IDEAS home Printed from https://ideas.repec.org/a/eee/jetheo/v213y2023ics0022053123001321.html
   My bibliography  Save this article

On sustainable equilibria

Author

Listed:
  • Govindan, Srihari
  • Laraki, Rida
  • Pahl, Lucas

Abstract

Following the ideas laid out in Myerson (1996), Hofbauer (2003) defined a Nash equilibrium of a finite game as sustainable if it can be made the unique Nash equilibrium of a game obtained by deleting/adding a subset of the strategies that are inferior replies to it. This paper proves a result about sustainable equilibria and uses it to provide a refinement as well. Our result concerns the Hofbauer-Myerson conjecture about the relationship between the sustainability of an equilibrium and its index: for a generic class of games, an equilibrium is sustainable iff its index is +1. von Schemde and von Stengel (2008) proved this conjecture for bimatrix games; we show that the conjecture is true for all finite games. More precisely, we prove that an equilibrium is isolated and has index +1 if and only if it can be made unique in a larger game obtained by adding finitely many strategies that are inferior replies to that equilibrium. It follows in a straightforward way from our result that sustainable equilibria fail the Decomposition Axiom for games as formulated by Mertens (1989a). In order to rectify this problem we propose a refinement, called strongly sustainable equilibria, which is shown to exist for all regular games.

Suggested Citation

  • Govindan, Srihari & Laraki, Rida & Pahl, Lucas, 2023. "On sustainable equilibria," Journal of Economic Theory, Elsevier, vol. 213(C).
    • Handle: RePEc:eee:jetheo:v:213:y:2023:i:c:s0022053123001321
    DOI: 10.1016/j.jet.2023.105736
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0022053123001321
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jet.2023.105736?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868.
    2. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384, April.
    3. Andrew McLennan, 2018. "Advanced Fixed Point Theory for Economics," Springer Books, Springer, number 978-981-13-0710-2, January.
    4. P. Herings & Ronald Peeters, 2010. "Homotopy methods to compute equilibria in game theory," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 119-156, January.
    5. Faruk Gül & David Pearce & Ennio Stacchetti, 1993. "A Bound on the Proportion of Pure Strategy Equilibria in Generic Games," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 548-552, August.
    6. Hauk, Esther & Hurkens, Sjaak, 2002. "On Forward Induction and Evolutionary and Strategic Stability," Journal of Economic Theory, Elsevier, vol. 106(1), pages 66-90, September.
    7. Roger Myerson & Jörgen Weibull, 2015. "Tenable Strategy Blocks and Settled Equilibria," Econometrica, Econometric Society, vol. 83(3), pages 943-976, May.
    8. MERTENS, Jean-François, 1989. "Stable equilibria - a reformulation. Part I. Definition and basic properties," LIDAM Reprints CORE 866, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. Hillas, John, 1990. "On the Definition of the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 58(6), pages 1365-1390, November.
    10. Arndt Schemde, 2005. "Index and Stability in Bimatrix Games," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-29102-2, July.
    11. Ritzberger, Klaus, 1994. "The Theory of Normal Form Games form the Differentiable Viewpoint," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 207-236.
    12. Demichelis, Stefano & Ritzberger, Klaus, 2003. "From evolutionary to strategic stability," Journal of Economic Theory, Elsevier, vol. 113(1), pages 51-75, November.
    13. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Peter Wikman, 2022. "Nash blocks," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(1), pages 29-51, March.
    2. Demichelis, Stefano & Ritzberger, Klaus, 2003. "From evolutionary to strategic stability," Journal of Economic Theory, Elsevier, vol. 113(1), pages 51-75, November.
    3. Carlos Alós-Ferrer & Klaus Ritzberger, 2020. "Reduced normal forms are not extensive forms," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(2), pages 281-288, October.
    4. 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.
    5. 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).
    6. Geir B. Asheim & Mark Voorneveld & Jörgen W. Weibull, 2016. "Epistemically Robust Strategy Subsets," Games, MDPI, vol. 7(4), pages 1-16, November.
    7. Reuben Bearman, 2023. "Signaling Games with Costly Monitoring," Papers 2302.01116, arXiv.org.
    8. Pahl, Lucas, 2023. "Polytope-form games and index/degree theories for extensive-form games," Games and Economic Behavior, Elsevier, vol. 141(C), pages 444-471.
    9. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2015. "The refined best-response correspondence in normal form games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(1), pages 165-193, February.
    10. Stefano Demichelis & Klaus Ritzberger & Jeroen M. Swinkels, 2004. "The simple geometry of perfect information games," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(3), pages 315-338, June.
    11. Demichelis, Stefano & Germano, Fabrizio, 2002. "On (un)knots and dynamics in games," Games and Economic Behavior, Elsevier, vol. 41(1), pages 46-60, October.
    12. Norman, Thomas W.L., 2018. "Inefficient stage Nash is not stable," Journal of Economic Theory, Elsevier, vol. 178(C), pages 275-293.
    13. Anne Balthasar, 2010. "Equilibrium tracing in strategic-form games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 39-54, January.
    14. Lucas Pahl, 2022. "Polytope-form games and Index/Degree Theories for Extensive-form games," Papers 2201.02098, arXiv.org, revised Jul 2023.
    15. Man, Priscilla T.Y., 2012. "Efficiency and stochastic stability in normal form games," Games and Economic Behavior, Elsevier, vol. 76(1), pages 272-284.
    16. Srihari Govindan & Robert Wilson, 2009. "On Forward Induction," Econometrica, Econometric Society, vol. 77(1), pages 1-28, January.
    17. Satoru Takahashi, 2020. "Non-equivalence between all and canonical elaborations," The Japanese Economic Review, Springer, vol. 71(1), pages 43-57, January.
    18. Hauk, Esther & Hurkens, Sjaak, 2002. "On Forward Induction and Evolutionary and Strategic Stability," Journal of Economic Theory, Elsevier, vol. 106(1), pages 66-90, September.
    19. Battalio,R. & Samuelson,L. & Huyck,J. van, 1998. "Risk dominance, payoff dominance and probabilistic choice learning," Working papers 2, Wisconsin Madison - Social Systems.
    20. John Hillas & Mathijs Jansen & Jos Potters & Dries Vermeulen, 2001. "On the Relation Among Some Definitions of Strategic Stability," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 611-635, August.

    More about this item

    Keywords

    Sustainable equilibria; Index of equilibria; Refinements of equilibria;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jetheo:v:213:y:2023:i:c:s0022053123001321. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/622869 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.