Approximately Envy-Free Spectrum Allocation with Complementarities

TL;DR

This paper introduces a polynomial-time auction mechanism for spectrum allocation that achieves approximate envy-freeness and efficiency, even with agents' valuations exhibiting complementarities and limited bundle sizes.

Contribution

It presents a novel auction mechanism combining linear programming relaxation and iterative rounding to ensure approximate envy-freeness and strategy-proofness in spectrum auctions with complementarities.

Findings

The mechanism is efficient and approximately envy-free.

Allocations over-allocate goods by at most k-1 units.

Dual prices are approximately envy-free regardless of allocation.

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.