Hypergraph conditions for the solvability of the ergodic equation for zero-sum games

TL;DR

This paper introduces a new hypergraph-based condition that guarantees the solvability of the ergodic equation in mean-payoff stochastic games with finite states, extending previous bounded payment results.

Contribution

It provides a general sufficient condition involving hypergraphs for the ergodic equation's solvability, applicable to games with arbitrary action spaces and unbounded payments.

Findings

Hypergraph conditions ensure ergodic equation solvability.

Refines previous bounded payment results.

Applicable to games with unbounded payments.

Abstract

The ergodic equation is a basic tool in the study of mean-payoff stochastic games. Its solvability entails that the mean payoff is independent of the initial state. Moreover, optimal stationary strategies are readily obtained from its solution. In this paper, we give a general sufficient condition for the solvability of the ergodic equation, for a game with finite state space but arbitrary action spaces. This condition involves a pair of directed hypergraphs depending only on the ``growth at infinity'' of the Shapley operator of the game. This refines a recent result of the authors which only applied to games with bounded payments, as well as earlier nonlinear fixed point results for order preserving maps, involving graph conditions.