Common Knowledge and State-Dependent Equilibria.

Many puzzling social behaviors, such as avoiding eye contact, using innuendos, and insignificant events that trigger revolutions, seem to relate to common knowledge and coordination, but the exact relationship has yet to be formalized. Herein, we present such a formalization. We state necessary and sufficient conditions for what we call state-dependent equilibria --- equilibria where players play different strategies in different states of the world. In particular, if everybody behaves a certain way e.g. does not revolt in the usual state of the world, then in order for players to be able to behave a different way e.g. revolt in another state of the world, it is both necessary and sufficient for it to be common p-believed that it is not the usual state of the world, where common p-belief is a relaxation of common knowledge introduced by Monderer and Samet [16]. Our framework applies to many player r-coordination games --- a generalization of coordination games that we introduce --- and common r,p-beliefs --- a generalization of common p-beliefs that we introduce. We then apply these theorems to two particular signaling structures to obtain novel results.