Implementable allocation rules

Abstract

Let A be a finite set of goods. R is the set of all valuation functions on A, that is the set of all real valued functions defined on A. The value of a for a buyer with valuation v is thus va. It is convenient to represent each good a by its associated unit vector e ∈ R, where ea = 1 and eb = 0 for every b 6= a. Let Z(A) be the set of all sub-probability vectors z ∈ R. That is, Z(A) = {z ∈ R| z ≥ 0 ∀a, ∑ a∈A z a ≤ 1}. Let D ⊆ R, and let f : D → Z(A). We think of D as the set of all possible valuations of a given buyer with quasi-linear utility function, and f is interpreted as a randomized allocation rule in some direct mechanism (D, f, c), where c : D → R; If a buyer with valuation v declares w she receives a with probability f(w), and therefore she evaluates f(w) by the inner product, 〈v, f(w)〉 = ∑ a∈A vaf (w) minus c(w). If ∑ a∈A f (w) < 1, there is a positive probability that the buyer does not receive any good. In such case the utility of the outside option is assumed to equal zero. A randomized allocation rule satisfying f(v) ∈ {e|a ∈ A} for every v ∈ D is called in this manuscript a pure allocation rule. A randomized allocation rule f is finite-valued if its range {f(v)|v ∈ D} is a finite set. We say that a randomized allocation rule f is implementable if there exists a function c : D → R such that truth telling is a dominant strategy in the direct mechanism (D, f, c). That is,