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On uniqueness and stability of symmetric equilibria in differentiable symmetric games

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  • Andreas Hefti

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.

Suggested Citation

  • Andreas Hefti, 2011. "On uniqueness and stability of symmetric equilibria in differentiable symmetric games," ECON - Working Papers 018, Department of Economics - University of Zurich.
  • Handle: RePEc:zur:econwp:018
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    File URL: http://www.econ.uzh.ch/static/wp/econwp018.pdf
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    References listed on IDEAS

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    1. Carlos Alós-Ferrer & Ana Ania, 2005. "The evolutionary stability of perfectly competitive behavior," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(3), pages 497-516, October.
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    Cited by:

    1. Martin Grossmann, 2015. "Evolutionarily Stable Strategies in Sports Contests," Journal of Sports Economics, , vol. 16(1), pages 108-121, January.
    2. Martin Grossmann, 2011. "Endogenous Liquidity Constraints in a Dynamic Contest," Working Papers 0148, University of Zurich, Institute for Strategy and Business Economics (ISU).
    3. Martin Grossmann & Andreas Hefti & Markus Lang, 2012. "Aggregative Contests and Ex-post Heterogeneity: the Case of the UEFA Champions League," Working Papers 0161, University of Zurich, Institute for Strategy and Business Economics (ISU).
    4. Andreas M. Hefti, 2013. "Local contraction-stability and uniqueness," ECON - Working Papers 112, Department of Economics - University of Zurich.
    5. Andreas M. Hefti, 2011. "Attention competition," ECON - Working Papers 028, Department of Economics - University of Zurich.
    6. Oliver Guertler & Markus Lang & Tim Pawlowski, 2011. "On the Release of Players to National Teams," Working Papers 0047, University of Zurich, Center for Research in Sports Administration (CRSA).

    More about this item

    Keywords

    Symmetric games; Nash equilibrium; uniqueness; stability;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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