Sequential Equilibria with Infinite Histories

A fundamental non-stationarity of infinitely repeated games as usually studied is that the length of the history of play gets longer each period. With private actions (and mixed strategies) or private signals, this introduces a particular difficulty with common solution concepts such as sequential equilibria: At the beginning of the game, each player knows every other player's continuation strategy (which is simply his strategy), but this is no longer true after the game begins. When continuation strategies are functions of privately observed variables, each player is now uncertain regarding the continuation strategy of the other players. This study considers infinitely repeated games with mixed strategies, and private and public signals where the game is assumed to have been going on forever. We introduce a new solution concept: Stationary Nash Equilibrium with Infinite Histories. An equilibrium is a joint mixed strategy $\pi$ mapping infinite histories of private actions, and public and private signals to action probabilities, along with a probability measure $\mu$ by which infinite histories are drawn such that strategies are mutual best responses and the probability measure over infinite histories $\mu$ replicates itself given $\pi$.