Theorem of the maximin and applications to Bayesian zero-sum games

Consider a family of zero-sum games indexed by a parameter that determines each players payoff function and feasible strategies. Our first main result characterizes continuity assumptions on the payoffs and the constraint correspondence such that the equilibrium value and strategies depend continuously and upper hemicontinuously (respectively) on the parameter. This characterization uses two topologies in order to overcome a topological tension that arises when players strategy sets are infinite-dimensional. Our second main result is an application to Bayesian zero-sum games in which each players information is viewed as a parameter. We model each players information as a sub-sigma-field, so that it determines his or her feasible strategies: those that are measurable with respect to the players information. We thereby characterize conditions under which the equilibrium value and strategies depend continuously and upper hemicontinuously (respectively) on each players information. This clarifies and extends related results of Einy et al. (2008).

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