Wardrop Equilibrium in Discrete-Time Selfish Routing with Time-Varying Bounded Delays

This paper presents a multi-commodity, discrete- time, distributed and non-cooperative routing algorithm, which is proved to converge to an equilibrium in the presence of heterogeneous, unknown, time-varying but bounded delays. Under mild assumptions on the latency functions which describe the cost associated to the network paths, two algorithms are proposed: the former assumes that each commodity relies only on measurements of the latencies associated to its own paths; the latter assumes that each commodity has (at least indirectly) access to the measures of the latencies of all the network paths. Both algorithms are proven to drive the system state to an invariant set which approximates and contains the Wardrop equilibrium, defined as a network state in which no traffic flow over the network paths can improve its routing unilaterally, with the latter achieving a better reconstruction of the Wardrop equilibrium. Numerical simulations show the effectiveness of the proposed approach

Sensitivity of wardrop equilibria

We study the sensitivity of equilibria in the well-known game theoretic traffic model due to Wardrop. We mostly consider single-commodity networks. Suppose, given a unit demand flow at Wardrop equilibrium, one increases the demand by ε or removes an edge carrying only an ε-fraction of flow. We study how the equilibrium responds to such an ε-change. Our first surprising finding is that, even for linear latency functions, for every ε> 0, there are networks in which an ε-change causes every agent to change its path in order to recover equilibrium. Nevertheless, we can prove that, for general latency functions, the flow increase or decrease on every edge is at most ε. Examining the latency at equilibrium, we concentrate on polynomial latency functions of degree at most p with nonnegative coefficients. We show that, even though the relative increase in the latency of an edge due to an ε-change in the demand can be unbounded, the path latency at equilibrium increases at most by a factor of (1 + ε) p . The increase of the price of anarchy is shown to be upper bounded by the same factor. Both bounds are shown to be tight. Let us remark that all our bounds are tight. For the multi-commodity case, we present examples showing that neither the change in edge flows nor the change in the path latency can be bounded

Models and applications of Optimal Transport in Economics, Traffic and Urban Planning

Some optimization or equilibrium problems involving somehow the concept of optimal transport are presented in these notes, mainly devoted to applications to economic and game theory settings. A variant model of transport, taking into account traffic congestion effects is the first topic, and it shows various links with Monge-Kantorovich theory and PDEs. Then, two models for urban planning are introduced. The last section is devoted to two problems from economics and their translation in the language of optimal transport

A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks

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 distribution of traffic, but they do not forecast what will happen at later times; 3. Drivers take into account the current and future distribution of vehicles. Notice that, in the latter case, we enter the field of differential games, and, if a solution exists, it likely represents a global equilibrium among drivers. Numerical simulations highlight the differences between the three behaviors and suggest the existence of multiple Wardrop equilibria

Matroids are Immune to Braess Paradox

The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur. In a nutshell, bases of matroids are exactly this maximal structure. We prove our characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest.Comment: 21 page