Wardrop Model Research Articles

Approximation and 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 games in which 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 second setting, we consider an increasing number of players with a unit weight that participate in the game with a decreasingly small probability. In this case, the Nash equilibrium flows converge in total variation toward Poisson random variables whose expected values are Wardrop equilibria of a different nonatomic game with suitably defined costs. The latter can be viewed as symmetric equilibria in a Poisson game in the sense of Myerson, establishing a plausible connection between the Wardrop model for routing games and the stochastic fluctuations observed in real traffic. In both settings, we provide explicit approximation bounds, and we study the convergence of the price of anarchy. Beyond the case of congestion games, we prove a general result on the convergence of large games with random players toward Poisson games.Funding: R. Cominetti gratefully acknowledges the support of Proyecto Anillo [Grant ANID/PIA/ACT192094]. M. Scarsini’s work was partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni, Istituto Nazionale di Alta Matematica [Grant 2020 “Random walks on random games”] and the Progetti di Rilevante Interesse Nazionale 2017 [Grant “Algorithms, Games, and Digital Markets”]. This research also received partial support from the European Cooperation in Science and Technology [Action European Network for Game Theory].

A Continuous Model of Transportation Revisited

We review two models of optimal transport where congestion effects during the transport can possibly by 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., which is in turn a 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's decomposition theorem for normal 1-currents.

Fast Convergence to Wardrop Equilibria by Adaptive Sampling Methods

We study the question of whether a large population of agents in a traffic network is able to converge to an equilibrium quickly. To that end, we consider a round-based variant of the Wardrop model. Every agent is allowed to reroute its traffic once in a while with the aim of finding a path with minimal latency. As a first result we find that using a replication policy which allows agents to imitate others gives rise to a bicriterial approximate equilibrium very quickly. In particular, the time bound depends logarithmically on the ratio between minimum and maximum latency but is otherwise independent of the network size. In the single-commodity case, this bicriteria approximate equilibrium has an intuitive interpretation as a state in which almost all agents are almost happy. This kind of approximate equilibrium, however, is transient. In order to reach a global approximation, we need to add an exploration component which enables the agents to explore the strategy space independently of the other agents. Although it can be shown that, when used exclusively, exploration policies imply an exponential lower bound, applying exploration carefully allows the population to approximate the global Wardrop equilibrium in polynomial time. Since the distributed and concurrent fashion of our policies bears the risk of oscillating behavior, we must take into account the steepness of the latency functions. We show that the relevant parameter is elasticity, a parameter closely related to the polynomial degree. This improves significantly over earlier results which depend on the absolute slope and therefore have a pseudopolynomial flavor.

Nash equilibria in discrete routing games with convex latency functions

In a discrete routing game, each of n selfish users employs a mixed strategy to ship her ( unsplittable) traffic over m parallel links. The ( expected) latency on a link is determined by an arbitrary non-decreasing, non-constant and convex latency function ϕ. In a Nash equilibrium, each user alone is minimizing her ( Expected) Individual Cost, which is the (expected) latency on the link she chooses. To evaluate Nash equilibria, we formulate Social Cost as the sum of the users' (Expected) Individual Costs. The Price of Anarchy is the worst-case ratio of Social Cost for a Nash equilibrium over the least possible Social Cost. A Nash equilibrium is pure if each user deterministically chooses a single link; a Nash equilibrium is fully mixed if each user chooses each link with non-zero probability. We obtain: For the case of identical users, the Social Cost of any Nash equilibrium is no more than the Social Cost of the fully mixed Nash equilibrium, which may exist only uniquely. Moreover, instances admitting a fully mixed Nash equilibrium enjoy an efficient characterization. For the case of identical users, we derive two upper bounds on the Price of Anarchy: For the case of identical links with a monomial latency function ϕ ( x ) = x d , the Price of Anarchy is the Bell number of order d + 1 . For pure Nash equilibria, a generic upper bound from the Wardrop model can be transfered to discrete routing games. For polynomial latency functions with non-negative coefficients and degree d, this yields an upper bound of d + 1 . For the case of identical users, a pure Nash equilibrium (and thereby an optimum pure assignment) can be computed in time O ( m log m log n ) . For the general case, computing the best or the worst pure Nash equilibrium is NP -complete, even for identical links with an identity latency function.

Adaptive routing with stale information

We investigate the behaviour of load-adaptive rerouting policies in the Wardrop model where decisions must be made on the basis of stale information. In this model, each one of an infinite number of agents controls an infinitesimal amount of flow, thus contributing to a network flow which induces latency. In our dynamic extension of this model, agents are activated in a concurrent and asynchronous fashion and may reroute their flow with the aim of reducing their sustained latency. It is a well-known problem that in settings where latency information is not always up-to-date, such behaviour may lead to oscillation effects which seriously harm network performance. Two quantities determine the difficulty of avoiding oscillation: the steepness of the latency functions and the maximum possible age of the information T.In this work we ask for conditions that the rerouting policies must adhere to in order to converge to an equilibrium despite the information being stale. We consider simple policies which sample another path in a first step and then migrate from the current path to the new one with a probability that is a function of the anticipated latency gain. In fact we can show that our class of policies guarantees convergence if the latter migration probability function satisfies a certain smoothness condition that resembles Lipschitz continuity. It turns out that for smooth adaptation policies where the migration probability is chosen small enough relative to the inverse of the steepness of the latency functions and T, the population actually converges to an equilibrium.In addition, we analyse the speed of convergence towards approximate equilibria of two specific variants of smooth adaptive routing policies, e.g., for a replication policy adopted from evolutionary game theory.

The price of anarchy for polynomial social cost

In this work, we consider an interesting variant of the well studied KP model for selfish routing on parallel links, which reflects some influence from the much older Wardrop model [J.G. Wardrop, Some theoretical aspects of road traffic research, Proc. Inst. of Civil Eng. Part II 1 (1956) 325–378]. In the new model, user traffics are still unsplittable and links are identical. Social cost is now the expectation of the sum, over all links, of latency costs; each latency cost is modeled as a certain polynomial latency cost function evaluated at the latency incurred by all users choosing the link. The resulting social cost is called polynomial social cost, or monomial social cost when the latency cost function is a monomial. All considered polynomials are of degree d, where d ⩾ 2 , and have non-negative coefficients. We are interested in evaluating Nash equilibria in this model, and we use the monomial price of anarchy (MPoA) and the polynomial price of anarchy (PPoA) as our evaluation measures. Through establishing some remarkable relations of these costs and measures to some classical combinatorial numbers such as the Stirling numbers of the second kind and the Bell numbers, we obtain a multitude of results: • For the special case of identical users: ∘ The fully mixed Nash equilibrium, where all probabilities are strictly positive, maximizes polynomial social cost. ∘ The MPoA is no more than B d , the Bell number of order d. This immediately implies that the PPoA is no more than ∑ 1 ⩽ t ⩽ d B t . For the special case of two links, the MPoA is no more than 2 d - 2 ( 1 + ( 1 / n ) d - 1 ) , and this bound is tight for n = 2 . • The MPoA is exactly ( ( 2 d - 1 ) d / ( d - 1 ) ( 2 d - 2 ) d - 1 ) ( ( d - 1 ) / d ) d for pure Nash equilibria. This immediately implies that the PPoA is no more than ∑ 2 ⩽ t ⩽ d ( ( 2 t - 1 ) t / ( t - 1 ) ( 2 t - 2 ) t - 1 ) ( ( t - 1 ) / t ) t .

Benefit estimation of transport projects—a representative consumer approach

In this paper, by focusing on the forms of the utility function of a representative consumer, we explain the three typical models of benefit estimation for transport projects: the basic model, the Wardrop model, and the logit model. The basic model is a representative consumer model with a quasi-linear utility function that is additively separable between numeraire and transport services. The paper clarifies the relationship between these models and derives their implications for benefit estimation. We find that the Wardrop and logit models are special cases of the basic model and that the logit model degenerates to the Wardrop model in a limiting case. Although this relationship implies that one can apply the benefit estimation method for the basic model whatever method is used to estimate transport demand, the Wardrop and logit models have useful features for conducting benefit estimation for new routes.