Alan Deckelbaum

Mechanism design via optimal transport

Optimal mechanisms have been provided in quite general multi-item settings [Cai et al. 2012b, as long as each bidders 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 [Daskalakis et al. 2013]. Even for item distributions with special structure, optimal mechanisms have been surprisingly rare [Manelli and Vincent 2006] and the problem is challenging even in the two-item case [Hart and Nisan 2012]. 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 [Manelli and Vincent 2006]), 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.

Strong Duality for a Multiple‐Good Monopolist

We characterize optimal mechanisms for the multiple‐good monopoly problem and provide a framework to find them. We show that a mechanism is optimal if and only if a measure μ derived from the buyers type distribution satisfies certain stochastic dominance conditions. This measure expresses the marginal change in the sellers revenue under marginal changes in the rent paid to subsets of buyer types. As a corollary, we characterize the optimality of grand‐bundling mechanisms, strengthening several results in the literature, where only sufficient optimality conditions have been derived. As an application, we show that the optimal mechanism for n independent uniform items each supported on [c,c+1] is a grand‐bundling mechanism, as long as c is sufficiently large, extending Pavlovs result for two items Pavlov, 2011. At the same time, our characterization also implies that, for all c and for all sufficiently large n, the optimal mechanism for n independent uniform items supported on [c,c+1] is not a grand‐bundling mechanism.

Strong Duality for a Multiple-Good Monopolist

We provide a duality-based framework for revenue maximization in a multiple-good monopoly. Our framework shows that every optimal mechanism has a certificate of optimality, taking the form of an optimal transportation map between measures. Using our framework, we prove that grand-bundling mechanisms are optimal if and only if two stochastic dominance conditions hold between specific measures induced by the buyers type distribution. This result strengthens several results in the literature, where only sufficient conditions for grand-bundling optimality have been provided. As a corollary of our tight characterization of grand-bundling optimality, we show that the optimal mechanism for n independent uniform items each supported on [c; c + 1] is a grand-bundling mechanism, as long as c is sufficiently large, extending Pavlovs result for 2 items [Pavlov 2011]. Surprisingly, our characterization also implies that, for all c and for all sufficiently large n, the optimal mechanism for n independent uniform items supported on [c; c + 1] is not a grand bundling mechanism. The necessary and sufficient condition for grand bundling optimality is a special case of our more general characterization result that provides necessary and sufficient conditions for the optimality of an arbitrary mechanism for an arbitrary type distribution.

Optimal pricing is hard

We show that computing the revenue-optimal deterministic auction in unit-demand single-buyer Bayesian settings, i.e. the optimal item-pricing, is computationally hard even in single-item settings where the buyers value distribution is a sum of independently distributed attributes, or multi-item settings where the buyers values for the items are independent. We also show that it is intractable to optimally price the grand bundle of multiple items for an additive bidder whose values for the items are independent. These difficulties stem from implicit definitions of a value distribution. We provide three instances of how different properties of implicit distributions can lead to intractability: the first is a #P-hardness proof, while the remaining two are reductions from the SQRT-SUM problem of Garey, Graham, and Johnson [14]. While simple pricing schemes can oftentimes approximate the best scheme in revenue, they can have drastically different underlying structure. We argue therefore that either the specification of the input distribution must be highly restricted in format, or it is necessary for the goal to be mere approximation to the optimal schemes revenue instead of computing properties of the scheme itself.

Collusion, efficiency, and dominant strategies

Green and Laffont proved that no collusion-resilient dominant-strategy mechanism, whose strategies consist of individual valuations, guarantees efficiency in multi-unit auctions. Chen and Micali bypassed this impossibility by slightly enlarging the strategy spaces, yet assuming knowledge of the maximum value a player may have for a copy of the good, and the ability of imposing high fines on the players. For unrestricted combinatorial auctions, efficiency in collusion-resilient dominant strategies has remained open, with or without the above two assumptions. We fully generalize the notion of a collusion-resilient dominant-strategy mechanism by allowing for arbitrary strategy spaces; construct one such mechanism for multi-unit auctions, without relying on the above two assumptions; and prove that no such mechanism exists for unrestricted combinatorial auctions, with or without any additional assumptions. Our results hold when the mechanism does not know who colludes with whom, and players in the same coalition can perfectly coordinate their strategies.

Simulating one-reversal multicounter machines by partially blind multihead finite automata

This work is concerned with simulating nondeterministic one-reversal multicounter automata (NCMs) by nondeterministic partially blind multihead finite automata (NFAs). We show that any one-reversal NCM with k counters can be simulated by a partially blind NFA with k blind heads. This provides a nearly complete categorization of the computational power of partially blind automata, showing that the power of a (k+1)-NFA lies between that of a k-NCM and a (k+1)-NCM.