47 research outputs found
On the Price of Anarchy of Highly Congested Nonatomic Network Games
We consider nonatomic network games with one source and one destination. We
examine the asymptotic behavior of the price of anarchy as the inflow
increases. In accordance with some empirical observations, we show that, under
suitable conditions, the price of anarchy is asymptotic to one. We show with
some counterexamples that this is not always the case. The counterexamples
occur in very simple parallel graphs.Comment: 26 pages, 6 figure
Price of Anarchy in Bernoulli Congestion Games with Affine Costs
We consider an atomic congestion game in which each player participates in
the game with an exogenous and known probability , independently
of everybody else, or stays out and incurs no cost. We first prove that the
resulting game is potential. Then, we compute the parameterized price of
anarchy to characterize the impact of demand uncertainty on the efficiency of
selfish behavior. It turns out that the price of anarchy as a function of the
maximum participation probability is a nondecreasing
function. The worst case is attained when players have the same participation
probabilities . For the case of affine costs, we provide an
analytic expression for the parameterized price of anarchy as a function of
. This function is continuous on , is equal to for , and increases towards when . Our work can be interpreted as
providing a continuous transition between the price of anarchy of nonatomic and
atomic games, which are the extremes of the price of anarchy function we
characterize. We show that these bounds are tight and are attained on routing
games -- as opposed to general congestion games -- with purely linear costs
(i.e., with no constant terms).Comment: 29 pages, 6 figure
The Price of Anarchy in Routing Games as a Function of the Demand
Most of the literature on the price of anarchy has focused on worst-case
bounds for specific classes of games, such as routing games or more general
congestion games. Recently, the price of anarchy in routing games has been
studied as a function of the traffic demand, providing asymptotic results in
light and heavy traffic. In this paper we study the price of anarchy in
nonatomic routing games in the intermediate region of the demand. We begin by
establishing some smoothness properties of Wardrop equilibria and social optima
for general smooth costs. In the case of affine costs we show that the
equilibrium is piecewise linear, with break points at the demand levels at
which the set of active paths changes. We prove that the number of such break
points is finite, although it can be exponential in the size of the network.
Exploiting a scaling law between the equilibrium and the social optimum, we
derive a similar behavior for the optimal flows. We then prove that in any
interval between break points the price of anarchy is smooth and it is either
monotone, or unimodal with a minimum attained on the interior of the interval.
We deduce that for affine costs the maximum of the price of anarchy can only
occur at the break points. For general costs we provide counterexamples showing
that the set of break points is not always finite.Comment: 22 pages, 6 figure
The price of anarchy in routing games as a function of the demand
The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smoothness properties of Wardrop equilibria and social optima for general smooth costs. In the case of affine costs we show that the equilibrium is piecewise linear, with break points at the demand levels at which the set of active paths changes. We prove that the number of such break points is finite, although it can be exponential in the size of the network. Exploiting a scaling law between the equilibrium and the social optimum, we derive a similar behavior for the optimal flows. We then prove that in any interval between break points the price of anarchy is smooth and it is either monotone (decreasing or increasing) over the full interval, or it decreases up to a certain minimum point in the interior of the interval and increases afterwards. We deduce that for affine costs the maximum of the price of anarchy can only occur at the break points. For general costs we provide counterexamples showing that the set of break points is not always finite
Boltzmann meets Nash: Energy-efficient routing in optical networks under uncertainty
Motivated by the massive deployment of power-hungry data centers for service
provisioning, we examine the problem of routing in optical networks with the
aim of minimizing traffic-driven power consumption. To tackle this issue,
routing must take into account energy efficiency as well as capacity
considerations; moreover, in rapidly-varying network environments, this must be
accomplished in a real-time, distributed manner that remains robust in the
presence of random disturbances and noise. In view of this, we derive a pricing
scheme whose Nash equilibria coincide with the network's socially optimum
states, and we propose a distributed learning method based on the Boltzmann
distribution of statistical mechanics. Using tools from stochastic calculus, we
show that the resulting Boltzmann routing scheme exhibits remarkable
convergence properties under uncertainty: specifically, the long-term average
of the network's power consumption converges within of its
minimum value in time which is at most ,
irrespective of the fluctuations' magnitude; additionally, if the network
admits a strict, non-mixing optimum state, the algorithm converges to it -
again, no matter the noise level. Our analysis is supplemented by extensive
numerical simulations which show that Boltzmann routing can lead to a
significant decrease in power consumption over basic, shortest-path routing
schemes in realistic network conditions.Comment: 24 pages, 4 figure