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34
Selfish Routing In Capacitated Networks
 MATHEMATICS OF OPERATIONS RESEARCH
, 2003
"... According to Wardrop's first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason wh ..."
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Cited by 100 (6 self)
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According to Wardrop's first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason why it should be close to a solution of minimal total travel time, i.e. the system optimum. In this paper, we offer extensions of recent positive results on the efficiency of Nash equilibria in traffic networks. In contrast to prior work, we present results for networks with capacities and for latency functions that are nonconvex, nondifferentiable and even discontinuous. The inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows. In this more general model, multiple Nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient. Nonetheless, our main result shows that the best equilibrium is as efficient as in the model without capacities. Moreover, this holds true for broader classes of travel cost functions than considered hitherto.
TimeExpanded Graphs for FlowDependent Transit Times
 Proc. 10th Annual European Symposium on Algorithms
, 2002
"... Motivated by applications in road tra#c control, we study flows in networks featuring special characteristics. Firstly, there are transit times on the arcs of the network which specify the amount of time it takes for flow to travel through an arc; in particular, flow values on arcs may change over t ..."
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Cited by 25 (3 self)
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Motivated by applications in road tra#c control, we study flows in networks featuring special characteristics. Firstly, there are transit times on the arcs of the network which specify the amount of time it takes for flow to travel through an arc; in particular, flow values on arcs may change over time. Secondly, the transit time of an arc varies with the current amount of flow using this arc. The latter feature is crucial for various reallife applications; yet, it dramatically increases the degree of di#culty of the resulting optimization problems. While almost all flow problems with constant transit times on the arcs can be solved e#ciently by applying classical (static) flow algorithms in a corresponding timeexpanded network, no such approach was known for flowdependent transit times, up to now. One main contribution of this paper is a timeexpanded network with flowdependent transit times to which the whole algorithmic toolbox developed for static flows can be applied. Although this approach does not entirely capture the behavior of flows over time with flowdependent transit times, we present approximation results which provide evidence of its surprising quality.
Optimal routing of traffic flows with length restrictions in networks with congestion
, 1999
"... When traffic flows are routed through a road network it is desirable to minimize the total road usage. Since a route guidance system can only recommend paths to the drivers, special care has to be taken not to route them over paths they perceive as too long. This leads in a simplified model to a n ..."
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Cited by 9 (3 self)
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When traffic flows are routed through a road network it is desirable to minimize the total road usage. Since a route guidance system can only recommend paths to the drivers, special care has to be taken not to route them over paths they perceive as too long. This leads in a simplified model to a nonlinear multicommodity flow problem with constraints on the available paths. In this article an algorithm for this problem is given, which combines the convex combinations algorithm by Frank and Wolfe with column generation and algorithms for the constrained shortest path problem. Computational results stemming from a cooperation with DaimlerChrysler are presented.
Mathematical Models for Transportation Demand Analysis
, 1996
"... this paper, we will concentrate on the overspeci#cation arising from the ASCs and will not consider other possible errors sources. ..."
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Cited by 7 (1 self)
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this paper, we will concentrate on the overspeci#cation arising from the ASCs and will not consider other possible errors sources.
Relaxation Criteria for Iterated Traffic Simulations
, 1997
"... Iterated transportation microsimulations adjust their travelers' route plans by iterating between the microsimulation and the route planner and adjusting the route choice of individuals based on the preceeding microsimulations. Empirically, this process give good results; but it is usually ..."
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Cited by 7 (6 self)
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Iterated transportation microsimulations adjust their travelers' route plans by iterating between the microsimulation and the route planner and adjusting the route choice of individuals based on the preceeding microsimulations. Empirically, this process give good results; but it is usually unclear when to stop the iterative process, when one wants to model realworld traffic. This paper investigates several criteria to judge relaxation of the iterative process. The paper concentrates on criteria that are related to the decisionmaking process of the drivers.
StierMoses. Stochastic selfish routing
 In SAGT
, 2011
"... Abstract. We embark on an agenda to investigate how stochastic delays and risk aversion transform traditional models of routing games and the corresponding equilibrium concepts. Moving from deterministic to stochastic delays with riskaverse players introduces nonconvexities that make the network ga ..."
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Cited by 6 (2 self)
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Abstract. We embark on an agenda to investigate how stochastic delays and risk aversion transform traditional models of routing games and the corresponding equilibrium concepts. Moving from deterministic to stochastic delays with riskaverse players introduces nonconvexities that make the network game more difficult to analyze even if one assumes that the variability of delays is exogenous. (For example, even computing players ’ best responses has an unknown complexity [24].) This paper focuses on equilibrium existence and characterization in the different settings of atomic vs. nonatomic players and exogenous vs. endogenous factors causing the variability of edge delays. We also show that succinct representations of equilibria always exist even though the game is nonadditive, i.e., the cost along a path is not a sum of costs over edges of the path as is typically assumed in selfish routing problems. Finally, we investigate the inefficiencies resulting from the stochastic nature of delays. We prove that under exogenous stochastic delays, the price of anarchy is exactly the same as in the corresponding game with deterministic delays. This implies that the stochastic delays and players’ risk aversion do not further degrade a system in the worstcase more than the selfishness of players. Keywords: Nonadditive nonatomic congestion game, stochastic Nash equilibrium, stochastic Wardrop equilibrium, risk aversion. 1
Wardrop equilibria
 Encyclopedia of Operations Research and Management Science
, 2010
"... A common behavioral assumption in the study of transportation and telecommunication networks is that travelers or packets, respectively, choose routes that they perceive as being the shortest under the prevailing traffic conditions [1]. The situation resulting from these individual decisions is one ..."
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Cited by 5 (0 self)
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A common behavioral assumption in the study of transportation and telecommunication networks is that travelers or packets, respectively, choose routes that they perceive as being the shortest under the prevailing traffic conditions [1]. The situation resulting from these individual decisions is one in which drivers cannot reduce their journey times by unilaterally choosing another route, which prompted Knight [2] to call the resulting traffic pattern an equilibrium. Nowadays, it is indeed known as the Wardrop (or user) equilibrium [3], and it is effectively thought of as a steady state evolving after a transient phase in which travelers successively adjust their route choices until a situation with stable route travel costs and route flows has been reached [4]. In a seminal contribution, Wardrop [5, p. 345] stated two principles that formalize this notion of equilibrium and the alternative postulate of the minimization of the total travel costs. His first principle reads: The journey times on all the routes actually used are equal, and less than those which would be experienced by a single vehicle on any unused route. Wardrop’s first principle of route choice, which is identical to the notion postulated by Kohl [1] and Knight [2], became accepted as a sound and simple behavioral principle to describe the spreading of trips over alternate routes due to congested conditions [6]. Since its introduction in the context of transportation networks in 1952 and its mathematical formalization by Beckmann
Approximating earliest arrival flows with flowdependent transit times
 IN 29TH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2004, PRAGUE, CZECH REPUBLIC, AUGUST 2004, LNCS 3153
, 2004
"... For the earliest arrival flow problem one is given a network G = (V, A) with capacities u(a) and transit times τ(a) on its arcs a ∈ A, together with a source and a sink vertex s, t ∈ V. The objective is to send flow from s to t that moves through the network over time, such that for each point in ..."
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For the earliest arrival flow problem one is given a network G = (V, A) with capacities u(a) and transit times τ(a) on its arcs a ∈ A, together with a source and a sink vertex s, t ∈ V. The objective is to send flow from s to t that moves through the network over time, such that for each point in time θ ∈ [0, T) the maximum possible amount of flow reaches t. If, for each θ ∈ [0, T) this flow is a maximum flow for time horizon θ, then it is called earliest arrival flow. In practical applications a higher congestion of an arc in the network often implies a considerable increase in transit time. Therefore, in this paper we study the earliest arrival problem for the case that the transit time of each arc in the network at each time θ depends on the flow on this particular arc at that time θ. For constant transit times it has been shown by Gale that earliest arrival flows exist for any network. We give examples, showing that this is no longer true for flowdependent transit times. For that reason we define an optimization version of this problem where the objective is to find flows that are almost earliest arrival flows. In particular, we are interested in flows that, for each θ ∈ [0, T), need only αtimes longer to send the maximum flow to the sink. We give both constant lower and upper bounds on α; furthermore, we present a constant factor approximation algorithm for this problem. Finally, we give some computational results to show the practicability of the designed approximation algorithm.