Some Limitations of Virtual Bayesian Implementation

Ž Ž .. AS IS WELL KNOWN, Maskin monotonicity Maskin 1977 is a necessary condition for a social choice correspondence to be Nash implementable. In economic environments with at least three agents and a private good, this condition is also sufficient for Nash Ž . Ž . implementation. Remarkably, Abreu and Sen 1991 and Matsushima 1988 showed that this condition can be entirely dispensed with if the following modifications are introduced: first, random allocations are permitted; and second, the notion of implementation is weakened to virtual Nash implementation, requiring the implementation of a social choice function that is arbitrarily close to the given function. While Nash implementation pertains to environments with complete information, there is also an extensive literature on Bayesian Nash implementation in environments Ž . with incomplete information; see, for example, Postlewaite and Schmeidler 1986 , Ž . Ž . Palfrey and Srivastava 1987, 1989 , Mookherjee and Reichelstein 1990 , and Jackson Ž . Ž 1991 . It has been shown that, apart from incentive compatibility which is clearly a . necessary condition for implementation in any solution concept , a Bayesian implementable social choice set must also satisfy Bayesian monotonicity a suitable analog of Maskin monotonicity. In economic environments with at least three agents, these Ž . conditions are also sufficient; see, for example, Jackson 1991 . Bayesian monotonicity is Ž an involved and sometimes quite strong condition as will be apparent from Example 1 . below . It is therefore natural to examine whether the virtual approach can help dispense with it, or replace it with weaker conditions. Besides incentive compatibility, two sufficient conditions for virtual Bayesian impleŽ . mentation have been identified in the literature. Abreu and Matsushima 1992b , in analyzing virtual implementation in iteratively undominated strategies, introduce a meaŽ . surability condition referred to as A-M measurability in the sequel , which they show is necessary for implementation in their solution concept. Further, they show that under weak domain restrictions this is sufficient for virtual implementation in iteratively undominated strategies and, a fortiori, in the weaker notion of implementation in Bayesian Nash equilibrium. They also show that this condition is necessary for virtual Bayesian Nash implementation if one insists on ‘regular game forms’, which rule out unattractive features such as integer games, and is in this sense close to being necessary. Ž . On the other hand, Duggan 1997 introduces a condition termed incentive consistency, which he argues is weak under standard topological and informational assumptions. Our aim is to clarify the strength of these sufficient conditions. We do so with the help of two examples.