The menu-size complexity of auctions

Hart, Sergiu; Nisan, Noam

doi:10.1145/2492002.2482544

Cited by 94 publications

(165 citation statements)

References 35 publications

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“…The result is tight. These results are in sharp contrast to Hart and Nisan's recent result that there is some distribution where finite number of menu items cannot guarantee any fraction of revenue [Hart and Nisan 2013]. Here we show that, for several wide classes of distributions, the optimal mechanisms have a finite and even extremely simple menus.…”

Section: Introductioncontrasting

confidence: 99%

“…Under some technical assumptions, Daskalakis et al [2013] show close relation between mechanism design and transport problem and use techniques there to solve for optimal mechanisms in a few special distributions. Hart and Nisan [2013] investigate how the "menu size" of an auction can effect the revenue and show that revenue of any finite menu-sized auction can be arbitrarily far from optimal (thus confirm an earlier consensus that restricting attention to deterministic auctions, which has an exponentiallysized (at most) menu, indeed loses generality). On the economic front, Manelli and Vincent [2006,2007], Pavlov [2011a,b] obtain the optimal mechanisms in several specific distributions (such as both items are distributed according to uniform [0,1]).…”

Section: Introductionsupporting

confidence: 59%

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Optimal mechanisms with simple menus

Tang

Wang

2017

Journal of Mathematical Economics

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We consider optimal mechanism design for the case with one buyer and two items. The buyer's valuations towards the two items are independent and additive. In this setting, optimal mechanism is unknown for general valuation distributions. We obtain two categories of structural results that shed light on the optimal mechanisms. These results can be summarized into one conclusion: under certain conditions, the optimal mechanisms have simple menus.The first category of results state that, under certain mild condition, the optimal mechanism has a monotone menu. In other words, in the menu that represents the optimal mechanism, as payment increases, the allocation probabilities for both items increase simultaneously. This theorem complements Hart and Reny's recent result regarding the nonmonotonicity of menu and revenue in multi-item settings. Applying this theorem, we derive a version of revenue monotonicity theorem that states stochastically superior distributions yield more revenue. Moreover, our theorem subsumes a previous result regarding sufficient conditions under which bundling is optimal [Hart and Nisan 2012]. The second category of results state that, under certain conditions, the optimal mechanisms have few menu items. Our first result in this category says, for certain distributions, the optimal menu contains at most 4 items. The condition admits power (including uniform) density functions. Our second result in this category works for a weaker condition, under which the optimal menu contains at most 6 items. This condition includes exponential density functions. Our last result in this category works for unit-demand setting. It states that, for uniform distributions, the optimal menu contains at most 5 items. All these results are in sharp contrast to Hart and Nisan's recent result that finite-sized menu cannot guarantee any positive fraction of optimal revenue for correlated valuation distributions.

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Section: Introductioncontrasting

confidence: 99%

Section: Introductionsupporting

confidence: 59%

Optimal mechanisms with simple menus

Tang

Wang

2017

Journal of Mathematical Economics

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“…The second class is additive preferences, for this class we give sufficient conditions under which the posting a price for the grand bundle is optimal. This result generalizes a recent result of Hart and Nisan (2013) and relates to work of Armstrong (1999). Similarly to an approach of Alaei et al (2013), these results for single-agent pricing problems described above can be generalized naturally to multi-agent auction problems.…”

Section: Introductionsupporting

confidence: 76%

Reverse Mechanism Design

Haghpanah

Hartline

2015

Proceedings of the Sixteenth ACM Conference on Economics and Computation

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Myerson's 1981 characterization of revenue-optimal auctions for single-dimensional agents follows from an amortized analysis of the incentives of a single agent. To optimize revenue in expectation, he maps values to virtual values which account for expected revenue gain but can be optimized pointwise. For single-dimensional agents the appropriate virtual values are unique and their closed form can be easily derived from revenue equivalence. A main challenge of generalizing the Myersonian approach to multi-dimensional agents is that the right amortization is not pinned down by revenue equivalence.For multi-dimensional agents, the optimal mechanism may be very complex. Complex mechanisms are impractical and rarely employed. We give a framework for reverse mechanism design. Instead of solving for the optimal mechanism in general, we assume a (natural) specific form of the mechanism. As an example of the framework, for agents with unit-demand preferences, we restrict attention to mechanisms that sell each agent her favorite item or nothing. From this restricted form, we will derive multi-dimensional virtual values. These virtual values prove this form of mechanism is optimal for a large class of item-symmetric distributions over types. As another example of our framework, for bidders with additive preferences, we derive conditions for the optimality of posting a single price for the grand bundle.

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“…Example 3 shows that the optimal mechanism for two Beta distributed items offers a continuum of lotteries, thereby having infinite menu-size complexity [HN13]. Still, using our techniques we can obtain a succinct and easily-computable description of the mechanism.…”

Section: Deriving the Optimal Mechanism For A Concrete Setting Of Parmentioning

confidence: 99%

Strong Duality for a Multiple-Good Monopolist

Daskalakis

Deckelbaum

Tzamos

2015

Proceedings of the Sixteenth ACM Conference on Economics and Computation

148

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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 buyer's type distribution satisfies certain stochastic dominance conditions. This measure expresses the marginal change in the seller's 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 Pavlov's result for 2 items [Pav11]. 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.

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The menu-size complexity of auctions

Cited by 94 publications

References 35 publications

Optimal mechanisms with simple menus

Optimal mechanisms with simple menus

Reverse Mechanism Design

Strong Duality for a Multiple-Good Monopolist

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