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Implementation in Mixed Nash Equilibrium

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  • Claudio Mezzetti

    (Department of Economics, Warwick University,)

  • Ludovic Renou

    (Department of Economics, University of Leicester,)

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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Paper provided by University of Warwick, Department of Economics in its series The Warwick Economics Research Paper Series (TWERPS) with number 902.

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Date of creation: 2009
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Handle: RePEc:wrk:warwec:902

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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. Maskin, Eric, 1999. "Nash Equilibrium and Welfare Optimality," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 66(1), pages 23-38, January.
  2. Bergemann, Dirk & Morris, Stephen & Tercieux, Olivier, 2011. "Rationalizable implementation," Journal of Economic Theory, Elsevier, Elsevier, vol. 146(3), pages 1253-1274, May.
  3. Matthew O. Jackson, 2001. "A crash course in implementation theory," Social Choice and Welfare, Springer, Springer, vol. 18(4), pages 655-708.
  4. Abreu, Dilip & Matsushima, Hitoshi, 1992. "A Response [Virtual Implementation in Iteratively Undominated Strategies I: Complete Information]," Econometrica, Econometric Society, Econometric Society, vol. 60(6), pages 1439-42, November.
  5. Bochet, Olivier & Maniquet, François, 2010. "Virtual Nash implementation with admissible support," Journal of Mathematical Economics, Elsevier, vol. 46(1), pages 99-108, January.
  6. Kim-Sau Chung & Jeffrey C. Ely, 2001. "Implementation with Near-Complete Information," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 1332, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  7. Antonio Cabrales & Roberto Serrano, 2007. "Implementation in adaptive better-response dynamics," Working Papers, Instituto Madrileño de Estudios Avanzados (IMDEA) Ciencias Sociales 2007-16, Instituto Madrileño de Estudios Avanzados (IMDEA) Ciencias Sociales.
  8. Benoît, Jean-Pierre & Ok, Efe A., 2008. "Nash implementation without no-veto power," Games and Economic Behavior, Elsevier, Elsevier, vol. 64(1), pages 51-67, September.
  9. Matsushima, Hitoshi, 1988. "A new approach to the implementation problem," Journal of Economic Theory, Elsevier, Elsevier, vol. 45(1), pages 128-144, June.
  10. Palfrey, Thomas R & Srivastava, Sanjay, 1991. "Nash Implementation Using Undominated Strategies," Econometrica, Econometric Society, Econometric Society, vol. 59(2), pages 479-501, March.
  11. Maskin, Eric & Sjostrom, Tomas, 2002. "Implementation theory," Handbook of Social Choice and Welfare, Elsevier, 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.
  12. BOCHET, Olivier, 2005. "Nash implementation with lottery mechanisms," CORE Discussion Papers, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) 2005072, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  13. Roberto Serrano & Rajiv Vohra, 2009. "Multiplicity of Mixed Equilibria in Mechanisms: a Unified Approach ot Exact and Approximate Implementation," Working Papers 2009-11, Brown University, Department of Economics.
  14. M. Sanver, 2006. "Nash implementing non-monotonic social choice rules by awards," Economic Theory, Springer, Springer, vol. 28(2), pages 453-460, 06.
  15. Moore, John & Repullo, Rafael, 1990. "Nash Implementation: A Full Characterization," Econometrica, Econometric Society, Econometric Society, vol. 58(5), pages 1083-99, September.
  16. Hannu Vartiainen, 2007. "Subgame perfect implementation of voting rules via randomized mechanisms," Social Choice and Welfare, Springer, Springer, vol. 29(3), pages 353-367, October.
  17. Sjostrom, T., 1991. "Implementation in Undominated Nash Equilibria without Integer Games," Papers, Stockholm - International Economic Studies 491, Stockholm - International Economic Studies.
  18. Abreu, Dilip & Matsushima, Hitoshi, 1992. "Virtual Implementation in Iteratively Undominated Strategies: Complete Information," Econometrica, Econometric Society, Econometric Society, vol. 60(5), pages 993-1008, September.
  19. Saijo, Tatsuyoshi, 1987. "On constant maskin monotonic social choice functions," Journal of Economic Theory, Elsevier, Elsevier, vol. 42(2), pages 382-386, August.
  20. Gibbard, Allan, 1977. "Manipulation of Schemes That Mix Voting with Chance," Econometrica, Econometric Society, Econometric Society, vol. 45(3), pages 665-81, April.
  21. Dutta, Bhaskar & Sen, Arunava, 1991. "A Necessary and Sufficient Condition for Two-Person Nash Implementation," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 58(1), pages 121-28, January.
  22. Danilov, Vladimir, 1992. "Implementation via Nash Equilibria," Econometrica, Econometric Society, Econometric Society, vol. 60(1), pages 43-56, January.
  23. Eichberger, Jurgen & Kelsey, David, 2000. "Non-Additive Beliefs and Strategic Equilibria," Games and Economic Behavior, Elsevier, Elsevier, vol. 30(2), pages 183-215, February.
  24. Jackson, Matthew O, 1991. "Bayesian Implementation," Econometrica, Econometric Society, Econometric Society, vol. 59(2), pages 461-77, March.
  25. Barbera, Salvador & Bogomolnaia, Anna & van der Stel, Hans, 1998. "Strategy-proof probabilistic rules for expected utility maximizers," Mathematical Social Sciences, Elsevier, Elsevier, vol. 35(2), pages 89-103, March.
  26. Abreu, Dilip & Sen, Arunava, 1991. "Virtual Implementation in Nash Equilibrium," Econometrica, Econometric Society, Econometric Society, vol. 59(4), pages 997-1021, July.
  27. Marion Oury & Olivier Tercieux, 2009. "Continuous Implementation," Economics Working Papers, Institute for Advanced Study, School of Social Science 0090, Institute for Advanced Study, School of Social Science.
  28. Aumann, Robert & Brandenburger, Adam, 1995. "Epistemic Conditions for Nash Equilibrium," Econometrica, Econometric Society, Econometric Society, vol. 63(5), pages 1161-80, September.
  29. Cabrales, Antonio & Serrano, Roberto, 2011. "Implementation in adaptive better-response dynamics: Towards a general theory of bounded rationality in mechanisms," Games and Economic Behavior, Elsevier, Elsevier, vol. 73(2), pages 360-374.
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Citations

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Cited by:
  1. Roberto Serrano & Rajiv Vohra, 2009. "Multiplicity Of Mixed Equilibria In Mechanisms: A Unified Approach To Exact And Approximate Implementation," Working Papers, CEMFI wp2009_0908, CEMFI.
  2. Bergemann, Dirk & Morris, Stephen & Tercieux, Olivier, 2011. "Rationalizable implementation," Journal of Economic Theory, Elsevier, Elsevier, vol. 146(3), pages 1253-1274, May.
  3. Ludovic Renou & Karl H. Schlag, 2009. "Implementation in Minimax Regret Equilibrium," Discussion Papers in Economics, Department of Economics, University of Leicester 09/24, Department of Economics, University of Leicester.

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