Foundations of Game Theory

The English edition differs only slightly from the Russian original. The main struc­ tural difference is that all the material on the theory of finite noncooperative games has been collected in Chapter 2, with renumbering of the material of the remain­ ing chapters. New sections have been added in this chapter: devoted to general questions of equilibrium theory in nondegenerate games, subsections 3.9-3.17, by N.N. Vorob'ev, Jr.; and
4, by A.G. Chernyakov; and
5, by N.N. Vorob'ev, Jr., on the computational complexity of the process of finding equilibrium points in finite games. It should also be mentioned that subsections 3.12-3.14 in Chapter 1 were written by E.B. Yanovskaya especially for the Russian edition. The author regrets that the present edition does not reflect the important game-theoretical achievements presented in the splendid monographs by E. van Damme (on the refinement of equilibrium principles for finite games), as well as those by J.e. Harsanyi and R. Selten, and by W. Giith and B. Kalkofen (on equilibrium selection). When the Russian edition was being written, these direc­ tions in game theory had not yet attained their final form, which appeared only in quite recent monographs; the present author has had to resist the temptation of attempting to produce an elementary exposition of the new theories for the English edition; readers of this edition will find only brief mention of the new material.

Historical notes and references for the Foreword and Introduction.- 1 Noncooperative games.- 1 Noncooperative games and their components.- 2 Optimality principles in noncooperative games.- 3 Realizability of A-optimality principles.- 4 Realizability of A-principles in metastrategies.- 5 Realizability of equilibrium situations in mixed strategies.- 6 Natural topology in games.- Notes and references for Chapter 1.- 2 Finite noncooperative games.- 1 Finite noncooperative games.- 2 Dyadic games.- 3 Solution of general finite noncooperative games.- 4 On the structure of the set of equilibria in finite noncooperative games.- 5 The complexity of solution of finite noncooperative games.- 6 Reduction to three-person games.- Notes and references for Chapter 3.- 3 Two-person zero-sum games.- 1 Optimality in two-person zero-sum games.- 2 Basis of the maximin principle.- 3 Minimax theorems.- 4 Finitely additive strategies.- 5 Analytic games on the unit square.- 6 Separable games.- 7 Convex games.- 8 Games with a simple payoff function.- 9 Games of timing.- Notes and references for Chapter 3.- 4 Matrix games.- 1 Basic concepts and propositions.- 2 Solution of matrix games of small format.- 3 Matrix games and linear programming.- 4 Description of all equilibrium situations in matrix and bimatrix games.- 5 Solution of matrix games with matrix payoffs of a special form.- 6 Approximate methods for solving matrix games.- 7 Structure of the set of a matrix game.- Notes and references for Chapter 4.- References.- Collections.- List of Joint Authors.- Index of Notations.