Optimization Issues in Combinatorial Auctions

Auctions are used more and more to sell a large variety of goods. In this chapter, it is our objective to concentrate on applications of auctions in telecommunication, which possess a part or a feature that can be optimized. Optimization methods are necessary, in particular, when auctions are used to sell or purchase goods which consist of combinations of items, and where combinations have higher or lower value than the sum of values of individual items: combinatorial auctions. In the first part, we review the theory on combinatorial auctions, starting with the various properties and mechanisms found in the literature on combinatorial auctions. Then the allocation decision is identified as the winner determination problem (WDP), which is the central subject of this chapter. The winner determination problem is formulated as an Integer Linear Program (ILP) with the structure of a set-packing problem. Therefore, complexity results, polynomial special cases, and general solution methods for the WDP are often obtained from results for the set-packing problem. In the second part of this chapter, we turn to applications from telecommunications. First, a model for bandwidth allocation in networks is discussed. The problem is translated into a formulation that has close relations to multi-commodity flow and network synthesis. This guides us to alternative formulations and to solution methods. Second, the auctions of radio spectrum in the US and Europe are reviewed. The WDP of these multi-round auctions can be modeled using the XOR-of-OR bidding language, and solved by methods originally developed for set-packing.