270 research outputs found
Making the Most of Your Samples
We study the problem of setting a price for a potential buyer with a
valuation drawn from an unknown distribution . The seller has "data"' about
in the form of i.i.d. samples, and the algorithmic challenge is
to use these samples to obtain expected revenue as close as possible to what
could be achieved with advance knowledge of .
Our first set of results quantifies the number of samples that are
necessary and sufficient to obtain a -approximation. For example,
for an unknown distribution that satisfies the monotone hazard rate (MHR)
condition, we prove that samples are
necessary and sufficient. Remarkably, this is fewer samples than is necessary
to accurately estimate the expected revenue obtained by even a single reserve
price. We also prove essentially tight sample complexity bounds for regular
distributions, bounded-support distributions, and a wide class of irregular
distributions. Our lower bound approach borrows tools from differential privacy
and information theory, and we believe it could find further applications in
auction theory.
Our second set of results considers the single-sample case. For regular
distributions, we prove that no pricing strategy is better than
-approximate, and this is optimal by the Bulow-Klemperer theorem.
For MHR distributions, we show how to do better: we give a simple pricing
strategy that guarantees expected revenue at least times the maximum
possible. We also prove that no pricing strategy achieves an approximation
guarantee better than
Extreme-Value Theorems for Optimal Multidimensional Pricing
Original manuscript: June 2, 2011We provide a Polynomial Time Approximation Scheme for the multi-dimensional unit-demand pricing problem, when the buyer's values are independent (but not necessarily identically distributed.) For all ϵ >; 0, we obtain a (1 + ϵ)-factor approximation to the optimal revenue in time polynomial, when the values are sampled from Monotone Hazard Rate (MHR) distributions, quasi-polynomial, when sampled from regular distributions, and polynomial in n[superscript poly(log r)] when sampled from general distributions supported on a set [u[subscript min],ru[subscript min]]. We also provide an additive PTAS for all bounded distributions. Our algorithms are based on novel extreme value theorems for MHR and regular distributions, and apply probabilistic techniques to understand the statistical properties of revenue distributions, as well as to reduce the size of the search space of the algorithm. As a byproduct of our techniques, we establish structural properties of optimal solutions. We show that, for all ϵ >; 0, g(1/ϵ) distinct prices suffice to obtain a (1 + ϵ)-factor approximation to the optimal revenue for MHR distributions, where g(1/ϵ) is a quasi-linear function of 1/ϵ that does not depend on the number of items. Similarly, for all ϵ >; 0 and n >; 0, g(1/ϵ · log n) distinct prices suffice for regular distributions, where n is the number of items and g(·) is a polynomial function. Finally, in the i.i.d. MHR case, we show that, as long as the number of items is a sufficiently large function of 1/ϵ, a single price suffices to achieve a (1 + ϵ)-factor approximation. Our results represent significant progress to the single-bidder case of the multidimensional optimal mechanism design problem, following Myerson's celebrated work on optimal mechanism design [Myerson 1981].National Science Foundation (U.S.) (Award CCF-0953960)National Science Foundation (U.S.) (Award CCF-1101491)Alfred P. Sloan Foundation (Fellowship
Budget Constrained Auctions with Heterogeneous Items
In this paper, we present the first approximation algorithms for the problem
of designing revenue optimal Bayesian incentive compatible auctions when there
are multiple (heterogeneous) items and when bidders can have arbitrary demand
and budget constraints. Our mechanisms are surprisingly simple: We show that a
sequential all-pay mechanism is a 4 approximation to the revenue of the optimal
ex-interim truthful mechanism with discrete correlated type space for each
bidder. We also show that a sequential posted price mechanism is a O(1)
approximation to the revenue of the optimal ex-post truthful mechanism when the
type space of each bidder is a product distribution that satisfies the standard
hazard rate condition. We further show a logarithmic approximation when the
hazard rate condition is removed, and complete the picture by showing that
achieving a sub-logarithmic approximation, even for regular distributions and
one bidder, requires pricing bundles of items. Our results are based on
formulating novel LP relaxations for these problems, and developing generic
rounding schemes from first principles. We believe this approach will be useful
in other Bayesian mechanism design contexts.Comment: Final version accepted to STOC '10. Incorporates significant reviewer
comment
Optimal Auctions vs. Anonymous Pricing: Beyond Linear Utility
The revenue optimal mechanism for selling a single item to agents with
independent but non-identically distributed values is complex for agents with
linear utility (Myerson,1981) and has no closed-form characterization for
agents with non-linear utility (cf. Alaei et al., 2012). Nonetheless, for
linear utility agents satisfying a natural regularity property, Alaei et al.
(2018) showed that simply posting an anonymous price is an e-approximation. We
give a parameterization of the regularity property that extends to agents with
non-linear utility and show that the approximation bound of anonymous pricing
for regular agents approximately extends to agents that satisfy this
approximate regularity property. We apply this approximation framework to prove
that anonymous pricing is a constant approximation to the revenue optimal
single-item auction for agents with public-budget utility, private-budget
utility, and (a special case of) risk-averse utility.Comment: Appeared at EC 201
Lower Bounds on Revenue of Approximately Optimal Auctions
We obtain revenue guarantees for the simple pricing mechanism of a single
posted price, in terms of a natural parameter of the distribution of buyers'
valuations. Our revenue guarantee applies to the single item n buyers setting,
with values drawn from an arbitrary joint distribution. Specifically, we show
that a single price drawn from the distribution of the maximum valuation Vmax =
max {V_1, V_2, ...,V_n} achieves a revenue of at least a 1/e fraction of the
geometric expecation of Vmax. This generic bound is a measure of how revenue
improves/degrades as a function of the concentration/spread of Vmax.
We further show that in absence of buyers' valuation distributions,
recruiting an additional set of identical bidders will yield a similar
guarantee on revenue. Finally, our bound also gives a measure of the extent to
which one can simultaneously approximate welfare and revenue in terms of the
concentration/spread of Vmax.Comment: The 8th Workshop on Internet and Network Economics (WINE
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