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Erschienen in:
Ordinal Approximation for Social Choice, Matching, and Facility Location Problems Given Candidate Positions Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

Optimal Pricing for MHR Distributions

verfasst von : Yiannis Giannakopoulos, Keyu Zhu

Erschienen in: Web and Internet Economics

Verlag: Springer International Publishing

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Abstract

We study the performance of anonymous posted-price selling mechanisms for a standard Bayesian auction setting, where n bidders have i.i.d. valuations for a single item. We show that for the natural class of Monotone Hazard Rate (MHR) distributions, offering the same, take-it-or-leave-it price to all bidders can achieve an (asymptotically) optimal revenue. In particular, the approximation ratio is shown to be \(1+O(\ln {\ln {n}}/ \ln {n})\), matched by a tight lower bound for the case of exponential distributions. This improves upon the previously best-known upper bound of \(e/(e-1)\approx 1.58\) for the slightly more general class of regular distributions. In the worst case (over n), we still show a global upper bound of 1.35. We give a simple, closed-form description of our prices which, interestingly enough, relies only on minimal knowledge of the prior distribution, namely just the expectation of its second-highest order statistic.

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Vorheriges Kapitel Pricing Multi-unit Markets
Nächstes Kapitel Learning Convex Partitions and Computing Game-Theoretic Equilibria from Best Response Queries
Fußnoten
1
In this paper we will avoid discussing such subtler issues as implementability and truthfulness, since our goal is to study the performance of specific and very simple pricing mechanisms. The interested reader is pointed to [25] as a good starting point for a deeper investigation of those ideas.
 
2
As a matter of fact, one can use the techniques of Blumrosen and Holenstein [6] to get a slightly better lower bound of (at least) 1.4: a corollary of their work is that, for any \(k>1\), if the valuations are drawn from a Pareto distribution with cdf \(F(x)=1-1/x^k\) and the number of bidders grows arbitrarily large, then the separation between the optimal revenue and that of anonymous pricing is \(\varGamma \left( \frac{k-1}{k}\right) \left( 1-\frac{1}{k}\right) /\frac{k}{e^{\eta (k)}}\eta (k)^{1-1/k}\), where \(\varGamma \) is the standard gamma function and \(\eta (k)\) is the unique positive solution of equation \(e^x=1+k\cdot x\). Optimizing this ratio over \(k\in (1,2)\), we can get a lower bound greater than 1.403.
 
3
This lower bound was very recently improved to 2.62 by Jin et al. [22].
 
4
Regularity a la Myerson [24] requires the virtual valuation \(x-\frac{1-F(x)}{f(x)}\) to be nondecreasing. Notice that this is a (strictly) weaker condition than MHR. For example, some Pareto distributions \(\propto x^{-\alpha }\) with \(\alpha \ge 2\) are regular but not MHR.
 
5
As a matter of fact, this bound holds even for the more general class of regular distributions (see also the discussion in Sect. 2.).
 
6
See (6) below.
 
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Metadaten
Titel
Optimal Pricing for MHR Distributions
verfasst von
Yiannis Giannakopoulos
Keyu Zhu
Copyright-Jahr
2018
Buch
Web and Internet Economics

Print ISBN: 978-3-030-04611-8

Electronic ISBN: 978-3-030-04612-5

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-030-04612-5_11