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On the indices of zeros of nash fields

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  • DE MICHELIS, Stefano
  • GERMANO, Fabrizio
Abstract
Given a game and a dynamics on the space of strategies it is possible to associate to any component of Nash equilibria, an integer, this is the index, see Ritzberger (1994). This number gives useful information on the equilibrium set and in particular on its stability properties under the given dynamics. We prove that indices of components always coincide with their local degrees for the projection map from the Nash equilibrium correspondence to the underlying space of games, so that essentially all dynamics have the same indices. This implies that in many cases the asymptotic properties of equilibria do not depend on the choice of dynamics, a question often debated in recent litterature. In particular many equilibria are asymptotically unstable for any dynamics. Thus the result establishes a further link between the theory of learning and evolutionary dynamics, the theory of equilibrium refinements and the geometry of Nash equilibria.The proof holds for very general situations that include not only any number of players and strategies but also general equilibrium settings and games with a continuum of pure strategies such as Shapley-Shubik type games, this case will be studied in a forthcoming paper.

Suggested Citation

  • DE MICHELIS, Stefano & GERMANO, Fabrizio, 2000. "On the indices of zeros of nash fields," LIDAM Discussion Papers CORE 2000017, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2000017
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    References listed on IDEAS

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    3. Takahashi, Satoru & Tercieux, Olivier, 2020. "Robust equilibrium outcomes in sequential games under almost common certainty of payoffs," Journal of Economic Theory, Elsevier, vol. 188(C).
    4. Jason Milionis & Christos Papadimitriou & Georgios Piliouras & Kelly Spendlove, 2022. "Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics," Papers 2203.14129, arXiv.org.
    5. Bich, Philippe & Fixary, Julien, 2022. "Network formation and pairwise stability: A new oddness theorem," Journal of Mathematical Economics, Elsevier, vol. 103(C).
    6. Gaël Giraud, 2000. "Notes sur les jeux stratégiques de marchés," Cahiers d'Économie Politique, Programme National Persée, vol. 37(1), pages 257-272.
    7. Dieter Balkenborg & Dries Vermeulen, 2016. "Where Strategic and Evolutionary Stability Depart—A Study of Minimal Diversity Games," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 278-292, February.
    8. Anna Rubinchik & Roberto Samaniego, 2013. "Demand for contract enforcement in a barter environment," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(1), pages 73-97, June.
    9. Demichelis, Stefano & Ritzberger, Klaus, 2003. "From evolutionary to strategic stability," Journal of Economic Theory, Elsevier, vol. 113(1), pages 51-75, November.
    10. DE MICHELIS, Stefano, 2000. "On the index and asymptotic stability of dynamics," LIDAM Discussion Papers CORE 2000018, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    11. Sandholm, William H., 2015. "Population Games and Deterministic Evolutionary Dynamics," Handbook of Game Theory with Economic Applications,, Elsevier.
    12. Lucas Pahl, 2022. "Polytope-form games and Index/Degree Theories for Extensive-form games," Papers 2201.02098, arXiv.org, revised Jul 2023.
    13. Demichelis, Stefano & Germano, Fabrizio, 2002. "On (un)knots and dynamics in games," Games and Economic Behavior, Elsevier, vol. 41(1), pages 46-60, October.
    14. Hefti, Andreas, 2016. "On the relationship between uniqueness and stability in sum-aggregative, symmetric and general differentiable games," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 83-96.

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