Linear Programming Solutions to Ratio Games

Abstract

This paper develops a method for finding the minimax solutions to a game whose payoff function is of the form X t AY / X t BY , where X , Y are mixed-strategy vectors and A , B > 0 are matrices. Such a payoff function arises in stochastic games, economic models of an expanding economy, and some nonzero-sum game formulations. The game is transformed into a linear program with a parameter in the constraint set. By successive solutions of this program with appropriate values of the parameter, the value and optimal strategies of the game can be approximated to any desired degree of accuracy. A new approach of perturbing linear programs is used to prove convergence of the computational technique and to bound the rate of convergence.