A 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 player’s payoff function and feasible strategies. The authors' 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. The second main result is an application to Bayesian zero-sum games in which each player’s information is viewed as a parameter. The authors model each player’s information as a sub-σ-field, so that it determines her feasible strategies: those that are measurable with respect to the player’s information. The authors thereby characterize conditions under which the equilibrium value and strategies depend continuously and upper hemicontinuously (respectively) on each player’s information.