403 research outputs found
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
Envy Freedom and Prior-free Mechanism Design
We consider the provision of an abstract service to single-dimensional
agents. Our model includes position auctions, single-minded combinatorial
auctions, and constrained matching markets. When the agents' values are drawn
from a distribution, the Bayesian optimal mechanism is given by Myerson (1981)
as a virtual-surplus optimizer. We develop a framework for prior-free mechanism
design and analysis. A good mechanism in our framework approximates the optimal
mechanism for the distribution if there is a distribution; moreover, when there
is no distribution this mechanism still performs well.
We define and characterize optimal envy-free outcomes in symmetric
single-dimensional environments. Our characterization mirrors Myerson's theory.
Furthermore, unlike in mechanism design where there is no point-wise optimal
mechanism, there is always a point-wise optimal envy-free outcome.
Envy-free outcomes and incentive-compatible mechanisms are similar in
structure and performance. We therefore use the optimal envy-free revenue as a
benchmark for measuring the performance of a prior-free mechanism. A good
mechanism is one that approximates the envy free benchmark on any profile of
agent values. We show that good mechanisms exist, and in particular, a natural
generalization of the random sampling auction of Goldberg et al. (2001) is a
constant approximation
Constant-Competitive Prior-Free Auction with Ordered Bidders
A central problem in Microeconomics is to design auctions with good revenue
properties. In this setting, the bidders' valuations for the items are private
knowledge, but they are drawn from publicly known prior distributions. The goal
is to find a truthful auction (no bidder can gain in utility by misreporting
her valuation) that maximizes the expected revenue.
Naturally, the optimal-auction is sensitive to the prior distributions. An
intriguing question is to design a truthful auction that is oblivious to these
priors, and yet manages to get a constant factor of the optimal revenue. Such
auctions are called prior-free.
Goldberg et al. presented a constant-approximate prior-free auction when
there are identical copies of an item available in unlimited supply, bidders
are unit-demand, and their valuations are drawn from i.i.d. distributions. The
recent work of Leonardi et al. [STOC 2012] generalized this problem to non
i.i.d. bidders, assuming that the auctioneer knows the ordering of their
reserve prices. Leonardi et al. proposed a prior-free auction that achieves a
approximation. We improve upon this result, by giving the first
prior-free auction with constant approximation guarantee.Comment: The same result has been obtained independently by E. Koutsoupias, S.
Leonardi and T. Roughgarde
Prior-free multi-unit auctions with ordered bidders
Prior-free auctions are robust auctions that assume no distribution over bidders' valuations and provide worst-case (input-by-input) approximation guarantees. In contrast to previous work on this topic, we pursue good prior-free auctions with non-identical bidders.
Prior-free auctions can approximate meaningful benchmarks for non-identical bidders only when sufficient qualitative information about the bidder asymmetry is publicly known. We consider digital goods auctions where there is a total ordering of the bidders that is known to the seller, where earlier bidders are in some sense thought to have higher valuations. We use the framework of Hartline and Roughgarden (STOC'08) to define an appropriate revenue benchmark: the maximum revenue that can be obtained from a bid vector using prices that are nonincreasing in the bidder ordering and bounded above by the second-highest bid. This monotone-price benchmark is always as large as the well-known fixed-price benchmark , so designing prior-free auctions with good approximation guarantees is only harder. By design, an auction that approximates the monotone-price benchmark satisfies a very strong guarantee: it is, in particular, simultaneously near-optimal for essentially every Bayesian environment in which bidders' valuation distributions have nonincreasing monopoly prices, or in which the distribution of each bidder stochastically dominates that of the next. Even when there is no distribution over bidders' valuations, such an auction still provides a quantifiable input-by-input performance guarantee.
In this paper, we design a simple -competitive prior-free auction for digital goods with ordered bidders. We also extend the monotone-price benchmark and our -competitive prior-free auction to multi-unit settings with limited supply
Optimal Mechanism Design with Flexible Consumers and Costly Supply
The problem of designing a profit-maximizing, Bayesian incentive compatible
and individually rational mechanism with flexible consumers and costly
heterogeneous supply is considered. In our setup, each consumer is associated
with a flexibility set that describes the subset of goods the consumer is
equally interested in. Each consumer wants to consume one good from its
flexibility set. The flexibility set of a consumer and the utility it gets from
consuming a good from its flexibility set are its private information. We adopt
the flexibility model of [1] and focus on the case of nested flexibility sets
-- each consumer's flexibility set can be one of k nested sets. Examples of
settings with this inherent nested structure are provided. On the supply side,
we assume that the seller has an initial stock of free supply but it can
purchase more goods for each of the nested sets at fixed exogenous prices. We
characterize the allocation and purchase rules for a profit-maximizing,
Bayesian incentive compatible and individually rational mechanism as the
solution to an integer program. The optimal payment function is pinned down by
the optimal allocation rule in the form of an integral equation. We show that
the nestedness of flexibility sets can be exploited to obtain a simple
description of the optimal allocations, purchases and payments in terms of
thresholds that can be computed through a straightforward iterative procedure.Comment: 8 pages. arXiv admin note: text overlap with arXiv:1607.0252
On Revenue Monotonicity in Combinatorial Auctions
Along with substantial progress made recently in designing near-optimal
mechanisms for multi-item auctions, interesting structural questions have also
been raised and studied. In particular, is it true that the seller can always
extract more revenue from a market where the buyers value the items higher than
another market? In this paper we obtain such a revenue monotonicity result in a
general setting. Precisely, consider the revenue-maximizing combinatorial
auction for items and buyers in the Bayesian setting, specified by a
valuation function and a set of independent item-type
distributions. Let denote the maximum revenue achievable under
by any incentive compatible mechanism. Intuitively, one would expect that
if distribution stochastically dominates .
Surprisingly, Hart and Reny (2012) showed that this is not always true even for
the simple case when is additive. A natural question arises: Are these
deviations contained within bounds? To what extent may the monotonicity
intuition still be valid? We present an {approximate monotonicity} theorem for
the class of fractionally subadditive (XOS) valuation functions , showing
that if stochastically dominates under
where is a universal constant. Previously, approximate monotonicity was
known only for the case : Babaioff et al. (2014) for the class of additive
valuations, and Rubinstein and Weinberg (2015) for all subaddtive valuation
functions.Comment: 10 page
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