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 (that induces the topology of setwise convergence on beliefs). This is unlike the case of general Bayesian games, where lower semi-continuity of Bayesian equilibrium payoffs rests on the convergence of conditional beliefs (Engl (1994), Kajii and Morris (1998)). We also show upper, and approximate lower, semi-continuity of the optimal strategy correspondence with respect to the total variation norm, and discuss approximate lower semi-continuity of the Bayesian equilibrium correspondence in the context of zero-sum games.