Continuity Properties of Value Functions in Information Structures for Stochastic Team Problems and Zero-Sum and General Games

Abstract

We study continuity properties of value functions in information structures for stochastic team, zero-sum, and general game problems. We establish continuity properties of the value function under total variation, setwise, and weak convergence of information structures. Our analysis reveals that the value function for a bounded game is continuous under total variation convergence of information structures in both zero-sum games and team problems. However, continuity fails under setwise or weak convergence of information structures. The value function exhibits upper semicontinuity properties under weak and setwise convergence of information structures for team problems, and upper or lower semicontinuity properties hold for zero-sum games when such convergence is through a Blackwell-garbled sequence of information structures. Finally, a counterexample reveals that value functions for players may not be continuous even under total variation convergence of information structures in general non-zero-sum games.