Psychological Nash Equilibria under Ambiguity

Psychological games aim to represent situations in which players have belief-dependent motivations or believe that their opponents have belief-dependent motivations. In this setting, utility functions are directly dependent on the entire hierarchy of beliefs of each player. On the other hand, the literature on strategic ambiguity in classical games highlights that players may have ambiguous (or imprecise) beliefs about opponents' strategy choices. In this paper, we look at the issue of strategic ambiguity in the framework of psychological games by taking into account ambiguous hierarchies of beliefs and we study the natural generalization of the psychological Nash equilibrium concept to this framework. We give an existence result for this new concept of equilibrium and provide examples that show that even an infinitesimal amount of ambiguity may alter significantly the equilibria of the game or can work as an equilibrium selection device. Finally, we look at the problem of stability of psychological equilibria with respect to ambiguous trembles on the entire hierarchy of correct beliefs and we provide a limit result that gives conditions so that sequences of psychological equilibria under ambiguous perturbation converge to psychological equilibria of the unperturbed game.