Budget Feasible Mechanism Design: From Prior-Free to Bayesian

Budget feasible mechanism design studies procurement combinatorial auctions where the sellers have private costs to produce items, and the buyer(auctioneer) aims to maximize a social valuation function on subsets of items, under the budget constraint on the total payment. One of the most important questions in the field is "which valuation domains admit truthful budget feasible mechanisms with `small' approximations (compared to the social optimum)?" Singer showed that additive and submodular functions have such constant approximations. Recently, Dobzinski, Papadimitriou, and Singer gave an O(log^2 n)-approximation mechanism for subadditive functions; they also remarked that: "A fundamental question is whether, regardless of computational constraints, a constant-factor budget feasible mechanism exists for subadditive functions." We address this question from two viewpoints: prior-free worst case analysis and Bayesian analysis. For the prior-free framework, we use an LP that describes the fractional cover of the valuation function; it is also connected to the concept of approximate core in cooperative game theory. We provide an O(I)-approximation mechanism for subadditive functions, via the worst case integrality gap I of LP. This implies an O(log n)-approximation for subadditive valuations, O(1)-approximation for XOS valuations, and for valuations with a constant I. XOS valuations are an important class of functions that lie between submodular and subadditive classes. We give another polynomial time O(log n/loglog n) sub-logarithmic approximation mechanism for subadditive valuations. For the Bayesian framework, we provide a constant approximation mechanism for all subadditive functions, using the above prior-free mechanism for XOS valuations as a subroutine. Our mechanism allows correlations in the distribution of private information and is universally truthful.Comment: to appear in STOC 201

Budget-Feasible Mechanism Design for Non-Monotone Submodular Objectives: Offline and Online

The framework of budget-feasible mechanism design studies procurement auctions where the auctioneer (buyer) aims to maximize his valuation function subject to a hard budget constraint. We study the problem of designing truthful mechanisms that have good approximation guarantees and never pay the participating agents (sellers) more than the budget. We focus on the case of general (non-monotone) submodular valuation functions and derive the first truthful, budget-feasible and $O(1)$-approximate mechanisms that run in polynomial time in the value query model, for both offline and online auctions. Prior to our work, the only $O(1)$-approximation mechanism known for non-monotone submodular objectives required an exponential number of value queries. At the heart of our approach lies a novel greedy algorithm for non-monotone submodular maximization under a knapsack constraint. Our algorithm builds two candidate solutions simultaneously (to achieve a good approximation), yet ensures that agents cannot jump from one solution to the other (to implicitly enforce truthfulness). Ours is the first mechanism for the problem where---crucially---the agents are not ordered with respect to their marginal value per cost. This allows us to appropriately adapt these ideas to the online setting as well. To further illustrate the applicability of our approach, we also consider the case where additional feasibility constraints are present. We obtain $O(p)$-approximation mechanisms for both monotone and non-monotone submodular objectives, when the feasible solutions are independent sets of a $p$-system. With the exception of additive valuation functions, no mechanisms were known for this setting prior to our work. Finally, we provide lower bounds suggesting that, when one cares about non-trivial approximation guarantees in polynomial time, our results are asymptotically best possible.Comment: Accepted to EC 201

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

Privacy-Preserving Adversarial Networks

We propose a data-driven framework for optimizing privacy-preserving data release mechanisms to attain the information-theoretically optimal tradeoff between minimizing distortion of useful data and concealing specific sensitive information. Our approach employs adversarially-trained neural networks to implement randomized mechanisms and to perform a variational approximation of mutual information privacy. We validate our Privacy-Preserving Adversarial Networks (PPAN) framework via proof-of-concept experiments on discrete and continuous synthetic data, as well as the MNIST handwritten digits dataset. For synthetic data, our model-agnostic PPAN approach achieves tradeoff points very close to the optimal tradeoffs that are analytically-derived from model knowledge. In experiments with the MNIST data, we visually demonstrate a learned tradeoff between minimizing the pixel-level distortion versus concealing the written digit.Comment: 16 page

Selling Privacy at Auction

We initiate the study of markets for private data, though the lens of differential privacy. Although the purchase and sale of private data has already begun on a large scale, a theory of privacy as a commodity is missing. In this paper, we propose to build such a theory. Specifically, we consider a setting in which a data analyst wishes to buy information from a population from which he can estimate some statistic. The analyst wishes to obtain an accurate estimate cheaply. On the other hand, the owners of the private data experience some cost for their loss of privacy, and must be compensated for this loss. Agents are selfish, and wish to maximize their profit, so our goal is to design truthful mechanisms. Our main result is that such auctions can naturally be viewed and optimally solved as variants of multi-unit procurement auctions. Based on this result, we derive auctions for two natural settings which are optimal up to small constant factors: 1. In the setting in which the data analyst has a fixed accuracy goal, we show that an application of the classic Vickrey auction achieves the analyst's accuracy goal while minimizing his total payment. 2. In the setting in which the data analyst has a fixed budget, we give a mechanism which maximizes the accuracy of the resulting estimate while guaranteeing that the resulting sum payments do not exceed the analysts budget. In both cases, our comparison class is the set of envy-free mechanisms, which correspond to the natural class of fixed-price mechanisms in our setting. In both of these results, we ignore the privacy cost due to possible correlations between an individuals private data and his valuation for privacy itself. We then show that generically, no individually rational mechanism can compensate individuals for the privacy loss incurred due to their reported valuations for privacy.Comment: Extended Abstract appeared in the proceedings of EC 201