Zero-Sum Games

Abstract

Zero-sum games are to the theory of games what the twelve-bar blues is to jazz: a polar case, and a historical point of departure. A game is a situation in which (i) each of a number of agents (players) has a set of alternative courses of action (strategies) at his disposal; (ii) there are outcomes which depend on the combination of the players’ actions and give rise to preferences by the players over these combinations; (iii) the players know, and know that each other knows, these preferences. (Strictly, such a situation is a game of complete information in normal form: these qualifications should henceforth be understood.) In the case which dominates the literature of zero-sum games there are two players, A and B say, each with a finite set of strategies, and their preferences can be represented by von Neumann–Morgenstern utilities. The preference structure can then be displayed in a payoff matrix, whose (i, j)th entry (u_ij, υ_ij) gives the expected utilities or payoffs of A and B respectively for A using his ith strategy and B using his jth. A game of this type in which u_ij + υ_ij = 0 for all i, j is known as a zero-sum matrix game (henceforth simply zero-sum game). In a zero-sum game the players have exactly opposed preferences over strategy-pairs. Hence there is no scope for the pair of them to act as a pair – there is nothing for them to cooperate about. The theory of cooperative zero-sum games is thus an empty box; zerosum games are non-cooperative games, and each player must choose in uncertainty of the other’s choice.