A Learning Theory for the Harsanyi's Doctrine in Repeated Games

This paper investigates simultaneous learning about both nature and others' actions in repeated games, and identifies a set of sufficient conditions assuring that equilibrium actions converge to a Nash equilibrium. Players have each an utility function over infinite histories continuous for the product topology. Nature' drawing after any history can depend on any past actions, or can be independent of them. Provided that 1) every player maximizes her expected payoff against her own beliefs, 2) every player updates her beliefs in a Bayesian manner, 3) prior beliefs about both nature and other players' strategies have a grain of truth, and 4) beliefs about nature are independent of actions chosen during the game, we show that after some finite time the equilibrium outcome of the above game is arbitrarily close to a Nash equilibrium. Those assumptions are shown to be tight.