Coordination and the Relative Cost of Distinguishing Nearby States

We study a coordination game where players simultaneously acquire information prior to the play of the game. We allow general information acquisition technologies, modeled by a cost functional mapping from information structures. Costly local distinguishability is a property requiring that the cost of distinguishing nearby states is hard relative to distinguishing distant states. This property is not important in decision problems, but is crucial in determining equilibrium outcomes in games. If it holds, there is a unique equilibrium; if it fails, there are multiple equilibria close to those that would exist if there was complete information. We study these issues in the context of a regime change game with a continuum of players. We also provide a common belief foundation for equilibria of this game. This allows us to distinguish cases where the players could (physically) acquire information giving rise to multiple equilibria, but choose not to, and situations where players could not physically have acquired information in a way consistent with multiple equilibria. Our analysis corresponds to the former case, while the choosing precision of additive noise corresponds to the latter case.