Ergodic and thermodynamic games

Let [Formula: see text] and [Formula: see text] be compact sets, and [Formula: see text], [Formula: see text] be continuous maps. Let [Formula: see text] where [Formula: see text] is [Formula: see text]-invariant and [Formula: see text] is [Formula: see text]-invariant, be payoff functions for a game (in the usual sense of game theory) between players that have the set of invariant measures for [Formula: see text] (player 1) and [Formula: see text] (player 2) as possible strategies. Our goal here is to establish the notion of Nash equilibrium for the game defined by these payoffs and strategies. The main tools come 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 deals with 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 shows connections between cooperative games and ergodic transport.