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The Algebraic Geometry of Perfect and Sequential Equilibrium

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  • Lawrence E. Blume

    (Cornell University)

  • William R. Zame

    (The Johns Hopkins University and UCLA)

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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Bibliographic Info

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
Date of revision:
Handle: RePEc:wpa:wuwpga:9309001

Note: paper = 16 pages (including title & abstract): LaTeX file; macros at top
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  1. KOHLBERG, Elon & MERTENS, Jean-François, . "On the strategic stability of equilibria," CORE Discussion Papers RP -716, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  2. Kreps, David M & Wilson, Robert, 1982. "Sequential Equilibria," Econometrica, Econometric Society, vol. 50(4), pages 863-94, July.
  3. Simon, Leo K., 1987. "Basic Timing Games," Department of Economics, Working Paper Series qt8kt5h29p, Department of Economics, Institute for Business and Economic Research, UC Berkeley.
  4. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Choice under Uncertainty," Econometrica, Econometric Society, vol. 59(1), pages 61-79, January.
  5. McLennan, Andrew, 1989. "Consistent Conditional Systems in Noncooperative Game Theory," International Journal of Game Theory, Springer, vol. 18(2), pages 141-74.
  6. Leo K. Simon., 1987. "Basic Timing Games," Economics Working Papers 8745, University of California at Berkeley.
  7. Roger B. Myerson, 1986. "Axiomatic Foundations of Bayesian Decision Theory," Discussion Papers 671, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  8. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January.
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