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# Contributions to the Theory of Games (AM-24), Volume I

H. F. BOHNENBLUST
G. W. BROWN
M. DRESHER
D. GALE
S. KARLIN
H. W. KUHN
J. C. C. MCKINSEY
J. F. NASH
J. VON NEUMANN
L. S. SHAPLEY
S. SHERMAN
R. N. SNOW
A. W. TUCKER
H. WEYL
Pages: 404
Stable URL: http://www.jstor.org/stable/j.ctt1b9rzq1

1. Front Matter (pp. i-iv)
2. PREFACE (pp. v-xiv)
H. W. Kuhn and A. W. Tucker
4. Part I. Finite Games
• 1. THE ELEMENTARY THEORY OF CONVEX POLYHEDRA (pp. 3-18)
H. Weyl

If S is a closed bounded point set in (n - 1)-dimensional affine space with the coordinates x1, x2, ..., xn-1, then the points of the convex closure of S can be characterized in two ways: (1) they are the centers of gravity of points from S; (2) they belong to all of the “supports” of S. A support of S is a halfspace

$\mathrm{\alpha _{1}x_{1}+\cdots +\alpha _{n-1}\; x_{n-1}+\alpha \geq 0}$,

in which lie all of the points of S. The fundamental theorem concerning convex closures asserts that the two definitions are equivalent. Moreover, (1) can be sharpened so that only centers of gravity of...

• 2. ELEMENTARY PROOF OF A MINIMAX THEOREM DUE TO VON NEUMANN (pp. 19-26)
Hermann Weyl

J. von Neumann’s minimax problem in the theory of games belongs to the theory of linear inequalities and can be approached in the same elementary way in which I proved the fundamental facts about convex pyramids. As elementary are considered such operations in an ordered field K of numbers as require nothing but addition, subtraction, multiplication and division, and the decision whether a given number is > 0 or = 0 or < 0. Decisions about a set of numbers are elementary only if they concern a finite set, the members of which are exhibited one by one. In such a sequence...

• 3. BASIC SOLUTIONS OF DISCRETE GAMES (pp. 27-36)
L. S. Shapley and R. N. Snow

All solutions of the general two-person zero-sum game can be represented by means of a finite number of “basic solutions,” which may be visualized as pairs of vertices from two convex sets in the spaces of all mixed strategies for the two players. Each basic solution has associated with it one or more sub-matrices of the whole game-matrix, called the kernels of the solution. Once a basic kernel has been located in the game-matrix, the associated basic solution may be computed by a formula.

The game Γ is represented by the m x n matrix$\mathrm{M=||a_{j}^{i}||}$where$\mathrm{a_{j}^{i}}$is the...

• 4. SOLUTIONS OF FINITE TWO-PERSON GAMES (pp. 37-50)
D. Gale and S. Sherman

The fundamental theorem of the finite two-person game asserts the existence of a value and optimal mixed strategies for both players in any game Γ. The optimal strategies need not be unique. It is easy to show, however, that the optimal strategies for an m by n game form in a natural way polyhedral subsets of an (m - 1)- and (n - 1)-dimensional simplex respectively, but that not all such pairs of polyhedra can be obtained as solutions of a game. The purpose of this paper is two-fold.

1. To give a simple characterization of the sets of optimal strategies...

• 5. SOLUTIONS OF DISCRETE, TWO-PERSON GAMES (pp. 51-72)
H. F. Bohnenblust, S. Karlin and L. S. Shapley

In this paper we propose to investigate the structure of solutions of discrete, zero-sum, two-person games. For a finite game-matrix it is well known that a solution (i.e., a pair of frequency distributions describing the optimal mixed strategies of the two players) always exists (see [2]2, Chapter III, Section 17). Moreover, the set of solutions is known to be a convex polyhedron, each of whose vertices corresponds to a submatrix with special properties [3].

In Part I of the present paper we prove a fundamental relationship between the dimensions of the sets of optimal strategies, and devote particular attention to...

• 6. SOLUTIONS OF GAMES BY DIFFERENTIAL EQUATIONS (pp. 73-80)
G. W. Brown and J. von Neumann

The purpose of this note is to give a new proof for the existence of a “value” and of “good strategies” for a zero-sum two-person game. This proof seems to have some interest because of two distinguishing traits:

(a) Although the theorem to be proved is of an algebraical nature, a very simple proof is obtained by analytical means.

(b) The proof is “constructive” in a sense that lends itself to utilization when actually computing the solutions of specific games. The procedure could be “mechanized” with relative ease, both for “digital” and for “analogy” methods. In the latter case it...

• 7. ON SYMMETRIC GAMES (pp. 81-88)
D. Gale, H. W. Kuhn and A. W. Tucker

A symmetric game is a game with a skew-symmetric payoff matrix; informally, the two players play the same role in a symmetric game and have the same set of available pure strategies. Possessed of natural interest because of their special character, symmetric games are given additional importance by the computational procedures which are discussed by G. W. Brown and J. von Neumann in their contribution to this Study. In this short note we will investigate two methods for symmetrizing an arbitrary game; by this, we mean constructing a symmetric game whose solution yields a solution to the original game by...

• 8. REDUCTIONS OF GAME MATRICES (pp. 89-96)
D. Gale, H. W. Kuhn and A. W. Tucker

In attempting to solve games with a large number of pure strategies it is natural to group together strategies that are similar or are subject to some intrinsic connection such as symmetry. In many cases, one feels that the optimal probabilities within such a grouping of pure strategies can be fixed without regard to the game as a whole. Computationally, this is certainly desirable, since it replaces a set of strategies by a single new strategy and thus reduces the size of the game. In this note we shall investigate the possibility of such reductions; related results have been obtained...

• 9. A SIMPLIFIED TWO-PERSON POKER (pp. 97-104)
H. W. Kuhn

A fascinating problem for the game theoretician is posed by the common card game, Poker. While generally regarded as partaking of psychological aspects (such as bluffing) which supposedly render it inaccessible to mathematical treatment, it is evident that Poker falls within the general theory of games as elaborated by von Neumann and Morgenstern [1]. Relevant probability problems have been considered by Borel and Ville [2] and several variants are examined by von Neumann [1] and by Bellman and Blackwell [3].

As actually played, Poker is far too complex a game to permit a complete analysis at present; however, this complexity...

• 10. A SIMPLE THREE-PERSON POKER GAME (pp. 105-116)
J. F. Nash and L. S. Shapley

In the study of games Poker, in its varied forms, has become a popular source of models for mathematical analysis. Various simple Pokers have been investigated by von Neumann,³ Bellman and Blackwell,⁴ and Kuhn.⁵,⁶ Our paper is the first to consider a three-person model. This version has just two kinds of hands, no drawing or raising, and only one size of bet. We suppose that the game is non-cooperative and solve for “equilibrium points.” The game turns out to have a well-defined value if the ante does not exceed the amount of the bet, or is more than four times...

• 11. ISOMORPHISM OF GAMES, AND STRATEGIC EQUIVALENCE (pp. 117-130)
J. C. C. McKinsey

We shall be concerned here with the notion of strategic equivalence in the theory of games.² The intuitive notion of strategic equivalence is rather vague; but it nevertheless happens to be sufficiently sharp, that one can specify a precise mathematical condition A, which is intuitively recognized to be necessary for strategic equivalence, and a precise mathematical condition B, which is intuitively recognized to be sufficient. It turns out, however, that B is a consequence of A: so, in actuality, we can say that A (or B) is a necessary and sufficient condition for strategic equivalence. This paper will be devoted...

5. Part II. Infinite Games
• 12. OPERATOR TREATMENT OF MINMAX PRINCIPLE (pp. 133-154)
Samuel Karlin

In recent years due to the stimulus provided by the theory of games developed by J. von Neumann² much interest has been given to the question of when

$\mathrm{(*)\; \; \underset{g}{min}\; \underset{f}{max}\int_{0}^{1}\int_{0}^{1}K(x,y)df(x)dg(y)=\underset{f}{max}\; \underset{g}{min}\int_{0}^{1}\int_{0}^{1}K(x,y)df(x)dg(y)}$,

where f and g denote distributions over the unit interval. In the case where K(x, y) is a simple finite valued function, the integral reduces to a matrix. This case has been investigated by von Neumann. Contributions to the continuous case have been given by J. Ville [2, 3] and A. Wald [4]. Furthermore, J. Ville has shown that relation (*) does not hold for every function K(x, y) defined...

• 13. ON A THEOREM OF VILLE (pp. 155-160)
H. F. Bohnenblust and S. Karlin

Among the several procedures which lead to the existence of the value of a discrete two person, zero sum game — one is based on a theorem of Ville [1] and another is based on a fix point theorem of Kakutani [2]. In the present paper these two theorems are extended under certain conditions to infinite dimensional spaces. The results are useful tools in the theory of non-discrete games.

The theorem of Ville deals with matrices aik, i = 1, ..., m; k = 1, ..., n. It states that if to each qk≥ 0, ∑ qk= 1...

• 14. POLYNOMIAL GAMES (pp. 161-180)
M. Dresher, S. Karlin and L. S. Shapley

A basis is laid in this paper for a theory of two-person zero-sum games in which the payoff is a polynomial function P(x, y) of the two strategy variables x and y, the latter taking their values from closed, one-dimensional intervals. A somewhat more general category of "polynomiallike" games is examined first: games whose payoff has the form

$\mathrm{K(x,y)=\sum_{i=1}^{m}\; \sum_{j=1}^{n}a_{ij}\; r_{i}(x)s_{j}(y)}$,

riand sj. being any continuous functions. A general discussion of games with continua of strategies appears elsewhere in this volume [2].

Polynomial games are important as a bridge, leading from the discrete games, whose theory has been well explored,...

• 15. GAMES WITH CONTINUOUS, CONVEX PAY-OFF (pp. 181-192)
H. F. Bohnenblust, S. Karlin and L. S. Shapley

In the “normal form” of a two-person, zero-sum game, as the theory has been set forth by von Neumann [3], there are just two moves. They are the choices of strategy, made simultaneously by each player. One player is then required to pay to the other an amount (positive or negative) determined by the pay-off function, which is a function only of the strategy-choices. The theory is best known at present for games in which the number of strategies available to each player is finite. This article will explore a rather special class of games in which the strategies of...

6. BIBLIOGRAPHY (pp. 193-201)
7. Back Matter (pp. 202-202)