Combinatorial auctions with decreasing marginal utilities

• Published in: Games and economic behavior Vol. 55; no. 2; pp. 270 - 296
• Main Authors: Lehmann, Benny, Lehmann, Daniel, Nisan, Noam
• Format: Journal Article
• Language: English
• Published: Duluth Elsevier Inc 01.05.2006; Elsevier; Academic Press
• Series: Games and Economic Behavior
• Subjects: Algorithms; Approximation; Auctions; Bidders; Bidding; Combinatorial auctions; Consumer behavior; Decreasing marginal utilities; Economic models; Game theory; Mathematical methods; Microeconomics; Studies; Valuation; Winner determination; Combinatorial auctions; Winner determination; C44; D44; Decreasing marginal utilities
• ISSN: 0899-8256; 1090-2473
• DOI: 10.1016/j.geb.2005.02.006

In most of microeconomic theory, consumers are assumed to exhibit decreasing marginal utilities. This paper considers combinatorial auctions among such submodular buyers. The valuations of such buyers are placed within a hierarchy of valuations that exhibit no complementarities, a hierarchy that includes also OR and XOR combinations of singleton valuations, and valuations satisfying the gross substitutes property. Those last valuations are shown to form a zero-measure subset of the submodular valuations that have positive measure. While we show that the allocation problem among submodular valuations is NP-hard, we present an efficient greedy 2-approximation algorithm for this case and generalize it to the case of limited complementarities. No such approximation algorithm exists in a setting allowing for arbitrary complementarities. Some results about strategic aspects of combinatorial auctions among players with decreasing marginal utilities are also presented.