Observing Each Other's Observations in a Bayesian Coordination Game

We study a Bayesian coordination game where agents receive private information on the game's payoff structure. In addition, agents receive private signals that inform them of each other's private information. We show that once agents possess these different types of information, there exists a coordination game in the evaluation of this information. Even though the precisions of both signal types is exogenous, the precision with which agents forecast each other's actions in equilibrium turns out to be endogenous. As a consequence, there exist multiple equilibria which differ with regard to the way that agents weight their private information to forecast each other's actions. Finally, even though all players' signals are of identical quality, it turns out that efficient equilibria are asymmetric.