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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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  • Herings, P. Jean-Jacques & van den Elzen, Antoon, 2002. "Computation of the Nash Equilibrium Selected by the Tracing Procedure in N-Person Games," Games and Economic Behavior, Elsevier, vol. 38(1), pages 89-117, January.
  • Handle: RePEc:eee:gamebe:v:38:y:2002:i:1:p:89-117
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    1. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702.
    2. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384, January.
    3. 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.
    4. van den Elzen, A.H., 1996. "Constructive Application of the Linear Tracing Procedure to Polymatrix Games," Research Memorandum 738, Tilburg University, School of Economics and Management.
    5. Talman, A.J.J. & van den Elzen, A.H., 1991. "A procedure for finding Nash equilibria in bi-matrix games," Other publications TiSEM 14df3398-1521-43ad-8803-a, Tilburg University, School of Economics and Management.
    6. B. Curtis Eaves, 1971. "The Linear Complementarity Problem," Management Science, INFORMS, vol. 17(9), pages 612-634, May.
    7. Jean-Jacques Herings & Dolf Talman & Zaifu Yang, 1996. "The Computation of a Continuum of Constrained Equilibria," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 675-696, August.
    8. Talman, A.J.J. & van der Laan, G., 1980. "A new subdivision for computing fixed points with a homotopy algorithm," Other publications TiSEM d702630e-5e0d-4c31-bd1e-1, Tilburg University, School of Economics and Management.
    9. Talman, A.J.J. & Doup, T.M., 1987. "A continuous deformation algorithm on the product space of unit simplices," Other publications TiSEM 0f7c777f-9ae5-4218-b3bd-2, Tilburg University, School of Economics and Management.
    10. G. van der Laan & A. J. J. Talman, 1982. "On the Computation of Fixed Points in the Product Space of Unit Simplices and an Application to Noncooperative N Person Games," Mathematics of Operations Research, INFORMS, vol. 7(1), pages 1-13, February.
    11. 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. 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).
    2. 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.
    3. 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.
    4. Theodore L. Turocy, 2002. "A Dynamic Homotopy Interpretation of Quantal Response Equilibrium Correspondences," Game Theory and Information 0212001, EconWPA, revised 16 Oct 2003.
    5. Haake, Claus-Jochen & Su, Francis Edward, 2011. "A simplicial algorithm approach to Nash equilibria in concave games," Center for Mathematical Economics Working Papers 382, Center for Mathematical Economics, Bielefeld University.
    6. P. Herings & Ronald Peeters, 2010. "Homotopy methods to compute equilibria in game theory," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 119-156, January.
    7. 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.
    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. Yin Chen & Chuangyin Dang, 2016. "A reformulation-based smooth path-following method for computing Nash equilibria," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(2), pages 231-246, October.
    10. 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.

    More about this item

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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