Beliefs correspondences and equilibria in ambiguous games

The Nash equilibrium concept combines two fundamental ideas. First, rational players choose the most preferred strategy given their beliefs about what other players will do. Second, it imposes the consistency condition that all players' beliefs are correct. This consistency condition has often been considered too strong and different solution concepts have been introduced in the literature in order to take into account ambiguous beliefs. In this paper, we show, by means of examples, that in some situation beliefs might be dependent on the strategy profile and that this kind of contingent ambiguity affects equilibrium behavior differently with respect to the existing models of ambiguous games. Hence we consider a multiple prior approach and subjective beliefs correspondences which depend on the strategy profile; we investigate existence of the equilibrium concepts corresponding to different attitudes towards ambiguity (namely optimism and pessimism). Finally we analyze particular beliefs correspondences: beliefs given by correlated equilibria and by ambiguity levels on events