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The Algebraic Geometry of Perfect and Sequential Equilibrium

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Author Info
Lawrence E. Blume (Cornell University)
William R. Zame (The Johns Hopkins University and UCLA)

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Abstract

Two of the most important refinements of the Nash equilibrium concept for extensive form games with perfect recall are Selten's (1975) {\it perfect equilibrium\/} and Kreps and Wilson's (1982) more inclusive {\it sequential equilibrium\/}. These two equilibrium refinements are motivated in very different ways. Nonetheless, as Kreps and Wilson (1982, Section 7) point out, the two concepts lead to similar prescriptions for equilibrium play. For each particular game form, every perfect equilibrium is sequential. Moreover, for almost all assignments of payoffs to outcomes, almost all sequential equilibrium strategy profiles are perfect equilibrium profiles, and all sequential equilibrium outcomes are perfect equilibrium outcomes. \par We establish a stronger result: For almost all assignments of payoffs to outcomes, the sets of sequential and perfect equilibrium strategy profiles are identical. In other words, for almost all games each strategy profile which can be supported by beliefs satisfying the rationality requirement of sequential equilibrium can actually be supported by beliefs satisfying the stronger rationality requirement of perfect equilibrium. \par We obtain this result by exploiting the algebraic/geometric structure of these equilibrium correspondences, following from the fact that they are {\em semi-algebraic sets\/}; i.e., they are defined by finite systems of polynomial inequalities. That the perfect and sequential

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Paper provided by EconWPA in its series Game Theory and Information with number 9309001.

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Length: 16 pages
Date of creation: 30 Sep 1993
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Handle: RePEc:wpa:wuwpga:9309001

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Find related papers by JEL classification:
C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
D8 - Microeconomics - - Information, Knowledge, and Uncertainty

References listed on IDEAS
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  1. Leo K. Simon., 1987. "Basic Timing Games," Economics Working Papers 8745, University of California at Berkeley.
  2. McLennan, Andrew, 1989. "Consistent Conditional Systems in Noncooperative Game Theory," International Journal of Game Theory, Springer, vol. 18(2), pages 141-74.
  3. Roger B. Myerson, 1986. "Axiomatic Foundations of Bayesian Decision Theory," Discussion Papers 671, Northwestern University, Center for Mathematical Studies in Economics and Management Science. [Downloadable!]
  4. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Choice under Uncertainty," Econometrica, Econometric Society, vol. 59(1), pages 61-79, January. [Downloadable!] (restricted)
  5. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-37, September. [Downloadable!] (restricted)
  6. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January. [Downloadable!] (restricted)
  7. Kreps, David M & Wilson, Robert, 1982. "Sequential Equilibria," Econometrica, Econometric Society, vol. 50(4), pages 863-94, July. [Downloadable!] (restricted)
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  1. Carlos Pimienta, 2007. "Generic Determinacy of Nash Equilibrium in Network Formation Games," Discussion Papers 2007-31, School of Economics, The University of New South Wales. [Downloadable!]
  2. J. Carlos Gonzalez-Pimienta & Cristian M. Litan, 2005. "On The Equivalence Between Subgame Perfection And Sequentiality," Economics Working Papers we052616, Universidad Carlos III, Departamento de Economía. [Downloadable!]
  3. Borm, P. & Vermeulen, D. & Voorneveld, M., 1998. "The structure of the set of equilibria for two person multicriteria games," Discussion Paper 75, Tilburg University, Center for Economic Research. [Downloadable!]
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  4. Carlos Pimienta & Cristian Litan, 2008. "Conditions for equivalence between sequentiality and subgame perfection," Economic Theory, Springer, vol. 35(3), pages 539-553, June. [Downloadable!] (restricted)
  5. Eddie Dekel & Drew Fudenberg & David K. Levine, 1999. "Payoff Information and Self-Confirming Equilibrium," Levine's Working Paper Archive 172, UCLA Department of Economics. [Downloadable!]
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  6. Mark Voorneveld, 2006. "Probabilistic Choice in Games: Properties of Rosenthal’s t-Solutions," International Journal of Game Theory, Springer, vol. 34(1), pages 105-121, April. [Downloadable!] (restricted)
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  7. Predtetchinski,Arkadi, 2004. "A General Structure Theorem for the Nash Equilibrium Correspondence," Research Memoranda 023, Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization. [Downloadable!]
  8. John Hillas & Elon Kohlberg, 1996. "Foundations of Strategic Equilibrium," Game Theory and Information 9606002, EconWPA, revised 18 Sep 1996. [Downloadable!]
  9. Fabrizio Germano, 2006. "On some geometry and equivalence classes of normal form games," International Journal of Game Theory, Springer, vol. 34(4), pages 561-581, November. [Downloadable!] (restricted)
  10. Francesco De Sinopoli & Giovanna Iannantuoni, 2003. "On the Generic Strategic Stability of Nash Equilibria if Voting is Costly," CEIS Research Paper 41, Tor Vergata University, CEIS. [Downloadable!]
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  11. Vermeulen,Dries & Jansen,Mathijs, 2004. "On the computation of stable sets for bimatrix games," Research Memoranda 020, Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization. [Downloadable!]
  12. Fabrizio Germano & Gábor Lugosi, 2004. "Global Nash Convergence of Foster and Young's Regret Testing," Economics Working Papers 788, Department of Economics and Business, Universitat Pompeu Fabra. [Downloadable!]
  13. Raymond Wladimir & Mohnen Pierre & Palm Franz & Schim van der Loeff Sybrand, 2006. "Persistence of Innovation in Dutch Manufacturing: Is it Spurious?," Research Memoranda 009, Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization. [Downloadable!]
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  14. Carlos Pimienta, 2007. "Generic Finiteness of Outcome Distributions for Two Person Game Forms with Three Outcomes," Discussion Papers 2007-20, School of Economics, The University of New South Wales. [Downloadable!]
  15. Felix Kuber & Karl Schmedders, 2007. "Competitive Equilibria in Semi-Algebraic Economies," PIER Working Paper Archive 07-013, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania. [Downloadable!]
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