Continuity and sensitivity analysis of parameterized Nash games

TL;DR

This paper investigates the continuity properties of Nash equilibria in parameterized games, especially addressing cases with multiple equilibria, and establishes conditions for continuity using approximate Nash equilibria.

Contribution

It provides a comprehensive analysis of equilibrium continuity in parameterized Nash games, including cases with non-unique equilibria, and introduces methods to prove continuity via approximate equilibria.

Findings

Continuity of equilibrium mappings holds for unique Nash equilibria.

Discontinuities can occur when equilibria are not unique.

Approximate Nash equilibria can be used to establish continuity in complex cases.

Abstract

In this paper we consider continuity of the set of Nash equilibria and approximate Nash equilibria for parameterized games. For parameterized games with unique Nash equilibria, the continuity of this equilibrium mapping is well-known. However, when the equilibria need not be unique, there may exist discontinuities in the equilibrium mapping. The focus of this work is to summarize continuity properties for parameterized Nash equilibria and prove continuity via the approximate Nash game with uniformly continuous objective functions over potentially non-compact strategy spaces.