Revenue Loss in Shrinking Markets

We analyze the revenue loss due to market shrinkage. Specifically, consider a simple market with one item for sale and $n$ bidders whose values are drawn from some joint distribution. Suppose that the market shrinks as a single bidder retires from the market. Suppose furthermore that the value of this retiring bidder is fixed and always strictly smaller than the values of the other players. We show that even this slight decrease in competition might cause a significant fall of a multiplicative factor of $\frac{1}{e+1}\approx0.268$ in the revenue that can be obtained by a dominant strategy ex-post individually rational mechanism. In particular, our results imply a solution to an open question that was posed by Dobzinski, Fu, and Kleinberg [STOC'11]

Optimal Auctions for Correlated Buyers with Sampling

Cr\'emer and McLean [1985] showed that, when buyers' valuations are drawn from a correlated distribution, an auction with full knowledge on the distribution can extract the full social surplus. We study whether this phenomenon persists when the auctioneer has only incomplete knowledge of the distribution, represented by a finite family of candidate distributions, and has sample access to the real distribution. We show that the naive approach which uses samples to distinguish candidate distributions may fail, whereas an extended version of the Cr\'emer-McLean auction simultaneously extracts full social surplus under each candidate distribution. With an algebraic argument, we give a tight bound on the number of samples needed by this auction, which is the difference between the number of candidate distributions and the dimension of the linear space they span

Public projects, Boolean functions and the borders of Border's theorem

Border's theorem gives an intuitive linear characterization of the feasible interim allocation rules of a Bayesian single-item environment, and it has several applications in economic and algorithmic mechanism design. All known generalizations of Border's theorem either restrict attention to relatively simple settings, or resort to approximation. This paper identifies a complexity-theoretic barrier that indicates, assuming standard complexity class separations, that Border's theorem cannot be extended significantly beyond the state-of-the-art. We also identify a surprisingly tight connection between Myerson's optimal auction theory, when applied to public project settings, and some fundamental results in the analysis of Boolean functions.Comment: Accepted to ACM EC 201

Near-Optimal and Robust Mechanism Design for Covering Problems with Correlated Players

We consider the problem of designing incentive-compatible, ex-post individually rational (IR) mechanisms for covering problems in the Bayesian setting, where players' types are drawn from an underlying distribution and may be correlated, and the goal is to minimize the expected total payment made by the mechanism. We formulate a notion of incentive compatibility (IC) that we call {\em support-based IC} that is substantially more robust than Bayesian IC, and develop black-box reductions from support-based-IC mechanism design to algorithm design. For single-dimensional settings, this black-box reduction applies even when we only have an LP-relative {\em approximation algorithm} for the algorithmic problem. Thus, we obtain near-optimal mechanisms for various covering settings including single-dimensional covering problems, multi-item procurement auctions, and multidimensional facility location.Comment: Major changes compared to the previous version. Please consult this versio

Optimal Parametric Auctions

We study the problem of profit maximization in auctions of one good where the buyers' valuations are drawn from independent distributions. When these distributions are known to the seller, Myerson's optimal auction is a well-known mechanism for maximizing revenue. In many cases, however, the seller may not know the buyers' distributions. We propose an alternative model where the seller only knows the mean and the variance of each distribution. We call parametric an auction whose mechanism only uses these parameters. We construct parametric auctions both when the seller only has one copy of the good to sell, and when she has an infinite number of identical copies (i.e., when the good is digital). For a very large class of distributions, including (but not limited to) distributions with a monotone hazard rate, our auctions achieve a constant fraction of the revenue of Myerson's auction. When the seller has absolutely no knowledge about the distributions, it is well known that no auction can achieve a constant fraction of the optimal revenue when the players are not identically distributed. Our parametric model gives the seller a small amount of extra information, allowing her to construct auctions for which (1) no two bidders need to be drawn from identical distributions and (2) the revenue obtained is a constant fraction of the revenue in Myerson's optimal auction

Crowdsourced Bayesian auctions

We investigate the problem of optimal mechanism design, where an auctioneer wants to sell a set of goods to buyers, in order to maximize revenue. In a Bayesian setting the buyers' valuations for the goods are drawn from a prior distribution D, which is often assumed to be known by the seller. In this work, we focus on cases where the seller has no knowledge at all, and "the buyers know each other better than the seller knows them". In our model, D is not necessarily common knowledge. Instead, each buyer individually knows a posterior distribution associated with D. Since the seller relies on the buyers' knowledge to help him set a price, we call these types of auctions crowdsourced Bayesian auctions. For this crowdsourced Bayesian model and many environments of interest, we show that, for arbitrary valuation distributions D (in particular, correlated ones), it is possible to design mechanisms matching to a significant extent the performance of the optimal dominant-strategy-truthful mechanisms where the seller knows D. To obtain our results, we use two techniques: (1) proper scoring rules to elicit information from the players; and (2) a reverse version of the classical Bulow-Klemperer inequality. The first lets us build mechanisms with a unique equilibrium and good revenue guarantees, even when the players' second and higher-order beliefs about each other are wrong. The second allows us to upper bound the revenue of an optimal mechanism with n players by an n/n--1 fraction of the revenue of the optimal mechanism with n -- 1 players. We believe that both techniques are new to Bayesian optimal auctions and of independent interest for future work.United States. Office of Naval Research (Grant number N00014-09-1-0597