The present state of the theory of games

Abstract

A great amount of information about the theory of games of very varied mathematical content has been accumulated in recent years. The account by Karlin in his monograph [64] of only some of the most highly developed branches of the theory of antagonistic games took up about 500 pages. A common approach to the theory of games as a whole has not yet been worked out. This article attempts to survey systematically the basic branches and directions of the theory of games in its present state. The general definition of a game as a formalized representation of a conflict is taken as a basis. All the “forms” of games considered earlier can be obtained from this definition as particular cases. A systematic look at the theory of games, as in the case of normative theory, enables us to place many of the results of the theory of games in fairly natural groupings. Without being able to give either an exhaustive or even a fairly full description of these results the author has restricted himself to an account of the most typical of them. The amount of detail is not uniform and is inversely proportional to the accessibility of the original material. Facts that can be found in Russian publications are merely noted or just mentioned. In particular, questions considered in the author's survey article [32] are only very briefly touched upon. Specific assertions given in this paper are mainly illustrative in character and some could be replaced by others without detriment. We consider practically all the significant branches of the theory of games with the exception of differential games. Although the theory of differential games has a well-defined place in a number of directions of the theory of games, its methods and problems are becoming increasingly independent. Detailed surveys of the theory of differential games are given in [59], [97]. The author does not touch upon the history of the theory of games and refers the interested reader to the addendum to the Russian edition of von Neumann and Morgenstern's basic monograph “The Theory of Games and Economic Behaviour” [82]. The present article is similar in concept to a paper of the same name given by the author at the First All-Union Conference on the Theory of Games at Erevan in November 1968 (see [123] and [37]), but there are essential differences in the selection of material and the presentation.