The Biased Sampling Profit Extraction Auction

We give an auction for downward-closed environments that generalizes the random sampling profit extraction auction for digital goods of Fiat et al. (2002). The mechanism divides the agents in to a market and a sample using a biased coin and attempts to extract the optimal revenue from the sample from the market. The latter step is done with the downward-closed profit extractor of Ha and Hartline (2012). The auction is a 11-approximation to the envyfree benchmark in downward-closed permutation environments. This is an improvement on the previously best known results of 12.5 for matroid and 30.4 for downward-closed permutation environments that are due to Devanur et al. (2012) and Ha and Hartline (2012), respectively. Economic mechanisms that are less dependent on the assumptions of the environment are more likely to be relevant (cf. Wilson, 1987). The area of prior-free mechanism design attempts to remove the distributional assumption on agents while, at the same time, guaranteeing a good approximation of the optimal revenue. The performance of a prior-free mechanism is measured with respect to a benchmark. Recently, Hartline and Yan (2011) proposed the envy-free benchmark, denoted by EFO(v) where v is the valuation vector of the agents. This benchmark is the maximum revenue attainable given that the allocation and payment vectors are envy-free: no agent prefers another’s outcome to her own. A downward-closed environment is one where given a feasible set of agents, all subsets are feasible. A permutation environment is one where the agent identities are randomly permuted with respect to the feasibility constraint. In downward-closed permutation environments, Hartline and Yan (2011) provide detailed justification for the approximation of the envy-free benchmark.1 The main approaches to prior-free auctions for digital goods generalize to downward-closed permutation environments. Hartline and Yan (2011) generalized the random sampling auction; Ha and Hartline (2012) generalized the consensus estimate profit extraction auction; and in the present paper we generalize the random sampling profit extraction auction from Fiat et al. (2002). The random sampling profit extraction auction splits the agents into a market and a sample, estimates the optimal profit from the sample, and then attempts to extract that profit from the market. Ha and Hartline (2012) give a profit extractor for the envy-free benchmark in downward-closed permutation environments. This profit extractor is parameterized by a target valuation profile ṽ and on actual valuation profile v is able to extract at least the profit of the envy-free benchmark EFO(ṽ) when v pointwise dominates ṽ, i.e., vi ≥ ṽi for all i, denoted v ≥ ṽ where both v and ṽ are sorted in non-increasing order. ∗Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL. Email: {bach,hartline}@u.northwestern.edu. For technical reasons the benchmark considered is EFO(v) where v = (v2, v2, v3, . . . , vn) is same as v but with the highest value lowered to the second highest value.