# Zero-Sum Games

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1658

## 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 *i*th strategy and B using his *j*th. 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.

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