51 research outputs found
An O(log log m) prophet inequality for subadditive combinatorial auctions
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here n agents with subadditive valuation functions compete for the assignment of m items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of O(log m). We make major progress on this question by providing an O(log log m) prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an O(log log m) approximation to the optimal revenue for subadditive valuations under an item-independence assumption
Improved Prophet Inequalities for Combinatorial Welfare Maximization with (Approximately) Subadditive Agents
m-1} measures the maximum number of items that complement each other, and (3) as a byproduct, an O(1)-competitive prophet inequality for submodular or fractionally subadditive (a.k.a. XOS) agents, matching the optimal ratio asymptotically. Our framework is computationally efficient given sample access to the prior and demand queries
Learning Multi-item Auctions with (or without) Samples
We provide algorithms that learn simple auctions whose revenue is
approximately optimal in multi-item multi-bidder settings, for a wide range of
valuations including unit-demand, additive, constrained additive, XOS, and
subadditive. We obtain our learning results in two settings. The first is the
commonly studied setting where sample access to the bidders' distributions over
valuations is given, for both regular distributions and arbitrary distributions
with bounded support. Our algorithms require polynomially many samples in the
number of items and bidders. The second is a more general max-min learning
setting that we introduce, where we are given "approximate distributions," and
we seek to compute an auction whose revenue is approximately optimal
simultaneously for all "true distributions" that are close to the given ones.
These results are more general in that they imply the sample-based results, and
are also applicable in settings where we have no sample access to the
underlying distributions but have estimated them indirectly via market research
or by observation of previously run, potentially non-truthful auctions.
Our results hold for valuation distributions satisfying the standard (and
necessary) independence-across-items property. They also generalize and improve
upon recent works, which have provided algorithms that learn approximately
optimal auctions in more restricted settings with additive, subadditive and
unit-demand valuations using sample access to distributions. We generalize
these results to the complete unit-demand, additive, and XOS setting, to i.i.d.
subadditive bidders, and to the max-min setting.
Our results are enabled by new uniform convergence bounds for hypotheses
classes under product measures. Our bounds result in exponential savings in
sample complexity compared to bounds derived by bounding the VC dimension, and
are of independent interest.Comment: Appears in FOCS 201
Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard
State-of-the-art posted-price mechanisms for submodular bidders with
items achieve approximation guarantees of [Assadi and
Singla, 2019]. Their truthfulness, however, requires bidders to compute an
NP-hard demand-query. Some computational complexity of this form is
unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an
-approximation for any [Dobzinski and
Vondr\'ak, 2016]. Together, these establish a stark distinction between
computationally-efficient and communication-efficient truthful mechanisms.
We show that this distinction disappears with a mild relaxation of
truthfulness, which we term implementation in advised strategies, and that has
been previously studied in relation to "Implementation in Undominated
Strategies" [Babaioff et al, 2009]. Specifically, advice maps a tentative
strategy either to that same strategy itself, or one that dominates it. We say
that a player follows advice as long as they never play actions which are
dominated by advice. A poly-time mechanism guarantees an -approximation
in implementation in advised strategies if there exists poly-time advice for
each player such that an -approximation is achieved whenever all
players follow advice. Using an appropriate bicriterion notion of approximate
demand queries (which can be computed in poly-time), we establish that (a
slight modification of) the [Assadi and Singla, 2019] mechanism achieves the
same -approximation in implementation in advised
strategies
Combinatorial Auctions via Posted Prices
We study anonymous posted price mechanisms for combinatorial auctions in a
Bayesian framework. In a posted price mechanism, item prices are posted, then
the consumers approach the seller sequentially in an arbitrary order, each
purchasing her favorite bundle from among the unsold items at the posted
prices. These mechanisms are simple, transparent and trivially dominant
strategy incentive compatible (DSIC).
We show that when agent preferences are fractionally subadditive (which
includes all submodular functions), there always exist prices that, in
expectation, obtain at least half of the optimal welfare. Our result is
constructive: given black-box access to a combinatorial auction algorithm A,
sample access to the prior distribution, and appropriate query access to the
sampled valuations, one can compute, in polytime, prices that guarantee at
least half of the expected welfare of A. As a corollary, we obtain the first
polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular
combinatorial auctions, given access to demand query oracles. Our results also
extend to valuations with complements, where the approximation factor degrades
linearly with the level of complementarity
Asymptotically Optimal Welfare of Posted Pricing for Multiple Items with MHR Distributions
We consider the problem of posting prices for unit-demand buyers if all
buyers have identically distributed valuations drawn from a distribution with
monotone hazard rate. We show that even with multiple items asymptotically
optimal welfare can be guaranteed.
Our main results apply to the case that either a buyer's value for different
items are independent or that they are perfectly correlated. We give mechanisms
using dynamic prices that obtain a -fraction of the optimal social welfare in expectation. Furthermore,
we devise mechanisms that only use static item prices and are -competitive compared to the
optimal social welfare. As we show, both guarantees are asymptotically optimal,
even for a single item and exponential distributions.Comment: To appear at the 29th Annual European Symposium on Algorithms (ESA
2021
Prophet Secretary for Combinatorial Auctions and Matroids
The secretary and the prophet inequality problems are central to the field of
Stopping Theory. Recently, there has been a lot of work in generalizing these
models to multiple items because of their applications in mechanism design. The
most important of these generalizations are to matroids and to combinatorial
auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and
Feldman et al. \cite{feldman2015combinatorial} show that for adversarial
arrival order of random variables the optimal prophet inequalities give a
-approximation. For many settings, however, it's conceivable that the
arrival order is chosen uniformly at random, akin to the secretary problem. For
such a random arrival model, we improve upon the -approximation and obtain
-approximation prophet inequalities for both matroids and
combinatorial auctions. This also gives improvements to the results of Yan
\cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who
worked in the special cases where we can fully control the arrival order or
when there is only a single item.
Our techniques are threshold based. We convert our discrete problem into a
continuous setting and then give a generic template on how to dynamically
adjust these thresholds to lower bound the expected total welfare.Comment: Preliminary version appeared in SODA 2018. This version improves the
writeup on Fixed-Threshold algorithm
Pricing for Online Resource Allocation: Intervals and Paths
We present pricing mechanisms for several online resource allocation problems
which obtain tight or nearly tight approximations to social welfare. In our
settings, buyers arrive online and purchase bundles of items; buyers' values
for the bundles are drawn from known distributions. This problem is closely
related to the so-called prophet-inequality of Krengel and Sucheston and its
extensions in recent literature. Motivated by applications to cloud economics,
we consider two kinds of buyer preferences. In the first, items correspond to
different units of time at which a resource is available; the items are
arranged in a total order and buyers desire intervals of items. The second
corresponds to bandwidth allocation over a tree network; the items are edges in
the network and buyers desire paths.
Because buyers' preferences have complementarities in the settings we
consider, recent constant-factor approximations via item prices do not apply,
and indeed strong negative results are known. We develop static, anonymous
bundle pricing mechanisms.
For the interval preferences setting, we show that static, anonymous bundle
pricings achieve a sublogarithmic competitive ratio, which is optimal (within
constant factors) over the class of all online allocation algorithms, truthful
or not. For the path preferences setting, we obtain a nearly-tight logarithmic
competitive ratio. Both of these results exhibit an exponential improvement
over item pricings for these settings. Our results extend to settings where the
seller has multiple copies of each item, with the competitive ratio decreasing
linearly with supply. Such a gradual tradeoff between supply and the
competitive ratio for welfare was previously known only for the single item
prophet inequality
A Bridge between Liquid and Social Welfare in Combinatorial Auctions with Submodular Bidders
We study incentive compatible mechanisms for Combinatorial Auctions where the
bidders have submodular (or XOS) valuations and are budget-constrained. Our
objective is to maximize the \emph{liquid welfare}, a notion of efficiency for
budget-constrained bidders introduced by Dobzinski and Paes Leme (2014). We
show that some of the known truthful mechanisms that best-approximate the
social welfare for Combinatorial Auctions with submodular bidders through
demand query oracles can be adapted, so that they retain truthfulness and
achieve asymptotically the same approximation guarantees for the liquid
welfare. More specifically, for the problem of optimizing the liquid welfare in
Combinatorial Auctions with submodular bidders, we obtain a universally
truthful randomized -approximate mechanism, where is the number
of items, by adapting the mechanism of Krysta and V\"ocking (2012).
Additionally, motivated by large market assumptions often used in mechanism
design, we introduce a notion of competitive markets and show that in such
markets, liquid welfare can be approximated within a constant factor by a
randomized universally truthful mechanism. Finally, in the Bayesian setting, we
obtain a truthful -approximate mechanism for the case where bidder
valuations are generated as independent samples from a known distribution, by
adapting the results of Feldman, Gravin and Lucier (2014).Comment: AAAI-1
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