Bayesian Implementation

The subject of this paper is the decentralization of decision making when agents have
information which is incomplete and possibly exclusive. The first theorem states that in
economic environments with three or more individuals, there exists a mechanism whose
Bayesian equilibria coincide with a desired collection of social choice functions if and only
if closure, incentive compatibility, and Bayesian monotonicity conditions are satisfied.
With regards to the previous literature, this closes the gap between necessary and
sufficient conditions by using a slightly stronger definition of Bayesian monotonicity, and
extends the definition of economic environments to permit externalities. The second
theorem extends the analysis to noneconomic environments. It states that closure,
incentive compatibility, and a combination of monotonicity and no-veto conditions, are
sufficient for implementation, when there are at least three individuals. An example
shows that in a Bayesian setting, the second theorem does not hold for separate
monotonicity and no-veto style conditions.
KEYWORDS:Implementation, decentralization, incomplete information, Bayesian equilibrium.
1. INTRODUCTION
THE DISTRIBUTION OF INFORMATION among the members of a society plays a
critical role in determining their actions. This has important implications for the
decentralization of collective decision making. In particular, the structure of
information in a society is a factor in identifying the collections of social choice
rules (functions from states to allocations) which can be decentralized. Ultimately
we wish to know which collections of social choice rules can be decentralized
given various informational structures.
With this issue in mind, environments with incomplete information can be
separated into two categories as defined by Postlewaite and Schmeidler (1986).
The first is comprised of the environments in which information is nonexclusive:
each agent's information is redundant given the collective information of the
other agents. The second is comprised of the environments in which agents may
possess exclusive information. Blume and Easely (1988) show that this is an
important distinction by showing that the Walrasian allocation correspondence
is only generally implementable in environments with nonexclusive information.
The intuition behind this result is that if information is nonexclusive, it is
possible to have each set of N - 1 agents (where N is the total number of
agents) police the behavior of the remaining agent, and thus incentive compatibility
constraints need not apply. (This construction is similar to a "tweed ring."