Mechanism Design for Subadditive Agents via an Ex Ante Relaxation

Abstract

We consider the problem of maximizing revenue for a monopolist offering multiple items to multiple heterogeneous buyers. We develop a simple mechanism that obtains a constant factor approximation under the assumption that the buyers' values are additive subject to a matroid feasibility constraint and independent across items. Importantly, different buyers in our setting can have different constraints on the sets of items they desire. Our mechanism is a sequential variant of two-part tariffs. Prior to our work, simple approximation mechanisms for such multi-buyer problems were known only for the special cases of all unit-demand or all additive value buyers.Our work expands upon and unifies long lines of work on unit-demand settings and additive settings. We employ the ex ante relaxation approach developed by Alaei (2011) for reducing a multiple-buyer mechanism design problem with an ex post supply constraint into single-buyer problems with ex ante supply constraints. Solving the single-agent problems requires us to significantly extend a decomposition technique developed in the context of additive values by Li and Yao (2013) and its extension to subadditive values by Rubinstein and Weinberg (2015). This paper belongs to a long line of research on finding simple and approximately optimal mechanisms for multi-parameter settings under various assumptions on the buyers' value functions and type distributions, and on the seller's supply constraint. The first breakthrough along these lines was made by Chawla et al. (2007) who showed that the revenue of an optimal mechanism for a single unit-demand buyer can be approximated within a factor of 3 by an item pricing, 1 a mechanism that allows the buyer to choose any item to buy at fixed per-item prices. More recently, Babaioff et al. (2014) developed a similar result for a single buyer with additive values. 2 They showed that the revenue of an optimal mechanism in this case is approximated within a factor of 6 by one of two simple mechanisms: an item pricing that fixes a price for each item and allows the buyer to choose any subset of items to buy, and a bundle pricing that allows the buyer to buy the grand bundle of all items at a fixed price. Observe that item pricing and bundle pricing are both two-part tariffs (with the entry fee or the per-item prices being zero, respectively).Unit-demand and additive types are two extremes within a broader class of value functions that we call constrained additive values. A constrained additive buyer has a value (drawn independently) for each item under sale; he is interested in buying a set of items that satisfies a certain downward-closed constraint; his value is additive over any such set. We have only recently begun to understand optimal mechanism design for a single agent with constrained additive values. Rubinstein and Weinberg (2015) proved that in this setting, as in the additive case, either item pricing or bundle pricing gives a constant-factor approximation to the optimal revenue. 3 There are many similarities between the t...

Simple and Approximately Optimal Pricing for Proportional Complementarities

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A duality-based unified approach to Bayesian mechanism design

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