Ittai Abraham, Moshe Babaioff, Shaddin Dughmi, and Tim Roughgarden
Complements between goods — where one good takes on added value in the presence of another — have been a thorn in the side of algorithmic mechanism designers. On the one hand, complements are common in the standard motivating applications for combinatorial auctions, like spectrum license auctions. On the other, welfare maximization in the presence of complements is notoriously difficult, and this intractability has stymied theoretical progress in the area. For example, there are no known positive results for combinatorial auctions in which bidder valuations are multi-parameter and non-complement-free, other than the relatively weak results known for general valuations.
To make inroads on the problem of combinatorial auction design in the presence of complements, we propose a model for valuations with complements that is parameterized by the “size” of the complements. The model permits a succinct representation, a variety of computationally efficient queries, and non-trivial welfare-maximization algorithms and mechanisms. Specifically, a hypergraph-r valuation v for a good set M is represented by a hypergraph H=(M,E), where every (hyper-)edge e in E contains at most r vertices and has a nonnegative weight we. Each good j in M also has a nonnegative weight wj. The value v(S) for a subset S of goods is defined as the sum of the weights of the goods and edges entirely contained in S.
We design the following polynomial-time approximation algorithms and truthful mechanisms for welfare maximization with bidders with hypergraph valuations.
|Published in||ACM Conference on Electronic Commerce (EC'12)|