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A General Structure Theorem for the Nash Equilibrium Correspondence

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  • Predtetchinski Arkadi

    (METEOR)

Abstract

We consider n--person normal form games where the strategy set of each player is a non--empty compact convex subset of a Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in player i''s strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. We demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well--known structure theorem in Kohlberg and Mertens (On the Strategic Stability of Equilibria, Econometrica, 54, 1003--1037, 1986). It is supplemented by an extension analogous to the unknottedness theorems in Demichelis and Germano (On (Un)knots and Dynamics in Games, Games and Economic Behavior, 41, 46--60, 2002): the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.

Suggested Citation

  • Predtetchinski Arkadi, 2006. "A General Structure Theorem for the Nash Equilibrium Correspondence," Research Memorandum 010, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  • Handle: RePEc:unm:umamet:2006010
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    1. DeMichelis, Stefano & Germano, Fabrizio, 2000. "Some consequences of the unknottedness of the Walras correspondence," Journal of Mathematical Economics, Elsevier, vol. 34(4), pages 537-545, December.
    2. Blume, Lawrence E & Zame, William R, 1994. "The Algebraic Geometry of Perfect and Sequential Equilibrium," Econometrica, Econometric Society, vol. 62(4), pages 783-794, July.
    3. Govindan, Srihari & Wilson, Robert, 2001. "Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes," Econometrica, Econometric Society, vol. 69(3), pages 765-769, May.
    4. Zhou, Yuqing, 1997. "Genericity Analysis on the Pseudo-Equilibrium Manifold," Journal of Economic Theory, Elsevier, vol. 73(1), pages 79-92, March.
    5. Stefano Demichelis & Klaus Ritzberger & Jeroen M. Swinkels, 2004. "The simple geometry of perfect information games," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(3), pages 315-338, June.
    6. Balasko, Yves, 1978. "Economic Equilibrium and Catastrophe Theory: An Introduction," Econometrica, Econometric Society, vol. 46(3), pages 557-569, May.
    7. Demichelis, Stefano & Germano, Fabrizio, 2002. "On (un)knots and dynamics in games," Games and Economic Behavior, Elsevier, vol. 41(1), pages 46-60, October.
    8. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
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    mathematical economics;

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