On Behavior Strategy Solutions in Two-Person Zero-Sum Finite Extended Games with Imperfect Information. Part I: A Method for Determination of Minimally Complex Behavior Strategy Solutions

Abstract

In any two-person zero-sum finite extended game with imperfect information, an upper bound on the quality of each player’s strategies is established by his choice of a scheme for gathering and retaining the information that becomes available to him as the game progresses. Further, for a given game, strategies of the maximum possible quality may exist for each player on a number of information schemes of widely differing complexities.By analyzing the relationship between any given two-person zero-sum finite extended game with given information schemes for its players, and an associated N-person noncooperative game, a heuristic scheme is uncovered for searching for minimally complex information schemes which support behavior strategies having within $\varepsilon $ of the maximum possible quality, for any given $\varepsilon $.In the course of this analysis, a minimax theorem in behavior strategies is obtained, which is similar to von Neumann’s minimax theorem in mixed strategies.