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

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  • Herings, P.J.J.
  • Elzen, A.H. van den

    (Tilburg University, Center for Economic Research)

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

Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 1998-04.

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Date of creation: 1998
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Handle: RePEc:dgr:kubcen:199804

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Web page: http://center.uvt.nl

Related research

Keywords: Computation of equilibria; Non-cooperative game theory; Tracing procedure;

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References

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  1. Van Den Elzen,A. & Talman,D., 1995. "An Algorithmic Approach Towards the Tracing Procedure of Harsanyi and Selten," Papers 95111, Tilburg - Center for Economic Research.
  2. B. Curtis Eaves, 1971. "The Linear Complementarity Problem," Management Science, INFORMS, vol. 17(9), pages 612-634, May.
  3. Elzen, A.H. van den & Talman, A.J.J., 1988. "A procedure for finding Nash equilibria in bi-matrix games," Research Memorandum 334, Tilburg University, Faculty of Economics and Business Administration.
  4. 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.
  5. 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.
  6. 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.
  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. Robert Wilson, 2010. "Computing Equilibria of n-person Games," Levine's Working Paper Archive 402, David K. Levine.
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Citations

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Cited by:
  1. P. Herings & Ronald Peeters, 2010. "Homotopy methods to compute equilibria in game theory," Economic Theory, Springer, vol. 42(1), pages 119-156, January.
  2. Herings, P. Jean-Jacques & Peeters, Ronald J. A. P., 2004. "Stationary equilibria in stochastic games: structure, selection, and computation," Journal of Economic Theory, Elsevier, vol. 118(1), pages 32-60, September.
  3. 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.
  4. 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.
  5. 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).
    • Klaus Abbink & Jordi Brandts, 0000. "24," Working Papers 62, Barcelona Graduate School of Economics.
    • Jordi Brandts & Klaus Abbink, 2004. "24," Levine's Bibliography 122247000000000073, UCLA Department of Economics.
  6. Theodore L. Turocy, 2002. "A Dynamic Homotopy Interpretation of Quantal Response Equilibrium Correspondences," Game Theory and Information 0212001, EconWPA, revised 16 Oct 2003.
  7. 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.
  8. 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).
  9. 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.

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