Theory of Correspondences and Games

Abstract

In this thesis, we will consider games with large structures in the sense that the player/state space can be uncountable. We propose an appropriate condition called ?setwise coarseness? and prove several regularity properties (convexity, compactness and preservation of upper hemicontinuity) of conditional distributions/expectations of correspondences in various contexts as our mathematical preparations. Based on this condition, new results on large games/economies, Bayesian games and stochastic games are presented. Firstly, we show that the setwise coarseness condition is both necessary and sufficient for the determinateness property, equilibrium existence result and closed graph property in large games and economies. Secondly, we analyze Bayesian games and prove the existence of pure strategy Bayesian equilibria in context with finite/uncountable actions. Thirdly, we demonstrate the existence of stationary Markov perfect equilibria in discounted stochastic games via establishing a new connection between the equilibrium payoff correspondence and a general result on the conditional expectations of correspondences.