Opponent Games

Opponent Games 

(mathematics), a concept in game theory. Opponent games are games in which two players (usually denoted as I and II) with opposing interests participate. It is characteristic in opponent games for the victory of one player to be equivalent to the loss of the other and vice versa; therefore, concerted actions, negotiations, and agreements between the two players are meaningless. Most gambling and athletic games with two participants (teams) can be considered as opponent games. Making decisions under unspecified conditions—for instance, making statistical determinations—can also be interpreted as an opponent game. Opponent games are characterized by a multiplicity of player strategies and by the victory of player I in each situation that consists in the choice by the players of their respective strategies. In this way, opponent games are formally functions of a set of three variables: ⟨A, B, H⟩ in which A and B are the multiplicity of player strategies and H(a, b) is the material function (winning function) for the pair (a, b) where a ∊ A, b ∊ B. Player I, choosing a, tries to maximize H (a, b); and player II, choosing b, tries to minimize H (a, b). Opponent games with a finite number of player strategies are called matrix games.

The basis of expedient behavior for players in opponent games is considered the principle of “minimax,” according to which I is assured of victory

max	min H (a, b)
a	b

just as II may deny I more than

max	min H (a, b)
b	a

If these “minimaxes” are equal, then their common value is called the value of the game; strategies in which the external extremes are reached are called optimal player strategies. If the “minimaxes” are different, then the players must use mixed strategies—that is, choose their original (“pure”) strategies randomly with specific probabilities. In this case the value of the winning function is a random quantity, but its mathematical expectation is taken as the victory of player I (and thus, as player II’s loss). In games played against nature, nature’s optimal mixed strategy can be taken as the least favorable a priori distribution of the probabilities of its states. In opponent games the players, using their optimal strategies, expect to achieve completely predictable victories (for example, on an average basis when the game is repeated many times). This is the basis for the recurrent approach to dynamic games in those cases where they are reduced to the consistency of opponent games, the solution of which can be found directly (for example, if these opponent games are matric). Opponent games constitute a class of games in which the main bases of the players’ conduct are sufficiently clear. Therefore, any analysis of more general games based on opponent games is useful in the area of theory. An example of such an analysis is provided by classical cooperative game theory, which studies general coali-tionless games by using opponent game systems with each of the coalitions of players against the coalition composed of all the remaining players.

REFERENCE

Beskonechnye antagonisticheskie igry. Edited by N. N. Vorob’ev. Moscow, 1963.

N. N. VOROB’EV