381,336 research outputs found
Average-case Approximation Ratio of Scheduling without Payments
Apart from the principles and methodologies inherited from Economics and Game
Theory, the studies in Algorithmic Mechanism Design typically employ the
worst-case analysis and approximation schemes of Theoretical Computer Science.
For instance, the approximation ratio, which is the canonical measure of
evaluating how well an incentive-compatible mechanism approximately optimizes
the objective, is defined in the worst-case sense. It compares the performance
of the optimal mechanism against the performance of a truthful mechanism, for
all possible inputs.
In this paper, we take the average-case analysis approach, and tackle one of
the primary motivating problems in Algorithmic Mechanism Design -- the
scheduling problem [Nisan and Ronen 1999]. One version of this problem which
includes a verification component is studied by [Koutsoupias 2014]. It was
shown that the problem has a tight approximation ratio bound of (n+1)/2 for the
single-task setting, where n is the number of machines. We show, however, when
the costs of the machines to executing the task follow any independent and
identical distribution, the average-case approximation ratio of the mechanism
given in [Koutsoupias 2014] is upper bounded by a constant. This positive
result asymptotically separates the average-case ratio from the worst-case
ratio, and indicates that the optimal mechanism for the problem actually works
well on average, although in the worst-case the expected cost of the mechanism
is Theta(n) times that of the optimal cost
Improved Revenue Bounds for Posted-Price and Second-Price Mechanisms
We study revenue maximization through sequential posted-price (SPP)
mechanisms in single-dimensional settings with buyers and independent but
not necessarily identical value distributions. We construct the SPP mechanisms
by considering the best of two simple pricing rules: one that imitates the
revenue optimal mchanism, namely the Myersonian mechanism, via the taxation
principle and the other that posts a uniform price. Our pricing rules are
rather generalizable and yield the first improvement over long-established
approximation factors in several settings. We design factor-revealing
mathematical programs that crisply capture the approximation factor of our SPP
mechanism. In the single-unit setting, our SPP mechanism yields a better
approximation factor than the state of the art prior to our work (Azar,
Chiplunkar & Kaplan, 2018). In the multi-unit setting, our SPP mechanism yields
the first improved approximation factor over the state of the art after over
nine years (Yan, 2011 and Chakraborty et al., 2010). Our results on SPP
mechanisms immediately imply improved performance guarantees for the equivalent
free-order prophet inequality problem. In the position auction setting, our SPP
mechanism yields the first higher-than approximation factor. In eager
second-price (ESP) auctions, our two simple pricing rules lead to the first
improved approximation factor that is strictly greater than what is obtained by
the SPP mechanism in the single-unit setting.Comment: Accepted to Operations Researc
Budget Feasible Mechanisms
We study a novel class of mechanism design problems in which the outcomes are
constrained by the payments. This basic class of mechanism design problems
captures many common economic situations, and yet it has not been studied, to
our knowledge, in the past. We focus on the case of procurement auctions in
which sellers have private costs, and the auctioneer aims to maximize a utility
function on subsets of items, under the constraint that the sum of the payments
provided by the mechanism does not exceed a given budget. Standard mechanism
design ideas such as the VCG mechanism and its variants are not applicable
here. We show that, for general functions, the budget constraint can render
mechanisms arbitrarily bad in terms of the utility of the buyer. However, our
main result shows that for the important class of submodular functions, a
bounded approximation ratio is achievable. Better approximation results are
obtained for subclasses of the submodular functions. We explore the space of
budget feasible mechanisms in other domains and give a characterization under
more restricted conditions
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
Fast Convex Decomposition for Truthful Social Welfare Approximation
Approximating the optimal social welfare while preserving truthfulness is a
well studied problem in algorithmic mechanism design. Assuming that the social
welfare of a given mechanism design problem can be optimized by an integer
program whose integrality gap is at most , Lavi and Swamy~\cite{Lavi11}
propose a general approach to designing a randomized -approximation
mechanism which is truthful in expectation. Their method is based on
decomposing an optimal solution for the relaxed linear program into a convex
combination of integer solutions. Unfortunately, Lavi and Swamy's decomposition
technique relies heavily on the ellipsoid method, which is notorious for its
poor practical performance. To overcome this problem, we present an alternative
decomposition technique which yields an approximation
and only requires a quadratic number of calls to an integrality gap verifier
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