79 research outputs found
An Incentive Compatible, Efficient Market for Air Traffic Flow Management
We present a market-based approach to the Air Traffic Flow Management (ATFM)
problem. The goods in our market are delays and buyers are airline companies;
the latter pay money to the FAA to buy away the desired amount of delay on a
per flight basis. We give a notion of equilibrium for this market and an LP
whose solution gives an equilibrium allocation of flights to landing slots as
well as equilibrium prices for the landing slots. Via a reduction to matching,
we show that this equilibrium can be computed combinatorially in strongly
polynomial time. Moreover, there is a special set of equilibrium prices, which
can be computed easily, that is identical to the VCG solution, and therefore
the market is incentive compatible in dominant strategy.Comment: arXiv admin note: substantial text overlap with arXiv:1109.521
Smoothed Efficient Algorithms and Reductions for Network Coordination Games
Worst-case hardness results for most equilibrium computation problems have
raised the need for beyond-worst-case analysis. To this end, we study the
smoothed complexity of finding pure Nash equilibria in Network Coordination
Games, a PLS-complete problem in the worst case. This is a potential game where
the sequential-better-response algorithm is known to converge to a pure NE,
albeit in exponential time. First, we prove polynomial (resp. quasi-polynomial)
smoothed complexity when the underlying game graph is a complete (resp.
arbitrary) graph, and every player has constantly many strategies. We note that
the complete graph case is reminiscent of perturbing all parameters, a common
assumption in most known smoothed analysis results.
Second, we define a notion of smoothness-preserving reduction among search
problems, and obtain reductions from -strategy network coordination games to
local-max-cut, and from -strategy games (with arbitrary ) to
local-max-cut up to two flips. The former together with the recent result of
[BCC18] gives an alternate -time smoothed algorithm for the
-strategy case. This notion of reduction allows for the extension of
smoothed efficient algorithms from one problem to another.
For the first set of results, we develop techniques to bound the probability
that an (adversarial) better-response sequence makes slow improvements on the
potential. Our approach combines and generalizes the local-max-cut approaches
of [ER14,ABPW17] to handle the multi-strategy case: it requires a careful
definition of the matrix which captures the increase in potential, a tighter
union bound on adversarial sequences, and balancing it with good enough rank
bounds. We believe that the approach and notions developed herein could be of
interest in addressing the smoothed complexity of other potential and/or
congestion games
Nash Social Welfare Approximation for Strategic Agents
The fair division of resources is an important age-old problem that has led
to a rich body of literature. At the center of this literature lies the
question of whether there exist fair mechanisms despite strategic behavior of
the agents. A fundamental objective function used for measuring fair outcomes
is the Nash social welfare, defined as the geometric mean of the agent
utilities. This objective function is maximized by widely known solution
concepts such as Nash bargaining and the competitive equilibrium with equal
incomes. In this work we focus on the question of (approximately) implementing
the Nash social welfare. The starting point of our analysis is the Fisher
market, a fundamental model of an economy, whose benchmark is precisely the
(weighted) Nash social welfare. We begin by studying two extreme classes of
valuations functions, namely perfect substitutes and perfect complements, and
find that for perfect substitutes, the Fisher market mechanism has a constant
approximation: at most 2 and at least e1e. However, for perfect complements,
the Fisher market does not work well, its bound degrading linearly with the
number of players.
Strikingly, the Trading Post mechanism---an indirect market mechanism also
known as the Shapley-Shubik game---has significantly better performance than
the Fisher market on its own benchmark. Not only does Trading Post achieve an
approximation of 2 for perfect substitutes, but this bound holds for all
concave utilities and becomes arbitrarily close to optimal for Leontief
utilities (perfect complements), where it reaches for every
. Moreover, all the Nash equilibria of the Trading Post mechanism
are pure for all concave utilities and satisfy an important notion of fairness
known as proportionality
Prophet Inequalities for Cost Minimization
Prophet inequalities for rewards maximization are fundamental to optimal
stopping theory with several applications to mechanism design and online
optimization. We study the cost minimization counterpart of the classical
prophet inequality, where one is facing a sequence of costs in an online manner and must stop at some point and take the last cost
seen. Given that the 's are independent, drawn from known distributions,
the goal is to devise a stopping strategy that minimizes the expected cost.
If the 's are not identically distributed, then no strategy can achieve
a bounded approximation if the arrival order is adversarial or random. This
leads us to consider the case where the 's are I.I.D.. For the I.I.D.
case, we give a complete characterization of the optimal stopping strategy, and
show that, if our distribution satisfies a mild condition, then the optimal
stopping strategy achieves a tight (distribution-dependent) constant-factor
approximation.
Our techniques provide a novel approach to analyze prophet inequalities,
utilizing the hazard rate of the distribution. We also show that when the
hazard rate is monotonically increasing (i.e. the distribution is MHR), this
constant is at most , and this is optimal for MHR distributions.
For the classical prophet inequality, single-threshold strategies can achieve
the optimal approximation factor. Motivated by this, we analyze
single-threshold strategies for the cost prophet inequality problem. We design
a threshold that achieves a
-factor approximation,
where the exponent in the logarithmic factor is a distribution-dependent
constant, and we show a matching lower bound.
We note that our results can be used to design approximately optimal posted
price-style mechanisms for procurement auctions which may be of independent
interest.Comment: 38 page
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