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

1 MAXPLUS - Max-plus algebras and mathematics of decision
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
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.
Document type :
Conference papers

https://hal.inria.fr/hal-01249321
Contributor : Stephane Gaubert <>
Submitted on : Thursday, December 31, 2015 - 3:26:01 PM
Last modification on : Tuesday, December 8, 2020 - 9:45:37 AM

### Identifiers

• HAL Id : hal-01249321, version 1
• ARXIV : 1510.05396

### Citation

Marianne Akian, Stephane Gaubert, Antoine Hochart. Hypergraph conditions for the solvability of the ergodic equation for zero-sum games. 54th IEEE Conference on Decision and Control (CDC 2015), Dec 2015, Osaka, Japan. ⟨hal-01249321⟩

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