A Truthful-in-Expectation Mechanism for the Generalized Assignment Problem

Abstract

We propose a truthful-in-expectation, $(1-\frac{1}{e})$-approximation mechanism for the generalized assignment auction. In such an auction, each bidder has a knapsack valuation function and bidders’ values for items are private. We present a novel convex optimization program for the auction which results in a maximal-in-distributional-range (MIDR) allocation rule. The presented program contains at least a $(1-\frac{1}{e})$ ratio of the optimal social welfare. We show how to implement the convex program in polynomial time using a fractional local search algorithm which approximates the optimal solution within an arbitrarily small error. This leads to an approximately MIDR allocation rule which in turn can be transformed to an approximately truthful-in-expectation mechanism. Our contribution has algorithmic importance, as well; it simplifies the existing optimization algorithms for the GAP while the approximation ratio is comparable to the best given approximation.