GAMELPP: A PASCAL program for solving two-person zero-sum games on a microcomputer

Abstract

The solution of a two-person zero-sum game provides optimal strategies for both players. In game-theoretic terms, the solution specifies a minimax strategy for one player and a maximin strategy for the other. These strategies can be pure strategies, that is, single strategies selected from the set ofstrategies available to each player, or they may be mixed strategies that are weighted com­ posites of the single strategies. The weights in these com­ posites consist of probabilities for playing the various strategies. The probabilities in a pair of mixed strategies are selected so that the expected value of the game is optimal for each player. The solution of a two-person zero-sum game, therefore, rests on fmding a vector of probabilities x that would allow one player to maximize the game value and another vector of probabilities y enabling the other player to minimize the game value. Solutions consisting of pure strategies can be readily obtained by examination of the game matrix, and the mixed strategies in a 2 by 2 game matrix can be deter­ mined by applying straightforward computational formulas (e.g., Coombs, Dawes, & Tversky, 1970). However, to obtain mixed-strategy solutions for game matrices with dimensionality greater than two, linear programming is required (Luce & Raiffa, 1957). Linear programming is a technique for maximizing and mini­ mizing linear functions that are subject to a number of linear constraints and nonnegativity conditions. Since the value of a game can easily be expressed as a linear function, and the rows and columns of a game matrix can serve as constraints, linear programming may be utilized to determine the vectors of probabilities x and y that, respectively, maximize and minimize the value of a two-person zero-sum game. Although linear programming computer programs have been available for some time, they are typically written for general linear programming applications. Many of the programs (such as MPOS,t for example) are large canned packages capable of solving a wide variety of linear programming problems. Since the primary application of linear programming in psychol­ ogy is for solving zero-sum games, the rationale for implementing a large general-purpose optimization package for solving a small class of problems is question­ able. These general programs require manual conversion of a game problem to a linear programming problem and manual conversion of a linear programming solution back to a game solution. Since linear programming is frequently used in business applications, many programs