Some Tractable Combinatorial Auctions

Auctions are the most widely used strategic gametheoretic mechanism in the Internet. Auctions have been mostly studied from a game-theoretic and economic perspective, although recent work in AI and OR has been concerned with computational aspects of auctions as well. When faced from a computational perspective, combinatorial auctions are perhaps the most challenging type of auctions. Combinatorial auctions are auctions where agents may submit bids for bundles of goods. Given that finding an optimal allocation of the goods in a combinatorial auction is intractable, researchers have been concerned with exposing tractable instances of combinatorial auctions. In this work we introduce polynomial solutions for a variety of non-trivial combinatorial auctions, such as combinatorial network auctions, various sub-additive combinatorial auctions, and some restricted forms of multi-unit combinatorial auctions. The emergence of electronic commerce has led to increasing interest in the design of protocols for noncooperative environments (see e.g. (Rosenschein & Zlotkin 1994; Kraus 1997; Tennenholtz 1999; Durfee 1992)). The wide-spread of auctions in the Internet, and the fact auctions are basic building blocks for a variety of economic protocols have attracted many researchers to tackle the challenge of efficient auction design (e.g. (Wellman et al. 1998; Monderer & Tennenholtz 2000; Lehmann, O’Callaghan, & Shoham 1999; Sandholm 1996; Parkes 1999)). The design of auctions introduces deep problems and challenges both from the game-theoretic and from the computational perspectives. This paper mainly concentrates on computational aspects of auctions. More specifically, we concentrate on addressing computational problems of combinatorial auctions, extending upon previous work on this basic topic (Rothkopf, Pekec, & Harstad 1998; ∗Current address: Computer Science Department, Stanford University, Stanford, CA 94305, USA Copyright c ©2000, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. Nisan 1999; Sandholm 1999; Fujishima, Leyton-Brown, & Shoham 1999). In an auction, a seller sells several goods to several potential buyers. In typical single-object auctions, determining the auction’s winner and its payment is a computationally tractable problem. This is also true when agents’ valuations for the different objects are additive, i.e. determined in an additive manner by their valuations for the single goods. However, consider a situation where a VCR, a TV, and a Microwave are sold; an agent may be willing to pay $200 for the TV, $300 for the VCR, and $150 for the microwave, but might be willing pay only $500 for getting all of them. In order to allocate the goods in a satisfactory manner, bids for bundles of goods should be allowed; given these bids, we need to find an optimal, revenue maximizing, allocation of the goods. This problem is referred to as the combinatorial auction problem, and it is in general intractable. One can partition previous work on computational aspects of combinatorial auctions into two parts. One part deals with heuristics for the solution of combinatorial auctions (see e.g. (Sandholm 1999; Fujishima, Leyton-Brown, & Shoham 1999)), while the other part deals with the identification of tractable cases of the combinatorial auctions problem (see (Rothkopf, Pekec, & Harstad 1998; Nisan 1999)). Our work fits into the latter category. Previous results on that category can be obtained in a relatively straightforward manner by a linear programming [LP] relaxation of the combinatorial auctions problem (which can be stated as an integer programming [IP] problem). This paper extends on these results, by exposing non-trivial tractable instances of the combinatorial auctions problem. In Section 2 we present some preliminaries. In Sections 3–4 we expose the use of b-matching techniques for the solution of combinatorial auctions, and present The use of b-matching techniques in the solution of other auctions is discussed in (Penn & Tennenholtz 1999). From: AAAI-00 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.