Introduction

The need to buy or sell multiple objects arises in areas such as industrial procurement, logistics, or when governments allocate spectrum licenses or other assets. It is a fundamental topic and the question how multiple indivisible objects should be allocated via an auction has enjoyed renewed interest in recent years (Airiau and Sen 2003; Cramton et al. 2006; Day and Raghavan 2007; Xia et al. 2004). One of the key goals in this research literature is to develop mechanisms that achieve high (allocative) efficiency with a strong game-theoretical solution concept such as a dominant strategy or an ex-post Nash strategy, such that bidders have no incentive to misrepresent their valuations. In other words, the strategic complexity for bidders is low as they do not need prior information about other bidders’ valuations. Allocative efficiency measures whether the auctioned objects finally end up with the bidders who value them the most, thus, representing a measure of social welfare.

Combinatorial auctions are the most general types of multi-object market mechanisms, as they allow selling or buying a set of heterogeneous items to or from multiple bidders (Cramton et al. 2006). Bidders can specify package (or bundle) bids, i.e., prices are defined for the subsets of items that are auctioned. The price is only valid for the entire bundle, and the bid is indivisible. For example, in a combinatorial auction a bidder might want to buy a bundle, consisting of item x and item y, for a bundle price of $100, which might be more than the sum of the item prices for x and y, when sold individually. We will say that bidder valuations for both items are complementary in this case.

Many publications have focused on the computational complexity of the allocation problem in combinatorial auctions (Lehmann et al. 2006; Rothkopf et al. 1998). Computational complexity is often manageable in real-world applications with a low number of items and bidders.