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.
Bibliographical noteFunding Information:
✩ A preliminary version of this paper has been presented at EC-2001. Supported by grants from the Israeli Ministry of Science and the Israeli Academy of Sciences. Corresponding author. E-mail addresses: email@example.com (D. Lehmann), firstname.lastname@example.org (N. Nisan). § Benny passed away on July 1st, 2004.
- Combinatorial auctions
- Decreasing marginal utilities
- Winner determination