108 research outputs found
Price of Anarchy for Non-atomic Congestion Games with Stochastic Demands
We generalize the notions of user equilibrium and system optimum to
non-atomic congestion games with stochastic demands. We establish upper bounds
on the price of anarchy for three different settings of link cost functions and
demand distributions, namely, (a) affine cost functions and general
distributions, (b) polynomial cost functions and general positive-valued
distributions, and (c) polynomial cost functions and the normal distributions.
All the upper bounds are tight in some special cases, including the case of
deterministic demands.Comment: 31 page
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
Atomic congestion games with random players : network equilibrium and the price of anarchy
In this paper, we present a new model of congestion games with finite and random number of players, and an analytical method to compute the random path and link flows. We study the equilibrium condition, reformulate it as an equivalent variational inequality problem, and establish the existence and non-uniqueness of the equilibria. We also upper bound the price of anarchy with affine cost functions to characterize the quality of the equilibria. The upper bound is tight in some special cases, including the case of deterministic players. Finally a general lower bound is also provided
The Anarchy-Stability Tradeoff in Congestion Games
This work focuses on the design of incentive mechanisms in congestion games,
a commonly studied model for competitive resource sharing. While the majority
of the existing literature on this topic focuses on unilaterally optimizing the
worst case performance (i.e., price of anarchy), in this manuscript we
investigate whether optimizing for the worst case has consequences on the best
case performance (i.e., price of stability). Perhaps surprisingly, our results
show that there is a fundamental tradeoff between these two measures of
performance. Our main result provides a characterization of this tradeoff in
terms of upper and lower bounds on the Pareto frontier between the price of
anarchy and the price of stability. Interestingly, we demonstrate that the
mechanism that optimizes the price of anarchy inherits a matching price of
stability, thereby implying that the best equilibrium is not necessarily any
better than the worst equilibrium for such a design choice. Our results also
establish that, in several well-studied cases, the unincentivized setting does
not even lie on the Pareto frontier, and that any incentive with price of
stability equal to 1 incurs a much higher price of anarchy.Comment: 27 pages, 1 figure, 1 tabl
Convergence of Large Atomic Congestion Games
We consider the question of whether, and in what sense, Wardrop equilibria
provide a good approximation for Nash equilibria in atomic unsplittable
congestion games with a large number of small players. We examine two different
definitions of small players. In the first setting, we consider a sequence of
games with an increasing number of players where each player's weight tends to
zero. We prove that all (mixed) Nash equilibria of the finite games converge to
the set of Wardrop equilibria of the corresponding nonatomic limit game. In the
second setting, we consider again an increasing number of players but now each
player has a unit weight and participates in the game with a probability
tending to zero. In this case, the Nash equilibria converge to the set of
Wardrop equilibria of a different nonatomic game with suitably defined costs.
The latter can also be seen as a Poisson game in the sense of Myerson (1998),
establishing a precise connection between the Wardrop model and the empirical
flows observed in real traffic networks that exhibit stochastic fluctuations
well described by Poisson distributions. In both settings we give explicit
upper bounds on the rates of convergence, from which we also derive the
convergence of the price of anarchy. Beyond the case of congestion games, we
establish a general result on the convergence of large games with random
players towards Poisson games.Comment: 34 pages, 3 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
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