Comparison and Continuity Properties of Equilibrium Values in Information Structures for Stochastic Games

Abstract

In stochastic games where players measure a cost-relevant exogenous state variable through measurement channels, an information structure is the joint probability measure induced on the state space and player measurement spaces. For single-player decision problems in finite spaces, a theorem due to Blackwell leads to a complete characterization of when one information structure is "better" than another. For zero-sum games with finite state, measurement, and action spaces, Peski produced necessary and sufficient conditions for ordering information structures. In this thesis, we obtain an infinite dimensional (standard Borel) generalization of Peski's result. A corollary is that more information cannot hurt a decision maker taking part in a zero-sum game. We also establish two supporting results which are essential and explicit, though modest contributions to the literature: (i) a partial converse to Blackwell's ordering in the standard Borel setup and (ii) an existence result for equilibria in zero-sum games with incomplete information. Then we study continuity properties of stochastic game problems with respect to various notions of convergence of information structures. For zero-sum games, team problems, and general games, we will establish continuity properties of the value function under total variation, setwise, and weak convergence of information structures.