On the Ratio of Revenue to Welfare in Single-Parameter Mechanism Design

What fraction of the potential social surplus in an environment can be extracted by a revenue-maximizing monopolist? We investigate this problem in Bayesian single-parameter environments with independent private values. The precise answer to the question obviously depends on the particulars of the environment: the feasibility constraint and the distributions from which the bidders' private values are sampled. Rather than solving the problem in particular special cases, our work aims to provide universal lower bounds on the revenue-to-welfare ratio that hold under the most general hypotheses that allow for non-trivial such bounds. Our results can be summarized as follows. For general feasibility constraints, the revenue-to-welfare ratio is at least a constant times the inverse-square-root of the number of agents, and this is tight up to constant factors. For downward-closed feasibility constraints, the revenue-to-welfare ratio is bounded below by a constant. Both results require the bidders' distributions to satisfy hypotheses somewhat stronger than regularity; we show that the latter result cannot avoid this requirement.Comment: 15 page

Bayesian Incentive Compatibility via Fractional Assignments

Very recently, Hartline and Lucier studied single-parameter mechanism design problems in the Bayesian setting. They proposed a black-box reduction that converted Bayesian approximation algorithms into Bayesian-Incentive-Compatible (BIC) mechanisms while preserving social welfare. It remains a major open question if one can find similar reduction in the more important multi-parameter setting. In this paper, we give positive answer to this question when the prior distribution has finite and small support. We propose a black-box reduction for designing BIC multi-parameter mechanisms. The reduction converts any algorithm into an eps-BIC mechanism with only marginal loss in social welfare. As a result, for combinatorial auctions with sub-additive agents we get an eps-BIC mechanism that achieves constant approximation.Comment: 22 pages, 1 figur

Prophet Inequalities with Limited Information

In the classical prophet inequality, a gambler observes a sequence of stochastic rewards $V_1,...,V_n$ and must decide, for each reward $V_i$, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value $V_i$. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose $k$ items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards $V_1,...,V_n$ are drawn. The assumption that the gambler knows the distribution from which $V_1,...,V_n$ are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, \emph{even with full knowledge of the distribution}. Specifically, we provide a novel single-sample algorithm when the gambler can choose any $k$ elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related \emph{secretary problem} to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. We apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders

Truthful Learning Mechanisms for Multi-Slot Sponsored Search Auctions with Externalities

Sponsored search auctions constitute one of the most successful applications of microeconomic mechanisms. In mechanism design, auctions are usually designed to incentivize advertisers to bid their truthful valuations and to assure both the advertisers and the auctioneer a non-negative utility. Nonetheless, in sponsored search auctions, the click-through-rates (CTRs) of the advertisers are often unknown to the auctioneer and thus standard truthful mechanisms cannot be directly applied and must be paired with an effective learning algorithm for the estimation of the CTRs. This introduces the critical problem of designing a learning mechanism able to estimate the CTRs at the same time as implementing a truthful mechanism with a revenue loss as small as possible compared to an optimal mechanism designed with the true CTRs. Previous work showed that, when dominant-strategy truthfulness is adopted, in single-slot auctions the problem can be solved using suitable exploration-exploitation mechanisms able to achieve a per-step regret (over the auctioneer's revenue) of order $O(T^{-1/3})$ (where T is the number of times the auction is repeated). It is also known that, when truthfulness in expectation is adopted, a per-step regret (over the social welfare) of order $O(T^{-1/2})$ can be obtained. In this paper we extend the results known in the literature to the case of multi-slot auctions. In this case, a model of the user is needed to characterize how the advertisers' valuations change over the slots. We adopt the cascade model that is the most famous model in the literature for sponsored search auctions. We prove a number of novel upper bounds and lower bounds both on the auctioneer's revenue loss and social welfare w.r.t. to the VCG auction and we report numerical simulations investigating the accuracy of the bounds in predicting the dependency of the regret on the auction parameters

Reallocation Mechanisms

We consider reallocation problems in settings where the initial endowment of each agent consists of a subset of the resources. The private information of the players is their value for every possible subset of the resources. The goal is to redistribute resources among agents to maximize efficiency. Monetary transfers are allowed, but participation is voluntary. We develop incentive-compatible, individually-rational and budget balanced mechanisms for several classic settings, including bilateral trade, partnership dissolving, Arrow-Debreu markets, and combinatorial exchanges. All our mechanisms (except one) provide a constant approximation to the optimal efficiency in these settings, even in ones where the preferences of the agents are complex multi-parameter functions