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Computation of the Nash Equilibrium Selected by the Tracing Procedure in N-Person Games

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  • Herings, P. Jean-Jacques
  • van den Elzen, Antoon

Abstract

Harsanyi and Selten (1988) have proposed a theory of equilibrium selection that selects a unique Nash equilibrium for any non-cooperative N-person game. The heart of their theory is given by the tracing procedure, a mathematical construction that adjusts arbitrary prior beliefs into equilibrium beliefs. The tracing procedure plays an important role in the definition of risk-dominance for Nash equilibria. Although the term "procedure" suggests a numerical approach, the tracing procedure itself is a non-constructive method. In this paper we propose a homotopy algorithm that generates a path of strategies. By employing lexicographic pivoting techniques it can be shown that for the entire class of non-cooperative N-person games the path converges to an approximate Nash equilibrium, even when the starting point or the game is degenerate. The outcome of the algorithm is shown to be arbitrarily close to the beliefs proposed by the tracing procedure. Therefore, the algorithm does not compute just any Nash equilibrium, but one with a sound gametheoretic underpinning. Like other homotopy algorithms, it is easily implemented on a computer. To show our results we apply methods from the theory of simplicial algorithms and algebraic geometry.

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Bibliographic Info

Article provided by Elsevier in its journal Games and Economic Behavior.

Volume (Year): 38 (2002)
Issue (Month): 1 (January)
Pages: 89-117

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Handle: RePEc:eee:gamebe:v:38:y:2002:i:1:p:89-117

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Web page: http://www.elsevier.com/locate/inca/622836

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  1. B. Curtis Eaves, 1971. "The Linear Complementarity Problem," Management Science, INFORMS, vol. 17(9), pages 612-634, May.
  2. Doup, T.M. & Talman, A.J.J., 1986. "A continuous deformation algorithm on the product space of unit simplices," Research Memorandum 219, Tilburg University, Faculty of Economics and Business Administration.
  3. Elzen, A.H. van den & Talman, A.J.J., 1995. "An algorithmic approach towards the tracing procedure of Harsanyi and Selten," Discussion Paper 1995-111, Tilburg University, Center for Economic Research.
  4. Talman, A.J.J. & Elzen , A.H. van den, 1991. "A procedure for finding Nash equilibria in bi-matrix games," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153117, Tilburg University.
  5. Herings, P.J.J. & Talman, A.J.J. & Zang, Z., 1994. "The computation of a continuum of constrained equilibria," Discussion Paper 1994-38, Tilburg University, Center for Economic Research.
  6. Robert Wilson, 2010. "Computing Equilibria of n-person Games," Levine's Working Paper Archive 402, David K. Levine.
  7. Talman, A.J.J. & Laan , G. van der, 1980. "A new subdivision for computing fixed points with a homotopy algorithm," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153017, Tilburg University.
  8. Talman, A.J.J. & Laan , G. van der, 1982. "On the computation of fixed points on the product space of unit simplices and an application to noncooperative N-person games," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153028, Tilburg University.
  9. van den Elzen, Antoon & Talman, Dolf, 1999. "An Algorithmic Approach toward the Tracing Procedure for Bi-matrix Games," Games and Economic Behavior, Elsevier, vol. 28(1), pages 130-145, July.
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Cited by:
  1. Klaus Abbink & Jordi Brandts, 0000. "24," Working Papers 62, Barcelona Graduate School of Economics.
    • Klaus Abbink & Jordi Brandts, 2002. "24," UFAE and IAE Working Papers 523.02, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
    • Jordi Brandts & Klaus Abbink, 2004. "24," Levine's Bibliography 122247000000000073, UCLA Department of Economics.
  2. Abbink, Klaus & Brandts, Jordi, 2008. "24. Pricing in Bertrand competition with increasing marginal costs," Games and Economic Behavior, Elsevier, vol. 63(1), pages 1-31, May.
  3. Theodore L. Turocy, 2002. "A Dynamic Homotopy Interpretation of Quantal Response Equilibrium Correspondences," Game Theory and Information 0212001, EconWPA, revised 16 Oct 2003.
  4. P. Herings & Ronald Peeters, 2010. "Homotopy methods to compute equilibria in game theory," Economic Theory, Springer, vol. 42(1), pages 119-156, January.
  5. Keyzer, Michiel & van Wesenbeeck, Lia, 2005. "Equilibrium selection in games: the mollifier method," Journal of Mathematical Economics, Elsevier, vol. 41(3), pages 285-301, April.
  6. Herings,P. Jean-Jacques & Peeters,Ronald J.A.P, 2000. "Stationary Equilibria in Stochastic Games: Structure, Selection, and Computation," Research Memorandum 004, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  7. Claus-Jochen Haake & Francis Edward Su, 2006. "A simplicial algorithm approach to Nash equilibria in concave games," Working Papers 382, Bielefeld University, Center for Mathematical Economics.
  8. Turocy, Theodore L., 2005. "A dynamic homotopy interpretation of the logistic quantal response equilibrium correspondence," Games and Economic Behavior, Elsevier, vol. 51(2), pages 243-263, May.
  9. Wheatley, W. Parker, 2003. "Survival And Ownership Of Internet Marketplaces For Agriculture," 2003 Annual meeting, July 27-30, Montreal, Canada 22214, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).

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