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

TL;DR

This paper investigates how the value functions in various stochastic game models behave under different types of convergence of information structures, establishing conditions for their continuity or semicontinuity.

Contribution

It provides a comprehensive analysis of the continuity properties of value functions in zero-sum and team games under various topologies on information structures, including new results on total variation and weak convergence.

Findings

Value functions are continuous under total variation convergence in bounded games.

Continuity may fail under setwise or weak convergence, but upper semicontinuity often holds.

Independence of channels and total variation conditions ensure continuity under weak convergence.

Abstract

We study continuity properties of stochastic game problems with respect to various topologies on information structures, defined as probability measures characterizing a game. We will 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. Continuity may fail to hold under setwise or weak convergence of information structures, however, 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. If the individual channels are independent, fixed, and satisfy a total variation continuity condition, then the value functions are continuous under weak convergence of priors. We finally show that value functions for players may not be continuous even under total variation convergence of information structures in general non-zero-sum games.