Contributions to the Theory of Games (AM-28), Volume II 9781400881970

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Annals o f Mathematics Studies Number 28

ANNALS OF MATHEMATICS STUDIES Edited by Emil Artin and Marston Morse 1. Algebraic Theory of Numbers, by H e r m a n n W e y l 3. Consistency of the Continuum Hypothesis, by K u r t G o d e l 6.

The Calculi of Lambda-Conversion, by A lonzo C hurch

7. Finite Dimensional Vector Spaces, by P a u l R. H a l m o s 10. Topics in Topology, by S o lo m o n L efsc h etz 11.

Introduction to Nonlinear Mechanics, by N. K r y l o f f and N. B o g o liu bo ff

14. Lectures on Differential Equations, by S o lo m o n L e fsc h e tz 15. Topological Methods in the Theory of Functions of a Complex Variable, by M arsto n M orse 16 . Transcendental Numbers, by C a r l L u d w ig S iegel

17. Probleme General de la Stabilite du Mouvement, by M. A. L i a p o u n o f f 19. Fourier Transforms, by S. B ochn er and K. C h a n d r a se k h a r a n 20.

Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. L e f s c h e t z

21. Functional Operators, Vol. I, by J ohn

von

22. Functional Operators, Vol. II, by J ohn 23.

Ne u m a n n von

Ne u m a n n

Existence Theorems in Partial Differential Equations, by D orothy L . B ern stein

24. Contributions to the Theory of Games, Vol. I, edited by H. W. K uhn and A. W . T u c k e r 2 5.

Contributions to Fourier Analysis, by A. Zy g m u n d , W. T r a n su e , M. M o r se , A. P. C a ld e r o n , and S. B ochner

26.

A T h eo ry o f Cross-Spaces, by R o bert S ch atte n

27. Isoperimetric Inequalities in Mathematical Physics, by G. P o l y a and G. S zego 28. Contributions to the Theory of Games, Vol. II, edited by H. K uh n and A. W . T u c k e r 29. Contributions to the Theory of Nonlinear Oscillations, Vol. II, edited by S. L efsc h etz 30.

Contributions to the Theory of Riemann Surfaces, edited bu L . A h lfo r s et al.

31. Order-Preserving Maps, by E d w a r d J. M c S h an e 32. Curvature and Betti Numbers, by K. Y an o and S. B ochner

CONTRIBUTIONS TO THE THEORY OF GAMES VOLUME I I K. J . ARROW

H. W . K U H N

E. W . B A R A N K I N

J . P. M A Y B E R R Y

D. B L A C K W E L L

J . W . MILNOR

R. B O T T

T. S. M O T Z K I N

N. D A L K E Y

J . YON N E U M A N N

M . DRESHER

H. RAIFFA

D. G A L E D. B. G I L L I E S I. G L I C K S B E R G O. GROS S S. K A R L I N

L. S. S H A P L E Y M . SH IFFM AN F. M . S T E W A R T G. L . T H O M P S O N R. M . T H R A L L

Edited by H. W. Kuhn and A. W. Tucker

Princeton, New Jersey Princeton U niversity Press

1953

London:

Geoffrey Cumberlege, Oxford University Press Printed in the United States of America

Copyright 1953 by Princeton University Press

Papers k, 6, 7, 8, 9, 12, 16, 17, and 20 are published by permission of The RAND Corporation

Republication in whole or in part for any purpose of the United States Government will be permitted

iv

PREFACE

It has been said of the THEORY OF GAMES AM) ECONOMIC BEHAVIOR by John von Neumann and Oskar Morgenstern that "posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century."1

At the beginning of the second half of the twentieth

century the theory of games continues to be an object of vigorous and ex­ panding research, as the papers assembled in the present Study amply demonstrate.

The authors of these papers have not been content just to

solve outstanding problems and elaborate existing results but have gone on to raise fresh problems and extend the theory in new directions. This Study is a sequel to CONTRIBUTIONS TO THE THEORY OF GAMES, Volume I (Annals of Mathematics Study No. 2b, Princeton, 1950).

The

continuity of research bridging the two volumes Is evidenced by the fact that some of the problems posed in the Preface to Study 2k are now solved in this Study.

On the other hand, some reorganization has taxen place.

The simple division of Study 2k into two parts, dealing with finite and with infinite games*, has been replaced by a division of this Study into .four parts:

finite zero-sum two-person games, infinite zero-sum two-person

games, games in extensive form, and general

n-person games.

(The mounting

Interest in n-person games and extensive games is quite striking; scarcely touched in Study 2k, they now occupy fully half of this Study.) Each of the four parts is prefaced by an Introduction that describes the papers therein and their interconnections. At the same time these Introductions are designed to indicate how the four parts relate to one another.

The

newcomer to game theory is referred to the Preface in Study 2k for a gen­ eral exposloory Introduction to the problems of contemporary research in the field.

1A. H. Copeland, Bulletin of the American Mathematical Society 51 (19^5) p. b9o .

v

PREFACE The Bibliography at the end of this Study supplements the Bibli­ ography in Study 2b.

Recent publications have been added and some omis­

sions filled in, but the previous listings have not been repeated.

Many of

the new items, particularly those in economics journals, have been obtained from a bibliography on the theory of games prepared by

0 . Morgenstern.

The editing and preparing of this Study have been done at Princeton University, in the Department of Mathematics, as part of the work of a Logistics Project sponsored by the Office of Naval Research. Members of the Project who have participated in the task have been C. H. Bernstein, D. B. Gillies, I. Glicksberg, R. C. Lyndon, H- Mills, H. Rogers Jr., R. J. Semple, L. S. Shapley, G- L. Thompson, and the under­ signed.

Papers for the Study were refereed by members of the Project with

the generous assistance of D- Gale, B. R. Gelbaum, G- K. Kalisch, J. Laderman, J. P. Mayberry, J. F- Nash, E. D- Nering, and J. von Neumann. The typing of the master copy has been the painstaking work of Mrs. S. H. Robinson.

The good services of the Princeton University Press

have been ever available through its Science Editor, H. S. Bailey, Jr. To all these individuals, for their friendly cooperation, we express our sincere thanks.

H. W. Kuhn December 1952 A. W. Tucker

CONTENTS Preface

v Part I.

FINITE ZERO-SUM TWO-PERSON GAMES

1

Introduction Paper

1.

A Certain Zero-sum Two-person Game Equivalent to the Optimal Assignment Problem By John von Neumann

2.

Two Variants of Poker By D. B. Gillies, J. P. Mayberry and J. von Neumann

13

3.

The Double Description Method By T. S. Motzkin, H- Raiffa, G- L. Thompson and R. M. Thrall

51

Solutions of Convex Games as Fixed-points By M. Dresher and S. Karlin 5. Admissible Points of Convex Sets By K. J- Arrow, E. W. Barankin and D. Blackwell

75

k.

Part II.

5

87

INFINITE ZERO-SUM TWO-PERSON GAMES

Introduction

93

Paper

97

6.

Games of Timing By Max Shiffman

7.

Reduction of Certain Classes of Games to Integral Equations By Samuel Karlin

125

8. On a Class of Games By Samuel Karlin

159

9•

173

10.

Notes on Games over the Square By I. Glicksberg and 0 . Gross On Randomization in Statistical Games with inal Actions By David Blackwell

vii

k

Term­ 183

CONTENTS Part III.

GAMES IN EXTENSIVE FORM

Introduction Paper

189

1 1.

Extensive Games and the Problem of Information By H. W. Kuhn

12.

Equivalence of Information Patterns and Essentially Determinate Games By Norman Dalkey

13-

Infinite Games with Perfect Information By David Gale and F- M- Stewart

1k . Signaling Strategies in By G- L. Thompson

n-Person Games

217 2b5 267

15•

Bridge and Signaling By G- L. Thompson

279

16.

Sums of Positional Games By John Milnor

291

Part IV.

GENERAL

n-PERSON GAMES

Introduction Paper

193

17-

A Value for n-Person Games By L. S. Shapley

18.

Symmetric Solutions to Majority Games By Raoul Bott

19-

Discriminatory and Bargaining Solutions to a Class of Symmetric n-Person Games By"D. B. Gillies

20.

Quota Solutions of By L. S. Shapley

21.

Arbitration Schemes for Generalized Two-person Games By Howard Raiffa

n-Person Games

303 307 319 325 3*0 361

389

Bibliography

viii

Part I

FINITE ZERO-SUM TWO-PERSON GAMES

Finite zero-sum two-person games, keystone of the theory of games, now pose few problems that do not pertain to methods of computing optimal mixed strategies.

(See Study 2k, Preface, Problems 1 and 2.)

Part I of the present Study provides striking confirmation of this. The first paper deals with the equivalence of the solution of a certain game with the solution of a seemingly unrelated combinatorial problem; the second determines the solutions of two variants of two-person Poker; the third and fourth papers present computational methods applicable to all the games under consideration; while the fifth paper contains a theorem which is closely related to the criterion that a pure strategy be an active part of some optimal mixed strategy. The zero-sum two-person game considered by J. von Neumann in PAPER 1 might be called Hide-and-Seek. in some cell of an and columns by

n

by

n

In the first move, player I "hides”

grid, with rows indexed by

j = 1, 2, ..., n.

i = 1, 2, ..., n

In the second move, player II attempts

to find I's hiding place by guessing either its row

i

or its column

If player I is discovered, he must pay the positive penalty pending on the cell

(i,j)

the payoff is zero.

It is clear that there are

player I and

2n

j.

he-

in which he was hiding; if he is not discovered, p n pure strategies for

pure strategies for player II.

Von Neumann shows that

the solution of this game is equivalent to the "optimal assignment problem," which calls for the assignment of

n

persons to

n

jobs in such a way as

to maximize the total value of the assignment, given that the value of the ith person in the jth job is

a^j =

•

It

shown that the

game is simpler computationally than examination of the

n!

n

by

2n

possible

assignments. PAPER 2 presents at long last the material promised in a footnote on page 196 of the THEORY OF GAMES AND ECONOMIC BEHAVIOR.

It gives the

exhaustive solutions of the original discrete forms of the two variants of Poker considered by J. von Neumann and 1

0 . Morgenstern,

ibid., Chapter IV.

2

FINITE ZERO-SUM TWO-PERSON GAMES

This paper was translated from German notes of J. von Neumann (made In

1929)

and arranged for publication by J. P. Mayberry and D. B* Gillies with the active collaboration of Professor von Neumann. The "double description method" treated in PAPER 3 is a procedure devised by T. S. Motzkin for calculating the full set of solutions of a system of linear inequalities and developed independently by H. Raiffa, G. L- Thompson and R. M. Thrall as an algorithm for computing the solutions of a zero-sum two-person game. to explain their common method. player I in a game with matrix j = 1, ..., n,

Here the four authors have joined forces To find the optimal mixed strategies of

A = (a^j) , where

they seek those points of the

I, 1 0. over which the

> °;

t

i = 1, ..., m

and

(m - 1)-dimensional simplex

+ ••• + sm = 1

(m - 1)-dimensional hypersurface z = min

attains its maximum elevation.

The vertices (extreme points) of this

concave polyhedral surface are effectively calculated by the double description method.

The virtues of the method lie in the simplicity of

the individual steps, each of which amounts to the calculation of the intersection of a straight line with a hyperplane, and in the fact that all optimal mixed strategies are obtained.

In a final section the method is

applied to a general system of linear inequalities. The fact that solutions of zero-sum two-person games can be obtained as fixed points of appropriate mappings has been known since the original existence proof of von Neumann.

M. Dresher and S. Karlin utilize

this fact in PAPER b to construct an algorithm for the solution of any "convex game:" A(r,s)

namely, a game determined by a bilinear payoff function

of mixed strategies

r

and

s,

that range over compact convex

sets in Euclidean m-space and n-space, for players I and II, respectively. To such games the authors extend theorems known for finite zero-sum twoperson games and polynomial-like games, both of which are readily seen to constitute subclasses of the class of convex games. In PAPER 5 K. J. Arrow, E. W. Barankin, and D. Blackwell establish the following theorem concerning a closed convex set S in Euclidean n-space. Call a point s = (s^ ..., sn ) of S "admissible" if there is no point t = (t^ ..., tR ) in S that is distinct from s and such that

t^ < s^

all points of

S

for

i = 1 , ..., n,

and denote by

B

the set of

through which pass at least one supporting hyperplane

whose normal has all components positive.

Then every point of

B

is

FINITE ZERO-SUM TWO-PERSON GAMES admissible and every admissible point of

S

3

is a limit of points in

B.

It is a simple by-product of this theorem that every pure strategy which achieves the value of a zero-sum two-person game against all opposing optimal mixed strategies appears with positive probability in some optimal mixed strategy.

H. W. K. A. W. T.

A CERTAIN ZERO-SUM TWO-PERSON GAME EQUIVALENT TO THE OPTIMAL ASSIGNMENT PROBLEM1 John von Neumann The optimal assignment problem is as follows: ana

n

given

n

persons

jobs, and a set of real numbers a. ., each representing the value th th . ^ i person in the j job, what assignments of persons to jobs

of the

will yield maximum total value? tion of

n

A solution can be expressed as a permuta­

objects, or, equivalently, as an

(Such a matrix can be expressed by P symbol and i is the image of i

n x n

permutation matrix.

J p , where . is the Kronecker i i 9 J under permutation P .) The value

of a particular assignment (i.e., permutation)

P

will be

V

out further investigation, a direct solution of the problem to require

ni

p . Withi i 9 appears

steps, -- the testing of each permutation to find the

optimal permutations giving the maximum

V

a i i

9

We observe that the solution is transformation a.. — y a.. +

J

constants.

a

-L

j

u. + v., where

It is clear that

u^ +

5

J-

u.

jv j

invariant under the matrix and

Jv. are

any sets of

be added to each assign­

ment value, and that thus the order of values, particularly the maxima, will be preserved.

This enables us to transform a given assignment problem

with possibly negative aij,

a^j

to an equivalent one with strictly positive

by adding large enough positive

u^

and

v •.

We shall now construct a certain related 2-person game and we shall show that the extreme optimal strategies can be expressed in terms of the optimal permutation matrices in the assignment problem. (The game 2 matrix for this game will be 2n x n . From this it is not difficult to infer

how manysteps are needed to get significant approximate

with the method of G- W. Brown and J. von Neumann.

solutions

[C f .: "Contributions

to the Theory of Games," Annals of Mathematics Studies, No. 2b, Princeton University Press, 1950 -- pp.

7 3 -7 9 ,

this number is a moderate power of "obvious” estimate

ni

especially $

n,

5 -]

It turns out that

i.e., considerably smaller than the

mentioned earlier.)

Editors1 Note: This is a transcript, prepared under Office of Naval Research sponsorship by Hartley Rogers, Jr., of a seminar talk given by Professor von Neumann at Princeton University, October 26, 1951.

5

6

VON NEUMANN We firstconstruct a

game:

We may think of the

indexed

simple preliminary game, the 1-dimensional

game as

played with a set of

n cells or boxes

i = 1 , ..., n .

Move 1:

Player I 'hioes' in a ceil.

Move 2:

Player II, ignorant of I's choice, attempts to 'find1 piayer This is a play.

I by similarly choosing a cell.

The payoff is determined by a set of

(positive).

If player I is 'found* in cell

amount

otherwise he pays

d^;

i,

he pays player II the

0.

What are the optimal strategies for I? choose cell payoff °*tLxi

clLj_x j_

i

with probability

by choosing

i.

maximuin* The value Now let

cL.x . < max

J J

j_

x = (xjJ

(d .x.). Choose

1 1

x^.

Let his strategy be to

Then player II will obtain expected

Hence he will choose a cell for him will thus be be optimal for I.

£ >0

such that

i

for which

max (oLj_xj_) •

Assume that an d. (x *+ £) = max (d.x.).

J J

i

Define , ( = x. + £ xi \ = x^ Then

max (d-x^) = max (d^x^) ,

and

for

i = j

otnerwise x± =

Xj_ xp + £ = 1 + £•

Hence the

can be used as probabilities, and max (d.x!) i max (d.x!') = ---------- < max (d.x.) , i 11 1 + £ i 11 i.e.,

x = (x.) was not optimal. Thus necessarily all d.x. = max (d.x.), i j J ^ i i i.e., d 1x 1 = ... = dnxn = A. Now Xj_ xj_ = 1 implies A = 1/ X ± cT~ > A and, of course, xj_ = • The value of the game (for II) is clearly A. We now introduce the game in which our particular interest lies. We call it the 2-dimensional game; it is a generalization of the 1-dimen­ sional game as follows: Thecells are doubly indexed from 1 to n. (They may be thought of as fieldsin an n x n matrix.) Move 1: Player I hides as before. Move 2: Player II now attempts to'find' I byguessing either of the indices of the cell in which player I has hidden. He must state which index he is guessing. (I.e., II attempts to pick the row, or the column, of I.) Player I, if so 'found' in cell i,j pays to

A CERTAIN ZERO-SUM TWO-PERSON GAME II the amount

ck

where theoUj

otherwise he pays 0. Thus player I has

n

2

7

are a given set of positive

numbers;

pure strategies and player II has

We now discuss optimal strategies for player I.

2n.

Let his mixed

strategy be x = (x^.), xij = 1* ^here each x^j represents the probability of his hiding in cell i,j. Then player II's pure strategies will give a return of for column choice

J -1-J

J

^ . ol. . x. . for row choice

j.As in the

x

or

2 .

Jj x. J•

1-dimensional game, he can nowsimply play

pure strategies giving the maximum such return. choose

i,

minimising this return.

Player I will try to

Thus the value of the game (for II)

will be:

isj-

’2i v

xij')

•

The characterization of I's strategies is not quite as easy as before.

The

simple direct compensatory adjustment of the 1-dimensional game cannot be miade. For further progress, we obtain certain results on the geometry of convex bodies. We define: R = Set of such that

all

vectors

Zi j ^ ° ’ S = Set of

all

z = (zj_j)

(in

Zi j = 1 ’

vectors

n

2

dimensions),

Zi j = 1 ’

z = (zj_j)

(in

n2

dimensions),

such that

zij I °> T = Set of

ail

zij i 1 ’

vectors

Zij < 1 •

z = (zj_j)

sions) , such that z^. = N(z,f) > ...,

can only terminate with a

z ^

G R.

z £ z r;

ana

Hence N(z) > N(z’). / / / sequence z \ z f < z 11 < ...

in

which therefore must terminate. Hence

w = z ^

It

has all desired

properties. LEMMA 2.

R = Convex of

(This theorem is due to Series A, Vol.

5

[19^6], pp.

T.

G-

117 - 1 ^8 .

Birkhoff, Rev. Univ. Nac. Tacuman, Cf. also

0 * Birkhoff,

"Lattice

Theory," Revised Edition, Amer. Math. Soc. Coll. Series, Vol. 21 [19^8], example k* on p . 266.

The proof that follows is more direct than

Birkhoff’s .) PROOF.R R

Convex

is clearly convex.

R

is Immediate. Hence

T.

R C Convex T is demonstrated, if it is established, that all extreme points of the convex R belong to T. Actually theyform pre­ cisely the set T . That every point of T is an extreme point of R is clear: A z G T belongs to R, and if it were not extreme, then z = tz1 + (1 - t) z ’1 with z 1, z 11 G R ; z ’^ z '1; 0 y say z£j < zjj.Then z±j = tzj j + (1 t)z{j, hence zj_j < z±j < z ^ y Now either z±j = 0, implying implying zjj > 1 -- and both are impossible.

z|j < o,

or

z±j = 1,

9

A CERTAIN ZERO-SUM TWO-PERSON GAME

There remains, therefore, only this: point of

R belongs to For a z £ R

z £ T (for a

z £ R)means that all

inner.

z $ T

Hence

If a line

(or a column

i,j j)

i,j

is not inner either.

i,j

is

exist.

contains at most one inner

element z ^ y then z^. = 1 - Xy z ^ y (or necessarily = 1, 0, -1, -2, ... . Since z ^ 1.e.,

^ 0, 1. Clearly

z ^ . = 0, 1,i.e., that no

means that inner i

To prove that every extreme

T . This is shown as follows: call a pair i,j inner, if z^-

-

Xy i,j) z is therefore z^ - = 0, 1,

z^- = 1 > 0,

In other words:

If

i,j

is inner, then

there exists an inner i ,j 1 (i',j) with j ’ ^ j (i* ^ i). Now consider a z£ R, such that z ( T. Let i,jbe inner. Choose

j' ^ j

such that

is inner, then

j11 ^ j'

is inner, then such that

i 1,j11

if ^ i

such that

is inner, then

i ’,j 1

i 1' ^ i*

such

that i 11,j11 is inner, etc. In this way two sequences i, i', i ,f, ... and j, j1, j1', ... arise, such that i ^ , j ^ is inner, j_(m) ^j(m+l ) is inner, and i ^ ^ ^(m+1)^ j(m) ^ j(m+l) ^all t h i s for all m = o, 1, 2, ... ).

Hence

i ^

= i ^

p / q. Choose such a pair with and with p i(q-l)^ j_(q)

or p
-L -l

\ J

f •••

q_p _i

x

Gr ^o

1 y Jq_p_ i —

^

j(q“1}. For i = ±(P) (= o 1 ^

hence necessarily i ^ = i ^ , define i = i(c3 } i = i(P+1> i = i(P+1> i } ’ Jo J * -H * J1 J > •••> ^-q-p-i

i(q_1t Jq-p-1 = l q_1t Thus tw0 sequences iQ, ±1 , ig_1 and j 0 > j,. J's_1 (s = q - p, also define jg = jQ) arise, with the following properties: iO , i-, I ..., i o— .l are pairwise different, also J0 > - > js_1 are pairwise different, it,Jt is inner, it,Jt+1 is inner (all this for t = 0, 1 , ..., s - 1). I.e., the quantities (1) z

are all

. , z. . , ..., z. . , oJo 1 s-1Js-1 (2) z± ., z. z. , O 1 11 J 2 1S- 1 JS )> o, 0.

zij^zij^

£

(j S

= j ) °

be the minimum of the quantities in the lines (1),

Define

z' = (zj_j)

z !l = ^zij^

r = z^j + £ (z^. - £)

for the

| = zij “ £ ^zij + ^ L =

for the otherwise

Then z 1 £ R and z ’1 £ R and z = Tj- z 1 + ~ z 1*.

i,j

are readily verified.

as f°l^0WS:

of line (i) line (2)

Also clearly

z* ^ z 11

VON NEUMANN Hence

z

is not an extreme point in

R,

q.e.d.

We now return to the 2-dimensional game and a characterization of player I's optimal strategies. A

let

Let

x = (xj_j)

be an optimal strategy and

be the value of the game (for player II). We define

Clearly all < A and all z. . < 1 and all z. . < 1 . Hence z = j tj = j— Now Lemma 1implies the existenceof a

°^j_jxij < A, i.e., all belongs to S.

X* R,

with

z < w.

Then all

= A

U j i uij:

Hence

• - - 0 = - u

JL ^

x^j =

1,

S X * ljUlj = A Also

Vj_.

z . • < w. ,, -LJ = - LJ

xi / - i j uij
0 . Y —

may be assumed.

, XT = (xj.)

A

,

Form

.

xv . Also

zT . = cf v iP ,j

A o ®*ij .PV . J 1 ,J

In other words: All optimal optimal strategies of the special form

strategies are centers of gravity of

A CERTAIN ZERO-SUM TWO-PERSON GAME (*)

x. . = 1J

ij

-p i >j

( P a permutation)

Consider, therefore, the strategies of the form (*). strategy all

= A

3X1(1 a11

S i ^ij^ij =

For such a

Hence

Max_ ( X j h ' j xi'j > S i h j ' xlj' ) = A ' Hence the optimal ones among these strategies are those that give the minimum S-t to

A.

Now, since the x. . are probabilities, J i i xii = ^ i.e., A 1 1 — = 1 , i.e., A = l/Si H • Hence the minimum A corresponds . .P . ,P the maximum V - dT"— • I*e., precisely those permutations P

i iP give the optimal strategies 9 in question, for which its maximum value.

1 Si 3;--P i,i

assumes

To sum up: THEOREM.

The extreme optimal strategies (i.e.,

those, of which all others are centers of gravity) of the 2-dimensional game are precisely the following ones: Consider those permutations

P

which maximize

i,i For each P of this class form the strategy according to (*) above.------

x = (xij)

Note, that this means that player I plays only those cells where the permutation matrix (of

p

P)

has a

1.

(Here line guesses

(i)

and

column guesses

(j)

tion

His play among these cells is then determined by the

j = i .)

correspond to each other equivalently under the rela-

1-dimensional game. -----The condition expressed in the above Theorem for the optimal assignment problem with

a^j =

where the

P

is exactly a^j

are the

elements of the assignment matrix (which, we saw, could be considered as all positive). Several further remarks can be made. 1) A transformation of a^j — > a^ j + u^ + Vj in the assignment matrix leaves the solution unchanged, and hence the game will be invariant (in its P's) under the corresponding

VON NEUMANN

cL.

.-> x

ij

1 +

r.

+ Vj)

That the game should be so invariant is not at all clear initially from the game itself.

(Note, this is not complete invariance.

solutions for a particular

P

may change, though the

The 1-dimensional P

remains the

same.) 2)

Various extensions of the optimum assignment problem are

possible and can be settled in essentially the same manner.

Thus one can

specify certain many-to-many assignment patterns between persons and jobs, and the like. In addition, certain formal generalizations of the game are possible —

to various k-dimensional forms with

seem to be interesting, but present

k = 3, ^ , ••• •

These

serious difficulties.

J . von Neumann

The Institute for Advanced Study

TWO VARIANTS OF POKER1 D- B. Gillies, J. P. Mayberry, and J. von Neumann INTRODUCTION Although the minimax theorem for zero-sum two-person games asserts that there always exist good strategies for such games, their calculation may be a most formidable problem.

Methods for simplifying this problem are

certainly necessary if the complex games which simulate economic situations are to be attacked.

In "The Theory of Games and Economic Behavior" (here­

after designated as [1 ], in accordance with the bibliography at the end of this paper) several idealizations of actual games of Poker are discussed, complete solutions are given to two of them, and some information is given about the solutions to others.

This paper supplements that discussion

with the solutions of two of the other variants mentioned there. deals with the discrete case of sections

1 9 .4 -1 9 . 6

Part I

of [1 ] while Part II

completes the discussion of the continuous variant of section 19-13-

PART I $ 1 . DESCRIPTION OF THE GAME We deal 196.

The

with the game treated in [1 ], sections

1 9 .4 - 1 9 .6 ,

playersV and 2 each obtain by a chance device

an integer from among

1, 2, ..., S);

for either player, each of these

handsis to have equal probability, independently of the Then each player,

opponent’s hand.

being informed of his own hand but not of his opponent’s,

elects to bet either the amount low bet), where

pp. 190-

a "hand," (i.e.,

a > b > 0.

a

(the high bet) or the amount

If both bet

a

[both bet

the higher hand (larger integer) receives the amount

Editors’ Note: This is a supplement to THE THEORY BEHAVIOR, pp. 186-219 (as promised in a footnote on pared under Office of Naval Research sponsorship by and J. P. Mayberry (Part I) from notes of Professor advice and collaboration. 13

b], a

b

(the

the holder of

[the amount

b]

OF GAMES AND ECONOMIC page 196). It was pre­ D. Gillies (Part II) von Neumann, with his

GILLIES, MAYBERRY, VON NEUMANN from his opponent} (if the hands are equal, no payment is made) .

If one has

bet high and the other low, the latter may choose either to "see,” or to "pass."

If he chooses to see, payment is made as if both had bet high

originally; if he chooses to pass, he must forfeit the sum of the hands held. For any s

with

s = 1, ..., S

choices, described by a numerical index "high" bid;

i

= 2

regardless

the player has three strategic

ig = 1, 2, 3 ; ig = 1

meaning a

meaning a "low" bid with subsequent "seeing" (if the

ig = 3

occasion arises);

b,

meaning a "low" bid with subsequent "passing"

(if the occasion arises). Thus the (pure) strategy is a specification of such an index

i

for every

s = 1, ..., S

-- i.e., of the sequence

i,, •••, is This applies to both players. Accordingly we shall denote the above strategy by by

S 2 (1 i’

^(i-j, ^

•••, ig)

for player 1, and a corresponding one

for player 2. ^ 2.

DEFINITIONS

Mixed strategies are introduced as in [1], pp. 192-19k. instead of introducing a separate probability

*. *ii

I.e.,

. [>>. . ] ... is (i1 ... Ig

for

player 1 's [player 2's] using the pure strategy ••• ig) [ 2 (±i ... ig)], it suffices to use the probabilities[ 0 [ o]

for all

s,i,

and

= 1 [

Tf = 1 ] for all sT If players 1 and 2 use the mixed strategies

= (cr^),

then the expected payoff of the game is the

(19:6) in [1], p. 195 (there designated Clearly (cf. also (1 9 :7 ) in [1], p. (2.1)

p = (/>?) K(/>|o-)

and of formula

K(/^ , ..., ^ S |

one

i°r

s, *

say

s^s

-- and

S

p^ = o

for

= 0

/

for

.

too (cf. above), 'K'

s y s .

Summarizing:

occurs at all, then it occurs for precisely * s s = s . In this case f> = 0 for

g

s < s ; p^ = 0 />® = 1 )

^

*

s > s ,

/>

for

Next define

for

*

S

S

s = s ; />2 = f>^ = 0

(hence

s > s*.

p = (^?)

n +el (4.2)

as follows:

for

s = s , i = 1 ,

for

s = s*, 1 = 2

,

otherwise. (Note, that

p^* = 0

= 0 .)

p2*=

(2 .2 .c) give

?? ^

this case tf2 that ?? ) ^

3”

[cf. (4.1)] implies, that actually

Let this substitution take for all

s

implies that one of

C If

J

into

s,i, except for

p^* = 1 , Now (2 .2 .a) -

?^.

s > s*, i =

P

p.

-

p

(i-e., the

y)

jg , such y) im-

-- i.e., the goodness of

P

Summarizing:

is a good strategy according to (4.1) (i.e., with

^ 2 *^ then the P of (5 -2 ) (which has all is also a good strategy.

p^ = °)

Thus the statement in the title of (4) is correct. throughout (5 ) - (8 ),

= °>

2 , and in

^ • I-e -> for ©very s and igthere exists a . Hence the validity of (2 .5 ) for P (i-e., the

plies its validity for

(4.3)

jr?

p = (f?)

Accordingly,

will be assumed to be a strategy with all

an^ its goodness will be the subject of the investigation. $ 5 • ALGEBRAIC CRITERIA FOR GOOD STRATEGIES Consider the criteriura of goodness (2 . k ).

Since

p 2 = 0,

it

need not be applied for 1 = 2 . Consider now j = 2. (2.2.a), (2.2.b) show that always Hence the case i = 1, j = 2 need not be considered. The above also implies * Hence the case i = 3, j = 2 is disposed of if the case 1 = 3 , j = 1 is. Thus there remain only the cases with i, j / 2, i.e., i = 1 , j = 3 and i = 3, j = 1 . These cases state, that

implies

= o,

and that

TWO VARIANTS OF POKER y® < !Tj

Implies

^

= 0,

p

i.e.,

(5 -1)

17

= 1 . Put

uS = i f b

- O

•

Then these conditions become

(5 -2 )

us > o

implies

p^ = 1 ,

us < 0

implies

p^ = o .

^

P

Thus the goodness of

is expressed by (5.2).

Next, (2.2.a), (2 .2 .c ) give

“a = s t3 ( -

■

/>?

p\ = 1 “ p \ > this becomes us = £ ts p \ + ^

(5.3)

(2 (S - s) + 1) .

From (5 •3 ) (5 .M

u s+1 - u s = ^

k 6.

uS (5-3) p f r = 1. Thus

/>3 = 1

u^ > 0, (for some

gives

.

s

hence by (5-2) p^ = 1. s = 1, ..., S - 1j,

us+1 - us > 0, hence

with

the smallest one, then

/>3 = 1 s = R +

'Hence they are precisely (6 1) < [

^ +1 - J j g

DERIVATION OF THE BASIC STRUCTURAL PROPERTIES

(5-3) yields If

+

us+1 > 0,

exist (e.g., 1, ..., S,

then(5-2) gives and so by (6.2)

s = S),

and if s = R

s = R, R + 1, ..., S:

= 1 occurs precisely for s = R, is a suitable number = 1, ..., S .

R + 1, ..., S,

where

For s = 1, . .., R - 1 P^ ^ 1 , hence by (5•2) is therefore of interest to determine those s = 1 , . . . , R - 1 , us = 0. Let */ be the set of these s . Consider an u3"1 < 0

is

too, belong to this class.

s £ */.

(cf. above), hence

Assume =0

s - 1 £ ^ (s ^ 1) . (by 5*2) and

Then

R

us < 0. It for which us = 0,

us - u3”1 > 0,

18

GILLIES, MAYBERRY, YON NEUMANN

-1 0 (by (5-JO ), i.e., />?> 2b hence Hence a+b > ' 1 x a+b /■r rts s+1 s+1 s s+1 «s+1 = 1 P t + ^i - a+b x , i.e., u ~ ’’ - u~ > 0 (by (5**0), ua+l > 0, p* (by (5-2)). Hence s + 1 y R, and since s < R - 1 , this gives s = R - 1 . Thuss G ft/ implies s - 1 G / if s ^ R - 1 (and s ^ 1 ). Now let P be the largest numberamong 0, 1, ..., R - i, such that all

s = 1,

consists of the

..., P belong to j/.

s = 1, .. , ?,and possibly

1

then the addendum s = R addendum

s = R - 1

only occur for

P

Assume u, - d

Then,according

is

s

not needed;

would require

= R -* 1 .

P = R - 1.

< o,

R ^ 1 ,2.

p\

If

if P = R - 2,

^ R - 2, R - 1. P ^ 0 and

to theabove, P = R -

If

P ^ 0

with

and

p \

Next, assume If

f ? £ af^ ’

then

=

i ?

this addendum can

I.e.

1

Then

u i £ °» hence 2b (by (5-^)), hence f .1 Y < a+b

1

then there exists an

0

£ )

for

This means P = 1, R = 2. R = 1, 2 . b ( 1 + £) with - 1 £ < 1, Sives " 3 a+b and

If

we redefine

/ />f »“ > ^

P : P = 0,

P = 0

hence

p^

(6.2) is vacuous.

Summarizing: For

1

necessarilyus < 0.

Let

s =

1 , ..., R - 1 with

us = 0.

s = 1, ..., R -

be the set of all The elements of

J

*/

can be represented in this way:

(a) Ail s = 1, ..., P . (b) Possibly s = R - 1. (6 .3 )

U P

is a certain number curs, then P / R - 2,

R = 2

is not excluded.

= 0 , 1, ..., R - 1. If (b) oc­ R - 1, except that P = 1 , The

resented as follows: (a) There exists a fixed such that for

p>^,

s G J

for

£with

s = 1, ..., P

-

1

and

Thus this addendum,

as the addendum referred to above 2b Also /> < 1, p^ > a+b 0 < T[ < 1 .

Finally, for

£

s fodci 1 . leven{

has at any rate

with

Also

s = 1, ..., P

s = 1 = R - 1

s = R - 1

f ^ 7!

-

( 1

P ^ 0 d

2,

such that for

b

i.e., (6.2) still applies. view

1,

R /

- 1 < £ < 1,

1,

then the

-sje< 0 b with hence />1 1 < I < 1 • Next, let P\ l 0 , a+b ( 1 + g ) s+1 ~ "*S+1~ s, s + 1 = 1 , ...,P. Then us = u' = 0, u - u = o, i.e., f>® + />®+1 - ^’2b* = 0 (by 5 •*0 ), hence ^ - (>*. Summarizing:

( 6 .2)

j/

are rep­

< £
—

cl

S -

1,

i.e.,

S S with

=

if

Tj ^ 0.

a a-b ,

Hence necessarily L < 4 S + -th ^a+b a+b

( 8 . 6)

L >

S -

4

a-b

and if these inequalities are fulfilled, then the original ones are, too, with 7| = 0 . Thus (8.6) (for an Note, that if an L unique. Indeed, the opposite of (8.6) be ) 1. This means

^a+b 4 which gives

S
(v S a+b

£r-) + 1 ' , a-b'

. Then the first relation of (8.6) gives

L < 1,

TWO VARIANTS OF POKER

23

hence no solution other than L = 1 exists. Note further, that (8.6) necessitates

I s -sa>

b j a+b "

a-b q a+b D

i.e.

iPF III:

L

Q

(8.3)

=

0 --

1

i.e.,

L |°vgn j •

+ 2ab - b a-(5 : w

0,Q

P /

=

2

0:

3

- | ig) - S=& n ,

(8 .4 ) becomes

S - I l 0} " 1 »

L> for

s i

P

becomes: L - S?

for

a

L {even) ' wlth = for ^ ^ 0 ' The proviso made in (6.5) refers to this case and implies

rj = 0.

Consequently we have the conditions t a-b q b r0 ■» L = "a" S - a l?) ’ L

for

>

(odd )f • For an odd L fVGv fcyl1J

(8.8)

a ^ b

L

s

_

|

{ ? }

_

1

>

the first condition becomes

L = ^

S ,

while the second one is then automatically satisfied. a —b q b a a

(8.9)

S?

S - | < L < §?

For an even

L

the

s + ^>

and the second one is again automatically satisfied. L = 2, k,

Thus (8.8) (for an L = 1 , 3 , ■•• ) ... ) characterizes this case.

or (8.9) (for an

Note, that if an L according to (8.8) or (8.9) exists at all, it is unique, because of ^ < 1 : If (8.8) gives an odd L, (8.9) cannot

2k

GILLIES, MAYBERRY, VON NEUMANN

contain an even IV:

L,

and (8.9) cannot contain two different even

L^ft^O

--i.e.,

Since

1,

Q =

Hence (8.3) implies

L < —

2, ...,

(8 .1 0 )

Q ^

Q + (^}

and (9 -t) implies

L = ^

L.

0:

therefore

S

cl

0,

P ^

and

L >

Q - (q) 9-

S -

>

0.

1.

Hence

If

d

S - K ,

where (8 .1 1 )

2,

S - K =

0

3, ... ;

Clearly (8.11) canbe satisfied by a suitable

< K
2,

i.e. (8.12) Also, the

S 2|!bK

of (8.11) is unique, unless

3. ——

S =

3 , k,

...,

in which case

K = 0, 1 will do. In this case, however, (8.8) (with L = 3 , 5 , ••• ) or (8.9) (with L = 2, 1 , ... ) can be fulfilled. Hence we canignore it. Now (8.3) becomes K =

^ ( Q +(£)) +

J

(8 .1 3 ) L

^

{even) •

(8 .t) becomes

K £ - I (Q- (§)) + 1 for

L - Q (even) 5

wlth

=

if

V

It is best to discuss (8.13), (8.it) separately for for

T] ^ 0 . T| = 0 :

Tj = 0

and

(8.13) becomes

Q + (0 ) = | K

(8.15)

0 '

for L - Q { ° ^ n) .

(8. It) becomes (8 .1 6 )

Q -(£) § K - -b

(8.1 1 0 becomes (8.18)

Q - (§) = |

(1

f'or

- K)

The second relation in (8 .1 7 ) can be transformed by adding (8.18) to it. This gives 2Q + i > 1 . Since for

Q = 1 , 2, .. .,

L | ^ n} .

this is always true, except when

Q = 1, £ = -

forbidding this. Hence all these cases (expressed by (8.15), (8.16) for and by (8.17), (8.18) for

(8 .1 9 )
a 2 > ... > am _, > a^ = b are allowed. The whole discussion that follows will be more informal, and in parts (2 ) and

(*0

more heuristic, than the discussions in "Theory of Games"

or in the preceding paper.

The definitely formulated conclusions and their

proofs are, nevertheless, rigorous. $

2.

CASE (I) : A CONTINUUM OF ALLOWED BIDS

2 .1 .

Let us analyze a potential good strategy -- since the game

is symmetric, it does not matter for which player. Assume that for all hands

x > xQ ,

where

determined, the strategy provides always betting sible bid.

For the other bids

(i.e.,

a,

xQ

is a constant to be

i.e., the highest pos­

a > oi1 > b) ,

let (|)( y )> x 1

y £ I

of the open interval

Then (3-1 -1 ) implies the existence (y) 7^0have

r-

in

^ 0.

Consider an open sub-interval x

x

y

of a jjlsuch that the 0.Henceu ^ y.

measure^

Now the goodness of the opponent's strategy excludes on a set with measure ^o where 0v (y) ^ 0, hence y' £ I with %A y') V a^L0yU ” ay^Lt1 a^ V i.e.(since

~

©^ = 0 for

ai9 0. Hence the goodness of the strategy allows yy (x) )> Yy - 1 (x) these x only for a set of measure 0. I.e., with such exceptions

for

TWO VARIANTS OF POKER (3 -1*-*0

*v (x)
b

be given.

Choose

av > dJ > av+1 . Then by (5 -T -^+")

S

e a >c*.'

=Y /*

_

“ e

/

e

^ < v

.5 ;

A>v

= TT\ /*■

----- ---

A>v aA-l + a A

ln aA-JL .+. aA 2a

A

y

with

m

TWO VARIANTS OF POKER

ln(l + \

Replacing

0 ((aA _ 1

- aA )2),

0 (g (ay

changed by

----- - )

i.e. by

£

- aA )).

i.e. by

0 (g(a

- am )),

_ e

= e"

'

---= e

-

Now

affl = b,

- ln( 1 +

by

0 (s(aA_ 1

o

^ ln ^

5 111

a

----- )

changes it by

Hence* the entire

0 (g).

- b))j i.e. by

+

A *A

A )

'is Thus

+ ° (£)

0(O

m

. o (S) i ( i a )

= „.

I.e. :

2

ik.,.2)

45

The right hand side of (4 .1 .2 ) is equal to (4.1 .3)

a f (cOdcl +

(1

- x0) ,

01' J

with the

$(cO

probabilities (for

£ — >

0)

of (2 .1 .6 ) and the

9^

for the bids

a^

xQ

of (2.1.7).

I.e. the a priori

define a distribution that converges

to the distribution with the cumulative distribution func­

tion (^.1.3) -- i.e. with the probability density

(])(cO

in

a > oc > b

and

the point probability 1 - x at ot = a. According to (2.3) this is precisely the distribution of a priori probabilities of bids that character­ izes the good strategies that were obtained in (2) for the continuum case, i.e. (I) in (1 ). In other words:

If the mesh-width of the bid system (cf. (^.1 .1 ))

GILLIES, MAYBERRY, VON NEUMANN converges to

0,

the distribution of a priori probabilities of the bid

system for case (II) converges to that one for case (I). To this extent, then, we have continuity.

4 .2 . ev

The behavior of the a priori probabilities, i.e. of the

(or, rather, of the

us now consider the

0 ^),

XQ

©v (x)

turned out to be continuous.

Let

themselves, or, which in view of (3.3.2)

amounts to the same thing, the Consider again the set up of the

£

(4.1 .1).

of

4 .1 ,

more specifically

Wewill prove, that this is

variety of behaviors for the

£— > 0

for

compatible with

|v .

We observe first, that (3 -7 -3 ) implies

Y

o

r^ y p

S ! l i i V

2ay

^

A

1 /*• '

i.e.

av ^ f K v

^ av+i ^/* £1^.

^

S / , 9ll < a ,

Since

Replace

1.

..., m -

(3 -1 -3 )), (t.2 .2 ) implies

there follows

f°r in (b-3-1)-

Then (t.2.1) gives

— . 7T (a - a , ) < ( a 9 •••>

1, ..., m,

Hence (1 -3 .7 )

h = (—)/°

V(A)

•••, n. Again, we plot these marginal lines in P 2 . As a typical example let us consider the diagram for a

2x5

case:

The minimum function for player I is given by the broken dark line, the optimal strategy is and value the g81116 v * In the above diagram there are nineteen critical points involved, some of which are obviously unimportant. Since in higher dimensional cases we cannot have the support of a diagram and hence cannot see which points are important, we want now to devise a computational procedure from purely algebraic considerations. In Part II we will discuss the formal details of the computational procedure, but for the present we use the above diagram to Illustrate our procedure. Since and x^(£) take on minimum values on the boundary

59

THE DOUBLE DESCRIPTION METHOD planes

= O

and

f2 = 0

we

consider the minimum function of

these two marginal lines, viz:

Now evaluating (0,1)

and

at the critical points

be discarded.

But

x2

at

of

and

lies below the critical point and

hence we must find the points of intersection of and

(1,0),

we note that these lines lie above the minimum function and so may x1

and

x2

and of

x2

x ^ . We then discard the critical point arising from the intersection x1

and

x ^ . The critical point with the largest

x

value then gives

the optimal strategy and the value of the game.

)5 • In the

3 x n

THE 3 x n GAME

game

12

In

*21

22

x2n

*31

32

3n

we consider the game space

viz:

Each marginal function x j($) can represented as a plane in • It is now possible to represent geometrically I's minimum function and to find the optimal strategies and value, v, of the game. A basic strategy, is an optimal solution such that (^°^,v) is a point of inter­ section of 3 planes (we include boundary planes 7^; £ = 0 ,

60

MOTZKIN, RAIFFA, THOMPSON, AND THRALL

The geometrical model for the three dimensional case can be constructed analogously to the two dimensional model with planes playing the role of straight lines.

It is suggested the the reader keep this model

in mind while reading the next part of this paper. PART II.

THE COMPUTATIONAL PROCEDURE

$ 6.

SUMMARY OP PROBLEM

As we have seen, the goal of player I is to find those maximize his minimum function hyperplanes in

f(£).

Since the marginals

the minimum surface

x = f(f)

in

£

x. = £Cj

which are

Effl is a polyhedron

and so can be completely described by giving its vertices which are, of course, finite in number.

We now develop a systematic computational pro­

cedure for determining these vertices. Suppose the vertices are Q 1 = (^ 1^ ; x ^1^), ..., Qg = (^ S ^; x ^s ^). ( i) £ J

value of the game and the

Then

v = max x ^

( i) x J = v

for which

is the

^ are basic optimal

strategies in the sense that the set of all optimal strategies is just the set of points in the smallest convex subspace of see [8], page 2 8 , 3 0 .

maximizing

Sffi which contains these

It can be shown that the two

definitions of basic strategies are equivalent, see [6], appendix. $ 7•

THE SOLUTIONS FOR PLAYER I

A basic component of the computational technique, to be discussed, will be to find the intersection of the line segment joining two points of Tn] and a hyperplane in LEMMA. P1 =

P^.

To this end, we introduce the following lemma.

The line segment joining the two points x^)

and

P. =

intersects the hyperplane: only if and

d[k) d^.k) < 0

djk ^ =

d|k ^

< 0

x^)

in

^(£;£Ck ); ^ £ S ^ }

where

- x ^ .

if ana

d[k) = £(l)Ck - x (l)

In particular, if

then the point of intersection,

P,

is

unique, and given by the relation:

p — p

Proof.

-d(,k) P. + d(k) P. J 1 d

-

The line segment joining

P^

and

P . can be parametrized

61

THE DOUBLE DESCRIPTION METHOD by the relations

t P. + (l - t)P . where

0 < t < 1.

This line segment

will intersect the hyperplane if and only if there exists a interval

[0,1]

t

in the

such that

(t

+

(1

- t)£^)CL. - (t x ^

+ (1 - t ) x ^ )

=0

,

or if (7-1 ) If

t d(k ^ + (1 - t ) d ^ - o,

then the point

P^

= 0 .

lies in the hyperpiane.

then the point Pj lies in the hyperplane. If d P ^ = line segment lies in the hyperplane. If both d^ and dj

If

= 0

= 0 then the are different

from zero and of the same sign then there is no solution of 7-1 for 0 < t < 1.

If, however,

different sign (i.e.,

d ^

dP^

and

djk ^

10/21, 0, 0), (3/7, 0, 5/1^, 0, 3/l *0,

and

(2 1 / 6 3 , 0 , 22 / 6 3 , 20 / 6 3 , 0 ). PART III.

GENERAL INEQUALITIES AND SECOND VARIANT OF THE COMPUTATIONAL PROCEDURE $10.

Let of

h

ROUGH PROCEDURE

S a ^ x ^ > 0 , j = 1 , ..., h; k = 1 , ..., n;

inequalities in

n

-unknowns.

the system formed hy the first fixed Then the

h ”^

inequalities is given hy

taking on ali non-negative values.

inequality will determine a half-space (or the whole space)

hoth in x-space and A-space. the half space

h - 1

Pkl >

Sb-^A^ > o

Putting

h1 =

Sa^p^,

cone generated hy the rays from 0

toward

h-, > o, and

11

and 12

qk m, m = hy

with

the common part of

and of the positive A-orthant will he the convex

unit vector) for every every

be a system

Suppose that the general solution of

1

with h^ b^

•••> s >

and

0.

stants and adding the inequality

The points with

| = 0

lie in

the hyperplane at infinity. Now suppose that a full double description has been obtained at a certain stage, and a new plane to

H.

step is to be taken corresponding to a hyper­

We write the coefficients

C and

If the products involving

H

by

of

by

H

and the hyperplanes

C and

of

H

as an additional column next

D and compute the inner products with the rows of

A

and

B.

are all non-negative, then the cone defined C

and

D

is the same as the cone defined

D above, which by supposition Is fully described.

Hence

H

is superfluous (it either gives no face at all or a face already indicated) and should be omitted. Secondly, if the products involving zero, while there exist rows of such row should be thrown out.

B

H

and rows of

A

are all

for which the product is negative, each

However, before throwing a row out it

should be combined with each row of

B

corresponding to an adjacent face,

as explained before, and giving a positive product.

By combining we mean

determining the Intersection ray of the plane corresponding to the two rows and of

H

and writing the row obtained as an additional row to

determination can

be effected by the formula

Section 1 0 , or as

in the computing instruction below.

Finally rows of

A

cription by

H

vanish.

We choose as normal form of

A

pairs of opposite vectors; hence the product with

positive for a certain row

a

of

A

I of

2 ) delete every row

a* ^ a

of

A

a

and

-a

and

adjoin a

II is to

negative of that combination; 3) replace every row of or positive product by its combination with SPECIAL CASE:

a

or

-a,

B

and by the giving a negative

respectively.

x = 0

If the inequalities x 1 = o, xn = o occur in the given system, as for instance in the game case treated in Parts I and II, these inequalities can be taken first and since their solution is the positive orthant,

d

B;

giving a positive product, while re­

-a* by its "combination" defined as before witha

412.

of

a des­

(if there are several such rows

choose one). Then one has to: 1 ) delete placing

I b^

we have to consider the case where not all products

involving d

pkl | b^ |+ p ^

B. The

is already zero, the central part

A

is void and the last-

TO

MOTZKIN, RAIFFA, THOMPSON, AND THRALL

mentioned possibility in the procedure of Section 11 does not occur.

If

we suppose that the given system is non-degenerate in the sense that no n + 1

of the (inhomogeneous) linear functions vanish at the same point,

which is always the case after a small change of the coefficients, then also the condition for adjacency takes on an especially simple form and a computing instruction for such a system would be as given below. The computation at the end (see Tables II and III) concerns a submatrix of the diet-matrix in [9]• C

In this example the matrices

of the schematic diagram above are absent.

(L i

represents the

ties

x1

i-th

row below

0, ..., xn > 0,

L),

The rows

A

and

L 1, ..., L n

corresponding to the inequali­

are the initial entries in the matrix

D

which

is written as rows rather than columns for convenience in tabulation. additional rows

L n + 1, ..., L n + h

of D,

given inequalities, are used successively in the computation. in

L n + h + 1

Column

o

The

corresponding to the other The entries

are the coefficients of the function to be minimized.

of the table contains the constants of the linear inequalities

and, below

P,

through

contain the coefficients of the linear inequalities and of the

6

the homogenizing coordinates of the vertices.

Columns

1

function to be minimized and the coordinates of the vertices. The rows P 1, constitute the matrix computation rows

..., P n + 1 ( P i means the

B

i-th

row under

at the beginning of the computation.

P n + 2, ...

are incorporated in

B,

P)

During the

while some P-rows

may cease to belong to

B- The scalar products in the matrix

the computations appear

in columns

n + 1, ..., n+ h + 1 .

M

used in

The arrows

indicate the correspondence between the L-rows and these latter columns. Columns

-3, -2, and

-1

are used to record labels and side calculations.

The final polyhedron is described by B namely, it has vertices P 12, P 20; sides

L 0

and by to

L 10

D

P1

to

P 35

except

in its final form,

P 1, P 8, P 10, P 11,

in its final form, namely, by the five-dimensional except

L 8,

where

L o

denotes the hyperplane at

infinity. Since row

P 19

gives the smallest value in column

11,

it is

the only solution of the minimization problem. ABBREVIATIONS.

L 3

means

the third row below

L.

P 3 means the third row below P . P 3 7 means the number in row P 3 and column ORDER OFSTEPS. Perform step 1 A, then all steps 1 B, then step 2 A, then steps 2 B, step 3 A, ..., until step h B. STEP s A. Compute entries P k n+s in the n+sth far down as possible by formula 1, except that if P k 0 = 0 P k n+s = oo; and omit P k n+s if P k n+h+1 is an x. for which

7.

column as write

STEP s B. For every k for which P k n+s < 0 and for every P 1 n+s > 0 or P I n+s = oo, and for which P k and P 1

1

THE DOUBLE DESCRIPTION METHOD

-k

-3

-2

-1

0

1

2

...

n

L 1

0

1

0

...

0

2

0

0

1

...

0

n

0

0

0

...

1

...

1

71

n+1

...

n+h

n+h+1

n+1 given n+h 0

1

P 1

1

0

0

...

0

2

0

1

0

...

0

n+1

0

0

0

...

1

n+h+1

1

n+2 to be computed

n+3

TABLE II

have

n-1

common zeros perform substep n+s k 1.

stepsn+s k 1 all), to

belonging

make an

n+h+1

column

1000 |P k

x in P k n+h+1

in row P k. SUBSTEP n+s k 1-

-k.

Wrtie

k

in

o o );

tions by

1000

umns in which

write

Start a new row by its number, say column

-3, 1 in column

1000

in column

in column 0

and

P 1

-2,

-1

n+1

m,

in

and

(to be omitted if

(n.b., we multiply our computa­

to avoid decimals); and write P k

all sub­

(there may be- none at

and in all free places in columns

n+s|/(|P k n+s| + P 1 n+s)

P 1 n+s =

"When through

to the same P k n+s < 0

0

in each of the

have common zeros and in column

n-T col­

n+s. The

remaining

P m j, j = 1, ..., n,

P 1 n+s =

oo,

umn

are unnecessary for subsequent calculation and are marked by an

n+s

compute by formula 2, except that if

use formula 3-(Other entries in row

FORMULA 1.

P k

x.

1/1000[P m (-1) (P 1 j - P k j)] .

3. P m j = 1000 |L n+s 0 1 / L n+s j

in which case write

coming before col­

n+s = (P k o .L n+s 0) + (P k 1 . L n+s 1)

+ ... + (P k n . L n+s n) . FORMULA 2. P m j = P k j + FORMULA

P m

P m j = 1000 |L n+s o | .

unless

L n+s j = 0

THE DOUBLE DESCRIPTION METHOD

72

-4 -3 -2 -1 0 L 1 0 2 0 0 3 4 0 0 5 6 0 7 -3 8 -70 -1 9 10 -1 2 0 11 P

26

27

28

29 30 31 32 33 34 35

3

k

5

6

1 0 0 0 0 0

0 1 0 0 0 0 12 393

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0 6

0 0 0 0 0 1 2

283

94

25 786

1

15 203 183

1

1

26 18 655 651

1

245 1 1 1 72

10 6

1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1000 120 0 0 0 0 0 250 0 0 0 1000 0 0 115 0 0 1000 0 0 0 166 0 1000 1000 0 0 0 0 500 0 0 0 0 0 1000 1000 1000 0 0 0 0 0 0 0 1000 91 61 1000 98 0 0 0 90 1000 96 0 0 0 0 0 0 1000 0 0 1000 0 61 87 0 0 1000 0 0 94 0 91 1000 0 0 92 0 0 1000 0 1000 0 0 0 250 0 11*8 0 0 27 1000 0 0 0 153 39 1000 0 0 0 0 152 1000 1000 0 0 0 0 2000 0 53 0 0 59k 1000 1000 0 0 0 0 380 1000 51 0 0 0 288 0 0 0 1+0 381 1000 1000 35 0 60 0 91 1000 0 0 159 0 81+ 0 28 91 0 49 1000 0 0 93 0. 34 1000 0 0 80 2k 83 1000 TABLE III

7

8

9

10

11

-3 00 00

X 00 00 00 00 00 00

X 00 00 00 00 00 00

X 00

X 00 00 00 00 00 00 X

3 1

27

0 0 0 0 0 0 1 0 0 0 0 0 1500

0 0 0

300

0 0 0

303

0 0 0

131

0 0

360

0 0 0 0 0

189

0

00

00 00 00 00 00 00 00 0 24320 -880 X 0 28250 2750 33750 250 0 5325 -885 X X 38066 0 -336 X X 0 71500 4000 -9000 X 0 71000 3500 1500 X 0 191 1000 X 0 X 0 17636 152 0 X 0 8434 188 0 X 0 15588 396 X 0 60000 1000 X 0 X 0 5427 148 0 X 0 -4686 X 0 X 0 2832 395 X 0 49250 250 X 0 X 0 29201 175 0 X 0 25719 192 0 X 0 28777 283 0 2000 X X X 0 447 0 X X 0 740 0 X X 0 339 0 X X 0 421 0 X X 0 0 186 X X 0 243 X 0 X 0 0 0 168 X 0 0 31 6 X 0 0 0 187 X 0 00

0

16

17 18 19 20 21 22 23 24 25

1 2 1 3 1 4 1 5 1 6 1 7 8 2 8 9 242 8 12 180 8 13 200 10 4 10 9 243 10 12 181 10 13 202 11 5 11 9 108 11 12 077 11 13 087 12 6 12 9 21 1 12 13 2k0 12 16 51 6 12 24 259 20 16 357 20 18 072 20 19 463 20 21 627 20 2k 15k

2

0 CCir\OVJ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

THE DOUBLE DESCRIPTION METHOD

73

BIBLIOGRAPHY J.-B., Oeuvres II (Paris, 1 8 9 0 ), p. 3 2 5 .

[1]

FOURIER, J.

[2]

MOTZKIN, T. S., Beitrage zur Theorie der linearen Ungleichungen (Dissertation, Basel, 1933) Jerusalem, 1936.

[3 ]

MOTZKIN, T. S., "The double description method of maximization,11 Notes of Seminar on Linear Programming at the Institute for Numerical Analysis, National Bureau of Standards (Los Angeles, December, 1950).

[b]

MOTZKIN, T. S., "Two consequences of the transposition theorem on linear inequalities," Econometrica 19 (1951), pp • 1 8 ^- 1 8 5 .

[5 ]

von NEUMANN, J. and MORGENSTERN, 0-, Theory of Games and Economic Behavior, Princeton 1 9 ^ , 2nd ed. 19^7-

[6]

RAIFFA, H., THOMPSON, G. L., and THRALL, R. M., "An algorithm for the determination of all solutions of a two-person zero-sum game with a finite number of strategies," Engineering Research Institute, University of Michigan, Report No. M 7 2 0 - 1 , R 28 (September, 1950).

[7 ]

RAIFFA, H., THOMPSON, G- L-, and THRALL, R. M., "Determination of all solutions of a two-person zero-sum game," Symposium on Linear In­ equalities and Programming, Department of the Air Force and National Bureau of Standards, Washington D. C. (June, 1951)-

[8]

SHAPLEY, L. S. and SNOW, R. N-, "Basic solutions of discrete games," Annals of Mathematics Study No. 2k (Princeton, 1950) pp. 2 7 -3 5 .

STIGLER, G- F-, "The cost of subsistence," Journal of Farm Economics 27 (19^5), pp. 303-3U. [10] WEYL, H., "Elementary proof of a minimax theorem due to von Neumann," Annals of Mathematics Study No. 2k (Princeton, 1950) pp. 1 9 -2 5 . [9]

T. S. Motzkin H. Raiffa G- L. Thompson R. M- Thrall

National Bureau of Standards, University of California, Los Angeles, and University of Michigan

SOLUTIONS OF CONVEX GAMES AS FIXED-POINTS1 M- Dresher and S. Karlin

jj 1 . INTRODUCTION In a game with a finite number of pure strategies the choice of a mixed strategy is equivalent to the selection of a point in a simplex. If the strategies are constrained in any way, then the choice is no longer made from a simplex but from an arbitrary convex set.

Many infinite games,

e.g., polynomial and polynomial-like, are essentially finite games over general convex sets since the choosing of a mixed strategy is equivalent to choosing a point in a finite dimensional convex set. In this paper we study games played over arbitrary convex sets. Interpreting the solutions of a game as the fixed-points in a continuous mapping, we obtain some general results on the dimensionality and continui­ ty of solutions. Dimensional relationships for games played over simplices were first derived in [1] and [3].

Some dimensional and continuity rela­

tionships for polynomial-like games were first obtained in [2].

The gen­

eral convex game treated here can be formulated as a polynomial-like game by spanning the convex set with a Peano space-filling curve.

However, the

complicated nature of such a curve makes this formulation impractical for theoretical and computational purposes. We also describe a method of com­ puting the solutions by mapping one convex set onto another.

The method is

applicable to both finite games and infinite games with polynomial or polynomial-like payoffs.

j 2.

CONVEX GAMES

We define a finite dimensional convex game as follows: Player I chooses a point

r = (r1, r2 , ..., r )

from a convex set

R

lying in

Euclidean m-space. Player II chooses a point s = (s^ s2, ..., sn ) from a convex set S in Euclidean n-space. R and S are bounded and closed. The payoff from Player II to Player I is given by a bilinear form

1The preparation of this paper was sponsored by The RAND Corporation.

75

76 (1 ) where

DRESHER AND KARLIN A(r,s) = f(r)

f j(r)sj + fo (r) = X i = l

and

g(s)

are linear functions of

If Player I has strategies, then

R

and

m S

r

and

s,

respectively.

pure strategies and Player II has are

(m - 1)-dimensional and

sional simplices, respectively, and to Player I.

S1 (s)r*1 + g0 (s) =

A(r,s)

n

pure

(n - 1)-dimen­

is the mixed strategy payoff

However, if a game has a finite number of pure strategies and

the mixed strategies are subject to some finite number of linear constraints, then the sets

R

and

S

are polyhedral convex sets.

If a game is con­

tinuous and the payoff is polynomial-like, i.e., the payoff to Player I from Player II if they choose pure strategies

x

and

y,

respectively,

is given by M(x,y) =

where

r^

and

Sj

a±J.r1 (x)Sj(y)

are continuous functions, then

R

is the convex set

spanned by the curve r± = r1 (x), traced out in

m

0 < x < 1,

dimensions, and

s j = s i y ^’ traced out in

n

S

1=1,2,

is the convex set spanned by the curve

0 < y < 1.

dimensions [2]. $3-

j =

The payoff

A(r,s)

2 , •••, n, Is given by (l).

SOLUTIONS OP CONVEX GAMES

The existence of optimal strategies

r°,s°

and a game value

v,

such that (2)

max A(r,s°) = min A(r°,s) = v r£R sGS

can be established in two fundamental ways.

We may use properties of con­

vex sets, specifically the result that two non-overlapping convex sets can be separated by a hyperplane. It also follows from the Kakutani fixed-point theorem [ k ]. The optimal strategies (one set for each player), or solutions are the generalized fixed-points in the upper semicontinuous mapping de­ scribed as follows: Let r° £ R. Define the image of r° on S to be the set of points S(r°) C S where min A(r°,s) is assumed- S(r°), which is sGS the intersection of a hyperplane with the boundary of S or coincides with

SOLUTIONS OF CONVEX GAMES S,

is

a convex set.Let

s° G S

77

and let the image of

s°

on

R be the

set of

points R(s°)C R where max A(r,s°) is assumed. If r° £ R(s°) o o ~~ o o s £ S(r ), then r ,s satisfies (2) and is therefore a solution of

and

the game. r°, r°

Further,since and

s°

clear that if S°

and

R

and

and S,

s° is an image of

respectively.

are the sets of optimal strategies of the R°

is an

image of everypoint

It is two players

ofS° and every point

is an image of every point of R ° . We can also formulate the solutions as fixed-points of the

point-set mapping R (x) S

is an image of s°

arefixed points in

R°,S°

then every point of of

r°

F

which takes a point

into the non-void set

S(r)

(r,s)

in the product space

(R(s),S(r)) = F(r,s)

in

R(x)S,

where

R(s)

are defined above. CONTINUITY OF SOLUTIONS To develop the theory of convex games, we first study the solu­

tions for continuity.

The following theorem, first proven in [2], is

repeated for completeness. THEOREM 1.

The solution of a convex game is a

lower semi-continuous function of the payoff. PROOF.

Let

G

be an open set containing the optimal strategies

R°

of player Iin the

of

A is perturbed by at most

game with payoff £,

let

A = A(r,s).

and R^denote the set of optimal strategies of player for

asufficiently small

£, R^

Then there exists a sequence sequence of payoffs rn

be in R£

rn

tends to

A^

n butnot in r

is in (£n )

as n —> oo,

Since where

G.

I.

We assert that,

For, let us assume the contrary

tending to zero with a corresponding

such that each G*

Suppose each element

A^ denote the resulting payoff

R

R& is not contained in G- Let n is compact we may suppose that

ris not in

G

but in

R.

Further,

it is readily verified that a limit point of optimal strategies for the payoffs G

A£ is an optimal strategy for the payoff A. But r n 0 and hence not in R and so we arrive at a contradiction.

is not in

By an identical argument one can show that if the strategy spaces Rn

converge to

R,

then the sets of solutions are lower semi-continuous.

A set of spaces Rr is said to converge to R if every point of R is a limit point of points of R r and there exist no other limit points of points of R .

78

DRESHER AND KARLIN $ 5 • GAMES WITH UNIQUE SOLUTIONS In this section we derive a dimensional relationship for games

with unique solutions.

It is shown that the relevant property of the

polyhedral face containing the optimal strategy is the dimension of the face rather than the number of vertices. Let

r°,s°

be

the unique solution of the game with payoff

A(r,s)over the strategy space

(R,S). Assume that

hedral convex sets. Let

be interior to a k-dimensional face

R,

and

s°

R

i

Since

o

R . Similarly, LEMMA 1•

s° r

0

tion

R0 ^ r 0 ,

strategy space

of

are poly­

S. R°

R°

of

and

1

maps onto some maximal face

The game with payoff

t

i

(R ,S )

Since

r°

maps onto

it follows that

t *

(R ,S ).

r°,s°

S°

A(r,s)

0

S ^ S .

over the

has the unique solu­

S ^ S°

s°

and

s°

maps onto

is a solution of the game over the

To show it is unique, let us assume that

is another optimal strategy for Player II. containing R ° .

S°

S

r°,s°.

PROOF. p

and

is optimal, it maps onto some maximal

reduced strategy spaces

R*

R

be interior to an 1-dimensional face

are polyhedral faces. face

r°

Let s = £s

Then

s

t

therefore maps onto a face containing

t

G S

t

maps onto some face

(1 - e ) s°be a solution close

+

s

to

s°,

and

Now for the original game over i o we defined R as the maximal face upon which s maps. Hence a 0 f s sufficiently close to s will map into some part of R con­

(R,S) point taining

r ° . Therefore

s

R°.

is another solution of the full game, which

contradicts the uniqueness assumption.

t

t

(R ,S ). Similarly,

We may now confine ourselves to the reduced polyhedral game over In this game

r

s°is interior

o

is interior to to

LEMMA 2.

S° and

If r°,s° t

polyhedral game

is the

maps

o

and

r

onto

maps onto

S

t

R .

unique solution ofthe

i

(R ,S ),

R° = R

s°

o

R ,

then and

S° = S .

o 1 o 1 PROOF. Suppose s is on the boundary of S , or S C S . 1 0 Consider a sequence S of polyhedra interior to S excluding S which -•-i i expand out to all of S as n increases. We can construct S by using i any inner point c of S and taking the set of all points on the segment

Ac + (1 - A)x

0 < * < 1 - £n

SOLUTIONS OP CONVEX GAMES ana

x

79

I

any point of S . Consider the game over the spaces

note the solutions to this game.

(R ,Sn ) . Let

the solutions

r° r

Therefore, for

is interior to

R°,

then for

large,

maps into all of S , it follows that t game over (R ,Sn ) . It also follows that terior to

sn

tends to

sufficiently large R°

as a face. o R . But

sn maps into a polyhedron containing

r°

game over

n

lie interior to a polyhedron having n

(r°,s ) is a solution of the o (r ,sn ) is a solution of the

(R ,Sl). This contradicts the hypothesis, since s * ’ 0 1 0 S . Therefore S = S , and similarly R = R . REMARK.

de­

Then from the lower semi-continuity of

the solutions it follows that every sequence of solutions s °. Now since

(XR , Yn )

is in-

Using a similar argument, we can generalize Lemma 2 to

games with non-unique solutions.

Let

R°

he the smallest polyhedral face

containing the set of solutions of Player I.

Now every optimal strategy

of Player II will map onto some polyhedral face of R; some of the optimal o * strategies will map onto R . Let R be the maximal intersection of these polyhedral faces. Then, by an argument identical to above, it can be » o 0 R = R . Similarly, if S is the smallest polyhedral i face containing the optimal strategies of Player II, and S is the inter­

readily shown that

section of all polyhedral faces into which are mapped Player I ’s optimal • o S = S . Again, we may confine ourselves to the reduced

strategies, then

game over the space

(R°,S°).

THEOREM 2.

If a polyhedral game has a unique solu­

tion, then the two optimal strategies lie in polyhedra of the same dimension. PROOF. we must have

Since

r°

maps into an 1-dimensional polyhedron in

fj(r) = 0 or some

1

least,

J = 1, 2, ..., 1,

linear relations must be satisfied.

a manifold of points in m-1.

R

S,

mapping onto

S°

These relations determine

and having dimension, at

Now the manifold and the k-dimensional polyhedron have only

r° in common, otherwise the uniqueness of the solution would be contra­ dicted . Therefore, m - 1 + k < m or k < 1

80

DRESHER AND KARLIN

Similarly we can show

l . For any polyhedral game, the set of solutions and their containing polyhedra satisfy the dimensional relationship u - k = v - L. PROOF.

We haveshown that

X

maps onto

R ° . Consider the reauced game over the spaces

S°

and

Y

maps onto

R°, S ° . The payoff now

becomes A(r,s) = The common zeros

g d ) s j + f0 (r) = 2 k " of

f^r), f0(r), ..., x\.(r)

E j U ) ^ + g0 (s). in

R°

correspond to the

optimal strategies of Player I and tho* common zeros of gu (s)

in

S°

Form the factor space of space containing R° anc taking a cross section “T polyhedron

R

becomes a

to R°, then X way we construct the polyhedron terior point and

g^s), g2(s),

correspond to the optimal strategies of Player II. Eu/ Xu ,

where

becomes

is the Euclidean

Xu is the linear extension-of X in Eu . By inEu perpendicular to the manifold X, the nowpolyhedron in

T.

Since X

becomes a unique interior point the factor space E / and by S°

Eu

a polyhedron

U

was interior

r° of T. In a similar taxing a cross section,

anaY

s° . We obtain the induced mappings of

becomes a unique in­ f1 , f'2, ..., f

gpi gP, d '-mi on T to U, which are qwellq defined. For this mapping it is clear that r , s are optimal strategies,

as Xu and Yv constitute the zeros of f. and g., respectively. They are also unique, since any other strategy r 1 of Player I must cover all of T and hence must be a common zero of theinduced mappings f1, f2 , ..., f . In terms of Eu , this implies that r 1 belongs to the coset Xu . From Theorem 2 it follows that dim T = dim U

81

SOLUTIONS OF CONVEX GAMES or u - k = v - 1.

t) 7 • UNIQUENESS OF SOLUTION AND PERTURBATION OF PAYOFF We shall demonstrate that if a finite polyhedral game has a unique solution, then for any sufficiently small change in the payoff the solution remains unique. Let the solution r°,s° a

k

and 1

dimensionalface

that

k = 1.

Let

S ° . Then

T

T

where b^j T

be

is

gamebe interior to

S ,respectively.

be the manifold of all points in

is defined by Xi=i

of apolyhedral R and

1

R

We have proven mapping into

linear relations, say

Dijri = 0 '

j = 1, 2, ..., 1

a linear function of the matrix

A.

Let the dimension of

w, then w > m - 1.

Since

T

anaR°

intersect in a

unique point

r°,

we have

m > v + k = w + 1, ana thus w = m - 1. The last relationship implies that the matrix rank.

If thepayoff is perturbed by a sufficiently

mension of

T

(bpj)

has fall

small amount, the di­

will not change -- the rank is preserved.

preserve the rank of the intersection of the manifold

We can also T

and

R°.

There­

fore we can obtain an (m - 1)-dimensional manifold which intersects R° t t in a unique point r . In a similar way we can obtain a unique point s o 1 1 in S . Since r and s are unique points of intersections, it follows t t that r ,s is the unique solution of the game with the perturbed payoff. If the game is a general convex game, not necessarily polyhedral, then the uniqueness of a solution is no longer preserved under small perturbations. For example, in the game with payoff

M(x,y) = xy - x ,

both

players have unique optimal strategies lying on the boundaries of their o respective spaces. If the payoff is perturbed to xy - x + £x, the uniqueness is destroyed.

However, if both players possess interior unique

82

DRESHER AND KARLIN

optimal strategies, then the planes terior point and the planes

^

g^vs) = o

= 0

intersect in a unique in­

also intersect in a unique interior

point, and small perturbations of the payoff preserving this intersection property will preserve the uniqueness as well. $8.

INTERIOR AND "IDENTICALLY v" SOLUTIONS

We can interpret geometrically two general types of solutions: 1.

Solutions interior to

R

and

S.

2.

Strategies which yield identically

v

to a

player independent of strategies of the other player. Let

r°

be an interior optimal strategy of Player I and

some optimal strategy of Player II. containing no interior point of

R

Then unless

s°

maps onto a set

gj_(s°) = o

for all

s°

R(s°) i.

There­

fore every optimal strategy of Player II is on the intersection of the planes tion,

gn-(s) = 0, and thereby yields identically v. The interior soluo r , of Player I need not have any special position relative to the

planes ^j(r ) = °- However, an optimal strategy r yields identically v if and only if it lies on the common portion of all the planes f*(r) = 0.

J

9 ■ SYMMETRIZING A CONVEX GAME A finite convex game can be symmetrized in a manner similar to the simplex game.

Let the payoff be represented by

(AT..) -

( Si:,

*ij-i)»r

Define p(rn ,sn ) = max (Ar,sn) - min (Ar0 ,s). rGR sGS It can be verified that if and only if (r0 ,sQ )

p(r0 ,sQ ) > 0 , for all r0 ,so and p(r0 ,sQ) = 0 is a solution of the game. Furthermore,

p(rn ,s ) = max (Ar,s ) + max (rn ,-A's) r£R U sGS U = max [(Ar,s ) + (rn ,-As)]. r£R sGS Form the product space

En+m = En ® Em

and the Product convex

SOLUTIONS OF CONVEX GAMES set

Rg£.

Consider the linear operator

83

(A,-A } lr,s}

defined over

Ln+rn*

The payoff over the product space becomes ( [A ,-A'} (r,s), ts0 >ro ^ = (Ar,sQ) + (rQ ,-A s) . This is a symmetric game in which Player I picks picks

(s,r),

game are those

each from the space (r0 ,sQ)

Every finite in (n + m)-dimensional

for which

R®S.

(r,s)

and Player II

The solutions of this symmetric

p(r0 ,sQ) = 0.

convex game can be

symmetrized to a newconvexgame

space whose payoff matrix is given by

r\ 0

° )

-A' /

and where both players choose points from the product set and

S

are both simplices, then

R@S

RgS.

is no longer a simplex.

If

R

In order

to play on a simplex set it is necessary to add an extra dimension. $10.

COMPUTATION

We can compute the set of solutions nation of the spaces

R

The method consists of vex subsets S

R

boundary of

S,

= °>

and similarly dividing the

R.

S

and each of the subsets

The subsets, R.

by means

R^,

S 1, S2, ...,

onto

S . maps onto some maximal

will overlap, but their union is the

From the previous discussion, it follows that

are the fixed-points in the mappings of for some

S 1, S2, ..., Sp

g^(s) = 0, i = 1, 2, ..., m, and the boundaries of R. is such that each of its subsets R^ maps onto some

full strategy space,

S° = S.

into a finite number of con­

into a finite numbe;? of convex subsets

maximal boundary of

and

for fixed-points in a continuous mapping.

dividing the space

and the boundaries of

of the hyperplanes The division of R

R°,S°

S

by an orderly exami-

R 1, R 2 , .. ., R We now divide the spaces

R

and

S

into the following convex

sets : R,(P,

> 0. P 2> 0),

R2 (p1> 0, p 2
0 ,q 2 < 0 ),

S^q, < 0 , q 2

s t (q i

= q2 = ° ) •

>

0),

0 ),

into

S

the later mappings. Some of the

and each

SJ

R1

*(S,)-- »(R3),

R^

^

where

^

(S.)

Therefore

(

R

into

R,

R

= R,_

P 2 = 0).

they will not

sets overlap.Mapping eachR^

R 2 ---X S , ) -- >(R3),

2 )>

and

Kg(P1 =

we have R 3 -- K S 2) -- v(R2)

> R^.

represents a point in 0

0, p 2 > 0)

S 3 (q, = 0 , q2 < 0 )

Some of these sets may be void, in which case appear in

Rjtp,
aj ,

in case

There results

lim + U(a) = a — >a

f a A(o,y)fa (y)dy

1

But

f a, A ^0,y)fa 1^y)dy i J^a, A(al,y)fai(y)dy »

M a1) = 1,

GAMES OF TIMING

119

and this second integral is zero since it is the negative of the corre­ sponding quantity W in Case I, which was proved to be zero in Case I. This proves the second limiting relation. To obtain the third relation set a = 0 in (6.10 and (6 .5 ). The two integrals on the right-hand sides are then identical, and their elimina­ tion gives U( 0 ) = — 1 +

I0

T7fy>— 0.

solution of the game with

But we have already established in a

discussion near the beginning of $ 3, that

a

cannot be

0

in the case

A( 0 ,0) = 0 . There would therefore be no optimal strategy and if we suppose that in this case an optimal strategy must exist, the resulting contra0

.diction would establish the lemma. We now give another proof of the lemma independent of the general theory of games. A(a) < 1

Again suppose that the lemma were false, so that

for all

a in

game with the same kernel £ < x < 1, £ < y < since

A(£,fc) > 0.

l.

0

< a 0.

Select a fixed

0. We have, from (6 .7 ) for sufficiently small

j

£

A(o,y) < 0 b > 0

for so tha

8,

A(g,y)Yg (y)dy + J ^ A(£,y)Y£ (y)ay = 0

or (A(g,y) - A(£,afc))Y£ (y)dy +

= -A(£,a£) (l ~ \

The left-hand side is

( 0,

-

j h

(A(g,y) - A(£,b) )V£ (y)dy

- A(£,b)

Y£(y)dy .

while the right-hand sicie approaches

-A(0,b) lim f} Y (y)dy which is > 0 unless the limit expression is 1 C_^ Q C 8 zero. Therefore lim /, Vc(y)dy = o. £ ^0 ^ From (6.8), we now have

c*^A(£,1 ) + J

+j l

^

(A(x,1 ) - A(b,1 ) )'Y^(x)dx + A(b,1 ) ^ 1 -

y^(x)dx^

(A(x, 1) - A(1,1))Vt(x)dx + A( 1 ,1 )J ’ Y g x ) d x = 0 .

Taxe the limit as £ — > 0. All the integrals above except for the last are < o, with g f(x ) d x — > o, and A(0,1) < o, A(b,D < o. if

SHIFFMAN

— y dQ ,

y

satisfying the following conditions: (a)

The functions

and have continuous triangles

x y

The value

respectively.

]§(1 )

lies between

K( 1 ,1 )

and

M( 1,1 )

and

5(0) lies between K(o,o) and M(o,o) while the value assigned to for 0 0

and

Mx (x,y) > 0

for

x< 1

Ky(x,y) < 0

and

M^(x,y) < 0

for

y< 1

(in their respective domains of definition). In particular,

K

and

tion, are strictly increasing in

M, x

in their respective regions of defini­ and strictly decreasing in

A solution to the game defined by the payoff kernel pairof distribution functions F(x) and G(y) whereo < x < 1 0 v

for all

y and

L(x,y)aG(y) < v

for all

x.

y. L(x,y)

is a

and

If F(x) isa strategy of the form ((x)dx *£(0) ,

for

y > b.

For

y = 0, dK(o,o)

must be replaced by

and for y = 1 , /5M( 1 ,1 ) by ySj( 1 ) . The procedure we follow will be to try to find a solution of the

following form: an absolutely continuous distribution with continuous derivative on the interior of the unit interval with additional possible jumps at the ends of the interval.

Such solutions will be exhibited and

afterwards their uniqueness will be established. only as a guide to the subsequent theory.

The following lemma serves

It will not be explicitly used

hereafter. LEMMA 1.1.

If both players possess optimal strat­

egies of the form F = G = (tfl0 , Y cd(y) , cfl-j ), is given by

$ab = $al

^ab^x ^ ^ 1 ^ and then the form of the density and

ycd = 'V^1

i.e., the spec­

trum of the absolutely continuous part of both distri­ butions begin at a common value

0 < a < 1

and extend

to the upper end of the unit interval. PROOF.

For any distribution of the type indicated above, we see

that

/q L(x,y)dF(x)

is continuous and non-increasing for

$(x)

is zero in any interval, then

in that portion by condition (c). strategy and yield

c

is in the spectrum of

/ L(x,y)dF(x) = v

includes the interval establish that

/ L(x,y)dF(x)

Since

for

F(x) G,

c < y < 1,

0 < y < i.

If

is strictly decreasing

represents an optimal

it therefore follows that the and hence that the spectrum of

[c ,1] . A similar argument applies to

G

and we

c = a. THE MAIN THEOREM

The main theorem of Part I deals with a complete description of the optimal strategies for the payoff kernel

(1.1).

Unfortunately, many

different type of solutions may occur depending upon the nature of M(x,x)

and the values of

L(x,y)

K(x,x),

at the extreme points of the unit square.

It is therefore necessary to present a mutually exclusive classification of the various possibilities. The analysis of the kernel L(x,y) of (1.1) subdivides into three main parts. In the case where there exists an x o such that K ( x q ,x o ) = M(x0 ,xQ), then the type of optimal strategies that

KARLIN appear occur under the headings of B, C and D of Theorem 1.

More precisely,

the essential feature is to study the spectral radius

and

A(a)

the two integral equations (1.2) and (1. l) given below.

a

(x

o both

,x

for some

) = M(x ,x ) O O O A(a) > 1 and

a

in the unit interval. for some

jul( a) > 1

ana

assumption of

A(o) M(x,x)

ana

K(0,l) > M(1,0)

respectively.

plays a very fundamental role.

The statement of the theorem to be established in Part I is summarized in the following array: THEOREM 1.1. off kernel

(1 .1 )

The optimal strategies for the pay­ are unique and are enumerated in the

following table:

Optimal P

Kernel A

K(l,l) < M(1,1) K(x,x) > M(x,x), a < x < 1

Optimal G

t

B A(a) = 1 A a) < 1 K(x,x) > M(x,x), a < x < 1

n a>

C A(a) = 1 /x(a) = 1 K(x,x) > M(x,x), a < X < 1 D

M&) < 1 >i(a) = 1 If

K ( x q ,x o )

for some

= M(xqJ'xo}

B, C, or D

o < xQ M(x,x), 0 < X < 1 A(o) < i m ( o ) < 1 K(0,1 ) < M(1,0) E

§(0 ) = K(o,o)

or

xr with 0 1

Furthermore, if

second and third parts of Theorem 1 suppose that t3th 0 < x < 1

of

The type of solu­

tions as described in B, C, arm D always appear when either A(a) > 1

/t(a)

(*I ) v o', $ yo/

B, C, or D

In

for

REDUCTION TO INTEGRAL EQUATIONS

Optimal G

Optimal F

Kernel

( n 0 , g , rfi,)

F

SQ < 5(0) < K(o,o)

G

i(o) = S0

Klo. g )

H

M(o,o) < 5(o) < SQ

(*I0 , g , W

(yio> r &)

I

M(o,o) = 5(o)

( g , ,81,)

(*i0 ,

Io

h

K(x,x) > M(x,x)j 0 < x < 1

A(O) < 1 / * ( o ) < 1 K(o,1) > M(1,o) j

K(o,1) > 5(0) > M(i ,o )

K

K(o,o) > 5(0) > K(0,1)

L

K(o,o) = 5(0)

ug,

M

M(1,o) > 5(o) > M(o,o)

(*I0 , g > /SI,)

N

5(o) = M(o,o)

(g, ,91,)

Ug,

/I,)

(Y0 ,

H o

i I

Fur thermor e , the densities tion to certain integral equations.

$

and

ug,

Y

g)

are obtained as the solu-

These solutions are either Neumann

series or eigenfunctions of integral operators. The proof lemmas.

The aim of

of this theorem shall be divided into a series of the first series of lemmas, Lemma 1.2 - 1.16 is purely

to show the existence of the solutions indicated in the shorthand of Theorem 1.1, in the various cases

A

through

N.

The uniqueness question

is settled by Lemmas 1.17 - 1.20. LEMMA 1.2. (1 ,1 ) PROOF. and (c).

If

K( 1 ,1 ) < M(1,1),

is a saddle point of the kernel

then the point L(x,y).

The statement is an immediate consequence of conditions (b)

130

KARLIN THE INTEGRAL EQUATIONS On account of Lemma 1.2 we can suppose in all that follows that

K( 1 ,1 ) > M( 1,1 ) . The continuity of val

a < x < 1

for which

K

and

M

provides a non-empty inter­

K(x,x) > M(x,x). We introduce now the following

integral equations: Let

(1 - 2 )

f( V

~

Ja T(x>f)f (x)dx = f - xQ) .

ana integration over

(a, 1 - £)

° “

«t ^ A(ai / r *

becomes unbounded as

a — > 1,

- [K(xo ,X(J

?),

*• j v * a as

K

for which

a

tends to

by Property II we have an LEMMA 1.1.

- *•> x ,

and since

for which

a

A(a) — > 0

A(a) = 1 .

Under tne assimption of Lemma 1 .3 there

exist optimal strategies for both players of the form: an absolutely continuous distribution over an interval [a,l]

with a possible jump at

1

for one of the

players. PROOF. which

either

A(a) a (x)dx +

f lS 3(b* - 1

a,

(a > xQ)

Since

for which

M(x,y)a (x)dx

is a constant depending only on

a

yu.(a) = 1 ,

/^(a) = 1 . We begin with the first case.

we obtain a $Q (x) > 0 This implies that (1.5)

and

for

for

or A(a) = 1,

cl cl

cl

T_c|)_ = $ .

a < y < l

since differentiation of

equation (1-5) yields precisely the relation Ta = 0. Let /a Y ?(y)dy = c. Define J 1+c so that f J Y S ( y)dy = i 0 . An analogous remark applies to (I U& ) . If we set (I - Ta )_ 1p ± = 0,

+ a (x))

is now straightforward and similar to the last

part of the argument in Lemma 1.10. LEMMA 1 .1 6 . Statement M of Theorem 1.1. PROOF. Similar to Lemma 10. Uniqueness. The optimal strategies of the specified types have now been exhibited and there remains only the question of uniqueness. In Cases A and J it is easily verified via condition (c) that the solutions are unique. From the remaining cases we shall select Case (F) as typical, and show uniqueness in this case.

Any other case can be handled similarly

REDUCTION TO INTEGRAL EQUATIONS LEMMA 1.17.

If

F

represents any optimal strate­

gy for player I, then the spectrum of

F

on the in­

terior of the unit interval is continuous. PROOF.

Since we already have an optimal strategy

G = (jfl^Y^cfl,) for player II for which / L(x,y)dG(y) < v for o a i 0 < x < a, the spectrum of F must be confined to the interval [a,l] and

the end point

/ L(x,yQ)dF = v

0.

Consider any

while

yQ

with

/ L(x,y)dF(x) = v

a < yQ < 1 ;

then

for a neighborhood of' yQ . By

bounded convergence, we get lim y-*y0+ Similarly

f L(x,y)dF =

J

/ L(x,y~)dF(x) = v.

[K(y0 ,y0 ) - M(y0 ,yo )] cry

o' ^o has

F = 0.

[ M(y0 ,yQ),

we have

can be handled similarly. [(J)(a)

- M(a,a) ] a*olF < 0. —

we deduce from these two relations that

In fact, one Since