A note on local uniqueness of equilibria: How isolated is a local equilibrium?

TL;DR

This paper investigates how singular values influence the local uniqueness of equilibria, providing a method to estimate the size of neighborhoods around regular equilibria to assess their isolation in various economic and game-theoretic models.

Contribution

It introduces a theorem that constructs neighborhoods around regular equilibria, linking singular values to local uniqueness, applicable across diverse economic and strategic settings.

Findings

Neighborhood size estimates for local equilibrium uniqueness

Singular values determine the degree of equilibrium isolation

Applicable to exchange economies and non-cooperative games

Abstract

The motivation of this note is to show how singular values affect local uniqueness. More precisely, Theorem 3.1 shows how to construct a neighborhood (a ball) of a regular equilibrium whose diameter represents an estimate of local uniqueness, hence providing a measure of how isolated a (local) unique equilibrium can be. The result, whose relevance in terms of comparative statics is evident, is based on reasonable and natural assumptions and hence is applicable in many different settings, ranging from pure exchange economies to non-cooperative games.