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Implementation in Mixed Nash Equilibrium

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  • Claudio Mezzetti
  • Ludovic Renou

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Abstract

A mechanism implements a social choice correspondence f in mixed Nash equilibrium if at any preference profile, the set of all pure and mixed Nash equilibrium outcomes coincides with the set of f-optimal alternatives at that preference profile. This definition generalizes Maskin’s definition of Nash implementation in that it does not require each optimal alternative to be the outcome of a pure Nash equilibrium. We show that the condition of weak set-monotonicity, a weakening of Maskin’s monotonicity, is necessary for implementation. We provide sufficient conditions for implementation and show that important social choice correspondences that are not Maskin monotonic can be implemented in mixed Nash equilibrium.

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Bibliographic Info

Paper provided by Department of Economics, University of Leicester in its series Discussion Papers in Economics with number 09/10.

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Date of creation: Apr 2009
Date of revision: Jan 2010
Handle: RePEc:lec:leecon:09/10

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Keywords: implementation; Maskin monotonicity; pure and mixed Nash equilibrium; weak set-monotonicity; social choice correspondence;

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References

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  1. Dirk Bergemann & Stephen Morris & Olivier Tercieux, 2010. "Rationalizable Implementation," Cowles Foundation Discussion Papers 1697R, Cowles Foundation for Research in Economics, Yale University.
  2. Abreu, D. & Matsushima, H., 1991. "Virtual Implementation in Iteratively Undominated Strategies: Incomplete Information," Working Papers e-91-2, Hoover Institution, Stanford University.
  3. Sjostrom Tomas, 1994. "Implementation in Undominated Nash Equilibria without Integer Games," Games and Economic Behavior, Elsevier, vol. 6(3), pages 502-511, May.
  4. Eichberger, Jurgen & Kelsey, David, 2000. "Non-Additive Beliefs and Strategic Equilibria," Games and Economic Behavior, Elsevier, vol. 30(2), pages 183-215, February.
  5. Saijo, Tatsuyoshi, 1987. "On constant maskin monotonic social choice functions," Journal of Economic Theory, Elsevier, vol. 42(2), pages 382-386, August.
  6. Moore, John & Repullo, Rafael, 1990. "Nash Implementation: A Full Characterization," Econometrica, Econometric Society, vol. 58(5), pages 1083-99, September.
  7. Benoît, Jean-Pierre & Ok, Efe A., 2008. "Nash implementation without no-veto power," Games and Economic Behavior, Elsevier, vol. 64(1), pages 51-67, September.
  8. BOCHET, Olivier, 2005. "Nash implementation with lottery mechanisms," CORE Discussion Papers 2005072, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  9. Roberto Serrano & Antonio Cabrales, 2007. "Implemetation In Adaptive Better-Response Dynamics," Working Papers wp2007_0708, CEMFI.
  10. Barbera, S & Bogomolnaia, A & van der Stel, H, 1996. "Strategy-Proof Probabilistic Rules for Expected Utility Maximizers," UFAE and IAE Working Papers 330.96, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
  11. Dutta, Bhaskar & Sen, Arunava, 1991. "A Necessary and Sufficient Condition for Two-Person Nash Implementation," Review of Economic Studies, Wiley Blackwell, vol. 58(1), pages 121-28, January.
  12. Olivier, BOCHET & François, MANIQUET, 2006. "Virtual Nash implementation with admissible support," Discussion Papers (ECON - Département des Sciences Economiques) 2006043, Université catholique de Louvain, Département des Sciences Economiques.
  13. Maskin, Eric & Sjostrom, Tomas, 2002. "Implementation theory," Handbook of Social Choice and Welfare, in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 1, chapter 5, pages 237-288 Elsevier.
  14. Hannu Vartiainen, 2007. "Subgame perfect implementation of voting rules via randomized mechanisms," Social Choice and Welfare, Springer, vol. 29(3), pages 353-367, October.
  15. Roberto Serrano & Rajiv Vohra, 2009. "Multiplicity of mixed equilibria in mechanisms: A unified approach to exact and approximate implementation," Working Papers 2009-08, Instituto Madrileño de Estudios Avanzados (IMDEA) Ciencias Sociales.
  16. Abreu, D. & Sen, A., 1989. "Virtual Implementation In Nash Equilibrium," Working Papers e-90-13, Hoover Institution, Stanford University.
  17. Kim-Sau Chung & Jeffrey C. Ely, 2001. "Implementation with Near-Complete Information," Discussion Papers 1332, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  18. Matsushima, Hitoshi, 1988. "A new approach to the implementation problem," Journal of Economic Theory, Elsevier, vol. 45(1), pages 128-144, June.
  19. Palfrey, Thomas R & Srivastava, Sanjay, 1991. "Nash Implementation Using Undominated Strategies," Econometrica, Econometric Society, vol. 59(2), pages 479-501, March.
  20. Danilov, Vladimir, 1992. "Implementation via Nash Equilibria," Econometrica, Econometric Society, vol. 60(1), pages 43-56, January.
  21. Maskin, Eric, 1999. "Nash Equilibrium and Welfare Optimality," Review of Economic Studies, Wiley Blackwell, vol. 66(1), pages 23-38, January.
  22. Marion Oury & Olivier Tercieux, 2009. "Continuous Implementation," Economics Working Papers 0090, Institute for Advanced Study, School of Social Science.
  23. Aumann, Robert & Brandenburger, Adam, 1995. "Epistemic Conditions for Nash Equilibrium," Econometrica, Econometric Society, vol. 63(5), pages 1161-80, September.
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Citations

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Cited by:
  1. Dirk Bergemann & Stephen Morris, 2009. "Rationalizable Implementation," Cowles Foundation Discussion Papers 1697, Cowles Foundation for Research in Economics, Yale University.
  2. Serrano, Roberto & Vohra, Rajiv, 2010. "Multiplicity of mixed equilibria in mechanisms: A unified approach to exact and approximate implementation," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 775-785, September.
  3. Ludovic Renou & Karl H. Schlag, 2009. "Implementation in Minimax Regret Equilibrium," Discussion Papers in Economics 09/24, Department of Economics, University of Leicester.

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