In the classical Monge-Kantorovich problem, the transportation cost only depends on the amount of mass sent from sources to destinations and not on the paths followed by this mass. Thus, it does not allow for congestion effects. Using the notion of traffic intensity, we propose a variant taking into account congestion. This leads to an optimization problem posed on a set of probability measures on a suitable paths space. We establish existence of minimizers and give a characterization. As an application, we obtain existence and variational characterization of equilibria of Wardrop type in a continuous space setting.
Modeling Network Congestion under Demand Uncertainty Using Wardrop Principles
Eindhoven University of Technology（埃因霍温理工大学）；Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, Consiglio Nazionale delle Ricerche（意大利国家研究委员会系统与信息分析研究所）；Dipartimento di Ingegneria Informatica, Automatica e Gestionale, Sapienza Università di Roma（罗马大学信息与自动化与工商管理系）；ESSEC Business School（欧洲高等商学院）
In this paper we propose a LWR-like model for traffic flow on networks which allows one to track several groups of drivers, each of them being characterized only by their destination in the network. The path actually followed to reach the destination is not assigned a priori, and can be chosen by the drivers during the journey, taking decisions at junctions. The model is then used to describe three possible behaviors of drivers, associated to three different ways to solve the route choice problem: 1. Drivers ignore the presence of the other vehicles; 2. Drivers react to the current distributio…
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 games where each player's weight is small. We prove that when the number of players goes to infinity and their weights to zero, the random flows in all (mixed) Nash equilibria for the finite games converge in distribution to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the seco…
We study uniqueness of Nash equilibria in atomic splittable congestion games and derive a uniqueness result based on polymatroid theory: when the strategy space of every player is a bidirectional flow polymatroid, then equilibria are unique. Bidirectional flow polymatroids are introduced as a subclass of polymatroids possessing certain exchange properties. We show that important cases such as base orderable matroids can be recovered as a special case of bidirectional flow polymatroids. On the other hand we show that matroidal set systems are in some sense necessary to guarantee uniqueness of e…
In this paper, we compute $ε$-approximate Nash equilibria in atomic splittable polymatroid congestion games with convex Lipschitz continuous cost functions. The main approach relies on computing a pure Nash equilibrium for an associated integrally-splittable congestion game, where players can only split their demand in integral multiples of a common packet size. It is known that one can compute pure Nash equilibria for integrally-splittable congestion games within a running time that is pseudo-polynomial in the aggregated demand of the players. As the main contribution of this paper, we decide…
We study transportation networks controlled by dynamical feedback tolls. We consider a multiscale transportation network model whereby the dynamics of the traffic flows are intertwined with those of the drivers' route choices. The latter are influenced by the congestion status on the whole network as well as dynamic tolls set by the system operator. Our main result shows that a broad class of decentralized congestion-dependent tolls globally stabilise the transportation network around a Wardrop equilibrium. Moreover, using dynamic marginal cost tolls, stability of the transportation network ca…
Solving the road congestion problem is one of the most pressing issues in moderncities since it causes time wasting, pollution, higher industrial costs and huge roadmaintenance costs. Advances in ITS technologies and the advent of autonomousvehicles are changing mobility dramatically. They enable the implementation of acoordination mechanism, called coordinated traffic assignment, among the sat-navdevices aiming at assigning paths to drivers to eliminate congestion and to re-duce the total travel time in traffic networks. Among possible congestion avoidance methods, coordinated traffic assignm…
The objective of this paper is to analyze the existence of equilibria for a class of deterministic mean field games of controls. The interaction between players is due to both a congestion term and a price function which depends on the distributions of the optimal strategies. Moreover, final state and mixed state-control constraints are considered, the dynamics being nonlinear and affine with respect to the control. The existence of equilibria is obtained by Kakutani's theorem, applied to a fixed point formulation of the problem. Finally, uniqueness results are shown under monotonicity assumpt…
We review two models of optimal transport, where congestion effects during the transport can be possibly taken into account. The first model is Beckmann's one, where the transport activities are modeled by vector fields with given divergence. The second one is the model by Carlier et al. (SIAM J Control Optim 47: 1330-1350, 2008), which in turn is the continuous reformulation of Wardrop's model on graphs. We discuss the extensions of these models to their natural functional analytic setting and show that they are indeed equivalent, by using Smirnov decomposition theorem for normal 1-currents.
UC Santa Barbara（加州大学圣巴巴拉分校）；Stanford University（斯坦福大学）
Road congestion induces significant costs across the world, and road network disturbances, such as traffic accidents, can cause highly congested traffic patterns. If a planner had control over the routing of all vehicles in the network, they could easily reverse this effect. In a more realistic scenario, we consider a planner that controls autonomous cars, which are a fraction of all present cars. We study a dynamic routing game, in which the route choices of autonomous cars can be controlled and the human drivers react selfishly and dynamically. As the problem is prohibitively large, we use d…
Credit-based congestion pricing (CBCP) has emerged as a mechanism to alleviate the social inequity concerns of road congestion pricing - a promising strategy for traffic congestion mitigation - by providing low-income users with travel credits to offset some of their toll payments. While CBCP offers immense potential for addressing inequity issues that hamper the practical viability of congestion pricing, the deployment of CBCP in practice is nascent, and the potential efficacy and optimal design of CBCP schemes have yet to be formalized. In this work, we study the design of CBCP schemes to ac…
How can we design mechanisms to promote efficient use of shared resources? Here, we answer this question in relation to the well-studied class of atomic congestion games, used to model a variety of problems, including traffic routing. Within this context, a methodology for designing tolling mechanisms that minimize the system inefficiency (price of anarchy) exploiting solely local information is so far missing in spite of the scientific interest. In this manuscript we resolve this problem through a tractable linear programming formulation that applies to and beyond polynomial congestion games.…
This paper considers a mean field game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena. After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the mean field game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a furthe…
The University of Tokyo（东京大学）；NTT Communication Science Laboratories（NTT通信科学实验室）
We address Stackelberg models of combinatorial congestion games (CCGs); we aim to optimize the parameters of CCGs so that the selfish behavior of non-atomic players attains desirable equilibria. This model is essential for designing such social infrastructures as traffic and communication networks. Nevertheless, computational approaches to the model have not been thoroughly studied due to two difficulties: (I) bilevel-programming structures and (II) the combinatorial nature of CCGs. We tackle them by carefully combining (I) the idea of \textit{differentiable} optimization and (II) data structu…
Stanford University（斯坦福大学）；University of California, Santa Barbara（加州大学圣巴巴拉分校）
Traffic congestion has large economic and social costs. The introduction of autonomous vehicles can potentially reduce this congestion, both by increasing network throughput and by enabling a social planner to incentivize users of autonomous vehicles to take longer routes that can alleviate congestion on more direct roads. We formalize the effects of altruistic autonomy on roads shared between human drivers and autonomous vehicles. In this work, we develop a formal model of road congestion on shared roads based on the fundamental diagram of traffic. We consider a network of parallel roads and …
We consider a large population dynamic game in discrete time. The peculiarity of the game is that players are characterized by time-evolving types, and so reasonably their actions should not anticipate the future values of their types. When interactions between players are of mean-field kind, we relate Nash equilibria for such games to an asymptotic notion of dynamic Cournot-Nash equilibria. Inspired by the works of Blanchet and Carlier for the static situation, we interpret dynamic Cournot-Nash equilibria in the light of causal optimal transport theory. Further specializing to games of potent…
We study oligopolistic competition in service markets where firms offer a service to customers. The service quality of a firm - from the perspective of a customer - depends on the congestion and the charged price. A firm can set a price for the service offered and additionally decides on the service capacity in order to mitigate congestion. The total profit of a firm is derived from the gained revenue minus the capacity investment cost. Firms simultaneously set capacities and prices in order to maximize their profit and customers subsequently choose the services with lowest combined cost (cong…
In this work, we consider the problem of minimising the social cost in atomic congestion games. For this problem, we provide tight computational lower bounds along with taxation mechanisms yielding polynomial time algorithms with optimal approximation. Perhaps surprisingly, our results show that indirect interventions, in the form of efficiently computed taxation mechanisms, yield the same performance achievable by the best polynomial time algorithm, even when the latter has full control over the agents' actions. It follows that no other tractable approach geared at incentivizing desirable sys…
Columbia University（哥伦比亚大学）；National Technical University of Athens（雅典国立技术大学）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Criteo AI Lab（Criteo人工智能实验室）；Singapore University of Technology and Design（新加坡科技设计大学）
Understanding the behavior of no-regret dynamics in general $N$-player games is a fundamental question in online learning and game theory. A folk result in the field states that, in finite games, the empirical frequency of play under no-regret learning converges to the game's set of coarse correlated equilibria. By contrast, our understanding of how the day-to-day behavior of the dynamics correlates to the game's Nash equilibria is much more limited, and only partial results are known for certain classes of games (such as zero-sum or congestion games). In this paper, we study the dynamics of "…
