Belief-Based Equilibrium

We introduce a new solution concept: a Belief-Based-Equilibrium. In a BBE, players optimize myopically given their beliefs which are not necessarily correct, but are not contradicted by the data (i.e., players' beliefs are calibrated). We show that if the stage game has a unique correlated equilibrium then the play of a BBE resembles a Nash-equilibrium play (i.e, it is a Nash-Normal play). However, a BBE may not be a Nash-equilibrium. In particular, in a BBE players may play deterministically when the only Nash equilibrium is in mixed strategies. It is easier to develop learning models leading to a BBE than to a Nash-equilibrium because, after many interactions, players beliefs may become calibrated, but not correct. We show that there exist forecasting schemes that lead to calibrated beliefs (but not to correct beliefs) for all sequences of data.