Theory of Games and Economic Behavior [Sixth printing ed.]

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UNIVERSAL

Osmania University Library Call No.

Author Title

”7^ O'

t

Accession No. C?

3

^

»* t i>

(11:4)

•

• •

,

=

rn )

0

forallro.n,

•

• • ,

t„.

*-i

And if we finally introduce 3C*(n,

•

• • ,

t„), in

the sense of 11.2.3.,

we obtain

n

(11:5)

2)

,

t„)

=

0

for all

n,

•

•

• ,

rn

.

*-i Conversely,

it is

clear that the condition (11:5)

makes the game

T,

which we

defined in 11.2.3., one of zero sum. 1 Reverting to the definition of a strategy as given in 11.1.1.: In this game a player & has one and only one personal move, and this independently of the course of the play, the move 911*. And he must make his choice at 911* with nil information. So his strategy is simply a definite choice for the move 9TI*, no more and no less; i.e. precisely

—

n-

1

•

,

•

,

0 *.

We leave it to the reader to describe this game in terms of partitions, and to compare the above with the formalistic definition of a strategy in (11 :A) in 11.1.3.

CHAPTER III ZERO-SUM TWO-PERSON GAMES: THEORY 12.

Preliminary Survey

12.1.

General Viewpoints

In the preceding chapter we obtained an all-inclusive formal game of n persons (cf. 10.1.). We followed up by developing an exact concept of strategy which permitted us to replace the rather complicated general scheme of a game by a much more simple special one, which was nevertheless shown to be fully equivalent to the former (cf. 11.2.). In the discussion which follows it will sometimes be more convenient to use one form, sometimes the other. It is therefore desirable to give them specific technical names. We will accordingly call them the extensive and the normalized form of the game, respectively. Since these two forms are strictly equivalent, it is entirely within our province to use in each particular case whichever is technically more convenient at that moment. We propose, indeed, to make full use of this possibility, and must therefore re-emphasize that this does not in the least affect the absolutely general validity of all our considerations. Actually the normalized form is better suited for the derivation of general theorems, while the extensive form is preferable for the analysis of special cases; i.e., the former can be used advantageously to establish properties which are common to all games, while the latter brings out characteristic differences of games and the decisive structural features which determine these differences. (Cf. for the former 14., 17., and for the latter 12.1.1.

characterization of the general

e.g. 15.)

12.1.2. Since the

we must now turn

formal description of

up a

all

games has been completed,

It is to be expected that a systematic procedure to this end will have to proceed from simpler games It is therefore desirable to establish an to more complicated games. ordering for all games according to their increasing degree of complication. We have already classified games according to the number of participants a game with n participants being called an n- person game and Thus we must also according to whether they are or are not of zero-sum. It will distinguish zero-sum n-person games and general n-person games. be seen later that the general n- person game is very closely related to the zero-sum (n +» l)-person game, in fact the theory of the former will (Cf. 56.2.2.) obtain as a special case of the theory of the latter.

to build

positive theory.

—

—

—

12.2.

The One-person Game

12.2.1. We begin with some remarks concerning the one-person game. In the normalized form this game consists of the choice of a number

85

ZERO-SUM TWO-PERSON GAMES: THEORY

86 r

=

•

•

•

1,

,

/?,

after

which the (only) player

obviously void 2 and there

gets the

1

amount 3C(r).

1

The

nothing to say concerning it. 3C(r) and the “best” corresponds to general function a general case The of acting playing consists obviously of this: way of “rational” i.e. or

zero-sum case

is

—

=

is

—

make 5C(r) a maximum. 1, This extreme simplification of the one-person game is, of course, due to the fact that our variable r represents not a choice (in a move) but the player’s strategy; i.e., it expresses his entire “theory” concerning the handling of all conceivable situations which may occur in the course of the

The player

1 will

choose r

•

•

•

,

$ so as to

play. It should be remembered that even a one-person game can be of a very complicated pattern: It may contain chance moves as well as personal moves (of the only player), each one possibly with numerous alternatives, and the amount of information available to the player at any particular personal move may vary in any prescribed way. 12 2 2 Numerous good examples of many complications and subtleties .

.

.

may

way are given by the various games of “Patience” There is, however, an important possibility for which, to the best of our knowledge, examples are lacking among the customary one-person games. This is the case of incomplete information, i.e. of nonequivalence of anteriority and preliminarity of personal moves of the unique player (cf. 6.4.). For such an absence of equivalence it would be necessary that the player have two personal moves SfTC* and 9Tt x at neither of which he is informed about the outcome of the choice of the other. Such a state of lack of information is not easy to achieve, but we discussed in 6.4.2. how it can be brought about by “splitting” the player into two or more persons of identical interest and with imperfect communications. We saw loc. cit. that Bridge is an example of this in a two-person game; it would be easy to construct an analogous one-person game but unluckily the known forms of “solitaire” are not such. 8 This possibility is nevertheless a practical one for certain economic setups: A rigidly established communistic society, in which the structure of the distribution scheme is beyond dispute (i.e. where there is no exchange, but only one unalterable imputation) would be such since the interests of all the members of such a society are strictly identical 4 this setup must be But owing to the conceivable imperfections treated as a one-person game. among the members, all sorts of incomplete information communications of can occur. This is then the case which, by a consistent use of the concept of strategy (i.e. of planning), is naturally reduced to a simple maximum problem. On the basis of our previous discussions it will therefore be apparent now which

arise in this

or “Solitaire.”

—

—

1

Cf. (ll:D:a), (ll:D:b) at the

end of

Then 3C(r) = 0, cf. 11.4. 3 The existing “ double solitaires”

11.2.3.

We suppress the index

1.

*

pants, 4

i.e.

The

are competitive

games between the two

partici-

two-person games. individual

members themselves cannot be considered as players, since all among them, as well as coalitions of some of them against the

possibilities of conflict

others, are excluded.

PRELIMINARY SURVEY that this lation

—

—and

i.e.

—

87

this only is the case in which the simple maximum formuthe “ Robinson Crusoe” form of economics is appropriate.

— —

These considerations show also the limitations of the pure maximum i.e. the “Robinson Crusoe” approach. The above example of a society of a rigidly established and unquestioned distribution scheme shows that on this plane a rational and critical appraisal of the distribution scheme itself is impossible. In order to get a maximum problem it was necessary to place the entire scheme of distribution among the rules of the game, which are absolute, inviolable and above criticism. In order to bring them into the sphere of combat and competition i.e. the strategy of the game it is necessary to consider n- person games with n ^ 2 and 12 2 3 .

.

.

—

—

—

maximum

thereby to sacrifice the simple 12.3.

aspect of the problem.

Chance and Probability

Before going further, we wish to mention that the extensive literature of “mathematical games” which was developed mainly in the 18th and 19th centuries deals essentially only with an aspect of the 12 3 .

.

—

—

matter which we have already left behind. This is the appraisal of the This was, of course, effected by the discovery and

influence of chance.

appropriate application of the calculus of probability and particularly

concept of mathematical expectations.

of the

In our discussions, the

operations necessary for this purpose were performed in 11.2.3. 1,2

Consequently we are no longer interested in these games, where the mathematical problem consists only in evaluating the role of chance i.e. Such games in computing probabilities and mathematical expectations. lead occasionally to interesting exercises in probability theory; 8 but we hope that the reader will agree with us that they do not belong in the theory of games proper.

—

12.4.

12 4 .

The

.

We now

proceed to the analysis of more complicated games. game having been disposed of, the simplest one

general one-person

of the remaining

we

The Next Objective

games

are going to discuss

is

the zero-sum two-person game.

Accordingly

it first.

Afterwards there is a choice of dealing either with the general twogame or with the zero sum three-person game. It will be seen that our technique of discussion necessitates taking up the zero-sum three-person person

We

1 do not in the least intend, of course, to detract from the enormous importance It is just because of their great power that we are now in a position of those discoveries. are interested in those aspects of to treat this side of the matter as briefly as we do. the problem which are not settled by the concept of probability alone; consequently these and not the satisfactorily settled ones must occupy our attention. 2 Concerning the important connection between the use of mathematical expectation

We

and the concept of numerical utility, cf. 3.7. and the considerations which precede it. 8 Some games like Roulette are of an even more peculiar character. In Roulette the mathematical expectation of the players is clearly negative. Thus the motives for participating in that game cannot be understood if one identifies the monetary return with

utility.

—

,

ZERO-SUM TWO-PERSON GAMES: THEORY

88

we

extend the theory to the zero-sum n- person (for all 1, 2, 3, ) and only subsequently to this will it be found convenient to investigate the general n- person game.

game game

After that

first.

n =

shall •

•

•

13. Functional Calculus 13.1. Basic Definitions

Our next

13.1.1.

objective

is

—as stated

in 12.4.

—the exhaustive

dis-

In order to do this adequately,

cussion of the zero-sum two-person games.

—

be necessary to use the symbolism of the functional calculus or at more extensively than we have done thus far. it The concepts which we need are those of functions of variables, of maxima and minima, and of the use of the two latter as functional operations. All this necessitates a certain amount of explanation and illustration, which it

will

least of certain parts of

—

be given here.

will

is done, we will prove some theorems concerning maxima, minima, and a certain combination of these two, the saddle value. These theorems will play an important part in the theory of the zero-sum twoperson games. is a dependence which states how certain entities 13.1.2. A function determine an entity u called the called the variables of > x, y, and by the x, y, Thus u is determined by and this value of determination i.e. dependence will be indicated by the symbolic equation

After that

•

•

•

—

the value 0(x, Vi These choices i.e. these combinations of ) is defined at all. •

*

form the domain of 0. The examples (a)-(e) show some •

x, y,

•

•

—

•

of functions:

They may

well as of others. (a)

•

—

•

*

We may

of the

many possibilities for the domains

consist of arithmetical or of analytical entities, as

Indeed: consider the

or equally well of

all real

domain

to consist of all integer numbers,

numbers.

(b) All pairs of either category of

numbers used

in (a),

form the domain.

The domain is the set ft of all objects ir which represent the the game T (cf. 9.1.1. and 9.2.4.). (d) The domain consists of pairs of a positive integer k and a set (e) The domain consists of certain systems of positive integers.

plays

(c)

of

A

D«.

function 0 is an arithmetical function if its variables are positive it is a numerical function if its variables are real numbers; it is a

integers;

if its variables are sets (as, e.g., D* in (d)). For the moment we are mainly interested in arithmetical and numerical

set-function

functions.

We conclude this section by an observation which is a natural consequence of our view of the concept of a function. This is, that the number of variables, the domain, and the dependence of the value on the variables, constitute the function as such: i.e., if two functions 0, 0 have the = same variables x, y, and the same domain, and if 0(x, y ) 1 0(x, y, ) throughout this domain, then 0,0 are identical in all respects. •

•

•

•

•

•

f

•

•

•

13.2.

13.2.1.

The Operations Max and Min

Consider a function

which has

(*,

Assume

first

chosen, say as x

real

numbers

for values

»,•••)•

If its variable can be that 0 is a one-variable function. x 0 so that 0(x o ) ^ 0(x') for all other choices x', then we

=

say that 0 has the maximum 0(x o) and assumes it at x = xo. Observe that this maximum 0(x o ) is uniquely determined; i.e., the maximum may be assumed at x = x 0 for several x 0 but they must all furnish the same value 0(x o). 2 We denote this value by Max 0(x), the maxi,

mum value

of 0(x).

then the concept of 0’s minimumy 0(x o), obtains, and of xo where 0 assumes it. Again there may be several such x 0 but they must all furnish the same value 0(x o ). We denote this value by Min 0(x), If

we

replace

^ by

,

minimum

the 1

The concept

of a function

is

closely allied to that of a set,

with the exposition of 8.2. Proof Consider two such xo, say xj and xj'.

viewed *

value of 0.

Hence

and the above should be

in parallel :

0(xj)

*

(xJ)

£

^(x®

)

and ^(x?)

£

^(xj).

>

—

s

1 i

ZERO-SUM TWO-PERSON GAMES: THEORY

90

Observe that there

Min

no a

is

Max

guarantee that either

'priori

or

(x)

1

4>(x) exist.

—

over which the variable x may run however, the domain of consists only of a finite number of elements, then the existence of both Max (x) and Min (x) is obvious. This will actually be the case for most functions which we shall discuss. 2 For the remaining ones it will be a consequence of their continuity together with the geometrical limitations of At any rate we are restricting our considerations to such their domains. 8 If,

Max

functions, for which

and Min

exist.

By sinhave any number of variables x, y z Let now as z, gling out one of the variables, say x, and treating the others, y of the function, constants, we can view (x, y, z, ) as a one- variable (x y z variable x. Hence we may form Max (x y z ) ), Min as in 13.2.1., of course with respect to this x. But since we could have done this equally well for any one of the other 13.2.2.

•

•

f

•

.

y

•

•

*

,

y

•

•

•

•

•

y

y

•

*

y

y

y

•

•

y

it becomes necessary to indicate that the operations Max, Min were performed with respect to the variable x. We do this by writing Max* 4>(x, y z, ), Min* (f>(x, y z ) instead of the incomMin Thus we can now apply to the function plete expressions Max (x Vj z ) any one of the operators Max*, Min*, Max„, Min„, Max*, Min*, They are all distinct and our notation is unambiguous. This notation is even advantageous for one variable functions, and we will use it accordingly; i.e. we write Max* (x) Min* (x) instead of the

variables y

z

y

•

•

•

y

•

•

y

•

y

.

y

*

•

•

•

y

*

*

i

y

•

•

•

.

y

Max

{x) than to express (x) as a function. We may then write say a, b •

•

y

1

E.g.

if {x)

•

—

®

x with

all

real

numbers

Max

as domain, then neither

(

x)

nor Min

exist. 2

Typical examples: The functions

function 8

3Ck\ri y

•

•

•

,

r„) of 11.2.3. (or of (e) in 13.1.2.), the

3C(ri, r 2 ) of 14.1.1.

Typical examples: The functions

K(

£

,

17

),

Max-> K(

£

,

17

),

Min-> K(

|

,

17

)

in

v fix

17.4.,

the functions

Min Tj

fit

^

3C(n, t 2 )£

r i *“ 1

—

—

v

Max fi 2) ^(r U-

i»

T *)v

r

in 17.5.2.

The

vari-

all these functions are £ or or both, with respect to which subsequent maxima and minima are formed. Another instance is discussed in 46.2.1. espec. footnote 1 on p. 384, where the mathematical background of this subject and its literature are considered. It seems unnecessary to enter upon these here, since the above examples are entirely elementary.

ables of

17

FUNCTIONAL CALCULUS Max

&,-••)> [Min

(a,

13 2 3 .

.

x, y, z,

tions,

Observe that while

. •

•

•

,

•