Virtual Bayesian Implementation

Allowing for incomplete information, this paper characterizes the social choice functions
that can be approximated by the cquilibrium outcomes of a mechanism: incentive
compatibility is necessary and almost sufticient for virtual Bayesian implementability. In
conjunction with a second condition, Buyesiun incentice consistency, inccntive compatibility
is also sufficient. This new condition is weak-under standard topological and informational
assumptions, it is satisfied by eiwy social choice function. Thc type sets of the
agents are taken to be arbitrary (possibly infinite) measurable spaces. An example shows
that thcre arc virtually (in fact, exactly) Bayesian implementablc social choice functions
that are not virtually implementablc in iteratively undominated strategies.
KEYWOKDS:Virtual implementation, decentralization, Bayesian equilibrium, measurability,
social choice.
1. INTRODUCTION
THE BAYESIANI~ IPLEMEXTATION literature seeks to characterize the way in
which achievable outcomes can depend on the private information of agents.
This dependence is critical for the solution of many collective choice problems:
the revenue maximizing sale in an auction depends on which bidder has the
highest valuation, the optimal provision of insurance depends on which consumers
are high or low risk, and the socially desirable outcomes of an election
depend on the preferences of the electorate. Because the nature of this
information is private, variation in outcomes must be achieved endogenouslythrough
the equilibrium behavior of the agents within a fixed institutional
framework. We model the private information of the agents with a set of
possible states of the world and define a social choicc function as a rule
associating an outcome to each state. A social choicc function is said to be
Bayesian implementable if there exists a mechanism that, at each state, yields the
outcome of the social choice function as its unique Bayesian equilibrium
outcome.