Ergodic and Thermodynamic Games

Let $T:X\to X$ and $S:Y \to Y$ be continuous maps defined on compact sets. Let $$\varphi_i(\mu,\nu)=\int_{X \times Y} A_i(x,y) d\mu(x) d\nu(y)\;\;{for} \;\; i=1,2,$$ where $\mu$ is $T$-invariant and $\nu$ is $S$-invariant, be pay-off functions for a game (in the usual sense of game theory) between players that have the set of invariant measures for $T$ (player 1) and $S$ (player 2) as possible strategies. Our goal here is to establish the notion of Nash equilibrium point for the game defined by this pay-offs and strategies. The main tools came from ergodic optimization (as we are optimizing over the set of invariant measures) and thermodynamic formalism (when we add to the integrals above the entropy of measures in order to define a second case to be explored). Both cases are ergodic versions of non-cooperative games. We show the existence of Nash equilibrium points with two independent arguments. One of the arguments works for the case with entropy, and uses only tools of thermodynamical formalism, while the other, that works in the case without entropy but can be adapted to deal with both cases, uses the Kakutani fixed point. We also present examples and briefly discuss uniqueness (or lack of uniqueness). In the end we present a different example where players are allowed to collaborate. This final example show connections between cooperative games and ergodic transport.