Worst-Case Mechanism Design via Bayesian Analysis

Abstract

Budget feasible mechanism design is the study of procurement combinatorial auctions in which the sellers have private costs to produce items, and the buyer (auctioneer) aims to maximize her valuation function on a subset of purchased 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 to the social optimum?" Singer [Proceedings of the 51st FOCS, IEEE Press, Piscataway, NJ, 2010, pp. 765-774] showed that submodular functions have a constant approximation mechanism. Dobzinski, Papadimitriou, and Singer [Proceedings of the 12th ACM Conference on Electronic Commerce, ACM, New York, 2011, pp. 273-282] gave an O(log 2 n) approximation mechanism for subadditive functions and remarked that "A fundamental question is whether, regardless of computational constraints, a constant-factor budget feasible mechanism exists for subadditive functions." In this paper, we give an affirmative answer to this question. To this end we relax the prior-free mechanism design framework to the Bayesian mechanism design framework (these are two standard approaches from computer science and economics, respectively). Then we convert our results in the Bayesian setting back to the prior-free framework by employing Yao's minimax principle. Along the way, we obtain the following results: (i) a polynomial time constant approximation for XOS valuations (a.k.a. fractionally subadditive valuations, a superset of submodular functions), (ii) a polynomial time O(log n/ log log n)-approximation for general subadditive valuations, (iii) a constant approximation for general subadditive functions in the Bayesian framework-we allow correlation in the distribution of sellers' costs and provide a universally truthful mechanism, (iv) the existence of a prior-free constant approximation mechanism via Yao's minimax principle. . Results (i)-(iii) were presented at STOC 12 [8]; result (iv) is new.

Optimal Budget-Feasible Mechanisms for Additive Valuations

“…We hope that our results may lead to new mechanisms and improved analysis for more general valuation classes. Indeed, given the same factor 2-approximation result of [BCGL17] for the promise version of the problem for a subadditive buyer, we are even so bold as to conjecture that the true approximation guarantee for a subadditive buyer is still 2 (leaving all computational considerations aside).…”

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

“…All of our mechanisms are randomized and, in fact, random sampling is an essential building block in our approach, as in previous related works [2,6,10,34]. Designing deterministic mechanisms seems very challenging and, beyond additive functions [17,40], no deterministic, polynomialtime, budget-feasible O(1)-approximation mechanisms are known, except for some special cases [1,2,18,29,41].…”

“…For subadditive functions, Dobzinski et al [18] suggested a O(log 2 n)-approximation mechanism, and gave the first constant factor mechanisms for a special case of non-monotone objectives, namely cut functions. The factor for subadditive functions was later improved to O(log n/log log n) by Bei et al [10], who also gave a randomized O(1)-approximation mechanism for XOS functions, albeit in exponential time in the value query model, and further initiated the Bayesian analysis in this setting. 1 Amanatidis et al [2] suggested O(1)-approximation mechanisms for a subclass of non-monotone submodular objectives, namely symmetric submodular objectives, however their approach does not seem to generalize beyond this subclass.…”

Budget-Feasible Mechanism Design for Non-Monotone Submodular Objectives

“…All of our mechanisms are randomized and, in fact, random sampling is an essential building block in our approach. Obtaining a good estimate of the optimal value via random sampling has been crucial in previous works on budget-feasible mechanism design for monotone objectives as well [2,5,10,35]. Designing deterministic budget-feasible mechanisms seems very challenging.…”

“…For settings with additional combinatorial constraints, Amanatidis et al [1] and Leonardi et al [35] gave O(1)-approximation mechanisms for additive valuation functions subject to independent system constraints. There is also a line of related work under the large market assumption (where no participant can significantly 2 Bei et al [10] propose an O (1)-approximation mechanism for non-decreasing XOS objectives that runs in polynomial time in the much stronger demand query model. However, they discuss how to extend their result to general XOS functions via the use ofˆ (S ) = max T ⊆S (T ).…”