Ergodicity conditions for zero-sum games

Marianne Akian; Stéphane Gaubert; Antoine Hochart

doi:10.3934/dcds.2015.35.3901

Article Contents

2015, Volume 35, Issue 9: 3901-3931. Doi: 10.3934/dcds.2015.35.3901

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Ergodicity conditions for zero-sum games

1.
INRIA and CMAP, Ecole polytechnique, CNRS, CMAP, Ecole polytechnique, Route de Saclay, 91128 Palaiseau Cedex

Received: May 2014

Revised: December 2014

Published: May 2017

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Abstract

A basic question for zero-sum repeated games consists in determining whether the mean payoff per time unit is independent of the initial state. In the special case of ``zero-player'' games, i.e., of Markov chains equipped with additive functionals, the answer is provided by the mean ergodic theorem. We generalize this result to repeated games. We show that the mean payoff is independent of the initial state for all state-dependent perturbations of the rewards if and only if an ergodicity condition is verified. The latter is characterized by the uniqueness modulo constants of nonlinear harmonic functions (fixed points of the recession function associated to the Shapley operator), or, in the special case of stochastic games with finite action spaces and perfect information, by a reachability condition involving conjugate subsets of states in directed hypergraphs. We show that the ergodicity condition for games only depends on the support of the transition probability, and that it can be checked in polynomial time when the number of states is fixed.

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Mathematics Subject Classification: Primary: 47H25; Secondary: 91A20, 05C65, 06A15.

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