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

Brown University Economics Working Paper Series

23 Pages Posted: 21 Sep 2009

See all articles by Roberto Serrano

Roberto Serrano

Brown University

Rajiv Vohra

Brown University - Department of Economics

Date Written: September 21, 2009

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.

Keywords: Exact implementation, approximate implementation, incomplete information

JEL Classification: C72, D78, D82

Suggested Citation

Serrano, Roberto and Vohra, Rajiv, Multiplicity of Mixed Equilibria in Mechanisms: A Unified Approach to Exact and Approximate Implementation (September 21, 2009). Brown University Economics Working Paper Series, Available at SSRN: https://ssrn.com/abstract=1476374 or http://dx.doi.org/10.2139/ssrn.1476374

Roberto Serrano (Contact Author)

Brown University ( email )

64 Waterman Street
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Rajiv Vohra

Brown University - Department of Economics ( email )

Box B
Providence, RI 02912
United States
401-863-3030 (Phone)
401-863-1970 (Fax)

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