Sealed Bid Mulit-object Auctions with Necessary Bundles and Its Application to Spectrum Auctions

In this paper, we consider multi-object auctions in which each bidder has a positive reservation value for only one special subset of objects, called a necessary bundle. In the auction, each bidder reports its necessary bundle and its reservation value. The seller solves the assignment problem of objects which maximizes its revenue and decides the winning bidders who can purchase their necessary bundles for their reporting prices. We show that this auction leads to an ecient allocation through Nash equilibria under complete information when the bid-grid size is suciently small. We apply our results to spectrum auctions satisfying the conditions that necessary bundles are intervals of discretized radio spectrum. We show that the revenue maximization problem for the seller can be solved in polynomial time for the above auctions. The algorithm also indicates a method to choose an accepted bidder randomly when the revenue maximization problem has multiple optimal solutions. Lastly, we introduce a linear inequality system which characterizes the set of Nash equilibria.