Finite uniform approximation of two-person games defined on a product of staircase-function infinite spaces

Abstract

A method of finite uniform approximation of two-person games in staircase-function infinite spaces is presented. A pure strategy of the player is a staircase function defined on a time interval. The method consists in uniformly sampling the player's pure strategy value set and finding equilibria in “smaller” bimatrix games, each defined on a subinterval where the pure strategy value is constant. Then the equilibria are successively stacked so that the stack is an approximate solution to the initial staircase game. The (weak) consistency, equivalent to the approximate solution acceptability, is studied by how much the players' payoff and equilibrium strategy change as the sampling density minimally increases. The consistency is decomposed into the payoff, equilibrium strategy support cardinality, equilibrium strategy sampling density, and support probability consistency. The most important parts are the payoff consistency and equilibrium strategy support cardinality (weak) consistency. However, it is practically reasonable to consider a relaxed payoff consistency, by which the player's payoff change in an appropriate approximation may grow at most by ε 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. An example is presented to show how the approximation is fulfilled for a case of when “smaller” bimatrix games have multiple equilibria.