Finite uniform approximation of zero-sum games defined on a product of staircase-function continuous spaces

Abstract

A method of finite approximation of zero-sum games defined on a product of staircase-function continuous spaces is presented. The method consists in uniformly sampling the player’s pure strategy value set, solving “smaller” matrix games, each defined on a subinterval where the pure strategy value is constant, and stacking their solutions if they are consistent. The stack of the “smaller” matrix game solutions is an approximate solution to the initial staircase game. The (weak) consistency, equivalent to the approxi-mate solution acceptability, is studied by how much the payoff and optimal situation change as the sampling density minimally increases. The consistency is decomposed into the payoff, optimal strategy support cardinality, optimal strategy sampling density, and support probability consistency. The most important parts are the payoff consistency and optimal strategy support cardinality (weak) consistency. However, it is practically reasonable to consider a relaxed payoff consistency, by which the game optimal value change in an appropriate approximation may grow at most by epsilon as the sampling density minimally increases. The weak consistency itself is a relaxation to the consistency, where the minimal decrement of the sampling density is ignored.