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Contributions to the theory of games. (English) Zbl 0041.25302

Annals of Mathematics Studies. 24. Princeton, NJ: Princeton University Press. 201 p. (1950).
The editors wrote a preface which describes the importance of the papers for the development of the theory, and contains notes on problems which are still unsolved.
Part I: “Finite Games” consists of 11 papers, the first 9 of which concern games played by two players which are “zero-sum”, i. e. any payment made by one player is received by the other. Nos. 10 and 11 deal with \(n\)-person games (n > 2).
In the four papers of Part II: “Infinite Games” again zero-sum two-person games are studied, but the set of pure strategies (see below) is not finite. The volume ends with a bibliography.
It is convenient to mention here some of the fundamental concepts of zero-sum two-person games which occur in the papers reviewed below. From an algebraic point of view the theory of these games considers the “pay-off matrix” \((a_{ij})\) \((i = 1, \ldots, m;\ j = 1, \ldots, n)\) and the function \(\Phi(x, y) = \sum \sum x_i a_{ij} y_j\), where all \(x_i\) and all \(y_j\) are non-negative and \(\sum x_i = \sum y_j = 1\). Such vectors \(x\) and \(y\) are called “mixed strategies” of players I and II, respectively; they are composed of “pure strategies” \((1, 0, \ldots, 0), \ldots (0, 0, \ldots, 1)\). Any pair of strategies \(x\) and \(y\) is a solution if \(\sum_i x_ia_{ij}\ge v\) for all \(j\) and \(\sum_j a_{ij} y_j\le v\) for all \(i\). \(v\) is called
Contents: Preface;
Part I. Finite Games
1. H. Weyl, The elementary theory of convex polyhedra (3-18) (translation by H. W. Kuhn of [Comment. Math. Helv. 7, 290–306 (1935; Zbl 0011.41104; JFM 61.1382.01);
2. H. Weyl, Elementary proof of a mlnimax theorem due to von Neumann (19–26);
3. L. S. Shapley and R. N. Snow, Basic solutions of discrete games (27–36);
4. D. Gale and S. Sherman, Solutions of finite two-person games (37–50);
5. H. F. Bohnenblust, S. Karlin and L. S. Shapley; Solutions of discrete two-person games (51–72);
6. G. W. Brown and J. von Neumann, Solutions of games by differential equations (73–80);
7. D. Gale, H. W. Kuhn and A. W. Tucker, On symmetric games (81–88);
8. D. Gale, H. W. Kuhn and A. W. Tucker, Reductions of game matrices (89–96);
9. H. W. Kuhn, A simplified two-person poker (97–104);
10. J. F. Nash and L. S. Shapley, A simple three-person poker game (105–116);
11. J. C. C. McKinsey, Isomorphism of games and strategic equivalence (117–131).
Part II. Infinite Games
12. S. Karlin, Operator treatment of minmax principle (133–154);
13. H. F. Bohnenblust and S. Karlin, On a theorem of Ville (155–160);
14. M. Dresher, S. Karlin and L. S. Shapley, Polynomial games (161–180);
15. H. F. Bohnenblust, S. Karlin and L. S. Shapley, Games with continuous, convex pay-off (181–192).
Bibliography.
The articles will be reviewed individually, see “Contrib. Theory of Games, Ann. Math. Stud. 24”.

MSC:

91-06 Proceedings, conferences, collections, etc. pertaining to game theory, economics, and finance
00B15 Collections of articles of miscellaneous specific interest
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