On Uniqueness and Stability of Symmetric Equilibria in Differentiable Symmetric Games

University of Zurich Department of Economics Working Paper No. 18

46 Pages Posted: 30 May 2011

See all articles by Andreas Hefti

Andreas Hefti

University of Zurich - Department of Economics; Zurich University of Applied Sciences

Date Written: February 1, 2011

Abstract

Higher-dimensional symmetric games become of more and more importance for applied micro- and macroeconomic research. Standard approaches to uniqueness of equilibria have the drawback that they are restrictive or not easy to evaluate analytically. In this paper I provide some general but comparably simple tools to verify whether a symmetric game has a unique symmetric equilibrium or not. I distinguish between the possibility of multiple symmetric equilibria and asymmetric equilibria which may be economically interesting and is useful to gain further insights into the causes of asymmetric equilibria in symmetric games with higher-dimensional strategy spaces. Moreover, symmetric games may be used to derive some properties of the equilibrium set of certain asymmetric versions of the symmetric game. I further use my approach to discuss the relationship between stability and (in)existence of multiple symmetric equilibria. While there is an equivalence between stability, inexistence of multiple symmetric equilibria and the unimportance of strategic effects for the comparative statics, this relationship breaks down in higher dimensions. Stability under symmetric adjustments is a minimum requirement of a symmetric equilibrium for reasonable comparative statics of symmetric changes. Finally, I present an alternative condition for a symmetric equilibrium to be a local contraction which is more general than the conventional approach of diagonal dominance and yet simpler to evaluate than the eigenvalue condition of continuous adjustment processes.

Keywords: Symmetric games, Nash equilibrium, uniqueness, stability

JEL Classification: C72

Suggested Citation

Hefti, Andreas M., On Uniqueness and Stability of Symmetric Equilibria in Differentiable Symmetric Games (February 1, 2011). University of Zurich Department of Economics Working Paper No. 18, Available at SSRN: https://ssrn.com/abstract=1855663 or http://dx.doi.org/10.2139/ssrn.1855663

Andreas M. Hefti (Contact Author)

University of Zurich - Department of Economics ( email )

Zürich
Switzerland

Zurich University of Applied Sciences ( email )

Institut fuer Angewandte Medienwissenschaft
Zur Kesselschmiede 35
Winterthur, CH 8401
Switzerland

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
57
Abstract Views
461
rank
484,555
PlumX Metrics