Generalized Belief Operator and the Impact of Small Probability Events on Higher Order Beliefs

Abstract

This paper studies the robustness of an equilibrium to incomplete information in binary-action supermodular games. Using a generalized version of belief operator, we explore the restrictions that prior beliefs impose on higher order beliefs. In particular, we obtain a non-trivial lower bound on the probability of a common belief event, uniform over type spaces, when the underlying game has a monotone potential. Conversely, when the game has no monotone potential, we construct a type space with an arbitrarily high probability event in which players never have common belief about that event. As an implication of these results, we show for generic binary-action supermodular games that an action profile is robust to incomplete information if and only if it is a monotone potential maximizer. Our study offers new methodology and insight to the analysis of global game equilibrium selection.