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Axiomatic Equilibrium Selection for Generic Two‐Player Games

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  • Srihari Govindan
  • Robert Wilson

Abstract

We apply three axioms adapted from decision theory to refinements of the Nash equilibria of games with perfect recall that select connected closed sub- sets called solutions. No player uses a weakly dominated strategy in an equilibrium in a solution. Each solution contains a quasi-perfect equilibrium and thus a sequential equilibrium in strategies that provide conditionally admissible optimal continuations from information sets. A refinement is immune to embedding a game in a larger game with additional players provided the original players' strategies and payoffs are preserved, i.e. solutions of a game are the same as those induced by the solutions of any larger game in which it is embedded. For games with two players and generic payoffs, we prove that these axioms characterize each solution as an essential component of equilibria in undominated strategies, and thus a stable set as defined by Mertens (1989).
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  • Srihari Govindan & Robert Wilson, 2012. "Axiomatic Equilibrium Selection for Generic Two‐Player Games," Econometrica, Econometric Society, vol. 80(4), pages 1639-1699, July.
    • Handle: RePEc:ecm:emetrp:v:80:y:2012:i:4:p:1639-1699
    DOI: ECTA9579
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    1. Srihari Govindan & Robert Wilson, 2009. "Axiomatic Theory of Equilibrium Selection for Games with Two Players, Perfect Information, and Generic Payoffs," Levine's Working Paper Archive 814577000000000125, David K. Levine.
    2. Govindan, Srihari & Wilson, Robert, 2001. "Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes," Econometrica, Econometric Society, vol. 69(3), pages 765-769, May.
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    5. Mailath, George J & Samuelson, Larry & Swinkels, Jeroen M, 1993. "Extensive Form Reasoning in Normal Form Games," Econometrica, Econometric Society, vol. 61(2), pages 273-302, March.
    6. Srihari Govindan & Robert Wilson, 2009. "On Forward Induction," Econometrica, Econometric Society, vol. 77(1), pages 1-28, January.
    7. 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).
    8. Van Damme, Eric, 2002. "Strategic equilibrium," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 41, pages 1521-1596, Elsevier.
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    10. Srihari Govindan & Robert Wilson, 2006. "Metastable Equilibria," Levine's Bibliography 122247000000001211, UCLA Department of Economics.
    11. van Damme, E.E.C., 1984. "A relation between perfect equilibria in extensive form games and proper equilibria in normal form games," Other publications TiSEM 3734d89e-fd5c-4c80-a230-5, Tilburg University, School of Economics and Management.
    12. Srihari Govindan & Tilman Klumpp, 2003. "Perfect equilibrium and lexicographic beliefs," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(2), pages 229-243.
    13. Srihari Govindan & Robert Wilson, 2008. "Axiomatic Theory of Equilibrium Selection in Signalling Games with Generic Payoffs," Levine's Working Paper Archive 122247000000002381, David K. Levine.
    14. Srihari Govindan & Jean-François Mertens, 2004. "An equivalent definition of stable Equilibria," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(3), pages 339-357, June.
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    18. , & , B., 2006. "Sufficient conditions for stable equilibria," Theoretical Economics, Econometric Society, vol. 1(2), pages 167-206, June.
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    20. Hillas, John & Kohlberg, Elon, 2002. "Foundations of strategic equilibrium," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 42, pages 1597-1663, Elsevier.
    21. MERTENS, Jean-François, 1991. "Stable equilibria - a reformulation. Part II. Discussion of the definition, and further results," LIDAM Reprints CORE 960, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

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    2. Man, Priscilla T.Y., 2012. "Forward induction equilibrium," Games and Economic Behavior, Elsevier, vol. 75(1), pages 265-276.
    3. Gatti, Nicola & Gilli, Mario & Marchesi, Alberto, 2020. "A characterization of quasi-perfect equilibria," Games and Economic Behavior, Elsevier, vol. 122(C), pages 240-255.
    4. Sun, Lan, 2016. "Hypothesis testing equilibrium in signaling games," Center for Mathematical Economics Working Papers 557, Center for Mathematical Economics, Bielefeld University.
    5. Yildiz, Muhamet, 2015. "Invariance to representation of information," Games and Economic Behavior, Elsevier, vol. 94(C), pages 142-156.
    6. Xiao Luo & Xuewen Qian & Yang Sun, 2021. "The algebraic geometry of perfect and sequential equilibrium: an extension," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(2), pages 579-601, March.
    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.

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

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