Better mechanisms for combinatorial auctions via maximal-in-range algorithms?

Abstract

Algorithmic Mechanism Design attempts to design algorithms that handle the strategic behavior of selfish players. Many of the problems considered in the field involve allocation of resources to players, and the paradigmatic abstraction is that of combinatorial auctions. In a combinatorial auction we have n bidders and m items. Each bidder i has a valuation function vi that gives some non-negative value to each possible subset of the items. We assume that the valuations are monotone, and that for each vi we have that vi(;) = 0. In this note our goal is to find a partition of the items S1,...,Sn such that the total social welfare, ivi(Si), is maximized. Similarly to most recent work in algorithmic mechanism design, our goal is to design truthful mechanisms. I.e., mechanisms where the dominant strategy of each bidder is to reveal his true valuation. We require our algorithms to run in time that is polynomial in n and m, the natural parameters of the problem. From a purely computational point of view, there exists an O( p m)-approximation algorithm for this problem. This is the best ratio that can be obtained in polynomial time. See [Blumrosen and Nisan 2007] for a recent survey. From a game theoretic point of view, the classic VCG mechanism can be used to allocate the items in a truthful manner. However, the VCG mechanism involves finding an optimal solution ‐ a computationally unfeasible task. One of the most important questions in algorithmic mechanism design is to determine the best approximation ratio for this problem that is achievable by polynomial time truthful mechanisms 1 . We suggest tackling this question by using VCG-based mechanisms. Recall that in the VCG payment scheme we obtain an optimal solution (O1,...,On), and allocate accordingly. Each bidder i is being paid j6ivi(Oi). Thus, the utility of each bidder i equals to the value of the optimal solution: vi(Oi)+ j6ivi(Oi). It is not hard to see that truthfulness is a dominant strategy for each bidder. What about the running time? At a first glance, it looks that VCG does not help much in constructing polynomial-time mechanisms, as we have already mentioned that VCG requires finding the optimal solution, and that finding the optimal solution