5,878 research outputs found
Welfare Maximization and Truthfulness in Mechanism Design with Ordinal Preferences
We study mechanism design problems in the {\em ordinal setting} wherein the
preferences of agents are described by orderings over outcomes, as opposed to
specific numerical values associated with them. This setting is relevant when
agents can compare outcomes, but aren't able to evaluate precise utilities for
them. Such a situation arises in diverse contexts including voting and matching
markets.
Our paper addresses two issues that arise in ordinal mechanism design. To
design social welfare maximizing mechanisms, one needs to be able to
quantitatively measure the welfare of an outcome which is not clear in the
ordinal setting. Second, since the impossibility results of Gibbard and
Satterthwaite~\cite{Gibbard73,Satterthwaite75} force one to move to randomized
mechanisms, one needs a more nuanced notion of truthfulness.
We propose {\em rank approximation} as a metric for measuring the quality of
an outcome, which allows us to evaluate mechanisms based on worst-case
performance, and {\em lex-truthfulness} as a notion of truthfulness for
randomized ordinal mechanisms. Lex-truthfulness is stronger than notions
studied in the literature, and yet flexible enough to admit a rich class of
mechanisms {\em circumventing classical impossibility results}. We demonstrate
the usefulness of the above notions by devising lex-truthful mechanisms
achieving good rank-approximation factors, both in the general ordinal setting,
as well as structured settings such as {\em (one-sided) matching markets}, and
its generalizations, {\em matroid} and {\em scheduling} markets.Comment: Some typos correcte
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
Designing Networks with Good Equilibria under Uncertainty
We consider the problem of designing network cost-sharing protocols with good
equilibria under uncertainty. The underlying game is a multicast game in a
rooted undirected graph with nonnegative edge costs. A set of k terminal
vertices or players need to establish connectivity with the root. The social
optimum is the Minimum Steiner Tree. We are interested in situations where the
designer has incomplete information about the input. We propose two different
models, the adversarial and the stochastic. In both models, the designer has
prior knowledge of the underlying metric but the requested subset of the
players is not known and is activated either in an adversarial manner
(adversarial model) or is drawn from a known probability distribution
(stochastic model).
In the adversarial model, the designer's goal is to choose a single,
universal protocol that has low Price of Anarchy (PoA) for all possible
requested subsets of players. The main question we address is: to what extent
can prior knowledge of the underlying metric help in the design? We first
demonstrate that there exist graphs (outerplanar) where knowledge of the
underlying metric can dramatically improve the performance of good network
design. Then, in our main technical result, we show that there exist graph
metrics, for which knowing the underlying metric does not help and any
universal protocol has PoA of , which is tight. We attack this
problem by developing new techniques that employ powerful tools from extremal
combinatorics, and more specifically Ramsey Theory in high dimensional
hypercubes.
Then we switch to the stochastic model, where each player is independently
activated. We show that there exists a randomized ordered protocol that
achieves constant PoA. By using standard derandomization techniques, we produce
a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu
Approximating Generalized Network Design under (Dis)economies of Scale with Applications to Energy Efficiency
In a generalized network design (GND) problem, a set of resources are
assigned to multiple communication requests. Each request contributes its
weight to the resources it uses and the total load on a resource is then
translated to the cost it incurs via a resource specific cost function. For
example, a request may be to establish a virtual circuit, thus contributing to
the load on each edge in the circuit. Motivated by energy efficiency
applications, recently, there is a growing interest in GND using cost functions
that exhibit (dis)economies of scale ((D)oS), namely, cost functions that
appear subadditive for small loads and superadditive for larger loads.
The current paper advances the existing literature on approximation
algorithms for GND problems with (D)oS cost functions in various aspects: (1)
we present a generic approximation framework that yields approximation results
for a much wider family of requests in both directed and undirected graphs; (2)
our framework allows for unrelated weights, thus providing the first
non-trivial approximation for the problem of scheduling unrelated parallel
machines with (D)oS cost functions; (3) our framework is fully combinatorial
and runs in strongly polynomial time; (4) the family of (D)oS cost functions
considered in the current paper is more general than the one considered in the
existing literature, providing a more accurate abstraction for practical energy
conservation scenarios; and (5) we obtain the first approximation ratio for GND
with (D)oS cost functions that depends only on the parameters of the resources'
technology and does not grow with the number of resources, the number of
requests, or their weights. The design of our framework relies heavily on
Roughgarden's smoothness toolbox (JACM 2015), thus demonstrating the possible
usefulness of this toolbox in the area of approximation algorithms.Comment: 39 pages, 1 figure. An extended abstract of this paper is to appear
in the 50th Annual ACM Symposium on the Theory of Computing (STOC 2018
- …