# Unique ergodicity of deterministic zero-sum differential games

## TL;DR

This paper investigates the conditions under which deterministic zero-sum differential games exhibit unique ergodicity, characterized by the convergence of value functions, extending classical dynamical systems concepts.

## Contribution

It provides necessary and sufficient conditions for unique ergodicity in such games, involving symmetric criteria and the concept of dominions.

## Key findings

- Necessary and sufficient conditions for ergodicity are established.
- The notion extends classical ergodicity to game-theoretic settings.
- Conditions involve symmetric properties and invariant subsets called dominions.

## Abstract

We study the ergodicity of deterministic two-person zero-sum differential games. This property is defined by the uniform convergence to a constant of either the infinite-horizon discounted value as the discount factor tends to zero, or equivalently, the averaged finite-horizon value as the time goes to infinity. We provide necessary and sufficient conditions for the unique ergodicity of a game. This notion extends the classical one for dynamical systems, namely when ergodicity holds with any (suitable) perturbation of the running payoff function. Our main condition is symmetric between the two players and involve dominions, i.e., subsets of states that one player can make approximately invariant.