Continuity of Equilibria for Two-Person Zero-Sum Games with Noncompact Action Sets and Unbounded Payoffs

TL;DR

This paper generalizes Berge's maximum theorem to noncompact action sets and unbounded payoffs, analyzing continuity and existence of solutions in two-player zero-sum games with various information structures.

Contribution

It extends Berge's maximum theorem to broader settings and applies these results to establish continuity and existence of solutions in complex zero-sum games.

Findings

Continuity of value functions in perfect information games

Existence of lopsided values in simultaneous move games

Continuity of solution multifunctions

Abstract

This paper extends Berge's maximum theorem for possibly noncompact action sets and unbounded cost functions to minimax problems and studies applications of these extensions to two-player zero-sum games with possibly noncompact action sets and unbounded payoffs. For games with perfect information, also known under the name of turn-based games, this paper establishes continuity properties of value functions and solution multifunctions. For games with simultaneous moves, it provides results on the existence of lopsided values (the values in the asymmetric form) and solutions. This paper also establishes continuity properties of the lopsided values and solution multifunctions.