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Multiplicity of Mixed Equilibria in Mechanisms: A Unified Approach to Exact and Approximate Implementation

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We characterize full implementation of social choice sets in mixed strategy Bayesian equilibrium. Our results concern both exact and virtual mixed implementation. For exact implementation, we identify a strengthening of Bayesian monotonicity, which we refer to as mixed Bayesian monotonicity. It is shown that, in economic environments with at least three agents, mixed Bayesian implementation is equivalent to mixed Bayesian monotonicity, incentive compatibility and closure. For implementing a social choice function, the case of two-agents is also covered by these conditions and mixed Bayesian monotonicity reduces to Bayesian monotonicity. Following parallel steps, mixed virtual implementation is shown to be equivalent to mixed virtual monotonicity, incentive compatibility and closure. The key condition, mixed virtual monotonicity, is argued to be very weak. In particular, it is weaker than Abreu-Matsushima’s measurability, thereby implying that: (1) virtual implementation in mixed Bayesian equilibrium is more permissive than virtual implementation in iteratively undominated strategies, and (2) non-regular mechanisms are essential for the implementation of rules in that gap.

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Paper provided by CEMFI in its series Working Papers with number wp2009_0908.

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Date of creation: Sep 2009
Handle: RePEc:cmf:wpaper:wp2009_0908
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  1. Palfrey, Thomas R & Srivastava, Sanjay, 1989. "Implementation with Incomplete Information in Exchange Economies," Econometrica, Econometric Society, vol. 57(1), pages 115-134, January.
  2. Amengual, Dante & Sentana, Enrique, 2010. "A comparison of mean-variance efficiency tests," Journal of Econometrics, Elsevier, vol. 154(1), pages 16-34, January.
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  4. Bentolila, Samuel & Dolado, Juan J. & Jimeno, Juan F, 2007. "Does Immigration Affect the Phillips Curve? Some Evidence for Spain," CEPR Discussion Papers 6604, C.E.P.R. Discussion Papers.
  5. Gagliarducci, Stefano & Nannicini, Tommaso & Naticchioni, Paolo, 2008. "Electoral Rules and Politicians’ Behavior: A Micro Test," IZA Discussion Papers 3348, Institute for the Study of Labor (IZA).
  6. repec:cmf:wpaper:wp2008_0809 is not listed on IDEAS
  7. Mencía, Javier & Sentana, Enrique, 2009. "Multivariate location-scale mixtures of normals and mean-variance-skewness portfolio allocation," Journal of Econometrics, Elsevier, vol. 153(2), pages 105-121, December.
  8. Hospido, Laura, 2010. "Modelling Heterogeneity and Dynamics in the Volatility of Individual Wages," IZA Discussion Papers 4712, Institute for the Study of Labor (IZA).
  9. Roberto Serrano & Rajiv Vohra, 2002. "A Characterization of Virtual Bayesian Implementation," Working Papers 2002-11, Brown University, Department of Economics.
  10. Rafael Repullo & Jesús Saurina & Carlos Trucharte, 2009. "Mitigating the Procyclicality of Basel II," Working Papers wp2009_0903, CEMFI.
  11. Robert J. Aumann & Roberto Serrano, 2006. "An Economic Index of Riskiness," Levine's Bibliography 321307000000000585, UCLA Department of Economics.
  12. Matsushima Hitoshi, 1993. "Bayesian Monotonicity with Side Payments," Journal of Economic Theory, Elsevier, vol. 59(1), pages 107-121, February.
  13. Claudio Mezzetti & Ludovic Renou, 2009. "Implementation in Mixed Nash Equilibrium," Discussion Papers in Economics 09/10, Department of Economics, University of Leicester, revised Jan 2010.
  14. Roberto Serrano, 2009. "On Watson’s Non-Forcing Contracts and Renegotiation," Working Papers wp2009_0907, CEMFI.
  15. 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.
  16. Enrique Moral-Benito, 2010. "Determinants of economic growth: A Bayesian panel data approach," Working Papers 1031, Banco de España;Working Papers Homepage.
  17. Cabrales, Antonio & Serrano, Roberto, 2007. "Implementation in adaptive better-response dynamics," UC3M Working papers. Economics we075731, Universidad Carlos III de Madrid. Departamento de Economía.
  18. Georgy Artemov & Takashi Kunimoto & Roberto Serrano, 2007. "Robust virtual implementation with incomplete information: Towards a reinterpretation of the Wilson doctrine," Working Papers 2007-14, Instituto Madrileño de Estudios Avanzados (IMDEA) Ciencias Sociales.
  19. Palfrey, Thomas R. & Srivastava, Sanjay., 1986. "On Bayesian Implementable Allocations," Working Papers 624, California Institute of Technology, Division of the Humanities and Social Sciences.
  20. Aguirregabiria, Victor & Mira, Pedro, 2010. "Dynamic discrete choice structural models: A survey," Journal of Econometrics, Elsevier, vol. 156(1), pages 38-67, May.
  21. Matsushima, Hitoshi, 1988. "A new approach to the implementation problem," Journal of Economic Theory, Elsevier, vol. 45(1), pages 128-144, June.
  22. Matthew O. Jackson, 1992. "Implementation in Undominated Strategies: A Look at Bounded Mechanisms," Review of Economic Studies, Oxford University Press, vol. 59(4), pages 757-775.
  23. Chakravorti, Bhaskar, 1992. "Efficiency and Mechanisms with No Regret," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 33(1), pages 45-59, February.
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  25. Dilip Mookherjee & Stefan Reichelstein, 1990. "Implementation via Augmented Revelation Mechanisms," Review of Economic Studies, Oxford University Press, vol. 57(3), pages 453-475.
  26. Eric Maskin, 1998. "Nash Equilibrium and Welfare Optimality," Harvard Institute of Economic Research Working Papers 1829, Harvard - Institute of Economic Research.
  27. Postlewaite, Andrew & Schmeidler, David, 1986. "Implementation in differential information economies," Journal of Economic Theory, Elsevier, vol. 39(1), pages 14-33, June.
  28. repec:cmf:wpaper:wp2008_0801 is not listed on IDEAS
  29. Enrique Sentana, 2009. "The econometrics of mean-variance efficiency tests: a survey," Econometrics Journal, Royal Economic Society, vol. 12(3), pages 65-101, November.
  30. Max Bruche & Javier Suarez, 2009. "The Macroeconomics of Money Market Freezes," Working Papers wp2009_0901, CEMFI.
  31. Manuel Arellano & Lars Peter Hansen & Enrique Sentana, 2009. "Underidentification? (Resumen)," Working Papers wp2009_0905, CEMFI.
  32. 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.
  33. Bhaskar Dutta & Arunava Sen, 1994. "2-person Bayesian implementation," Review of Economic Design, Springer;Society for Economic Design, vol. 1(1), pages 41-54, December.
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  35. Jackson, Matthew O, 1991. "Bayesian Implementation," Econometrica, Econometric Society, vol. 59(2), pages 461-477, March.
  36. 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.
  37. Max Bruche, 2009. "Bankruptcy Codes, Liquidation Timing, and Debt Valuation," Working Papers wp2009_0902, CEMFI.
  38. Mendez, Ildefonso, 2008. "Promoting Permanent Employment: Lessons from Spain," MPRA Paper 7933, University Library of Munich, Germany.
  39. Bergemann, Dirk & Morris, Stephen, 2009. "Robust virtual implementation," Theoretical Economics, Econometric Society, vol. 4(1), March.
  40. Javier Mencía & Enrique Sentana, 2012. "Distributional Tests in Multivariate Dynamic Models with Normal and Student-t Innovations," The Review of Economics and Statistics, MIT Press, vol. 94(1), pages 133-152, February.
  41. Serrano, Roberto & Vohra, Rajiv, 2001. "Some Limitations of Virtual Bayesian Implementation," Econometrica, Econometric Society, vol. 69(3), pages 785-792, May.
  42. John Duggan, 1997. "Virtual Bayesian Implementation," Econometrica, Econometric Society, vol. 65(5), pages 1175-1200, September.
  43. 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.
  44. repec:cmf:wpaper:wp2007_0710 is not listed on IDEAS
  45. Abreu, Dilip & Sen, Arunava, 1991. "Virtual Implementation in Nash Equilibrium," Econometrica, Econometric Society, vol. 59(4), pages 997-1021, July.
  46. repec:cmf:wpaper:wp2007_0713 is not listed on IDEAS
  47. 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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