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Multiplicity of mixed equilibria in mechanisms: A unified approach to exact and approximate implementation

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  • Serrano, Roberto
  • Vohra, Rajiv
Abstract
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.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:mateco:v:46:y:2010:i:5:p:775-785
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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. , & ,, 2012. "Mechanism design and communication networks," Theoretical Economics, Econometric Society, vol. 7(3), September.
    3. Mezzetti, Claudio & Renou, Ludovic, 2012. "Implementation in mixed Nash equilibrium," Journal of Economic Theory, Elsevier, vol. 147(6), pages 2357-2375.
    4. Takashi Kunimoto & Rene Saran & Roberto Serrano, 2020. "Interim Rationalizable Implementation of Functions," Working Papers 2020-23, Brown University, Department of Economics.
    5. Bergemann, Dirk & Morris, Stephen & Takahashi, Satoru, 2017. "Interdependent preferences and strategic distinguishability," Journal of Economic Theory, Elsevier, vol. 168(C), pages 329-371.
    6. 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).
    7. Eccles, Peter & Wegner, Nora, 2016. "Robustness of subgame perfect implementation," Bank of England working papers 601, Bank of England.
    8. Peralta, Esteban, 2019. "Bayesian implementation with verifiable information," Games and Economic Behavior, Elsevier, vol. 116(C), pages 65-72.
    9. Kunimoto, Takashi & Serrano, Roberto, 2011. "A new necessary condition for implementation in iteratively undominated strategies," Journal of Economic Theory, Elsevier, vol. 146(6), pages 2583-2595.
    10. Kym Pram, 2020. "Weak implementation," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 69(3), pages 569-594, April.
    11. Ashraf-Ball, Hezlin & Oswald, Andrew J. & Oswald, James I., 2009. "Hydrogen Transport and the Spatial Requirements of Renewable Energy," The Warwick Economics Research Paper Series (TWERPS) 903, University of Warwick, Department of Economics.
    12. Chen, Yi-Chun & Kunimoto, Takashi & Sun, Yifei & Xiong, Siyang, 2022. "Maskin meets Abreu and Matsushima," Theoretical Economics, Econometric Society, vol. 17(4), November.
    13. Soumen Banerjee & Yi-Chun Chen, 2022. "Implementation with Uncertain Evidence," Papers 2209.10741, arXiv.org.

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    More about this item

    Keywords

    Exact implementation Approximate implementation Incomplete information Incentive compatibility Monotonicity;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D78 - Microeconomics - - Analysis of Collective Decision-Making - - - Positive Analysis of Policy Formulation and Implementation
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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