Continuity of the value and optimal strategies when common priors change

We show that the value of a zero‐sum Bayesian game is a Lipschitz continuous function of the players’ common prior belief with respect to the total variation metric on beliefs. This is unlike the case of general Bayesian games where lower semi‐continuity of Bayesian equilibrium (BE) payoffs rests on the ‘almost uniform’ convergence of conditional beliefs. We also show upper semi‐continuity (USC) and approximate lower semi‐continuity (ALSC) of the optimal strategy correspondence, and discuss ALSC of the BE correspondence in the context of zero‐sum games. In particular, the interim BE correspondence is shown to be ALSC for some classes of information structures with highly non‐uniform convergence of beliefs, that would not give rise to ALSC of BE in non‐zero‐sum games.