Approximately Envy-Free Spectrum Allocation with Complementarities

Abstract: With spectrum auctions as our prime motivation, in this paper we analyze combinatorial auctions where agents' valuations exhibit complementarities. Assuming that the agents only value bundles of size at most $k$ and also assuming that we can assess prices, we present a mechanism that is efficient, approximately envy-free, asymptotically strategy-proof and that has polynomial-time complexity. Modifying an iterative rounding procedure from assignment problems, we use the primal and dual optimal solutions to the linear programming relaxation of the auction problem to construct a lottery for the allocations and to assess the prices to bundles. The allocations in the lottery over-allocate goods by at most $k-1$ units, and the dual prices are shown to be (approximately) envy-free irrespective of the allocation chosen. We conclude with a detailed numerical investigation of a specific spectrum allocation problem.