Acyclic Gambling Games

Abstract

We consider two-player, zero-sum stochastic games in which each player controls the player’s own state variable living in a compact metric space. The terminology comes from gambling problems in which the state of a player represents its wealth in a casino. Under standard assumptions (e.g., continuous running payoff and nonexpansive transitions), we consider for each discount factor the value vλ of the λ-discounted stochastic game and investigate its limit when λ goes to zero. We show that, under a new acyclicity condition, the limit exists and is characterized as the unique solution of a system of functional equations: the limit is the unique continuous excessive and depressive function such that each player, if the player’s opponent does not move, can reach the zone when the current payoff is at least as good as the limit value without degrading the limit value. The approach generalizes and provides a new viewpoint on the Mertens–Zamir system coming from the study of zero-sum repeated games with lack of information on both sides. A counterexample shows that under a slightly weaker notion of acyclicity, convergence of (vλ) may fail.