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Implementation in Mixed Nash Equilibrium

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  • Claudio Mezzetti
  • Ludovic Renou

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.

Suggested Citation

  • Claudio Mezzetti & Ludovic Renou, 2009. "Implementation in Mixed Nash Equilibrium," Discussion Papers in Economics 09/10, Division of Economics, School of Business, University of Leicester, revised Jan 2010.
    • Handle: RePEc:lec:leecon:09/10
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    References listed on IDEAS

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    1. Dirk Bergemann & Stephen Morris & Olivier Tercieux, 2012. "Rationalizable Implementation," World Scientific Book Chapters, in: Robust Mechanism Design The Role of Private Information and Higher Order Beliefs, chapter 11, pages 375-404, World Scientific Publishing Co. Pte. Ltd..
    2. Saijo, Tatsuyoshi, 1987. "On constant maskin monotonic social choice functions," Journal of Economic Theory, Elsevier, vol. 42(2), pages 382-386, August.
    3. Marion Oury & Olivier Tercieux, 2012. "Continuous Implementation," Econometrica, Econometric Society, vol. 80(4), pages 1605-1637, July.
    4. Barbera, Salvador & Bogomolnaia, Anna & van der Stel, Hans, 1998. "Strategy-proof probabilistic rules for expected utility maximizers," Mathematical Social Sciences, Elsevier, vol. 35(2), pages 89-103, March.
    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. Robert Aumann & Adam Brandenburger, 2014. "Epistemic Conditions for Nash Equilibrium," World Scientific Book Chapters, in: The Language of Game Theory Putting Epistemics into the Mathematics of Games, chapter 5, pages 113-136, World Scientific Publishing Co. Pte. Ltd..
    7. Matsushima, Hitoshi, 1988. "A new approach to the implementation problem," Journal of Economic Theory, Elsevier, vol. 45(1), pages 128-144, June.
    8. Eric Maskin, 1999. "Nash Equilibrium and Welfare Optimality," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 66(1), pages 23-38.
    9. Kim-Sau Chung & Jeffrey C. Ely, 2003. "Implementation with Near-Complete Information," Econometrica, Econometric Society, vol. 71(3), pages 857-871, May.
    10. Gibbard, Allan, 1977. "Manipulation of Schemes That Mix Voting with Chance," Econometrica, Econometric Society, vol. 45(3), pages 665-681, April.
    11. 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.
    12. Jackson, Matthew O, 1991. "Bayesian Implementation," Econometrica, Econometric Society, vol. 59(2), pages 461-477, March.
    13. Antonio Cabrales & Roberto Serrano, 2007. "Implemetation in Adaptive Better-Response Dynamics," Working Papers wp2007_0708, CEMFI.
    14. Hannu Vartiainen, 2007. "Subgame perfect implementation of voting rules via randomized mechanisms," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 29(3), pages 353-367, October.
    15. 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.
    16. Danilov, Vladimir, 1992. "Implementation via Nash Equilibria," Econometrica, Econometric Society, vol. 60(1), pages 43-56, January.
    17. Abreu, Dilip & Matsushima, Hitoshi, 1992. "A Response [Virtual Implementation in Iteratively Undominated Strategies I: Complete Information]," Econometrica, Econometric Society, vol. 60(6), pages 1439-1442, November.
    18. Abreu, Dilip & Sen, Arunava, 1991. "Virtual Implementation in Nash Equilibrium," Econometrica, Econometric Society, vol. 59(4), pages 997-1021, July.
    19. Palfrey, Thomas R & Srivastava, Sanjay, 1991. "Nash Implementation Using Undominated Strategies," Econometrica, Econometric Society, vol. 59(2), pages 479-501, March.
    20. Olivier Bochet, 2007. "Nash Implementation with Lottery Mechanisms," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(1), pages 111-125, January.
    21. Bhaskar Dutta & Arunava Sen, 1991. "A Necessary and Sufficient Condition for Two-Person Nash Implementation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 58(1), pages 121-128.
    22. Matthew O. Jackson, 2001. "A crash course in implementation theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(4), pages 655-708.
    23. M. Sanver, 2006. "Nash implementing non-monotonic social choice rules by awards," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 28(2), pages 453-460, June.
    24. Sjostrom Tomas, 1994. "Implementation in Undominated Nash Equilibria without Integer Games," Games and Economic Behavior, Elsevier, vol. 6(3), pages 502-511, May.
    25. Eichberger, Jurgen & Kelsey, David, 2000. "Non-Additive Beliefs and Strategic Equilibria," Games and Economic Behavior, Elsevier, vol. 30(2), pages 183-215, February.
    26. Moore, John & Repullo, Rafael, 1990. "Nash Implementation: A Full Characterization," Econometrica, Econometric Society, vol. 58(5), pages 1083-1099, September.
    27. 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.
    28. Roberto Serrano & Antonio Cabrales, 2007. "Implementation in Adaptive Better-Response Dynamics," Working Papers 2007-10, Brown University, Department of Economics.
    29. 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.
    30. 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, vol. 73(2), pages 360-374.
    31. 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.
    32. Abreu, Dilip & Matsushima, Hitoshi, 1992. "Virtual Implementation in Iteratively Undominated Strategies: Complete Information," Econometrica, Econometric Society, vol. 60(5), pages 993-1008, September.
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    Citations

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    Cited by:

    1. Dirk Bergemann & Stephen Morris & Olivier Tercieux, 2012. "Rationalizable Implementation," World Scientific Book Chapters, in: Robust Mechanism Design The Role of Private Information and Higher Order Beliefs, chapter 11, pages 375-404, World Scientific Publishing Co. Pte. Ltd..
    2. Laslier, Jean-François & Núñez, Matías & Remzi Sanver, M., 2021. "A solution to the two-person implementation problem," Journal of Economic Theory, Elsevier, vol. 194(C).
    3. Takashi Kunimoto & Roberto Serrano, 2019. "Rationalizable Implementation of Correspondences," Management Science, INFORMS, vol. 44(4), pages 1326-1344, November.
    4. Jain, Ritesh, 2021. "Rationalizable implementation of social choice correspondences," Games and Economic Behavior, Elsevier, vol. 127(C), pages 47-66.
    5. Takashi Kunimoto & Rene Saran & Roberto Serrano, 2020. "Interim Rationalizable Implementation of Functions," Working Papers 2020-23, Brown University, Department of Economics.
    6. Korpela, Ville & Lombardi, Michele & Vartiainen, Hannu, 2021. "Implementation in largest consistent set via rights structures," Games and Economic Behavior, Elsevier, vol. 128(C), pages 202-212.
    7. Chen, Yi-Chun & Kunimoto, Takashi & Sun, Yifei & Xiong, Siyang, 2022. "Maskin meets Abreu and Matsushima," Theoretical Economics, Econometric Society, vol. 17(4), November.
    8. Korpela, Ville & Lombardi, Michele & Vartiainen, Hannu, 2019. "Implementation with foresighted agents," MPRA Paper 102496, University Library of Munich, Germany.
    9. Ritesh Jain & Michele Lombardi, 2019. "Virtual implementation by bounded mechanisms: Complete information," IEAS Working Paper : academic research 19-A001, Institute of Economics, Academia Sinica, Taipei, Taiwan.
    10. Takahashi, Satoru & Tercieux, Olivier, 2020. "Robust equilibrium outcomes in sequential games under almost common certainty of payoffs," Journal of Economic Theory, Elsevier, vol. 188(C).
    11. 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.
    12. Jain, Ritesh & Lombardi, Michele, 2022. "Continuous virtual implementation: Complete information," Journal of Mathematical Economics, Elsevier, vol. 99(C).
    13. Renou, Ludovic & Schlag, Karl H., 2011. "Implementation in minimax regret equilibrium," Games and Economic Behavior, Elsevier, vol. 71(2), pages 527-533, March.
    14. Takashi Kunimoto & Roberto Serrano, 2016. "Rationalizable Implementation of Correspondences," Working Papers 2016-4, Brown University, Department of Economics.
    15. Saran, Rene, 2016. "Bounded depths of rationality and implementation with complete information," Journal of Economic Theory, Elsevier, vol. 165(C), pages 517-564.
    16. Peralta, Esteban, 2019. "Bayesian implementation with verifiable information," Games and Economic Behavior, Elsevier, vol. 116(C), pages 65-72.
    17. Siyang Xiong, 2022. "Nash implementation by stochastic mechanisms: a simple full characterization," Papers 2211.05431, arXiv.org.
    18. Jianxin Yi, 2021. "Nash implementation via mechanisms that allow for abstentions," Theory and Decision, Springer, vol. 91(2), pages 279-288, September.
    19. Mezzetti, Claudio & Renou, Ludovic, 2017. "Repeated Nash implementation," Theoretical Economics, Econometric Society, vol. 12(1), January.

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    More about this item

    Keywords

    implementation; Maskin monotonicity; pure and mixed Nash equilibrium; weak set-monotonicity; social choice correspondence;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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