Reducing two person, zero sum games with underlying symmetry

Abstract

We consider two person, zero sum games with several symmetries. Where such symmetries are present there is a group acting on the strategies of the game. We show how to use this action to produce a reduced game with a smaller matrix, but having the same value as the original game, and how to obtain optimal strategies for the original game from optimal strategies of the reduced game. An analysis of a simplified version of the popular game Mastermind is given to illustrate the theory developed. The rules of many games give them a number of symmetries. When analysing such a game one keeps meeting cases which seem very much like ones already encountered and the natural approach is to try to link together somehow strategies that appear similar to each other.In this paper we formalize these feelings for two person, zero sum games (with imperfect information) with several symmetries. Where such symmetries are evident, there is in fact a group acting on the strategies of the game. This action yields homogeneous optimal strategies (that is, optimal strategies in which linked strategies are used with equal probability) for each player, and makes it possible to guarantee the existence of a smaller matrix game with the same value as the original one. Further, all homogeneous optimal strategies in the original game can be obtained from a knowlege of all optimal strategies in the smaller or reduced game. The greater the number of symmetries, the smaller will this reduced game be, in general.