Combinatorial Auctions: Complexity and Algorithms

A combinatorial auction allows bidders to submit bids on bundles of objects and can be considered the pivotal example of a multiple object auctions. They also constitute a paradigmatic problem in algorithmic mechanism design. We provide an overview of both the computational complexity and strategic complexity inherent in the design of such auctions, and discuss how these challenges are addressed in various combinatorial auction formats. An auction can be defined as ”a market institution with an explicit set of rules determining resource allocation and prices on the basis of bids from the market participants” [1]. The competitive process serves to aggregate the scattered information about bidders’ valuations and to dynamically set a price. The auction format determines the rules governing when and how a deal is closed [2]. Auctions are typically evaluated using two main criteria, (allocative) efficiency and revenue. The first one measures whether the objects end up with those bidders who value them most, while the latter focuses on the expected selling price. Multiple object auctions can be divided in those, where multiple units of a single item are sold or where multiple items are sold. Of course also combinations are possible, where large quantities (multiple units) of different items get sold or bought, such as large quantities of different types of hard disk drives. Combinatorial auctions are a means to buy or sell multiple