Mechanism design via optimal transport

Abstract

Optimal mechanisms have been provided in quite general multi-item settings [4], as long as each bidder's type distribution is given explicitly by listing every type in the support along with its associated probability. In the implicit setting, e.g. when the bidders have additive valuations with independent and/or continuous values for the items, these results do not apply, and it was recently shown that exact revenue optimization is intractable, even when there is only one bidder [8]. Even for item distributions with special structure, optimal mechanisms have been surprisingly rare [13] and the problem is challenging even in the two-item case [10]. In this paper, we provide a framework for designing optimal mechanisms using optimal transport theory and duality theory. We instantiate our framework to obtain conditions under which only pricing the grand bundle is optimal in multi-item settings (complementing the work of [13]), as well as to characterize optimal two-item mechanisms. We use our results to derive closed-form descriptions of the optimal mechanism in several two-item settings, exhibiting also a setting where a continuum of lotteries is necessary for revenue optimization but a closed-form representation of the mechanism can still be found efficiently using our framework.

A duality-based unified approach to Bayesian mechanism design

“…Moreover, there exist instances where the buyer's two values are drawn from a correlated distribution where the optimal revenue achieves infinite revenue while the best deterministic mechanism achieves revenue ≤ 1 [BCKW10,HN13]. Even when the two item values are drawn independently, the optimal mechanism might offer uncountably many different randomized options for the buyer to choose from [DDT13]. Additionally, revenue-optimal multi-item auctions behave non-monotonically: there exist distributions F and F + , where F + stochastically dominates F , such that the revenue-optimal auction when a single additive buyer's values for two items are drawn from F × F achieves strictly larger revenue than the revenue-optimal auction when a single additive buyer's values are drawn from F + × F + [HR12].…”

Robustness and Separation in Multidimensional Screening

“…A second difficulty is that even if the boundary is identified correctly, typically the pointwise virtual value maximizer still is not incentive compatible because ironing is necessary; giving reasonable assumptions on primitives to rule out ironing is apparently more difficult in the multidimensional model than the one-dimensional model. Several significant contributions (Rochet and Choné (1998), Daskalakis, Deckelbaum, and Tzamos (2013), Daskalakis, Deckelbaum, and Tzamos (2015)) directly address ironing by offering "guess-and-check" methods that can be sometimes used to find the ironing regions, to thereby identify the optimal mechanism, and to find a dual solution to certify optimality. The arguments in these papers optimize over indirect utility functions (specifying the utility that each type of agent obtains in the mechanism), rather than working directly in the space of mechanisms, as in the present paper.…”

“…In the natural case where the values for each good are independent uniform on [0 1], the optimal mechanism is relatively simple (prices for each good plus a price for the bundle), but known proofs of optimality are involved (Manelli and Vincent (2006)). In other cases, the optimum may involve probabilistic bundling, and may even require a menu of infinitely many such bundles (Thanassoulis (2004), Manelli and Vincent (2007), Daskalakis, Deckelbaum, and Tzamos (2013)). Moreover, revenue can be nonmonotone: moving the distribution of buyer's types upward (in the stochastic dominance sense) may decrease optimal revenue (Hart and Reny (2015)).…”

“…These phenomena are not just artifacts of the possibilities for correlation in the multidimensional problem, since there are examples even when the values for the two goods are independently distributed. Furthermore, if the values are allowed to be correlated, then it can happen that restricting the seller to any fixed number k of bundles can lead to an arbitrarily small fraction of the optimal revenue (Hart and Nisan (2013)), and finding the optimal mechanism is computationally intractable (Conitzer and Sandholm (2002), Daskalakis, Deckelbaum, and Tzamos (2014)). With these challenges arising even for the simple monopoly problem, the prospects for more complex screening problems seem even more daunting.…”

Optimal Mechanism Design with Risk-Loving Agents

“…Finally, Chawla et al [2018] study the design of robust mechanisms under the cumulative prospect theory model.To the best our knowledge, the only work on mechanism design under risk-loving behavior is by Hinnosaar [2017], who shows that in the absence of regulations, the seller can extract infinite revenue from the buyer with asymptotically risk-loving behavior under both the expected utility theory and prospect theory models.Recently, the duality theory framework has drawn attention in the mechanism design community for understanding optimal mechanisms for selling multiple items. For example, Daskalakis et al [2017Daskalakis et al [ , 2013 and Koutsoupias [2014, 2015] discovered the connection between the dual problem and the optimal transport (bipartite matching) problem. Cai et al [2016] consider a duality framework via linear programming, and identify a connection between the virtual valuations and the dual variables.…”