Bayesian Implementation

In this chapter we will be concerned with the theory of resource allocation under uncertainty. We will assume that uncertainty only affects the preferences of the members of the society but not their endowments or the productive capabilities that determine the set of feasible allocations. In section 7.2 we will present the main concepts under which the theory will be built later on, namely the notions of a type and a state. We will see how the concept of an allocation has to be redefined in our new setting. Section 7.3 reviews the main game-theoretical concepts that will be used in this chapter. We will see that the notion of strategy must also be revised. With this new notion in hand, we define our main equilibrium concept in this chapter, namely that of a Bayesian equilibrium. We also define what we mean by Bayesian implementation. Section 7.4 studies necessary and sufficient conditions for Bayesian implementation. We will see that a form of the revelation principle holds in our framework: Bayesian implementation implies that truthful revelation is a Bayesian equilibrium (Proposition 1). The latter property is known as Bayesian incentive compatibility. Proposition 2 states that a form of monotonicity, known as Bayesian monotonicity, is also necessary for Bayesian implementation. Proposition 3 shows that, in the case of exchange economies, Bayesian incentive compatibility and Bayesian monotonicity are also sufficient for Bayesian implementation. The last section studies a case in which Bayesian monotonicity can be replaced by Maskin monotonicity because of the special structure of information (Proposition 4). This implies that full information (constrained) Walrasian allocations are Bayesian-implementable.