7. Correlated Equilibria, Learning, Bayesian Equilibria

This chapter starts with the concept of correlated equilibrium due to Aumann, an extension of Nash equilibrium with interesting properties from a strategic, geometric and dynamics viewpoint. In a learning framework, we then introduce no-regret procedures and calibrated strategies, and prove existence results. Next we show that in a repeated game: (1) if a player follows a strategy with no external regret, the empirical distribution of moves converges a.s. to its corresponding Hannan set, and (2) if each player follows a procedure with no internal regret, convergence to the set of correlated equilibrium distributions occurs. We conclude this chapter with games with incomplete information (Bayesian games), where the players have different information on the game they have to play, and we introduce the corresponding extension of Nash equilibrium (called Bayesian equilibrium).