# Acyclic Gambling Games

## TL;DR

This paper studies the limiting behavior of two-player zero-sum stochastic gambling games with controlled wealth states, establishing conditions for convergence of discounted values to a unique solution characterized by functional equations.

## Contribution

It introduces a strong acyclicity condition ensuring the existence and characterization of the limit of discounted game values as the discount factor approaches zero.

## Key findings

- Limit exists under strong acyclicity condition.
- Limit characterized as a unique solution to functional equations.
- Counterexample shows convergence can fail under weaker conditions.

## Abstract

We consider 2-player zero-sum stochastic games where each player controls his own state variable living in a compact metric space. The terminology comes from gambling problems where the state of a player represents its wealth in a casino. Under natural assumptions (such as continuous running payoff and non expansive transitions), we consider for each discount factor the value v $\lambda$ of the $\lambda$-discounted stochastic game and investigate its limit when $\lambda$ goes to 0. We show that under a strong 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 his opponent does not move, can reach the zone when the current payoff is at least as good than 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 (v $\lambda$) may fail.