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

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

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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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
Contact details of provider: Web page: http://www.elsevier.com/locate/inca/622836

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  1. 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, June.
  2. van den Elzen, A.H. & 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.
  3. Talman, A.J.J. & van der Laan, G., 1982. "On the computation of fixed points on the product space of unit simplices and an application to noncooperative N-person games," Other publications TiSEM ba74b902-87c8-43d5-8471-6, Tilburg University, School of Economics and Management.
  4. 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.
  5. 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.
  6. B. Curtis Eaves, 1971. "The Linear Complementarity Problem," Management Science, INFORMS, vol. 17(9), pages 612-634, May.
  7. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1996. "The computation of a continuum of constrained equilibria," Other publications TiSEM ba4c6d83-befb-4a25-8477-c, Tilburg University, School of Economics and Management.
  8. 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.
  9. 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.
  10. repec:cup:cbooks:9780521265140 is not listed on IDEAS
  11. 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.
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