Multi-Unit Auctions: Beyond Roberts

arXiv:1004.1449v2 [cs.GT] 8 Nov 2010 Multi-Unit Auctions: Beyond Roberts Shahar Dobzinski Department of Computer Science Cornell Unversity shahar@cs.cornell.edu Noam Nisan ∗ School of Computer Science and Engineering Hebrew University noam@cs.huji.ac.il November 2, 2018 Abstract We exhibit incentive compatible multi-unit auctions that are not affine maximizers (i.e., are not of the VCG family) and yet approximate the social welfare to within a factor of 1 + ǫ. For the case of two-item two-bidder auctions we show that these auctions, termed Triage auctions, are the only scalable ones that give an approximation factor better than 2. “Scalable” means that the allocation does not depend on the units in which the valuations are measured. We deduce from this that any scalable computationally-efficient incentive-compatible auction for m items and n ≥2 bidders cannot approximate the social welfare to within a factor better than 2. This is in contrast to arbitrarily good approximations that can be reached under computational constraints alone, and in contrast to the fact that the optimal social welfare can be obtained under incentive constraints alone. ∗Supported by a grant from the Israeli Academy of Sciences. 1 Introduction Background The field of Algorithmic Mechanism Design [27] designs mechanisms for achieving various computa- tional goals, under the assumption of rational selfishness of the involved parties. The notions used are taken from the economic field of Mechanism Design, and a basic notion is that of incentive- compatibility– where rational players are motivated to act truthfully. For background and survey see part II of [28]. This paper will consider only the simplest and most robust notion of incentive compati- bility, that of dominant strategies in quasi-linear settings with independent private values. The typical question in the field asks for a computationally-efficient incentive compatible mechanism that imple- ments a certain type of outcome, usually the approximate optimization of some target “social” goal. There are two variants of this challenge, the first considers situations where incentive compatibility itself is hard to achieve and the computational efficiency is just an additional burden, with the prime example being approximate minimization of the makespan in scheduling problems [27]. The second variant focuses on cases where each of the two constraints of incentive compatibility and computational efficiency can be achieved separately, and the challenge is to get them simultaneously, with the prime example being approximate welfare maximization in various types of combinatorial auctions [24]. While there has been much work and some progress on these types of challenges, with particular emphasis on the problems mentioned above of combinatorial auctions (e.g., [22, 20, 3, 15, 23, 16, 13, 6]) and scheduling (e.g., [7, 21, 2]), the basic challenge remains mostly unanswered. As noted in [22], the main issue turns out to be the richness of the domain of player’s valuations, i.e., of their private information. On one extreme are single-dimensional domains where the private information of each participant is captured by a scalar (or domains very close to it, e.g., [24]). For these types of problems, incentive-compatible mechanisms are well characterized by a certain monotonicity condition and, in most cases, the challenge of reconciling incentives with computational efficiency has been met [24, 1, 5, 9, 8]. On the other extreme are problems which are “fully dimensional” (or close to fully dimensional, e.g., [29, 17]) where there is no structure on valuations, in which case a key theorem of Roberts [30] characterizes incentive compatible mechanisms as “affine maximizers” “on a sub-range” – simple generalizations of the VCG mechanism. While such affine maximizers on a sub-range are not completely powerless in polynomial time, in most cases this characterization implies impossibility of good computationally efficient truthful mechanisms. Most interesting problems, including those mentioned above, lie in an intermediate range where the valuation spaces are neither single dimensional nor fully dimensional, a range for which very little is known. The main problem seems to be the lack of a good characterization of incentive compatibility in these intermediate ranges1. In particular, the key unknown is whether any useful truthful non-VCG mechanisms exist in the intermediate range2. Multi-unit Auctions As mentioned, the paradigmatic problems for the reconciliation of computational constraints with incentive constraints are the various subclasses of combinatorial auctions. In this paper we consider the simplest variant which exhibits this tension: multi-unit auctions. In this problem there are m identical items for sale among n bidders, where each bidder i has a valuation function vi : {0...m} →ℜ, where vi(k) denotes player i’s value for receiving k items. The valuations vi are assumed to be monotone non-decreasing (free dis

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