents a significant change from past presidential primaries in which the Iowa caucuses were the crucial first contest, closely followed by New Hampshire, and then Nevada and South Carolina. The intent of these changes is to ensure that, at least in South Carolina, the voters participating in the first contest(s) are more reflective of the diversity of the U.S. and representative of the Democratic Party as a whole. The Democratic National Committee’s Rules and Bylaws Committee has given the New Hampshire Democratic Party until early June to work out the details for the change in date. However, state officials in both Iowa and New Hampshire have indicated that they will continue to hold their primaries at the beginning of the season. There will, no doubt, be further changes both to the calendar and to the Delegate

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Selection Plans which, for both parties, determine how delegates are allocated to presidential candidates leading to the selection of the parties’ presidential nominee. The analysis in this book, therefore, provides a thorough investigation of past primaries and a starting point for the analysis of future elections. Ann Arbor, MI, USA Liberty, MO, USA New York, NY, USA November 2022

Michael A. Jones [email protected] David McCune [email protected] Jennifer M. Wilson [email protected]

Acknowledgments

We wish to thank Tony Roza of The Green Papers for his willingness to share presidential primary data with us. Without this sharing and several helpful conversations via email about the data, this book would be missing an important empirical component.

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Contents

Part I

Description of Delegate Allocation Rules

1 Apportionment in the US Presidential Primaries . . . . . . . . . . . . . . . . . . 1.1 Apportionment and the Primaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What is the Apportionment Problem? . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Apportionment Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Review of the US Primary Process and the History of US Presidential Primaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Why Apportionment in Primaries is Different . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4 7 10

2 The Democratic Party Primary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Democratic Delegate Plan and Primaries . . . . . . . . . . . . . . . . . . . . . . . 2.2 Geometry of Hamilton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Democratic Primaries and Ranked Choice Voting . . . . . . . . . . . . . . . 2.3.1 What Is Ranked Choice Voting? . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Ranked Choice Voting and Apportionment . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 24 32 34 34 35 37

3 The Iowa and Nevada Democratic Caucuses . . . . . . . . . . . . . . . . . . . . . . 3.1 The 2020 Iowa Caucuses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The 2020 Nevada Caucuses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 40 48 50

4 The Republican Party Primary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Republican Party Delegates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Thresholds in the Republican Primary . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Republican Selection Rules and Quota-Based Methods . . . . . . . . . . 4.3.1 Methods Based on Nearest-Integer Rounding . . . . . . . . . . . . 4.3.2 Methods Based on Lower Quotas . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 A Method Based on Upper Quotas . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Methods that Cannot Be Disentangled from Thresholds . . . . 4.3.5 Which States Used Which Methods . . . . . . . . . . . . . . . . . . . . .

51 52 54 55 57 60 61 62 63

13 18 21

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4.4 Geometry of Republican Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 District Delegates in Republican State Primaries . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II

65 68 70 71

Analysis of Delegate Allocation Rules

5 Properties of the Apportionment Methods Used in the Primaries . . . 5.1 Properties of the Delegate Apportionment Methods . . . . . . . . . . . . . . 5.1.1 Proportional Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Relationship to Quota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Quota Violations in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Majorization and Pairwise Comparisons of Bias . . . . . . . . . . 5.3.2 Delegate (Seat) Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Delegate Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Extreme Outcomes in Close Elections . . . . . . . . . . . . . . . . . . . 5.3.5 Bias in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Sensitivity to Small Perturbations in Vote Totals . . . . . . . . . . . . . . . . 5.4.1 Sensitivity to Vote Totals in Practice . . . . . . . . . . . . . . . . . . . . 5.5 Majority and Leader Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Majority Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Leader Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Majority and Leader Criteria in Practice . . . . . . . . . . . . . . . . . 5.6 Support for Candidate Coalitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 The Cumulative Effect: Comparing Vote Share to Delegate Share . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 78 81 85 90 92 94 97 100 103 105 108 110 111 112 115 116 119

6 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Elimination Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Geometry of the Elimination Paradox . . . . . . . . . . . . . . . . . . . 6.1.2 Likelihood of the Elimination Paradox . . . . . . . . . . . . . . . . . . 6.1.3 Who Is Affected by the Elimination Paradox? . . . . . . . . . . . . 6.1.4 The Effect of Threshold Level on the Elimination Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Aggregation Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Geometry of the Aggregation Paradox . . . . . . . . . . . . . . . . . . 6.2.2 Likelihood of the Aggregation Paradox . . . . . . . . . . . . . . . . . . 6.2.3 Who Is Affected by the Aggregation Paradox? . . . . . . . . . . . 6.2.4 The Effect of Threshold Level on the Aggregation Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 No-Show Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Geometry of the No-Show Paradox . . . . . . . . . . . . . . . . . . . . . 6.3.2 Likelihood of the No-Show Paradox Occurring . . . . . . . . . . . 6.3.3 Who Is Affected by the No-Show Paradox? . . . . . . . . . . . . . .

127 128 130 132 136

121 126

140 141 146 151 159 163 164 166 166 167

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6.3.4 The Effect of Threshold Level on the No-Show Paradox . . . 6.3.5 Weak No-Show Paradox: The Effects of Aggregation . . . . . 6.4 Alabama Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Geometry of the Alabama Paradox . . . . . . . . . . . . . . . . . . . . . 6.4.2 Likelihood of the Alabama Paradox Occurring . . . . . . . . . . . 6.4.3 Who Is Affected by the Alabama Paradox? . . . . . . . . . . . . . . 6.4.4 The Effect of Threshold Level on the Alabama Paradox . . . . 6.4.5 The Effect of Aggregation on the Alabama Paradox . . . . . . . 6.5 Population Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168 168 169 171 172 173 175 175 178 182 184 185

7 Exploring Alternative Ways to Allocate Delegates . . . . . . . . . . . . . . . . . 7.1 Other Apportionment Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Shift-Quota Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Divisor Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Measures of Fairness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Degressive Representation in the European Union . . . . . . . . . . . . . . . 7.4 Regressive Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Adapting Divisor Methods Using Power Functions . . . . . . . . 7.4.2 Adapting Divisor Methods Using Weighted Vote Totals . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 189 189 192 194 197 200 202 203 204 206

Appendix A: Description of the Monte Carlo Simulations . . . . . . . . . . . . . 207 Appendix B: Descriptions of the Databases of Primary Election Data . . . 209 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Abbreviations and Acronyms

CD cedm CEDM DEM_DATA DNC HAM HAM* HNH ILQ LAR LEAD MAJ NIE NIS PLEO

REP_DATA RNC SUQ

Congressional district Close-election delegate minimum Definition 5.14 in Sect. 5.3.4 Close-election delegate maximum Definition 5.14 in Sect. 5.3.4 Database of Democratic primary data Appendix B Democratic National Committee Hamilton’s method Definition 1.1 in Sect. 1.3 Hamilton’s method, conceptualized using nearest integer rounding Definition 1.2 in Sect. 1.3 Hamilton-NIE hybrid method Definition 4.4 in Sect. 4.3.1 Iterated lower quota method Definition 4.7 in Sect. 4.3.2 Large method Definition 4.6 in Sect. 4.3.2 Leader criterion Definition 5.16 in Sect. 5.5 Majority criterion Definition 5.15 in Sect. 5.5 Nearest Integer Extremes method Definition 4.1 in Sect. 4.3.1 Nearest Integer Sequential method Definition 4.2 in Sect. 4.3.1 Party Leaders and Elected Officials A type of delegate in a Democratic primary, allocated based on the statewide vote distribution Database of Republican primary data Appendix B Republican National Committee Sequential upper quota method Definition 4.8 in Sect. 4.3.3

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Part I

Description of Delegate Allocation Rules

The subject of this book is the mathematics behind the delegate allocation process in the US presidential primaries. The primaries consist of a series of state elections held every four years that determine the Democratic Party and Republican Party nominees for president in the general election. Each party creates a delegate selection plan which determines how many delegates each state receives and how the delegates are allocated to each candidate after the state primary based on the candidates’ shares of the vote. At the end of the process, each party holds a national convention. The candidate with a majority of state delegates is nominated at the convention to be the party’s nominee for president. The structure of the US presidential primaries has undergone intense scrutiny, both within the sphere of public debate and in the academic literature. Previous work has focused on the history and political consequences of the current primary system, the role of the primary calendar, the changing demographics of voters, the differences between primaries and caucuses, and the effect of the primaries on political campaigns and on the selection of winners. But there has been little detailed analysis of the actual rules that govern how delegates are awarded, rules that in modern primaries are almost entirely mathematical. The allocation of delegates is dictated by each party’s delegate selection plan, which determines how many delegates each state receives, and how they should be divided into the two categories of statewide and district delegates. It also provides guidance for how delegates should be awarded to the candidates after the election, based on the resulting votes. Significant differences exist between Democratic and Republican parties: the Democratic party prescribes uniform rules for each state primary while the Republican party allows each state to determine their own rules, within some general parameters. For both parties, the allocation of delegates at all stages is closely tied to apportionment. Thus, we analyze the delegate selection rules through the lens of apportionment methods (which, in a few cases, requires some interpretation), and we evaluate and compare the rules in comparison to how apportionment is applied in other contexts. In Part I we lay out the landscape, providing a complete mathematical description for how delegates are allocated in the presidential primaries and identifying the

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Part I: Description of Delegate Allocation Rules

different apportionment methods used at various stages of the allocation process. Chapter 1 introduces the subject of delegate apportionment, including key definitions and notation, and provides some historical context for the selection of party nominees in the US primaries. We also discuss why the analysis of the apportionment methods used for delegate allocation may require different approaches from those used in other settings such as apportionment in the US House of Representatives. In Chap. 2, we look at delegate allocation in the Democratic primary, describing how delegates are awarded at each stage of the process. We also present simplicial geometry as a means to visualize and to analyze apportionment methods. In Chap. 3, we describe delegate allocation in the Democratic caucuses, focusing particularly on the 2020 Iowa Democratic caucuses. The Iowa caucuses are especially interesting because it is the first state contest to occur in the primary season. However, the delegate selection process for the Iowa Democratic caucuses is also uniquely complex. For comparison, this chapter concludes with an examination of the 2020 Nevada Democratic caucuses, whose simpler rules more closely mirror those of a typical Democratic caucus election. In Chap. 4, we turn our attention to how delegates are awarded in the Republican Party’s state primaries, a process which differs significantly from that of the Democratic Party. Throughout Part I, we illustrate the delegate selection process using data from several recent primaries, especially those from 2016 and 2020. Part I is largely descriptive; in-depth analysis of the delegate allocation process is reserved for Part II.

Chapter 1

Apportionment in the US Presidential Primaries

The US presidential primaries are a series of state elections held every four years to determine the Democratic Party and Republican Party candidates for president. During the roughly nine-month period preceding the general presidential election in November, each state holds a primary or caucus in which voters select their preference for one of their party’s presidential candidates. Based on the results of these contests, delegates are awarded to each candidate. The candidate with a majority of delegates at the end of the process is officially endorsed as the party’s nominee for president at the party’s national convention held during the summer prior to the November presidential election. If no candidate receives a majority of the delegates, a further process is enacted during the national conventions to determine the party’s final nominee. In this book, we examine how this delegate allocation process occurs, from the initial division of delegates among states, to the classification of state and local delegates, to the awarding of delegates to the presidential candidates. Throughout, we focus on the role that apportionment plays in the many stages of delegate allocation. Apportionment problems arise any time a finite resource, such as a set of delegates, must be divided among several constituencies based on some criterion of proportionality, under the restriction that each share must be a nonnegative whole number. In primaries, apportionment problems occur when states are initially allocated delegates based on a combination of their population and previous voting patterns and when these delegates are then split among smaller geographic regions, such as congressional districts. Apportionment problems also arise after ballots are tallied for a state primary election, when candidates receive delegates based on their percentage of the vote share.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9_1

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1 Apportionment in the US Presidential Primaries

Mathematically, the allocation of delegates is similar to other well-known apportionment problems such as determining representation in the US House of Representatives or party seats in a legislative body using proportional representation. However, the implementation of apportionment methods for delegate allocation in the primaries, as well as the criteria by which methods should be evaluated, are sufficiently different to warrant a close examination. In this chapter, we provide an introduction to the primary process. Section 1.1 gives a brief overview of how delegates are allocated in the US presidential primaries. In Sect. 1.2, the apportionment problem is described more generally, and some of the historical considerations for determining the number of representatives each state receives in the US House of Representatives are summarized. In Sect. 1.3, we characterize formally the apportionment problem and define several well-known apportionment methods. Section 1.4 provides a brief history of the delegate selection process in the US presidential primaries; this section also includes a description of how the primaries evolved into their current structure. In Sect. 1.5, we summarize some of the ways in which delegate apportionment raises different questions from apportionment in other contexts.

1.1 Apportionment and the Primaries The modern system by which the Republican Party and Democratic Party select their presidential candidates dates roughly from the late 1960s. Prior to this, decisions were largely made during national conventions, where compromises were brokered among party officials and leaders who had influence over large blocs of state delegates (Coleman 2012). This process reached a head during the 1968 Democratic Convention when party officials nominated former Vice President and Minnesota Senator Hubert Humphrey despite wide-spread support for former Attorney General Robert Kennedy. The ensuing protests are largely credited with initiating widespread changes, leading ultimately to the system currently in place. Since the 1960s, first the Democratic Party, and then the Republican Party, enacted a series of reforms designed broadly to increase public input into the candidate selection process while ensuring that the eventual nominee had the best chance of winning the general election. Overall, these reforms have had the effect of standardizing the timeline and the way in which states hold contests, and how delegates are selected. While some of these reforms concern internal party structure or procedural rules, many of them affect not only how delegates are allocated among states and districts before the election, but also how delegates are distributed among the candidates following the primary elections. The current delegate allocation process is a product of these reforms. While changes continue to be made, the last several primaries have been sufficiently stable to allow an analysis of the current rules. Prior to the start of the primary season, the national parties determine how many delegates to award each state party. A state party’s number of delegates is generally based on a combination of how many party members reside in the state, the number

1.1 Apportionment and the Primaries

5

of registered voters, and prior election results. In the Democratic primary, a portion of these delegates is allocated to each congressional district (CD) based on similar factors within the district. In the Republican primary, each CD automatically receives three delegates. After a state’s primary or caucus is held, an apportionment method is used to award delegates to each candidate based on the proportion of the vote the candidate received. Delegates are awarded at both the state and district level. In the Republican primary, state parties decide on what apportionment method to use, often creating their own methods. In the Democratic primary, each state party uses the same apportionment method, but there is variability in how delegates are awarded in the caucuses. Apportionment methods are widely studied for their roles in determining political and party representation. In the US, the 435 members of the House of Representatives are redistributed among the 50 states every ten years following the decennial census based on each state’s share of the population. Apportionment methods are also used to allocate members of the European Parliament among member countries. Both of these examples illustrate how apportionment is used to determine representation across different geographic constituencies. In a different setting, apportionment methods are also used to allocate seats in legislative bodies among different political parties based on their shares of the vote. In this context, apportionment methods determine representation across different political constituencies. The apportionment problems that arise in the presidential primaries share commonalities with these situations, but also exhibit differences. Historically, apportionment methods used to allocate seats in the US House of Representatives have been evaluated for their degree of bias—the degree to which they favor either large or small states. Observations from 200 years of data from the census have given rise to an understanding of different methods’ bias, and their strengths and weaknesses more generally. The strengths include desirable properties, such as population monotonicity, where states which grow quickly in relative terms cannot lose seats to states that grow more slowly. The weaknesses are often framed as counterintuitive properties (or paradoxes), such as the Alabama paradox, where the number of House seats a state receives decreases despite the number of total representatives to be allocated increasing (while each state’s population remains fixed). In contrast to geographic-based apportionment, studies of apportionment for proportional representation have been less concerned with bias than with questions about how seat allocation impacts whether parties are able to form a majority government or to build stable party coalitions. In this situation, bias may be viewed favorably– whether in favor of large parties to encourage coalition building, or in favor of small parties to encourage a diversity of opinions. Apportionment methods are also compared based on the percentage of votes necessary or sufficient to receive a certain number of seats. This latter idea is embodied in the thresholds of inclusion (the vote shares below which a party cannot possibly receive a certain number of seats) and exclusion (the vote shares above which a party cannot fail to receive a certain number of seats). Other comparisons between methods are complicated by the fact that many proportional representation systems also employ a cutoff—a minimal percentage of votes a party must receive to have their votes count.

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1 Apportionment in the US Presidential Primaries

For the delegate allocation process in the presidential primaries, the first step of allocating delegates among the states is similar in spirit to apportionment in the US House of Representatives, with the proportionality criterion based not on state population but on compound measures related to the strength of the party in the state. In this step, a lack of bias is preferable. However, the final step in which delegates are allocated to candidates is more similar to apportionment in a legislative body, where bias is less of a concern. Similar to the case of party representation, bias may actually be desirable in delegate apportionment. Bias toward stronger candidates in the primaries helps narrow the field of candidates and builds consensus in the selection of a party’s final nominee, while bias toward weaker candidates encourages a broad range of party members’ views. In addition, presidential primaries often apply cutoffs to keep inconsequential votes (sometimes for a candidate from another party) from affecting the calculations. Applications of apportionment methods to delegate allocation also have some unique features. One concerns the effects of aggregation, as when delegates are awarded to candidates in each CD in a state and then summed, rather than awarded in a single state-wide contest. Another feature is the use of apportionment at different points in the process, resulting in a composition or nesting of apportionment methods whose outcome may deviate greatly from true proportionality. The evaluation of apportionment methods in presidential primaries also differs in their susceptibility to paradoxes. Indeed, many of the well-known paradoxes discovered when apportioning the US House of Representatives are less relevant when assigning delegates to candidates. In the context of delegate allocation, the Alabama paradox corresponds to the situation when an increase in the total number of delegates results in a candidate receiving fewer delegates. This situation does not arise because the number of delegates is fixed during the primary season. On the other hand, the incorporation of cutoffs for delegate allocation may cause some apportionment methods to suffer the elimination paradox—when the elimination of a weaker candidate results in a stronger candidate receiving fewer delegates. We explore these ideas further in Chap. 6. A comprehensive historical and mathematical account of apportionment to the US House can be found in Balinski and Young (2001). A thorough analysis of apportionment in European parliaments is considered in Pukelsheim (2017). While the use of apportionment both in the US House of Representatives and in proportional representation has been well-studied, there has been little analysis of the applications of apportionment in the US presidential primaries or in the mathematical framework of delegate allocation more broadly. Geist et al. (2010) examine the effect of using Hamilton’s method in the Democratic Party Delegate Selection Rules. Jones, McCune, and Wilson look at the effect of cutoffs on the Democratic selection process (Jones et al. 2019) and analyze the methods used in the state Republican primaries (Jones et al. 2020). This book takes a more comprehensive approach, examining the allocation of delegates in both parties throughout the whole primary process.

1.2 What is the Apportionment Problem?

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1.2 What is the Apportionment Problem? In the US, apportionment is best known for its role in determining the composition of the House of Representatives. The number of House seats has increased from 65 representatives in 1789 to the current 435 representatives, fixed as part of the Apportionment Act of 1911. This act set the cap at 433 seats but included increases of one seat each for when Alaska and Hawaii were granted statehood. Each state receives a number of representatives based on its proportion of the country’s population. Mathematically, this is the same problem as allocating delegates to candidates based on their shares of the popular vote. In either case, an apportionment problem arises because the number of seats or delegates awarded must be integers. There are multiple approaches to solving these apportionment problems. Because the question of congressional apportionment is familiar in the US, we use this setting to explore some of these approaches. House apportionment traces its origin to the following passage from Article I Section 2 of the US Constitution: Representatives [to the US House] and direct taxes shall be apportioned among the several states which may be included in this union, according to their respective numbers . . .

In simpler terms, if a state has p% of the nation’s population then that state should receive p% of the seats in the US House of Representatives. Proportional representation as a model for fair representation is seemingly universal: governmental structures across the world are built on this principle. But while the concept is intuitive and easy to understand, it is not always easy to implement. At the heart of the problem is that representation is usually based on individuals, who are indivisible. To illustrate the kind of difficulties that can arise, consider the following example. Example 1.1 Suppose a small country has 100 seats in its house of representatives and contains five states A, B, C, D, and E, whose populations are shown in Table 1.1, totaling 100,000 people. The second row identifies each state’s quota—the percentage of a state’s population multiplied by the total number of seats. A state’s quota represents the number of seats that each state would receive in a purely proportional allocation if house seats were divisible. Given these quotas, how should the 100 seats be distributed to the states? The intuitive solution to the question posed in Example 1.1 is to round each quota to the nearest integer. This leads to an apportionment of (54, 19, 17, 6, 3), resulting in an under allocation of 1 seat. Thus, apportioning seats are not merely a matter

Table 1.1 Example with 100 seats to apportion among 5 states with a total population of 100,000 State A B C D E Population Quota

54,440 54.44

19,380 19.38

17,290 17.29

6370 6.37

2520 2.52

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of standard rounding. What if we accepted the apportionment of (54, 19, 17, 6, 3), thereby changing the size of the house to 99? There are two problems with this solution. First, the size of the house is often prescribed by law. Second, and more importantly, if the house size changes, then the quotas do as well. In this example, if the house size were changed to 99 then the quota of state E falls to 2.4948, in which case standard rounding allocates only 2 seats to E, for a total of 98 seats. This kind of reasoning can result in a non-ending cycle where changing the house size to accommodate nearest integer rounding leads to a change in quotas which leads in turn to another potential change in the house size. Although this is probably not the reason why the size of the house is fixed in practice, it certainly supports that the house size should be fixed in apportionment theory. Assuming then that we must allocate all 100 seats in Example 1.1, there are several possible apportionments that might be appropriate depending on the context including the following: • (55, 19, 17, 6, 3): This apportionment would be advocated by Alexander Hamilton, who proposed a method to allocate seats in the first US House following the 1790 census. He stipulated that each state should be apportioned the integer part of its quota, and the remaining k seats should be allocated one at a time to the k states with largest fractional parts. In this example, rounding down the states’ quotas allocates a total of 98 seats. States A and E have the two largest fractional parts and thus each is awarded an additional seat. As an added bonus, (55, 19, 17, 6, 3) is the integer lattice point in R5 that is closest to (54.44, 19.38, 17.29, 6.37, 2.52) in the standard Euclidean metric, and thus there is a natural distance argument for this apportionment because it is the closest apportionment to the vector of quotas. Hence, one could argue that (55, 19, 17, 6, 3) is the most fair or most proportional apportionment. • (54, 19, 17, 7, 3): Beginning with each state receiving the integer value of their quotas, this allocation gives the two remaining seats one each to the states with the smallest populations. For state A, having 55 versus 54 seats probably does not make much of a difference in terms of the state’s representation. For a smaller state like E, on the other hand, the difference between 2 and 3 seats creates a sizable difference in the state’s quality of representation, an argument for smaller states to be given a boost at the expense of larger states. • (55, 20, 17, 6, 2): As before, each state receives initially the integer value of its quota, leaving two seats to be allocated. In this allocation, the two remaining seats are awarded to the states with the largest populations, ensuring that the proportional advantage given to any state by rounding up is minimized. This approach may be reasonable if this country also has a senate-type body that operates like the US Senate where each state receives 2 senators regardless of size. Since that body has an explicit bias in favor of small states, it might make sense to favor the large states in the house of representatives as this apportionment does. • (56, 19, 17, 6, 2): This apportionment takes a different approach, based on the idea that each representative should represent roughly 10,000/100 = 1000 people. Thus, we consider each state’s population divided by 1000. Of course these

1.2 What is the Apportionment Problem?

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ratios yield the quotas. But by altering the divisor and then rounding the ratios in some predetermined way, we can determine an allocation that adds to 100. In this instance, using a divisor of 970 and rounding the ratios down yields the apportionment given. In the context of delegate apportionment, the last allocation method focuses on the district size—the number of people in each CD—rather than the quota; it is one of a family of methods called divisor methods. This particular method was first advocated in the US by Thomas Jefferson as an alternative to the method proposed by Hamilton and stems from the language of the third sentence of Article 2 in the Constitution, which states: The Number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative.

As the example makes clear, apportionments using a divisor method can vary considerably from methods that are based on rounding the quotas; the latter are referred to as quota-based methods. Applying Jefferson’s method to Example 1.1 results in the largest state being awarded more representatives than the value of its quota rounded up. We take up the discussion of adherence to quota in Chap. 5 and the general question of comparing quota methods with divisor methods in Chap. 7. While breaking quota—when a state receives either more than its quota rounded up or less than its quota rounded down—may seem fundamentally unfair, it does raise questions about using quotas as a basis for determining fair representation, even in cases where standard rounding gives a legitimate apportionment. Suppose, for example, that a small state has a quota of 1.51 while a large state has a quota of 52.49. Does it make sense to round the 1.51 up to 2 and the 52.49 down to 52? Because the states have such different sizes, this rounding has a greater effect proportionally on the small state than on the large one, ensuring that the small state’s citizens will have greater representation than their neighbors. Different considerations underlie the trade-offs between different allocations when apportioning party seats in a proportional representation system. Many countries, particularly in Latin America and Europe, are governed by legislative houses whose members are at least partly determined by their political parties’ shares of the popular vote. Imagine, in the simplest case, that the data in Table 1.1 represent vote totals for five parties, A, B, C, D, and E, instead of state populations, and that the size of the legislative body is 100. A quota represents a party’s fair share of seats, and a state’s average district size (ratio of population to apportionment) corresponds to a party’s average constituency size (ratio of party vote to number of its representatives). Unlike the situation outlined in the US Constitution, exact equality of constituency size may not be an overriding consideration. Many federal systems require parties to achieve a minimal level of support (frequently 5 or 10%) in order to receive any members in the legislature. Moreover, methods that are biased in favor of large parties may be desirable as a way to ensure that the party that receives a plurality of the vote gains enough seats to govern effectively. Put simply, a political party is seen as earning the votes it receives in a way that states do not earn their population sizes, and thus a bias in favor of large parties may be desirable.

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This suggests that apportionment used to determine party representation may have different goals from apportionment to determine representation in the US House of Representatives, where methods have historically been analyzed for their ability to minimize bias toward large states or small states. Likewise, different considerations may underlie the choice of apportionment methods in cases when the data in Table 1.1 represents vote totals for candidates A, B, C, D, and E in a state primary with 100 delegates to allocate. As in the party representation context, many state primaries employ cutoffs which dictate minimal vote strength for a candidate to be awarded delegates. The application of an apportionment method is also complicated by the fact that in most states, the delegates are divided into different groups, with some at-large delegates awarded based on candidates’ shares of the popular vote in the state, and other district delegates awarded analogously in each CD. Adding up these allocations to determine state-wide delegate totals results in additional deviations from proportionality. We consider these factors in Part II. Regardless of the context, however, the selection and implementation of an apportionment method for allocating delegates can have major consequences. As the discussion of Example 1.1 suggests, choosing an apportionment involves weighing a series of trade-offs, each of which has implications for what we mean by fair representation. We conclude this section with a restatement of the fundamental apportionment problem, which appears repeatedly in the primaries, at both national and local scales. How do we proportionally distribute something that is indivisible? Every apportionment problem has three components: • a set of n constituents (candidates, parties or states), {1, . . . , n}; • a total number of D delegates or H representatives to be distributed to the n constituents; and • a vector of v = (v1 , . . . , vn ) votes or P = (P1 , . . . , Pn ) populations which determines the proportions used as the basis of the allocation. An apportionment method determines an allocation of the total D delegates or H representatives among the n constituents based on their proportion v or P of the vote or population in such a way that each allocation is a non-negative integer. In the next section, we introduce the apportionment problem formally and survey some of the best-known apportionment methods.

1.3 Apportionment Fundamentals Apportionment problems arise in the US presidential primaries in a variety of contexts. To avoid a multiplicity of language and symbols, we define our notation in terms of the problem of allocating delegates to presidential candidates based on their share of the popular vote in a particular jurisdiction. Alternative interpretations will be made explicit where appropriate.

1.3 Apportionment Fundamentals

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Let D be the number of delegates to be awarded and let Ci denote the ith candidate for i = 1 to n. Assume Ci receives vi votes and let v = (v1 , . . . , vn ) be the vector of Σn vi the total number of votes cast in the primary election. When votes with V = i=1 convenient, we will also refer to the distribution of vote percentages p = ( p1 , . . . , pn ) where pi = vi / V . Unlike in the US House of Representatives, apportionment in both parties’ primaries is based on quotas. Let candidate Ci ’s quota be given by qi = (vi / V )D; this is the number of delegates Ci would receive if fractional delegates were permitted. Let q = (q1 , . . . , qn ) denote the vector of quotas. Let [qi ] be the quota rounded to the nearest integer (assuming that quotas with a fractional part of 0.5 are rounded up) and [q] = ([q1 ], . . . , [qn ]) be the corresponding vector of rounded quotas. Similarly, we define [qi ] (resp. [qi ]) and [q] (resp. [q]) as the lower (resp. upper) quota, or the value of qi rounded down (resp. up), of Ci and its corresponding vector. . . . , vn ) to For a fixed n, an apportionment method F maps a vote vector v = (v1 ,Σ n di = a set of apportionment vectors {d = (d1 , . . . , dn )} with di ∈ Z+ ∪ {0} and i=1 D. For convenience, we sometimes consider F as a mapping from a quota vector q into the set of apportionment vectors. Because of the possibility of ties at some point in the apportionment process, F is a set-valued function and the set of possible apportionments for v (resp. q) under F is denoted F(v) (resp. F(q)). When F(v) is a singleton, we will write F(v) = d instead of F(v) = {d}. When F(v) is not a singleton, then we assume the apportionment is selected at random from F(v), according to a uniform distribution. At times, we consider D as an argument of the set-valued F and denote an apportionment as F(D, v). Let’s consider how an apportionment method is applied to the problem of allocating delegates to candidates based on their shares of the vote. The method most commonly used in the presidential primaries is known in the US as Hamilton’s method, and more generally as the Method of Largest Remainders. As described in Sect. 1.2, this method gets its name in the US from its advocacy in 1792 by Alexander Hamilton for the country’s first congressional apportionment. (Since the method has been “re-discovered” multiple times in different contexts, we avoid the term “invented.”) Congress approved the use of this method in the Apportionment Act of 1792; however, the act was vetoed by President George Washington as his first use of the veto. Consequently, Hamilton’s method was not used to apportion the US House of Representatives until 1852 (Balinski and Young 2001). each candidate is initially awarded Definition 1.1 (HAM) Under Hamilton’s method, Σ This leaves D − [q ] [qi ] = [Dpi ] delegates. i delegates remaining which are i Σ [q ] candidates in order of awarded to the D − Σ i i Σ the decreasing size of their fractional remainder qi − i [qi ]. (Note that D − i [qi ] ≤ n − 1, so this is always possible.) Many state primaries use Hamilton’s method to allocate delegates but define it in a different, although equivalent way, using Σ Σ nearest integer rounding. This formulaare made based to [qi ] if [qi ] = D. If [qi ] / = D then adjustments tion maps qiΣ Σ on whether [qi ] > D (an over allocation of delegates) or [qi ] < D (an under allocation of delegates).

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Table 1.2 Comparing Hamilton’s method for Example 1.1 to four well-known divisor methods State Population Quota HAM J (970) W (994) HH (995) A (1025) A B C D E

54,440 19,380 17,290 6370 2520

54.44 19.38 17.29 6.37 2.52

55 19 17 6 3

56 19 17 6 2

55 20 17 6 2

55 19 17 6 3

54 19 17 7 3

The divisor appears in parentheses beside J, W, HH, and A

DefinitionΣ 1.2 (HAM*) Under HAM*, each candidate isΣ initially awarded [qi ] delIf [qi ] < D then an addiegates. If [qi ] = D, then this is the apportionment. Σ with the largest fractional delegate is awarded to the k = D − [qi ] candidates Σ ] for which [q ] = [q ]. If [q ] > D then a delegate is tional remainder qi − [q i i i Σi removed from the k = [qi ] − D candidates with the smallest fractional remainder qi − [qi ] for which [qi ] = [qi ]. Example 1.2 (Example 1.1 revisited) Consider again Example 1.1 with 5 states and quotas as shown in Table 1.1. Under HAM, the states would be initially allocated 54, 19, 17, 6 and 2 seats, summing to 98. The final 2 seats would be awarded to states A and E for a final apportionment of (55, 19, 17, 6, 3). Because all of the quotas in Example 1.1 have a fractional piece less than 0.5, the initial rounding under HAM∗ is the same as under HAM. Of course the final result is the same under HAM∗ . For comparison, we define the best-known class of apportionment methods, the divisor methods. A divisor method corresponds to a rounding rule f that assigns to each positive integer k, a value f (k) ∈ [k − 1, k] such that there do not exist positive integers a and b for which f (a ) = a − 1 and f (b) = b. The value f (k) acts as a bifurcation point in which positive numbers greater than k − 1 and less than f (k) are rounded down to k − 1; numbers greater than or equal f (k) and less than k are rounded up to k. By convention, f (0) = 0, so that positive numbers below f (1) are rounded to 0. Definition 1.3 For a rounding rule f , give each candidate di delegates if there exists ) ≤ vi /x < f (di + 1) for each i, where the di are nona divisor x > 0 such that f (diΣ negative integers that satisfy i ≤n di = D. Note that if f (1) = 0, then each candidate with a positive share of the vote receives at least one delegate as long as D ≥ n. If f (1) = 0 and D < n, then by convention each of the D candidates with the largest vi receives a delegate. The most commonly used divisor methods include (Thomas) Jefferson’s method (also known as the D’Hondt method), (Daniel) Webster’s method (or Sainte-Lague method), Hill–Huntington’s method, and (John√ Quincy) Adams’s method with rounding rules f (k) = k, f (k) = k − 1/2, f (k) = k(k − 1) and f (k) = k − 1, respectively. With the exception of Adams’s method, each of these four divisor methods has

1.4 Review of the US Primary Process and the History …

13

been used to apportion the US House from 1791–1842; 1842–1852 and 1901–1941; and 1941 to the present, respectively. Balinski and Young (2001) provide the definitive account of both the mathematics and the history of the use of apportionment methods for assigning seats in the US House of Representatives. Example 1.3 Table 1.2 compares the apportionment from Example 1.1 under Hamilton’s method (HAM) to the apportionments under four well-known division methods: Jefferson (J), Webster (W), Hill–Huntington (HH), and Adams (A). The numbers in the parentheses indicate non-unique divisors x used in each instance. Note that in Example 1.3, Jefferson’s method allocates 56 seats to State A, despite its quota being 54.44. This illustrates that Jefferson’s method can break quota by awarding a state a number different from either its lower or upper quota. Historically, Jefferson’s method was criticized for its bias in favor of large states; Adams’ method was similarly criticized for its bias in favor of smaller states. (In Example 1.3, under Adams’ method, both states D and E are awarded their quotas rounded up, while the remaining states are awarded their quotas rounded down.) While this issue is contentious in the discussion of house seat apportionment, it may be less so in the context of delegates. Suppose this example referred instead to vote counts for 5 presidential candidates with 100 delegates to allocate. Should the apportionment favor candidate A, in hopes of providing momentum for A’s campaign and consolidation around a winner? Or should the apportionment favor candidate E to support a broader array of nominees? We consider this question in Chap. 5; further discussion of divisor methods occurs in Chap. 7.

1.4 Review of the US Primary Process and the History of US Presidential Primaries The modern methods by which the US Republican and Democratic Parties select their presidential candidates date from the late 1960s. Prior to this, decisions were largely made during national conventions, where compromises were brokered among party officials and leaders who had influence over large blocs of state delegates (Coleman 2012). There is an extensive literature that examines the recent history of the presidential primary process since the 1968 efforts for reform. Jewitt (2019) provides a comprehensive overview of the evolution of the rules of the primaries and their impacts on elections. As she notes, reform efforts have brought to the fore a number of conflicting demands: the sometimes contradictory aims of the national and state parties; the natural tensions between the calls for broader participation and fairness within the party and the influence of the party elite; and the desire to embrace a wider ideological spectrum while ensuring that a viable (i.e., electable) candidate is ultimately chosen. These differences often manifest as conflicts between party leaders and members of the national committees, who want to ensure that experienced senior party members have influence over the choice of the nominee, and other members of the party, who want to empower the voice of the voters. In addition, each series of

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reforms led to consequences—some expected and some unexpected—with impacts on the following general election cycle. Thus changes in one election cycle would generate calls for reform in which new rules were created or amended, adjusting for the lessons learned in the previous election. Changes were also made based on the success of the current candidate; a loss in the general election would energize calls for reform within the party while periods in which the nominee occupied the White House would decrease interest in tinkering with the rules. The shifting national mandates had a direct impact on how states allocated delegates. Jewitt (2019) cites Kansas as an example of the frequency and kinds of changes made during this period. In 1976, the state Republican Party held a caucus. In 1980, they switched to a closed primary (in which only Republicans were allowed to participate) with delegates allocated proportionally. Eight years later, they returned to a caucus format. In 1992, 1996, and 2000, they held a semi-open primary (in which independents could also participate) but with no formal allocation rule. In 2004, they instituted a winner-take-all rule. They returned again to a caucus in 2008, allowing only Republicans to participate. In 2012, the caucus rules were altered so that at-large delegates were allocated proportionally while district-level delegates were allocated using winner-take-all. In 2016, the state party lowered the minimum vote share required to earn delegates, and chose to allocate district-level as well as statewide delegates on a proportional basis. Finally, in 2020, they were one of several states that canceled their presidential primaries. As noted previously, prior to the 1970s the selection of presidential nominees was governed largely by party leaders. The number of state primaries fluctuated, and although candidates frequently ran in select primaries, the results did not have a large influence on the final nomination. In 1952, for example, Adlai Stevenson became the Democratic Party’s nominee despite winning only 1.6% of the primary vote, compared to Estes Kefauver’s 64%. In 1968, protests erupted at the Democratic Party National Convention to demand more transparency and a greater voice for the rank and file in the nomination process. In response, the party formed the Commission on Party Structure and Delegate Selection (the McGovern Fraser Commission) whose goal was to ensure all voters had “full, meaningful and timely opportunity to participate” (Democratic National Committee and Others 1970). This was one of several commissions that the Democratic Party formed over the ensuing decades to make changes to party procedures and delegate selection rules. The McGovern Fraser Commission issued a series of recommendations for the 1972 election to make the primaries more uniform across states and to encourage broader participation among rank and file party members. One recommendation stipulated that each state establish written rules governing how delegates would be chosen and ensuring that all caucus meetings be held on the same day across the state. Another required state parties to apportion delegates “on a basis of representation which fairly reflects the population and Democratic strength of the State” (Democratic National Committee and Others 1970). The commission left the decision to hold a primary or caucus up to the state. A second commission, the Commission on the Rules (known as the O’Hara Commission), was formed simultaneously to make recommendations about several inter-

1.4 Review of the US Primary Process and the History …

15

nal operating procedures as well to address the question of how to apportion delegates among the states. Recommendations from both commissions were put in place for the 1972 election. The Mikulski Commission followed, with recommendations for the 1976 election. These recommendations included forbidding winner-take-all elections and mandating proportional allocation with a 10% cutoff, in which candidates who receive less than 10% of the popular vote were assured to receive no delegates. The mandate of the Winograd Commission, established in 1974 as the Commission on the Role and Future of Presidential Primaries, was to wrestle with the consequences of the general shift from caucuses to primaries, and to consider the balance of power between party insiders and the broader Democratic Party. The commission reaffirmed that the choice of primary or caucus should be left to individual states but established the category of pledged delegates reserved for party leaders and elected officials (PLEOs). These were assigned to candidates in proportion to the outcome of the vote. Thus, when the PLEOs were first added to the 1980 primary, they did not change the distribution of the delegates among the candidates but did alter the composition of the delegates (who could serve as a delegate). Several other recommendations were put into place with an eye to ensuring Jimmy Carter’s reelection. These included allowing winner-take-all allocation at the district level and an increased cutoff of 15–20%. Following Carter’s 1980 loss, the Hunt Commission was formed to consider again the conflicting demands between openness (allowing a wider array of candidates and their supporters) and winnability (ensuring that the final nominee has a strong chance of winning the general election to become President). The repeated debates about the role of party elders and the broader party members centered in part around the question of whether the PLEOs should be pledged or unpledged. Changing the status of PLEOs to unpledged would allow party officials to weigh in at the national convention, independent of the results of the primaries, possibly delaying the emergence of a leading candidate until later in the primary season. Ultimately, the commission recommended retaining the pledged PLEOs but added an additional roughly 550 unpledged delegates or “superdelegates” to the total delegate count. They also established bonus delegates for the leading candidate. Both of these reforms were intended to protect strong candidates and make it more difficult for long-shot candidates to gain traction. The combination of reforms enacted as a result of the Hunt Commission led to widespread complaints about the fairness of the nomination process at the 1984 convention. Jesse Jackson testified that he had received 21% of the vote, but only 11% of the delegates. Gary Hart won the Ohio primary with 43% of the vote but received fewer delegates than Mondale, who had received 41% of the vote. In response, the Democratic Party established the Fairness Commission which, despite articulating a range of ultimate goals for the party primary, ultimately declared its top priority as creating a process that would ensure nominating a candidate who could win the national election and as a second priority to build consensus within the party. Ultimately, the commission lowered the cutoff threshold to 15%, justifying the change by saying that with no threshold, the results would be inconclusive, leading to a decision in “backrooms” by party officials. They continued to allow bonus delegates

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1 Apportionment in the US Presidential Primaries

and increased the number of superdelegates. There was also a general sense among party leaders that the constant revision of rules following each election should cease, while the party concentrated on winning elections (Jewitt 2019). During this 20-year period, the Republican Party, in contrast to the Democratic Party, was much less occupied in amending its rules. This was in part because there was no faction calling for reform, and in part because of its preference for state decision-making (Jewitt 2019). Procedurally, too, the national Republican Party faced more burdens in implementing changes since all changes to the rules had to be approved by a series of committees as well as at the convention. Thus, any new rules would not take effect until the following election cycle. Despite these hurdles, the Republican Party made a number of changes, often as the result of corresponding changes by the Democratic Party. For instance, to accommodate the Democratic Party shift from caucuses to primaries, many states enacted legislation to hold simultaneous primaries for both parties. Additionally, changes to the calendar were made at the state level for both parties. As with the Democratic Party, a number of small reforms generally increased the move toward primaries and an embrace of wider participation. In 1976, a new allocation formula in the Republican Party gave each state a fixed number of delegates per congressional district as well as a number of at-large delegates with bonus delegates based on the results of recent elections. During the 1980s, however, with the continued electoral success of Ronald Reagan and George H. W. Bush, there was little pressure to change delegate allocation rules. And in general, the national party left most of the details for how delegates were to be allocated to the state parties which largely continued to use winner-take-all rules to allocate delegates. During the 1990s, both parties placed renewed attention on the primary calendar— both the sequencing of state primaries and their timing. Of particular concern was the increased emphasis on front-loading, the practice of states moving their elections earlier in the primary calendar in hopes of having a stronger voice in the early winnowing out of candidates. The Republican Party’s Task Force on Primaries and Caucuses, created after the 1996 convention, tried to address this issue by creating incentives, offering bonus delegates to states holding their contests later. Their efforts were largely unsuccessful, with five states, in addition to Iowa and New Hampshire (which were granted permission to run early Republican primaries), holding elections in February, in advance of the Democrat contest. This ensured the Republican Party a monopoly of news coverage during the early part of the primary season. Not wanting the other party to gain an advantage during the 2004 primary, both parties allowed states to hold their primaries in February, resulting in a highly front-loaded contest. For the 2008 election, the Democratic Party’s Commission on Presidential Nomination Timing and Scheduling allowed South Carolina and Nevada to join Iowa and New Hampshire in early primaries. They also followed the Republican Party by introducing a system of bonus delegates to encourage states to delay their primaries. States holding contests after April 1 received an additional 10% of their delegates; states holding contests after May 1 received an additional 20%. They also added a 15% bonus for neighboring states if they held contests on the same day. The results were similarly unsuccessful. Florida and Michigan scheduled their primaries early,

1.4 Review of the US Primary Process and the History …

17

and the party retaliated by allowing each delegate only half a vote. The Republicans similarly removed half the delegates from their offenders: New Hampshire, Michigan, South Carolina, Florida, and Wyoming. Overall, the 2008 election was the most compressed yet with roughly half of the states holding primaries by February 5. While the Democratic Party’s nomination process was protracted (with Barack Obama and Hillary Clinton remaining nearly tied in total number of delegates), on the Republican side, the winner-take-all allocation process allowed John McCain to win a substantial lead early against his nearest rival, Mitt Romney. Following the 2008 election, the Republican Party formed the Temporary Delegate Selection Committee to make more sweeping changes. These changes were adopted in time for the 2012 election and included a condition that all states hold their caucuses or primaries in March or later, with the exception of the same four states (Iowa, New Hampshire, Nevada, and South Carolina). Further, states holding primaries or caucuses before April were required to use a method of proportional allocation to award their state-wide delegates (district-wide delegates could be winner-take-all). States holding primaries after April 1 could continue to use whatever method they desired. (Unlike the Democratic Party, which specified that Hamilton’s method be used to allocate delegates proportionally, the Committee did not specify a particular method. Suggestions for “rounding rules” were provided by the RNC’s counsel’s office (Coleman 2015), but states that used proportional allocation were allowed to devise their own rules.) Any state not abiding by these new rules was penalized by losing 50% of their delegates. Like the 2008 Democratic reforms, the changes to the Republican primary process were only somewhat successful in modifying the calendar for 2012. (Many states canceled their primaries on the Democratic side because Obama was running as an incumbent.) Most states delayed their primaries until March. However, Arizona, Florida and Michigan did not, which caused the exempted four states to move their primaries into January. In response, the national party stripped half the delegates from these states. In addition, Florida and Michigan violated the delegate allocation rules by continuing to use a winner-take-all approach. Neither state received an additional penalty for the double infraction. In the 2016 election cycle, the Republican Party continued to exert pressure on the calendar but moved up their date before which winner-take-all rules were forbidden to the middle of March. They also increased the penalty for states (with the exception of the four early states) that scheduled their contests earlier, threatening to reduce their delegate size to 9 or 12 delegates. This time, no state violated the rules. Among the states holding later primaries, many (New Jersey, Delaware, Maryland, etc.) continued to use a winner-take-all method of assigning at-large delegates; there were others, however, who adopted a proportional system or a hybrid model, in which a winner-take-all method applied if a candidate received a majority of the vote, and a proportional system was used if not. For a more detailed discussion of these changes, see Putnam (2012), Jewitt (2013), Coleman (2015). On the Democratic side, the 2016 race proved protracted again, this time between Hillary Clinton and Bernie Sanders. After Clinton’s eventual nomination, Bernie

18

1 Apportionment in the US Presidential Primaries

Sanders and his supporters argued that the inclusion of the superdelegates continued to advantage “establishment” candidates. In response, the party formed the Unity Reform Commission to reconsider their role. Ultimately, the party voted to approve an amended view of their recommendations. Superdelegates would now be prohibited from voting on the first ballot at the National Convention, unless their vote would not affect the outcome; they could continue to vote in subsequent ballots. This change was discussed at length during the early stages of the 2020 primary when, amidst a crowded field of nominees, Sanders appeared to be taking the lead, despite concern among many of the party elite. The fact that Joe Biden eventually won the Democratic nomination will likely not have an effect on the ongoing debate about the role of superdelegates or the balance of power between party officials and the broader members of the party. Few changes were made on the Republican side in 2020, since Donald Trump ran largely unopposed. For both parties, future election cycles will likely bring additional changes to the delegate selection process. The evolution of delegate selection rules in both parties has been the subject of extensive research with respect to its effect on elections, e.g., Ansolabehere and King (1990), Putnam (2012), Jewitt (2013). Meinke et al. (1990) examine an empirical model to explain why parties might be motivated to adopt more open processes. Cooper (2002) uses simulations to study the effect of diverse voter preferences and the implementation of proportional rules on the length of the nomination process. Ross et al. (2018) examines why states chose winner-take-all versus proportional delegate allocation in the Republican primary. Jewitt (2019) provides an extensive analysis of the impact of the rules on the primary process, focusing on four key factors in which the state parties largely have control to make decisions within the framework of national rules: delegate allocation rule (winner-take-all versus proportional), the openness of the contest, the type of contest (primary or caucus), and the date of each state’s contest.

1.5 Why Apportionment in Primaries is Different As discussed in Sect. 1.2, the delegate allocation process is complex, involving multiple nested apportionment problems across different geographical constituencies, application of cutoffs, and an array of different apportionment methods. In this section, we summarize some of the key features that make the problem of delegate allocation unique, and explain how the use and evaluation of delegate apportionment methods differs from methods used in other contexts. The structure of delegate apportionment adds to the complexity of the problem. Broadly speaking, both parties use apportionment to allocate delegates to states. State Democratic parties also use apportionment to allocate some delegates to the states’ congressional districts. Furthermore, both parties use apportionment to award delegates to candidates based on their share of the vote. Such a process creates nested apportionments and aggregation of delegates across geographic constituen-

1.5 Why Apportionment in Primaries is Different

19

cies, which can lead to the final allocation of delegates differing markedly from true proportionality (see Sects. 5.7 and 6.2). The use of cutoffs that eliminate weaker candidates also causes the results to diverge from proportionality. They may lead to unintended consequences. For example, there may be instances when candidates do less well because other candidates are eliminated, leading to a possible no-show paradox in which it can be better for an individual candidate if fewer supporters turn up to vote (see Sects. 6.1 and 6.3). Many of the apportionment methods used to allocate delegates are new, which invites analysis. We argue that the criteria for their evaluation are different from methods applied in other contexts, particularly for the US House or other geographicbased representation. Below, we enumerate some of the factors that establish this difference. First, when apportioning delegates there is less concern with bias toward large states/candidates or against small states/candidates. Elimination of bias has been a crucial goal in determining and assessing apportionment in the US House of Representatives in order to guarantee equal representation among citizens from both small and large states. However, in presidential primaries it is allowable, and possibly even desirable, to use a method that is biased in favor of top candidates rather than an unbiased method or a method that is biased in favor of candidates with small vote totals. Votes are typically viewed as being “earned” in a way that state populations definitely are not earned, and thus there is more justification for bias in favor of candidates with large vote totals. For example, it may not be offensive if the leading presidential candidate receives 60% of the delegates in a state primary while earning only 40% of the vote. In contrast, such a disparity between population share and representation share in the House of Representatives would not be tolerated. Second, and related to bias, there is less of a concern in what may be termed “strict proportionality.” In House seat apportionment, it is desirable for a state’s population share to stay as close as possible to its seat share. The exception would be to loosen slightly this requirement to protect small states. For example, the method currently used for House seat apportionment in the US, the method of Hill–Huntington, has a slight bias in favor of small states (Balinski and Young 2001). This is less of a concern in proportional representation systems where there is often a rationale for deviating from proportionality. Favoring small parties allows for more participation in the legislative process, while favoring large parties helps to create governing coalitions. However, legal constraints, as well as shared local norms for democratic representation, prevent these deviations from being too large. But in a presidential primary there are no such laws or built-in expectations. We may be willing to tolerate large deviations from proportionality as long as these deviations favor the top candidates or disadvantage the bottom candidates. Such disproportionality also helps a leading candidate gain momentum, encouraging other candidates to drop out, and builds a sense of consensus in favor of the party’s eventual nominee. Likewise, it may be desirable to favor the bottom candidate at the expense of stronger candidates, if the goal is to encourage a broader range of voices. Third, there is no need to guarantee representation. The US Constitution stipulates that every state is entitled to at least one seat in the House regardless of the size

20

1 Apportionment in the US Presidential Primaries

of its population. Likewise, the European parliament has minimal requirements for each country’s representation. In a presidential primary such a requirement makes little sense, although some state parties guarantee delegates to candidates that have achieved a high enough level of vote share. In fact, the use of thresholds in the presidential primaries provides a mechanism for explicitly denying guaranteed representation. Fourth, a presidential primary has an aspect of “winning” that is absent from geographic-based apportionment. California has the largest population and therefore the largest number of seats in the US House, but no particular value is attached to this. In a primary, winning the most delegates in a state primary has its own intrinsic value, especially if this victory occurs in a state with an early primary date. In addition, having a majority, rather than just a plurality, of delegates overall means that the top candidate is guaranteed to become the party’s presidential nominee. (See Sect. 5.5 for a fuller discussion.) Fifth, there are some classical apportionment paradoxes that are important for house representation or proportional representation but may not be important for primary apportionment. For example, the Alabama paradox (see Sect. 6.4 for a discussion of the Alabama paradox) caused much rancor and bitterness in the US House in the late nineteenth and early twentieth centuries, but it is not clear that these paradoxes have the same importance when evaluating allocation formulas for delegates. On the other hand, there are other paradoxes such as the elimination and aggregation paradoxes (see Sects. 6.1 and 6.2) that are important in the delegate context (and to a lesser extent in proportional representation), but are of no importance in the US House seat context. Sixth, the ideal of “one person one vote” is arguably not relevant in the presidential primary context. Much of the classical literature is concerned with trying to ensure that each congressperson represents the same number of people with as little variation as possible, thereby trying to achieve the ideal of equal representation for all citizens. Equal representation in this sense is not a goal of delegate allocation. Seventh, a presidential primary unfolds over the course of many months, and therefore an apportionment method used in a state with an early primary date may not be appropriate for a state with a later primary date. This can clearly be seen in the Republican primary, in which the national party requires that states with early primary dates use a proportional method, whereas states with later primary dates are permitted to use the winner-take-all method. There is no analogue of this rolling calendar in other apportionment problems involving representation. The proportional allocation of delegates earlier in the primary process allows voters to support potentially weaker candidates in an effort to affect the party’s platform by keeping these candidates in the race and on debate stages. (See Chap. 7 for a discussion of how delegate apportionment methods could be altered to respond to the primary calendar.) There are other differences, but many of these are less mathematical. As discussed in Sect. 1.4, the current delegate selection system is a product of many years of political reforms, compromises, and strategic considerations. These factors determine the overall calendar, structure, and process for candidates to accrue delegates and dictate the choice of apportionment method and cutoffs in each state. This in turn affects

References

21

campaign strategy, party member turnout, and application of the rules, which also affects opinions about the credibility and effectiveness of the delegation selection process among the voters, candidates, and party leaders. Thus the problem of allocating delegates occupies a unique place in the landscape of apportionment theory. In the remainder of Part I, we provide a complete description of the mathematical aspects of the apportionment process in each party beginning with the Democratic party in Chap. 2.

References Ansolabehere S, King G (1990) Measuring in presidential nominations the consequences of delegate selection rules. J Polit 52(2):609–621. https://doi.org/10.2307/2131908 Balinski ML, Young HP (2001) Fair representation: meeting the ideal of one man, one vote, 2nd edn. Brookings Institution Press, Washington, DC Coleman KJ (2012) Presidential nominating process: current issues. Congressional Research Service RL34222 Coleman KJ (2015) The presidential nominating process and the national party conventions, 2016: frequently asked questions. Congressional Research Service Cooper AL (2002) The effective length of the presidential primary season: the impact of delegate allocation rules and voter preferences. J Theor Polit 14(1):71–92 Democratic National Committee and Others (1970) Commission on party structure and delegate selection. In: Mandate for reform, p 40 Geist KA, Jones MA, Wilson JM (2010) Apportionment in the Democratic primary process. Math Teacher 104:214–220. https://doi.org/10.2307/20876835 Jewitt CE (2013) The Republican Party and the unsuccessful 2012 presidential nomination reforms. In: Presented at state of the parties conference, Akron, OH. https://www.uakron.edu/dotAsset/ 79292ea0-70b6-4375-8ecd-80f79a3cff04.pdf Jewitt CE (2019) The primary rules: parties, voters, and presidential nominations. University of Michigan Press, Ann Arbor, MI. https://doi.org/10.3998/mpub.10020994 Jones MA, McCune D, Wilson JM (2019) The elimination paradox: apportionment in the Democratic party. Public Choice 178(1):53–65. https://doi.org/10.1007/s11127-018-0608-3 Jones MA, McCune D, Wilson JM (2020) New quota-based apportionment methods: the allocation of delegates in the Republican presidential primary. Math Soc Sci 108:122–137. https://doi.org/ 10.1016/j.mathsocsci.2020.05.001 Meinke SR, Staton JK, Wuhs ST (1990) State delegate selection rules for presidential nominations, 1972–2000. J Polit 68(1):180–193. https://doi.org/10.1111/j.1468-2508.2006.00379.x Pukelsheim F (2017) Proportional representation: apportionment methods and their applications, 2nd edn. Springer, Cham (With a foreword by Andrew Duff) Putnam JT (2012) The impact of rules changes on the 2012 Republican presidential primary process. Society 49(5):400–404. https://doi.org/10.1007/s12115-012-9573-5 Ross RE, Cann DM, Burt J (2018) Polls and elections without rhyme or reason? Understanding presidential nomination delegate allocation rules. President Stud Quart 48(4):804–816

Chapter 2

The Democratic Party Primary

The allocation of delegates to presidential candidates in the Democratic Party presidential primaries occurs in several stages. In this chapter, we provide a broad overview of the process, beginning with how delegates are divided among the states and subsequently apportioned to candidates in proportion to their vote share in state primary elections. A few states hold caucuses with delegate allocation rules that are more complicated than the rules used in the state primaries; the allocation process for state caucuses is analyzed in Chap. 3. The rules dictating delegate allocation in the 2020 primaries are outlined in the 2020 DNC Delegate Selection Plan (DEL) (Democratic National Committee 2018), Call for the Convention (CALL) (Democratic National Committee 2018), and Regulations of the Rules and Bylaws Committee (REGS) (Democratic National Committee 2018). Similar documents exist for primaries from previous election cycles. Our discussion focuses on the 2020 primary, but generally the process we describe has remained unchanged for the last several Democratic primaries. Our goal is not to include a complete description of the Democratic primary process. We mostly omit non-mathematical details such as committee membership and rules related to the organizational structure and procedures of the party. Instead, we focus on the items of mathematical interest, providing an overview of the quantitative factors that directly affect delegate apportionment. Additional information can be found in the three rules documents listed above, as well as at The Green Papers (The Green Papers 2022) and other online sources. As noted in Chap. 1, there has been little examination of these rules from a mathematical standpoint. Jewitt (2019) analyzes the impact of factors such as whether the primary is open or closed, the choice of winner-take-all or proportional allocation, and the timing of the primary season in the calendar. However, she does not consider mathematical factors such as the apportionment methods themselves or the consequences of using thresholds. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9_2

23

24

2 The Democratic Party Primary

Geist et al. (2010) describes mathematically the process used in the Democratic primary, while Jones and Wilson (2016) discuss the role of aggregating delegates across multiple districts. Further, Jones et al. (2019) discuss how the application of minimal thresholds for delegate apportionment can lead to paradoxical outcomes in Democratic state primaries. However, these three prior papers focus only on translating the candidates’ popular votes into delegate counts and not the apportionment that takes place before the election. In Sect. 2.1, we outline the four stages in which apportionment methods are used to determine the number of delegates per state, the number of Pledged Leaders and Election Officials delegates (PLEOs), the number of delegates per congressional district, and the number of delegates a candidate receives based on the popular vote. Throughout this section, we use data from the 2020 presidential primary. We introduce the apportionment method known as Hamilton’s method, which is used by all state Democratic parties to apportion delegates to candidates based on the results of the state primary. In Sect. 2.2, we discuss geometric interpretations of Hamilton’s method and examine the consequences of the geometry. In Sect. 2.3, we discuss the new role of ranked choice voting in several state primaries. This chapter is mostly descriptive; theoretical analyses of these processes can be found in Chaps. 5 and 6.

2.1 Democratic Delegate Plan and Primaries The rules that guided delegate apportionment in the 2020 Democratic Party primaries are described in the three rules documents distributed by the Democratic National Committee (DNC). For clarity, we break our description of the delegate allocation process into stages. The apportionment rules apply to all Democratic state primaries; many of them apply to caucuses as well, but some caucuses have important differences that are described in Chap. 3. We illustrate the apportionment process using data from the 2020 New Hampshire Democratic Party primary, which was the first primary election in the 2020 Democratic presidential primary, preceded only by the Iowa caucuses. Stage I: Determining each state party’s number of delegates. Apportionment in the Democratic presidential primary begins by determining a base number of delegates that each state party receives to apportion in its state primary. Thirty-two hundred delegates are apportioned to the state parties in proportion to the state’s population and how “Democratic” the state is. The DNC uses the following formula, which we write in terms of the 2020 primary, to determine what portion of the 3200 delegates each state party should receive: A=

1 2

(

S DV (2008) + S DV (2012) + S DV (2016) SEV + T DV (2008) + T DV (2012) + T DV (2016) 538

) .

(2.1)

The number A is the state party’s portion (called the allocation factor) of the 3200 base delegates that the state party should receive. S DV (i ) is the total state vote for the

2.1 Democratic Delegate Plan and Primaries

25

Democratic presidential candidate in election year i, T DV (i) is the total national vote for the Democratic presidential candidate in election year i, and S E V is the state’s number of electoral votes.1 Thus, a state party’s portion of the 3200 delegates is a function of how supportive the state has been of Democratic presidential candidates in the last three presidential elections and the state’s population. Specifically, the allocation factor is the average of the state’s portion of all electoral votes for the upcoming presidential election and the average portion of all Democratic votes cast in the state in the last three presidential elections. An analogous formula has been used in the last several presidential primaries. This allocation factor is multiplied by 3200, resulting in a quota of qi = 3200 Ai , where Ai is state i’s allocation factor. The result is then rounded to the nearest integer, which we indicate using the notation [ ], so that each state receives a base number of delegates di = [qi ]. Example 2.1 To determine allocation factor for New Hampshire (NH) in the 2020 Democratic primary, S E V = 4 (the number of NH’s electoral votes), S DV (2008) = 384,826, S DV (2012) = 369,561 and S D V (2016) = 348,526. The Democratic vote totals for the previous three elections were T DV (2008) = 69,498,516, T DV (2012) = 65,915,795, and T DV (2016) = 65,853,514. Thus, the allocation factor for NH in 2020 was ( ) 1 384,826 + 369,561 + 348,526 4 A= = 0.00646 + 2 69,498,516 + 65,915,795 + 65,853,514 538 and NH was entitled to di = [3200(0.00646)] = 21 delegates. As discussed in Chap. 1, one reason why the apportionment problem exists is that nearest integer rounding of quotas does not always yield apportionments that sum to the desired number of delegates. This occurred in the 2020 primary, when nearest integer rounding yielded 3198 delegates rather than 3200. Rather than adjust the apportionment so that the number of delegates sums to 3200, the DNC Σ accepted the values di = [qi ] = [3200 Ai ], resulting in a total allocation of D , = i [3200 Ai ] delegates, which, in general, may not equal D = 3200. Thus, this stage of delegate allocation is not a classical apportionment problem. The decision to allocate D , and not D delegates means that the apportionment of the 3200 base delegates is susceptible to an issue discussed in Chap. 1. If D , were used to determine the quotas, then qi, = D , Ai would also change, resulting perhaps in a different apportionment di, = [qi, ] for some state and necessitating further changes to D. For example, the 2020 allocation factor for Pennsylvania (PA) was A = 0.04143 (CALL mistakenly lists the number as 0.04145), resulting in the PA Democratic Party being awarded [3200(0.04143)] = 133 base delegates. However, if 3198 delegates were being allocated, then the party would receive [3198(0.04143)] = 132 delegates, which would necessitate another change to D and result in further changes to the quotas. In apportionment theory, this is one reason to fix the number of delegates (or house size). 1

In some election years, S E V was the average number of the state’s electoral votes in the current election year and the previous two presidential elections.

26

2 The Democratic Party Primary

In some cases, a state party may receive additional bonus delegates. The rules governing when this happens have varied in the last few primaries, but generally extra delegates are awarded to state parties that hold their election later in the primary season, or that coordinate the date of their primary with neighboring states. (Section 1.4 discusses some of the inducements adopted by both parties over the last several decades to influence when and how state parties schedule their primaries.) Since New Hampshire has historically maintained its place as the first primary in the calendar and did not geographically cluster with neighboring states, its number of base delegates remained at 21 for the 2020 primary. Once the total number of base delegates has been calculated, each state party receives a number of PLEO delegates, which equals 15% of the number of base delegates, rounded to the nearest integer. In 2020, NH received [21(0.15)] = 3 PLEO delegates, for a total of 24 delegates in its primary.2 New Hampshire is a Democratic-leaning state, but its small population and special place at the beginning of the primary mean that it received relatively few delegates in 2020. The state with the largest number delegates in 2020 was California with 415. California is the most populous state, and it is firmly Democratic in presidential elections. California was allocated this large number of delegates while receiving no bonus delegates because its primary occurred early in the primary calendar on March 3. New York had the next largest delegate total of 274; it received a large number of bonus delegates due to its late primary date of June 23. Texas, the second most populous state in the country, received only 228 delegates, meaning that California received 1.8 times more delegates than Texas even though California has 1.5 times more electoral votes than Texas. This discrepancy occurs by design, as Texas has been a Republican-leaning state in the three previous presidential elections. The smallest numbers of delegates were given to Wyoming and North Dakota, each of which received 14 delegates. These states are sparsely populated and are Republican leaning, so their allocation factors are much smaller. The addition of bonus and PLEO delegates means that the total number of (pledged) delegates going to the National Convention is much larger than the original base number of delegates 3200. In 2020, the total number of delegates for the 50 states and Washington, DC, was 3889. In summary, the number of delegates that a state party receives to apportion in its primary is a function of the state’s population, the state’s level of support for Democratic presidential candidates in recent elections, the state’s primary date, and geographical clustering. Although the initial part of this process uses proportional allocation, it is not formally an apportionment problem since the total number of delegates is not fixed. An interesting question is how much the apportionment at this stage differs from what would have been allocated had a traditional apportionment method been used. 2

Each state party also receives a certain number of unpledged PLEO delegates, or superdelegates. These delegates played a large role in the 2008 and 2016 primaries, but in 2020 the rules were changed so that these delegates were not allowed to cast a vote in the initial round of voting at the Democratic National Convention. Thus, unpledged PLEO delegates vote only if a candidate does not secure a majority of the pledged delegate vote, which did not occur.

2.1 Democratic Delegate Plan and Primaries

27

Percentage Difference

10

5

0

-5

Jun 2

May 19

May 12

Date

May 5 May 2 Apr 28

Apr 7 Apr 4

Mar 24

Mar 17

Mar 10

Mar 3 Feb 29

Feb 22

Feb 11

Feb 3

-10

Fig. 2.1 Percentage differences between the number of delegates each state received in 2020 and what it would have received using Hamilton’s method

Suppose, for example, that the 3889 delegates were distributed using Hamilton’s method which, as we discuss below, is the Democratic apportionment method of choice, without regard for primary date or geographic clustering. The results are shown in Fig. 2.1. Each point in the scatterplot shows a given state’s percentage difference between the number of delegates that the state actually received and what it would have received from applying Hamilton’s method. The scatterplot clearly shows the effect of the bonus delegates received for waiting until later in the primary calendar. The DNC also allocates delegates to six other geographic/demographic entities for which the allocation factor cannot be computed because they do not participate in presidential elections: American Samoa (6 base delegates), Democrats Abroad (12 base delegates and 1 pledged PLEO), Guam (7 base delegates), the Northern Mariana Islands (6 base delegates), Puerto Rico (44 base delegates and 7 pledged PLEOs), and the Virgin Islands (7 base delegates). Thus, the total number of pledged delegates in the 2020 Democratic presidential primary was 3979.

28

2 The Democratic Party Primary

Stage II: Determining the number of delegates of each type. Once a state party receives its allotment of base delegates and PLEO delegates, the party divides the base delegates into two categories: district and at-large delegates. After the state primary, district delegates are apportioned to candidates based on results in individual congressional districts (CDs) while the at-large and PLEO delegates are apportioned to candidates based on the statewide election results. To determine the number of district delegates, each state party calculates 75% of its base delegates prior to the addition of the PLEO delegates, rounded to the nearest integer. The remaining 25% of base delegates become at-large delegates.3 Note that this calculation can be interpreted as a two-state apportionment problem in which 75% and 25% are the proportions used to allocate the total number of state delegates into district- and state-level delegates. Since there are only two categories, nearest integer rounding provides a unique apportionment except in the case when both quotas result in a decimal term of 0.5 (in which case the quota corresponding to the district delegates is rounded up, as indicated in CALL). Example 2.2 In NH, there were 21 base delegates before the PLEO addition. Thus, [21(0.75)] = 16 became district delegates, and the remaining 5 were at-large delegates in the 2020 primary. Stage III: Determining the number of district delegates received by each congressional district. Once a state party has calculated its number of district delegates, the next step is to determine how to divide these delegates among the state’s CDs. This is a classical apportionment problem since the number of district delegates in the state is fixed and these delegates are allocated proportionally among the districts. Each state party has limited freedom over how to determine the proportionality criterion for allocating district delegates. Appendix A of REGS outlines four possible allocation factor formulas, all of which are similar in spirit to Eq. 2.1: 1. A formula that gives equal weight to the total population of the district and to the average vote for the Democratic candidates in the two most recent presidential elections. This formula has the form ( ) 1 DP D DV (2012) + D DV (2016) + (2.2) A= 2 SP S DV (2012) + S DV (2016) where D P and S P are the district and state populations, respectively, and D DV (i ) and S DV (i ) are the number of district and state votes for the Democratic presidential candidate in year i, respectively. 2. A formula that gives equal weight to the vote in the district for the Democratic candidates in the most recent presidential and gubernatorial elections. For 2020, this formula has the form ( ) 1 D DV (2016) D DG(2016) + (2.3) A= 2 S DV (2016) S DG(2016) 3

This rule was used by all 50 states as well as Puerto Rico, which divided its 44 base delegates into 33 district, 11 at-large delegates.

2.1 Democratic Delegate Plan and Primaries

29

where D DV (i) and S DV (i ) are as before and D DG(i ) and S DG(i ) are the number of district and state votes for the Democratic gubernatorial candidate, respectively. 3. A formula that gives equal weight to the average of the vote for the Democratic candidates in the two most recent presidential elections and to Democratic Party registration or enrollment as of January 1, 2020. This is defined similarly to Eqs. 2.2 and 2.3. 4. A formula that gives equal weight to each of the numbers calculated from (1), (2), and (3) above. Thus, regardless of the formula that a state party chooses to use, each CD is entitled to delegates in proportion to some combination of the district’s Democratic lean and its population. The state parties have some freedom in the data used in the above formulas. For example, formula (1) requires district populations to be used but does not specify the source from which the population data should come. The 2020 New York Democratic Party used 2010 census numbers (meaning that all of the districts have essentially the same size), while the 2020 Georgia Democratic Party used 2017 population estimates provided by the American Community Survey. Furthermore, state parties are not required to provide details of their calculations when using these formulas. The majority of state party delegate selection plans (DSPs) simply state which of the four formulas they use and then give the resulting allocation among districts. For example, NH used formula (3); its DSP states “New Hampshire’s district-level delegates are apportioned among the districts based on a formula giving equal weight to the average of the vote for the Democratic candidates in the 2012 and 2016 presidential elections and to Democratic Party registration or enrollment as of January 1st, 2020.” (New Hampshire Democratic Party 2019). The NH DSP then states, without data or explanation, that each of the two congressional districts receives 8 delegates. Of the 39 states with at least three congressional districts (states in which the apportionment problem is interesting), we were able to obtain 33 Democratic state party DSPs from 2020. Of these, only 2 (GA and TN) included their calculations when reporting the number of delegates allocated to each district. Without more information, it is difficult to know precisely how state parties solve this apportionment problem. Since the national DSP explicitly requires states to use HAM (see Definition 1.1) in apportioning delegates to candidates, it is likely that this method or HAM∗ (see Definition 1.2) is also used at this stage. Applying publicly available information regarding district population, Democratic registration, and election results to the state’s given allocation factor formula provides some confirmation, although it is difficult to determine a general rule since nearest integer rounding of quotas frequently gives a legitimate apportionment. Thus, for many states, we can infer that nearest integer rounding is their primary methodology, even if it is not clear how the apportionment would be adjusted if such rounding did not work. Several party officials confirmed that they would use a version of HAM∗ , but we did not verify this for every state party.4 4

This was confirmed with state party officials from Colorado and county party officials from Iowa.

30

2 The Democratic Party Primary

Table 2.1 Apportioning district delegates among Georgia’s 14 districts using HAM∗ CD Population 2012 2016 AF Quota [Quota] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total

747,334 674,100 738,066 765,854 784,370 756,389 807,492 702,262 755,181 736,838 768,968 722,937 752,631 716,857 10,429,279

111,949 110,191 153,998 136,455 97,746 102,155 218,499 224,907 241,217 259,805 155,017 155,077 101,065 132,012 99,677 91,360 54,388 57,467 107,039 112,691 94,447 116,457 117,085 108,937 202,794 213,936 58,906 56,513 1,773,827 1,877,963

0.06624 0.07209 0.06275 0.09743 0.10620 0.07324 0.07063 0.05982 0.05152 0.06541 0.06574 0.06561 0.09314 0.05017 1

4.50458 4.90187 4.26731 6.62506 7.22185 4.9806 4.80253 4.06806 3.50336 4.44793 4.47050 4.46119 6.33358 3.41160 68

5 5 4 7 7 5 5 4 4 4 4 4 6 3 67

Appt 5 5 4 7 7 5 5 4 4 4 5 4 6 3 68

Example 2.3 The DSP of the 2020 Georgia Democratic Party provides details for how the 68 district delegates were apportioned among its 14 districts. The data, reproduced in Table 2.1, confirm that the state used HAM∗ . Next to the apportionment for CD 11, they provide an asterisk and note, “Needed another one due to rounding” (Georgia Democratic Party 2019). As with the choice of apportionment method, state parties are also free to perform their calculations as accurately as they wish. In Table 2.1, GA rounded its allocation factors to five decimal places. Had the allocation factor for CD 1 been recorded more accurately as 0.06593653, the quota for CD 1 would have been approximately 4.484, putting it on the other side of the rounding threshold. In this case, CD 1 would still receive 5 delegates under HAM∗ , and the apportionment would be unchanged. In general, however, these kinds of distinctions can make a difference, particularly for the apportionment methods discussed in Chap. 5. Finally, it seems likely that in this apportionment process, no CD should be left without a delegate. Since the number of delegates is large enough for this issue not to arise, it is not clear what real role this plays. However, we could argue that the apportionment method used in Stage III is HAM* with the constraint that each CD receives at least one delegate. Stage IV: Apportioning delegates to candidates. Stages I–III take place before the primary election occurs. The apportionment in Stage IV occurs after the primary election and is used to allocate delegates to candidates based on the percentage share each candidate receives at district and state levels. The RULES document outlines how each type of delegate is to be allocated (see Table 2.2). Steps 3–5 comprise HAM

2.1 Democratic Delegate Plan and Primaries

31

Table 2.2 Democratic Delegate Selection Rules, rule 14.D (DEL) Steps Action 1 2

3 4 5

Tabulate the percentage of the vote that each presidential preference (including uncommitted status) receives in the congressional district to three decimals Retabulate the percentage of the vote to three decimals, received by each presidential preference excluding the votes of presidential preferences whose percentage in Step 1 falls below 15% Multiply the number of delegates to be allocated by the percentage received by each presidential preference Delegates shall be allocated to each presidential preference based on the whole numbers which result from the multiplication in Step 3 Remaining delegates, if any, shall be awarded in order of the highest fractional remainders in Step 3

(see Definition 1.1), while steps 1 and 2 ensure that HAM is applied only to candidates who receive at least 15% of the vote. Thus, the delegate apportionment problem in a state Democratic primary consists of two steps: (1) eliminating candidates (and their votes) that fall beneath a viability threshold of 15%, and (2) applying HAM to the remaining candidates. The delegate apportionment problem in this stage is not treated like a house seat apportionment problem where strict proportionality is the goal, as evidenced by the 15% threshold. However, because the Democrats use HAM for the remaining candidates, it seems they endorse the principle of proportional allocation for candidates with strong enough support. The rules in Table 2.2 stipulate how district delegates are to be apportioned in a CD. They do not explicitly describe how the at-large and PLEO delegates are to be allocated; however, the RULES document strongly suggests that this methodology be used when apportioning these delegates as well, and state parties consistently appear to follow this procedure for statewide as well as district apportionment. We note that even though both at-large and PLEO delegates are apportioned based on the same statewide data, each type of delegate is allocated in its own separate calculation. Example 2.4 In the 2020 NH Democratic primary, only the candidates Sanders, Buttigieg, and Klobuchar surpassed the 15% threshold at the statewide level and in each of the CDs. The data and corresponding HAM apportionments are given in Table 2.3, which displays only the votes that remain after the elimination of other candidates. Summing the allocations for each candidate from the four separate apportionments, the total is 9, 9, and 6 delegates for Sanders, Buttigieg, and Klobuchar, respectively. Note that if all 24 delegates were allocated based solely on the statewide vote, then Sanders, Buttigieg, and Klobuchar would have received 9, 8, and 7 delegates, respectively. Breaking the delegates into categories and then aggregating together the apportionments from each category caused a delegate to be transferred from Klobuchar to Buttigieg. This is an example of an aggregation paradox, which we

32

2 The Democratic Party Primary

Table 2.3 Results of the 2020 NH Democratic primary CD 1 Candidate

Votes

Sanders

CD 2 Quota

Appt

Votes

37,257 2.904

3

Buttigieg

35,940 2.801

Klobuchar

29,442 2.295

PLEO Quota

Appt

Votes

39,127 2.984

3

3

36,514 2.784

2

29,272 2.232

At-Large Quota

Appt

Votes

76,384 1.104

1

76,384 1.840

Quota

Appt 2

3

72,454 1.047

1

72,454 1.745

2

2

58,714 0.849

1

58,714 1.414

1

explore more fully in Sect. 6.2. Because all Democratic state primaries allocate some delegates within individual CDs and break the statewide delegates into the two categories of at-large and PLEO, such aggregation issues are common in these elections. In summary, delegate allocation in a state Democratic primary occurs in four stages. First, delegates are allocated to each state as base and PLEO delegates. Second, the base delegates are divided into the two categories of district and at-large delegates. Third, district delegates are allocated to each CD. Fourth, after the primary election, delegates are apportioned to candidates in proportion to vote totals, both within individual congressional districts and statewide. Hamilton’s method is used at two stages of Democratic delegate apportionment: when apportioning district delegates to districts HAM∗ is generally applied without a threshold, and when apportioning delegates to candidates at the district or statewide level HAM is applied to candidates’ votes that surpass a 15% viability threshold.

2.2 Geometry of Hamilton’s Method In this section, we introduce the geometric view of apportionment methods used by Bradberry (1992), Lucas (1983) and Balinski and Young (2001). This geometry can help illuminate certain properties of an apportionment method and can also provide a means for visually comparing different apportionment methods. We focus on the situation when delegates must be apportioned among three candidates (or three CDs, as in Stage III from Sect. 2.1). Given a fixed number of delegates D, the candidates’ quotas satisfy q1 + q2 + q3 = D. Allowing the quotas to be any set of nonnegative numbers that satisfy the equality constraint Σ yields a set of points q = (q1 , q2 , q3 ) which form a 2-simplex Σ2D = {q ∈ R3 | i qi = D; qi ≥ 0}. This simplex can be visualized in R3 as the portion of the plane x + y + z = D lying in the first octant. The resulting triangle is shown for D = 5 in Fig. 2.2 (Top Left) with integer lattice points labeled. Under a particular apportionment method, Σ2D can be subdivided into regions for which the points in the interior of a given region are precisely the quotas that are apportioned to the same integer lattice point. We refer to these regions as apportionment regions. In the case of HAM (or HAM∗ ), this subdivision corresponds to a hexagonal tiling of the simplex; see Fig. 2.2 (Bottom Left). Under HAM, each

2.2 Geometry of Hamilton’s Method

33

0, 0, 5 . 1, 0, 4 .

. 0, 1, 4 1, 1, 3 .

2, 0, 3 .

. 0, 2, 3

2, 1, 2 1, 2, 2 . .

3, 0, 2 .

. 0, 3, 2

3, 1, 1 2, 2, 1 1, 3, 1 . . .

4, 0, 1 .

. 0, 4, 1

. . . . . . 5, 0, 0 4, 1, 0 3, 2, 0 2, 3, 0 1, 4, 0 0, 5, 0 0, 0, 5 . 1, 0, 4 . 2, 0, 3 . 3, 0, 2 . 4, 0, 1 .

0, 0, 5 .

a : 3, 1, 1 b : 1, 3, 1 1, 0, 4 .

. 0, 1, 4 1, 1, 3 .

2, 0, 3 .

. 0, 2, 3

2, 1, 2 1, 2, 2 . . 3, 1, 1 2, 2, 1 1, 3, 1 . . .

. 0, 3, 2 . 0, 4, 1

. . . . . . 5, 0, 0 4, 1, 0 3, 2, 0 2, 3, 0 1, 4, 0 0, 5, 0

3, 0, 2 . 4, 0, 1 .

. 0, 1, 4 . 0, 2, 3

.

1,1,3

.

.

2,1,2 1,2,2

.a

2,2,1

.

b.

. 0, 3, 2 . 0, 4, 1

. . . . . . 5, 0, 0 4, 1, 0 3, 2, 0 2, 3, 0 1, 4, 0 0, 5, 0

Fig. 2.2 Illustrating the geometry of Hamilton’s method. The top left image shows the simplex Σ25 before being divided into apportionment regions. The top right shows the region in which, after rounding quotas down, there are two (shaded) or one (unshaded) delegates remaining. The bottom left (resp. right) is an apportionment diagram for HAM with no threshold (resp. 15% threshold)

quota point lying in a hexagon is mapped to the integer lattice point at the center of the hexagon. The boundaries of the hexagons correspond to quota vectors in which two candidates have the same fractional remainder, resulting in multiple possible apportionments due to this “tie.” A justification for this hexagonal subdivision can be found in Bradberry (1992). We note that in this hexagonal tiling, each quota point is mapped to the nearest integer lattice point under the standard Euclidean metric. This geometric view of HAM provides an argument that the method is the “fairest” or “most proportional” method; a method that is not HAM will map some q to an integer point that is not the closest apportionment when using Euclidean distance. When n = 3, after HAM allocates the floors of the quotas (i.e., the quotas rounded down to the nearest integer) to the candidates, there are either one or two delegates remaining to allocate. The shaded regions in Fig. 2.2 (Top Right) show where in the simplex Σ25 there are two delegates remaining; the unshaded regions show where

34

2 The Democratic Party Primary

there is only one delegate left. Note that each Hamilton hexagon consists of six congruent triangles, three of which are shaded and three of which are not. Thus, the simplex shows that an interior integer lattice point has an apportionment region consisting of equal parts one or two delegates remaining in the final step of HAM. Under the Democratic rules for Stage IV of apportionment, any candidate who receives less than 15% of the vote at the state or district level is eliminated. This rule necessitates changes in the apportionment diagram, as some quota vectors will now correspond to different integer points. This is visualized in Fig. 2.2 (Bottom Right) for a threshold of 15%. Applying the threshold creates regions near the boundary in which only one or two candidates remain, thereby reducing a three-candidate apportionment problem to a problem involving only two candidates or, at the corners of the diagram, only one candidate. We conclude by noting that the 2-dimensional simplex can be generalized to fixed n and D, higher dimensions when there are more than three candidates. For Σ qi = D} can the (n − 1)-dimensional simplex ΣnD = {(q1 , . . . , qn ) | qi ≥ 0 and be partitioned into solids such that every point q in the interior of a given solid is mapped to the same integer lattice point. The higher-dimensional simplices cannot be easily visualized but may still be used for investigating the properties of a given method.

2.3 Democratic Primaries and Ranked Choice Voting In 2020, the Democratic Parties of AK, HI, KS, and WY (the WY Democratic Party describes its election as a caucus, but in 2020 it operated, functionally, like a primary due to COVID-19 concerns) introduced an innovation into the primary process taken from social choice theory: ranked choice voting (RCV). It is not surprising that RCV would be used in Democratic primaries, as the voting method has seen a surge in support and usage in the US since 2016. For example, the state of Maine began using RCV in 2020 in primaries for statewide offices and for federal offices such as US House Representative, US Senator, and President. Additionally, New York City used RCV in primaries for mayor, city council members, and other municipal offices in 2021. Arguably, RCV was used in the Democratic primary prior to 2020 because some caucus states like IA and NV already mimicked a two-round RCV process; however, 2020 marks the first year in which RCV was explicitly married with apportionment theory in the presidential primary process.

2.3.1 What Is Ranked Choice Voting? RCV is a voting method designed for determining the winner of an election, or for determining a ranking of the candidates from first place to last place. In elections that use this method, voters provide a ranking of the candidates by using preference

2.3 Democratic Primaries and Ranked Choice Voting

35

Table 2.4 Voter profile in a 4-candidate RCV election with 10,000 voters Number 2200 2000 3700 750 650 500 of voters 1st Choice 2nd Choice 3rd Choice 4th Choice

A B D C

A C B D

B C A D

C B A D

C A B D

D C B A

200 D B A C

ballots. In most US jurisdictions that use RCV, voters are limited in the number of candidates they rank and often cannot express a ranking of all the candidates. For example, in municipal elections in Minneapolis, MN, voters can rank up to three candidates, regardless of how many candidates are in the race. The RCV method proceeds in a series of rounds. If a candidate receives a majority of first-place votes, that candidate is declared the winner. Otherwise, the candidate with the fewest first-place votes is eliminated from the race and, in the next round, the votes from voters whose first-place candidate was eliminated are transferred to the second-place candidates on their ballots. This process of eliminating candidates and transferring votes continues, until a candidate has secured a majority of first-place votes and this candidate is declared the winner. Example 2.5 Consider a hypothetical election with 4 candidates A, B, C and D and 10,000 voters with preferences as given in Table 2.4. This kind of table is called a voter profile. The numbers across the top row indicate the number of voters with the corresponding ranking. For example, 2200 voters ranked A first, B second, D third, and C fourth. Since no candidate received a majority of first-place votes, D is eliminated and 500 votes (from voters with preferences D ≻ C ≻ B ≻ A) are transferred to C and 200 (from D ≻ B ≻ A ≻ C) are transferred to B. This results in 4200, 3900, and 1800 votes for A, B, and C, respectively. There is still no candidate with a majority, and thus C is eliminated and 1250 votes are transferred to B and 650 to A. In this last round, B has secured a majority and is declared the RCV winner. RCV is well studied in the social choice and voting theory literature. We do not discuss the method in depth or the large literature surrounding it because we are only concerned with the apportionment aspect of RCV in the 2020 primary.

2.3.2 Ranked Choice Voting and Apportionment The 2020 Democratic primaries that employed RCV did not use the method to declare a “winner” of the primary election because the purpose of a state primary is to apportion delegates. Instead, RCV was used to winnow down the field of candidates to a set of candidates who all receive at least 15% of the first-place votes. Once this winnowing is complete, HAM is used to calculate the apportionment. In Round 1,

36

2 The Democratic Party Primary

the first-place votes are tallied and the candidate with the fewest number is identified. If this candidate has earned at least 15% of the vote, then HAM is used to calculate the apportionment; otherwise, this candidate is eliminated and their votes are redistributed to other candidates. This process is continued until a round is reached in which all candidates have earned at least 15% of the remaining votes, at which point HAM is applied. In states where RCV was used, it was applied to apportion delegates in three categories (CD, at-large, and PLEO). We note that the Democratic primaries that used RCV placed a limit on the number of candidates that a voter could rank, similar to how RCV elections are implemented in many other American elections. For example, in the HI Democratic primary, voters were allowed to rank only three candidates even though there were 11 candidates. Example 2.6 We illustrate this RCV-HAM apportionment method for the voter profile in Table 2.4, assuming there are 20 delegates to allocate. Candidate D has the fewest first-place votes and does not surpass the 15% threshold, and thus, this candidate is eliminated and the votes are transferred as in Example 2.5. After this transfer, each of the candidates has enough first-place votes to surpass the 15% threshold, and thus HAM is applied to apportion the 20 delegates to candidates A, B, and C based on vote totals of 4200, 3900, and 1800. The result is 8, 8, and 4 delegates, respectively. Note that if we used the standard apportionment rules for the election data in Table 2.4, both C and D would be eliminated because neither of them surpass the 15% threshold for (first choice) votes, and each of A and B would receive 10 delegates. The purpose of using RCV in this context is to allow weaker candidates like C the opportunity to receive delegates if enough support comes from the voters of previously eliminated candidates. Furthermore, this system allows voters to rank their true favorite candidate as their first choice, without having to worry as much about “wasting” their votes on eliminated candidates who receive no delegates. Example 2.7 In the 2020 Hawaii Democratic primary, there were 14,583 valid ballots cast for 11 Democratic candidates in the state’s first CD, where D = 7. The only two candidates who surpassed the 15% threshold for first-place votes were Joe Biden, who received 8580 first-place votes, and Bernie Sanders, who received 4044. Elizabeth Warren came in a distant third with 730 first-place votes. The candidate with the fewest first-place votes was Deval Patrick, who received only 4. Patrick was eliminated, and these 4 votes were redistributed to the candidates ranked second on these ballots. One ballot did not have any other candidate ranked, so this ballot was discarded and became exhausted (the number of ballots is then reduced to 14,582 for the next round). Of the other three ballots, one had Biden ranked second, one had Sanders ranked second, and one had Yang ranked second, so each of these three candidates received an additional vote, completing Round 1. This process lasted 10 rounds, at which point there were only two candidates with at least 15% of the vote:

References

37

Biden (now with 9306 votes) and Sanders (now with 4700), resulting in an apportionment of 5 delegates for Biden and 2 for Sanders.5 Note that this result would have been unchanged had RCV not been used. As with Hawaii, the use of RCV in the remaining state primaries in 2020 made no difference in the final apportionment.6 The reason is that these primaries occurred late in the primary season, at which time Joseph Biden and Bernie Sanders were the only viable candidates left in the race. While RCV is an interesting innovation in presidential primaries, if it is used only for state primaries that occur at the end of the primary calendar, then the method will most likely have no effect on the apportionments in the state primaries or on the national primary. If RCV were used in earlier primaries, it is less clear what the effects would be. As far as we are aware, prior to the 2020 Democratic primaries RCV had never been used in conjunction with apportionment methods. This potentially opens an avenue of interesting research. For example, RCV is famously susceptible to voting paradoxes such as monotonicity and no-show paradoxes; how does this susceptibility interact with apportionment methods? Conversely, an apportionment method like HAM is famously susceptible to apportionment paradoxes such as the population and Alabama paradoxes (see Chap. 6); how does this susceptibility interact with voting methods like RCV? If RCV remains confined to the end of the primary season, then perhaps these questions are not immediately relevant, but this combination of RCV with HAM calls for further investigation.

References Balinski ML, Young HP (2001) Fair representation: meeting the ideal of one man, one vote, 2nd edn. Brookings Institution Press, Washington, DC Bradberry BA (1992) A geometric view of some apportionment paradoxes. Math Mag 65(1):3–17. https://doi.org/10.2307/2691355 Democratic National Committee (2018a) Call for the 2020 Democratic National Committee. https://democrats.org/wp-content/uploads/2019/02/2020-Call-for-Convention-WITHAttachments-2.26.19.pdf. Accessed 2021 Dec 2015 Democratic National Committee (2018b) Delegate selection rules for the 2020 Democratic National Committee. https://democrats.org/wp-content/uploads/2019/01/2020-DelegateSelection-Rules-12.17.18-FINAL.pdf. Accessed 2021 Dec 15 Democratic National Committee (2018c) Regulations of the rules and bylaws committee for the 2020 Democratic National Committee. https://democrats.org/wp-content/uploads/sites/2/2019/07/ Regulations-of-the-RBC-for-the-2020-Convention-12.17.18-FINAL.pdf. Accessed 2021 Dec 15 5

We note that the HI primary allows for voters to vote for uncommitted. This shadow candidate was not eliminated until Round 4, having lasted longer than Deval Patrick, Tom Steyer, and Amy Klobuchar. In HI’s second CD, uncommitted lasted until Round 6. 6 We obtained the ranked choice vote data for HI, KS, and WY and verified this ourselves. We were unable to obtain the AK data but were told in a private communication by an election official that the use of RCV also made no difference in AK.

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2 The Democratic Party Primary

Geist KA, Jones MA, Wilson JM (2010) Apportionment in the Democratic primary process. Math Teacher 104:214–220. https://doi.org/10.2307/20876835 Georgia Democratic Party (2019) Georgia delegate selection plan for the 2020 Democratic National Committee. https://www.georgiademocrat.org/wp-content/uploads/2019/10/ DPG-2020-Delegate-Selection-Plan-FINAL.pdf. Accessed 2021 Dec 15 Jewitt CE (2019) The primary rules: parties, voters, and presidential nominations. University of Michigan Press, Ann Arbor, MI. https://doi.org/10.3998/mpub.10020994 Jones MA, Wilson J (2016) The geometry of adding up votes. Math Horizons 24(1):5–9. https:// doi.org/10.4169/mathhorizons.24.1.5 Jones MA, McCune D, Wilson JM (2019) The elimination paradox: apportionment in the Democratic party. Public Choice 178(1):53–65. https://doi.org/10.1007/s11127-018-0608-3 Lucas WF (1983) The apportionment problem. In: Brams SJ, Lucas WF, Straffin PJ Jr (eds) Political and related models, modules in applied mathematics. Springer, New York, Chapter 14, pp 358– 396 New Hampshire Democratic Party (2019) New Hampshire delegate selection plan for the 2020 Democratic National Committee. https://www.nhdp.org/_files/ugd/696cec_ 039185b132f1438994df5d7a3140d353.pdf. Accessed 2021 Dec 15 The Green Papers (2022) The Green Papers. http://www.TheGreenPapers.com. Accessed 2021 Dec 15

Chapter 3

The Iowa and Nevada Democratic Caucuses

In the 2020 Democratic presidential primary, only two states, Iowa and Nevada, held caucuses in the traditional sense of the term. This chapter describes the mathematical structure of those two caucuses. Two other states, North Dakota and Wyoming, also held elections which were termed “caucuses,” but those contests functioned essentially like primaries.1 Three territories also held caucuses: American Samoa, Guam, and the Virgin Islands. Caucuses differ from primaries in a number of ways. While no formal definition of caucus exists, generally caucuses require participants to meet in person to discuss and vote on candidates. The process may take place over several hours, and the meeting places are usually fewer in number than polling places. For both of these reasons, caucuses have been seen by both parties as less “open” than primaries. Participation is more likely to be concentrated among only those party members who are highly active and motivated and who have the ability to attend the meetings. For these reasons, the national Democratic Party has long been interested in encouraging states to switch to primaries. (In 2016, caucuses were held in 13 states: Iowa, Nevada, Alaska, Colorado, Minnesota, North Dakota, Wyoming, Kansas, Kentucky, Maine, Hawaii, Idaho, Washington, in addition to the three territories.) Prior to the 2020 election, the national party approved reforms to make the caucuses more accessible by requiring states to allow absentee voting. For a history of state primaries and caucuses in the presidential primary process, see Jewitt (2019).

1 As described in Chap. 2, the WY Democratic Party (along with three other state parties) used ranked choice voting for its caucuses, which was an innovation in delegate apportionment. However, the election process itself in Wyoming functioned like a standard primary.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9_3

39

40

3 The Iowa and Nevada Democratic Caucuses

By 2020, there were only two traditional caucuses remaining in the Democratic presidential primary. However, both of these contests wield an outsized influence on the nomination process because of their early slots in the primary. The Iowa (IA) caucuses are of particular importance because they always occur first in the primary calendar. The complex delegate selection rules used in Iowa are also interesting mathematically. From the perspective of apportionment theory, the 2020 IA caucuses in particular may have been the most interesting state election in the history of presidential primaries because of the mathematical complexity of its rules and the number of apportionment paradoxes observed (see Chap. 6 and McCune 2023). Hence, in this chapter we focus mainly on the caucuses in IA and then briefly describe the Nevada (NV) caucuses. We note that many of the mathematical details outlined in this chapter have not been previously explored, as far as we are aware.

3.1 The 2020 Iowa Caucuses Delegate allocation in the IA caucuses is governed by the rules outlined in the Iowa Delegate Selection Plan or IADEL (Iowa Democratic Party 2019). Unlike other states, the process is based on the notion of state delegate equivalency, which is defined below. Allocating in this manner requires additional layers of apportionment, which are used to calculate each candidate’s state delegate equivalents (SDEs). Put simply, each individual voting precinct is allocated a number of SDEs (which is usually a fraction less than one) and a number of county delegates. Candidates compete for county delegates at the precinct, and the number of county delegates earned by each candidate determines the amount of SDEs they earn at the precinct. The candidates’ precinct SDEs are then summed across a congressional district or across the state, forming the basis for the final apportionment using HAM∗ . At each stage, the calculations involve an apportionment problem: determining the number of SDEs at each precinct, determining the number of county delegates at each precinct, and determining how many county delegates are apportioned to each candidate at each precinct. Most of these apportionment problems are solved using HAM or HAM∗ although there are some interesting exceptions and constraints at several levels. Before describing the mathematics, we define the types of delegates that are relevant to the IA caucuses. In addition to the national delegates allocated by the DNC, IA also has state and county delegates. A state delegate is a delegate that attends the IA State Democratic Convention. There were 2107 of these delegates in the 2020 IA caucuses. Each county in IA also has its own convention; a county delegate is a delegate that attends a county convention. In 2020, there were a total of 11,402 of these delegates across the state. Thus, the allocation of the 11,402 county delegates to candidates at individual precincts determines the number of state delegates received by each candidate, which in turn determines the number of national delegates received by each candidate, which is the apportionment with which we are primarily concerned.

3.1 The 2020 Iowa Caucuses

41

We break our description of the delegate selection process into stages, some of which are similar to the stages of a Democratic primary as detailed in Chap. 2. The first three stages are dictated by the national rules and are implemented in the same manner as for a primary. Only the final two stages occur after the election, and thus, several apportionment problems are solved before any votes are cast. I. II. III. IV. V. VI.

The state party receives its national delegates (base and PLEO) from the DNC. The state party divides the base delegates into district and at-large delegates. The district delegates are apportioned to the CDs. The state delegates are apportioned to counties. The county delegates are apportioned to precincts within the county. The county delegates are apportioned to candidates in proportion to vote totals at each precinct. VII. The candidates are apportioned delegates in proportion to state delegate equivalents. Stages I–III. These stages are virtually identical to the first three stages for state primaries. In 2020, the IA Democratic Party was allocated 49 total national delegates: 36 base delegates, 5 PLEOs, and 8 additional unpledged party leaders. Of the 36 base delegates, 27 were calculated to be district level and 9 were at-large. The 27 district delegates were allocated to the 4 CDs based on an allocation factor, A∗ =

DDV(2016) + DDG(2018) , SDV(2016) + SDG(2018)

(3.1)

that is a variant of Eq. 2.3, but still gives equal weight to the results of the 2016 presidential and 2018 gubernatorial elections. In 2020, this apportionment resulted in an allocation of 7, 7, 8, and 5 delegates to the CDs. Stage IV: State delegates are apportioned to counties. As with Stage III, the allocation of state delegates to counties is a standard apportionment problem based on a fixed number of state delegates. In 2020, there were 2107 state delegates. This number is somewhat arbitrary as the number of state delegates is independent of the number of the state’s national delegates. Informal communication with caucus organizers in other states suggests that the number is chosen to be large enough to guarantee a robust state convention and ensure each county has at least one delegate, while not exceeding the maximum occupancy of the convention location. The IADEL indicates that state delegates are allocated to counties in proportion to the number of Democratic votes for the president in 2016 and the governor in 2018 using a formula analogous to Eq. 3.1, but with CD votes for the presidential and gubernatorial races being replaced by county votes for the presidential and gubernatorial races. The plan does not specify an apportionment method, however HAM∗ was used in prior caucuses.2 In 2020, state delegates appear to have been apportioned to the state’s 99 counties using a HAM-type method that incorporates the principle of degressive proportionality. This concept, developed formally for allocating seats 2

Party documents from 2008 clearly describe the use of HAM∗ .

42

3 The Iowa and Nevada Democratic Caucuses

Table 3.1 Number of Iowa state delegates for the 11 counties whose apportionment differed from HAM 2016 2018 A∗ Quota Appt HAM County Adams Allamakee Greene Ida Jasper Linn Osceola Polk Pottawattamie Taylor Wayne

549 2165 1675 713 7234 56,767 481 120,257 14,234 759 722

565 2421 1691 792 7109 58,935 552 119,804 15,355 758 719

0.00179 0.00357 0.00262 0.00117 0.01116 0.9006 0.0008 0.18687 0.02303 0.00118 0.00112

1.833 7.523 5.520 2.465 23.514 189.756 1.686 393.735 48.524 2.486 2.360

3 7 6 3 23 189 3 392 48 3 3

2 8 5 2 24 190 2 394 49 2 2

in the European Parliament, is intended to create an apportionment that is roughly proportional but also favors small states (or in this case, counties) at the expense of the large. Arguably, these two criteria are in tension. There is a literature dedicated to enacting the principle of degressive proportionality; for example, see Piotr (2011), Słomczy´nski and Zyczkowski (2012) and Pukelsheim and Grimmett (2018), all of which discuss degressive proportionality in the context of apportionment in the European Parliament. We discuss degressive proportionality as well as its opposite, regressive proportionality, in Sects. 7.3 and 7.4. In the 2020 IA caucuses, the results of this process favored small counties to some degree, but the overall apportionment was still fairly proportional. In particular, 88 counties received an apportionment consistent with HAM∗ . Table 3.1 shows the apportionments for the 11 counties in which the actual apportionment differed from a HAM apportionment. Note that the smaller counties are given a boost, sometimes receiving more than their upper quotas, while larger counties sometimes receive less than their lower quotas. The numbers suggest that the state party decided each county should receive a minimum of three delegates. It is not clear why the small Allamakee County was rounded down since there were larger counties with a similarly sized fractional part that had their quotas rounded up. This example illustrates that states holding caucuses are mostly free to interpret proportionality criteria, and sometimes apportionments are completed ad hoc without the use of a mathematical method (as we suspect was done here). Regardless, state parties are not required to justify this kind of apportionment. Stage V: County delegates are apportioned to precincts. Each county in IA has a convention attended by county delegates. Like state delegates, the total number of county delegates is arbitrary. The 2008 IADEL suggests it is based loosely on several factors including: (1) past convention size; (2) county party strength; and (iii) the number of registered Democrats in the county. In addition, the number needs to be

3.1 The 2020 Iowa Caucuses

43

Table 3.2 Apportionment of Calhoun County delegates among precincts Precinct 2016 2018 A∗ Quota [Quota] 1 2 3 4 5 6 7 8 9 10 Total

126 373 38 107 49 234 62 287 66 56 1398

147 337 35 114 46 262 69 332 70 52 1464

0.09539 0.24808 0.02551 0.07722 0.03319 0.17331 0.04577 0.21628 0.04752 0.03774

5.723 14.885 1.531 4.633 1.991 10.399 2.746 12.977 2.851 2.264

6 15 2 5 2 10 3 13 3 2 61

Table 3.3 Apportionment of Greene County delegates among precincts 2016 2018 A∗ Quota [Quota] Precinct 1 2 3 4 5 6 7 Total

172 245 258 276 397 210 133 1691

195 250 271 262 374 191 132 1675

0.10903 0.14706 0.15716 0.15983 0.22906 0.11913 0.07873

5.452 7.353 7.858 7.992 11.453 5.957 3.937

5 7 8 8 11 6 4 49

Appt 6 15 1 5 2 10 3 13 3 2 60

Appt 5 7 8 8 12 6 4 50

large enough to ensure that when the county delegates are apportioned among its precincts, each precinct receives at least one county delegate. As in Stage IV, apportionment is based on an allocation factor similar to Eq. 3.1, but with CD and state votes for the presidential and gubernatorial races being replaced by precinct and county votes for the presidential and gubernatorial races. Very few county parties compute the apportionment of county delegates to their precincts on their own.3 Rather, most send their total number of county delegates and voting data to the state party, which then calculates the apportionment for the county. No details are provided for how this apportionment should be conducted, but the data are generally consistent with the use of HAM∗ . Tables 3.2 and 3.3 provide the allocation of delegates to precincts for Calhoun and Greene Counties. Rounding quotas to nearest integers results in an over allocation and an under allocation, respectively; in both cases HAM∗ appears to have been used to obtain the apportionment. 3

Private conversation with county party leaders.

44

3 The Iowa and Nevada Democratic Caucuses

Table 3.4 Calculation of SDEs for the precincts in Adair County Precinct County delegates SDEs 1 2 3 4 5 Total

8 11 9 10 13 51

0.63 0.86 0.71 0.78 1.02 4

We note that since each county party chooses the size of its county delegation at the same time as it apportions the delegates, the value of D is not a priori fixed. Thus, this process need not be a traditional apportionment problem. In fact, at least one county chair determined the number of county delegates so that the nearest integer rounding of quotas would sum to the correct number.4 We can now define the notion of state delegate equivalent which is the conversion of county delegates to state delegates, which in turn allows for the final apportionment of the national delegates. Suppose a county has been awarded s state delegates and has C county delegates which it has allocated among its precincts. If precinct i receives ci county delegates, then it receives (ci /C)s state delegate equivalents. That is, each precinct is allocated (ci /C) · 100% of the county’s s state delegates. Note that SDEs are not integers in general. Example 3.1 For the 2020 caucus, Adair County had 4 state delegates, apportioned to it using the HAM∗ -based degressive proportionality criterion described in Stage IV. The total number of county delegates was 51 which was apportioned as described above among its 5 precincts resulting in the allocations shown in column 2 of Table 3.4. Since the 1st precinct has 8 of the 51 delegates, it receives 4(8/51) = 0.63 SDEs; hence, in the 1st precinct, candidates are competing for 0.63 of a single state delegate. The SDE equivalents of the remaining precincts in the county are as shown in column 3 of Table 3.4. Stage VI: Apportioning county delegates to candidates at each precinct. The previous stages occur before the date of the caucuses and may be viewed as preliminary as they are necessary to set up the caucus. In contrast, Stage VI occurs after the caucus when county delegates are apportioned to presidential candidates based on the results of the precinct caucus. As suggested by the national rules, candidates receive delegates in proportion to vote totals using HAM with a 15% threshold (see Chap. 2), although at the precinct level this method is conceptualized as HAM∗ . In contrast to the straightforward application of HAM in a primary election, in these precinct caucuses there are a number of additional details of mathematical interest when implementing this apportionment method. 4

Private conversation.

3.1 The 2020 Iowa Caucuses

45

Table 3.5 Results of the 2020 election in Adair County, precinct 4 (Left); and a hypothetical result if several voters choose not to realign (Right) Candidate First exp. Biden

Final exp.

Quota [q] q

Appt

SDE

Candidate

Final exp.

Quota [q] q

Appt

SDE

8

0

8

0

10

12

1.538 2

1

0.078

Buttigieg

10

12

1.538 2

2

0.156

Klobuchar 26

32

4.103 4

4

0.312

Klobuchar 26

26

3.333 3

4

0.312

Sanders

16

21

2.692 3

3

0.234

Sanders

16

16

2.051 2

2

0.156

Warren

6

0

Warren

6

0

Buttigieg

Biden

First exp.

Yang

12

13

1.667 2

2

0.156

Yang

12

12

1.538 1

Total

78

78

10

10

0.78

Total

78

66

8.46

11

7

2

0.156

10

0.78

In the IA caucuses, there are no secret ballots5 and voting is not anonymous; rather, to cast votes the voters at the precinct physically divide into groups. The resulting vote totals for each candidate are called the first expression. After the first expression results are tabulated, the voters for candidates who did not achieve the 15% threshold have a chance to realign their vote to support a different candidate. Voters for a non-viable candidate have three choices: They can realign to a viable candidate (which occurs the vast majority of the time); they can choose not to realign so that their vote does not count in favor of any candidate; or they can try to persuade voters of other non-viable candidates to support their candidate in hopes of making their candidate viable.6 If they are unable to persuade enough other voters to support their candidate in the allotted realignment time, then these voters must chose one of the other two options. Once voters have realigned and non-viable candidates have been eliminated, the resulting vote totals are termed the final expression. Table 3.5 (Left) shows the results of the 2020 election in the 4th precinct of Adair County which had 10 county delegates to apportion. After the first expression, Biden, Buttigieg, and Warren were all non-viable, as each candidate received less than 11.7 votes, or 15% of the 78 total votes. None of the voters left; instead the Buttigieg supporters convinced enough Biden and Warren supporters to realign for Buttigieg, so that Buttigieg became viable in the final expression. HAM∗ was then applied to the vote totals in the final expression, yielding the apportionment of (0, 1, 4, 3, 0, 2), as given in Table 3.5 (Left). At precinct 4, the candidates competed for a portion of the 0.78 SDEs allotted to this precinct; since Buttigieg received one of the precinct’s 10 county delegates, he received (1/10)(0.78) = 0.078 SDEs from this precinct. The others candidates’ SDEs are calculated similarly. The version of HAM∗ used at the precinct level has a few unusual features. First, if there are more viable candidates than there are delegates at the precinct, the voters of the last-place viable candidate must realign, even though their candidate is viable. This occurs in rare circumstances when, for example, five candidates surpass the 5

Historically, most caucuses have not used secret ballots. Votes for viable candidates are locked in and cannot be eliminated or changed, even if the individuals leave the caucus at this point.

6

46

3 The Iowa and Nevada Democratic Caucuses

15% threshold at a 4-delegate precinct. Second, after the initial rounding, candidates cannot have their only delegate taken away from them. Third, even though voters for non-viable candidates may choose not to realign, the quotas for each viable candidate are calculated using the number of caucus-goers for the first expression. For example, suppose, as in Table 3.5 (Right), that only 2 of Biden supporters realign to Buttigieg in the 4th precinct and the remaining 6 Biden supporters as well as the 6 Warren supporters refuse to realign. Those 12 voters who choose not to realign are still counted in part of the denominator for the calculation of quotas in the final expression. Thus, Klobuchar’s quota is (26/78)10 = 3.333, and not (26/66)10 = 3.940, as might be expected. Since the quotas are smaller, they sum to 8.46, which is less than the 10 delegates assigned to the 4th precinct. Rounding to the nearest integer results in an initial under allocation of 7 delegates and a final apportionment of (0, 2, 4, 2, 0, 2). Because the denominator is larger than the sum of the candidates’ votes, there may be cases when the number of delegates remaining to be allocated after the initial nearest integer rounding is greater than the number of viable candidates, meaning that even if every candidate received their quota rounded up, there would still be delegates left to allocate. The IA rules do not say what to do in this (highly unlikely) scenario. Since the vast majority of voters choose to realign, it is unlikely that this issue occurs in practice. An interesting consequence of the aforementioned feature of the IA rules is that a candidate that is viable in the first expression may get fewer delegates after the second expression even if the candidate does not lose any voters. This is an instance of the elimination paradox, discussed in Sect. 6.1. Because a precinct may have very few delegates to award, the threshold to receive a delegate may be adjusted. The 15% threshold is used only if a precinct has 4 or more delegates. The threshold is set at 1/6 ≈ 16.67% for a 3-delegate precinct and 25% for a 2-delegate precinct. Naturally, there is no threshold for a 1-delegate precinct, as the method reduces to winner-take-all. Stage VII: Apportioning national delegates to candidates in proportion to SDEs. At the completion of Stage VI, each candidate has a number of SDEs from each of Iowa’s 1678 precincts. These SDEs are added up across all of the precincts so that each candidate has a total number of SDEs from each CD and a total number of SDEs for the entire state. Apportionment then proceeds as in a Democratic primary, except delegates are apportioned in proportion to SDEs rather than vote totals. Per the standard Democratic rules, there is still a 15% threshold at this stage, even though a 15% threshold was already enforced at each individual precinct. Thus in order to receive delegates in a CD or at the statewide level, a candidate must cross two 15% thresholds: First, the candidate must exceed the 15% threshold at enough precincts to receive some SDEs, and second, the candidate must then have at least 15% of the SDE total at the district or statewide level. Example 3.2 In Iowa’s first congressional district, the viable candidates were Buttigieg, Sanders, Biden, and Warren; these candidates received 147.8162, 145.2917, 102.7936, and 95.5602 SDEs in the district, respectively. These values were obtained

3.1 The 2020 Iowa Caucuses

47

Table 3.6 2020 Iowa Democratic Party results in the case of a primary Biden Buttigieg Klobuchar Sanders Warren CD 1 Appt CD 2 Appt CD 3 Appt CD 4 Appt Statewide PLEO appt At-large appt Overall appt

Total votes

6853 1(2) 6305 0(0) 8179 0(1) 4772 1(1) 26,291 0(1) 0(1)

8962 2(2) 9755 2(2) 12,405 3(3) 6233 1(2) 37,572 2(2) 3(3)

5359 0(0) 5376 0(0) 6948 0(0) 4476 0(1) 22,454 0(0) 0(0)

10,100 2(2) 13,129 3(3) 13,071 3(2) 7094 2(1) 43,581 2(1) 3(3)

7260 2(1) 10,219 2(2) 10,062 2(2) 4800 1(0) 32,589 1(1) 3(2)

41,575 − 48,832 − 54,836 − 29,923 − 176,352 − −

2(6)

13(14)

0(1)

15(12)

11(8)

−

by summing the SDEs earned at each precinct, over all precincts in the district. Using HAM to apportion 7 delegates allocated to CD 1 yields an apportionment of (2, 2, 2, 1). (See www.thegreenpapers.com for the complete data from the 2020 IA caucuses.) We conclude this section by asking: What if the IA caucuses in 2020 had run as a primary? That is, how would the final apportionment differ if we remove all of the nested apportionment problems that occur in the caucuses and the multiple 15% threshold hurdles the candidates must meet? This question would be difficult to answer prior the 2020 caucuses because the IA Democratic Party previously only reported the SDE information and not the original vote totals. In 2020, however, the state party reported both the first and final expression data from each precinct, and thus, it is possible to calculate what would have happened if the voter information was interpreted as the results of a primary. Of course, it is impossible to truly measure the difference between a primary and a caucus in this way, since the circumstances of how individuals vote would be so different. Nevertheless, a quick comparison is instructive, giving insight into the effect of how votes at the precinct level are aggregated to districts and then to the state. We use the first expression data rather than the final expression, since the first expression more likely mirrors the vote totals we would see in a primary. We also include all first expression votes in our totals, including votes for “uncommitted.”7 The resulting apportionments are given in Table 3.6. For comparison, the actual results from the caucuses are shown in parentheses. Note that the 7

The IA rules allow for voters to vote for the category uncommitted. Thus, strangely enough, “uncommitted” behaves like an actual candidate and can receive SDEs. In the 2020 caucus, uncommitted actually received more SDEs than known candidates such as Michael Bloomberg and Tulsi Gabbard. If we discounted the uncommitted votes, the apportionment in a primary would have been

48

3 The Iowa and Nevada Democratic Caucuses

Table 3.7 Vote percentages for the top five candidates in the 2020 IA caucuses for the first and final expression of votes Biden Buttigieg Klobuchar Sanders Warren First expression 14.9 Final expression 13.8

21.3 25.3

12.7 12.4

24.7 26.7

18.5 20.4

statewide vote totals are not the sum of the district vote totals. In 2020, the IA Democratic Party held 87 satellite caucuses for registered Iowa Democrats who could not participate in person in the other caucuses. Sixty of these were in Iowa; the rest were for individuals residing outside the state. The votes from these caucuses were not assigned to any CD but were lumped into the statewide totals. The effects of using the voting data in a primary are pronounced. Biden receives only a third of the delegates that he actually received, and Sanders receives the largest number of delegates (which seems appropriate, as he received the most votes). The changes in the apportionment moving from primary to caucus cannot be explained by the movement from first expression to final expression in votes, as shown by the vote distributions in the first and final expressions in Table 3.7. In moving from a primary to a caucus, Biden’s delegate total is tripled; however, in moving from first expression to final expression his vote share actually decreases by approximately one percentage point. Similarly, the gap between Buttigieg and Sanders narrows between the first and final expression, but Buttigieg does not overtake Sanders in vote share, and thus, it is notable that he overtook Sanders in delegate share in the caucuses. These changes are most likely due to dividing county delegates among precincts and apportioning the precinct results separately and adding them up, rather than apportioning all the congressional districts in one group. We discuss counterintuitive behavior related to aggregation in Sect. 6.2. To conclude this section, we note that the 2020 IA Democratic caucuses are the most mathematically complex of any primary election of which we are aware. Democratic Party officials in Iowa have designed an interesting allocation process involving multiple nested layers of apportionment, creating an overall apportionment method that has no analogue in the context of apportioning seats in legislative bodies.

3.2 The 2020 Nevada Caucuses In this section, we describe the other prominent caucus of the 2020 Democratic primary, the Nevada (NV) caucuses. The mathematics of these caucuses is much less complicated than those of IA. The rules also seem to be more representative of

(2, 13, 1, 14, 11). The effect of keeping uncommitted votes is to transfer Klobuchar’s single delegate to Sanders.

3.2 The 2020 Nevada Caucuses

49

how caucus elections have run historically (Jewitt 2019). For example, most caucuses prior to 2020 did not involve notions such as state delegate equivalents. The main differences between the NV and IA caucuses are that national delegates are apportioned in proportion to the number of county delegates earned by the candidates. (State delegates play no role in NV, in contrast to IA.) In addition, the apportionment of delegates to counties and then county delegates to precincts are not traditional apportionment problems. The apportionment process in NV is laid out in two documents: Caucus Memo: Precinct Delegate Apportionment (NVDEMS Caucus Memo 2019) and Caucus Memo: Delegate Count Scenarios and Tie Breakers (NVDEMS Caucus Memo 2019), which we refer to as NVPDA and NVDCS, respectively. The process can be divided into the following stages. I. II. III. IV. V.

The state party receives its national delegates (base and PLEO) from the DNC. The state party divides the base delegates into district and at-large delegates. The district delegates are apportioned to the CDs. The number of county delegates is calculated for each precinct. The county delegates are apportioned to candidates in proportion to vote totals at each precinct. VI. The candidates are apportioned to national delegates in proportion to county delegate totals. Stages I–III are determined by the national rules, as in the IA caucuses. Stage IV is different from and more straightforward than what occurs in IA. Each precinct receives a number of county delegates based on how many registered Democrats live within the geographical boundary of the precinct and how many registered Democrats live in the county. However, it is not a formal apportionment problem since the total number of county delegates is not fixed. The NVPDA document describes the process through a series of cases: “In counties in which the total number of registered voters of that party has exceeded 2000 but has not exceeded 3000, each precinct is entitled to one delegate for each 30 registered voters or major fraction thereof” (NVDEMS Caucus Memo 2019), with similar statements for other county populations. An example illustrates that the “major fraction thereof” refers to rounding to the nearest integer. Given a fictional county with 2020 registered Democrats, a precinct in the county with 75 registered Democrats is entitled to [75/30] = 3 county delegates. Stage V is almost identical to the equivalent step (Stage VI) in IA with two small exceptions. First, if there are more viable candidates than there are delegates at the precinct and rounding dictates that every candidate receives a delegate, then the precinct simply increases its delegate size to accommodate this. (Recall that in IA, the supporters of the bottom candidate would be forced to realign, meaning that it is possible for a viable candidate to lose their only delegate.) In NV, if the rounding of HAM∗ gives a delegate to a viable candidate, then that delegate cannot be taken away in any circumstance. The NVDCS document provides an example of a 4-delegate precinct in which 4 candidates each receive approximately 17% of the vote and one candidate receives approximately 32% of the vote. In this case,

50

3 The Iowa and Nevada Democratic Caucuses

rounding gives each candidate one delegate, and so the precinct is allocated an extra delegate to accommodate this. Second, the NV rules are explicit about what occurs in the unlikely event that many voters do not realign, meaning that after rounding and allocating according to fractional sizes, there are still delegates leftover. In this case, the candidates are ordered by fractional part, and the excess delegates are handed out in rounds. For example, suppose the three surviving candidates are ordered A, B, and C by fractional parts and there are 5 delegates leftover. Then each candidate receives an extra delegate, followed by A and B each receiving a second extra delegate. Stage VI is similar to Stage VII in IA, except that instead of apportioning delegates in proportion to SDEs, candidates receive delegates in proportion to the number of county delegates they have accumulated. In order to receive any delegates in a CD, a candidate must receive at least 15% of the county delegates in that district; a similar restriction holds for the statewide delegates. If a candidate doesn’t achieve the 15% threshold, then their county delegates are removed from the calculations. Thus, just as in IA, candidates must pass the 15% thresholds at the precinct and then the district and state levels.

References Iowa Democratic Party (2019) Iowa delegate selection plan for the 2020 Democratic National Convention. https://iowademocrats.org/wp-content/uploads/2019/05/2020Iowa-Delegate-Selection-Plan-4.5.19-Final-1.pdf. Accessed 13 Jan 2022 Jewitt CE (2019) The Primary rules: parties, voters, and presidential nominations. University of Michigan Press, Ann Arbor, MI. https://doi.org/10.3998/mpub.10020994 McCune D (2023) The many apportionment paradoxes of the 2020 Iowa Democratic presidential caucuses. https://doi.org/10.1007/s00283-022-10196-9 (to appear) NVDEMS Caucus Memo (2019a) Nevada Dems caucus memo: delegate count scenarios and tie breakers. https://nvdems.com/delegate-selection-and-caucus-materials/. Accessed 1 Oct 2021 NVDEMS Caucus Memo (2019b) Nevada Dems caucus memo: precinct delegate apportionment. https://nvdems.com/delegate-selection-and-caucus-materials/. Accessed 1 Oct 2021 Piotr D (2011) Degressive proportionality-source, findings and discussion of the Cambridge Compromise. Math Econ 7(14):39–50 Pukelsheim F, Grimmett G (2018) Degressive representation of member states in the European Parliament 2019–24. Representation 54(2):147–158. https://doi.org/10.1080/00344893.2018. 1475417 Słomczy´nski W, Zyczkowski K (2012) Mathematical aspects of degressive proportionality. Math Social Sci 63(2):94–101. https://doi.org/10.1016/j.mathsocsci.2011.12.002

Chapter 4

The Republican Party Primary

The selection process and the apportionment of delegates in the Republican presidential primary are roughly similar to those of the Democratic primary with a couple of notable distinctions. The main difference is that state Republican parties have much more control over the methodology used for apportioning delegates among presidential candidates. The delegate apportionment rules from the Republican National Committee (RNC) stipulate only that primaries held before a certain date must use proportional allocation, but say nothing about how this proportionality should be enacted. By contrast, the rules for allocating delegates in the Democratic primary are relatively straightforward and not governed by the state parties. When apportioning delegates to candidates in congressional districts or statewide, state Democratic parties must use Hamilton’s method with a 15% threshold (see Chap. 2). There are other significant differences between the Republican and Democratic primaries which are of less mathematical interest, and therefore, we do not discuss them in depth. For example, some Republican state primaries that occur late in the primary calendar allocate delegates on a winner-take-all basis, which negates the goals of proportionality obtained through applying apportionment methods. Also, there has been no corresponding controversy over superdelegates as the number of state party leaders in the Republican primary forms a smaller fraction of the total number of delegates and they are bound to vote for the candidate that won their state. Historically, proportional allocation has not played a major role in Republican primaries. The use of proportional apportionment methods in Republican state primaries was not a significant factor until the 2012 primary: Even as late as 2008, most state primaries used a winner-take-all method in which the leading candidate received all of the delegates; for other states, delegates were chosen in a non-mathematical manner that did not correspond to election results. An interesting example of the latter occurred in the 2008 West Virginia Republican primary in which John McCain received 76.01% of the primary vote and Mike Huckabee received only 10.30% but, because at-large delegates were chosen through direct election by state delegates © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9_4

51

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4 The Republican Party Primary

at the state convention, McCain received only 9 delegates from the primary (three district delegates from three CDs) while Huckabee received all 18 at-large delegates. As discussed in Sect. 1.4, the Republican Party delegate selection process has seen a number of changes since the 1970s, including a regularization of election practices across the states and a gradual movement from caucuses to primaries. In the last twenty years, many parts of the delegate selection process, including the allocation of delegates among states, have not changed significantly. Recent alterations have focused instead on adjustments based on the primary calendar and a gradual move away from winner-take-all to proportional methods of apportioning delegates among candidates. In this chapter, we focus on the proportional allocation methods used by Republican state parties in the 2012, 2016, and 2020 primaries.1 This chapter is organized as follows. In Sect. 4.1, we provide a broad outline of how delegates are awarded to the state by the RNC and subsequently divided among district and statewide levels. Section 4.2 discusses the use of thresholds in Republican primaries. Section 4.3 defines and summarizes the different apportionment methods used by the state parties to allocate their at-large delegates based on the statewide results of the election. Section 4.4 discusses the geometry of the Republican methods, focusing on the methods that are used for at-large delegate allocation. Section 4.5 describes how district delegates are apportioned in Republican primaries.

4.1 Republican Party Delegates In this section, we describe the process of allocating delegates to state parties. These rules are outlined in the 2020 Call to the Convention (CALL) (Republican National Committee 2019). This document was published by the RNC and stipulates the procedures for all stages of the Republican primary. The first step is to allocate delegates to state parties. Like the Democrats, the allocation is based on how “red” or “Republican” the state has been in recent elections. However, the process is much more straightforward. Each state party is allocated 10 at-large delegates plus 3 district delegates for each CD. Recall from Chap. 2 that in the Democratic primary, each state is awarded a total number of district delegates which must be divided among the state’s CDs, resulting in the state primary’s first apportionment problem. This does not occur in the Republican primary since each CD automatically receives 3 delegates. (These district delegates may be allocated based on election results in each CD, but often they are grouped with the at-large delegates and apportioned based on the statewide vote. See Sect. 4.5.) If a state casts its electoral votes for the Republican presidential candidate in the preceding election, it receives a number of bonus at-large delegates equal to 4.5 plus 60% of the number of electoral votes of that state, rounded up to the next whole number. It also receives 1 Since Donald Trump was an incumbent in the 2020 primary it was essentially a one-candidate contest. However, many state parties still defined explicit proportional allocation rules for this year, even though the elections themselves were not close.

4.1 Republican Party Delegates

53

one additional at-large delegate for each of the following: (i) a Republican governor; (ii) a Republican senator; (iii) a Republican majority among the state members of the House of Representatives;and (iv) a majority of the members in one or more houses of the state legislature. At-large delegates are also assigned to other jurisdictions. In 2020, there were as follows: 6 from American Samoa; 16 to the District of Columbia; 6 to Guam; 6 to the Northern Mariana Islands; 20 to Puerto Rico; and 6 to the Virgin Islands. Washington, DC was also awarded a small number of bonus delegates in a similar manner as the fifty states. Finally, each jurisdiction (state or territory) is given an additional 3 delegates for the jurisdiction party leaders: (i) the national committeeman; (ii) the national committee woman; and (iii) the chairman of the jurisdiction’s Republican Party. The state parties are encouraged to try to ensure an equal number of women and men among its delegation. They are also required to provide an equal number of alternate delegates. Similar to the process used in the Democratic primary, states can lose delegates if they do not respect the calendar rules outlined by the RNC. Most recently, the CALL document has included delegate penalty rules which have served both to encourage adoption of proportional allocation methods (in place of a winner-take-all approach), as well as to curb front-loading. In 2008, for instance, states were required to hold primaries after February 5 or suffer a penalty of a loss of 50% of their delegates. Despite this stipulation, five states (Michigan, New Hampshire, South Carolina, Wyoming and Florida) held their primaries early and suffered this penalty. In 2012, states were required to hold their primaries after Tuesday 6 March 2012 with the exception of Iowa, New Hampshire, South Carolina, and Nevada who were allowed to begin on or after February 1, 2012. In addition, all states holding primaries or caucuses before April were required to use a method of proportional allocation to award their statewide delegates (district-wide delegates could be winner-takeall) to the candidates. States holding primaries after April 1 could continue to use whatever method they desired. Any state not abiding by these rules was penalized by losing 50% of their delegates. As a result, 5 states received the 50% reduction: New Hampshire, Michigan, Arizona (for using winner-take-all), and the states South Carolina and Florida (who violated both rules but only had a 50% reduction in their delegates). In the 2020 primary, all state contests, with the exception of the four carve-out states Iowa, New Hampshire, South Carolina, and Nevada, were required to hold their primaries on March 1 or later. Any state violating this rule had its delegation reduced to 9 plus its members from the RNC (if the delegation was at least 30) or 6 plus its members from the RNC (if the delegation was less than 30). Additionally, any state contest held prior to March 15 had to allocate delegates proportionally or be subject to a 50% reduction in delegates. Thus, for the 2020 convention, there were a total of 2550 delegates, consisting of 560 at-large, 1305 district level (3 × 435), 168 party leader delegates, and 517 bonus delegates. This was almost identical to the 2016 convention where there were 2472 total delegates, the difference stemming from the smaller number (39) of bonus

54

4 The Republican Party Primary

delegates. Comparing this process to the one in Sect. 2.1, it is evident that this stage of the presidential primary is much simpler for the Republican Party than for the Democratic Party.

4.2 Thresholds in the Republican Primary After each state party has been given its allotted number of delegates, a primary or caucus election is held and the delegates are apportioned to candidates in proportion to their vote totals at district and state levels. Unlike the Democratic primary, the method used to apportion delegates and the minimal threshold required for candidates to receive delegates are not dictated by the national party. As a result, state parties use a variety of approaches. In this section, we examine the threshold structures used in the different state primaries. In Sect. 4.3, we discuss the delegate apportionment methods. In contrast to the uniform rules stipulated by the Democratic Party, the rules laid by the RNC stipulate only two conditions regarding thresholds: A state may establish... [a] minimum threshold of the percentage of votes received by a candidate that must be reached, below which a candidate may receive no delegates, provided such threshold is no higher than twenty percent (20%). A state may establish... [a] minimum threshold of the percentage of votes received by a candidate that must be reached, above which the candidate may receive all the delegates, provided such threshold is no lower than fifty percent (50%). ((Republican National Committee 2019), Rule No. 16(c)(3)).

Thus, Republican state parties have a much greater latitude in whether and how they integrate thresholds into the delegate selection process, using a range of values or opting for no threshold at all. Some parties use a sequence of nested thresholds defined by a set of cutoffs, p1 %, p2 %, . . . , pn % . If any candidates surpass the p1 % threshold, then the apportionment method is applied only to those candidates and their votes; if no candidate surpasses the p1 % threshold, then the apportionment method is applied only to the candidates (and their votes) that surpass the p2 % threshold, and so on. If no candidate surpasses the pn % threshold, then the apportionment method is applied to all candidates. The rules listed in the CALL document do not mention sequences of thresholds; however, they are not explicitly forbidden and many state parties apply them. Table 4.1 summarizes the threshold structures applied in statewide delegate apportionment in the 2016 Republican primary for states using proportional allocation methods.2 The designation ∗ for a state means that even though candidates who do not surpass the threshold are eliminated, their votes are not eliminated. Thus, these thresholds apply to candidates but not to their votes, and the states use apportionment 2

As with all state delegate selection rules, this information is gathered from state delegate selection plans, where available, on Republican Party state websites. Information about the remaining states comes from The Green Papers (The Green Papers 2022), which obtained its information when these documents were still available on state party websites.

4.3 Republican Selection Rules and Quota-Based Methods

55

Table 4.1 The threshold structures used by Republican state parties in the 2016 primary Threshold sequence State Threshold sequence State AK AL CT DC GA HI IA ID KS KY LA* MA ME MI MN MO

13% 50%, 20% 50%, 20% 50%, 15%, 10%, 8% 50%, 20%, 15%, 10% 0% 0% 50%, 20% 10% 5% 20% 5% 50%, 10%, 5% 15%, p1 % − 5% 85%, 10% 50%

MS NC NH* NM NV NY OK* OR PR RI TN TX† UT† VA VT WA

15%, 10% 0% 10% 15% 3.33% 50%, 20% 50%, 15% 0% 50%, 20% 10% (2/3)100%, 20% 50%, 20% 50%, 15% 0% 50%, 20%, 15%, 10% 20%

The ∗ denotes instances where the vote totals are not adjusted after the candidates are eliminated. † TX and UT use “soft” thresholds

methods that take into account the extra non-eliminated votes (see Sect. 4.3.4). In MI, note that if no candidate surpasses the 15% threshold, then the new threshold is obtained by taking the percentage of the vote earned by the top candidate and subtracting 5% points. For example, if the top candidate receives only 14% of the vote then the threshold is set at 9%. Two state parties use “soft” thresholds that do not always eliminate candidates falling below the threshold. In TX, if only one candidate surpasses the 20% threshold and this candidate does not receive a majority of the vote then the delegates are allocated proportionally between the top two candidates, regardless of the vote percentage earned by the second candidate. In UT, if there is no majority candidate and at most 2 candidates surpass the 15% threshold, the 15% threshold is removed. Because the thresholds in many states are so high, these state primaries are best described as “winner-take-most” despite being nominally proportional. In fact, the frequent use of an initial 50% cutoff demonstrates that Republicans are often willing to jettison the notion of proportionality when there is a majority candidate.

4.3 Republican Selection Rules and Quota-Based Methods In this section, we describe how delegates are apportioned to candidates in the Republican state primaries after thresholds have been applied. We focus on the allocation of statewide delegates. The allocation of district delegates is addressed in Sect. 4.5. Not

56

4 The Republican Party Primary

all state parties allocate delegates proportionally: Some states use a winner-take-all approach, and a smaller number of states adopt a direct election of delegates (rather than voting for a particular candidate) or a caucus system in which the final apportionment need not reflect the election results. Since these allocation methods are less interesting mathematically, we confine ourselves to methods that allocate delegates proportionally. A complete list of the methods used by each state is provided in Sect. 4.3.5. As noted previously, the RNC does not specify a particular method for attaining proportional allocation. The 2020 CALL document says only (as did the 2012 and 2016 documents): Any presidential primary... that occurs prior to March 15 in the year in which the national convention is held shall provide for the allocation of delegates on a proportional basis. Proportional allocation of total delegates... shall be based upon the number of statewide votes cast or the number of Congressional district votes cast in proportion to the number of votes received by each candidate. ((Republican National Committee 2019), Rules 16(c)(2) and 16(c)(3).)

The RNC does not define or clarify the term proportional basis, and thus, there is wide latitude for interpretation. As a consequence, allocation rules vary greatly among the states with little coordination: often similar or identical rules are worded or conceptualized differently in different states. In addition, not all rules stipulate how to apportion delegates in every situation. For example, the rules for the 2016 MS Republican primary state are as follows: All fractional proportions of a delegate/alternate shall be rounded to the nearest whole number ((Mississippi Republican Party 2015), Rule 4).

This rule does not address what to do when nearest-integer rounding does not yield the correct number of delegates, so it is difficult to categorize this rule as an apportionment method because there is no way of knowing how the party would apportion delegates in these cases. Other state parties provide clearer rules but still fail to describe how delegates are to be apportioned in every case. The rules of the 2016 IA Republican party indicate the following: The proportional delegate allocation shall be rounded to the nearest whole delegate. In the event that a delegate is unallocated due to mathematical rounding, the unallocated delegate vote shall be cast in favor of the candidate closest to the rounding threshold. In the event that delegates are over-allocated due to mathematical rounding, the over-allocated delegate shall be removed from a candidate based on the rounding threshold ((Republican State Central Committee of Iowa 2019), Article VIII).

The term “threshold” refers to the standard rounding threshold 0.5. These rules assume that nearest integer rounding results in at most one delegate too many or too few (“the” candidate closest to or furthest from the rounding threshold), even though this may not always be the case. However, the intent suggests that these rules can reasonably be interpreted as HAM∗ (see Definition 1.2). We assume when situations arise that are not encompassed by the delegate selection rules, decisions are made ad hoc. But the examples illustrate how difficult it can

4.3 Republican Selection Rules and Quota-Based Methods

57

be to characterize all the rules as apportionment methods, particularly in states like MS, where the rules are incomplete or not well defined. Despite these caveats, the large majority of delegate selection methods can be classified as true apportionment methods. In this section, we focus on the methods that were used during the 2012, 2016, and 2020 primaries where we found seven different apportionment methods that can be defined independently of the use of thresholds. All of them are quota-based. Thus, they are often referred to as rounding rules—decisions about how to allocate delegates based on the candidates’ quotas. Of the seven methods, six of them appear to have been new when they were created by the different state Republican parties. The material in this section first appeared in (Jones et al. 2020). For clarity, we divide our analysis into categories based on whether the apportionment methods are based on nearest-integer, lower quota or upper quota rounding. We assume the candidates are ordered so that v1 > v2 > · · · > vn , and therefore, C1 receives the most votes and Cn receives the least. We first define the methods and then summarize which methods were used by which states.

4.3.1 Methods Based on Nearest-Integer Rounding Methods based on nearest-integer rounding involve a two-step process. First, round the quotas to the nearest integer and assign di = [qi ] delegates to Ci . Then, adjust the di values if the resulting sum does not equal the total number of delegates available. Σn di . The methods vary based on how they make this adjustment. Recall that D = i=1 Σn Definition remaining Σthe Σn 4.1 Nearest-Integer Extremes (NIE): If D > i=1 [qi ], n [q ] delegates are awarded to candidate C . If D < [q D − i 1 i=1 i=1 i ], then Σn [q ] − D delegates are taken from C if possible. If not possible because [qn ] < i n Σn Σi=1 n [q ] − D, then all the delegates from C are taken and [q ] − D − [q i n i i=1 i=1 Σn n ] del[qi ] − egates are taken from Cn−1 if possible. If not possible because [qn−1 ] < i=1 D − [qn ] then all the delegates from Cn and Cn−1 are taken and we move onto Cn−2 , etc. Note that this definition, as well as those following, do not address the event of a tie between at least two candidates. On a practical note, ties are extremely unlikely to occur, and thus, it seems logical that state party rules do not address this contingency. We address the question of ties from a mathematical perspective in Sect. 4.4. Example 4.1 Let n Σ = 6 and D = 101,Σand suppose the vote totals are as shown in Table 4.2. Note that i [qi ] = 99 and i [qi ] = 98. Hence, nearest-integer rounding results in 2 delegates unallocated. Under NIE, each candidate Ci receives [qi ] except for C1 who receives a bonus of two delegates. The NIE apportionment is (27, 17, 15, 15, 14, 13) as shown in column 7 of Table 4.2. The results of the different apportionment methods for this example are shown in the last 6 columns. This example will be referred to as each apportionment method is introduced.

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4 The Republican Party Primary

Table 4.2 A sample election with 6 candidates and 101 delegates i vi qi [qi ] [qi ] [qi ] NIE NIS 1 2 3 4 5 6

29,130 20,000 17,720 16,750 16,550 15,350

25.473 17.489 15.495 14.647 14.472 13.423

25 17 15 15 14 13

25 17 15 14 14 13

26 18 16 15 15 14

27 17 15 15 14 13

26 18 15 15 14 13

HAM LAR

ILQ

SUQ

25 18 16 15 14 13

28 17 15 14 14 13

26 18 16 15 15 11

26 18 16 14 14 13

The NIE method was used in 2016 Vermont Republican party, which described the method as follows. Fractional delegates will be rounded to the nearest whole number. If rounding results in a total number of delegates exceeding the number to which Vermont is entitled, the excess delegates will be subtracted from the total awarded to the candidate who received the fewest votes of all candidates entitled to delegates. If rounding results in a total number of delegates less than the number to which Vermont is entitled, the additional delegates will be added to the total awarded to the candidate who received the most votes of all candidates entitled to delegates ((Vermont Republican Party 2021), Rule 12 2.(h)).

As with the IA rules, this description does not address all situations: It is not clear what to do if the lowest ranked candidate does not have enough delegates to subtract to reach the required delegate total. However, because of VT’s threshold structure, it is extremely unlikely that this will be an issue. Thus, the method described can be characterized unambiguously as the NIE method. Sometimes, the same rules are articulated by different state parties using different language. The delegate selection process in AL is outlined more formally than in VT, but clearly describes the same NIE method. [B]ased on the relationship the number of votes received by each presidential candidate bears to the total number of votes cast in the Republican presidential primary election in the entire state, the Steering Committee of the Alabama Republican Executive Committee shall apportion pro rata the number of delegates from the state at large each of the presidential candidates is entitled to receive rounded to the nearest whole number... In the event the pro rata apportionment of delegates leaves one or more delegates unassigned by process of mathematical distribution, then any such delegate or delegates shall be authorized for the candidate with the largest portion of the vote in the State in said Republican presidential preference primary. In the event pro rata apportionment entitles candidates by process of mathematical distribution to more delegates than authorized hereunder pursuant to the Rules adopted by the 2012 Republican National Convention and subsequently amended, then the number of delegates authorized for the candidate receiving the least number of votes in the State, among those otherwise entitled to delegates, shall be decreased to the extent necessary to conform to the number of authorized delegate positions. ((Alabama Republican Party 2015), Rules 6 & 7)

Given the differences between how NIE is defined in VT and AL, it seems likely that the state parties did not confer. A similar situation exists for several of the other methods used in the Republican primary. For the sake of brevity, however, we do

4.3 Republican Selection Rules and Quota-Based Methods

59

not investigate the variations in language used to describe these methods by different state parties. The next method differs from NIE in that at most one delegate is given or taken away from a candidate when nearest-integer rounding results in delegates being under- or over-allocated, respectively. Σn [qi ], the remainDefinition Nearest-Integer Sequential (NIS): If D > i=1 Σ4.2 n [q ] delegates are assigned one each to candidates 1 through D − D − ing i=1 i Σ Σn n [q ]. If D < [q ], one delegate is taken away each from the weakest i i=1 i Σi=1 n [q ] − D candidates who were initially assigned a positive number of delei=1 i gates. For the election in Example 4.1, NIS results in the two remaining delegates being given to C1 and C2 , resulting in an apportionment (column 8 of Table 4.2) of (26, 18, 15, 15, 14, 13). Several Republican state parties use Hamilton’s method to apportion delegates. As with the Democratic Party (see Chap. 2), the method is conceptualized sometimes in terms of nearest-integer rounding and sometimes in terms of lower quotas. We provide both definitions for completeness. Σn [qi ], Definition 4.3 Hamilton’s Method (HAM ∗ , see Definition 1.2): If D > Σi=1 Σn n [qi ] the remaining D − i=1 [qi ] delegates are assigned one each to the D − i=1 − [q ] are closest to 0.5 but less than 0.5. If candidates whose fractional parts q i i Σn Σn [q ] > D, one delegate is taken from each of the [q ] − D candidates i=1 i i=1 i whose fractional remainders are closest to 0.5 but are at least 0.5. For the election in Example 4.1, HAM∗ results in the two remaining delegates being given to C2 and C3 , as these candidates have the largest fractional remainders qi − [qi ] that are less than 0.5. The resulting apportionment (Table 4.2, column 9) is (25, 18, 16, 15, 14, 13). The last method based on nearest-integer rounding combines aspects of NIE and HAM∗ . For this reason, we call it the HAM∗ -NIE Hybrid (HNH). Σn Definition 4.4 HAM ∗ -NIE Hybrid (HNH)∗ : If i=1 [qΣ i ] < D, the remaining D − Σ n n ] candidates whose i=1 [qi ] delegates are assigned one each to the D − i=1 [qiΣ n [qi ] > D, [q1 ] fractional parts qi − [qi ] are closest to 0.5 but less than 0.5. If i=1 are assigned to C , etc., until the delegates delegates are assigned to C1 , [q2 ] delegates 2 Σ Σ run out. (That is, if k is such that i≤k [qi ] < D k + 1 receive no delegates.) For the election in Example 4.1, since D , < D, the apportionment, (25, 18, 16, 15, 14, 13), agrees with that under HAM∗ . In fact, we have the following: Σn Proposition 4.1ΣThe HNH method is a combination of HAM∗ (when i=1 [qi ] ≤ D) n [qi ] > D). and NIE (when i=1

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4 The Republican Party Primary

Σn Proof If D ≥ i=1 [q ], then the apportionments under HNH and HAM∗ are clearly Σn i Σk Σk+1 the same. If D < i=1 [qi ], let k be such that i=1 [qi ] ≤ D but i=1 [qi ] > D. Then,Σunder HNH, Candidates 1 through k receive [qi ] delegates, Ck+1 receives k [qi ] delegates and Candidates k + 2 through n receive 0 delegates. But, D − i=1 this is precisely the apportionment given by NIE. ⟁

4.3.2 Methods Based on Lower Quotas In methods based on lower Σn quotas, each candidate Ci is initially assigned di = [qi ] [qi ] (unless all qi Σ are integers), these methods differ delegates. Since D > i=1 n [qi ] delegates. The traditional based on how they assign the remaining D − i=1 way of describing Hamilton’s method uses lower quota rounding. Definition 1.1): Assign the remaining Σn Σn 4.5 Hamilton’s Method (HAM, see Definition [qi ] delegates one each to the D − i=1 [qi ] candidates with largest D − i=1 fractional remainders qi − [qi ]. It is easy to see that this definition of Hamilton’s is equivalent to the preΣmethod n [qi ] = 101 − 98 = 3. Thus, vious one. For the election in Example 4.1, D − i=1 under this definition of Hamilton’s method, each candidate receives their lower quota and then the remaining 3 delegates are assigned to C2 , C3 , and C4 , who have the largest fractional remainder. The resulting apportionment is again (25, 18, 16, 15, 14, 13). Σn [qi ] delegates Definition 4.6 Large Method (LAR): Assign the remaining D − i=1 one each to the strongest candidates without integer quotas. (In most cases, this will Σn [qi ].) be candidates 1 through D − i=1 For the election in Example 4.1, the three remaining delegates are given one each to C1 , C2 , and C3 , resulting in an apportionment (column 10) of (26, 18, 16, 14, 14, 13). Definition 4.7 Iterated Lower Quota (ILQ): The ILQ method is defined by folΣthe n [qi ] lowing algorithm which determines how to allocate the remaining D − i=1 delegates. Σn [qi ] be the number of remaining delegates, and let j = 1. 1. Let R1 = D − i=1 R v j 2. Let qi = Vj 1 and then do the following: (a) (b) (c) (d)

j

Assign an additional [qi ] delegates to Ci . j If [q1 ] < 1, assign an additional R j delegates to C1 . Σ j Calculate the remaining delegates R j+1 = D − i [qi ]. Increment j.

4.3 Republican Selection Rules and Quota-Based Methods

61

Table 4.3 Demonstrating the ILQ method with 25 delegates and 6 candidates Cand. Vote Quota Rd. 1 Quota Rd. 2 Quota Rd. 3 total D = 25 R1 = 5 R2 = 3 1 2 3 4 5 6 Total

12,000 11,500 6000 3000 2800 2700 38,000

7.89 7.57 3.95 1.97 1.84 1.78 25

7 7 3 1 1 1 20

1.58 1.51 0.79 0.39 0.37 0.36 5

1 1

2

0.95 0.91 0.47 0.24 0.22 0.21 3

3

3

Total

11 8 3 1 1 1 25

The ILQ method is significantly different from the other methods in that it occurs over a series of rounds; see (Jones et al. 2023) for a more detailed analysis. We illustrate with an example requiring three rounds. Example 4.2 Suppose there are 25 delegates and six candidates with vote totals as indicated in Table 4.3. Under the ILQ method, each candidate is initially given their lower quota, resulting in an allocation of 20 delegates distributed to candidates C1 to C6 as (7, 7, 3, 1, 1, 1). We then recalculate the quotas using R1 = 5 (from 25 − 20) delegates, resulting in the adjusted quotas indicated in the “Quota R1 = 5” column. These quotas are then rounded down, leading to an additional distribution of (1, 1, 0, 0, 0, 0) delegates in the second round. The quotas are then recalculated once more for the remaining R2 = 3 delegates. Since all quotas are now less than 1, C1 receives the remaining three delegates in Round 3. Thus, the final delegate allocation is (11, 8, 3, 1, 1, 1). For the election in Example 4.1, R1 = 3. In Round 2, the adjusted quotas for all candidates are less than 1. Hence, C1 receives the remaining delegates, resulting in an apportionment (column 11) of (28, 17, 15, 14, 14, 13).

4.3.3 A Method Based on Upper Quotas Σn Since D < i=1 [qi ] unless all [qi ] are integers, the single method based on upper quotas determines which candidates receive less than their upper quota. Definition 4.8 Sequential Upper Quota (SUQ): Assign [q1 ] delegates to C1 , then C2 , etc., until (if there are enough delegates remaining) assign [q2 ] delegates Σto k [qi ] ≤ D but all delegates have been assigned. That is, let k be such that i=1 Σk+1 [qi ] > D. Then, assign [qi ] delegates to Candidates 1 through k and assign i=1Σ k D − i=1 [qi ] delegates to Ck+1 .

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4 The Republican Party Primary

For the election in Example 4.1, the SUQ method allocates [qi ] delegates to Candidates 1 through 5 leaving 11 delegates for C6 , which is less than their upper quota. Thus, the SUQ apportionment (column 12) is (26, 18, 16, 15, 15, 11). The results from the different apportionment methods for the 6 candidate election in Example 4.1 show six of the apportionment methods can yield different outcomes. (The HNH method will always agree with either NIE or HAM methods.) We note that n = 6 is the smallest number of candidates for which this can occur; see (Jones et al. 2020). Proposition 4.2 For fixed D, the minimum number of candidates for which an election can yield different apportionment for each of the six methods NIE, NIS, HAM, LAR, ILQ, and SUQ is n = 6.

4.3.4 Methods that Cannot Be Disentangled from Thresholds Some state parties use apportionment methods that cannot be conceptualized in the two-step framework defined by first applying a threshold by eliminating candidates and their votes and then applying an apportionment method. This is because the delegate selection rules involve applying a threshold to eliminate candidates but not their votes, meaning that the remaining candidates’ quotas are not readjusted after elimination. In this situation, the quotas of the surviving candidates will sum to a value less than D, and the party’s apportionment method must address this gap. In this section, we describe the rules of the three states in the 2016 primary that did not readjust quotas after elimination. The first of these, the 2016 NH primary, used the following method. Calculate the quotas of candidates before elimination and then apply a 10% threshold to eliminate lower-ranked candidates but not their votes. Round the non-adjusted quotas to the nearest integer. Give any remaining delegates to C1 (New Hampshire Republican Party 2015). This method embraces the spirit of the NIE method in that surplus delegates are given to the top candidate. However, the rules do not address the possibility of an over allocation of delegates, and thus, we do not know whether the rules would suggest removing delegates from Cn , as with the NIE method. In practice, the 10% threshold means that there likely would never be an over allocation of delegates. The 2016 LA Republican primary used a variation of the method used in the NH primary: (1) Calculate the quotas of candidates before elimination and then apply a 20% threshold to eliminate lower-ranked candidates but not their votes; (2) round the non-adjusted quotas to the nearest integer; and (3) allow surplus delegates to become unbound, meaning that they can attend the convention but are not required to support any particular candidate (Louisiana Republican Party 2015). This method is interesting because when rounding does not add up to the total number of delegates, the target D value is simply abandoned. As with NH, the rules do not indicate what to do in the event of an over allocation.

4.3 Republican Selection Rules and Quota-Based Methods

63

Table 4.4 The 2012 primary calendar and apportionment methods for at-large delegates for states and territories Date Method: Date Method: Date Method: State(s) State(s) State(s) Jan 3

NM: IA

Jan 10 Jan 21 Jan 29 Jan 31 Feb 4 Feb 7 Feb 28

NIE: NH† WIN: SC NM: ME WIN: FL UN: NV NM: CO, MN WIN: AZ NIE: MI NM: WA

Mar 3 Mar 6

Mar 10

Mar 13

Mar 15 Mar 18 Mar 20 Mar 24 Apr 3

LAR:GA Apr 24 NIE: ID, VT NM: MA, ND, WY UN: OH, OK SUQ: AK, TN, VA

DEL: GU, NMI, VI SUQ: KS NIE: AL, MS DEL: AS SUQ: KS NM: MO UN: PR NM: IL UN: LA WIN: DC, MD, WI NIE: CT† WIN:DE UN:NY

May 8

May 15 May 22 May 29 Jun 5

Jun 26

WIN: IN NM: NC DEL: WV UN: OR SUQ: AR† UN: KY SUQ: TX WIN: CA, NJ NM: NM SUQ: SD WIN: UT

DEL: PA NM: RI

AS American Samoa; GU Guam; NMI Northern Marianas Islands; PR Puerto Rico; VI Virgin Islands. States with a † indicate when assumptions have been made because the rules are incomplete

The 2016 OK Republican primary used the same method as the LA primary, but with a 15% threshold (Oklahoma Republican Party 2015).

4.3.5 Which States Used Which Methods In this section, we discuss the frequency and sequence with which each method was used. Table 4.4 summarizes the date and apportionment method used by each state in the 2012 Republican primary. Table 4.5 provides the same information for the 2016 Republican primary. We do not include information about the 2020 primary because it was almost identical to the 2016 primary, with a couple of exceptions that we note below. In both tables, WIN denotes a winner-take-all method and DEL denotes a method in which delegates are elected directly (i.e., voters cast votes for delegates, not for presidential candidates). UN denotes a method that is unclear, but the state’s delegate selection rules imply that a proportional method is intended even if not all the details are included. The 2016 MS primary is an example, where the rules state that nearest-

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4 The Republican Party Primary

Table 4.5 The 2016 primary calendar and apportionment methods for at-large delegates for states and territories Date Method: Date Method: Date Method: State(s) State(s) State(s) Feb 1 Feb 9

HAM∗ : IA NIE: NH†

Feb 20

WIN: SC

SUQ: HI

Feb 23 Mar 1

UN: NV HAM∗ : AR+ , VA† HNH: MN Mar 10 ILQ: GA Mar 12 NIE: AL, VT† Mar 15

UN: MS NM: ID

Mar 6 Mar 8

NIS: MA SUQ: TN, TX

Mar 5

UN: AK, ND Mar 22 NM:CO, OK, WY HAM∗ : KY† SUQ: KS, ME Apr 1–3 NM: LA Apr 5

UN: PR NIE: MI

DEL: VI DEL: GU WIN: FL, IL, MO, NMI, OH UN: NC DEL: AS SUQ: UT†

Apr 19 Apr 26

May 3 May 10

NIE: NY NIE: CT† , RI∗∗ WIN: DE, MD, PA WIN: IN WIN: NE

May 17 May 24

NM: WV HAM: OR HAM∗ : WA‡

Jun 7

Jun 24

HAM∗ : NM† WIN: CA, MT, NJ, SD HAM∗ : DC

WIN:AZ NM:ND WIN: WI

AS American Samoa; GU Guam; NMI Northern Marianas Islands; PR Puerto Rico; VI Virgin Islands. States with a † indicate when assumptions have been made because the rules are incomplete. + AR uses HAM after each candidates who receives 15% of the vote is given 1 delegate. ∗∗ RI had no formal method defined, but used the NIE method. ‡ denotes that the party allows for unbound delegates if there is an under allocation

integer rounding is to be used but says nothing about over or under allocation. NM denotes No Method which we use to denote two different situations. In the first, delegates are chosen at a state convention with no stated requirement to reflect the election results of the caucus or primary. In these cases, the delegate selection rules are almost entirely silent about the delegate allocation process. In the second situation, NM denotes a method that might be somewhat mathematical, but the rules explicitly allow for a deviation from mathematics under certain conditions, such as occurred in LA and OK, as described in Sect. 4.3.4, which may lead to an actual apportionment. As with the true apportionment methods, the table includes some judgment calls about when a method is classified as UN or NM. For example, in Table 4.5, we classify WA as using HAM∗ rather than NM because the method deviates from HAM∗ only in the case of an under allocation of delegates, in which case the delegates become unbound. If nearest-integer rounding yields a valid apportionment or an over allocation, the party uses HAM∗ .

4.4 Geometry of Republican Methods

65

Table 4.6 Number of times each apportionment method was used in the 2012, 2016, and 2020 primaries HAM/HAM∗ HNH ILQ LAR NIE NIS SUQ Method Number of times used

16

2

2

1

21

1

21

As noted above, the calendar and methods used in the 2020 Republican primary were almost the same as those indicated in Table 4.5. This is largely because the contest was essentially a one-candidate race with Donald Trump as the incumbent. The only differences between the 2016 and 2020 primaries are the dates of some of the state primaries and the fact that some states chose not to hold a primary. The only change of interest mathematically is that the MA Republican primary switched from NIS to SUQ. It is clear from the tables that some apportionment methods are more popular than others. Table 4.6 indicates the number of times each method was used in the approximately 84 state elections that applied an apportionment method during the 2012, 2016, and 2020 primaries. By far, the most popular methods are HAM, NIE, and SUQ. HNH (resp. ILQ) was used only twice, both times in MN (resp. GA); LAR (resp. NIS) was used only once, in the 2012 GA (resp. 2016 MA) primary. Note that the numbers in the table do not sum to 84; the elections classified as UN make up the gap. Finally, we note that even though state parties are allowed to use a winner-take-all approach after a certain point in the calendar, a number of states still chose to use proportional allocation after that date in 2016 and 2020. For example, in 2016, four of the last (latest in the calendar) eight state primaries used HAM to apportion their delegates.

4.4 Geometry of Republican Methods The simplicial geometry of Hamilton’s method that was introduced in Sect. 2.2 can also be used to visualize the Republican apportionment methods. In this section, we show images of simplices for the methods introduced in Sect. 4.3. Figure 4.1 illustrates the geometry of the seven methods for n = 3 and D = 12. Note that when n = 3, nearest-integer rounding results in at most one delegate underor over-allocated, and hence, the NIE and NIS methods will always produce the same apportionment in this case. To make the size of the images reasonable, we show only one-sixth of the simplex where v1 ≥ v2 ≥ v3 . In each case, dotted lines around the perimeter denote apportionment regions Rd that extend to other portions of the simplex. The behavior of the apportionment methods at the boundaries of each region Rd differs. For instance, the boundaries of the hexagonal regions for Hamilton’s method correspond to tie situations where the apportionment is multivalued. By contrast, the boundaries of the parallelogram regions for the ILQ method (Fig. 4.1) are single-

66

4 The Republican Party Primary a : 12, 0, 0 b : 8, 2, 2 c : 6, 3, 3

4, 4, 4

d : 5, 5, 2 e : 6, 6, 0

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Fig. 4.1 One-sixth of the simplex Σ312 (where n = 3 and D = 12, and v1 ≥ v2 ≥ v3 ) for the HAM, HNH, NIE/NIS, SUQ, LAR and ILQ methods

valued: Each Rd contains the lower and left-most boundaries of its corresponding parallelogram. In addition, the apportionments (6, 6, 0) and (5, 5, 2) occur only along the tie line where q1 = q2 . The varied behavior at the boundaries of apportionment regions Rd among the methods is reflective of their different behavior under ties. None of the methods described indicate what to do if two more more candidates have equal vote shares and the allocation rules need to be adjusted. Presumably, this is because the likelihood of this occurring in practice is extremely small. Theoretically, under Hamilton’s method, a set of quotas such as (5.5, 4.5, 3) is generally assumed to correspond to both apportionments (6, 4, 3) and (5, 5, 3), with the expectation that in practice the final apportionment would be decided by chance. We will take this approach with the other methods, although some interpretation may be necessary. For example, suppose n = 11, D = 41 and the quotas were (5.5, 5.5, 4.4, 4.4, 4.2, 3.4, 3.4, 3.2, 2.4, 2.4, 2.2).

4.4 Geometry of Republican Methods

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Fig. 4.2 The MN apportionment, using HNH and threshold sequence of 85%, 10% with n = 3 and D = 14 (Left); HAM and threshold p1 − 25%, based on the MI apportionment (Right)

Using nearest-integer rounding, this yields an apportionment of (6, 6, 4, 4, 4, 3, 3, 3, 2, 2, 2), an under allocation of 2 delegates. Under the NIE method, the remaining 2 delegates could go to C1 or C2 , or perhaps be split between them. Using thresholds before applying these apportionment methods can result in dramatic differences in the geometry. Some threshold structures cannot be visualized when there are only 3 candidates, since at least one candidate must always receive 1/3 of the vote share. Thus, the threshold sequence in Vermont, for example, cannot be illustrated past the 20% cutoff (if we wanted to see the full effects of VT’s threshold sequence, we would need more than 10 candidates). We demonstrate the geometric effects of thresholds by considering Minnesota, which used the threshold sequence of 85%, 10%, and HNH to apportion its 14 statewide delegates (Minnesota Republican Party 2015). Figure 4.2 (Left) illustrates how the simplex is carved into three regions, one region each for the number of candidates that remain after elimination. The bottom left region corresponds to elections in which C1 is the only surviving candidate, which occurs when C1 earns at least 85% of the vote or when C1 earns less than 85% but the other two candidates fall short of the 10% threshold. The other regions along the bottom boundary correspond to elections in which both C1 and C2 achieve the threshold but C3 does not, and thus, the apportionment in these regions corresponds to the integer lattice point along the bottom boundary. The remaining regions are the same as in the standard HNH simplex, corresponding to elections in which all three candidates remain after the elimination. This kind of picture is typical of the images generated by using thresholds with delegate apportionment methods. One of the more interesting threshold structures occurs in Michigan, where the threshold depends on the performance of C1 in the case when C1 receives less than 15% of the vote (Michigan Republican Party 2015). When n = 3, the top candidate always earns at least 15% of the vote, and thus, we cannot see this threshold structure play out geometrically. To get an idea of the geometric effects, Fig. 4.2 (Right) shows HAM with D = 14 (in the actual primary D was 59, which would be too large of an image) using a threshold of p1 % − 25%. We use 25% rather than 5% so that we

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can more fully see what happens with a floating threshold of this nature. Note that this image, like the MN simplex, contains three sections, each of which corresponds to the number of candidates remaining after elimination. The bottom left region corresponds to elections in which C2 does not earn enough votes to be within 25% points of C1 ’s vote share. The top right section corresponds to elections in which C3 ’s vote share is within 25% points of C1 , and thus, all three candidates remain after the elimination. The bottom right region corresponds to elections in which only C1 and C2 remain.

4.5 District Delegates in Republican State Primaries Up to this point in Chap. 4, we have addressed only the allocation of statewide (or at-large) delegates in Republican primaries. In this section, we describe the different ways that district delegates are apportioned in Republican primaries, focusing on the 2016 primary. We are not exhaustive in the descriptions; our purpose is to give a sense of the variety of methods used. Before examining the details of the apportionment, we note three structural differences between Democratic and Republican rules concerning district delegates. First, Republican state parties can decide not to have any district delegates: The parties can combine all of the district delegates with the at-large delegates and allocate all available delegates in a single statewide allocation. In 2016, there were 15 state parties which opted to do this. Second, Republican state parties do not need to use the same apportionment method for district delegates that they use for statewide delegates. In Democratic primaries, the parties must use HAM at both levels, but Republican state parties often use different methods to apportion district and statewide delegates. In fact, even in primaries that use proportional allocation, some state parties use a proportional method at the statewide level but use winner-take-all at the district level. This occurred in the 2016 CT primary, for example, where NIE was used at the statewide level but all district delegates were allocated using winner-take-all in their district elections. Third, while Democrats use HAM* to apportion the state’s district delegates among the districts (see Chap. 2), the RNC mandates that each CD receives three delegates to apportion.3 Because each CD is allocated three delegates, Republicans often use an apportionment method, which we call the 2–1 method, that is designed only for apportioning three delegates and which largely jettisons the notion of proportionality. Definition 4.9 The 2–1 method: Give two delegates to the candidate who receives the most votes and one delegate to the candidate who receives the second-most votes. 3 In 2016, there were two exceptions to this rule, and it is not clear why these states were allowed to deviate from the norm. The Missouri Republican Party allocated 5 delegates to each CD, giving each CD an additional two delegates which were taken from the statewide delegate pool. The Wyoming Republican Party was allowed to break its one CD into 12 districts, each of which was allocated a single delegate.

4.5 District Delegates in Republican State Primaries

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Fig. 4.3 Examples of Republican methods for allocating 3 district delegates. Upper left: the 2–1 method. Upper right: the 2–1 method with a 50% threshold. Bottom left: A majority candidate or a candidate who uniquely receives at least 15% of the vote is apportioned all 3 delegates. If all three candidates receive more than 15% of the vote, each candidate receives one delegate. Otherwise, if two candidates receive at least 15% of the vote, use the 2–1 method. Bottom right: The SUQ method with a 10% threshold

The 2–1 method used in conjunction with thresholds and the winner-take-all method are the most common way to allocate district delegates in Republican primaries. We illustrate the variety of other methods used to apportion district delegates through the following examples. Three of these methods are illustrated in Fig. 4.3 for three candidates.

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• Alabama: A majority candidate receives all 3 delegates. If there is no majority candidate, and at least 2 candidates surpass the 20% threshold, then the 2–1 method is used. If only one candidate surpasses the 20% threshold, they receive all 3 delegates. If no candidate surpasses the 20% threshold, the NIE method is used. • Arkansas: A majority candidate receives all 3 delegates. Otherwise, the 2–1 method is used (Upper right of Fig. 4.3). • Hawaii: SUQ with no threshold. • Kansas: SUQ with a 10% threshold (Bottom right of Fig. 4.3). • Louisiana: HAM with no threshold. • Minnesota: HNH with a 10% threshold. • Oklahoma: A majority candidate receives all 3 delegates. Otherwise, the threshold is 15%. If exactly two candidates surpass the 15% threshold, then the 2–1 method is used. If three or more candidates surpass the threshold, the top three candidates each receive a delegate (Bottom left of Fig. 4.3). As we saw in Sect. 4.3, this sample of district delegate apportionment methods shows that Republicans use the term “proportional” very loosely. Similarly, the use of thresholds or the adoption of a method like SUQ may cause a candidate’s final apportionment to deviate significantly from quota. We note that in rare circumstances district delegates are allocated using methods that fall outside of apportionment theory. For example, in the 2016 Illinois primary, the three district delegates in each CD were directly elected in a standard plurality election. Each delegate running in the election stated which presidential candidate they supported, and the three delegates that received the most votes were elected to attend the national convention. In New York, the district delegates were all given to a majority candidate if one existed; otherwise, the threshold was 20% and the 2–1 method was used. If no candidate surpassed the 20% threshold, the party directly elected the three delegates without regard for the election results of the primary.

4.6 Conclusion Republican state parties have much more freedom than Democratic state parties in how they apportion delegates in their primaries. Before a certain date in the primary calendar, Republican primaries must allocate delegates on a “proportional basis” but the RNC does not specify what is meant by proportional. Consequently, Republican state parties use a wide range of methods and threshold structures to apportion their delegates and, arguably, some of the overall methods used by some parties cannot meaningfully be called “proportional” once we take into account thresholds and methods like 2–1 at the district level. As far as we are aware, the apportionment methods described (considered separate from the thresholds) in this chapter constitute at least six apportionment methods that have not been implemented in any setting other than the presidential primaries. Ultimately, this chapter is a testament to the mathematical creativity of Republican state party officials, most of whom probably have no background in the mathematics of apportionment theory.

References

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References Alabama Republican Party (2015) 2016 presidential preference primary resolution. https:// ballotpedia.org/RNC_delegate_guidelines_from_Alabama,_2016. Accessed: 03 Jan 2020 Jones MA, McCune D, Wilson JM (2020) New quota-based apportionment methods: the allocation of delegates in the Republican presidential primary. Math Social Sci 108:122–137. https://doi. org/10.1016/j.mathsocsci.2020.05.001 Jones MA, McCune D, Wilson J (2023) An iterative procedure for apportionment and its use in the Georgia Republican primary. Contemp Math, Amer Math Soc, Providence, RI (to appear) Louisiana Republican Party (2015) Rules for convening of the state convention to elect delegates to the 2016 Republican national convention. https://ballotpedia.org/RNC_delegate_guidelines_ from_Louisiana,_2016. Accessed: 03 Jan 2022 Michigan Republican Party (2015) Republican party of Michigan rules for selection of delegates and alternates to the 2016 Republican county, state and national conventions. https://ballotpedia. org/RNC_delegate_guidelines_from_Michigan,_2016. Accessed: 03 Jan 2022 Minnesota Republican Party (2015) 2016 national delegate selection. https://ballotpedia.org/RNC_ delegate_guidelines_from_Minnesota,_2016. Accessed: 03 Jan 2022 Mississippi Republican Party (2015) Resolution of the state executive committee of the Mississippi Republican Party. https://ballotpedia.org/RNC_delegate_guidelines_from_Mississippi,_ 2016. Accessed: 04 Jan 2022 New Hampshire Republican Party (2015) New Hampshire revised statutes annotated. http://www. gencourt.state.nh.us/rsa/html/lxiii/659/659-93.htm. Accessed: 03 Jan 2022 Oklahoma Republican Party (2015) Oklahoma delegate and alternate delegate nomination forms and procedures. https://ballotpedia.org/RNC_delegate_guidelines_from_Oklahoma,_ 2016. Accessed: 03 Jan 2022 Republican National Committee (2019) Call of the 2020 Republican national convention. https://prod-cdn-static.gop.com/media/documents/2020_RNC_Call_of_the_Convention_ 1575665975.pdf. Accessed: 04 Jan 2022 Republican State Central Committee of Iowa (2019) Republican state central committee of Iowa bylaws. https://www.iowagop.org/wp-content/uploads/2019/01/RPI-Bylaws-Updated2019.pdf. Accessed: 04 Jan 2022 The Green Papers (2022) The green papers. http://www.TheGreenPapers.com. Accessed: 15 Dec 2021 Vermont Republican Party (2021) VTGOP: rules. https://www.vtgop.org/rules. Accessed: 03 Jan 2022

Part II

Analysis of Delegate Allocation Rules

In Part I, we provided a complete description of the delegate apportionment rules used by the Democratic and Republican parties in the presidential primaries. In Part II we investigate these methods. Discussions of apportionment are usually centered around the question: What makes an allocation “good”? Relatedly, what makes one apportionment method “better” than another? The answers depend on the context in which the methods are used. Apportionment methods that are appropriate for determining the number of seats in the US House of Representatives or for allocating party seats to legislatures may not be appropriate for apportioning delegates to candidates. Below, we summarize some of the primary characteristics that have been traditionally considered important in apportioning house or parliamentary seats. In Part II, we evaluate how the delegate methods perform relative to these properties, and we consider whether these properties are still relevant in the delegate context. 1. Proportionality. A method is proportional if, whenever the quotas are all integers, each candidate receives exactly their quota, and, if the total number of delegates is scaled down, the allocation is scaled down proportionally (assuming the results remain integer-valued). Colloquially, proportionality has also been understood to mean that when the quotas are not all integers, the allocations do not deviate too far from quota. There are many ways to formalize this notion. We explore several of these in Sects. 5.1, 5.7, and again in Sect. 7.2, and discuss to what degree they correspond to the somewhat looser notion of “proportional” described in the delegate selection plans. 2. Quota Condition. A method is said to satisfy quota if each candidate receives either [qi ] or [qi ] delegates for any vote distribution. A method that does not satisfy quota is said to break quota. Satisfying quota is generally less relevant for apportionment in the primaries than in other contexts. We analyze the various ways that delegate apportionment methods can break quota in Sect. 5.2. 3. Unbiasedness. A method is unbiased if it does not systematically favor the stronger candidates at the expense of the weaker ones or vice versa. Unbiasedness is also less important when apportioning delegates: Indeed, several of the methods used in the primaries are explicitly defined to be biased toward stronger candidates. As

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with proportionality, there are many ways to measure bias. In Sect. 5.3, we explore several of them to quantify and compare the bias inherent in the different delegate apportionment methods. 4. Majority and Leader Criteria. A method satisfies the majority criterion if whenever a candidate receives a majority of the votes, the candidate receives a majority of the delegates. Similarly, a method satisfies the leader criterion if the candidate with the most votes receives more delegates than any other candidate. We analyze these criteria in Sect. 5.5. The majority and leader criteria are not relevant for apportioning house seats, but are an important consideration in the parliamentary context. They are arguably relevant in apportioning delegates as well. 5. Avoidance of Paradoxes. There are many situations in which apportionment methods behave “paradoxically.” In Chap. 6, we define several well-known paradoxes and introduce a couple of less well-studied ones. We also discuss how and under what circumstances the delegate apportionment methods are susceptible to these problems, and how important they are for the presidential primaries. 6. Fairness (in an optimization sense). Many of the traditional apportionment methods can be justified as maximizing some function which quantifies the “fairness” of the allocation (or minimizes some measure of unfairness). For example, HAM minimizes the Euclidean distance between q and d. In Sect. 7.2, we review several of these fairness measures and analyze how the delegate apportionment methods behave according to these measures in comparison to other standard apportionment methods. 7. Degressiveness/Regressiveness. Degressive representation (favoring weaker constituencies) is a requirement for apportioning national representatives to the European Parliament. In contrast, delegate apportionment explicitly embraces a spirit of regressiveness (favoring stronger constituencies or candidates). In Chap. 7 we explore how the degressive methods proposed for the European Parliament could be adapted to create regressive methods for the presidential primaries. We begin Part II with a discussion in Chap. 5 of the basic properties satisfied by the delegate apportionment methods and an analysis of the criteria 1–4 listed above. In Chap. 6, we explore the paradoxes to which the delegate apportionment methods are susceptible. We supplement the theoretical discussion in both of these chapters with results obtained through simulations, allowing us to compare the delegate apportionment methods over a wide range of delegate sizes and numbers of candidates. We also contrast the theoretical analysis with the outcomes of the delegate selection plan that occur in practice, illustrating the impact of using the thresholds to eliminate weaker candidates and the effect of dividing delegate allocation into separate district and statewide contests. General information about the simulations and the data used for the empirical analyses can be found in Appendices A and B, respectively. We conclude in Chap. 7 with a discussion of alternative methods of apportionment, including divisor and shift-quota methods, and a comparison of how these methods behave compared to the delegate apportionment methods based on traditional measures of fairness. We examine how ideas for apportioning representatives in the European Parliament degressively can be applied to delegate apportionment, creating a uniform way to allocate delegates either degressively or regressively. We

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end by making a number of suggestions for improving the delegate selection plans which draw on the unique aspects of delegate apportionment and, in particular, the role of the primary calendar.

Chapter 5

Properties of the Apportionment Methods Used in the Primaries

The delegate allocation process described in Part I reveals the overarching structure, as well as the nuances, of how delegates are apportioned among the candidates in the presidential primaries. Substantial differences exist between the Democratic and Republican Parties’ approaches at the national level, between the Democratic primary and caucus states, and among the Republican state primaries. At the heart of delegate apportionment are the seven basic apportionment methods: Hamilton’s method (HAM), the nearest integer sequential method (NIS), the nearest integer extreme method (NIE), the Hamilton-NIE hybrid (HNH), the large method (LAR), the iterated lower quotamethod (ILQ) and the sequential upper quota method (SUQ), described in Chaps. 2 and 4. In this chapter, we analyze and compare these seven apportionment methods. Our discussion focuses on the properties that are most relevant to delegate apportionment in the presidential primaries. We preface the discussion in Sect. 5.1 with an analysis which summarizes a number of standard properties that apportionment methods are expected to satisfy. This includes a detailed discussion of the idea of proportional consistency, identified in (Balinski and Young 2001), which restricts how apportionment methods behave in response to scaling the total number of delegates. In Sect. 5.2, we analyze the methods’ relationships to quota, considering both how far from quota a candidate’s apportionment may be, as well as the frequency with which the methods break quota. In Sect. 5.3, we compare the methods by their degree of bias, either for or against the strongest or weakest supported candidates. The problem of quantifying and comparing the bias of apportionment methods has been of great interest historically since it is important to minimize bias in the context of political representation (for instance, minimizing bias against small or large states in apportioning for the US House of Representatives). Bias is not a priori a concern when allocating delegates in the primaries; however, defining and measuring bias helps to explain and to contrast the behavior of the different apportionment methods. To that end, we analyze bias in © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9_5

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multiple ways: by measuring expected deviation from quota; by comparing delegate thresholds (the percentage of the vote required to obtain a fixed number of delegates); and by comparing how the apportionment methods behave in close elections. In Sect. 5.4, we discuss the sensitivity of the apportionment methods to small changes of the vote totals. In Sect. 5.5, we analyze and compare the methods’ allocation of delegates to a leading candidate, and to a majority candidate. In Sect. 5.6, we consider candidate coalitions. Finally, in Sect. 5.7, we conclude our analysis by examining the overall impact of delegate selection plans including apportionment methods, thresholds, and structure of the state primaries through comparing vote share to delegate share during the course of the primaries. Throughout the chapter, we apply multiple approaches. On the theoretical side, we identify and compare the characteristics of each apportionment method through both analysis and Monte Carlo simulations, based on a uniform vote distribution. See Appendix A for more information on how the simulations were conducted. Where appropriate, we also investigate the methods’ behavior empirically using data from recent presidential primaries. The data, compiled from (The Green Papers 2022), includes district and statewide elections from the significantly contested Presidential primaries since 2000. This includes the 2004, 2008, 2016, and 2020 Democratic primaries and the 2008, 2012, and 2016 Republican primaries (since the remainder were essentially one-candidate contests). See Appendix B for complete information on how the data was collected and what kinds of elections are included. Throughout the text, we refer to these two databases as DEM_DATA and REP_DATA.

5.1 Properties of the Delegate Apportionment Methods We begin with a formal definition of apportionment for allocating delegates to candidates. Assume that D > 0 delegates are to be apportioned among a set of n candidates denoted by C1 , …, Cn . Suppose that candidate Ci receives vi > 0 votes with Σ i vi = V . Let v = (v1 , . . . , vn ). Definition 5.1 An apportionment method F is a set-valued function that maps D, v to a non-empty set of n-tuple vectors d = (d1 , . . . , dn ) of non-negative integers such Σ that i di = D. Because of the possibility of “ties” at some point in the apportionment process, F is a set-valued function. However, in most instances, F(D, v) is single valued. For convenience, we usually write F(D, v) = d rather than d ∈ F(D, v) with the understanding that when F is multivalued, we refer to one of the possible apportionments. Unless otherwise stated, we assume candidates are ordered so that v1 ≥ v2 ≥ · · · ≥ vn . The apportionment methods defined in Chaps. 2 and 4 by the delegate selection rules do not in general state what to do if two or more candidates share equal vote share and the allocation rules need to be adjusted. Presumably, this is because the likelihood of this occurring in practice is extremely small. Under Hamilton’s method,

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a set of quotas such as q = (5.5, 4.5, 3) corresponds to both apportionments (6, 4, 3) and (5, 5, 3), with the expectation that in practice the final apportionment would be decided by chance. We take this approach with the other methods, although some interpretation may be necessary. For example, suppose n = 11, D = 41, and q = (5.5, 5.5, 4.4, 4.4, 4.2, 3.4, 3.4, 3.2, 2.4, 2.4, 2.2). Under NIE, rounding each quota to its integer yields (6, 6, 4, 4, 4, 3, 3, 3, 2, 2, 2), which sums to 39. The remaining 2 delegates could be awarded to C1 or C2 or could be split between them. Similar problems exist under ILQ and under SUQ for lower-ranked candidates. The above definition of an apportionment method is broad. However, there are a number of basic properties that all apportionment methods are expected to satisfy. We outline them here, following (Balinski and Young 2001) and (Pukelsheim 2017). The first property is that the apportionment should not depend on the identity of the candidates: If the ordering of the candidates were rearranged, then the allocation should be rearranged in the same order. To formalize this, suppose that π : {1, 2, . . . , n} → {1, 2, . . . , n} is a permutation which defines a reordering of an n-dimensional vector w by w˜ = (wπ(1) , wπ(2) . . . , vπ(n) ). Definition 5.2 An apportionment method F is anonymous if for all d ∈ F(D, v) and for every permutation π , then d˜ ∈ F(D, v˜ ). The second property is that if one candidate receives more votes than another candidate receives, then the first candidate should receive at least as many delegates as the second candidate. Definition 5.3 An apportionment method F is order-preserving if whenever d ∈ F(D, v), then vi > vj implies di ≥ dj . The third property is if the votes for each candidate change by the same proportion, then the set of possible apportionments should remain the same. Definition 5.4 An apportionment method F is homogeneous if F(D, v) = F(D, λv) for every positive integer λ. Homogeneity allows the apportionment to be defined in terms of a vote distribution, v, the corresponding vector of vote shares p, where pi = vi /V , or the vector of quotas q where qi = D(vi /V ). Thus the domain of F can be extended to vectors v where vi is rational. While we will not need to consider more general vectors here, allowing vi to be real numbers requires the following property. Definition 5.5 An apportionment method F is complete if for every sequence of vectors vn → w, then F(D, vn ) → F(D, w). The next property is that if an apportionment method can be solved perfectly in integers without rounding because each candidate’s quota is an integer, then the vector of quotas should be the unique solution to the apportionment problem. Definition 5.6 An apportionment method F is weakly proportional if whenever qi is an integer for each candidate Ci then F(D, v) = q.

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The next property is a commonly cited requirement that restricts the allocations that are possible whenever two candidates receive the same number of votes. Definition 5.7 An apportionment method F is balanced, if d ∈ F(D, v) and vi = vj , then |di − dj | ≤ 1. All the delegate apportionment methods discussed in Chaps. 2 and 4 satisfy the properties of anonymity, order-preserving, homogeneity, and weak proportionality. Since the methods are ambiguous with respect to ties, we leave the question of whether they—with the exception of HAM—are balanced open. It will not be assumed in what follows; indeed, the range of allocations possible in the event of close elections contributes to the difference among the methods, which we discuss in Sect. 5.3. The next three properties are not satisfied by any of the delegate apportionment methods (or, more generally, by any quota-based methods of apportionment). The first of these stipulates that if the vote share of some candidate Ci increases while that of another candidate Cj decreases, then Ci should not get a smaller number of delegates while Cj gets a greater number, unless there is a tie. Definition 5.8 An apportionment method F is population monotone, if for every vote distributions v and w with d ∈ F(D, v) and e ∈ F(D, w), then for all i /= j, vi ≥ wwji implies either vj 1. di ≥ ei or dj ≤ ej ; or 2. vi /vj = wi /wj and e˜ ∈ F(D, w) where e˜ = e except with di and dj replacing ei and ej . Methods that do not satisfy population monotonicity are susceptible to the population paradox. Likewise, methods that do not satisfy delegate monotonicity (or house monotonicity for allocating representatives in the US House) defined below, are subject to what is commonly called the Alabama paradox. The behavior of the delegate apportionment methods with respect to these paradoxes is discussed further in Chap. 6. Definition 5.9 An apportionment method F is delegate monotone, if for every d ∈ F(D, v) and d, ∈ F(D + 1, v) then di, ≥ di . Balinski and Young (2001) prove that an apportionment method is population monotone if and only if it is a divisor method (see Definition 1.3). They also show that population monotonicity implies delegate (or house) monotonicity. The final property is uniformity or consistency. Loosely speaking, a method is uniform if, when any subset of candidates “takes” their allocated delegates, and the remaining delegates are collected together to reapportion among the other candidates, the resulting allocation remains the same. Moreover, if the allocation of the remaining delegates has an additional apportionment (perhaps because of a tie), then that second apportionment corresponds to a second apportionment of the initial problem. The following definition of uniformity accounts for all possible subsets of {1, 2, . . . , n} by truncating all possible permutations of {1, 2, . . . , n}.

5.1 Properties of the Delegate Apportionment Methods

81

Definition 5.10 An apportionment method F is uniform if for every 2 ≤ m ≤ n and for every permutation σ of {1, 2, . . . , n}, then Σ 1. d ∈ F(D, v) implies (dσ (1) , . . . , dσ (m) ) ∈ F( m i=1 dσ (i) , (vσ (1) , . . . , vσ (m) )); and Σm 2. d ∈ F(D, v) and (e1 , . . . em ) ∈ F( i=1 dσ (i) , (vσ (1) , . . . , vσ (m) )) implies (e1 , . . . , em , dσ (m+1) , . . . , dσ (n) ) ∈ F(D, (vσ (1) , . . . , vσ (n) )). Uniformity is also satisfied by divisor methods but not by quota-based methods (Balinski and Young 2001). See Chap. 6 for a discussion of the delegate apportionment methods and the failure of uniformity. Because divisor methods satisfy so many desirable properties, they are preferred for many apportionment problems. We discuss this further in Chap. 7. One disadvantage of divisor methods is that they sometimes fail to stay within quota—that is, the number of delegates allocated to a candidate by a divisor method is sometimes above the candidate’s quota rounded up or below the candidate’s quota rounded down. This is also a characteristic of several of the apportionment methods used in the primaries. This is discussed further in Sect. 5.2. We complete this analysis of apportionment properties with a detailed examination of how the delegate apportionment methods behave when scaling D by a constant allows for a corresponding scaling of d.

5.1.1 Proportional Consistency The property of proportional consistency, introduced informally by Balinski and Young (2001), reflects the fact that as D increases, the apportionment should get increasingly close to true proportionality. (Indeed, if D were large enough to equal V , then each voter would correspond to a single delegate.) Since allocations are integer-valued, this property cannot be addressed by requiring limD→∞ F(D, v) = q. Instead, we consider what happens as the number of delegates is scaled. As a reminder, if qi = Dvi /V is candidate Ci ’s quota, then [qi ], ]qi ], and [qi ] denote the quota rounded down, up, and to the nearest integer, respectively. We define the rounded quota vectors [q], ]q], and [q] analogously. Definition 5.11 An apportionment method F is proportionally consistent if whenever λd ∈ F(λD, v) for rational λ > 1 where λdi is an integer for every i, then d ∈ F(D, v). Note that we give the formal definition of the notion of proportional consistency. Balinski and Young (2001) define this property informally, stating that any method that is weakly proportional and is weakly consistent is “proportional.” As an example of proportional consistency, if an apportionment method allocates 24 delegates among 4 candidates according to d = (9, 6, 6, 3), then it should allocate 8 delegates according to (3, 2, 2, 1) (with λ = 1/3) and 16 delegates according to (6, 4, 4, 2) (with λ = 2/3). The reverse does not make sense: If 8 delegates are

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allocated according to d = (3, 2, 2, 1), then 24 delegates would not necessarily be allocated as (9, 6, 6, 3). Proportional consistency has not been previously studied because most apportionment methods, including divisor methods, satisfy it. However, not all of the seven delegate apportionment methods satisfy proportional consistency. We begin our analysis with a preliminary result about the effect of scaling on rounding functions. Proposition 5.1 Let λ > 1 and assume that di is an integer. (1) If [λqi ] = λdi then [qi ] = di . If ]λqi ] = λdi then ]qi ] = di . (2) If [λqi ] = λdi + t for some integer t ≥ 0 then di ≤ [qi ] ≤ di + t. If [λqi ] = λdi − t for some integer t ≥ 0 then di − t ≤ [qi ] ≤ di . In both cases, if λ ≥ 2t + 1 then [qi ] = di . Proof For (1), if [λqi ] = λdi then λdi ≤ λqi < λdi + 1, which implies that di ≤ qi < di + λ1 < di + 1. It follows that [qi ] = di . The proof of the second part of (1) is similar. For (2), if [λqi ] = λdi + t then λdi + t − 21 ≤ λqi < λdi + t + 21 . Dividing by λ ≤ qi < di + t+1/2 which implies di − 21 < qi < di + t+1/2 . If λ ≥ yields di + t−1/2 λ λ λ 2t + 1, the upper bound is less than 1/2, so di − 21 < qi < di + 21 and hence [qi ] = di . Otherwise, di − 21 < qi < di + t + 21 so di ≤ [qi ] ≤ di + t. The proof of the second part of (2) is similar. ▢ Notice in (2) that when t = 0, then [λqi ] = λdi implies that [qi ] = di . Σ = λD + k Note that as a consequence of Part (2) of Proposition 5.1, if i [λqi ]Σ Σ = D + k , for some 0 ≤ k , ≤ k. LikewiseΣif i [λqi ] = for some k ≥ 0 then i [qi ] Σ λD −Σ k for some k ≥ 0 then i [qi ] = D − k , for some 0 ≤ k , ≤ k. If i [λqi ] = λD then i [qi ] = D. Suppose F is an apportionment method that satisfies quota. Let Lλd = { i | λdi = [λqi ]} be the set of candidates whose quotas are rounded down or are integers and / Lλd | λdi = ]λqi ]} be the set of candidates whose quotas are rounded up. Uλd = { i ∈ Define Ld and Ud analogously. Then Proposition 5.1 implies if i ∈ Lλd then [qi ] = di , and if i ∈ Uλd , then [qi ] = di − 1. Proposition 5.2 LAR and HAM are proportionally consistent. Σ Σ Σ Proof Recall that λ > 1. It follows that i [λqi ] = i∈L Σλd λdi + Σj∈Uλd (λdj − 1) k = |U Σ Σλd |. But by Proposition 5.1, i [qi ] = i∈Lλd [qi ] + Σ = λD − k where j∈Uλd [qj ] = i∈Lλd di + j∈Uλd (dj − 1) = D − k. Thus, rounding down leaves the same number of delegates under allocated in both apportionment problems. Since LAR always allocates these delegates to the strongest k candidates, it is proportionally consistent. For HAM, the ordering of the remainders λqi − [λqi ] coincides with the ordering of the remainders qi − [qi ]. To see this, remainders for i ∈ Lλd and for j ∈ Uλd can be compared: λqi − [λqi ] ≤ λqj − [λqj ]. Dividing by λ gives qi − di ≤ qj − (dj − 1 ) ≤ qj − (dj − 1), or qi − [qi ] ≤ qj − [qj ]. This implies that HAM is proportionλ ally consistent. ▢

5.1 Properties of the Delegate Apportionment Methods

83

Proposition 5.3 The SUQ method is proportionally consistent. Proof Let the vi in v be ordered such that v1 ≥ v2 ≥ · · · ≥ vn . Suppose F(λD, v) = λd and F(D, v) = d, . We want to show that d, = d. Let c be the smallest positive integer such that if the first c candidates have their quotasΣλqi rounded up to ]λq Σ i ], then the number of delegates is exhausted. Equivalently, i 1 and m such that 0 ≤ m < [λqc ]. If i < c, then λdi Σ di = [qi ]. If i ≥ c then λdi ≤ [λqi ], which implies di ≤ [qi ]. Thus, i [qi ] ≥ D and hence d, = ([q1 ], . . . , [qc, −1 ], [qc, ] − m, , 0, . . . , 0) for some c, > 1 and 0 ≤ m, < [qc, ]. We claim c, = c. To see this, suppose c, < c so that d, has moreΣ0 entries then d. But di, = [qi ] = di for i ≤ c, − 1 and dc,, < [qc, ] = dc, . So D = i≤c, di,
c is shown analogously. This proves the claim. It follows that i≤c,

d, = ([q1 ], . . . , [qc−1 ], [qc ] − m, , 0, . . . , 0) = (d1 , . . . , dc−1 , [qc ] − m, , 0, . . . , 0), which implies [qc ] − m, = dc and d, = d.

▢

Propositions 5.2 and 5.4 imply that the HNH method is proportionally consistent. In contrast, we have the following. Proposition 5.5 The NIS and ILQ methods are not proportionally consistent. Proof To show that the NIS method is not proportionally consistent, let n = 8, D = 28, λ = 2.5, λD = 70, and λq = (10, 9.65, 9.65, 9.65, 9.65, 9.65, 6.25, 5.5). Under the NIS method, the apportionment is λd = (10, 10, 10, 10, 10, 10, 5, 5), which implies that d = (4, 4, 4, 4, 4, 4, 2, 2). For the NIS method to be proportionally consistent, then the apportionment of q = (4, 3.86, 3.86, 3.86, 3.86, 3.86, 3.86, 2.5, 2) must be d. However, the apportionment is d, = (4, 4, 4, 4, 4, 4, 3, 1) /= d under the NIS method. To show ILQ is not proportionally consistent, let n = 6, D = 13, λ = 2, λD = 26 and λq = (7.56, 7.48, 2.80, 2.76, 2.72, 2.68). Apportioning λq under the ILQ method yields λd = (10, 8, 2, 2, 2, 2), which implies that d = (5, 4, 1, 1, 1, 1). For the ILQ method to be proportionally consistent, then the apportionment of q = (3.78, 3.74, 1.4, 1.38, 1.36, 1.34) must be d. However, under the ILQ method, the ▢ apportionment is d, = (6, 3, 1, 1, 1, 1) /= d. While the ILQ method is not proportionally consistent for any range of λ, the NIS method is proportionally consistent if the number of candidates is sufficiently small or λ is sufficiently large, as proved in the next proposition. Proposition 5.6 If F is the NIS method such that F(λD, v) = λd and F(D, v) = d, , then d, = d if: (1) n ≤ 4; or (2) λ ≥ 3. Proof Let F(λD, v) = λd and F(D, v) = d, and suppose that Σn ≤ 4. Now, for each i, − 1/2 < [λq ] ≤ λq + 1/2, which implies λD − 2 < + 2. But, λq i i i i [λqi ] ≤ λDΣ Σ [λq ] = λD + 2 only if [λq ] = λq + 1/2 for each i. Otherwise, | i i i i i [λqi ] − λD| ≤ 1. In this case, by the paragraph following Proposition 5.1, it follows that Σ | i [qi ] − D| ≤ 1. Because NIE and NIS coincide when there is at most 1 delegate over- or underallocated and NIE is proportional, this implies ΣNIS is proportional. Finally suppose [λqi ] = λqi + 1/2 for each i so that i [λqi ] = D + 2. Then λdi = [λqi ] = λqi + 1/2 for i = 1, 2 and λdi = [λqi ] − 1 = λqi − 1/2 for i = 3, 4. (This follows because the v1 ≥ v2 ≥ v3 ≥ v4 .) However, λdi = λqi + 1/2 implies that di = qi + 1/2λ and, consequently, di − 1/2 ≤ di − 1/2λ = qi ≤ di . Thus [qi ] = di for i = 1, 2. Similarly, λdi = λqi − 1/2 implies di = qi − 1/2λ and so Σdi ≤ qi ≤ di + 1/2λ ≤ di + 1/2. Thus [qi ] = di for i = 3, 4. Putting it all together, i [qi ] = D and d, = d.

5.2 Relationship to Quota

85

Σ λ ≥ 3. If i [λqi ] = λD Σ To prove (2), suppose Σ then, as with the proof of NIE, , [q ] = D and d = d. Alternatively, suppose i i i [λqi ] = λD − k for some k ≥ 1 Σ (the proof when i [λqi ] = λD + k is analogous). Then λdi = [λqi ] + 1 for i ≤ k and λdi = [λqi ] for i ≥ k + 1. By (2) of Proposition 5.1, because λ ≥ 2(1) + 1 = 3, ▢ then [qi ] = di for all i. It follows that di, = [qi ] = di for all i.

5.2 Relationship to Quota An apportionment method is said to satisfy lower quota (respectively, satisfy upper quota) if Ci cannot receive fewer than [qi ] (respectively, more than ]qi ]) delegates for all i. A method satisfies the quota condition if Ci receives either [qi ] or ]qi ] delegates for all i. Historically, it has been considered important in the House seat context for apportionment methods to satisfy the quota condition, or at least to break it infrequently (Balinski and Young 2001). In this section, we describe which of the delegate apportionment methods satisfy lower or upper quota and, for the methods that do not satisfy the quota condition, include how badly quota can be broken under these methods and for which candidates. We also give estimates for how frequently quota is broken, and discuss briefly how these results change in practice when the apportionment methods are used in conjunction with threshold rules as prescribed by the delegate selection rules. Recall that the candidates’ votes are ordered so that v1 ≥ v2 ≥ · · · ≥ vn . HAM, LAR: Both HAM and LAR satisfy the quota rule by construction. HNH: Under the HNH method, if rounding to the nearest integer results in an under allocation then the method coincides with HAM; hence each candidate receives either their upper or lower quota in this case. If rounding results in an over allocation, the method coincides with the NIE method. Thus, HNH satisfies upper quota, but can break lower quota, as described for the NIE method below. If n = 3, then the HNH method satisfies the quota condition, as does the NIE method. ILQ: Under the ILQ method, all candidates receive at least [qi ] delegates; so it satisfies lower quota. The weakest two candidates receive exactly [qi ] delegates; the middle candidates may receive more and may break quota. The top vote-getting candidate C1 is guaranteed at least ]q1 ] delegates, and if there are at least 2 delegates remaining after the first round Σ of allocation, C1 will receive at least ]q1 ] + 1, breaking D − upper quota. Since i [qi ] ≤ n − 1, the maximum deviation occurs when C1 Σ receives i [q1 ] + (n − 1) = ]q1 ] + (n − 2) delegates. (This happens when C2 does not receive enough votes to achieve a quota of at least 1 after the initial round of allocation.) Thus, upper quota may be broken by a significant amount in favor of C1 . More information about the ILQ method appears in (Jones et al. 2023). NIE: Recall that under Σ NIE, if rounding to the nearest integer results in an under allocation of m = D − j [qj ] delegates, then C1 receives [q1 ] + m and the remaining candidates receive [qi ] delegates. Thus, the apportionment stays within quota, except

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possibly Σ for C1 . If rounding to the nearest integer results in an over allocation of m = i [qi ] − D delegates, then Cn receives [qn ] − m delegates, unless m > [qn ], in which case Cn receives 0 delegates and over-allocated delegates are removed from the next weakest candidate, and so on. The remaining candidates receive [qi ] delegates. Thus, the apportionment stays within quota for the stronger candidates; the weaker candidates may receive fewer than [qi ] or [qi ] delegates, and so may break quota. The largest deviation from quota occurs when all the over-allocated candidates are removed from Cn . To determine how large the deviation from quota can be, note that for Σ each candidate, −1/2 ≤ qi − [qi ] < 1/2. Summing, this implies −n/2 ≤ D − i [qi ] < n/2; the lower bound is strict unless all the fractional remainders of qi − [qi ] are equal to 1/2. Suppose n is even. If there is an under allocation, then C1 receives at most [q1 ] + (n − 2)/2 ≤ ]q1 ] + (n − 2)/2. If there is an over allocation and all the overΣ allocated delegates are removed from Cn , then either: (i) Σi [qi ] − D ≤ (n − 2)/2 and Cn receives at least [qn ] − (n − 2)/2 delegates; or (ii) i [qi ] − D = n/2 so Cn receives [qn ] − n/2 = [q1 ] − (n − 2)/2 delegates (because [qn ] = ]qn ]). In either case, the maximum deviation is (n − 2)/2. Alternatively, suppose n is odd. If there is an under allocation, then either: (i) Σ D − Σi [qi ] ≤ (n − 3)/2 and C1 receives at most [q1 ] + (n − 3)/2 delegates; or (ii) D − i [qi ] = (n − 1)/2. In this case, we claim that [q1 ] = [q1 ]. To see this, suppose [q1 ] = ]q1 ]. Let S be the set of candidates whose quotas are roundedΣup with |S| Σ integer and summing, we have i [qi ] = Σ = s. Rounding the quotas to the nearest [q ] + s = D − (n − 1)/2 or D − For each Σ i, let qi = i i i [qi ] = s + (n − 1)/2.Σ ] + x for some 0 ≤ i < 1. Since the quotas must sum to D, [q ] + [q i i i i∈S xi + Σ Σ Σ i Σ x = D. Note x < (n − s)/2, so x > D − [q ] − (n − s)/2 = i i i i i∈S / i∈S / i∈S i Σ s + (n − 1)/2 − (n − s)/2 = (3s − 1)/2. But i∈S xi < s, which implies (3s − 1)/ 2 < s or s < 1, an impossibility because 1 ∈ S. This proves the claim. Hence [q1 ] = [q1 ] and C1 receives [q1 ] + (n − 1)/2 = ]q1 ] + (n − 3)/2 delegates. If there is an over allocation, a similar argument shows that Cn receives at least [qn ] − (n − 3)/2 delegates. Hence, the maximum deviation is (n − 3)/2 delegates. Note that if n = 3, then NIE satisfies quota. NIS: Under the NIS method, if rounding to the nearest integer results in an under allocation, the weakest n − (]n/2] − 1) candidates are guaranteed [qi ] delegates; the remaining delegates may receive [qi ] + 1 delegates. If rounding results in an over allocation, the top n − [n/2] candidates are guaranteed [qi ] delegates; the remaining delegates may receive [qi ] − 1 delegates. In all cases, every candidate receives between [qi ] − 1 and [qi ] + 1 delegates; thus quota can be broken for a candidate by at most 1 delegate. When n = 3, NIS and NIE coincide, and thus NIS satisfies the quota rule. SUQ: Under the SUQ method, C1 receives ]q1 ] delegates. The remaining candidates receive at most ]qi ] delegates and may receive less than [qi ] delegates (or nothing at all). Unless all candidates have whole integer quotas, rounding up will result in at least one delegate over-allocated, meaning that Cn can never receive more than their lower quota and often receives less than that. At its most extreme, Cn

5.2 Relationship to Quota Table 5.1 Summary of quota violations Method Lower quota HAM HNH ILQ LAR NIE NIS SUQ

87

Upper quota

Largest quota violation

⦸∗

✓ ✓

✓ ✓

✓

⦸∗

⦸∗

⦸∗

⦸∗

⦸

✓

0 [(n − 2)/2] n−2 0 [(n − 2)/2] 1 n−2

✓

⦸

Entries with a ∗ indicate that violations occur only if n > 3

may have a quota of more than n − 2 and yet receive 0 delegates. This occurs, for example, when n = 5, D = 30, and q = (8.01, 7.01, 6.01, 5.01, 3.96). In this case, C5 has a quota of just under 4, yet does not receive any delegates. A summary of possible quota violations and how large they can be appears in Table 5.1. Note that of the above delegate apportionment methods only HAM and LAR satisfy both lower and upper quota for all n. However, their relationship to quota is slightly different. Under LAR, the strongest candidate is guaranteed ]q1 ] delegates. Allocations for the remaining candidates may be equal to [qi ] or ]qi ], as is true for all candidates under HAM. Under the other methods, the largest deviations from quota generally occur in relationship to the strongest and weakest candidates. Thus, the degree and frequency with which a method breaks quota is tied to its bias, the tendency of the apportionment method to favor strong candidates at the expense of the weak candidates.1 Of course, a method can satisfy the quota rule and still be biased, as the LAR method illustrates. We take up the subject of bias in Sect. 5.3. When there are three candidates, the only methods that break quota are ILQ and SUQ. Figure 5.1 illustrates when this occurs for D = 12. The geometry suggests the following proposition. Proposition 5.7 Assuming a uniform distribution over the simplex, the probability that quota is broken when n = 3 under ILQ or SUQ as D → ∞ is 1/2. Proof From Fig. 5.1, we see that in the interior of the simplex, the probability that quota is broken is roughly 1/2. As D → ∞, the anomalous shaded regions near the boundary have decreasing impact, leading to a limiting probability of a quota ▢ violation under either method of 1/2. For n > 3, we use simulations to estimate how frequently all the methods break quota both for specific D values, as well as asymptotically. The results are shown 1

In the context of house seat apportionment, bias may also be a result of a method’s tendency to favor the smaller states at the expense of larger states. This kind of bias does not arise in delegate apportionment, although it is conceivable that a party may wish to favor weaker candidates at the beginning of the primary season, as discussed in Chap. 7.

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5 Properties of the Apportionment Methods Used in the Primaries

Fig. 5.1 Shaded regions in the top (resp. bottom) image represent the quota distributions in which the ILQ (resp. SUQ) method breaks quota when D = 12

in Table 5.2 for n = 3, 4, 5, 6 and D = 20. The numbers in parentheses indicate the asymptotic probability that each method breaks quota as D → ∞, estimated by using D = 2,000,000. Differences between the D values in each entry illustrate both the sensitivity of the results with respect to D, as well as the effect of the boundary regions on the probabilities. The results in Table 5.2 indicate that the ILQ and SUQ methods break quota most often, with asymptotic probabilities greater than 0.95 for n ≥ 5: ILQ in favor of the top candidate and SUQ to the disadvantage of the weakest candidate(s). The NIE method, despite its emphasis on the strongest and weakest candidates, breaks quota significantly less often, although still quite frequently. The NIS and HNH methods

5.2 Relationship to Quota

89

Table 5.2 Likelihood of quota violations under each method for D = 20 and asymptotically as D → ∞ for n = 3, 4, 5, 6 Lower quota violation Upper quota violation Quota violation Method Likelihood of quotation violations for n = 3 candidates HNH 0.0 0.0 0.0 0.475 (0.500) ILQ 0.0 0.0 NIE 0.0 0.0 NIS 0.382 (0.500) 0.0 SUQ Likelihood of quotation violations for n = 4 candidates 0.014 (0.020) 0.0 HNH ILQ 0.0 0.806 (0.833) NIE 0.014 (0.021) 0.022 (0.021) NIS 0.016 (0.021) 0.022 (0.021) SUQ 0.520 (0.832) 0.0 Likelihood of quotation violations for n = 5 candidates HNH 0.020 (0.044) 0.0 0.0 0.945 (0.959) ILQ NIE 0.020 (0.042) 0.047 (0.041) 0.027 (0.039) 0.046 (0.039) NIS 0.578 (0.960) 0.0 SUQ Likelihood of quotation violations for n = 6 candidates HNH 0.022 (0.062) 0.0 0.0 0.987 (0.992) ILQ 0.021 (0.063) 0.073 (0.062) NIE NIS 0.032 (0.055) 0.065 (0.056) SUQ 0.601 (0.992) 0.0

0.0 0.475 (0.500) 0.0 0.0 0.382 (0.500) 0.014 (0.020) 0.806 (0.833) 0.037 (0.042) 0.038 (0.042) 0.520 (0.832) 0.020 (0.044) 0.945 (0.959) 0.067 (0.083) 0.072 (0.078) 0.578 (0.960) 0.022 (0.062) 0.987 (0.992) 0.094 (0.125) 0.098 (0.111) 0.601 (0.992)

break quota slightly less often than NIE. However there are some additional subtleties. Σ HNH never breaks upper quota since it coincides with HAM when i [qi ] < D. For smaller values of D, NIS and NIE break lower quota less often than upper quota. This is because if the quotas of the weaker candidates are less than 1, they cannot receive less than their quotas rounded down when adjustments are made in the case Σ i [qi ] > D. Table 5.2 also indicates that the probability of quota violations generally Σ increase of the simplices where with n, as expected, because the proportions i [qi ] / = D Σ Σ (for HNH, NIS, and NIE) or where i 3, and for more precise analysis of the bias exhibited by the different apportionment methods, we turn to alternative approaches. There are many ways to define, quantify, and compare bias. Balinski and Young (2001) include an extensive discussion of apportionment bias, particularly for divisor methods. Many of the ideas raised there rely on the fact that divisor methods are uniform: The allocation of delegates among pairs of candidates depends only on vi and vj and not the complete vote distribution. (None of the seven delegate apportionment methods satisfy uniformity, as discussed in Sect. 6.6.)

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In this section, we consider four separate approaches to measuring bias. The first is based on a pairwise comparison of apportionment methods from (Balinski and Young 2001), which induces a partial order of methods based on bias. This pairwise comparison method yields a complete linear ordering of the divisor methods. However, for the quota-based methods used in delegate apportionment, the pairwise comparison method highlights how some methods are not comparable and how some methods advantage stronger candidates while other methods may disadvantage weaker candidates. The second approach is based on the idea of seat bias, which was first introduced in (Schuster, Pukelsheim, Drton, and Draper 2003) to analyze bias in apportioning representatives to the US House of Representatives. A state’s seat bias is the expected deviation of its apportionment from its quota. Adapting this notion to delegate allocation, we investigate and compare the delegate bias for different candidates under each apportionment method. The third approach to analyzing bias is to compare the delegate threshold values of each apportionment method. Delegate thresholds, which are generalizations of the threshold of inclusion and threshold of exclusion, introduced in (Rokkan 1968), indicate the minimum and maximum percentage of the vote that candidates receive to obtain a fixed number of delegates. In our discussion, we focus on the situations when the election results are spread out with C1 receiving a large share of the vote or Cn receiving a small share of the vote. Our final approach examines close elections. We compare how delegate allocations differ between top and bottom candidates when all candidates receive similar vote totals based on the notion of close election delegate minimum and close election delegate maximum, introduced in (Jones et al. 2020). We close by considering how bias is exhibited when the apportionment methods are used in the presidential primaries, where thresholds are used to eliminate weaker candidates and delegate allocation is divided into separate problems of apportioning district and statewide delegates.

5.3.1 Majorization and Pairwise Comparisons of Bias To compare the bias exhibited by a pair of apportionment methods, we adapt a notion from (Balinski and Young 2001) for relative bias. For a fixed D and v, let Xi be the apportionment given to Ci under apportionment method X so that, for example, NIE1 is the apportionment given C1 under NIE. Definition 5.12 Apportionment method F 1 favors strong candidates relative to apportionment method F 2 (denoted F 1 ≻ F 2 ) if for all D and v, whenever qi > qj then either di ≥ ei or dj ≤ ej where d = F 1 (D, v) and e = F 2 (D, v). Relative bias is a reformulation of the idea of majorization of vectors. Briefly, given two vectors d = (d1 , . . . , dn ) and e = (e1 , e2 . . . , en ), we say d majorizes e if

5.3 Bias

95

Σ

Σ di ≥ i≤k ei for all k. If d and e are allocations of delegates and the candidates are ordered so that v1 ≥ v2 ≥ · · · ≥ vn then the statement d majorizes e means that stronger candidates are given more delegates in d as compared to e. If apportionment method F 1 favors strong candidates relative to apportionment method F 2 then d majorizes e. Balinski and Young (2001) use this definition to provide a complete ordering of the most common class of divisor methods based on their rounding functions. See Chap. 7 for a more complete discussion. In general, the binary operation ≻ does not provide a complete ordering of divisor methods, nor over the broader group of all apportionment methods. Additionally, the relationship may not even be transitive. Nonetheless, we can apply the notion to compare the seven apportionment methods used in the primaries, yielding the following proposition. (Note that in the proposition’s statement X ≻ Y ≻ Z means X ≻ Y, Y ≻ Z and X ≻ Z, so that transitivity holds in these instances.) i≤k

Proposition 5.8 For fixed D and n, 1. 2. 3. 4.

SUQ ≻ LAR ≻ HAM; ILQ ≻ LAR ≻ HAM; NIE ≻ NIS ≻ HAM; and NIE ≻ HNH ≻ HAM.

Proof Let D and v be given with qi > qj . (1) and (2): To show SUQ ≻ LAR, suppose SUQi < LARi . Then SUQi ≤ [qj ], which implies SUQj ≤ [qj ] ≤ LARj . A similar argument proves all the other pairwise comparisons. Σ (3): To show NIE ≻ NIS, suppose NIEi D and i = n which is impossible since i < j ≤ n; or i [qi ] < D, so that NIEi = [qi ] and NISi = [qi ] + 1. But then NIEj = [qj ] ≤ NISj . Σ To show NIS ≻ HAM, we consider two cases. If i [qi ] < D, then NISk ≥ [qk ] for all k. If NISi < HAMi , this implies NISi = [qi ] = [qj ] and HAMj = ]qi ]. If in addition, NISj > HAMj then NISj = [qj ] = ]qj ] and HAMj = [qj ]. But then qi − [qi ] < 1/2 ≤ qj − [qj ]. Since HAMi = ]qi ] and HAMΣ j = [qj ], this is impossible. Hence, NISj ≤ HAMj . A similar argument works for i [qi ] < D. The proof that NIE ≻ HAM is identical. (4): The statement NIE ≻ HNH ≻ HAM follows from the fact that HNH is a ▢ combination of NIE and HAM. Proposition 5.8 includes all the pairwise comparisons among the seven apportionment methods. For all other pairs, X and Y, there are counterexamples showing that X ⊁ Y and Y ⊁ X. We illustrate with the following example which identifies qi > qj such that Xi < Yi and Xj > Yj .

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5 Properties of the Apportionment Methods Used in the Primaries

Example 5.1 Suppose D = 30, n = 11 and q = (5.2, 4.3, 3.7, 2.1, . . . , 2.1). Then ⎧ (7, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2) ⎪ ⎪ ⎪ ⎪ ⎨ (6, 5, 4, 3, 3, 3, 3, 3, 0, 0, 0) d = (6, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2) ⎪ ⎪ (6, 5, 3, 2, 2, 2, 2, 2, 2, 2, 2) ⎪ ⎪ ⎩ (5, 5, 4, 2, 2, 2, 2, 2, 2, 2, 2)

under ILQ under SUQ under NIE and NIS under LAR under HAM and HNH.

Note that: (i) NIE1 = 6 < 7 = ILQ1 and NIE3 = 4 > 3 = ILQ3 , which implies NIE, NIS ⊁ ILQ; (ii) NIE2 = 4 < 5 = SUQ2 and NIE10 = 2 > 0 = SUQ10 , which implies NIE, NIS ⊁ SUQ; and (iii) NIE2 = 4 < 5 = LAR2 and NIE3 = 4 > 3 = LAR3 , which implies NIE, NIS ⊁ LAR; (iv) HAM1 = 5 < 6 = NIE1 and HAM2 = 5 > 4 = NIE2 , which implies HAM, HNH ⊁ NIE, NIS; and (v) LAR3 = 3 < 4 = SUQ3 and LAR10 = 2 > 0 = SUQ10 , which implies LAR ⊁ SUQ. Similar examples can be constructed to show, together with Example 5.1, that (i) ILQ cannot be ordered with respect to NIE, NIS, and HNH; (ii) NIE and NIS cannot be ordered with respect to ILQ, LAR, and SUQ; and (iii) HNH cannot be ordered with respect to NIS, SUQ, or LAR. One of the reasons that so many of the methods are incomparable is that under some methods, the advantage or disadvantage to candidates falls only on the strongest or weakest candidates, while in others it is spread more broadly. However, even by focusing on the allocations to C1 or Cn , the methods cannot be linearly ordered. Comparing the values of d1 for fixed D and v, we have ILQ1 , NIE1 ≥ LAR1 = SUQ1 ≥ HAM1 , and NIE1 ≥ NIS1 . Comparing the values of dn leads to SUQn ≤ LARn = LARn ≤ HAMn , and NIEn ≤ NISn . Neither relation provides a complete ranking. Overall, there is no single method that will always be best for the leading candidate. Both NIE1 and ILQ1 can be considerably greater than other X1 values, and these advantages increase with n. The ILQ method most often favors the leader: given a fixed D and n, ILQ1 ≥ NIE1 for most quota distributions, as occurred in Example 4.1 and Table 4.2. There are situations, however, in which the NIE method is better for C1 . This happens when v2 is close to v1 and the fractional parts of the quotas are less than but close to 0.5 as in the following example. Example 5.2 Let n = 10 and D = 100, and suppose the vote totals are as follows: v1 = 44600, v2 = 44400, and vi = 1400 for 2 ≤ i ≤ 10. Then the ILQ apportionment is (47, 45, 1, . . . , 1) while the NIE apportionment is (48, 44, 1, . . . , 1). Thus, there is no method that is consistently best for the strongest candidate. In contrast, the SUQ method always disadvantages the bottom candidate Cn more than any other method. Proposition 5.9 For any apportionment problem for which there are no ties among the last-place candidates, SUQn ≤ Xn where X is LAR, HAM, HNH, ILQ, NIE, or NIS.

5.3 Bias

97

Proof It suffices to show that SUQn ≤ NIEn . To see this, note that we can define SUQ as follows: Assign all candidates their upper quota, leading to an over allocation. Then remove as many delegates as needed from Cn ; if there are more delegates to be removed, remove them from Cn−1 , and so on. This is analogous to the NIE method except that the NIE method is based on nearest integer rounding which will always over-allocate fewer or the same number of delegates. Thus Cn will always receive ▢ fewer or the same number of delegates under SUQ than under NIE. The fact that the ordering induced by X1 is not the reverse of the ordering induced by Xn suggests a crucial difference between analyzing bias in delegate apportionment methods to previous analyses of bias: Notably, that bias toward stronger candidates is not necessarily linked to bias against weakest candidates. Thus, in the following sections, we consider the bias in allocations for C1 and Cn separately.

5.3.2 Delegate (Seat) Bias The notion of seat bias was first introduced by Schuster, Pukelsheim, Drton, and Draper (2003); they discussed seat biases for HAM and for the divisor methods of Webster and Jefferson. Drton and Schwingenschlögl (2005) calculate asymptotic seat biases for stationary divisor methods, and Pukelsheim (2017) provides formulas for exact seat biases for stationary divisor methods. (We consider divisor methods in Chap. 7.). In this section we examine the seat (or delegate) biases of each of the seven delegate apportionment methods used in the presidential primaries. Definition 5.13 For a fixed n and D, given an apportionment d, the delegate excess (or deviation from quota) of Ci is di − qi . Assuming the candidates are ordered from most support to least, the delegate bias of Ci is the expected delegate excess of this candidate assuming a uniform distribution over the simplex. The asymptotic delegate bias of Ci is the candidate’s delegate bias as D → ∞. Informally, a candidate’s delegate bias is a measure of how much that candidate’s apportionment deviates from their quota, on average, over the course of many elections, assuming that all quota vectors are equally likely. When n = 3, it is straightforward to calculate the asymptotic delegate bias for the delegate apportionment methods. As argued in the proof of Proposition 5.7, as D increases, the impact of the quota vectors near the boundary has less impact on the average apportionment. So the asymptotic bias can be determined by integrating each candidate’s deviation from quota over a typical interior region and dividing by the area of the region. See (Jones et al. 2020) for an example of this calculation. Determining exact delegate biases for specific D values is more complicated, and when n > 3, the geometry is intractable. Thus, we estimate delegate bias using simulations. Table 5.4 gives approximate delegate biases for the top two and bottom two candidates candidates under all seven delegate apportionment methods for n ∈ {3, 4, 5, 6} and D = 20, based on simulations of 1,000,000 runs. For the same n,

98

5 Properties of the Apportionment Methods Used in the Primaries

Table 5.4 Estimated delegate biases of the top two and bottom two candidates under each delegate apportionment method for D = 20 and n = 3, 4, 5, 6 HAM HNH ILQ LAR NIE NIS SUQ Ci n=3 1 2 3 n=4 1 2 3 4 n=5 1 2 4 5 n=6 1 2 5 6

0.01 0.01 −0.02

0.04 0.04 −0.09

0.98 −0.49 −0.49

0.50 −0.02 −0.48

0.13 0.00 −0.13

0.13 0.00 −0.13

0.50 0.36 −0.86

0.01 0.01 0.01 −0.03

0.05 0.05 0.04 −0.13

1.42 −0.47 −0.50 −0.50

0.50 0.31 −0.36 −0.45

0.18 0.00 −0.12 −0.17

0.18 0.00 0.00 −0.18

0.50 0.49 −0.05 −0.94

0.01 0.01 0.01 −0.05

0.05 0.05 0.01 −0.15

1.86 −0.44 −0.50 −0.42

0.50 0.45 −0.47 −0.42

0.22 0.00 −0.04 −0.19

0.22 0.00 0.00 −0.22

0.50 0.50 −0.60 −0.76

0.02 0.01 0.00 −0.07

0.05 0.05 −0.02 −0.16

2.28 −0.41 −0.50 −0.38

0.50 0.49 −0.49 −0.38

0.26 0.00 −0.07 −0.19

0.26 0.01 0.00 −0.27

0.50 0.50 −0.89 −0.56

Table 5.5 includes the asymptotic delegate bias as D → ∞, again, for the top two and bottom two candidates. For n = 3, the values are calculated directly and taken from (Jones et al. 2020); for n > 3, these are estimated using a D value of 2,000,000. As expected, the asymptotic delegate bias for each candidate under HAM is 0, as is the asymptotic delegate bias for the middle candidates under NIE and NIS where they receive [qi ] delegates. Methods such as LAR and SUQ which assign ]q1 ] delegates to C1 have asymptotic delegate bias equal to 0.5 for C1 for all n values. Likewise, methods such as LAR and ILQ which assign [qn ] delegates to Cn have asymptotic delegate bias equal to -0.5 for Cn for all n value. Generally, for all methods except HAM, the delegate bias decreases with candidate strength: The top two candidates have non-negative bias while the bottom two candidates have non-positive bias. The only exceptions to this rule are for C2 under LAR (when n = 3) and ILQ. Overall, ILQ is the most biased in favor of C1 and SUQ is the most biased against Cn , for all values of n, with the other methods falling somewhere in between. Interestingly, NIE, whose definition is noticeably favorable to the leader, has a delegate bias forΣC1 less than LAR for the n values shown, indicating the infrequency with which i [qi ] /= D. In fact not until n = 21 does the asymptotic delegate bias for C1 exceed 0.5. Similarly, the ILQ method, whose definition is somewhat favorable

5.3 Bias

99

Table 5.5 Asymptotic biases of the top two and bottom two candidates under each delegate apportionment method as D → ∞ for n = 3, 4, 5, 6 HAM HNH ILQ LAR NIE NIS SUQ Ci n=3 1 2 3 n=4 1 2 3 4 n=5 1 2 4 5 n=6 1 2 5 6

0 0 0

1/24 1/24 −1/12

1 −1/2 −1/2

1/2 0.00 −1/2

1/8 0 −1/8

1/8 0 −1/8

1/2 1/2 −1

0 0 0 0

0.04 0.04 0.04 −0.13

1.46 −0.46 −0.50 −1/2

1/2 0.33 −0.33 −1/2

0.17 0.00 0.00 −0.17

0.17 0.00 −1/8 −0.17

1/2 0.50 0.50 −1.50

0 0 0 0

0.04 0.04 0.04 −0.16

1.92 −0.42 −0.50 −1/2

1/2 0.46 −0.46 −1/2

0.20 0.00 0.00 −0.20

0.20 0.00 0.00 −0.20

1/2 0.50 0.50 −2.00

0 0 0 0

0.04 0.04 0.04 −0.20

2.36 −0.37 −0.50 −1/2

1/2 0.49 −0.49 −1/2

0.24 0.00 0.00 −0.23

0.22 0.00 0.00 −0.23

1/2 0.50 0.50 −2.50

Fractional values are exact, while decimal values are from simulations

to C2 , shows a negative delegate bias for C2 , indicating that most of the time, the second-strongest candidate does not receive any additional delegates beyond the [q2 ] delegates received in the first round of the ILQ allocation process. The bias for C2 is not positive until n = 11. Overall, the positive (resp. negative) bias of C1 (resp Cn ) increases with n under all methods that demonstrate bias. The most extreme examples are the asymptotic delegate bias for C1 under ILQ and for Cn under SUQ, which indicate that on average, these candidates receive a number of delegates that is significantly different from their quotas. This phenomenon is also observable in the delegate thresholds and behavior under close elections, examined in Sects. 5.3.3 and 5.3.4. In Sect. 5.3.5, we return to the question of quantifying bias by examining the overall bias exhibited by these apportionment methods when used within the context of the presidential primaries.

100

5 Properties of the Apportionment Methods Used in the Primaries

5.3.3 Delegate Thresholds The term threshold was first introduced in relation to apportioning representatives to the US House of Representatives. Rokkan (1968) defined the threshold of inclusion— the minimum percentage required to receive a single representative—and obtained explicit expressions for it under Hamilton’s, Jefferson’s, and Webster’s methods; Rae, Hanby, and Loosemore (1971) did the same for the threshold of exclusion—the minimum percentage required to be guaranteed a representative. Lipjhart and Gibberd (1977) generalized this work to thresholds for any number of seats, deriving values for Hamilton’s, Jefferson’s, and Webster’s methods. Jones and Wilson (2010) extended this to divisor methods with concave or convex rounding functions, including Hill– Huntington’s method. Because “threshold” has other meanings in the context of presidential primaries, as the minimum percentage of the vote for candidates to be eligible for delegates, we use the term delegate thresholds: The lower and upper delegate thresholds refer to the minimum and maximum percentage of the vote corresponding to a fixed number of delegates, respectively. For ease of notation, we define the lower and upper thresholds in terms of candidate C1 ; the definitions for the other candidates are analogous. For apportionment method F and fixed D, we define for each k ∈ [0, D], Ik = inf{v1 /V | there exists v2 . . . , vn such that d1 = k for some d ∈ F(D, v)} and Sk = sup{v1 /V | there exists v2 . . . , vn such that d1 = k for some d ∈ F(D, v)}. The interval [Ik , Sk ] determines the range of percentages p1 = v1 /V for which C1 may receive k delegates. If p1 < Ik , then C1 is guaranteed fewer than k delegates. If p1 > Sk , then C1 is guaranteed more than k delegates. (This depends in some cases on how ties are interpreted; we make this explicit in the following analysis.) If Ik ≤ p1 ≤ SK , then C1 may or may not receive k delegates depending on the vote distribution of all the candidates. Note that if p1 > Sk−1 , then C1 is guaranteed to receive at least k delegates. These definitions generalize the delegate threshold of inclusion (I1 ) and delegate threshold of exclusion (S0 ). Since the apportionment methods used in the presidential primaries (with the exception of HAM) are not symmetric, e.g., treating C1 differently than C2 , the values of Ik and Sk differ depending on which candidate is under consideration. For our discussion, we focus on delegate thresholds for the weakest and strongest candidates. We consider the values for C1 for large values of k and for Cn for small values of k. In the former case, Ik and Sk can be seen as measures of bias toward the strongest candidate: The smaller the values, the easier it is for C1 to receive a large number of delegates. In the latter case, Ik and Sk indicate bias toward the weakest candidate: The larger the values, the more difficult it is for Cn to receive delegates. Of particular

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101

Table 5.6 For D ≥ n ≥ 3 and k > D/n, the Ik and Sk values for the first-place candidate for the seven methods are given HAM LAR ILQ NIS HNH NIE SUQ Ik Sk

k−(n−1)/n a D k+(n−1)/n D

k−1 a D k D

k−(n−1) b D k D

k−(3/2) c D k+(1/2) D

k−(n−1)/2 d D k+(1/2) D

k−(n−1)/2 d k−1 a D D k+(1/2) k D D 2n(n − 1). c Valid for

a Valid for k > ]D/n] + 1. b Valid for k > ]D/n] + n + 1 and D > ]D/n] + 2 and n = 3. For n = 3, Ik = (k − 1)/D. d Valid for k > ]D/n] + ]n/2]

k>

interest are I[D/2]+1 and S[D/2] for C1 as they indicate the minimum percentages required by the leader to receive or be guaranteed a majority of the delegates. We start with the delegate threshold values for C1 , and consider values of k > D/n. For n = 3, we determine Ik and Sk by examining the geometry of the simplex from Fig. 4.1 and identifying the smallest and largest q1 values for given apportionment regions Rd and dividing by D (since the simplex indicates q values and not v/V values). For LAR and SUQ, Ik = (k − 1)/D and Sk = k/D. For NIE and NIS, Ik = (k − 1)/D and Sk = (k + 1/2)/D. For HAM, Ik = (k − 2/3)/D and Sk = k/D. For HNH, Ik = (k − 2/3)/D and Sk = (k + 1/2)/D. And for ILQ, Ik = (k − 2)/D and Sk = k/D. The next proposition generalizes these results for n > 3. To simplify the statement of the proposition and its proof, we make additional assumptions on the size of k relative to D and n. For example, in deriving the expression for Ik under NIE, we assume that k > ] Dn ] + ]n/2] since otherwise, C1 would not be first-ranked if p1 = I k . Proposition 5.10 Let D ≥ n ≥ 3 and k > D/n. The values of Ik and Sk for C1 for the seven delegate apportionment methods appear in Table 5.6. Proof The proof is divided into cases for the different apportionment methods. HAM: See (Rokkan 1968) and (Rae, Hanby, and Loosemore 1971). LAR and SUQ: Under both methods, C1 receives ]q1 ] delegates. Thus it is clear that Ik = (k − 1)/D and Sk = k/D. NIS: Under NIS, C1 receives either [q1 ] or [q1 ] + 1 delegates. For Ik , if q1 < k − (3/2), then [q1 ] ≤ k − 2 and so d1 ≤ k − 1. Thus, Ik ≤ (k − (3/2))/D. Next, supε , i = 2, . . . , n − 1 and qn = pose n > 3 and let q1 = k − 1 − ε, qi = k − 3/2 − n−2 ] = k − 1, [qi ] = k − 2 and [qn ] = D − (n − 1)(k − 1) + (n − 2)/2 + 2ε. Then [q 1 Σ D − (n − 1)(k − 1) + [(n − 1)/2]. So j [qj ] − D = D − [(n − 2)/2]. Thus, if n > 3, there is an under allocation, leading to d1 = k, which implies Ik = (k − (3/2))/D. For Sk , suppose q1 = k + (1/2). Then [q1 ] = k + 1 so C1 is guaranteed to receive more than k delegates. If q1 = k + (1/2) − ε, qi = k, i = 2, . . . , n − 1 and qn = D Σ− (n − 1)k − (1/2) + ε. Then [q1 ] = [qi ] = k and [qn ] = D − (n − 1)k + 1 so implies d1 = k. Thus, Sk = (k + (1/2))/D. j [qj ] = D + 1, which Σ HNH: RecallΣ that if j [qi ] ≤ D then HNH reduces to HAM, and so d1 = [qn ] or [q1 ] + 1. If j [qj ] > D, then HNH reduces to NIE, and so d1 ≥ [q1 ]. For Ik , note that if q1 = k − (n − 1)/n + ε, qi = k − 1 − (n − 1)/n i = 2, . . . , n − 1

102

5 Properties of the Apportionment Methods Used in the Primaries

and qn = D − (n − 1)(k − (n − 1)/n) + (nΣ − 2) − ε then [q1 ] = k − 1, qi = k − 2 and [qn ] = D − (n − 1)(k − 1) + n − 3 so j [qj ] = D − 1, which implies that, by HAM, d1 = k (since its remainder is largest). Alternatively, if q1 = k − (n − 1)/n − ε = k − 1 + 1/n − ε then [q1 ] = k − 1, and under neither HAM or NIE can it receive more delegates. Thus Ik = (k − (n − 1)/n)/D. The argument for Sk is identical to that for NIS. Σ delegates (if j [qj ] ≥ D) or [q1 ] + m NIE: Under Σ NIE, C1 receives either [q1 ]Σ delegates (if j [qj ] < D), where m = D − j [qj ]. For Ik , suppose q1 < k − (n − 1)/2. From the discussion on quota, if n is odd then d1 ≤ ]q1 ] + (n − 3)/2 ≤ (k − (n − 1)/2) + (n − 3)/2 ≤ k − 1. If n is even, d1 ≤ ]q1 ] + (n − 2)/2 < (k − n/2) + (n − 2)/2 ≤ k − 1. To show Ik = (k − (n − 1)/2)/D, suppose n is even. Let q1 = k − (n − 1)/2 + 2ε, qi = k − (n − 1)/2 − ε/(n − 2), i = 2, . . . , n − 1 and qn = D − (n − 1)(k − (n − 1)/2) − ε. Then [q1 ] = k − (nΣ− 2)/2, [qi ] = k − n/2 and [qn ] = D − (n − 1)(k − (n − 2)/2) + (n − 2)/2. So j [qj ] = D − (n − 2)/2, which implies d1 = k. If n is odd, let q1 = k − (n − 1)/2 + 2ε, qi = k − (n − 1)/2 − (1/2) − ε/(n − 2), i = 2, . . . , n − 1 and qn = D − (n − 1)(k − (n − 1)/2) + (n − 2)/2 − ε. Then [q1 ] = k − (n − 1)/2, [qi ] = Σk − (n − 1)/2 − 1 and [qn ] = D − (n − 1)(k − (n − 1)/2) + (n − 1)/2 − 1. So j [qj ] = D − (n − 1)/2, which implies d1 = k. The argument for Sk is exactly the same as for the NIS method. ILQ: Under ILQ, ]q1 ] ≤ d1 ≤ [q1 ] + (n − 1). Suppose q1 = k − (n − 1) − ε. Then [q1 ] = k − (n − 1) − 1, which implies d1 < k. This implies Ik ≥ (k − (n − 1))/D. Alternatively, suppose D > 2n(n − 1). Since k > D/n, D − k < D − (D/n) = (D/n)(n − 1). Let D − k = a(n − 1) + b for some integers 0 ≤ a < D/n and 0 ≤ b ≤ (n − 1). Let q1 = k − (n − 1) + ε, qi = a + 2 − ε/(n − 1), i = 2, . . . , b + 1 j = b + 2, . . . , n. Then [q1 ] = k − (n − 1), [qi ] = and qj = a + 1 − ε/(n − 1), Σ a + 1, [qj ] = a so that D − l [ql ] = D − (n − 1). Thus, in round 2, there are n − 1 remaining delegates. Since n−1 n−1 n−1 qi < (a + 2) < D D D

(

) D n − 1 2(n − 1) + < 1, +2 = n n D

this implies C1 receives the remaining delegates. So d1 = k. Thus, Ik = (k − (n − 1))/D. For Sk , note that C1 receives at least ]q1 ], so if q1 > k, d1 > k. If q1 = k and the ▢ other quotas are integers, then d1 = k. Thus, Sk = k/D. The results from Proposition 5.10 largely confirm the simulation results in Sect. 5.3.2. They suggest that ILQ is the most biased in favor of C1 because the Ik value is the smallest and the Sk value is tied for smallest with LAR and SUQ. (Indeed, ILQ allocates at least as many delegates to C1 as LAR and SUQ.) Nextmost biased is NIE, with NIS, SUQ, LAR, and HNH somewhat less biased. To make these comparisons clear, we compare Ik for ILQ and HAM. Suppose D = 100 and n = 10. Under ILQ, it is possible for C1 to have just over 41% of the vote and get a

5.3 Bias

103

Table 5.7 For D ≥ n ≥ 3, and k < D/n the Ik and Sk values for the last-place candidate for the seven methods are given HAM LAR ILQ NIS HNH NIE SUQ Ik Sk a Valid

k−(n−1)/n D k+(n−1)/n a D ≤ [ Dn ] − 1.

k−(1/2) b k−(n−1)/n k−(1/2) 2 k k D D D D D k+(3/2) c k+(n−1)/2 d k+(n−1)/2 d k+n−1 e k+1 a D D D D D for k ≤ [ Dn ] − 1. b Valid for k ≤ [ Dn ] − 2 and n > 3. For n = D e [ Dn ] − [ n−1 2 ]. Valid for k ≤ [ n ] − (n − 1)

k D k+1 a D b Valid

for k Sk = k/D. d Valid for k ≤

3,

majority of the delegates (k = 50); however under HAM, C1 must get 49.1% of the vote to possibly receive that majority. We return to these issues in Sect. 5.5. Next, we consider threshold values for Cn , which are defined analogously to those for C1 . For n = 3 and k < D/n we can again determine the delegate thresholds by considering the geometry of the simplex from Fig. 4.1. For LAR and ILQ, Ik = k/D and Sk = (k + 1)/D. For NIE and NIS, Ik = (k − (1/2))/D and Sk = (k + 1)/D. For SUQ, Ik = k/D and Sk = (k + 2)/D. For HAM, Ik = (k − 2/3)/D and Sk = (k + 2/3)/D. And for HNH, Ik = (k − 2/3)/D and Sk = (k + 1)/D. The Proposition 5.11 generalizes these results for n > 3. We omit the proof since it is similar to that of Proposition 5.10. Proposition 5.11 Let D ≥ n ≥ 3 and k < D/n. The values of Ik and Sk for Cn for the seven delegate apportionment methods appear in Table 5.7. The results from Proposition 5.11 again align with those in Sect. 5.3.2, suggesting that SUQ is most biased against Cn , compared to the other methods. The S0 and S1 values for SUQ are the greatest, and the I1 value is tied for highest with LAR and ILQ. Unlike LAR and ILQ, however, which guarantee Cn at least [qn ] delegates, the SUQ method frequently gives Cn fewer delegates. NIE and HNH can also be quite disadvantageous to the last-place candidate, as is illustrated by the relatively high S1 values. The smallest S1 and I1 values belong to HAM, the least biased method of the seven. To illustrate the differences between largest and smallest values of S1 for SUQ and HAM, let D = 100 and n = 10. Under SUQ it is possible for the last-place candidate to have a quota just under 9 and yet receive no delegates. By contrast under HAM, if the last-place candidate has a quota just over 0.9 then C10 is guaranteed a delegate. This comparison of delegate threshold values has focused on situations where C1 receives a large number of votes, and where Cn receives a small number of votes. In Sect. 5.3.4, we consider what happens the results of the election are very close.

5.3.4 Extreme Outcomes in Close Elections We gain additional insight into the behavior of the different apportionment methods by considering how delegates are allocated when all the candidates receive a similar

104

5 Properties of the Apportionment Methods Used in the Primaries

number of votes. From a geometric point of view, these occur when the quota lies very close to the barycenter of the simplex. To this end, we provide two measures of bias, the close-election delegate maximum and close-election delegate minimum, introduced in (Jones et al. 2020). To motivate why these notions shed light on a method’s bias, we introduce several examples where candidates receive very different allocations despite the closeness of the election. The first two examples demonstrate bias toward the top candidate under the ILQ and NIE methods; the third example demonstrates bias against the bottom candidate under the SUQ method. Example 5.3 Let n = 10, D = 40, q1 = 4.01, and qi = 35.99/9 ≈ 3.99 for 2 ≤ i ≤ 10. Under HAM, d = (4, 4, . . . , 4). However, under ILQ, d = (13, 3, . . . , 3). Even though C1 receives approximately as many votes as all other candidates, C1 receives more than triple the number of delegates, turning a vote share of 10.03% into a delegate share of 32.5%. Example 5.4 Let n = 12, D = 54, q1 = 4.51, and qi = 49.49/11 ≈ 4.499 for 2 ≤ i ≤ 12. Under NIE, the apportionment is (10, 4, . . . , 4). Thus C1 receives an apportionment more than double their quota, turning a vote share of 8.4% into a delegate share of 18.5%. Example 5.5 Let n = 6, D = 19, qi = 3.1670 for 1 ≤ i ≤ 4, q5 = 3.1666, and q6 = 3.1654. Under SUQ, the apportionment is (4, 4, 4, 4, 3, 0). Thus, despite a quota of more than 3, Cn receives no delegates. These three examples illustrate that under some methods, even when an election is extremely close the strongest or weakest candidate can receive a significantly different apportionment than the other candidates. This observation leads to the following definitions which are based on the apportionment regions Rd which partition the simplex under each apportionment method (see Fig. 4.1). Definition 5.14 For a fixed apportionment method F, and delegate size D, consider the set of apportionment regions Rd generated by F which divide up the simplex of size D. Suppose that the tie point (D/n, . . . , D/n) is contained in the interior or boundary of each of the regions Rd1 , . . . , Rdm . Define the close-election delegate j maximum as CEDM = max1≤j≤m {d1 }. Define the close-election delegate minimum j as cedm = min1≤j≤m {dn }. To determine the CEDM for each method, we consider the maximum number of delegates C1 receives when their proportion of the vote is just over 1/n. For all methods, this occurs when the remaining candidates have vote shares just less than 1/n. Likewise, to determine the cedm for each method, we consider the minimum number of delegates Cn receives when their proportion of the vote is just under 1/n and the remaining candidates have vote shares just greater than 1/n. These observations lead to the following proposition, which is proved in (Jones et al. 2020). Proposition 5.12 (Jones et al. 2020) Suppose D = nk + r for non-negative integers k and r ≤ n − 1. The CEDM and cedm values for the seven apportionment methods are given in Table 5.8.

5.3 Bias

105

Table 5.8 CEDM and cedm values for the seven delegate apportionment methods Method CEDM cedm SUQ

k +1

LAR

k +1 { k r=0 k +1 r >0 { k r=0 k +1 r >0 ⎧ ⎪ r=0 ⎨k k + 1 r > 0, r /= n/2 ⎪ ⎩k + 2 r = n/2 { k + r r ≤ [n/2] k + 1 r > [n/2] { k +n−1 r =0 k +r r>0

HAM HNH

NIS

NIE ILQ

max{k + 1 − (n − r), 0} { k −1 r =0 k r>0 k { k max{k + 1 − (n − r), 0} { k r /= n/2 k − 1 r = n/2 { k max{k + 1 − (n − r), 0} { k −1 r =0 k r>0

r < n/2 r ≥ n/2

r < n/2 r ≥ n/2

The results of Proposition 5.12 mirror the previous results with ILQ, NIE, and SUQ showing the greatest variation. Table 5.8 also highlights the difference between the allocations for C1 and Cn when the elections are very close. Under HAM and LAR, the strongest and weakest candidates’ delegate allocations differ by only 1 (except under LAR in the case when D is divisible by n). Similarly, under NIS, the strongest and weakest candidates’ delegate allocations differ by at most 2 except when r = [n/2]. Under SUQ and HNH, C1 is not much advantaged, however, Cn can receive up to [n/2] fewer delegates under HNH and up to n fewer delegates under SUQ. Under NIE, C1 can be advantaged while Cn is disadvantaged, with allocations that can differ by up to [n/2] as occurred in Example 5.4, where D = 54 and n = 12, r = 6 and d1 = 10 and d10 = 4. The largest differences between CEDM and cedm values arise under SUQ and ILQ when D is a multiple of n. This is illustrated in Example 5.3, where D = 40 and n = 10 and C1 received n = 10 more delegates than the other candidates. More broadly, Proposition 5.12 highlights how small changes in candidates’ vote totals can result in large changes to their allocations. This sensitivity to the vote distribution and to other factors such as the relationship between D and n is discussed further in Sect. 5.4.

5.3.5 Bias in Practice We finish our discussion of delegate basis by considering how bias manifests in the actual presidential primaries. In particular, we look at how our previous estimates for

106

5 Properties of the Apportionment Methods Used in the Primaries

Table 5.9 Average delegate bias for C1 across individual contests in recent Presidential primaries under each method HAM HNH ILQ LAR NIE NIS SUQ T = 0.15 Unmodified Quota Modified Quota T = 0.1 Unmodified Quota Modified Quota T = 0.05 Unmodified Quota Modified Quota

0.656

0.666

1.164

1.073

0.671

0.671

1.073

0.003

0.013

0.513

0.422

0.020

0.020

0.422

0.396

0.415

1.085

0.879

0.450

0.450

0.879

0.001

0.020

0.691

0.485

0.056

0.056

0.485

0.247

0.268

1.077

0.727

0.356

0.356

0.727

0.028

0.048

0.857

0.507

0.137

0.136

0.507

delegate bias in Table 5.4 change when we examine the bias that occurred in recent past elections. We focus on the leading candidate, looking at data from the 2004, 2008, 2016, and 2020 Democratic presidential primaries. Table 5.9 indicates the average delegate bias for C1 across all individual contests (district and statewide) with at least three candidates in the primaries. To clarify, the numbers in the T = 0.15 section are calculated by determining d115% − q1 and d115% − q115% , the difference between the number of delegates allocated to C1 using a 15% threshold and the unmodified or modified quota in each contest, and then averaging over all 2736 elections in the database. The corresponding numbers for the other thresholds are calculated similarly, except that the total number of elections is 2631, reflecting some incomplete data at the district level. Table 5.9 summarizes the results, for HAM, the actual method used in the Democratic primaries, and for the other methods for comparison. As expected, the delegate bias using the modified quota is much smaller than the delegate bias using the unmodified quota. Additionally, the delegate bias decreases with threshold levels, for all methods. Similar to the results in previous sections, these results suggest that the ILQ method is most biased in favor of the leading candidate, with SUQ and LAR close behind. Under all three methods, C1 receives on average one additional delegate in each individual contest when a 15% threshold is used; the delegate bias under the remaining methods averages about two-thirds of a delegate. As the threshold decreases, the delegate biases all decrease, where the delegate bias for ILQ is the only one to exceed 1 delegate when T is reduced to 5%. To get a better sense of how the strongest candidate is advantaged in the actual primaries, we compute the same bias, this time considering each state’s primary as a

5.3 Bias

107

Table 5.10 Average seat bias for C1 across 183 Democratic state primaries in 2004, 2008, 2016, and 2020 under each method Delegate bias Method HAM HNH ILQ LAR NIE NIS SUQ

5.96 6.01 10.59 10.21 6.05 6.05 10.26

single election, making a total of 183 contests over the 2004, 2008, 2016, and 2020 Democratic presidential primaries. The results are shown in Table 5.10, which records the average of d1 − q1 where d1 is the total number of state delegates awarded to C1 and q1 is the unmodified quota based on the statewide vote. The first line shows the actual bias under HAM; the remaining lines show the corresponding hypothetical bias under other methods. Thus, on average in these state contests, the leading candidate received approximately 6 more delegates than their state vote percentage suggests they should receive. Had ILQ been used, this number would be increased to roughly 10.6. The fact that the numbers in Table 5.9 are so much smaller than those in Table 5.10 suggest that the delegate bias within individual contests is not that great: The larger numbers in Table 5.10 are reflective of the distortion created by allocating delegates in different categories separately (different contests in each district, as well as, in the Democratic primary, both PLEO and at-large statewide contests) and then aggregating the results to obtain a total number of state delegates. We study the effects of aggregation more fully in Chap. 6. Note that the values in Tables 5.9 and 5.10 do not provide estimates for the delegate bias held by any single candidate, as the identity of the leading candidate varies, both within a primary season and across different primaries. To determine how delegate bias can impact an individual, we analyze the average delegate bias for the eventual Democratic nominee during the 2008 and 2020 primaries. Neither Barack Obama nor Joe Biden, the individual winners, consistently received the most first place votes in the individual state primaries in which they participated, hence this calculation involves averaging delegate deviations of different candidates (C1 , C2 , etc.), depending on how well each individual did in the state. In the 2008 Democratic primary, comparing Obama’s unmodified quota with the total number of delegates he was awarded in each state contest, we see that he cumulatively received approximately 32 more delegates than his quota suggests he should, resulting in an average delegate bias of 32/56 = 0.57. By contrast, Biden received 489 more delegates in excess of his quota, resulting in an average delegate bias of 489/58 = 8.43. The smaller number in 2008 is likely reflective of the fact that

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the 2008 primary was essentially a two-candidate contest between Barack Obama and Hillary Clinton, ensuring that both candidates received delegates counts close to their quota. In 2020, however, there were a large number of active candidates early in the primary season, resulting in more distortions in the apportionment. We return to the question of how the strongest candidate fared in the most recent Democratic primaries in Sect. 5.7, where we compare vote shares to seat shares.

5.4 Sensitivity to Small Perturbations in Vote Totals The results from Sects. 5.3.3 and 5.3.4 suggest that there is a relationship between the bias of a method and the method’s sensitivity to candidates’ vote totals and to the relationship between D and n. A biased method may have the property that small perturbations in the vote totals lead to a large effect on the apportionment. In the political science literature, indices of volatility are used to measure changes in the outcome of two elections over time. In that case, small changes in voting patterns may dramatically alter the outcome of party representation or district representation. In this section, we consider the change in delegate allocation created by small perturbations of the vote distributions. Of course, it is not surprising if these changes result in a change to the ordering of the candidates: If a small relative increase in the votes for the second-place candidate, for instance causes C2 to overtake C1 , then, for a strongly a biased method like ILQ, we expect this change of vote leader to have a sizable effect on the apportionment. However, for some of the methods used in the primaries, small changes in vote totals can lead to large changes in the apportionment, even when the candidate ordering does not change. This subsection explores such outcomes. These outcomes occur because the methods are biased and also quota-based. We motivate our discussion with two examples. The first is based on the proof that under ILQ, Ik = (k − (n − 1))/D for the leading candidate. If p1 < Ik , then C1 is awarded at most k − (n − 1) delegates; if p1 ≥ Ik , then C1 may receive k delegates. Example 5.6 Let n = 5, D = 42 and v = (1096, 801, 801, 801, 701) so that q = (10.96, 8.01, 8.01, 8.01, 7.01). Under the ILQ method, the resulting apportionment is (11, 8, 8, 8, 7). However, if 2 supporters of each of the weaker candidates transfer their support to C1 , then v, = (1104, 799, 799, 799, 699) and q, = (11.04, 7.99, 7.99, 7.99, 6.99). Using the ILQ method on the perturbed problem yields an apportionment of (15, 7, 7, 7, 6). Thus, a shift of 8 voters out of a total of 4200 causes the leading candidate to earn an additional 4 delegates. Example 5.7 Let n = 3, D = 12 and v = (5999, 3999, 2002) so that q = (5.999, 3.999, 2.002) Under the SUQ method, the resulting apportionment is (6, 4, 2). However, if 4 supporters of C3 transfer their support, 2 each to C1 and C2 , so that v, = (6001, 4001, 1998) and q = (6.001, 4.001, 1.998), then the resulting SUQ apportionment is (7, 5, 0) and C3 loses all of their delegates.

5.4 Sensitivity to Small Perturbations in Vote Totals

109

In both examples, a sizable change in the apportionment occurs despite the change in vote totals being very small. Of course, under any reasonable method it is impossible (and undesirable) for apportionments never to change under small perturbations of vote totals: The portions of the simplex near the boundaries of apportionment regions will always be susceptible to an apportionment change if vote totals are slightly perturbed in the right direction. However, the sensitivity to vote totals displayed in the above two examples do seem undesirable because of the large change in apportionment, and such outcomes can become much more extreme as n increases. We can see this occurring in Example 5.7 by examining the geometry of the simplex. In Fig. 4.1, under SUQ, the boundary of the apportionment region corresponding to (6, 4, 2) has non-empty intersection with the boundary of the region corresponding to (7, 5, 0). In contrast, an examination of the HAM simplex in Fig. 4.1 shows that under HAM small perturbations in vote totals causes at most one delegate to be gained or lost by a given candidate when n = 3. This is because the regions of two apportionment points (d1 , d2 , d3 ) and (d1, , d2, , d3, ) have a non-empty boundary intersection only if |di − di, | ≤ 1 for all i. An analysis of the methods’ geometry suggests an approach for detecting potential sensitivity to vote totals. We say that two apportionment points d and d, are within a small perturbation of vote totals if the boundary of their apportionment regions have non-empty intersection, and if both points respect the same ordering of the candidates. Some apportionment methods have apportionment points that are within a small perturbation of vote totals, even though in moving from one point to another a candidate may gain or lose many delegates. This leads to the following proposition whose proof is omitted because it is similar to the delegate threshold proofs from Sect. 5.3.3. Proposition 5.13 • Let D > n(n − 1). Under the SUQ method, there exist apportionment points d and d, such that the points are within a small perturbation of vote totals and |dn − dn, | = n − 1. Furthermore, as we vary over such points d and d, , n − 1 is the maximum value of |dn − dn, |. • Let D > 1.5n. Under the NIE method, there exist apportionment points d and d, such that the points are within a small perturbation of vote totals and |d1 − d1, | = [n/2]. Furthermore, as we vary over such points d and d, , [n/2] is the maximum value of |d1 − d1, |. • Under the ILQ method, there exist apportionment points d and d, such that the points are within a small perturbation of vote totals and |d1 − d1, | = n − 1. Furthermore, as we vary over such points d and d, , n − 1 is the maximum value of |d1 − d1, |. The large swings in apportionment identified in Proposition 5.13 reflect not only of the methods’ bias, but, also the fact they are quota based. An implication of Proposition 5.13 is that, for a biased, quota-based method of apportionment, a small number of votes can cause a dramatic swing in the resulting apportionment, even

110

5 Properties of the Apportionment Methods Used in the Primaries

when those votes do not change the ordering of the candidates, as demonstrated in Examples 5.6 and 5.7. Like these examples, the next example demonstrates how a similar shift in quota determines whether C1 receives all the delegates or 60% of the delegates. Example 5.8 Suppose n = 5, D = 10 and q = (5.996, 1.001, 1.001, 1.001, 1.001). Perturbing the quotas slightly to q, = (6.004, 0.999, 0.999, 0.999, 0.999) leads to a change in apportionment under ILQ from d = (6, 1, 1, 1, 1) to d, = (10, 0, 0, 0, 0). It seems undesirable for the apportionment of delegates to depend so heavily on accidents of nearest integer rounding or the exact value of quotas, suggesting that methods less susceptible to this kind of behavior might be more appropriate. The results in Sect. 5.3.4 also make clear that the relationship between D and n has an outsized impact on the allocation of delegates when the candidates are closely tied. Because the values of n and D are determined separately (the value of n is not known at the start of the primary season, and in fact, changes throughout the calendar), it seems all the more important to select a method that is not sensitive in this way. We return to this question in Chap. 7.

5.4.1 Sensitivity to Vote Totals in Practice As noted previously, when the apportionment methods are used to allocate delegates in the primaries they are often used in conjunction with thresholds. They are also applied both at the statewide and districts levels, so that the overall apportionment for the state primary is aggregated across several different vote distributions. Either of these factors can compound the issue of sensitivity to small perturbations in vote totals. Indeed, a threshold is almost designed to do so: If a candidate’s vote share is near the threshold, then a small change in vote totals can place that candidate on the other side of the threshold, causing a large gain or loss of delegates for that candidate (and causing a large change in the overall apportionment). To see how this has affected an actual primary, consider Table 5.11. The top half shows the results of the 2020 Rhode Island Democratic primary in which Bernie Sanders received approximately 14.8% of the total vote and yet received only 1 delegate. The bottom table shows a primary election in which the vote totals have been altered slightly. In this hypothetical election, Sanders receives 15.2% of the statewide vote and receives 6 delegates, a six-fold increase. Admittedly, Sanders performs better in the hypothetical election than in the original election, but from a global perspective his two election performances are essentially the same; it’s not clear why increasing his overall vote share by 0.4% points should cause such a dramatic shift in the apportionment.

5.5 Majority and Leader Criteria

111

Table 5.11 Results of the the 2020 RI Democratic Primary (top) and the results of a hypothetical election that is a small perturbation of the original election (bottom) Candidate CD 1 CD 2 PLEO At-Large Votes Appt Votes Appt Votes Appt Votes Appt 2020 RI democratic primary Biden 40,455 9 39,273 8 Sanders 7730 − 7795 1 Others 4881 − 4435 − Perturbation of the 2020 RI democratic primary Biden 40,000 7 39,200 7 Sanders 8010 2 7900 2 Others 4881 − 4435 −

79,728 15,525 9316

3 − −

79,728 15,525 9316

5 − −

79,200 15,910 9316

2 1 −

79,200 15,910 9316

4 1 −

5.5 Majority and Leader Criteria Apportionment methods used to allocate delegates or seats in a parliamentary system are often chosen (or constructed) to be more biased toward the top candidate(s) than methods designed for allocating house seats in proportion to state population. Therefore, there are a number of criteria that may be desirable for delegate apportionment but that are unimportant or irrelevant in the classical house seat context. In this section, we explore two such criteria. The first, the majority criterion, was developed in Schwingenschlögl (2007); the second, the leader criterion, is new, as far as we are aware. In Sect. 5.6 we explore a third characteristic: the propensity of apportionment methods to reward candidate coalitions. Definition 5.15 (Schwingenschlögl 2007) An apportionment method is said to satisfy the majority criterion (MAJ) if whenever a candidate receives a majority of the votes, that candidate receives a majority of the delegates. Schwingenschlögl (2007) studies apportionment in the parliamentary context, in which it is (arguably) important that when a party receives a majority of votes, it also receives a majority of legislative seats. In our view, the majority criterion is similarly important in the delegate context. Definition 5.16 An apportionment method is said to satisfy the leader criterion (LEAD) if the candidate who receives the most votes receives more delegates than any other candidate, assuming a unique vote leader exists. As discussed in Sect. 5.1, all delegate apportionment methods are order-preserving, so that vi > vj implies di ≥ dj . LEAD is a stronger criterion, requiring the candidate with a plurality of votes receive a strict plurality of delegates. The LEAD criterion may sometimes be in conflict with the concept of proportionality, as when, D = 6 and v = (101, 100, 199). Here, clearly the “fairest” apportionment would be

112

5 Properties of the Apportionment Methods Used in the Primaries

d = (2, 2, 2). Thus, we would not expect or want apportionment methods to satisfy LEAD in house seat apportionment. In delegate apportionment, however, where methods are designed to be biased, LEAD may be important in maintaining the sense that the “winner” of a primary “wins the delegate count.” And even for the methods that do not satisfy LEAD, it may be useful to compare the circumstances and frequency with which they fail this criterion. Violations of the majority criterion can also occur using divisor methods, where the application of divisor methods to allocate party seats has been well-studied. Jefferson’s method is the only divisor method that satisfies MAJ, and that is only for odd n. In practice, apportionment methods are often supplemented with additional rules to avoid failures of MAJ. A number of these are documented in (Pukelsheim 2017) and include adding an additional number of seats for the strongest candidate until they achieve a majority, or initially assigning the strongest party a bare majority of the seats and then applying the apportionment method to allocate the remaining seats to the other parties. We return to this question in Chap. 7.

5.5.1 Majority Criterion Any method which gives C1 at least their upper quota satisfies MAJ, and thus ILQ, LAR, and SUQ all have the property that a candidate receiving a majority of the vote receives a majority of the delegates. The other methods do not satisfy MAJ, with the exception of NIS, NIE, and HNH when D is odd. It is easy to see the failure of this criterion geometrically when n = 3. Figure 5.2 illustrates where the majority criterion fails under NIE for D = 12. 4, 4, 4

12, 0, 0

NIE method

6, 6, 0

Fig. 5.2 Shaded areas represent outcomes in which a majority candidate does not receive a majority of the delegates under NIE when n = 3 and D = 12

5.5 Majority and Leader Criteria

113

Table 5.12 Values of I[D/2]+1 and S[D/2] for the leading candidate for the seven methods when n≥3 HAM

LAR

ILQ

NIS

HNH

NIE

SUQ

I[D/2]+1

[D/2]+1/n D

[D/2] D

[D/2]−(n−2) a D

[D/2]−1/2 b D

[D/2]+1/n D

[D/2]+1−(n−1)/2 D

[D/2] D

S[D/2]

[D/2]+(n−1)/n D

[D/2] D

[D/2] D

[D/2]+1/2 D

[D/2]+1/2 D

[D/2]+1/2 D

[D/2] D

formula for Ik under ILQ is valid for D > 2n(n − 1). b The formula for Ik under NIS is valid only for n > 3. For n = 3, I[D/2]+1 = [D/2]/D

a The

Table 5.13 Probability of a majority criterion violation when n = 3 HAM HNH PMAJ (D), D even PMAJ (D), D odd

6D − 4 3D2 D−1 6D2

NIE/NIS

11D − 6 6D2

3D − 2 2D2

0

0

For n > 3, failures of MAJ can be identified by determining the I[D/2]+1 and S[D/2] values for the leading candidate. (Recall that if p1 < I[D/2]+1 , C1 will receive fewer than [D/2] + 1 delegates; if p1 > S[D/2] , C1 is guaranteed to receive at least [D/2] + 1 delegates.) These are shown in Table 5.12, adapted from the threshold formulas in Table 5.6. From Table 5.12, we see that there are differences depending on the parity of D. If D is even, C1 is guaranteed to receive a majority of the delegates if p1 > 1/2 (under LAR, SUQ, and ILQ), p1 ≥ 1/2 + 1/(2D) (under NIS, NIE and HNH) and p1 ≥ 1/2 + (n − 1)/D (under HAM). If D is odd, C1 is guaranteed to receive a majority of the delegates if p1 ≥ 1/2 − 1/(2D) (under LAR, SUQ, and ILQ), p1 ≥ 1/2 (under NIS, NIE, and HNH), and p1 ≥ 1/2 + (n − 1)D/n under HAM. Thus, LAR, SUQ, and ILQ are the only methods that satisfy MAJ for all values of D, while NIS, NIE, and HNH only satisfy MAJ if D is odd. We can estimate how often failures of MAJ occur. For a fixed apportionment method, let PMAJ (D, n) be the probability that a violation of the majority criterion occurs, given that a majority candidate exists. If n = 3, we can use the geometry of the simplex. Proposition 5.14 Assuming a uniform distribution across the simplex, the probabilities PMAJ (D, 3) are as displayed in Table 5.13. For any n, if D is odd then HAM is the only method that violates the majority criterion. We note that the probabilities given for HAM in Table 5.13 do not match those listed in Schwingenschlögl (2007) because the latter are not conditioned on the existence of a majority candidate. Proof These statements can be proved through simple geometrical calculations based on Fig. 5.2. We illustrate for the NIE/NIS method and for D even. (As noted above, if D is odd, NIE/NIS satisfies the majority criterion.) Assume C1 receives a

114

5 Properties of the Apportionment Methods Used in the Primaries

Table 5.14 Estimated probability of a majority criterion violation when n ∈ {4, 5, 6} and D ∈ {10, 20}, assuming a majority candidate exists n HAM HNH NIE NIS D = 10 4 5 6 D = 20 4 5 6

0.262 0.330 0.386

0.247 0.311 0.370

0.180 0.210 0.227

0.181 0.209 0.227

0.140 0.180 0.220

0.130 0.169 0.207

0.098 0.119 0.137

0.098 0.119 0.138

majority of votes and suppose that D ≡ 0 mod 4. Figure 5.2 shows that failures of monotonicity occur in two different kinds of regions: interior and boundary. Consider an interior region corresponding to an apportionment of (d1 , d2 , d3 ). We make a change of coordinates by transforming points (a, b, c) on the simplex to points √ (x, y) by x = (2b + c)/D and y = 3c/D. (This transformation maps a simplex of size D to a simplex of size 2 centered at (D, 0, 0). ) Using this transformation, the vertices of the interior region, starting at the bottom right and going clockwise, are given by ) ( ) √ √ 2d2 + d3 + 1/2) 3(d3 − 1/2) 2d2 + d3 − 1/2 3(d3 − 1/2) , , , D D D D ( ) ( ) √ √ 2d2 + d3 − 1 3(d3 ) 2d2 + d3 − 1/2 3(d3 + 1/2) , and , . D D D D

(

√ Each of these regions has area 3 3/(4D2 ), and there are a total of (D/4 − 1) of them. √ be calculated similarly, summing √ The area of the of the other two regions can to 5 3/(8D2 ). Thus, the total area is (3D − 2) 3/(8D2 ). Dividing by the area of one-eighth √ of the simplex—the region in which C1 has a majority of the vote which has area 3/8—gives the desired result. The proof when D ≡ 2 mod 4 is similar, except that in this case there are (D + 2)/4 − 1 interior regions and only one boundary region along the bottom edge of the ▢ simplex. Proposition 5.14 shows that if n = 3, PMAJ (D, n) is greatest under HAM, followed by HNH and then NIE/NIS. This makes sense since HAM is the only method that is relatively unbiased. In addition, PMAJ (D, n) → 0 as D → ∞ under all methods. These observations also hold when n > 3. In Table 5.14, we estimate the frequency of MAJ violations for n ∈ {4, 5, 6} and D ∈ {10, 20}.

5.5 Majority and Leader Criteria

115

4, 4, 4

12, 0, 0

SUQ method

6, 6, 0

Fig. 5.3 Shaded areas represent outcomes in which a the leading candidate does not receive the most delegates under SUQ when n = 3 and D = 12

These results confirm that PMAJ (D, n) is greatest for HAM regardless of the n value, and that the probability of a MAJ violation decreases as D increases. In fact, it is easy to see, using a geometric argument, that limD→∞ PMAJ (D, n) = 0 under all methods. This is because the hyperplane q1 = D/2 along which violations of MAJ occur is of dimension n − 2, and thus of decreasing significance as D → ∞ and the simplex grows larger. Finally, for each method, PMAJ (D, n) increases with n, since as the number of candidates increase, so do the opportunities for close elections.

5.5.2 Leader Criterion As discussed previously, satisfying the LEAD criterion may not be always be desirable, particularly it is conflicts with the goals of proportionality. Notably, the only apportionment method used in the primaries that satisfies this criterion is ILQ, which is the most biased in favor of C1 . Any method for which d1 always satisfies quota will violate LEAD frequently. We illustrate this with the following example. Example 5.9 Suppose n = 3, D = 8, and v = (28, 600, 22, 000, 7900). Then, q = (3.91, 3.01, 1.08). When the apportionment is (4, 4, 0), such as under SUQ, then C1 receives 30% more votes than C2 and wins by 11% points (49% versus 38%), yet the two candidates receive the same number of delegates. Figure 5.3 illustrates where the leader criterion fails under SUQ when n = 3 and D = 12. We can determine the frequency that violations of the LEAD criterion occur, as we did for failures of MAJ using the geometry of the simplex for n = 3 and simulation

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5 Properties of the Apportionment Methods Used in the Primaries

Table 5.15 Probability of a leader criterion violation when n = 3 and D ≡ 0 mod 6 HAM HNH LAR NIE/NIS SUQ (4D − 3) (4D2 ) The probabilities for other D values are similar PLEAD (D, 3)

(D − 1)/D2

1/(2D)

(7D − 6) (8D2 )

1/D

Table 5.16 Probability of a leader criterion violation when n ∈ {4, 5, 6} and D ∈ {10, 20} D = 10

D = 20

n

HAM

HNH

LAR

NIE

NIS

SUQ

n

HAM

HNH

LAR

NIE

NIS

SUQ

3

0.090

0.095

0.060

0.083

0.083

0.100

3

0.052

0.053

0.025

0.046

0.046

0.048

4

0.139

0.137

0.116

0.111

0.112

0.144

4

0.072

0.072

0.060

0.059

0.060

0.073

5

0.187

0.187

0.164

0.143

0.144

0.176

5

0.095

0.095

0.090

0.074

0.075

0.096

6

0.214

0.218

0.218

0.158

0.160

0.222

6

0.117

0.117

0.116

0.088

0.089

0.117

for n > 3. For a fixed apportionment method, let PLEAD (D, n) be the probability of a LEAD violation, assuming a uniform distribution on the simplex. Since the regions in the simplex where the criterion is not met occur near the line v1 = v2 , the value of PLEAD (D, n) is highly dependent on the relationship between D and n. (The geometry of the simplex varies near the tie lines based on the value of D mod n). Thus, we consider only one case; the other cases are similar. Proposition 5.15 Assuming a uniform distribution across the simplex, the probabilities PLEAD (D, 3) are displayed in Table 5.15 for D ≡ 0 mod 6. We omit the proof as it is similar to the proof of Proposition 5.14. As Proposition 5.15 indicates, all methods except ILQ violate the LEAD criterion. There is not the kind of even/odd dependency that was observed with the majority criterion. SUQ violates LEAD most frequently; LAR violates LEAD least frequently. Similarly to the frequency of MAJ failures, PLEAD (D, n) → 0 as D → ∞. These observations are consistent with the results obtained by simulations as shown in Table 5.16 which contains estimates of PLEAD (D, n) for other values of D and n. These values corroborate the fact that SUQ violates LEAD most frequently and LAR least frequently. As with PMAJ (D, n), the probability that the LEAD criterion is not met decreases in D and increases in n.

5.5.3 Majority and Leader Criteria in Practice Using the DEM_DATA and REP_DATA databases, we can compare the frequency of MAJ and LEAD violations estimated in Sects. 5.5.1 and 5.5.2 to how frequently violations have actually occurred in the most recent presidential primaries. As with the discussion of how bias is manifest in the primaries, it is useful to analyze these

5.5 Majority and Leader Criteria

117

Table 5.17 Number of majority criterion violations in recent Democratic primaries under different methods, with and without thresholds HAM HNH NIE NIS Without T = 0.15 282 With T = 0.15 237

280 233

202 232

202 232

criteria both at the level of individual contests (allocating a single group of district or statewide delegate) and aggregated at the state primary level. Among the former, there are significant differences between the Republican and Democratic elections. We start with the majority criterion. On the Republican side, there is only one instance of a majority criterion violation among the 749 individual contests allocating district or statewide delegates in the REP_DATA database. The one instance was from the 2012 Nevada Republican caucuses, where Mitt Romney received a majority of the statewide vote but received 14 of the 28 total delegates available. These caucuses used a proportional allocation method to apportion all 28 delegates in a single statewide calculation.2 Romney’s bare majority of the vote share, at 50.01%, was not large enough to earn a majority of the delegates in this case. There are three main reasons for the scarcity of examples in Republican primaries. First, when Republican state parties allocate delegates in congressional districts, the number of district delegates available is almost always three and it is harder to observe violations when D is odd. Second, Republican state parties often do not allocate delegates at the district level, creating fewer opportunities to observe a violation. Third, Republican state parties often use methods such as winner-take-all or methods that are non-mathematical, eliminating the possibility of observing a violation from a proportional allocation method. The situation is very different in the Democratic primaries. Of the 1988 contests in the database DEM_DATA with three or more candidates, there were 237 instances of a majority criterion violation. These violations often occurred at the district level when there were only four or six delegates available; there were no instances of a violation for D odd. To see how this number would be affected by changes to the primary process, we also investigated the number of violations that would have occurred if other apportionment methods were used, or if the threshold were removed. The results are shown in Table 5.17. (Note that the numbers without threshold are based on a total of 1988 contests investigated, due to limitations in how the data was reported.) With the threshold, the number of MAJ violations when different methods are used is quite similar, implying that when a 15% threshold is in place, the choice of apportionment method makes little difference for the frequency of majority criterion violations.

2

The method itself was described too vaguely to categorize as one of the seven apportionment methods we study. The method is based on nearest integer rounding, but there is no indication how to adjust the apportionment if the rounding does not yield a legitimate apportionment.

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5 Properties of the Apportionment Methods Used in the Primaries

Without the threshold, the frequency of MAJ violations for HAM and HNH increases. This makes sense intuitively because the use of a threshold should advantage the stronger candidates, making it easier for a majority candidate to obtain a majority of delegates. It is interesting, therefore, that the frequency of MAJ violations under NIE and NIS decreases without the threshold. This decrease is partly due to the smaller number of elections, but also to a phenomenon known as an elimination paradox, where applying a threshold can sometimes cause the delegate count of one of the remaining candidates to decrease. In fact, it is not uncommon under the NIE and NIS methods for a majority candidate to receive a majority of delegates when there is no threshold, yet not receive a majority of delegates when there is a 15% threshold. We discuss the elimination paradox further in Chap. 6. At the overall level of a state primary, there were only four violations of the majority criterion among either Republican or Democratic primaries. These were the following. • In the 2008 Guam Democratic primary, Barack Obama received a slight majority of the vote but received two of the four delegates available in the primary. • In the 2016 South Dakota Democratic primary, Hillary Clinton received a slight majority of the vote but received 10 of the 20 pledged delegates available in the primary. (This election had only two candidates listed on the ballot.) • In the 2016 Wyoming Democratic caucuses, Bernie Sanders received approximately 57% of the statewide vote, which translated into 55.71% of the state convention delegates (these caucuses based the apportionment on each candidate’s number of delegates to the state party convention). Sanders received 7 of the 14 pledged delegates available in the caucuses. • In the 2012 Nevada Republican caucuses, Mitt Romney received a majority of the statewide vote but received 14 of the 28 total delegates available as discussed previously. The small number of violations indicates that when the results of individual allocations at district levels are aggregated, a majority candidate almost always receives the majority of the state delegates. There have been a greater number of violations of the leader criterion in recent primaries, as to be expected. As with the majority criterion, these were primarily on the Democratic side, and for many of the same reasons. On the level of individual contests, there were 19 violations in the REP_DATA database, of which 18 occurred in 2016, reflecting the shift of the state Republican parties toward methods that are more proportional. The number of leader criterion violations in the DEM_DATA database, with and without thresholds, is given in Table 5.18. Again, these numbers are based on a total of 1988 contests with the threshold and 1882 without. For comparison to HAM, the number of hypothetical leader criterion violations is also given. Table 5.18 indicates that although LAR and SUQ do not always satisfy LEAD as they do MAJ, the frequency of LEAD violations is considerably smaller under LAR and SUQ than under the other methods, by a factor of at least 2. This is due to the

5.6 Support for Candidate Coalitions

119

Table 5.18 Number of leader criterion violations in recent Democratic primaries under different methods, with and without thresholds HAM HNH LAR NIE NIS SUQ Without T = 0.15 With T = 0.15

378 356

382 354

160 33

275 349

283 349

165 58

fact that in most contests, the leading candidate is sufficiently ahead of their nearest rival so that ]q1 ] > ]q2 ]. At the overall level of a state primary, there were six instances of a violation of the leader criterion where the candidate with the largest number of votes statewide did not win the most delegates. These are described below. • In the 2008 Missouri Democratic primary, Obama received the most votes with 406,917 and Clinton received the second most votes with 395,185, but they both received 36 delegates from the primary. • In the 2008 New Hampshire Democratic primary, Clinton received the most votes with 112,404 and Obama received the second most votes with 104,815, but they both received 9 delegates from the primary. • In the 2020 New Hampshire Democratic primary, Sanders received the most votes with 76,384 and Buttigieg received the second most votes with 72,454, but they both received 9 delegates from the primary. • In the 2012 Alaska Republican primary, Romney received the most votes with 4554 and Santorum received the second most with 4254, but they both received 8 delegates from the primary. • In the 2016 Louisiana Republican primary, Trump received the most votes with 124,854 and Cruz received the second most votes with 113,968, but they both received 18 delegates from the primary. • In the 2016 Vermont Republican primary, Trump received the most votes with 19,974 and Kasich received the second most with 18,534, but they both received 8 delegates from the primary. These numbers are relatively small. However when violations occur, particularly in the early part of the primary season, they can have an outsized impact on the way that the primary results are communicated in the media, affecting who is considered the “leader” and directly affecting decisions about campaign fundraising and strategy.

5.6 Support for Candidate Coalitions As the primary season progresses and candidates withdraw from the nomination process, they sometimes endorse another candidate. A natural question to ask, therefore, is whether an apportionment method supports or impedes this process by correspondingly increasing the number of delegates allocated to the remaining candidate. This

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5 Properties of the Apportionment Methods Used in the Primaries

idea is important in party representation where different political parties frequently form alliances to run under the same platform. In this context, an apportionment method encourages coalitions if the combined party receives at least as many seats as the sum of the parties individually. An apportionment method encourages schisms if the combined party receives at most as many seats as the sum of the parties individually. Balinski and Young (2001) proved that among divisor methods, Jefferson’s method uniquely encouraged coalitions while Adams’ method uniquely encouraged schisms. They also proved that Webster’s method is the unique coalition-neutral divisor method. A similar situation occurs in the presidential primaries, where it is useful to know whether a candidate, by leaving the race and voicing their support to another candidate (in effect, forming a candidate coalition), is actually ensuring the other candidate receive their delegates in addition to those they would have received on their own. Otherwise, a candidate might choose not to endorse another candidate until the national party convention in hopes that, if no candidate receives a majority of delegates in the first round of voting, the first candidate’s delegates will have a bigger effect in the subsequent rounds of voting. We formalize the notion of two candidates forming a coalition, and its impact on the apportionment, as follows. Given D, n and an apportionment method F such that F(D, v) = d, if candidates Ci and Cj join together to create a “new” candidate Cα with vote total v˜ α = vi + vj with adjusted vote vector v˜ for n − 1 candidates, then let ˜ F(D, v˜ ) = d. Definition 5.17 An apportionment method F encourages candidate coalitions if for all D and v, if any two candidates Ci and Cj join together to create candidate Cα then ˜ An d˜ α ≥ di + dj where d˜ α is the apportionment for Cα according to F(D, v˜ ) = d. apportionment method encourages schisms if under the same circumstances d˜ α ≤ di + dj . Given that all the delegate apportionment methods used in the primaries (with the exception of HAM) are explicitly biased in favor of stronger candidates, it is easy to see that none of these methods encourage schisms. It is also true that in many cases, the delegate apportionment methods do not encourage coalitions. This can happen even when a weaker candidate joins the leading candidate, as demonstrated by the following examples. Example 5.10 Suppose D = 12, n = 5, and q = (4.1, 3.1, 1.8, 1.5, 1.5). Under LAR, d = (5, 4, 1, 1, 1). If C1 and C2 form a coalition resulting in new quota vector q˜ = (7.2, 1.8, 1.5, 1.5), the new allocation under LAR is d˜ = (8, 2, 1, 1). Since d˜ 1 = 8 but d1 + d2 = 5 + 4 = 9, the candidate coalition does not receive as many delegates as the candidates separately. The same is true under SUQ, except that the initial allocation is d = (5, 4, 2, 1, 0). Example 5.11 Suppose D = 12, n = 4, and q = (4.7, 4.6, 1.5, 1.2). Under NIE, NIS, and HNH, d = (5, 5, 2, 0). If C1 and C2 join to form a new quota vector q˜ = (9.3, 1.5, 1.2), the new allocation under NIE/NIS is d˜ = (9, 2, 1). Since d˜ 1 = 9 but

5.7 The Cumulative Effect: Comparing Vote Share to Delegate Share

121

d1 + d2 = 5 + 5 = 10, the candidate coalition does not receive as many delegates as the candidates separately. The same is true under HAM, except that the initial allocation is d = (5, 5, 1, 1). Examples 5.10 and 5.11 demonstrate that six of the seven delegate apportionment methods do not encourage candidate coalitions. The missing apportionment method is ILQ, and it is an exception in that it does encourage candidate coalitions. Proposition 5.16 ILQ encourages candidate coalitions. Proof To see this, suppose Ci and Cj join to form Cα . Then [qα ] ≥ [qi ] + [qj ] so Cα receives at least as many as Ci and Cj together in round 1. Moreover, we claim at each subsequent round Cα continues to receive at least as many delegates as Ci and Cj receive together. Suppose ai ≤ Rvi /V < ai + 1 and bj ≤ Rvj /V < bj + 1 for some integers ai and bj where R is the number of delegates remaining at some round. Then ai + bj ≤ R˜vα /V < ai + bj + 2, so Cα receives at least ai + bj delegates, proving ▢ the claim. The results in Examples 5.10 and 5.11 suggest that the consequences of one candidate “throwing” their support to another candidate may not always be predictable. Certainly, it has not been the expectation in recent primaries that votes for an eliminated candidate would lead to an automatic transfer of delegates to one of the remaining candidates. In fact, there is a risk that if a candidate drops out and not all the voters who support them show up to vote, then their failure to vote may contribute to the detriment of a remaining candidate. We investigate the elimination and no-show paradoxes in Chap. 6.

5.7 The Cumulative Effect: Comparing Vote Share to Delegate Share We conclude this chapter by examining the overall impact of the delegate selection process on the strongest candidates. Despite the ostensible goals of apportionment, the cumulative effect of dividing delegates into state primaries and then into district and statewide delegates, eliminating candidates through thresholds, and finally using apportionment methods with varying degrees of bias can be a delegate allocation that departs significantly from being proportional. In Sects. 5.2, 5.3, and 5.5, we have seen how adherence to quota, bias, and characteristics like the majority and leader criteria depend on whether the analysis is based on the apportionment method alone, or on the larger delegate selection process. In this section, we take a macroscopic view. We focus on the total number of delegates received by the candidates in each state primary and consider how this allocation compares to the candidates’ vote distributions statewide. To simplify the analysis, we focus only on the top two candidates. Since there may be only a very small number of candidates receiving delegates late

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5 Properties of the Apportionment Methods Used in the Primaries

Delegate Share vs Vote Share for C1 in Democratic State Primaries 1

0.9

Delegate Share

0.8

0.7

0.6

0.5

0.4

0.3 0.3

0.4

0.5

0.6 0.7 Vote Share

0.8

0.9

1

Fig. 5.4 Delegate share versus vote share for the leading candidate C1 in 183 Democratic state primaries from 2004, 2008, 2016, and 2020

in the primary season, this has the added advantage of allowing us to look at all the state primaries. To understand the impact of the delegate selection plan on the allocation of delegates, we compare vote share to delegate share for the top two candidates in each of the state primaries. We focus on the recent Democratic primaries. Since these primaries all used HAM with a 15% threshold, this removes factors such as the choice of apportionment method and threshold value, highlighting the consequence of eliminating very weak candidates (at most 6 candidates can exceed 15% of the vote) and aggregating over multiple small districts. Figures 5.4 and 5.5 show scatterplots for vote share versus delegate share for C1 and C2 from all primaries in 2004, 2008, 2016, and 2020 for states with at least two congressional districts for which we have vote data. As Fig. 5.4 indicates, the leading candidate’s share of the state delegates meets or exceeds their statewide vote share. Since HAM is relatively unbiased, the clear advantage to C1 is largely a product of the threshold and the aggregation: Each district apportionment holds a slight positive bias toward the leading candidate, which accumulates when all the delegates are aggregated. The advantage accrued by the leading candidate occurs at all levels of support, although when C1 achieves less than a majority or when it wins a large majority it frequently receives a significantly higher proportion of the delegates than its vote share would suggest.

5.7 The Cumulative Effect: Comparing Vote Share to Delegate Share

123

Delegate Share vs Vote Share for C2 in Democratic State Primaries 0.5

Delegate Share

0.4

0.3

0.2

0.1

0.1

0.2 0.3 Vote Share

0.4

0.5

Fig. 5.5 Delegate share versus vote share for the leading candidate C2 in 183 Democratic state primaries from 2004, 2008, 2016, and 2020

The situation with the second-place candidate is different. When C2 just exceeds the 15% threshold, C2 receives substantially fewer delegates than the vote share warrants. This is the mirror image of the situation for C1 which has a strong majority in these elections. However, if C2 does just a little bit better, then the associated delegate share, like C1 ’s delegate share, is at least as good if not greater than the vote share. This also is a result of the threshold and aggregation across congressional districts. Figures 5.4 and 5.5 show that the two strongest candidates consistently receive more than their “fair share” of delegates. However, this positive bias does not necessarily accrue to a single individual because, particularly in the early stages of the primary, the identity of the leading candidates in each primary may fluctuate. To get a better sense of the impact of the delegate selection rules on the actual winners of the Democratic primaries, we can compare the scatterplots of delegate share versus vote share for Barack Obama in 2008 (Fig. 5.6) and Jopseh Biden in 2020 (Fig. 5.7). Obama’s delegate share typically does not deviate significantly from his vote share; this is largely due to the fact that the 2008 Democratic primary was mostly a two-person contest between Obama and Hillary Clinton. Furthermore, in earlier primaries the candidate John Edwards was strong enough that he generally achieved more than 15% of the vote; therefore, his votes were not eliminated and the

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5 Properties of the Apportionment Methods Used in the Primaries

Delegate Share vs Vote Share for Barack Obama in 2008 1 0.9 0.8

Delegate Share

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 0.6 Vote Share

0.7

0.8

0.9

1

Fig. 5.6 Delegate share versus vote share for Barack Obama across 56 state primaries in the 2008 Democratic primary

delegate shares of Clinton and Obama did not receive a large boost. After the first few primaries, the remaining candidates (including Edwards) received a minuscule number of votes. Thus, the elimination of the weaker candidates’ votes had little effect, and the surviving candidates’ vote shares did not deviate significantly from their delegate shares. Biden’s situation is very different because the 2020 primary contained a large number of candidates, many of whom received a non-trivial number of votes late in the primary season, even when they were no longer viable presidential candidates. As a result, a large number of votes were eliminated in many state primaries causing Biden to receive many more delegates than his statewide vote share would suggest. For example, in the 2020 South Carolina primary, Biden received approximately 48.7% of the statewide vote but received 72.2% of the state delegates. This is because candidates Michael Bloomberg, Thomas Steyer, and Elizabeth Warren received 11.3, 8.2, and 7.1% of the statewide vote, respectively, and did not surpass the 15% threshold at the statewide level or in any of the congressional districts, resulting in the dramatic difference between Biden’s vote share and delegate share. In comparison, Barack Obama never achieved such a large difference between vote share and delegate share in any state primary in 2008. In the 2008 South Carolina primary, Obama received 55.4% of the statewide vote and 55.6% of the delegates, for example.

5.7 The Cumulative Effect: Comparing Vote Share to Delegate Share

125

Delegate Share vs Vote Share for Joseph Biden in 2020 1 0.9 0.8

Delegate Share

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 0.6 Vote Share

0.7

0.8

0.9

1

Fig. 5.7 Delegate share versus vote share for Joseph Biden across 57 state primaries in the 2020 Democratic primary

This contrast highlights something odd in the way delegates are allocated. Obama performed better in South Carolina in 2008 than Biden did in 2020, yet his delegate share was less than Biden’s. In fact, under the Democratic delegate selection plan (and under some Republican state plans), it is possible that a vote share of 48.7% could result in a delegate share significantly less than 48.7% (if, for instance, the candidate performs extremely well in one congressional district but less well in the others) or a delegate share of as much as 100%. Biden received a large boost in South Carolina because that election contained several candidates who received a large number of votes that fell short of the 15% threshold; Obama received no such boost because the 2008 South Carolina primary did not contain such candidates. It is perhaps less than optimal that the allocation of delegates to the leading candidate should depend so strongly on the electoral performance of non-viable candidates whose vote totals fall beneath the elimination threshold. The effect appears particularly large when each state primary is considered as a whole. In fact, the elimination of candidates can have an impact at the individual district level as well, including in unexpected ways. We consider the consequence of this elimination paradox and other instances of undesirable behavior that are exhibited by the apportionment methods–and the delegates selection plans as a whole–in Chap. 6.

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References Balinski ML, Young HP (2001) Fair representation: meeting the ideal of one man, one vote, 2nd edn. Brookings Institution Press, Washington, DC Drton M, Schwingenschlögl U (2005) Asymptotic seat bias formulas. Metrika 62(1):23–31. https:// doi.org/10.1007/s001840400352 Jones MA, Wilson JM (2010) Evaluation of thresholds for power mean-based and other divisor methods of apportionment. Math Social Sci 59(3):343–348. https://doi.org/10.1016/j.mathsocsci. 2010.02.002 Jones MA, McCune D, Wilson JM (2020) New quota-based apportionment methods: the allocation of delegates in the Republican presidential primary. Math Social Sci 108:122–137. https://doi. org/10.1016/j.mathsocsci.2020.05.001 Jones MA, McCune D, Wilson J (2023) An iterative procedure for apportionment and its use in the Georgia Republican primary. Contemp Math, Amer Math Soc, Providence, RI (to appear) Lipjhart A, Gibberd RW (1977) Thresholds and payoffs in list systems of proportional representation. Eur J Polit Res 5(3):219–244. https://doi.org/10.1111/j.1475-6765.1977.tb01289.x Pukelsheim F (2017) Proportional representation: apportionment methods and their applications. Springer, Cham, second edition. With a foreword by Andrew Duff Rae D, Hanby V, Loosemore J (1971) Thresholds of representation and thresholds of exclusion. Comparat Polit Stud 3(4):479–488. https://doi.org/10.1177/001041407100300406 Rokkan S (1968) Elections: electoral systems. In: Sills DL (ed) International Encyclopedia of the Social Sciences. Crowell, Collier, Macmillan, New York, pp 6–21 Schuster K, Pukelsheim F, Drton M, Draper NR (2003) Seat biases of apportionment methods for proportional representation. Electoral Stud 22(4):651–676. https://doi.org/10.1016/s02613794(02)00027-6 Schwingenschlögl U (2007) Probabilities of majority and minority violation in proportional representation. Stat Probab Lett 77(17):1690–1695. https://doi.org/10.1016/j.spl.2007.04.024 The Green Papers (2022) The Green Papers. http://www.TheGreenPapers.com, accessed: 2021-1215

Chapter 6

Paradoxes

Apportionment paradoxes have been well studied in the context of house seat apportionment, both in the US (Balinski and Young 2001) and in parliamentary systems (Pukelsheim 2017). Because delegate apportionment has not been well studied, paradoxes in this context have received little attention. Exceptions are the following: Jones et al. (2019) and Jones and Wilson (2016) who study paradoxes of Hamilton’s method in the Democratic primaries; McCune et al. (2019) who explore paradoxes of the SUQ method in the Kansas Democratic and Republican caucuses; and McCune (2023) who describes the multiple paradoxes that occurred in the 2020 IA Democratic presidential caucuses. The seven apportionment methods for allocating delegates that we identify in Sects. 2.1 and 4.3 are quota-based and therefore are susceptible to the classical paradoxes that feature prominently in the apportionment literature. In addition, these methods are also susceptible to new paradoxes that arise mainly in the context of presidential primaries. This chapter explores the susceptibility of the delegate selection methods to all of these paradoxes, beginning with the elimination paradox in Sect. 6.1 and the aggregation paradox in Sect. 6.2, which have received little previous attention. Section 6.3 relates the elimination paradox to the no-show paradox which is not particularly relevant in the US House seat context, but has received some attention in the parliamentary context (Danˇcišin 2017). Sections 6.4, 6.5, and 6.6 explore the susceptibility of the delegate selection methods to the Alabama paradox, the population paradox, and paradoxes of uniformity, respectively. These are classical paradoxes that have been well studied in both house seat and parliamentary contexts. With the exception of the last two paradoxes, which we discuss only briefly, each section addresses most or all of the following questions about the given paradox under different apportionment methods. 1. How can the paradox be visualized geometrically? 2. How frequently does the paradox occur?

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9_6

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6 Paradoxes

3. Which candidates are affected by the paradox, either positively or negatively? For example, does the paradox tend to harm the bottom candidate or top candidate? 4. How does varying the threshold level affect occurrences of the paradox? In some cases, we also investigate the effects of using the methods across multiple districts and aggregating the results to obtain the final apportionment, which occurs in all Democratic and some Republican primaries. We explore these questions using a variety of approaches including theoretical analyses, computational estimates using Monte Carlo simulations, and empirical summaries based on data from recent presidential primaries. Investigating these questions allows us to compare the performance of the different methods with respect to these paradoxes, rather than simply observing that all methods are susceptible to all of the paradoxes. Note that for the theoretical analyses, we assume that the candidates are rank-ordered so that C1 has the highest vote totals and Cn has the least.

6.1 The Elimination Paradox As discussed in Chaps. 2 and 4, all Democratic and some Republican state primaries incorporate thresholds into the apportionment process so that candidates who receive less than a certain percentage of the vote are eliminated (along with their votes). The Democratic National Committee mandates a uniform threshold of 15% in every state primary, while the Republican National Committee allows state parties wide latitude in setting their thresholds. In this section, we examine the potentially paradoxical effects of using these thresholds in conjunction with delegate apportionment methods. Once the votes of the eliminated candidates are removed, the percentage of votes received by each surviving candidate increases. Intuitively, we would expect the delegate count for these remaining candidates either to increase or to stay the same; indeed this is presumably one of the reasons for applying a threshold. However, under some circumstances the elimination of a set of candidates can cause a remaining candidate’s number of delegates to decrease, a phenomenon we term the elimination paradox that was introduced in (Jones et al. 2019). This phenomenon can happen when an eliminated candidate has little support, such as a write-in candidate who may receive few votes and may not have a chance of earning a delegate, or when a candidate’s percentage of support is closer to the threshold. Example 6.1 illustrates how Kasich was the victim of an elimination paradox in the 2016 Kansas Republican caucuses because he received fewer delegates when lower-ranked candidates were eliminated than he would have received had they remained in the race. Example 6.1 (2016 Kansas Republican Caucuses) The 2016 Kansas Republican caucuses used the SUQ method with a 10% threshold to award 25 at-large delegates. The caucus results appear in Table 6.1. The middle three columns of the table show the SUQ apportionment without applying the 10% threshold. The last three columns show the apportionment in which SUQ was applied with the 10% threshold. Without

6.1 The Elimination Paradox

129

Table 6.1 2016 Kansas Republican caucuses results Candidate Votes % Quota Appt Cruz Trump Rubio Kasich Carson Others

37,512 18,443 13,295 8741 582 405

47.50 23.35 16.83 11.07 0.74 0.51

11.88 5.84 4.21 2.77 0.18 0.13

12 6 5 2 0 0

Adj. %

Adj. Quota

Appt

48.10 23.65 17.05 11.21

12.03 5.91 4.26 2.80

13 6 5 1

the threshold, Kasich receives two delegates. Applying the 10% threshold, the votes for Carson and other bottom candidates are eliminated. As a consequence, Kasich loses one of his delegates that is gained by Cruz. The elimination paradox does not arise in geographic-based apportionment because thresholds are not used in these settings. (Indeed, it would be considered fundamentally undemocratic to deny house seats to a state because of its small population.) However, this little-studied paradox is relevant in both the primaries and in legislative seat apportionment. In fact, many proportional representation systems require political parties to receive a minimal percentage of the vote in order to be allocated seats. The Israeli Knesset requires a threshold of 3.25%; any party receiving less than 3.25% of the vote has their votes eliminated. As with delegate apportionment, the possibility that the elimination of a smaller party or candidate adversely affects a stronger party or candidate seems undesirable and suggests that it would be more appropriate to apply apportionment methods that are not susceptible to this paradox. To formalize the notion of the elimination paradox, let E ⊂ N = {C1 , . . . , Cn } be a set of candidates to be eliminated and v' the votes for the remaining candidates. In principle, the elimination paradox occurs when a candidate Ci ∈ N \ E receives fewer delegates after the elimination of the candidates in E than before their elimination. The formal definition of the paradox requires some subtlety, as the full definition of the paradox takes into account the fact that an apportionment method is a pointto-set mapping with F(D, v) representing the set of possible apportionments. (As discussed previously, the set is often a singleton, but may not be if, for instance, the fractional pieces of qi and q j are equivalent, or if vi = v j . In such a case, there may be k delegates left to award, but more than k equally deserving candidates, amounting to more than one way to apportion the delegates.) The following definition rules out the possibility that the decrease in delegates awarded to Ci is an artifact from there being more than one possible original apportionment. Definition 6.1 Given v, D, and N , let E ⊂ N be a subset of candidates with v' the votes for candidates not in E. The elimination paradox occurs if for some candidate Ci ∈ N \E and apportionment d ∈ F(D, v), there exists an apportionment

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d' ∈ F(D, v' ) with di' < di such that there does not exist d'' ∈ F(D, v) with di'' = di' . An apportionment method F is susceptible to the elimination paradox if there exists a v, D, N , and E ⊂ N which give rise to an elimination paradox. In presidential primaries, the set E is usually determined by a threshold c > 0 such that E = {Ci | pi < ci }; when the candidates are ordered by support then either E = {Ck+1 , . . . , Cn } or E = ∅. It is also possible that a candidate drops out of a race while still obtaining enough votes to meet the threshold. With a higher number of mail-in votes, this rare event may occur more frequently. Example 6.1, where the threshold was c = 10%, demonstrates that SUQ is susceptible to the elimination paradox. Because the apportionment methods used by both parties in the presidential primaries are quota-based, they are all susceptible to the elimination paradox. (In (Jones et al. 2019), it is shown that divisor methods are not susceptible to the elimination paradox.) Not all of the methods are affected the same way. For example, the maximum number of delegates that can be lost, which candidates can be negatively affected, and the frequency with which the elimination paradox may occur all depend on the apportionment method. In the remainder of the section, we investigate these differences.

6.1.1 Geometry of the Elimination Paradox We can use geometry to better understand how and when the elimination paradox arises, particularly when n = 3. Figure 6.1 illustrates, for D = 5, how a quota q = (q1 , q2 , q3 ) is projected (via the dotted lines) to a 2-candidate apportionment problem for C1 and C2 if C3 is eliminated. For instance, the point q = (1.3, 2.5, 1.2), depicted by Δ, is projected to the point q' = (1.7, 3.3). Under HAM, HNH, NIE, and NIS, twocandidate apportionment problems are solved using nearest integer rounding, and q' results in an apportionment of d' = (2, 3). Under LAR, SUQ, and ILQ, two-candidate apportionment problems are solved by giving the top (resp. bottom) candidate the ceiling (resp. floor) of their adjusted quota, and q' leads to an apportionment of d' = (1, 4). As another example, the point r = (3.35, 1.42, 0.23), depicted by ⊗ in Fig. 6.1, is projected to r' = (3.51, 1.49). The seven aforementioned apportionment methods map r' to the same outcome (4, 1). To see how this projection can lead to an elimination paradox, consider the following example based on the 2008 Democratic primary in New York. Example 6.2 In the 2008 Democratic primary, the 27th congressional district of New York was allocated 5 district delegates. Candidate Hillary Clinton received 45,276 votes, Barack Obama received 19,256 votes, and all other candidates combined received 3121 votes in the district election. Interpreting the remaining candidates as a single candidate C3 , this yields a quota vector q = (3.35, 1.42, 0.23),

6.1 The Elimination Paradox

131

5, 0, 0

Δ ⊗ •

•

HAM, HNH, • NIE/NIS

•

LAR, SUQ • ILQ

•

5, 0, 0

•

•

•

•

⊗

•

•

Δ

•

•

⊗

•

•

Δ

•

•

0, 5, 0

Fig. 6.1 The geometry of eliminating candidates when n = 3

which corresponds to ⊗ in Fig. 6.1. This point is reproduced again in Fig. 6.2, where it is clear from the geometry that, before elimination, under HAM, the delegate allocation would have been (3, 2, 0). However, since only the top two candidates exceeded the 15% Democratic threshold, the other candidates were eliminated leading to the allocation (4, 1). Thus, Obama was adversely affected by the elimination paradox. More generally, the shaded areas in Fig. 6.2 indicate the vote distributions for which the elimination paradox occurs under HAM if C3 is eliminated, when D = 5 and n = 3. These areas lie primarily along the bottom row where the second-placed candidate (either C1 or C2 ) loses a delegate. However, there are two smaller triangular regions which indicate areas where the elimination of the second-placed candidate (either C1 or C2 ) causes C3 to lose a delegate. The left one corresponds to a tiny triangular region where the 3-candidate appointment is d = (3, 1, 1), but after C2 is eliminated, C1 receives all five delegates. The region on the right is analogous. For the paradox to occur in these two regions, the candidate with the second highest number of votes would have to be eliminated, ostensibly by dropping out of the election. This is not likely to occur in a real primary. We can also use geometry to visualize when the elimination paradox occurs under the other apportionment methods. Figure 6.3 offers shading of quota distributions where the elimination paradox occurs if C3 is eliminated when n = 3 and D = 12

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Fig. 6.2 The elimination paradox when D = 5 and n = 3 under HAM

(showing only the bottom left sixth of the simplex where the candidates ordered by number of votes C1  C2  C3 ) for six apportionment methods; recall that NIE and NIS are equivalent when there are only three candidates. The areas of the shaded regions where the elimination paradox occurs under HAM, HNH and NIE/NIS are much smaller than the areas for SUQ, LAR, and ILQ. Perhaps, it is not surprising that the quota distributions that yield the elimination paradox are the same for HAM and HNH. The border of the shaded regions are formed in part by projection lines, like those from Fig. 6.1. Under HAM, HNH, SUQ, and LAR, it is C2 who suffers from the elimination paradox because the shaded area falls to the left of the projection lines. In contrast, under NIE and ILQ, it is C1 who is affected because the shaded area falls to the right of the projection lines. While it is difficult to visualize the geometry when there are more than 3 candidates, exhibited behavior for n = 3 candidates can be used to construct examples for n > 3 candidates.

6.1.2 Likelihood of the Elimination Paradox The geometry can be used to characterize and to compare the likelihood that the elimination paradox occurs among the different methods. For instance, Figs. 6.2 and 6.3 suggest that the elimination paradox occurs most often for quota vectors near

6.1 The Elimination Paradox

133

4, 4, 4

12, 0, 0

HAM method

6, 6, 0 12, 0, 0

4, 4, 4

SUQ Method

4, 4, 4

12, 0, 0

HNH method

6, 6, 0 12, 0, 0

4, 4, 4

LAR Method

4, 4, 4

12, 0, 0

NIE/NIS method

6, 6, 0 12, 0, 0

6, 6, 0

6, 6, 0 4, 4, 4

ILQ Method

6, 6, 0

Fig. 6.3 The regions in which elimination paradox occurs when C3 is eliminated when D = 12 under the seven apportionment methods

the boundary of the simplex and that the paradox occurs much more frequently for a method like SUQ than for HAM. We can test such observations using simulation. Table 6.2 compares the probability (expressed as a percentage) of the elimination paradox occurring when n = 3 under different methods for several values of D and a range of different thresholds. The simulation selects with equal probability a quota distribution from the sixth of the simplex where C1 receives the most votes, then C2 , and finally C3 ; this corresponds with the images from Fig. 6.3. The columns

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Table 6.2 Simulation data indicating the likelihood of the elimination paradox occurring when n=3 Thresholds Thresholds Susc. 5% 10% 15% 5% 10% 15% Susc. D

10 25 50

D

10 25 50

D

10 25 50

HAM 1.070 0.466 0.243 HNH 1.077 0.458 0.245 ILQ 5.938 2.989 1.442

2.324 1.305 0.604

1.289 0.459 0.251

0.493 0.257 0.128

2.337 1.302 0.615

1.275 0.451 0.259

0.495 0.255 0.129

5.398 10.671 5.128

10.589 5.609 2.669

7.732 3.971 1.865

LAR 20.368 11.682 7.387 NIE/NIS 1.857 0.693 0.369 SUQ 37.055 25.245 17.153

18.936 27.276 20.616

30.618 17.409 10.377

20.657 10.191 5.660

6.488 2.474 1.323

3.523 1.272 0.707

2.223 0.877 0.496

18.946 39.090 43.484

30.570 36.932 25.492

35.327 23.992 13.514

labeled Susc. (standing for “Susceptible”) give the likelihood that either C1 or C2 suffers from the elimination paradox if C3 is eliminated (regardless of whether C3 reaches a threshold). The other columns indicate the probability that the elimination paradox occurs conditional on candidate C3 or candidates C2 and C3 not receiving a percentage of the popular vote. For example, for Hamilton’s method in Table 6.2, when D = 10 with a threshold of 5%, then there is a 2.324% probability that the elimination paradox occurs if one or more of the candidates are eliminated for not receiving 5% of the popular vote. Notice that the values are greater for smaller D values because the associated simplices have a higher proportion of boundary points. Table 6.3 shows similar probabilities for n = 5. The column labeled Susc. represents the likelihood that the elimination paradox occurs when some subset of bottomranked candidates, ({C5 }, {C4 , C5 }, or {C3 , C4 , C5 }), is eliminated, assuming each quota distribution is equally likely from the region in which Ci receives at least as many votes as Ci+1 for all i. The remaining columns indicate the conditional probability that the elimination paradox occurs if at least one candidate is eliminated for not receiving the threshold percentage of the popular vote. When n = 3, the likelihood of the elimination paradox occurring is significantly greater for SUQ, ILQ, and LAR than for the other methods, as the geometry of the shaded regions demonstrated in Fig. 6.3. Under a uniform distribution, the simulations indicate that the likelihood of the paradox occurring increases as n increases. Because actual primary data are not uniformly distributed, we can get a better sense of the likelihood that the elimination paradox occurs by analyzing election results from recent primaries. Using the DEM_DATA and REP_DATA (see Appendix B), we can determine the number of instances of the elimination paradox in the 1880 Democratic and 739 Republican elections with n ≥ 3 and D > 1. Although

6.1 The Elimination Paradox

135

Table 6.3 Simulation data indicating the likelihood of the elimination paradox occurring when n=5 Thresholds Thresholds Susc. 5% 10% 15% Susc. 5% 10% 15% D

10 25 50

D

10 25 50

D

10 25 50

D

10 25 50

HAM 6.664 4.100 2.719 HNH 16.470 13.737 10.982 ILQ 36.763 22.327 12.671 LAR 51.835 36.415 25.589

5.415 3.411 1.880

1.623 0.710 0.366

0.460 0.127 0.061

11.524 10.924 9.743

5.543 5.378 2.657

3.900 1.003 0.681

15.196 21.344 14.075

27.856 13.967 5.824

21.990 7.117 2.521

23.460 34.556 27.438

42.759 23.244 11.245

27.759 9.404 3.433

NIE 18.424 14.318 10.823 NIS 18.264 14.191 10.706 SUQ 81.378 89.189 89.990

17.224 12.187 10.252

8.653 6.263 2.937

5.467 1.508 0.872

16.927 12.123 10.110

8.459 6.166 2.864

5.364 1.457 0.856

24.099 52.149 70.627

48.202 61.812 55.850

45.955 35.926 20.555

each of the elections used only one apportionment method, for comparison, we also determine the potential number of paradoxes under all seven quota-based methods used in the Democratic and Republican primaries. Table 6.4 provides a numerical summary of the results. The first entry in each cell corresponds to the number of instances of the elimination paradox among the Democratic elections; the second is the analogous number for the Republican elections. Thus, there were 21 and 19 cases, respectively, of the elimination paradox in each party using HAM and a 15% threshold. Note that the 21 actually occurred,1 since the Democratic Party used HAM and a 15% threshold in all of their elections; the 19 from the Republican data are a mixture of actual and hypothetical instances, since not all Republican elections used Hamilton’s method and a 15% threshold. Table 6.4 also includes the number of instances of the elimination paradox that did occur or could have occurred using the other six delegate apportionment methods. Note that methods like SUQ and LAR return many more cases of the paradox than HAM or HNH, just as the simplicial geometry suggested for quota distributions selected uniformly from the simplex. The

1 Included in the 21 instances is the at-large delegate apportionment in the 2020 Colorado primary. This elimination paradox did not occur in practice because some of the candidates, including Elizabeth Warren, withdrew from the primary, causing them to lose all of their delegates per the CO party rules. If no candidates had formally withdrawn, Warren would have lost an at-large delegate because of the elimination of the weaker candidates.

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Table 6.4 Number of instances of the elimination paradox in the DEM_DATA database (left coordinate) and REP_DATA database (right coordinate), if the given method and threshold were used for all of those elections Threshold HAM HNH ILQ LAR NIE NIS SUQ 15% 10% 5% 1%

21, 19 71, 62 115, 54 61, 13

34, 29 104, 83 155, 64 69, 19

252, 126 214, 96 161, 64 40, 33

420, 113 499, 111 417, 83 118, 43

139, 64 212, 61 199, 46 74, 27

124, 64 194, 62 191, 46 81, 27

431, 137 505, 145 421, 99 117, 42

data also show the likelihood of the elimination paradox occurring for thresholds of 1%, 5%, and 10%. We discuss the impact of different thresholds later in this section. Interestingly, three of the 21 instances of the elimination paradox in the Democratic primaries occurred in the same state primary: the 2008 California primary. The paradox occurred in CDs 41, 45, and 51, and favored Clinton in each case. Arguably, Clinton left the CA primary with three more delegates than she deserved because of the elimination paradox. Because state primaries often involve multiple elections at the CD and then the statewide level, the effects of the paradox can compound across different elections in the primary.

6.1.3 Who Is Affected by the Elimination Paradox? We start by considering the number of delegates a candidate could lose if other candidates are eliminated. Let qi' be Ci ’s adjusted quota after candidates have been eliminated and let di' be the adjusted delegate apportionment. Since qi' > qi , there are limitations to the loss a candidate can experience depending on the apportionment method. Under HAM and LAR, a candidate can lose at most one delegate. This occurs only if [qi' ] = [qi ], di = [qi ] and di' = [qi' ]. Under NIS, a candidate can lose at most two delegates. This happens when [qi' ] = [qi ], di = [qi ] + 1 and di' = [qi' ] − 1. Under NIE, SUQ, and ILQ, the losses can be greater, depending on the strength of the top vote-getting candidate. Because thresholds are designed to consolidate delegates to the top candidates, the effect of the elimination paradox on the top vote-getter is of particular interest. In the discussion of Fig. 6.3 and the geometry of the elimination paradox, it was noted that when n = 3 only C2 is negatively affected by the elimination paradox for HAM, HNH, SUQ, and LAR. For three of these methods, it is the case that C1 is never adversely affected by the elimination paradox, as described below. Proposition 6.1 Under the HAM, LAR, and SUQ methods, the apportionment of the leading candidate cannot decrease after an elimination.

6.1 The Elimination Paradox

137

Proof We start with HAM. Since p1 > p j (recall that candidates are numbered in order of decreasing vote strength), it is easy to see that q1' − q1 > q 'j − q j . Consequently, C1 ’s quota increases more than any other candidate’s after elimination. Now, suppose C1 loses a delegate because of the elimination of some candidates. Then, d1 = [q1 ] + 1, while d1' = [q1' ] = [q1 ]. Because C1 loses a delegate, there must be at least one candidate that receives an extra delegate after elimination. Let C j be such a candidate. For the first case, assume d j = [q j ]. Before elimination, C1 ’s fractional part is greater than C j ’s fractional part; but, after elimination, C1 ’s fractional part must be less than C j ’s fractional part if it C j gains an extra delegate and C1 loses one. So, q1 − [q1 ] ≥ q j − [q j ] and q1' − [q1' ] ≤ q 'j − [q 'j ]. Subtracting the left side of the first inequality from the left side of the second inequality and subtracting the right side of the first inequality from the right side of the second inequality and recalling that [q1' ] = [q1 ] yields q1' − q1 ≤ q 'j − q j − ([q 'j ] − [q j ]) ≤ q 'j − q j , which is a contradiction. For the second case, assume that d j = [q j ] + 1, which implies [q 'j ] = [q j ] + 1 and d 'j = [q 'j ] + 1. Then, q1' − [q1 ] ≤ q 'j − [q 'j ] = q 'j − [q j ] − 1 and q1 − [q1 ] + 1 > q j − [q j ]. Once again, subtracting the left side of the first inequality from the left side of the second inequality and subtracting the right side of the first inequality from the right side of the second inequality and using [q1' ] = [q1 ] gives q1' − q1 − 1 ≤ q 'j − q j − 1, which is another contradiction. This proves the result for HAM. Finally, under LAR and SUQ, C1 cannot lose a delegate under elimination since d1' = [q1' ] ≥ [q1 ] = d1 .   For the other four methods used in the Republican primaries, the leader’s delegate count may decrease after elimination, which seems undesirable since these methods are designed to favor the leading candidate. Proposition 6.2 After the elimination of a candidate and the candidate’s votes, the leading candidate can lose • at most 1 delegate under NIS; • at most [n/2] delegates under NIE and HNH; and • at most n − 2 delegates under ILQ.

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Proof Let qi' be Ci ’s quota after candidates have been eliminated, and let di' be the adjusted apportionment. Bounds on the number of delegates that the leader can lose for each of the three methods are determined. Then, it is shown that the bounds are tight by showing that the inequalities can be replaced by equalities for certain vote distributions. For the NIS method, d1' ≥ [q1' ] ≥ [q1 ] ≥ d1 − 1 so C1 can lose at most 1 delegate. Next, consider the NIE method. Recall that at most [n/2] delegates may be overallocated, so that d1 ≤ [q1 ] + [n/2]. But d1' ≥ [q1' ] ≥ [q1 ] ≥ d1 − [n/2]; hence, C1 can lose at most [n/2] delegates. This holds for HNH method because it is the NIE method when there is an over allocation of delegates (and HAM otherwise) and because the leading candidate cannot lose delegates after an elimination under HAM by Proposition 6.1. ∑ Finally, consider the ILQ method. Recall that D − i [qi ] ≤ n − 1 so C1 can receive at most [q1 ] + n − 1 delegates. After elimination, C1 must receive at least d1' = [q1' ] + 1 ≥ [qi ] + 1 delegates, which implies C1 can lose at most n − 2 delegates. To see that these bounds are tight for the NIS and NIE/HNH methods, suppose n is odd, D = (3n − 1)/2 and q satisfies q1 = 2 + 2∈, qi = 3/2 − ∈/(n − 2) for i = 2, . . . , n − 1, and qn = (1/2) − ∈ for some ∈ > 0. All of these methods∑begin with nearest integer∑ rounding that yield [q1 ] = 2, [qi ] = 1 and [qn ] = 0 so i [qi ] = n. Because D − i [qi ] = (3n − 1)/2 − n = (n − 1)/2, each of the methods have to allocate an additional (n − 1)/2 delegates. Under the NIS method, C1 receives an extra delegate so that d1 = [q1 ] + 1 = 3. Now suppose Cn is eliminated. Then qi' =

D qi D = qi for all i < n. D − qn D − ( 21 − ∈)

∑ For small enough ∈, it is easy to see that [qi' ] = 2, and hence, i r2 so that r1 > 1/2. Then, d j = (s + 1, t) under all seven of the delegate apportionment methods by Table 6.8, and d1 + d2 = (2s + 2, 2t). To apportion the combined district, q1 = 2s + 2r1 = (2s + 1) + (2r1 − 1) q2 = 2t + 2r2 . If (2r1 − 1) > 2r2 , then d1 = 2s + 2 and d2 = 2t under all apportionment methods. However, 2r1 − 1 > 2r2 is equivalent to r1 − r2 > 1/2. This happens whenever r1 > 3/4. Similarly, if 1/2 < r1 < 3/4, then (2r1 − 1) < 2r2 , Thus, the apportionment is d1 = 2s + 1 and d2 = 2t + 1 under HAM, HNH, NIE, and NIS, and d1 = 2s + 2 and d2 = 2t under ILQ, LAR, and SUQ. Because r1 is uniformly distributed over (1/2, 1), a type 2 aggregation paradox never occurs for ILQ, LAR, and SUQ and occurs half of the time for HAM, HNH, NIE, and NIS. Case 2. Assume that s > t and r1 < r2 so that r1 < 1/2. Under ILQ, LAR, and SUQ, the apportionment is d j = (s + 1, t) so that d1 + d2 = (2s + 2, 2t). Under HAM, HNH, NIE, and NIS, d j = (s, t + 1) so that d1 + d2 = (2s, 2t + 2). It follows that q1 = 2s + 2r1 q2 = 2t + 2r2 = (2t + 1) + (2r2 − 1). For ILQ, LAR and SUQ, because q1 > q2 , then d1 = 2s + 1 and d2 = 2t + 1. So a type 2 aggregation paradox always occurs. However, for HAM, NNH, NIE, and NIS, the apportionment depends on the inequality (2r2 − 1) > 2r1 which is equivalent to

6.2 The Aggregation Paradox

153

r2 > 3/4. Hence, if r2 > 3/4, d1 = 2s and d2 = 2t + 2, and there is no paradox. But, if 1/2 < r2 < 3/4, then d2 = 2t + 1. So the paradox occurs half of the time in Case 2. If D is even then r1 > 1/2 and r1 < 1/2, both occur with probability 1/2 and Cases 1 and 2 are equally likely. For the HAM family, the probability of the paradox is (1/2) · (1/2) + (1/2) · (1/2) = 1/2. For the ILQ family, the probability is 0 · (1/2) + 1 · (1/2) = 1/2. This matches the proposition statement for the ILQ family because [D/2]/D = 1/2 for D even. If D is odd then, because q1 ∈ [D/2, D], r1 > 1/2 with probability [D/2]/D and r1 < 1/2 with probability [D/2]/D. Under HAM, HNH, NIE, and NIS, since the probability of a type 2 paradox occurs with an equal probability of 1/2 in both Cases 1 and 2, then the overall probability of a paradox is still 1/2. For the ILQ   family, the probability is 0 · [D/2]/D + 1 · [D/2]/D = [D/2]/D. In contrast to type 2 paradoxes, finding the likelihood of a type 1 aggregation paradox is more difficult, as the geometry in Fig. 6.8 suggests. However, for twocandidate, two-district elections, the next example shows that under an additional assumption, the geometry is much simpler, allowing the probability of a type 1 paradox to be determined in some cases. This one additional assumption is that D 1 /V 1 = D 2 /V 2 , where V i = v1i + v2i , so that the ratio of the number of delegates to the number of voters in each district is the same. This is key because it implies j that D 1 / V 1 = D 2 /V 2 = (D 1 + D 2 )/(V 1 + V 2 ). Then, since qi = (vij /V j )D j , the additional assumption implies that qi =

(

( 1 2) ) +D D 1 + D 2 = (vi1 + vi2 ) D V 1 +V 2 ( 2) ( 1) v v = Vi1 D 1 + Vi2 D 2 = qi1 + qi2 ,

vi1 +vi2 V 1 +V 2

)(

where qi is the quota for Ci in the combined district. Example 6.7 Consider a two-district, two-candidate election in which D 1 = 6 and D 2 = 4 and D 1 /V 1 = D 2 /V 2 so that qi = qi1 + qi2 . Further, assume that delegates are apportioned under HAM. Then, the geometry of the type 1 aggregation paradox simplifies to Fig. 6.9. Instances in which a type 1 paradox occur appear in light or dark gray, as described in the caption. Each of the shaded regions forms an isosceles triangle with side length equal to 1/2. Thus, if a quota vector (q11 , q12 ) is selected at random, then the likelihood it is selected from a light gray region is (24 × (1/8))/24 = 1/8. Similarly, the likelihood of a quota vector being selected from a dark gray region is also 1/8. Hence, the probability of a type 1 aggregation paradox is 1/4. For all methods that coincide with HAM, similar geometry occurs for all twodistrict two-candidate elections in which D 1 /V 1 = D 2 /V 2 . Thus, under HAM, HNH, NIE, and NIS, it is straightforward to determine the likelihood of a type 1 aggregation paradox under these assumptions.

154 Fig. 6.9 For the election from Example 6.7, the instances in which a type 1 aggregation paradox occurs under HAM appear in gray below. The light (resp. dark) gray regions indicate that a candidate receives one fewer (resp. one more) delegate in the combined district than in the separate districts

6 Paradoxes

4

3

2

1

0 0

1

2

3

4

5

6

Proposition 6.5 Suppose there are 2 candidates and 2 districts and V 1 /D 1 = V 2 /D 2 . If one of the quotas is selected uniformly from [0, D j ] for each j, then a type 1 aggregation paradox (d /= d1 + d2 ) occurs with probability 1/4 under HAM, HNH, NIE, and NIS. Proof The proof follows from a counting argument applied to a generalization of   Fig. 6.9. The geometry of the aggregation paradox for ILQ, LAR and SUQ can also be visualized. However, it is more difficult to generalize, so we focus on the special case D 1 = D 2 , but still with the assumption that D 1 / V 1 = D 2 /V 2 . The following example is useful to develop intuition. Example 6.8 Consider a two-district, two-candidate election in which D 1 = D 2 = 6 and in which each district and the combined districts are apportioned using the ILQ method. Assume that D 1 /V 1 = D 2 / V 2 . The geometry of aggregation for ILQ appears in Fig. 6.10 (Left). The numbers inside unit squares indicate the number of delegates C1 receives if their quota vector q1 = (q11 , q12 ) is within the square. The combined district is divided into diagonal regions which yield the same apportionment. The possible apportionments for the combined district appear on the perimeter of the rectangle. Notice that the apportionment d1 = 6 is only possible if q1 = 6, which occurs on a set of measure zero, assuming a uniform distribution. The regions in which the aggregation paradox of type 1 occurs appear in Fig. 6.10 (Right), shaded in light or dark gray. The areas of dark gray consist of 32 + 2(1 + 2 + 3) triangles out of a total of 72 triangles. By symmetry, every dark gray region has a corresponding light gray region that is symmetric through the point (3, 3), so that the area of the light gray and dark gray regions are the same. It follows that the likelihood of the aggregation paradox of type 1 is 2[32 + 2(1 + 2 + 3)]/72 = 7/12. The key to generalizing Example 6.8 when D 1 = D 2 and the district sizes are even is the behavior exhibited in the four quadrants [0, 3] × [0, 3], [0, 3] × [3, 6],

6.2 The Aggregation Paradox

6 5 4 3 2 1

155

6

7

8

10

11

12 12

5

6

7

9

10

11 11

4

5

6

8

9

10 10

2

3

4

6

7

8

1

2

3

5

6

7

9 8

6 7 5 4 0 1 2 0 0 0 1 1 2 2 3 3 4 4 5 5 6

6 5 4 3 2 1 0

0

1

2

3

4

5

6

Fig. 6.10 For the election under the ILQ method from Example 6.8, (Left) shows the apportionment regions that are the sum of the apportionments from the separate districts; the possible apportionment values are the black numbers in the center of squares. The gray numbers along the perimeter of the rectangle indicate the possible apportionments for the diagonal regions for a single, combined district. In (Right), the light (resp. dark) gray regions indicate that a candidate receives one fewer (resp. one more) delegate in the combined district than in the separate districts

[3, 6] × [0, 3] and [3, 6] × [3, 6] from Fig. 6.10 (Right). The geometry when D 1 is odd is similar, as the following example demonstrates. Example 6.9 Consider a two-district, two-candidate election in which D 1 = D 2 = 7 and in which each district and the combined districts are apportioned using the ILQ method. Assume that D 1 /V 1 = D 2 /V 2 . The geometry of aggregation for ILQ appears in Fig. 6.11 (Top). The regions in which the aggregation paradox of type 1 occurs appear in Fig. 6.11 (Bottom). There are 32 + 2(1 + 2 + 3) = 21 dark gray isosceles right triangles with side length 1. There are an additional 14 triangles of side length 1/2. Hence, the total dark gray area is 21 · (1/2) + 14(1/8) = 12.25. By symmetry, the probability of a type 1 aggregation paradox is the area of the gray regions (24.5) divided by the area of the square (49), or 24.5/49 = 1/2. We generalize Examples 6.8 and 6.9 in the following proposition. Proposition 6.6 Suppose there are 2 candidates, 2 districts, D 1 =D 2 , and D 1 /V 1 = D 2 /V 2 . If one of the quotas is selected uniformly from [0, D j ] for each j, then a type 1 aggregation paradox (d /= d1 + d2 ) occurs under ILQ, LAR and SUQ with probability • 1/2 + 1/(2D) if D 1 is even; and • 1/2 if D 1 is odd.

156 Fig. 6.11 For the election under the ILQ method from Example 6.9, the (Top) shows the sum of the apportionments from the separate districts as the black numbers in the center of squares and the possible apportionments for the combined district, where gray numbers indicate the apportionment. (Bottom) shows the regions in which the aggregation paradox of type 1 occurs. The light (resp. dark) gray regions indicate that a candidate receives one less (resp. one more) delegate in the combined district than in the separate districts

6 Paradoxes 7

7

8

9 10 11 12

13

14 14

6

7

8

9 10 11

12

13 13

5

6

7

8 9 10

11

12 12

4 3

5 4

6 5

7 8 6 7

9 8

10 9

11 11 10

2

3

4

5 6

7

8

9 10

1

2

3

4 5

6

7

8

9

0

1

2

3 4

5

6

7

8

6 5 4 3 2 1 0

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7

7

14 6 5 4 3 2 1 0 0

1

2

3

4

5

6

7

Proof Figure 6.10 generalizes for any D 1 = D 2 even, which allows a counting argument to determine the likelihood of the type 1 aggregation paradox. Suppose D 1 = D 2 = 2m for some integer m. In each of the regions [0, m] × [0, m] and [m, 2m] × [m, 2m], the paradox occurs in m 2 isosceles right triangles with side length 1. In each of the regions [0, m] × [m, 2m] and [m, 2m] × [0, m], the paradox occurs in 2Tm such triangles, where Tm = m(m + 1)/2 is the mth triangular number. The total area in which the paradox occurs is (2m 2 + 4Tm ) · (1/2) = m 2 + 2Tm = 2 m + m 2 + m. The area of the square [0, 2m] × [0, 2m] is 4m 2 . Hence, the probability is (2m 2 + m)/4m 2 = 1/2 + 1/4m. If D 1 = D 2 = 2m + 1 for some m, the paradox occurs in 2m 2 + 4Tm isosceles right triangles with side length 1 and 4(2m + 1) isosceles right triangles with side length 1/2. Hence, the area of the regions in which the paradox occurs is

6.2 The Aggregation Paradox

157

Table 6.9 Likelihood of the aggregation paradox of type 2 and the net gain for C1 under the HAM method with and without a 15% threshold for a simulation of one million elections HAM with 15% threshold HAM with no threshold n Likelihood Net gain for n Likelihood Net gain for C1 C1 D 1 = 15; D 2 = 15 0.749021 3 0.874770 4 5 0.937269 0.968699 6 D 1 = 10; D 2 = 15 3 0.415046 0.544359 4 5 0.644312 0.722147 6 D 1 = 8; D 2 = 25 3 0.416694 4 0.542654 0.643504 5 6 0.722268

(

16,034 19,502 25,065 29,655

3 4 5 6

0.523688 0.592053 0.618612 0.628655

17,799 8063 10,758 17,207

21,948 25,416 31,250 38,930

3 4 5 6

0.269625 0.319989 0.338248 0.341545

−19,774 −18,859 −6631 8096

19,119 30,808 35,645 35,736

3 4 5 6

0.268465 0.323223 0.344080 0.341311

−6029 −14,047 −25,119 −32,656

) 1 1 1 2m 2 + 2m 2 + 2m · + (8m + 4) · = 2m 2 + 2m + . 8 2 2

Dividing this area by (2m + 1)2 = 4m 2 + 4m + 1 gives the likelihood of the aggregation paradox of type 1 occurring; this probability is 1/2.   As the geometry illustrates, instances of the aggregation paradox for 2 candidates and 2 districts are ubiquitous, even when constraints are placed on different facets of the apportionment problems. This applies to both type 1 and type 2 paradoxes. To determine how great this likelihood is more generally, without the additional assumptions and with more than 2 candidates, we turn to simulations. We look only at aggregation paradoxes of type 2, since we do not need to make any assumptions about the relationship between v1 and v2 . Table 6.9 indicates the likelihood of a paradox occurring for 2 districts under HAM, as well as the impact of a 15% threshold (as in the Democratic primary) on both the probability of an aggregation paradox and the apportionment for C1 . The net gain refers to the total number of delegates awarded to C1 in a single combined district minus the total number of delegates awarded to C1 in the two separate districts. The data indicate that the likelihood of an aggregation paradox of type 2 tends to increase with the number of candidates and that the imbalance between the number of delegates in the two districts has a big effect on the number of delegates awarded to C1 . As we did in Sect. 6.1, we can also use previous election data to investigate how often an aggregation paradox of either type would have occurred under the seven

158

6 Paradoxes

Table 6.10 Number of type 1 instances of the aggregation paradox from 151 (respectively, 183) elections without (respectively, with) thresholds applied from recent Democratic primaries HAM HNH ILQ LAR NIE NIS SUQ Number of type 1 paradoxes with no threshold Net gain by C1 across all of the data Number of type 1 paradoxes with 15% threshold Net gain by C1 across all of the data

141

140

149

148

132

133

148

195

200

1482

762

529

516

764

115

117

159

159

119

119

159

−11

−3

734

678

1

1

688

methods. Table 6.10 illustrates the number and effect of aggregation paradoxes of type 1 that would have occurred under the different methods using data from the 2008, 2016, and 2020 Democratic presidential primaries. Row 1 indicates the number of contests in which, when no threshold is applied, the sum of the apportionments of district, at-large, and PLEO delegates differs from the apportionment that would have arisen from allocating all the state delegates in a single allocation. Row 2 indicates the net gain by C1 . Rows 3 and 4 are analogous but with a 15% threshold. Note that since complete information was not obtained about all elections, these numbers are based on a total of 151 elections without a threshold and 183 elections without a 15% threshold.3 Table 6.10 shows that aggregation paradoxes occur often, regardless of whether a threshold is applied. Furthermore, when there is a small number of candidates, aggregation does not tend to favor the statewide vote leader for methods based on nearest integer rounding. Without a threshold, the use of aggregated districts is extremely favorable to C1 , regardless of the method used; although the effect is more extreme under ILQ, LAR, and SUQ where C1 is always guaranteed their quota rounded up. Note that although the net gains for C1 are larger under all methods when there is no threshold, the leader still does better overall when the threshold is applied. The majority of those gains for each method occurred in the 2020 primary, in which there were many more candidates than in prior primaries. Table 6.11 is similar to Table 6.10 except that it considers aggregation paradoxes of type 2 in which the sum of the apportionments of at-large, and PLEO delegates differs from the apportionment that arises from allocating these delegates in a single allocation. These numbers are based on a total of 210 contests. As with type 1 paradoxes, we see that using a threshold dampens the effects of the aggregation paradox.

3

Recall that we obtain all of our primary data from thegreenpapers.com. This website contains almost complete district vote data for the candidates that surpass the 15% threshold. There are several states for which the data for candidates who do not surpass the threshold is unavailable; in these states, it is impossible to measure the effects of aggregation when we remove the threshold.

6.2 The Aggregation Paradox

159

Table 6.11 Number of type 2 instances of the aggregation paradox from 210 elections in recent Democratic primaries HAM HNH ILQ LAR NIE NIS SUQ Number of type 2 paradoxes with no threshold Net gain by C1 across all the data Number of type 2 paradoxes with 15% threshold Net gain by C1 across all the data

101

103

127

132

103

106

127

28 58

28 61

167 110

101 108

72 62

70 62

101 104

−7

−5

119

104

−6

−6

104

6.2.3 Who Is Affected by the Aggregation Paradox? Tables 6.10 and 6.11 show that aggregation paradoxes can occur under any method. However, there are substantial differences between the methods with respect to frequency, size of the discrepancies, and the candidates who are affected. In this section, we focus on who is most affected by the aggregation paradox and by how much. We start with aggregation paradoxes of type 2 and consider the affect on C1 . As before, we assume that there are ∑ k districts and each district j has D j delegates, for j = 1 to k. Also, let D = j D j . Further, assume that F(v, D j ) = d j and F(v, D) = d. Under LAR and SUQ, C1 always receives a number of delegates equal to their quota rounded up. So the leading candidate will always be advantaged when apportionment is split into separate problems. Under the other methods, the leading candidate can be advantaged or ∑ disadvantaged by the aggregation. To see the extent of the j difference, let Δ(k, F) = j d1 − d1 be the difference in delegate totals C1 receives ∑ under the apportionment method F if D is divided into k groups with j D j = D. Proposition 6.7 If an election is divided into k districts instead of a single district, then the advantage to C1 , Δ(k, F), satisfies the following bounds, • • • • •

0 ≤ Δ(k, F) ≤ k − 1 when F is LAR method or SUQ method; −k ≤ Δ(k, F) ≤ k when F is the HAM method; −(n − 2) ≤ Δ(k, F) ≤ k(n − 1) − 1 when F is the ILQ method; ] ≤ Δ(k, F) ≤ [ k2 ] + k[ n−1 ] when F is the NIE method; and −[ k2 ] − [ n−1 2 2 k+2 −[ 2 ] ≤ Δ(k, F) ≤ [ 3k2 ] when F is the NIS method.

Proof Let d j = F(v, D j ) and d = F(v, D). By applying the general inequality [w] − 1 < w ≤ [w] to D j p1 , it follows that [D j p1 ] − 1 < D j p1 ≤ [D j p1 ] implies ∑ ∑ [D j p1 ] − k < Dp1 ≤ [D j p1 ]. (6.1) j

j

160

6 Paradoxes j

Under LAR and SUQ, d1 = [D j p1 ]. Hence, by (6.1), ∑ j ∑ j ∑ d1 = [D j p1 ] < Dp1 + k ≤ [Dp1 ] + k = d1 + k implies d1 ≤ d1 + k − 1. j

j

j

Alternatively,

∑

j

d1 ≥ Dp1 > [Dp1 ] − 1 = d1 − 1, which implies

j

Under HAM, [D j p1 ] − 1 ≤

j d1

∑

j

j

d1 ≥ d1 .

≤ [D j p1 ]. So by (6.1),

∑ j ∑ j ∑ d1 ≤ [D j p1 ] < Dp1 + k ≤ [Dp1 ] + k ≤ d1 + k + 1 implies d1 ≤ d1 + k. j

j

j

Alternatively, (k + 1). Hence,

∑ ∑

j

d1 ≥

j j d1

∑

j [D

j

p1 ] − k ≥ Dp1 − k > [Dp1 ] − (k + 1) ≥ d1 −

≥ d1 − k. j

Under ILQ, [D p1 ] ≤ d1 ≤ [D j p1 ] + n − 2. It follows that j

∑

j

d1 ≤

j

∑ [D j p1 ] + k(n − 2) < Dp1 + k(n − 1) j

≤ [Dp1 ] + k(n − 1) ≤ d1 + k(n − 1). So

∑

j

j

d1 ≤ d1 + k(n − 1) − 1. Alternatively, ∑

dj ≥

j

∑

[D j p1 ] ≥ Dp1 > [Dp1 ] − 1 ≥ d1 − (n − 1),

j

∑

j d j ≥ d1 − (n − 2). To prove the statements about NIS and NIE, we apply the general inequality [w] − 1/2 ≤ w < [w] + 1/2 to D j p1 to get

so

1 1 ≤ D j p1 < [D j p1 ] + , which implies that 2 2 ∑ ∑ k k [D j p1 ] − ≤ Dp1 < [D j p1 ] + . 2 2 j j [D j p1 ] −

j

Under NIE, [D j p1 ] ≤ d1 ≤ [D j p1 ] + [ n−1 ]. It follows that 2 ∑

| | | | ∑ k n−1 n−1 [D j p1 ] + k ≤ Dp1 + + k 2 2 2 j | | | | n−1 k+1 n−1 k+1 +k +k < [Dp1 ] + ≤ d1 + . 2 2 2 2

j

d1 ≤

j

Hence,

∑

j

j

d1 ≤ d1 + [ k2 ] + k[ n−1 ]. Alternatively, 2

6.2 The Aggregation Paradox

∑

j

d1 ≥

j

161

| | ∑ k k+1 n−1 k+1 ≥ d1 − . [D j p1 ] > Dp1 − ≥ [Dp1 ] − − 2 2 2 2 j

∑ j Therefore, j d1 ≥ d1 − [ n−1 ] − [ k2 ]. 2 j Under NIS, [D p1 ] ≤ d1 ≤ [D j p1 ] + 1. Thus, ∑

j

d1 ≤

j

Hence,

∑

∑ 3k + 1 3k + 1 3k < [Dp1 ] + ≤ d1 + . [D j p1 ] + k ≤ Dp1 + 2 2 2 j j

j

∑

d1 ≤ d1 + [ 3k2 ]. Alternatively, j

d1 ≥

j

∑ k+1 k+3 k [D j p1 ] > Dp1 − ≥ [Dp1 ] − ≥ d1 − . 2 2 2 j

∑ j Consequently, j d1 ≥ d1 − [ k−2 ]. 2 Finally, we show the bounds for HAM are tight; the proof for the other methods is similar. Assume k = 2; the example is easily modified for k > 2. Let n, D 1 and D 2 be fixed with D = D 1 + D 2 . p1 = 1 −

1 D1 D2

− ∈ and pi =

1 D1 D2

1 ∈ + for i > 2 for some ∈. n−1 n−1 j

Then it is easy to see that if 1/(D 1 D 2 ) + ∈ < (n − 1)/(D j n), then d1 = D j for both j HAM. under HAM. But if 1/(D 1 D 2 ) + ∈ > 2(n − 1)/(Dn), then d1 = D − 2 under∑ j Hence, if ∈ satisfies 2(n − 1)/(Dn) ≤ 1/(D 1 D 2 ) + ∈ < (n − 1)/(D j n) then j d1 − d1 = 2 under HAM. Similarly, let p1 = 1 − 1/D − ∈ and pi = (1/D) · 1/(n − 1) + ∈/(n − 1) for i > 2 for some ∈. Then, it is easy to see that for some ∈ under HAM, d11 = D 1 ∑ j and d12 = D 2 but d1 = D − 2. Thus, j d1 − d1 = −2.   Notice that under all the methods except LAR and SUQ, the leading candidate can be disadvantaged by splitting the apportionment into different pieces. However, the magnitude of the loss is less than the potential gain they could receive, particularly under ILQ or NIE. The exception is HAM, where the potential advantage and disadvantage are equal for any candidate. This advantage to the leading candidate is not surprising because disaggregating the apportionment into several pieces can allow any bias toward C1 to accumulate among the k allocations. We contrast∑ Proposition 6.7 with the potential losses of the weakest candidate. j Let δ(k, F) = j dn − dn be the difference in delegate totals Cn receives under the ∑ apportionment method F if D is divided into k groups such that j D j = D, where d j = F(v, D j ) and d = F(v, D). Similar reasoning as in the proof of Proposition 6.7 leads to the following.

162

6 Paradoxes

Proposition 6.8 If an n-candidate election is divided into k districts instead of a single district, then the disadvantage to Cn , δ(k, F), satisfies the following bounds, • • • • •

−(k − 1) ≤ δ(k, F) ≤ 0 when F is the LAR or the ILQ method; −k ≤ δ(k, F) ≤ k when F is the HAM method; ] ≤ δ(k, F) ≤ [ k2 ] + [ n−1 ] when F is the NIE method; −[ k2 ] − k[ n−1 2 2 k+3 3k −[ 2 ] ≤ δ(k, F) ≤ [ 2 ] when F is the NIS method; and −[ Dn ] ≤ δ(k, F) ≤ [ Dn ] when F is the SUQ method.

When there are only 2 candidates, the results are simpler. LAR, SUQ, and ILQ coincide, always apportioning d = ([q1 ], [q2 ]) delegates. HAM, NIE, and NIS coincide, apportioning d = ([q1 ], [q2 ]) delegates. Thus, we have the following. Proposition 6.9 For a 2-candidate election, • 0 ≤ Δ(k, F) ≤ k − 1 and −(k − 1) ≤ δ(k, F) ≤ 0 when F is LAR, SUQ or ILQ; and • − k2 ≤ Δ(k, F), δ(k, F) ≤ k2 when F is HAM, NIE or NIE. Although Propositions 6.7 and 6.9 place limits on the degree to which dividing into districts favors the leading candidate, in practice the leading candidate can be significantly advantaged, particularly when D is small. For example, in most district apportionments in the Republican primary, D = 3. If, as is frequently the case in early primaries, p1 < 1/3, then Dp1 < 1 and hence C1 is guaranteed at least 1, regardless of the apportionment method. A weaker candidate receiving, say, 5% of the vote, may get no district delegates at all. But in a state with k = 15 districts, this weaker candidate would have received 2 or 3 delegates had all 45 delegates been apportioned together. In general, for an election with many weaker (but not extremely weak) candidates and a relatively small D j , then using aggregation with any method can greatly favor the statewide vote leader(s), especially when there are many districts. This can be observed in the results from the NY primary in the 2008 and 2020 Democratic primaries, where we assume that the 15% threshold was not applied. Example 6.10 In the 2008 Democratic primary, NY had state 232 delegates, divided into 51 at-large, 30 PLEO, and 151 district delegates. Each of NY’s 29 congressional districts had either 5 or 6 delegates. The contest was essentially a two-way race between Hillary Clinton (56.50% of the statewide vote) and Barack Obama (39.71%) with John Edwards a distant third (1.16%). The remaining 1.52% of the vote was made up of blank or otherwise discarded ballots, as well as a small number of votes for the remaining candidates. The number of delegates received by each candidate is shown in Table 6.12 (column 2). By comparison, had all 232 NY delegates been apportioned together, Kucinich, and Richardson would each have received a delegate and Edwards’ total would have increased by 1 (see column 3). Thus, by splitting the

6.2 The Aggregation Paradox

163

Table 6.12 The 2008 and 2020 NY Democratic primaries with and without aggregation 2020 2008 Agg. No Agg. Candidate Agg. No Agg. Candidate Clinton Obama Edwards Kucinich Richardson

136 94 2 0 0

133 94 3 1 1

Biden Sanders Warren Bloomberg Buttigieg Yang Gabbard Klobuchar Patrick

199 50 17 4 2 1 0 0 0

192 48 14 7 4 4 2 2 1

delegates up and aggregating, Clinton received 3 more delegates than she might have otherwise. The situation was similar in 2020 where the top three candidates (Biden, Sanders, and Warren) each received more delegates than they would have had all the delegates been awarded together. Note that Joe Biden, the statewide vote leader in 2020, receives an extra 7 delegates from aggregation, more than doubling the gain experienced by Clinton in 2008. On the other hand, dividing delegates into different groups can work against the vote leader, especially if a threshold is applied. For instance, in a primary in which only the leading candidate meets the threshold in statewide votes, a weaker candidate may exceed the threshold in a specific district and receive a number of district delegates. But had the delegates been apportioned together, C1 would have received all the delegates. This happened in the 2020 CA Democratic primary where Sanders, Biden, Warren, and Bloomberg received 225, 172, 11, and 7 delegates, respectively. Warren and Bloomberg did not pass the statewide 15% threshold and so all of their delegates were earned at the district level. Had the delegates been apportioned at the statewide level in a single calculation, Sanders would have received 234 delegates. Thus, the process of aggregating results across distinct districts cost Sanders 9 delegates.

6.2.4 The Effect of Threshold Level on the Aggregation Paradox As can be seen in Tables 6.10 and 6.11, the presence of a threshold can have a large effect on the likelihood that the aggregation paradox occurs. Generally speaking, as we increase the threshold level the probability of the aggregation paradox occurring will decrease. The reason is that a higher threshold results in fewer candidates, pro-

164

6 Paradoxes

viding less of an opportunity for the paradox to occur. This observation is confirmed by the simulation results from Table 6.9, which shows an increase in the probability of the paradox as n increases. Therefore, the use of thresholds does not seem to be a problem with respect to the aggregation paradox; if anything, thresholds make the paradox less of a problem. This stands in sharp contrast to the elimination paradox from Sect. 6.1, which occurs only due to the existence of thresholds. Furthermore, the effect of the threshold level on the likelihood of the elimination paradox is more complicated. Increasing the threshold level can decrease the probability of the elimination paradox if the threshold is already set high, but increasing the threshold does not necessarily lower the probability of the paradox occurring, and often raises that probability.

6.3 No-Show Paradox The application of apportionment methods in conjunction with thresholds or aggregation can lead to paradoxical outcomes in which it is possible for a candidate to receive a better outcome in the primary election if that candidate’s vote total were unilaterally lowered. Such a situation is called a no-show paradox. In this section, we demonstrate how this paradox can arise either from using thresholds or from aggregation across CDs. We focus primarily on the standard notion of a no-show paradox, which arises in the delegate context due to the application of thresholds. We then discuss a weaker notion of the paradox that arises due to either thresholds or aggregation. Informally, the no-show paradox occurs if there exists a candidate Ci such that lowering Ci ’s vote total, while keeping the other vote totals constant, results in Ci receiving more delegates. This is considered paradoxical since candidates should not be disadvantaged by receiving more votes. An apportionment method susceptible to this paradox may lead to situations in which a candidate prefers that some of their voters not vote. We illustrate using data from the 2016 Hawaii Republican primary. Example 6.11 The 2016 Hawaii Republican primary involved 3 party delegates and 10 at-large delegates which were apportioned separately using the SUQ method with a 10% threshold based on the statewide vote distribution. For the purposes of this example, we combine both delegate categories into one apportionment problem with 13 delegates, as is the case in most Republican state primaries. (Atypically, the Hawaii Republican Party allocates the two categories separately.) Table 6.13 shows the state vote totals for the top four candidates. Carson receives less than 10% of the vote, and so is eliminated. Applying SUQ, the D = 13 delegates are allocated among the remaining three candidates resulting in the apportionment (7, 5, 1, 0).

6.3 No-Show Paradox

165

Table 6.13 Example of a no-show paradox based on the 2016 HI primary Trump Cruz Kasich Carson Votes

6805

5063

2068

1566

Others 206

Suppose now that 490 of Kasich supporters, instead of voting for Kasich, do not vote in the primary. Then, Kasich’s vote total decreases to 1578 and Carson’s votes are sufficient to cross the 10% threshold, leading to an apportionment of (6, 5, 2, 0). Thus, paradoxically, when Kasich loses 24% of his voters (while all other vote totals remain constant), Kasich’s delegate count increases. Example 6.11 typifies how the no-show paradox can arise in the presidential primaries. None of the apportionment methods used in the primaries are susceptible to the no-show paradox when used alone. However, when thresholds are applied, the situation changes. A candidate who is the victim of an elimination paradox could potentially counteract that paradox if some of their voters chose not to vote, causing other candidates to meet the threshold and change the allocation. This is the case in Example 6.11 where, based on the original vote totals, Kasich is the victim of an elimination paradox: If Carson is not eliminated then Kasich earns two delegates, but if Carson is eliminated Kasich earns only one. Kasich’s hypothetical loss of 490 voters causes Carson to surpass the 10% threshold, which results in Kasich regaining the delegate that he lost when Carson was eliminated. Both elimination and no-show paradoxes are related to the idea of a failure of monotonicity in voting theory. Recall that a voting system using ranked preference ballots is said to be monotonic if it is not possible to help (resp. hurt) a candidate by ranking the candidate lower (resp. higher) on some ballots. Informally, a monotonicity violation occurs if it is possible to help a candidate by lowering the candidate’s level of support or hurt a candidate by raising the level of support for the candidate. Example 6.11 illustrates exactly this kind of behavior in an apportionment context: If Kasich loses some of his supporters, then the apportionment outcome is better for him. Conversely, if the hypothetical election were the election that actually occurred, then Kasich would achieve a worse apportionment by gaining supporters. To define the no-show paradox formally, we introduce additional notation. For vote distributions v and vi' ≥ 0, let (v−i , vi' ) represent the vote distribution with vi replaced by vi' . Definition 6.3 Given N and D, a no-show paradox has occurred if for some candidate i, vote distribution v and vi' > vi , then di' < di for all d' ∈ F(D, (v−i , vi' )) and all d ∈ F(D, v). Since the no-show paradox occurs in the primaries only in conjunction with the elimination paradox, the geometry and frequency of the no-show paradox are closely related to that of the elimination paradox. We explore this fact in the next sections.

166

6 Paradoxes

Fig. 6.12 An example of the no-show paradox. If some supporters of C2 do not vote, then C3 is not eliminated and C2 gains a delegate

6.3.1 Geometry of the No-Show Paradox Because the no-show paradox is linked to the elimination paradox, it can be visualized by modifying the simplex pictures from Sect. 6.1. We illustrate in the following example. Example 6.12 Suppose n = 3, D = 5, v = (6625, 2775, 599) and the threshold is 6%. Figure 6.12 illustrates one of the shaded regions from Fig. 6.1, with v depicted by • (close to the point ⊗ in Fig. 6.1). Then C3 is eliminated, and under HAM, the apportionment is (4, 1). However, if 16 of C2 ’s voters stay home, the vote distribution is v' = (6625, 2259, 599) and the point • moves leftward along a projection line from (0, 5, 0) until it is above the threshold line and outside the shaded region (depicted as ◦). Then, C3 receives just over 6% of the vote and hence is not eliminated, resulting in an apportionment of (3, 2, 0). Thus, by voting, the 16 supporters of C2 cause their candidate to lose a delegate. Conversely, the absence of the 16 voters ensures that C2 is not a victim of the elimination paradox.

6.3.2 Likelihood of the No-Show Paradox Occurring As with the geometry, the link between the no-show and elimination paradoxes means that the frequencies with which these paradoxes occur are related. Thus, methods for which the elimination paradox is not likely to occur (such as HAM) are also less likely to produce a no-show paradox. While it is easy to construct examples similar to Examples 6.11 and 6.12 that demonstrate instances of the no-show paradox using any delegate apportionment method, in practice such examples are not likely to arise. This is because in order

6.3 No-Show Paradox

167

Table 6.14 Number of instances of the no-show paradox in the database DEM_DATA (left coordinate) and REP_DATA (right coordinate) HNH ILQ LAR NIE NIS SUQ Threshold HAM T T T T

= 15% = 10% = 5% = 1%

0, 1 0, 1 1, 0 0, 0

0, 1 0, 3 1, 2 0, 0

9, 2 7, 2 4, 3 1, 0

5, 7 12, 2 8, 1 3, 0

0, 4 3, 2 4, 2 0, 0

0, 4 3, 2 4, 1 0, 0

6, 7 11, 5 7, 3 3, 0

for a no-show paradox to occur, the election must have two necessary properties: (1) The elimination paradox occurs and (2) one of the eliminated candidates is very close to the elimination threshold. We can investigate the frequency of the no-show paradox by analyzing our database of primary elections. This is done as follows. First, we search the database to find all instances of the elimination paradox. For each of these occurrences, we determine all the candidates hurt by the paradox and unilaterally lower each of the vote totals of the eliminated candidates until the strongest of these now surpasses the threshold. Then, we check whether this causes the delegate count of any of the candidates effected by the elimination paradox to increase. The results of this analysis for both databases DEM_DATA and REP_DATA are shown in Table 6.14. Note that this paradox is extremely unlikely to occur for either database; however, the methods that are most susceptible to the elimination paradox are also the most susceptible to this paradox. One reason the no-show paradox occurs infrequently in practice is because the threshold is fixed. In the abstract, if thresholds are allowed to vary freely, almost any vote distribution and D value that can give rise to an elimination paradox can also give rise to the no-show paradox by varying the threshold value. For example, a no-show paradox would have occurred in the 2016 KY Republican primary (which used HAM rather than SUQ) if the threshold were decreased from the 5% threshold that was used to 0.081%. Despite the relative infrequency of the no-show paradox in Table 6.14, the possibility of it occurring is still disturbing. While the elimination paradox may strike many candidates (and voters) as hypothetical, the possibility that increasing voter support would work to the disadvantage of a candidate seems fundamentally undemocratic.

6.3.3 Who Is Affected by the No-Show Paradox? Because the no-show paradox arises in conjunction with the elimination paradox, the candidates hurt by the former are precisely those hurt by the latter. For example, candidate C1 cannot lose delegates under HAM, LAR, or SUQ as a result of the elimination paradox, and thus, the leading candidate is unaffected by the no-show

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6 Paradoxes

paradox under these methods. Under HNH, ILQ, NIE, or NIS, it is possible for C1 to lose a delegate after the elimination of the bottom candidates, and thus, it is possible using these methods to find examples in which C1 is better off losing votes in order to avoid the elimination paradox. In practice, Table 6.14 suggests that the question of which candidates are affected is largely moot because the paradox occurs so rarely.

6.3.4 The Effect of Threshold Level on the No-Show Paradox Varying the threshold can affect who is harmed by the no-show paradox and how frequently this occurs, although the size of these effects varies by apportionment method. Recall from Sect. 6.1.4 that for each method, it seems there is a threshold value that maximizes the likelihood of the elimination paradox occurring. This threshold then also maximizes the likelihood of the no-show paradox occurring. Generally speaking, if the threshold is very small or very large (a value that is close to 50%), then the probability of the elimination paradox occurring is very small, which in turn causes the probability of the no-show paradox occurring to be small. For other threshold values, we expect the probability of the no-show paradox occurring, like the elimination paradox, to be somewhat larger. However, given the infrequency demonstrated in Table 6.14, we expect this paradox to occur very rarely in practice, regardless of the threshold value.

6.3.5 Weak No-Show Paradox: The Effects of Aggregation The weak no-show paradox occurs when, if a candidate’s vote totals are unilaterally lowered, the candidate’s ranking among the other candidates improves. This may occur as a result of the candidate’s delegate count increasing (as with the no-show paradox). It may also arise as a result of changes to other candidates’ apportionments. Similar to the no-show paradox, the weak no-show paradox occurs only in connection with the use of thresholds or because of aggregation. In this subsection, we focus on issues arising from aggregation and hence restrict ourselves to cases when there are no thresholds. We define the weak no-show paradox as follows. Suppose there are k districts, j j with total delegates D 1 , . . . , D k and vote distributions v1 , . . . vk . Let (v−i , (vi )' ) j j ' denote the vote distribution in district j where vi is replaced with (vi ) . Definition 6.4 Given N , v1 , . . . vk , and D 1 , . . . , D k the weak no-show paradox j j , then Ci is higher has occurred if for some district j, candidate i and (vi )' < vi ∑ ranked (based on delegate total) for the sum of apportionments l/= j F(vl , D l ) + ∑ j j F((v−i , (vi )' ), D j ) than for the sum of apportionments l F(vl , D l ).

6.4 Alabama Paradox

169

Thus, a weak no-show paradox due to aggregation occurs if a candidate’s ranking based on delegate total is higher if some of the supporters in one of the districts do not vote. Unlike the no-show paradox, all candidates may be affected by the weak no-show paradox under any methods. Of most interest is when the ranking of the candidate with the most delegates changes due to non-voters of other candidates. Such a paradox may arise under many combinations of methods and delegate sizes. We illustrate one instance based on the rules of the Connecticut Republican Party. Example 6.13 The 2016 Connecticut Republican primary had 13 at-large delegates and 15 district delegates (3 each for the 5 CDs). Both statewide and district delegates were apportioned using NIE.4 Table 6.15 lists two possible vote distributions for a hypothetical election involving 3 candidates: A, B, and C. The total delegate count for the first vote distribution, (columns 2–4) is (14, 12, 2); the total delegate count for the second vote distribution, (columns 5–6) is (11, 12, 5). The only difference between the two vote distributions is that in the second election, candidate B has 1540 fewer voters across districts 1 and 2. This impacts the number of delegate allocated to A and C in both districts, as well as the statewide delegates, while leaving B’s delegate count unchanged. In aggregate, B’s ranking increases from second place to first despite the decrease in the number of votes. It is worth noting that if the 1540 voters who declined to vote for B in the second election in Example 6.13 had instead voted for C, the paradoxical shift in the apportionment would have been the same. This is an example of a migration paradox (see Bradberry 1992): If a set of voters switches from B to C, then it makes sense for the apportionment of C to increase at the expense of B. However, in this instance the shift in support from B to C actually helps B while hurting C.

6.4 Alabama Paradox The Alabama paradox is one of the best-known—and, in the US, one of the most historically important—apportionment paradoxes. In this section, we discuss how the paradox manifests in the presidential primaries and observe some of its consequences. Then, we analyze the paradox using the approaches adopted in the previous sections. In the apportionment literature, the Alabama paradox is said to occur when an increase in the number of house seats, while keeping the underlying state populations constant, causes a state to lose a seat. The paradox is named after the state of Alabama because in 1880, the US census clerk C. W. Seaton observed that under HAM, Alabama’s seat count would decrease from 8 to 7 if the size of the US House of Representatives was increased from 299 to 300 (Balinski and Young 2001). It is well known that HAM is susceptible to this paradox, and that this is the primary reason 4

As discussed in Chap. 4, some of the state party rules were not completely clear. The rules indicated using NIE if there was an under allocation of delegates, but was unclear about how to handle an over allocation. In the example given, we avoid over allocations.

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6 Paradoxes

Table 6.15 Two hypothetical elections using the rules of the 2016 CT Republican Party A B C A B C CD 1 votes CD 1 NIE appt CD 2 votes CD 2 NIE appt CD 3 votes CD 3 NIE appt CD 4 votes CD 4 NIE appt CD 5 votes CD 5 NIE appt Statewide votes At-large NIE appt Total appt

11,440 2

8640 1

3920 0

11,440 1

7600 1

3920 1

10,706 2

7700 1

3594 0

10,706 1

7200 1

3594 1

9940 1

10,500 2

560 0

9940 1

10,500 2

560 0

8867 1

9564 2

569 0

8867 1

9564 2

569 0

6907 1

4273 1

2820 1

6907 1

4273 1

2820 1

47,860

40,677

11,463

47,860

39,137

11,463

7

5

1

6

5

2

14

12

2

11

12

5

Candidate B loses 1540 voters but, as a result, moves from 2nd to 1st place in the primary

that HAM is no longer considered a viable method for apportioning House seats in the US. Arguments over this paradox grew so heated after the 1900 Census that Representative Charles Littlefield of Maine, a state whose HAM apportionment could have decreased from 4 to 3 had the House size been increased by the appropriate amount, exclaimed, “In Maine comes and out Maine goes . . . God help the State of Maine when mathematics reach for her and undertake to strike her down” (Balinski and Young 2001, p.41). In the context of delegate apportionment, the Alabama paradox occurs if an increase in the total number of delegates corresponds to a candidate receiving fewer delegates. The next example illustrates the paradox using data from the 2016 Hawaii Republican primary. Example 6.14 The state Republican Party of Hawaii used the SUQ method with no threshold in 2016 at both the statewide and district levels. Results from the election are shown in Table 6.16. There were 10 at-large delegates of which Rubio received 1. But had there been 9 at-large delegates then Rubio would have received two delegates. The Alabama paradox is defined formally as follows. Definition 6.5 Given v, D and an apportionment method F, the Alabama paradox has occurred if for some i, di' < di for all F(v, D) = d and F(v, D + 1) = d' . A

6.4 Alabama Paradox

171

Table 6.16 The allocation of 10 at-large delegates in the 2016 Hawaii Republican primary, which used the SUQ method Candidate Votes Quota Appt Quota Appt (D = 9) (D = 10) Trump Cruz Rubio Others

6805 5063 2068 1772

3.90 2.90 1.18 1.02

4 3 2 0

4.33 3.22 1.32 1.13

5 4 1 0

If only 9 delegates had been used, Rubio would have received an additional delegate

method F is susceptible to the Alabama paradox if the Alabama paradox occurs for some v and D. All the apportionment methods used in the primaries are quota-based; hence, it is easy to show they are all susceptible to the Alabama paradox. It could be argued that the Alabama paradox is not relevant in the context of delegate apportionment because a primary election is never rerun with the same vote totals and a higher value of D. When the 2016 Hawaii Republican primary occurred, the total number of at-large delegates was fixed at D = 10 and there was no question of changing it. However, the susceptibility of SUQ or any method to this paradox may be inappropriate for other reasons. For instance, recall that in the Republican Party primary, each state party receives a total number of delegates in proportion to how “red” the state has been in prior elections (see Chap. 4). Example 6.14 suggests that Rubio would have fared better if Hawaii had been less red, which is likely an inappropriate correlation—and certainly makes no sense in general. In a broader context, it is preferable for an apportionment method not to be sensitive to initial conditions such as D values. Small perturbations in D or other parameters such as vote distribution (assuming the small perturbations maintain the prior ordering of the candidates) should not cause large changes in the apportionment; otherwise the method is chaotic or unstable in an undesirable way (Saari 1999).

6.4.1 Geometry of the Alabama Paradox As with previous paradoxes, we can visualize the Alabama paradox using the simplex when n = 3. To see the effect of increasing the number of delegates by 1 for a given apportionment method, we overlay the simplex for D + 1 delegates on top of the simplex for D delegates. For a quota vector q, candidate Ci loses a seat when the number of seats increases from D to D + 1 if q is in the region corresponding to F(v, D) = (. . . , di , . . . ), but the quota vector for D + 1 delegates [(D + 1)/D]q is in the region corresponding to F(v, D + 1) = (. . . , di − 1, . . . ). Figure 6.13 shows the overlay of the HAM simplex for D = 4. Points in the gray regions correspond

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6 Paradoxes

Fig. 6.13 The shaded regions represent quota vectors that are susceptible to the Alabama paradox under HAM. The solid (resp. dashed) lines give a HAM simplex for D = 4 (resp. D = 5). The point ∗ maps to (2, 1, 1) when D = 4 but maps to (3, 2, 0) when D = 5, showing the loss of a delegate for C3 when D is increased. A version of this image appears in (Bradberry, 1992)

to instances in which the last-place candidate loses a delegate when D is increased from 4 to 5. Figure 6.14 illustrates the analogous picture (bottom left sixth only) as D increases from 4 to 5 under NIE (top left) and SUQ (top right). From the geometry, it is clear that the Alabama paradox is more likely to occur under SUQ than NIE in 3-candidate elections.

6.4.2 Likelihood of the Alabama Paradox Occurring Some apportionment methods are more susceptible to the Alabama paradox than others. In order to estimate the probability that a randomly chosen election is susceptible to the Alabama paradox, we use simulation based on a uniform distribution of vote totals with D = 25 and for n values from 3 to 7. The results are shown in Table 6.17. In general, most of the probabilities are not highly dependent on D; however, the probability of the paradox occurring for some methods does increase noticeably for very large D. The numbers in parentheses are probabilities derived from using a D value of 10,000, which gives an estimate of the value of the probability as D → ∞. Based on Table 6.17, the Alabama paradox is mostly likely to occur under SUQ and least likely to occur under HAM. As with previous paradoxes, we can investigate the probability of the Alabama paradox occurring empirically by querying our databases of election data. Table 6.18 shows the number of occurrences of the Alabama paradox in the databases DEM_DATA and REP_DATA. These results are based on 2206 elections with a threshold of 15% (where sufficient information is available) and 1981 elections with a variety of thresholds for which we have complete information. In each case, the

6.4 Alabama Paradox

173

Table 6.17 The probability, expressed as a percentage, that the Alabama paradox occurs for D = 25 assuming a uniform distribution on q HAM HNH ILQ LAR NIE NIS SUQ n 3

2.78

8.59

3.22

5.94

5.87

5.87

4

4.65

12.37

5.76

10.68

10.97

10.95

5

5.89

8.04

15.25

6

6.87

10.15

17.85

7

7.44

14.29 (15.4) 15.18 (16.6) 15.61 (17.3)

11.97

19.49

13.71 (15.1) 15.31 (16.9) 16.41 (18.1)

13.49 (14.9) 14.95 (16.6) 16.00 (17.8)

12.01 (15.7) 16.62 (20.7) 18.37 (22.9) 19.22 (24.0) 19.60 (24.6)

An estimate for a limiting value D → ∞ appears in parentheses if that value differs from the value in the table by more than a percentage point Table 6.18 Number of instances of the Alabama paradox from the elections in the DEM_DATA database (left coordinate) and REP_DATA database (right coordinate) under the given method and threshold HNH ILQ LAR NIE NIS SUQ Threshold HAM T T T T T

= 15% = 10% = 5% = 1% = 0%

1, 2 24, 16 58, 19 77, 31 85, 37

18, 20 95, 37 133, 41 125, 48 120, 47

29, 49 54, 48 51, 42 38, 39 31, 38

28, 16 55, 18 67, 27 95, 43 100, 51

17, 17 84, 25 107, 40 97, 56 94, 67

17, 17 84, 25 106, 40 88, 56 73, 67

40, 22 53, 35 75, 46 97, 51 104, 58

Alabama paradox was said to occur if the apportionment of any candidate would have decreased when moving from the D value in the election to D + 1. The data generally confirm the findings of Table 6.17 in terms of the relative likelihood of the paradox occurring for each method.

6.4.3 Who Is Affected by the Alabama Paradox? The answer to the question of whom is affected by the Alabama paradox and by how much varies depending on the apportionment method. As with the elimination paradox, methods that do not stray far from each candidate’s quota limit the size of the effect. For instance, under HAM and LAR, a candidate can lose at most one delegate when increasing from D to D + 1. This happens only if for D, then di = [Dpi ], and for D + 1, then di' = [(D + 1) pi ] and [Dpi ] = [(D + 1) pi ]. Similarly, under NIS, a candidate can lose at most two delegates if for D, then di = [Dpi ] + 1, and if

174

6 Paradoxes

for D + 1, then di' = [(D + 1) pi ] − 1 and [Dpi ] = [(D + 1) pi ]. Under SUQ, NIE or ILQ, however, a candidate can lose substantially more than one delegate. For a fixed n and method X , let M L n (X ) denote the maximum number of delegates that a candidate can lose under X when moving from D to D + 1 delegates, if the vote distributions are allowed to vary (we show below that this does not depend on D as long as D is suitably large). As above, let di' be the apportionment for Ci for a value of D + 1. Then, M L n (X ) = max{di' − di | i = 1, 2, . . . n and all q}. Proposition 6.10 The maximum numbers of delegates that a candidate can lose under the seven delegate apportionment methods when moving from D to D + 1, are • • • • •

M L n (HAM), M L n (LAR) = 1 for D ≥ n; M L n (NIS) = 2 for D ≥ n; M L n (ILQ) = n − 1 for D ≥ n − 2; M L n (SUQ) = n − 2 for D ≥ n 2 − 1; and M L n (NIE) = [n/2] for D ≥ n + 1.

Proof To prove the statement for ILQ, recall that when apportioning D delegates, each candidate is guaranteed at least [Dpi ] delegates and C1 receives at most [Dp1 ] + n − 1 delegates. When apportioning D + 1 delegates, C1 receives at least [(D + 1) p1 ] ≥ [Dp1 ] = [Dp1 ] + 1 delegates (if Dpi is not an integer ) or at least [(D + 1) p1 ] = [Dp1 ] + 1 = [Dp1 ] + 1 (if Dpi is an integer). Hence, C1 can lose at most n − 2 delegates, with the loss for other candidates necessarily smaller. This maximum loss occurs if p1 = 1 −

n−1 1 ∈ − ∈ and pi = + for i > 1. D+1 D+1 n−1

Consequently, F(p, D) = (D, 0, . . . , 0), but F(p, D + 1) = (D − (n − 2), 1, . . . , 1). Under SUQ, since [(D + 1) pi ] ≥ [Dpi ], the only way that Ci can lose delegates when D increases is for the upper quotas of some candidates C j , j < i, to increase, leaving fewer delegates for Ci . Since [(D + 1) pi ] − [Dpi ] ≤ 1 for all pi , the other candidates can gain at most one delegate when moving from D to D + 1. Thus, the largest number of delegates a candidate can lose occurs if i = n and [(D + 1) pi ] − [Dpi ] = 1 for i < n, leading to di − di' = n − 2. This occurs if ∈ 1 ∈ 1 [D − (n 2 − n − 1) − ], pi = [n − ] for i = 2, . . . , n − 1 and n−1 D n−1 D 1 pn = [n − 1 + ∈] for some ∈ > 0. D p1 =

Note that D ≥ n 2 − 1 implies pn = mini { pi }. Then, F(p, D) = d where d1 = D − (n 2 − n − 1), di = n, i = 2, . . . , n − 1 and dn = n − 1. But for ∈ small enough, [(D + 1) p1 ] = D − (n 2 − n − 1) + 1 and [(D + 1) pi ] = n + 1. Hence,

6.4 Alabama Paradox

175

∑

+ 1) pi ] = D, leading to F(p, D + 1) = d' where di' = [(D + 1) pi ] for i < n and dn' = 1.   The proof of the statement for NIE is similar. i d j and di' < di even though vi = vi' and vi = v'j , and thus, vi' /v'j = vi /v j . Therefore, in some sense the population paradox can be viewed as a generalization of the paradoxes examined in Sects. 6.1 and 6.3. Because all of the delegate apportionment methods used in the primaries are quota-based, they are all susceptible to the population paradox. Quantifying how frequently the paradox occurs is difficult, however, since vote distributions can vary so widely. In addition, the population paradox may affect all candidates in different circumstances. Recall from previous sections that, given a method and a paradox, some candidates cannot be hurt by the paradox under that method. For example, C1 cannot be hurt by the elimination, no-show, or Alabama paradoxes. The population paradox is more general than paradoxes like the no-show paradox, and thus, it is possible for any candidate to lose delegates in a hypothetical second election in which the population paradox occurs. Even using a method like SUQ which strongly favors the leading candidate can lead to C1 losing delegates through the population paradox, as the following example illustrates.

6.5 Population Paradox

181

Example 6.18 The 2016 TN Republican Party used the SUQ method with a 20% threshold to apportion its 31 at-large delegates. Columns 1–3 of Table 6.21 show the election results for the 3 viable candidates, and columns 4–6 show a hypothetical second election. In moving from the first election to the second, 1274 Rubio voters chose to vote for Trump instead, and an additional 20,000 Cruz supporters show up to vote for Cruz. Despite these changes, under SUQ, Rubio’s delegate count increases from 6 to 7 at the expense of the leading candidate, Trump. Thus, paradoxically, it would have been a good move for Rubio to persuade 1274 of his voters to vote for Trump and persuade an additional 20,000 people to vote for Cruz. As with previous paradoxes, the population paradox can occur and arguably be made worse by the use of thresholds or aggregation. It is unsurprising that thresholds can cause extreme examples of the population paradox because the application of a threshold can create distortions in small perturbations of vote totals when some of the vote totals are close to the threshold. Example 6.19 Recall from Chap. 4 that the 2016 NH Republican Party used an NIE-type method that did not re-adjust quotas after eliminating bottom candidates. To allocate the 23 delegates in the primary, quotas were calculated using the total statewide vote and were not recalculated after eliminating candidates who received less than 10% of the vote. The quotas of remaining candidates were rounded to the nearest integer, and any remaining delegates went to the leading candidate. Table 6.22 shows the primary results on the top and a second hypothetical election on the below it. In the actual election, the 45,613 votes for Others were spread over several other candidates, none of whom received more than 22,000 votes. In the second election, we assume that the extra votes for Others are distributed among the remaining candidates so that each of them receives no more than 34,000 votes. The second election should not produce a better outcome for Trump: He has lost votes while all of his opponents have gained votes. Yet Trump almost doubles his delegate total because in the second election only he and Kasich surpass the 10% threshold, and Kasich’s quota has decreased so that the fractional part is less than 0.5, resulting in Kasich receiving one less delegate. Examples like the above can be easily constructed for any of our apportionment methods. If an apportionment method that is susceptible to the population paradox is used in conjunction with aggregation, issues arising from this paradox can compound across the districts. We show how this occurs using the vote totals from the 2016 Massachusetts Democratic primary in CD 3 (see Example 6.17). Table 6.23 shows an election where the CD 3 data occur in every district for a hypothetical state with six districts, and a second election where the vote totals have been changed. Notice that C3 increases their overall delegate count by more than 50% by sending 400 of their voters to C1 and convincing an additional 6000 supporters of C2 to vote in each district.

182

6 Paradoxes

Table 6.22 The results of the 2016 NH Republican primary and a hypothetical second election outcome Trump Kasich Cruz Bush Rubio Others 2016 NH Republican primary Votes 100,735 44,932 8.103 3.614 Quota 11 4 Appt Hypothetical NH Republican primary 97,500 51,000 Votes Quota 6.586 3.445 20 3 Appt

33,244 2.674 3

31,341 2.521 3

30,071 2.419 2

45,613 – –

34,000 2.297 –

34,000 2.297 –

34,000 2.297 –

90,000 – –

Table 6.23 An example illustrating the effects of aggregation on the population paradox C1 C2 C3 C1 C2 C3 CDs 1–6 votes CDs 1–6 appt Statewide votes PLEO appt At-large appt Total appt

45,834

39,052

27,238

45,400

33,052

27,672

2

2

2

3

2

1

275,004

234,312

163,428

272,400

198,312

166,032

3 5

2 4

2 3

3 5

2 4

2 3

20

18

17

26

18

11

6.6 Uniformity In Sect. 6.1, we discussed the elimination paradox and why it is relevant in the context of delegate apportionment. Many of the examples of the elimination paradox that we examined can be understood as failures of uniformity, a notion introduced in (Balinski and Young, 1978). In this section, we introduce the notion of uniformity and discuss why it is desirable for delegate apportionment methods to have this property. Informally, an apportionment method is uniform if whenever an apportionment is determined, the apportionment of any subset of candidates would be unchanged if the apportionment were recalculated based only on the candidates in that subset, using the total number of delegates allocated originally to the candidates in that subset. The following example illustrates a failure of uniformity. Example 6.20 In the 2020 Colorado Democratic primary, only 4 candidates received more than 15% of the vote: Sanders, Biden, Bloomberg, and Warren with 355,293, 236,565, 177,727, and 168,695 votes, respectively. Under HAM, the 14 at-large delegates were apportioned as (5, 4, 3, 2). However, had the 11 delegates received

6.6 Uniformity

183

in total by Sanders, Biden, and Warren been reallocated using the same vote totals (using HAM), the apportionment would have been (5, 3, 3). Had HAM satisfied uniformity, the second apportionment would have been (5, 4, 2), agreeing with the apportionment for these three candidates when Bloomberg was included in the calculation. (We note that in the 2020 primary, neither Bloomberg nor Warren received any delegates because they withdrew from the primary after the election.) To define uniformity formally, we use the following notation. Suppose d ∈ F(v, D) is an apportionment of D delegates given vote distribution v for a set of N = {C1 , . . . , Cn } candidates. Given a subset of candidates, S ⊂ N , let v S (respectively d S ) be the restriction of v (respectively d) to the coordinates vi (respectively di ) where Ci ∈ S. This definition is framed differently than Definition 5.10 but is equivalent to it. Definition 6.7 A method F is uniform if the following condition is met. For all N , v and D with d ∈ F(v, D), and for all S ⊂ N , then ∑ (i) d S ∈ F(v S , D ' ) where D ' = Ci ∈S di ; and (ii) if e ∈ F(v S , D ' ) then dˆ ∈ F(v, D) where dˆi = ei for all Ci ∈ S and dˆi = di else. As Balinski and Young (1978) observe, uniformity is similar to the independence of irrelevant alternatives criterion from social choice theory, which (informally) states if an electorate prefers candidate A over candidate B then the electorate still prefers A over B if a third candidate C is introduced into the election. That is, the outcome between A and B should be unaffected by the presence of another candidate C. Similarly, in Example 6.20 the outcome of the apportionment for Sanders, Biden, and Warren should not be affected by the presence of Bloomberg; Bloomberg should be irrelevant to their sub-apportionment. In social choice theory, most voting methods fail the independence of irrelevant alternatives criterion. In contrast, among apportionment methods, the divisor methods do satisfy uniformity; however, quota-based methods do not (Balinski and Young 2001). This implies that the apportionment methods used in the primaries are subject to failures of uniformity. In fact, many instances of the elimination paradox can be interpreted as failures of uniformity, as illustrated in the following example. Example 6.21 The 2016 Kentucky Republican primary used HAM with a 5% cutoff, resulting in an apportionment of (17, 15, 7, 7) for Trump, Cruz, Rubio, and Kasich, respectively. If the bottom-ranked candidates had not been eliminated, then the apportionment would have been (17, 14, 8, 7, 0, . . . , 0), demonstrating that Rubio was a victim of the elimination paradox. Interpreted another way, however, had HAM satisfied uniformity, the apportionment of the subset of 4 candidates would have been (17, 14, 8, 7). One advantage of uniformity is that uniform apportionment methods are not susceptible to the Alabama paradox (Balinski and Young 2001). We use this to show the following.

184

6 Paradoxes

Proposition 6.11 Uniform apportionment methods are not susceptible to the elimination paradox. Proof Suppose an apportionment method is uniform and that d ∈ F(v, D). Further, suppose that a ∑ set of candidates is eliminated and that S is the set of candidates remaining. Let D ' = Ci ∈S di be their total delegates count. By uniformity if d' ∈ F(v, D ' ), then di' = di for all i ∈ S. Since D ≥ D ' and uniform apportionment methods are not susceptible to the Alabama paradox, this implies that if d S ∈ F(v S , D), than diS ≥ di' = di for all Ci ∈ S. Hence, no candidate Ci ∈ S can be the victim of an   elimination paradox. Thus, uniform methods, such as divisor methods, also avoid instances of the noshow paradox that arise due to thresholds. (The no-show paradox may still arise due to aggregation.) In contrast, the susceptibility of the quota-based methods used in the primaries to various paradoxes can be understood as stemming from their failure of uniformity. Furthermore, a method’s irrational behavior in response to elimination of candidates or changes in voter distribution can be seen as analogous to failures of the independence of irrelevant alternatives criterion in social choice theory.

6.7 Conclusion As the preceding sections have documented, all the methods used in the primaries are susceptible, to varying degrees, to a variety of well-known paradoxes such as the Alabama and population paradoxes. The methods are also subject to other paradoxes arising from their use in conjunction with thresholds or aggregation that are unique to the context of delegate apportionment in the presidential primaries. There are significant differences among the methods with respect to the frequency, extent, and the identity of candidates affected by these paradoxes. In addition, the paradoxes respond differently to changes in threshold. For instance, the frequency of the aggregation or Alabama paradoxes generally decreases as the threshold rises—primarily because there are fewer candidates. However, the frequency of the elimination, and hence, the no-show paradoxes, is affected in more complex ways to changing thresholds, with each apportionment method appearing to have a threshold or range of thresholds (for fixed D and n) that maximize the likelihood of the paradox arising. The widespread use of thresholds in the primaries suggests that it would be advantageous for the primaries to adopt methods that satisfy uniformity. In the next section, we explore the consequences of replacing the quota-based methods with divisor methods.

References

185

References Balinski ML, Young HP (1978) Stability, coalitions and schisms in proportional representation systems. Am Polit Sci Rev 72(3):848–858. https://doi.org/10.2307/1955106 Balinski ML, Young HP (2001) Fair representation: meeting the ideal of one man, one vote, 2nd edn. Brookings Institution Press, Washington, DC Bradberry BA (1992) A geometric view of some apportionment paradoxes. Math Mag 65(1):3–17. https://doi.org/10.2307/2691355 Danˇcišin V (2017) No-show paradox in Slovak party-list proportional system. Human Affairs 27(1):15–21 Jones MA, Wilson J (2016) The geometry of adding up votes. Math Horizons 24(1):5–9. https:// doi.org/10.4169/mathhorizons.24.1.5 Jones MA, McCune D, Wilson JM (2019) The elimination paradox: apportionment in the Democratic party. Public Choice 178(1):53–65. https://doi.org/10.1007/s11127-018-0608-3 McCune D (2023) The many apportionment paradoxes of the 2020 Iowa Democratic presidential caucuses. https://doi.org/10.1007/s00283-022-10196-9, to appear McCune D, McCune L, Nelson D (2019) The cutoff paradox in the Kansas presidential caucuses. UMAP J 40(1):21–45 Pukelsheim F (2017) Proportional representation: apportionment methods and their applications. Springer, Cham, second edition. (With a foreword by Andrew Duff) Saari DG (1999) Chaos, but in voting and apportionments? Proc Natl Acad Sci USA 96(19):10568– 10571. https://doi.org/10.1073/pnas.96.19.10568

Chapter 7

Exploring Alternative Ways to Allocate Delegates

In this last chapter, we consider whether there are ways to improve how delegates are allocated to candidates in the presidential primaries. More specifically, we ask: Are there aspects of the delegate selection process that can be altered to avoid some of the temperamental behavior and paradoxes exhibited by the apportionment methods described in Chaps. 5 and 6? Further, will more mathematically desirable solutions to the delegate apportionment problem align with the goals of the national and state Democratic and Republican parties? The convoluted processes laid out in the current delegate selection plans are a product of tradition, expediency and politically forged compromises. The processes differ substantially by party and, in the case of the Republican primary, by individual state. But the delegate selection processes as a whole—the allocation of delegates to states and districts and then to candidates—were seemingly not designed with an understanding of apportionment theory or indeed, of the consequences of the some of the methods chosen. In this chapter, we consider some alternatives to the current practices, particularly in how delegates are awarded to candidates. These alternate methods would simplify the process and make it more consistent. Ideally, too, the modifications would encompass the desire, explicit in most of the apportionment methods used for delegate apportionment, of allowing some bias toward stronger candidates. However, standardization and conformity run the risk of suppressing real differences between parties and among states. On the Democratic Party side, the priority is to uphold a fairly strict adherence to proportionality, because the apportionment method used by the Democratic Party (HAM) is unbiased (as measured by asymptotic delegate bias). The uniform 15% threshold introduces a measure of preference toward stronger candidates by ensuring at most 6 candidates receive delegates in any single contest. On the Republican side, the variety of state party delegate selection plans © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9_7

187

188

7 Exploring Alternative Ways to Allocate Delegates

suggest different approaches to balancing proportionality with a winner-take-most philosophy. At one end are state parties like those of KY and VA which use HAM and low thresholds of 5 and 0%, respectively, which allow for more candidates to receive delegates and for the allocation to be more closely proportional. At the other extreme are state parties like those of VT and AL which use NIE and complicated threshold structures that include the stipulation that a majority candidate receives all the delegates. The rules in TX exemplify how complicated the relationship between the apportionment method and threshold structure can be in manifesting bias: A majority candidate receives all the delegates; if no candidate receives a majority, then delegates are apportioned to candidates receiving at least 20% of the vote using SUQ, unless fewer than two candidates receive more than 20% of the vote; if only one candidate receives more than 20%, the delegates are apportioned to the top 2 candidates using SUQ; if no candidate receives more than 20% of the vote, the delegates are apportioned to all candidates using SUQ. Complicating the factors that determine how many delegates each candidate receives is the inconsistent behavior of some of the delegate apportionment methods. Methods such as NIE and ILQ are designed to favor the strongest candidate, yet do not always display this bias in a systematic or reasonable fashion. As an example, under NIE, if q = (9.2, 8.4, 7.4, 6.4, 6.4, 2.4, 2.4, 1.4) then C1 receives 12 delegates; but if q = (10.4, 7.9, 6.9, 6.4, 6.2, 2.5, 2.1, 1.6) then C1 receives only 10 delegates despite the stronger showing. Thus, the size of the “bonus” given to C1 is an accident of nearest integer rounding of quotas and has little to do with the leading candidate’s margin of victory. The susceptibility of the delegate apportionment methods to the paradoxes of Chap. 6 means, too, that delegate counts can change unexpectedly in response to changes in the delegate size D, to changes in the vote distribution v, or after a threshold is applied. These changes may be small, when there are few delegates at stake; they become more significant when the number of delegates is greater, as occurs when allocating at-large delegates in more populated states. The changes could also have a bigger cumulative effect if they occur in multiple state primaries as the calendar unfolds. This analysis suggests that if some degree of bias is to be endorsed, it would be better to rely on apportionment methods that are not based on quotas or, if the methods are quota-based, then the degree of disproportionality that methods represent should be less sensitive to small changes in the quota or to the relationship between D and n in close elections as discussed in Sect. 5.3.4. Additionally, the range of delegate selection plans adopted by the different states suggests that what is required is not a single method but a set of methods that allow for flexibility: a family of methods or a unified approach that can be adjusted to allow for a greater or a lesser amount of bias. One of the unique aspects of delegate apportionment is that the allocation of delegates takes place over time. This means that as the calendar unfolds, each state’s primary represents an entirely different apportionment problem: not only do the apportionment methods vary, but so do the number of delegates, and even the number of candidates. The current relationship between the delegate selection rules and the calendar is minimal. In the Democratic primary, the same apportionment method

7.1 Other Apportionment Methods

189

is used regardless of the date of the state’s electoral contest. In the Republican primary, the only distinction made for the calendar is the identification of a date before which delegate allocation must be “proportional.” Arguably, the Democratic and Republican Parties might want to tie the delegate selection plan more closely with the calendar. For instance, for state primaries earlier in the calendar, it might be advantageous to allocate delegates to more candidates in order to encourage more public interest and to allow the party to shape its message. In contrast, for primaries occurring later in the calendar, the parties may want the apportionment methods to be biased toward stronger candidates in order to build momentum for the final nominee. Thus, there are advantages to having a flexible allocation plan, or a family of apportionment methods, that allows the amount of bias to be adjusted by individual state parties—or by national committees. We discuss this further in Sect. 7.5. In the remainder of the chapter, we consider alternative apportionment methods and more systematic approaches to adjusting bias toward stronger or weaker candidates in delegate allocation. In Sect. 7.1, we introduce two other families of apportionment methods: the shift-quota methods and divisor methods, which were discussed briefly in Chap. 1. In Sect. 7.2, we consider traditional measures of fairness and analyze how the current delegate apportionment methods fare based on these measures in comparison with the shift-quota and divisor methods. In Sect. 7.3, we discuss how the principle of degressive representation, or bias toward smaller states, has been implemented and theorized about in the allocation of national representatives to the European Parliament in the European Union (EU). These ideas are adapted in Sect. 7.4, where we outline several ways in which regressive representation, or bias toward larger states (or stronger candidates), can be incorporated into new delegate apportionment methods. We conclude with an overview of the unique factors that characterize delegate apportionment and some further suggestions in Sect. 7.5.

7.1 Other Apportionment Methods In this section, we briefly describe two families of apportionment methods: shiftquota methods and divisor methods, and discuss the biases they exhibit.

7.1.1 Shift-Quota Methods Hamilton’s method is the most commonly used quota-based apportionment method. Yet it is only one of a family of quota methods, known as the shift-quota methods. Shift-quota methods share a similar methodology: round down a quota and then allocate the remaining delegates based on the size of the remainders. See (Pukelsheim 2017, Chap. 5) for a thorough discussion about shift-quota methods and their use.

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7 Exploring Alternative Ways to Allocate Delegates

Table 7.1 Example 4.1 from Chap. 4 reapportioned using shift-quota methods Cand. Votes qi0 HAM qi0.3 SQ qi0.9 (s = 0) (s = 0.3) 1 2 3 4 5 6

29,130 20,000 17,720 16,750 16,550 15,350

25.473 17.489 15.495 14.647 14.472 13.423

25 18 16 15 14 13

25.549 17.541 15.541 14.691 14.515 13.463

26 17 16 15 14 13

25.700 17.645 15.633 14.778 14.601 13.543

SQ (s = 0.9) 26 18 15 15 14 13

Definition 7.1 Shift-Quota Method (SQ): Under the shift-quota (SQ) method with s)(vi /V ) is determined. Each parameter s ∈ [−1, 1), an adjusted quota qis = (D + Σ s ] delegates. The remaining D − i [qis ] delegates are assigned candidate receives [qiΣ one each to the D − i [qis ] candidates with largest remainders qis − [qis ]. While shift-quota methods are defined for values of s < 0, we will generally be concerned only for s ∈ [0, 1). Note thatΣ unless all for each s, qis − 1 < [qis ] ≤ qis , soΣ s s [q ] ≤ D + s. It follows that the quotas are integers, D + s − n < i [qi ] > Σ s Σ is i D − s − n ≥ D − 1 − n. Because i [qi ] is an integer, then i [qi ] ≥ D − n. Consequently, after the initial allocation, there are at most n delegates remaining to apportion. If s = 0, then the SQ method reduces to HAM. We illustrate shift-quota methods by adapting Example 4.1 from Chap. 4. Example 7.1 Suppose that there are n = 6 candidates and D = 101 delegates. Table 7.1 shows the apportionment using three shift-quota methods. The allocation for s = 0.9 coincides with that under NIS; the allocation for s = 0.3 does not coincide with any of the current delegate apportionment methods. Like HAM, the shift-quota methods are subject to many of the paradoxes discussed in Chap. 6. However, by varying the value of the parameter s, they allow the apportionment to exhibit modest bias toward the stronger (or weaker) candidates. Since all SQ methods are based on the ordering of the remainders, they cannot be linearly ordered based on delegate majorization, as discussed in Sect. 5.3.1. Nonetheless, the bias can be observed through the formula for the asymptotic seat biases. Recall that the asymptotic delegate bias (see Sect. 5.3.2) is the expected deviation of a candidate’s delegate allocation from its quota, assuming that all distributions (in the simplex) are equally likely to occur as D → ∞. Pukelsheim (2017) provides an explicit formula for the delegate (seat) bias of a SQ method with parameter s and threshold value t, given by ⎛ ⎞ n Σ s 1 − 1⎠ · (1 − nt). Bis = ⎝ n j=i j

(7.1)

7.1 Other Apportionment Methods

191

Table 7.2 Asymptotic delegate biases of the top two and bottom two candidates under several shift-quota and stationary divisor methods for n = 3, 4, 5, 6 n=3

Shift-quota methods s=0

1

0

5 60 (0.08)

2

0

1 (−0.02) − 60

3

0

4 (−0.07) − 60

4 (−0.20) − 20

n=4

s = 0.3

Stationary divisor methods

Ci

s = 0.9

s=0

1

0

2

0

3

0

4

0

n=5

s = 0.3 13 160 (0.08) 1 160 (0.01) 5 − 160 (−0.03) 9 (−0.06) − 160

s=0

1

0

2

0

4

0

5

0

n=6

s = 0.3 77 1000 (0.08) 17 1000 (0.02) 33 (−0.03) − 1000 48 (−0.05) − 1000

s=0

1

0

2

0

5

0

6

0

s = 0.3 87 1200 (0.07) 27 1200 (0.02) 38 (−0.03) − 1200 50 (−0.04) − 1200

0.5

5 (−0.42) − 12

0

5 12 (0.42)

1 12 (0.08) 4 (0.33) 12

0

1 (−0.08) − 12

−0.5

0

4 (−0.33) − 12

−0.5

−1

s=1

ILQ

SUQ

Stationary divisor methods s = 0.9 39 160 (0.24) 3 160 (0.02) 15 − 160 (−0.09) 27 (−0.17) − 160

231 1000 (0.23) 51 1000 (0.05) 99 (−0.10) − 1000 144 (−0.14) − 1000

0.5

s=0

s = 0.5

s=1

13 (−0.54) − 24

0

1 (−0.04) − 24 5 24 (0.21) 9 24 (0.38)

0

13 (0.54) 24 1 24 (0.04)

0

5 (−0.21) − 24

−0.5

0.5

0

9 (−0.38) − 24

−0.5

−1.5

ILQ

SUQ

Stationary divisor methods s = 0.9

Shift-quota methods

Ci

1

5 20 (0.25) 1 (−0.05) − 20

Shift-quota methods

Ci

SUQ

s = 0.5

Shift-quota methods

Ci

ILQ

s=0

s=0

s = 0.5

s=1

77 (−0.64) − 120

0

17 (−0.14) − 120

0

33 120 (0.28) 48 120 (0.40)

0

77 120 (0.64) 17 120 (0.14) 33 (−0.28) − 120 48 (−0.40) − 120

0

Stationary divisor methods s = 0.9

s=0

s = 0.5

s=1

87 400 (0.22) 27 400 (0.68) 38 (−0.10) − 400 50 (−0.13) − 400

87 (−0.73) − 120

0

27 (−0.23) − 120 38 120 (0.32) 50 (0.42) 120

0

87 120 (0.73) 27 120 (0.23) 38 (−0.32) − 120 50 (−0.42) − 120

0 0

1.46

0.5

−0.46

0.5

1.92

0.5

−0.42

0.5

−0.5

0.5

−0.5

−2

ILQ

SUQ

2.36

0.5

−0.37

0.5

−0.5

−0.5

−0.5

−2.5

The numbers in parentheses are decimal approximations

Equation 7.1 shows that if s > 0 (resp. s < 0), the SQ method is biased in favor of strong (resp. weak) candidates, and that this bias is proportional to |s|. If s = 0, the asymptotic delegate bias for all candidates is equal to 0, which coincides with the results for HAM in Table 5.4. Table 7.2 indicates the biases for the top and bottom two candidates for s = 0, s = 0.3 and s = 0.9 (and threshold t = 0); it also includes the delegate biases for the divisor methods discussed in Sect. 7.1.2 and for ILQ and SUQ from Table 5.4 for comparison. The values in Table 7.2 show that the biases exhibited by different shift-quota methods are much smaller in absolute value than those under ILQ and SUQ.

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7 Exploring Alternative Ways to Allocate Delegates

Table 7.3 Example 4.1 from Chap. 4 reapportioned using divisor methods Cand.

Votes

vi /1181

Adams

vi /1143

HillHunt.

vi /1142.5 Webster

vi /1110

Jefferson

1

29,130

24.66

25

25.48

25

25.49

25

26.24

26

2

20,000

16.93

17

17.49

18

17.50

18

18.01

18

3

17,720

15.00

16

15.50

16

15.50

16

15.96

15

4

16,750

14.18

15

14.65

15

14.66

15

15.09

15

5

16,550

14.01

15

14.47

14

14.48

14

14.90

14

6

15,350

12.99

13

13.42

13

13.43

13

13.82

13

The decimal values are to two decimal places, but not rounded

7.1.2 Divisor Methods Divisor methods constitute the most frequently used class of apportionment methods. As discussed in Sect. 1.3, each divisor method corresponds to a rounding rule f that assigns to each positive integer k a value f (k) ∈ [k − 1, k] used as a dividing point to determine which numbers are rounded down and which are rounded up in the interval. Numbers between k − 1 and f (k) are rounded down to k − 1; numbers between f (k) and k are rounded up to k. In order to avoid ambiguity, there cannot exist positive integers a and b for which f (a) = a − 1 and f (b) = b. By convention, f (0) = 0, so that positive numbers below f (1) are rounded to 0. Definition 7.2 A divisor method F with rounding rule f satisfies F(D, v) = d if f (di ) ≤ vi /x ≤ f (di + 1) for each i, where di there exists a divisor x > 0 such thatΣ are non-negative integers satisfying i≤n di = D. Note that if f (1) = 0, then each candidate with a positive share of the vote receives at least one delegate as long as D ≥ n. If f (1) = 0 and D < n, then by convention each of the D candidates with the largest vi receives a delegate. The most commonly used divisor methods include the methods of Jefferson, Adams, and Webster; the first two are also referred to, primarily in Europe, as the D’Hondt and Sainte-Laguë methods, respectively. Jefferson’s method corresponds to the rounding rule f (k) = k, which rounds every quotient up. Adams’ method corresponds to the rounding rule f (k) = k − 1, which rounds every quotient down. Webster’s method with rounding rule f (k) = k − 1/2 corresponds to standard rounding in which decimals below 0.5 are rounded down and decimals equal to or above 0.5 are rounded up. All three methods are members of a larger class of divisor methods: the stationary divisor methods with rounding rule f (k) = k − s where 0 ≤ s ≤ 1. Another well-known class of divisor methods are those based on power means in which f p (k) = 21 [(k − 1) p + k p ]1/ p for some p. By convention, √ f −∞ (k) = k − 1, f ∞ (k) = k and f 0 (k) = k(k − 1). The latter, which in the US is called Hill-Huntington’s method and is used to allocate seats in the House of Representatives, corresponds to rounding based on the geometric mean. We illustrate these divisor methods by again adapting Example 4.1 from Chap. 4.

7.1 Other Apportionment Methods

193

Example 7.2 Suppose there are n = 6 candidates and D = 101 delegates. Table 7.3 shows the apportionment using four of the best-known divisor methods. The values of x used as the divisor for each method are indicated in the first row. As noted previously, Jefferson’s, Webster’s and and Hill-Huntington’s methods have all been used to apportion the US House (1791–1842; 1842–1852 and 1901– 1941; and 1941 to the present, respectively). Balinski and Young (2001) provide a thorough account of both the mathematics and the history of the use of apportionment methods in the US. Divisor methods are also widely used to apportion representatives to legislative bodies. Pukelsheim (2017) provides a detailed discussion of the implementation of divisor methods for party seat representation within different European countries and in the European Union. Divisor methods avoid all but one of the paradoxes discussed in Chap. 6. In particular, they are not subject to the Alabama paradox, the population and no-show paradoxes, and the elimination paradox, and they satisfy uniformity. Divisor methods are subject to the aggregation paradox, as all apportionment methods are. The primary disadvantage to divisor methods is that they sometimes break quota, which may not be a weakness in the presidential primary setting depending on which candidates are advantaged or disadvantaged. The behavior of divisor methods with regard to bias is also well understood. A divisor method F1 with rounding rule f 1 is majorized by a second divisor method F2 with rounding rule f 2 if and only if f 1 (k − 1) f 1 (k) < for all k ≥ 2. f 2 (k − 1) f 2 (k) This means that the apportionment F1 (D, v) will always favor weak candidates relative to F2 (D, v). (See Sect. 5.3 for a full discussion of majorization and bias among delegate apportionment methods.) Among stationary divisor methods, the divisor method with rounding rule f (k) = k − s1 will be majorized by the divisor method with rounding rule f (k) = k − s2 for all s1 > s2 . As a consequence, Adams’ method is the most biased in favor of weak candidates and Jefferson’s method is the most biased in favor of strong candidates. This behavior can be observed in Table 7.3 where the apportionments are listed in leftto-right in order of increasing bias toward candidates with larger vote totals. Among power-mean divisor methods, the divisor method with rounding rule f p1 (k) will be majorized by the divisor method with rounding rule f p2 (k) whenever p1 < p2 . The asymptotic delegate biases of the stationary divisor methods have also been explicitly determined, as presented in (Pukelsheim 2017). For stationary divisor method with rounding rule f (k) = k − s and threshold value t, the formula is given by ⎛ ⎞ ) Σ ( n 1 ⎝ 1 − 1⎠ · (1 − nt). Bis = s − 2 j j=i

194

7 Exploring Alternative Ways to Allocate Delegates

The ordering generated by the bias equation coincides with that generated by the majorization relationship. Thus, Adams’ method shows most bias in favor of weak candidates and Jefferson’s method shows most bias in favor of strong candidates, while the bias for Webster’s method is 0. The values of the asymptotic delegate biases are shown in Table 7.2 where it is clear that Jefferson’s method has more bias toward the stronger candidates than any of the shift-quota methods. The fact that the bias toward C1 is greater than 0.5 for n ≥ 3 indicates that on average, the strongest candidate gets more than their quota rounded up. Conversely, the fact that the bias against Cn is greater than −0.5 indicates that on average, the weakest candidate gets more than their quota rounded down. In fact, Jefferson’s method is known to be the unique divisor method that does not break lower quota. Both shift-quota and divisor methods provide examples of alternatives to the methods currently used to apportion delegates to candidates in the presidential primaries. The rationale for selecting one method over the other, particularly among the divisor methods, has traditionally been based on the kind of bias they exhibit, as well as how they perform on one of a number of measures of fairness, discussed in the next section.

7.2 Measures of Fairness Historically, there has been much debate about what the “best” or “fairest” apportionment method is. While bias plays a factor, so does an apportionment method’s “closeness” to true proportionality. However, this statement raises the question: How should closeness be measured? For each candidate i, should closeness be measured by the quantity di − qi , or di /D − vi /V , or possibly vi /di − V /D? What about for closeness for the apportionment of all candidates? Should this be calculated using a total deviation or a max-min approach? Because there are multiple of ways to measure closeness to proportionality, it is perhaps not surprising that different apportionment methods can be supported because they satisfy different criteria. In this section, we review how to evaluate and compare apportionment methods based on standard criteria. Except where a different context is crucial, we discuss these ideas using the terminology of delegate (rather than seat) allocation. One approach is to ask whether an allocation can be improved by transferring a delegate from one candidate to another. For instance, suppose we focus on minimizing the differences between the ratios di /vi . If di /vi is larger than d j /v j for some i and j, then candidate Ci is more favored than C j and we transfer one delegate from Ci to C j if (di − 1)/vi − (d j + 1)/v j < di /vi − d j /v j . This idea was introduced by Huntington (1928), who showed that after a finite number of such transfers, an allocation is obtained where no additional transfers can be made. Moreover, the resulting allocation coincides with Webster’s method. Swapping di /vi − d j /v j for other criteria, such as vi /di − v j /d j , leads to other apportionment methods or sometimes, as with vi /v j − di /d j , to no stable allocation. Huntington (1928) argued for the criterion (di /vi )/(d j /v j ) − 1, which leads to the Hill-Huntington method.

7.2 Measures of Fairness

195

This delegate-transfer approach makes the most sense when apportionment methods satisfy uniformity (see Sect. 6.6): The allocation remains unchanged if any pair of candidates reallocate the total number of delegates they are given using the apportionment method, while the other candidates’ shares are fixed. Uniform apportionment methods have the property that the allocation of delegates among any pair of candidates depends only on their relative vote distribution. All divisor methods satisfy uniformity, whereas none of the quota-based methods used in the primaries do, as discussed in Chap. 6. Another closely related approach used to justify one apportionment over another is based on optimization. An error function or a measure of disproportionality is selected and then an allocation is chosen that minimizes the error. As with the criteria for delegate transfers, there are a number of reasonable error functions to consider. Among the most commonly used measures are the following, where pi = vi /V and si = di /D are vote share and delegate share, respectively. • Maximum deviation: M D = maxi |si − pi | = maxi D1 |di − qi | Σ 1 Σ • Loosemore-Hanby Index: L H = 21 i |si − pi | = 2D i |di − qi | / Σ / Σ • Gallagher Index: G = 21 i (si − pi )2 = D1 21 i (di − qi )2 • Max delegate share to vote share: M D Q = maxi di /qi = maxi si / pi • Max vote share to delegate share: M Q D = maxi qi /di = maxi pi /si Σ Σ • Entropy: E = i di ln(si / pi ) = i di [ln(di /vi ) − ln(V /D)] More recently, Martínez-Panero et al. (2019) introduced the quota-disproportion index (QI). This index is designed to measure the amount of non-forced disproportionality of an allocation—the amount by which an apportionment method departs from quota. It is defined by QI =

⎧ ⎨ Σ

⎫ ⎬

Σ

1 max (di − [qi ]) , ([qi ] − di ) . ⎩ ⎭ D {i|d >q } {i|d v2 > · · · > vn , then an allocation is defined to be degressive if d1 /v1 < d2 /v2 < · · · < dn /vn . Similarly, an allocation is regressive if d1 /v1 > d2 /v2 > · · · > dn /vn . It is not always possible to achieve a reasonable allocation that satisfies strict degressiveness. In practice, allocations in the EU have required the apportionment to satisfy d˜1 /v1 < d˜2 /v2 < · · · < d˜n /vn where d˜i is a function of the population after dividing by a divisor specified by the rounding rule, before rounding; see (Pukelsheim and Grimmett 2017) and (Słomczy´nski and Zyczkowski 2012) for a fuller discussion. The problem of allocating seats in the EU while maintaining degressive proportionality was made more urgent with the signing of the 2007 Lisbon Treaty which capped the European Parliament at 751 seats and also set minimum and maximum caps on national representation: Countries must receive at least 6, but no more than 96 representatives. The need to find a systematic, transparent, and long-lasting way to apportion these representatives has given rise to a large literature. One approach is to divide countries into groups based on size and to determine what fraction of the seats should go to each group before apportioning. As an example,

7.3 Degressive Representation in the European Union

199

Pukelsheim (2007) suggested dividing the 27 member countries from the EU in 2007 into large and small classes and allocating 500 seats (or two-thirds) to the 7 largest countries and 250 (or one-third) to the smallest 20 countries. Since the large countries make up three-quarters of the population, this naturally builds in a level of degressive representation in which each member of the European Parliament from a large country would represent 540,000 individuals, while each member from a small country would represent 431,000 individuals (Pukelsheim 2007). Another approach is to use a method such as Adams’ method that has an innate bias toward smaller states, incorporating minimum and maximum caps as described in (Balinski and Young 2001). This is done as follows. Suppose a divisor method F has rounding rule f and minimum and maximum requirements, m and M. Then, d ∈ FmM (D, v) if there exists x > 0 such that di = med{ f (m), [ vxi ] f , f (M)} (the vi vi median Σ value of the set of three values) where [ x ] f = k if f (k) < x ≤ f (k + 1) and i≤n di = D. Applying this methodology to Adams’ method and the constraints of the EU yields ⎧ vi ⎪ ⎨6 if x ≤ 5 di = k if k − 1 < ⎪ ⎩ 96 if vxi > 96

vi x

≤ k where k ∈ [6, 96] .

To increase the degree of degressiveness, Adams’ method can be combined with an initial base assignment of seats. This solution was advocated as the Cambridge Compromise, one of a set of recommendations made during a 2017 workshop held by the Constitutional Affairs Committee (AFCO) of the European Parliament (Pukelsheim and Grimmett 2017). The Cambridge Compromise called for each country to receive 5 base seats and for the remainder to be allocated using Adams’ method, with the minimum and maximum constraints of 1 and 91 (since Adams’ method automatically allocates each country at least one representative). A third approach, which has been explored in the literature although not yet adopted, is to use a function to transform the population data before applying a divisor method. To ensure the apportionment is order-preserving and degressive, the function must be non-increasing and concave. One of the simplest ways to do this is through a power function, replacing vi in the quotient vi /x with vit for some 0 ≤ t ≤ 1. This is referred to as a power-weighted variant and is discussed in (Grimmett et al. 2012). There, the authors advocate for assigning each country 5 base seats and then selecting an exponent t so that the divisor method naturally allocates a total of 96 seats to the largest country. More generally, Słomczy´nski and Zyczkowski (2012) discuss families of non-decreasing and concave down functions that are indexed by 3 parameters which can be used to set minimum and maximum caps on representation and ensure that all delegates are allocated. These methods arguably allow for the incorporation of minimum and maximum caps in more natural ways than directly imposing caps as when applied by traditional divisor methods. However, they lack the clarity of the simpler approaches. Whereas a divisor ratio such as vi /x = 540,000, for instance, indicates that there is roughly

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one representative per 540,000 individuals, the expression vit /x = 540,000 has no such easy translation. This lack of transparency may be important for the perceived legitimacy of the European parliament. Overall, each of these recommendations provides a more systematic, enduring, and potentially more justifiable approach to the representative apportionment problem than past allocations. The fact that there are multiple ways to attack the problem reinforces the fact that they all contain some arbitrariness to them. Notably, few of the suggestions for apportionment to the EU parliament have involved quota methods, although there is a tradition of using modified quota methods in the national parliaments of several member countries (Pukelsheim 2017). This is perhaps because of the well-known susceptibility of such methods to the paradoxes discussed in Chap. 6. Certainly, the unusual behavior demonstrated by some of the delegate apportionment methods used in the primaries would no doubt be considered unacceptable in the representation structure for the European parliament, regardless of their behavior to produce regressive, rather than degressive, allocations. Yet, the wide range of approaches for incorporating degressive representation in the EU suggests that there may be an equally wide range of approaches for incorporating regressive representation in the presidential primaries, particularly since, as discussed in Chap. 1, there are fewer constraints on either fair representation or transparency. We consider this in the next section.

7.4 Regressive Representation Unlike the EU, the goals of delegate apportionment align more closely with regressive representation. The apportionment methods do not themselves satisfy strict regressiveness; nevertheless, the explicit bias embedded in the rules, as well as the threshold structure, support this interpretation. In this section, we briefly explore some alternatives for embedding regressive representation into delegate apportionment. We start by noting that just because a method exhibits bias in favor of strong candidates does not mean that it satisfies v1 /d1 > v2 /d2 > · · · > vn /dn . In Table 7.6, we recreate Example 4.1 from Chap. 4 with the allocations from the 6 different delegate apportionment methods and their vi /di ratios. It is easy to see that none of the methods results in an allocation that satisfies strict regressive proportionality. Indeed, given the relatively small number of delegates at stake in most of the districts or even when apportioning a state’s at-large delegates, we would expect that candidates would regularly receive an equal number of delegates, making strict regressiveness unlikely. Moreover, it is not clear that regressiveness is the goal of all of these methods. We focus on exploring some strategies for allocating delegates that would allow some amount of bias toward stronger candidates while avoiding some of the unexpected and possibly undesirable characteristics of the current delegate apportionment methods.

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Table 7.6 Example 4.1 from Chap. 4 for n = 6 and D = 101 delegates with di /vi values rounded to the nearest integer and listed in parentheses vi NIE NIS HAM LAR ILQ SUQ Cand. i 1 2 3 4 5 6

29,130 20,000 17,720 16,750 16,550 15,350

27 (1079) 17 (1176) 15 (1181) 15 (1117) 14 (1182) 13 (1181)

26 (1120) 18 (1111) 15 (1181) 15 (1117) 14 (1182) 13 (1181)

25 (1165) 18 (1111) 16 (1108) 15 (1117) 14 (1182) 13 (1181)

26 (1120) 18 (1111) 16 (1108) 14 (1196) 14 (1182) 13 (1181)

28 (1040) 17 (1176) 15 (1181) 14 (1196 14 (1182) 13 (1181)

26 (1120) 18 (1111) 16 (1108) 15 (1117) 15 (1103) 11 (1395)

One way to do this is to build regressive representation into the delegate selection plan before applying apportionment methods. Mimicking the suggestions for the EU, for instance, the candidates could be divided into strong and weak categories with the stronger candidates receiving an outsized share of the delegates; for example, the top two candidates could automatically be apportioned two-thirds of the delegate pool. Alternatively, a base or bonus number of delegates could be awarded to the leading candidate with the remainder allocated proportionally. Offering a bonus to the plurality party has been suggested by White (2021) as a way to improve the ability to govern since the largest party in proportional representation systems often has to cobble together enough votes to enact legislation. Offering such a bonus to the leading candidate(s) may mirror the effect of the current thresholds. In addition, adding a bonus would allow for more control in the amount of regressiveness in a primary: A smaller bonus for the strongest candidates would allow for minimal regressiveness and a larger bonus would promote more regressiveness. This would give the state party more control over the outcome rather than relying on the vagaries, for instance, of how quotas round when applying the ILQ method. Another way to create regressive representation is to change the delegate apportionment methods themselves. For example, the shift-quota methods, discussed in Sect. 7.1.1, exhibit a small amount of bias. Because the bias is small for these methods, for small values of D, the apportionments derived from using these methods is not likely to deviate much from HAM, which is unbiased. Moreover, shift-quota methods are susceptible to the same paradoxes as HAM, and in particular, they are susceptible to the elimination and no-show paradoxes which seem troubling given that the intent of the primary structure is to encourage individuals to vote sincerely (i.e., to express their true preferences). A more satisfactory alternative to the current apportionment methods would be to adopt a divisor method, which has the advantage that it is not susceptible to the elimination, no-show, population, or Alabama paradoxes discussed in Chap. 6. We take this up in Sect. 7.4.1.

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Table 7.7 Jefferson’s method applied with a power function g(vi ) = vi1.3 Ci

vi

qi

Trump Kasich Cruz Bush Rubio Christie Fiorina Carson Total

100,735 44,932 33,244 31,341 30,071 21,089 11,774 6527 279,713

8.28 3.69 2.73 2.58 2.47 1.73 0.97 0.54 23

(10,500)

Jeff. qˆi (t = 1.3) (260,000)

9 4 3 2 2 2 1 0 23

12 4 2 2 2 1 0 0 23

Jeff.

Jeff. (10,000)

Jeff. (t = 1.3) (247,000)

9.64 4.30 3.18 3.00 2.88

10 4 3 3 3

12 4 3 2 2

23

23

23

The numbers in parentheses indicate the divisors

7.4.1 Adapting Divisor Methods Using Power Functions The most logical divisor method to use for delegate apportionment is Jefferson’s method, which, among all divisor methods, exhibits the largest degree of bias in favor of stronger candidates. To increase the regressiveness, Jefferson’s method could be modified as was suggested for Adams’ method for apportionment in the EU by dividing candidates into classes based on the strength of the candidates or by including bonus delegates, as was discussed previously. A more direct approach is to transform the vote data by a function, as was suggested in Sect. 7.3, to support degressive representation. In this case, f would have to be a non-decreasing convex (concave up) function. We illustrate this using Jefferson’s method and the function g(vi ) = vit where t > 1. Since there is some arbitrariness to the choice of t, we select a t value such that the strongest candidate receives a majority of the delegates. Example 7.5 The 2016 New Hampshire Republican primary had 23 at-large delegates to allocate. The results for the top 8 candidates are shown in Table 7.7, along with their quotas qi (adjusted to eliminate any candidate receiving less than 1% of the vote). The 4th column shows that Jefferson’s method remains close to quota. The 5th column shows Jefferson’s method applied to the power function g(vi ) = vi1.3 . The 6th and 7th columns show the same result assuming a 10% threshold is applied. In both cases, the exponent t = 1.3 is chosen so that the strongest candidate (Trump) receives a majority of the delegates.

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Table 7.8 Using Jefferson’s method on weighted vote totals to apportion the 2016 New Hampshire Republican primary vi Jeff. g10 (vi ) Jeff. g2 (vi ) Jeff. g1 (vi ) Jeff. Ci Trump Kasich Cruz Bush Rubio Christie Fiorina Carson Total

100,735 44,932 33,244 31,341 30,071 21,089 11,774 6527 279,713

9 4 3 2 2 2 1 0 23

100,735 44,810 32,638 30,587 29,203 19,076 8377 3187

10 4 3 3 2 1 0 0 23

100,735 31,144 18,321 16,468 15,274 7906 2592 818

13 4 2 2 1 1 0 0 23

100,735 20,042 10,971 9751 8977 4415 1376 423

17 3 1 1 1 0 0 0 23

The gr (vi ) values are rounded to the nearest integer

7.4.2 Adapting Divisor Methods Using Weighted Vote Totals The selection of which function—a power function or another concave function—to use in conjunction with Jefferson’s method has an arbitrariness to it. An alternative approach, which still allows for flexibility but is more responsive to the vote distribution, is to use a function f that weights each candidate’s vote total based on how well the candidate performs. The idea of weighting is already built into the rules of delegate selection via the use of thresholds which eliminates some candidates based on how well they do relative to the top candidates. Essentially, the eliminated candidates receive a weight of 0 while the others receive a weight of 1. Creating a more nuanced weighting scheme allows for greater flexibility in how regressive the allocation is. Moreover, weighting is also used by both Democratic and Republican parties, at earlier stages in the delegate selection process, in apportioning delegates to states. State Democratic parties also use weighting when distributing district delegates to congressional districts, as outlined in Sect. 2.1. In these cases, however, the weighting is based on the number of registered party members and the strength of the party in recent past elections (see Sects. 2.1 and 4.1 for a fuller discussion of weighting in Democratic and Republican Party delegate selection rules). As an example of applying weights, suppose the vote totals for candidates Ci are based on how Ci performs relative to C1 . One way to do this is to give each of Ci ’s votes a weight vi /v1 , so that Ci ’s vote total is (vi /v1 )vi . More generally, for each r > 0 we give each vote for Ci weight 1 − (1 − vi /v1 )r , so that gr (vi ) = [1 − (1 − vi /v1 )r ]vi , which is a non-decreasing concave function. As r increases, gr (vi ) → f (vi ) = vi , leading to a traditional Jefferson’s method. As r decreases, the allocation becomes increasingly regressive. We illustrate the effect of different choices for r using the data from Example 7.5.

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Example 7.6 Recall the 2016 New Hampshire Republican primary had 23 at-large delegates to allocate. The apportionments using Jefferson’s method and Jefferson’s method modified using g10 , g2 , and g1 are shown in Table 7.8. As r decreases, the allocations become increasingly biased toward the strongest candidates, especially C1 . One advantage of using a power function like gt (vi ) = vit or a function based on relative weights such as gr (vi ) = [1 − (1 − vi /v1 )r ]vi is that the functions have parameters which can be altered to be regressive, proportional, or degressive as required. We discuss this further in Sect. 7.5.

7.5 Conclusion The range of strategies discussed in Sect. 7.4 provide alternatives to the current allocation of delegates in the presidential primaries. As noted at the beginning of Chap. 7, the problem of how to apportion delegates is different from other apportionment problems: The process involves multiple contests taking place over time amidst a constantly changing landscape of new voters, new sets of candidates, and, on the Republican side, new rules. Moreover, the goal of the presidential primaries is not to represent fairly all the presidential candidates but to select a final nominee. This means that delegate apportionment does not need to be evaluated based on traditional measures of proportionality. During the era of the brokered convention, which died out in the mid-twentieth century, the winners of the primaries would clinch the nomination at the Democratic and Republican National Conventions, often after multiple rounds of ballots. The last seriously contested race that was concluded at a party convention was the 1984 Democratic convention in which Walter Mondale entered with a plurality (but not a majority) of delegates. After urgent negotiations with his nearest rival, Gary Hart, Mondale won on the first ballot. More recently, the conventions have served as venues to formally endorse the presidential nominee, the leading candidate having secured either an outright majority or presumed majority of delegates long before the convention’s opening. Arguably, the process would be better and the candidates stronger if the primary season were more competitive—if more candidates continued their presidential bid longer during the primary calendar or even arrived at the convention as viable nominees. The Democratic Party’s consistent use of HAM shows that even the use of an unbiased method, when tied to thresholds, rarely results in the race being competitive late into the primary season. The Republican Party’s use of methods that lean regressive presumably speeds up the process of candidates dropping out of the race. Thus, adopting apportionment methods that are degressive earlier in the calendar might help prolong the competitiveness of the contest. Likewise, adopting apportionment methods that are regressive later in the season could help decrease the candidate pool and build support for the final nominee. Both the allocation of base delegates (either for weak or strong candidates) or the transformation of the vote vector by a

7.5 Conclusion

205

weight function such as gr (v) in conjunction with a divisor method would allow for apportionment methods to be adjusted, as on a dial, to make them more degressive or more regressive depending on the calendar and the goals of the state or national party. There are advantages and disadvantages to this approach. As discussed in Chap. 1, both parties have wrestled with the tension between encouraging broader participation among the party’s base, presumably by keeping the candidate pool wide, and ensuring that the eventual nominee is acceptable to the party establishment. Securing the nomination early can project an image of party strength and unified message; on the other hand there are real disadvantages. For example, Republican strategists were concerned in 2008 when John McCain clinched the Republican primary early in the season as media attention focused on the protracted battle between Barack Obama and Hillary Clinton in the Democratic primary. Another factor to consider is the impact of the rules governing who can vote in a primary. Is the contest open or closed (open just to party members), and can voters register as a party member on the spot? Having an open election may discourage party organizers from adopting rules that are more degressive. Zooming out beyond the individual contest, however, it is not likely that these changes would make a large difference unless the structure for how delegates are allocated is significantly changed. This is because so many delegates are assigned based on congressional district elections where the total number of delegates D is small. This is true in the Democratic primary; it is even more so in the Republican primary where districts generally have only 3 delegates at stake. Unless the district sizes becomes larger, in most cases only the strongest one or two candidates will ever receive district delegates. In addition, the more the delegate pools are divided into separate at-large and district allocations, the more frequently the aggregation paradox will arise (Sect. 6.2). There are compelling reasons, however, for parties to continue with the current allocation structure. Having local delegates encourages more widespread participation in the primaries, balancing the at-large delegates who represent broader statewide interests. For this reason, it seems that the disproportionality that arises due to the aggregation of separate allocations is likely to stay. Ultimately, decisions about how the primaries are organized and the details of the delegate selection plan are made by the national and state party committees and are a result of discussion, expediency, and hard-won compromise. Mathematical analysis can help to understand and explain the consequences of the how party presidential nominees are chosen but, as with most topics in mathematical political science, mathematics can only take us so far. Determining whether a delegate allocation system is “good” or “fair” depends on value judgments outside of mathematics which cannot be separated from politics. The parties and their voters are responsible for choosing how their presidential nominees are selected; our hope is that the analysis in this book provides necessary mathematical elucidation for making this choice.

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References Balinski ML, Young HP (2001) Fair representation: meeting the ideal of one man, one vote, 2nd edn. Brookings Institution Press, Washington, D.C Grimmett G, Oelbermann K, Pukelsheim F (2012) A power-weighted variant of the EU27 Cambridge compromise. Math Soc Sci 63(2):136–140. https://doi.org/10.1016/j.mathsocsci.2011.11. 001 Huntington EV (1928) The apportionment of representatives in Congress. Trans Amer Math Soc 30(1):85–110. https://doi.org/10.2307/1989268 Martínez-Panero M, Arredondo V, Peña T, Ramírez V (2019) A new quota approach to electoral disproportionality. Economies 7(1):17. https://doi.org/10.3390/economies7010017 Pukelsheim F (2007) A parliament of degressive representativeness? Institut für Mathemaatik, Universitätsstrasse. https://opus.bibliothek.uni-augsburg.de/opus4/frontdoor/index/ index/docId/510 Pukelsheim F (2017) Proportional representation: apportionment methods and their applications. Springer, Cham, 2nd edn. (With a foreword by Andrew Duff) Pukelsheim F, Grimmett G (2017) Linking the permanent system of the distribution of seats in the european parliament with the double-majority voting in the council of ministers. In: The composition of the European parliament. http://www.europarl.europa.eu/supporting-analyses Słomczy´nski W, Zyczkowski K (2012) Mathematical aspects of degressive proportionality. Math Social Sci 63(2):94–101. https://doi.org/10.1016/j.mathsocsci.2011.12.002 Taagepera R, Grofman B (2003) Mapping the indices of seats-votes disproportionality and interelection volatility. Party Politics 9:659–677. https://doi.org/10.1177/13540688030096001 White J (2021) What kind of electoral system sustains a politics of firm commitments? Representation 57(3):329–345. https://doi.org/10.1080/00344893.2019.1624601

Appendix A

Description of the Monte Carlo Simulations

This Appendix describes the mechanics of the Monte Carlo simulations that are used extensively throughout the book, particularly in Chaps. 5 and 6. All simulations occur under the impartial anonymous culture (IAC) condition, wherein a percentage distribution for the percentage of votes across the n candidates is chosen at random. For example, if n = 3 then under our simulations a vote percentage distribution of (0.2, 0.5, 0.3) is just as likely to occur as the distribution (0.01, 0.03, 0.96). Geometrically, this corresponds to choosing a point at random in the corresponding simplex. It is straightforward to choose a point in a simplex at random, and several simple algorithms exist to do so. We choose to use an algorithm which is sometimes referred to as the technique of the “broken stick” (Lepelley et al. 2000) and works as follows. To generate a percentage distribution ( p1 , . . . , pn ) of length n at random, we generate n − 1 numbers x1 , . . . xn−1 in [0, 1] via a uniform distribution on [0, 1], which we then rank from smallest to largest. Assume WLOG that x1 < x2 < · · · < xn−1 so that the numbers are already ordered. We then obtain the percentages by p1 = x1 , p2 = x2 − x1 , . . ., pn = 1 − xn−1 . Unless stated otherwise, a given Monte Carlo simulation uses 1,000,000 runs. Each run begins by choosing a percentage vote distribution at random under the IAC model using the broken stick technique and then checks the resulting election for whatever condition we are currently interested in. All simulations are done using code written in Python 3. Reference Lepelley D, Louichi A, Valognes F (2000) Computer simulations of voting systems. Adv Complex Syst 3(1):181–194. https://doi.org/10.1142/S0219525900000145

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9

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Appendix B

Descriptions of the Databases of Primary Election Data

All of the election data from presidential primaries used in this book is obtained from thegreenpapers.com. Throughout the book, we use two different databases of election data from the Green Papers; the purpose of this appendix is to describe these databases in detail. Generally speaking, our first database DEM_DATA consists of all Democratic state primary elections from the years 2004, 2008, 2016, and 2020 for which we have data. The second database, REP_DATA, contains all Republican elections for which we have data from the years 2008, 2012, and 2016. We choose these years because they are the years for which (1) data is available, and (2) the primary contained more than one viable candidate. We do not include the 2012 Democratic primary, for example, because that was essentially a one-candidate contest since Barack Obama was the incumbent presidential candidate. DEM_DATA: For a given state Democratic primary, the kind of vote data that is available falls into 4 categories: 1. We have complete vote data for each candidate in every CD and at the statewide level.1 2. We have vote data in every CD for the candidates who surpass the 15% threshold but not for the eliminated candidates, and we have complete vote data at the statewide level. 3. We have no data at the CD level, but we have complete statewide vote data. 4. We have no data either at the CD or statewide level. When thegreenpapers.com does not display complete CD vote data for a given state primary on the website, Tony Roza of the Green Papers often has access to the complete CD vote data. In such cases, the data was generously provided to us by Tony Roza. Before discussing how we query the database, we distinguish between a vote distribution v and an election (v, D), which is an ordered pair consisting of a vote 1

Primaries in some territories such as American Samoa and Guam; if we have complete statewide data, we put them in this category.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9

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distribution and a delegate value D that was actually used in the given primary election. The reason for this distinction is that sometimes the same vote distribution is used for multiple different D values, as occurs with at-large and PLEO apportionment in Democratic primaries, in which case there are multiple opportunities for a paradox such as the elimination paradox to arise when using the multiple D values. When we query the database looking for actual occurrences of a given paradox using the D values used in the primary election, we sometimes use the same vote distribution in multiple elections. On the other hand, sometimes we fix a hypothetical D value that we then apply to all possible vote distributions in the database, in which case we avoid double-counting and do not use the same vote distribution multiple times. Generally speaking, when we want to know how often a given paradox “actually” occurred in past primaries (or would have occurred if state parties hypothetically used a different apportionment method), we use all elections (v, D) in the databases. If we hypothetically mandate the same D value across the entire database, we use only the underlying vote distributions, resulting in fewer data points. For state primaries in Category 1, we count each CD vote distribution, along with the number of delegates used, as its own election, and we count the statewide vote distribution with the PLEO number of delegates and the statewide vote distribution with the at-large number of delegates as their own elections. Thus for a state primary in Category 1, the number of elections available to us is the number of CDs2 plus two. For state primaries in Category 2, we can investigate some questions at the CD level assuming that the 15% threshold is in place, but there are some questions (such as those involving the elimination paradox; see Sect. 6.1) which require knowledge of vote totals for candidates who fall below the 15% threshold and the CD elections from state primaries in this category are of no use for such questions. However, since we have complete data at the statewide level, we can use the PLEO and at-large elections. For state primaries in Category 3, we can use only the two statewide elections in our data-crunching. Of course, state primaries in Category 4 are of no use, but these are rare. Across entire database, there are 1981 total separate elections (v, D) for which we have complete vote data; 1882 of these satisfy n ≥ 3. When we query the database, we are usually interested in elections with three or more candidates, and thus we typically use 1882 elections from this database. We note that two elections satisfy D = 1; we include these two elections in the database. There are 2209 total elections in the database if we include the elections for which we have data for candidates exceeding the 15% threshold which occurs in all Democratic state primaries (but not the vote data for eliminated candidates); 2082 of these satisfy n ≥ 3. We include these elections when studying the aggregation paradox (Sect. 6.2) and Alabama 2

Recall from Chap. 2 that the DNC allows some states not to use literal CDs. Texas, for example, uses its state senatorial districts rather than CDs. The state of Delaware, which has only one CD, is allowed to break its state into three separate counties plus the City of Wilmington to create 4 mock “CD”s for their primary.

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211

paradox (Sect. 6.4), for example. There are 1766 vote distributions v in total across the database, with 1684 satisfying n ≥ 3. In summary, DEM_DATA contains all Democratic state primary elections for which we have data at least for those candidates who surpass the 15% threshold. In a given state primary, each CD, PLEO, and at-large vote distribution, along with the corresponding number of delegates, counts as its own election. If we query the database with a question for which we want to use the actual delegate values used in a state primary, we use all of these elections even though the underlying vote distributions are sometimes the same, as occurs for the PLEO and at-large elections.3 If we query the database with a question in which we use a hypothetical D value across all of the data, then we use distinct vote distributions, instead of elections, to avoid double-counting. We note that for some of the CD elections with complete vote data, the vote totals must be estimated. Secretary of State offices sometimes publish vote totals by county but not by CD, and for these elections it is often the case that neither the Secretary of State office nor the state Democratic Party is willing to provide the data by CD. In this case, the Green Papers estimates the vote totals in each CD by interpolating from results in the counties. The following statement is on their webpage: “When we are unable to acquire the official vote by CD, we estimate. We estimated the CD vote by apportioning the county vote using linear interpolation. If a county is 70% in district 1 and 30% in district 2, apportion the popular vote accordingly and compute the delegates.” It is unfortunate that they must sometimes estimate in this fashion, but it is the best we can do given the recalcitrance of Secretary of State offices and state party officials. REP_DATA: This database contains every Republican primary election and vote distribution from the 2008, 2012, and 2016 Republican primaries for which we could obtain complete vote data, including the vote totals for eliminated candidates. For this database, we make the same distinction between an election and a vote distribution that we use for the Democratic data. On the Republican side, there are 749 elections in this database with n ≥ 3 and D > 0 (sometimes Republican state parties do not allocate any delegates at the statewide level; we do not use elections of the form (v, 0) when querying the database). There are 758 vote distributions in the database with n ≥ 3. The reason that there are more vote distributions than elections in this database is that Republican state parties seldom perform multiple apportionments using the same statewide data, avoiding the potential double-counting of the PLEO and at-large issue on the Democratic side. Furthermore, for elections of the form (v, 0) which occur at the statewide level, which we do not count as an election when using the database, we count v as a vote distribution which can be used with a hypothetical D value.

3

If a state has only one CD, then the same vote distribution can be used three times: once for the CD allocation, once for PLEO, and once for at-large.

Index

A Adams’ method, 13, 120, 192–194, 199, 202 Aggregation paradox, 20, 31, 127, 141, 142, 144–159, 163, 164, 175, 193, 205, 210 Alabama paradox, 5, 6, 37, 80, 127, 169– 173, 175–178, 180, 183, 184, 193, 201, 211 Anonymous, 45 Asymptotic bias, 97, 99 At-large delegates, 10, 14, 16, 17, 28, 32, 51– 53, 63, 64, 68, 90, 128, 135, 144–146, 164, 169–171, 176, 178, 181, 182, 188, 200, 202, 204, 205

B Balanced, 80

C Cambridge Compromise, 199 Candidate coalitions, 78, 111, 119–121 Caucus, 3, 5, 14–18, 23, 24, 34, 39–42, 44– 49, 52–54, 56, 64, 77, 117, 127, 128, 142–145 Close election, 78, 80, 94, 99, 103, 115, 188 Close-election delegate maximum, 104 Close-election delegate minimum, 104

D Degressive proportionality, 41, 42, 44, 197, 198

Delegate, 3–7, 9–21, 23–37, 39–65, 67, 68, 70, 77–82, 85–87, 90–95, 97–113, 117–125, 127–131, 135–147, 149– 151, 153, 154, 157–159, 161–184, 187–190, 192–197, 199–205, 210, 211 Delegate bias, 94, 97–99, 106, 107, 187, 190, 191, 194 Delegate excess, 97 Delegate monotone, 80 Delegate selection plan, 54, 122, 187–189, 196, 201, 205 Delegate share, 78, 104, 121–123, 125, 195 Delegate threshold, 94, 99, 100, 103 Democratic National Committee (DNC), 23–25, 27, 40, 128, 177, 210 DEM_DATA, 78, 91, 92, 116–118, 134, 136, 140, 141, 167, 172, 173 District delegates, 10, 28, 30–32, 41, 52, 55, 68–70, 90, 91, 117, 130, 141–143, 145, 146, 162, 163, 169, 176–179, 203 Divisor methods, 9, 12, 13, 130, 183, 184, 189, 191–195, 198, 199, 201, 202, 205

E Elimination paradox, 6, 46, 118, 125, 127– 136, 139–141, 164–168, 173, 180, 182–184, 193, 210 Entropy (E), 7–10, 12, 13, 129, 130 European Union, 189, 193, 197, 198

G Gallagher Index (G), 195, 202, 203, 205

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. A. Jones et al., Delegate Apportionment in the US Presidential Primaries, Studies in Choice and Welfare, https://doi.org/10.1007/978-3-031-24954-9

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214 H HAM∗ -NIE Hybrid (HNH), 59, 60, 65, 67, 77, 84, 85, 88, 89, 92, 96, 101, 103, 105, 112, 113, 118, 130, 132, 135, 136, 138, 147, 152–154, 168, 197 Hamilton’s Method (HAM) and (HAM∗ ), 6, 11, 13, 17, 24, 27, 32, 51, 59, 60, 65, 66, 77 Homogeneous, 79

Index Nearest Integer Sequential method (NIS), 59, 65, 77, 84, 86, 88, 89, 92, 93, 96, 98, 101, 102, 105, 112–114, 118, 120, 130, 132, 136, 138, 139, 147, 152–154, 160, 162, 168, 173, 190, 197 No-show paradox, 19, 37, 121, 127, 164– 169, 175, 180, 184, 193, 201

O Order-preserving, 80, 111, 199 I Iterated Lower Quota method (ILQ), 61, 62, 65, 77, 79, 84, 85, 87, 92, 93, 98, 99, 102–110, 112, 113, 115, 116, 121, 130, 132, 134, 136, 138, 139, 147–149, 152– 155, 158, 161, 162, 168, 174, 175, 188, 191, 196, 197, 201

J Jefferson’s method, 9, 12, 13, 112, 120, 192– 194, 196, 197, 202–204

L Large method (LAR), 77 Leader criterion (LEA), 111, 115, 116, 118, 119 Loosemore-Hanby index (LH), 195, 196 Lower quota, 42, 57, 59–61, 77, 85, 86, 89, 91, 139, 194

M Majority criterion (MAJ), 111–118 Majorization, 94, 190, 193, 194 Max delegate share to vote share: (MDQ), 195, 196 Maximum Deviation (MD), 85, 86 Max vote share to delegate share: (MQD), 195–197 Monte Carlo simulations, 78, 128, 207

N Nearest Integer Extremes method (NIE) , 58–60, 62, 65, 67, 68, 77, 79, 83–86, 88, 89, 92, 93, 96–98, 101– 105, 109, 112, 118, 120, 130, 132, 136, 138, 139, 147, 152, 153, 161, 168, 172, 174–176, 188, 196, 197

P Pledged Leaders and Elected Officials (PLEO), 15, 24, 26–28, 31, 32, 36, 90, 92, 107, 144–146, 158, 162, 176, 210, 211 Population monotone, 80 Population paradox, 80, 127, 178–181, 184 Power functions, 199, 202–204 Proportional consistency, 77, 81, 82

Q Quota condition, 85 Quota violation, 87, 89–93, 196 Quota-disportin Index (QI), 195–197

R Regressive proportionality, 42, 200 REP_DATA, 78, 116–118, 134, 136, 140, 141, 167, 172, 173 Republican National Committee (RNC), 17, 51–54, 56, 68, 70, 128

S Sensitivity, 78, 88, 105, 108–110, 196 Shift-Quota methods, 189–191, 194, 201 Simplex, 32–34, 65, 67, 87, 89, 93, 97, 101, 103, 104, 109, 113–116, 132, 133, 135, 140, 166, 171, 175, 179, 190, 207 Simplicial geometry, 65, 135 Sequential Upper Quota method (SUQ), 62, 65, 70, 77, 79, 83, 87, 92, 93, 96–99, 103–106, 108, 109, 115, 116, 118, 120, 127, 128, 130, 132, 134, 136, 139, 140, 147, 152, 154, 155, 158, 159, 161, 164, 167, 171, 174, 175, 180, 181, 188, 191, 196, 197

Index T Threshold of exclusion, 94, 100 Threshold of inclusion, 94, 100 Thresholds, 23, 24, 30–32, 34, 36, 44–47, 50–52, 54–58, 62, 67–70, 78, 85, 90– 92, 94, 100, 101, 103, 106, 109, 110, 117, 118, 121–125, 127–131, 133–136, 140, 141, 143, 144, 149, 157, 158, 162– 168, 170, 172, 175–177, 179, 181, 184, 187, 188, 190, 191, 193, 196, 200–204, 210, 211

215 U Uniformity, 11, 14, 54, 78, 80, 81, 87, 93, 97, 113, 116, 128, 134, 140, 148, 149, 154, 172, 173, 175, 182–184, 187, 195 Upper quota, 13, 42, 57, 61, 62, 77, 85–87, 89–92, 97, 112

W Weakly proportional, 81 Weak no-show paradox, 168, 169 Webster’s method, 12, 100, 120, 192, 194