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

Learning an Unknown Network State in Routing Games

We study learning dynamics induced by myopic travelers who repeatedly play a routing game on a transportation network with an unknown state. The state impacts cost functions of one or more edges of the network. In each stage, travelers choose their routes according to Wardrop equilibrium based on public belief of the state. This belief is broadcast by an information system that observes the edge loads and realized costs on the used edges, and performs a Bayesian update to the prior stage's belief. We show that the sequence of public beliefs and edge load vectors generated by the repeated play converge almost surely. In any rest point, travelers have no incentive to deviate from the chosen routes and accurately learn the true costs on the used edges. However, the costs on edges that are not used may not be accurately learned. Thus, learning can be incomplete in that the edge load vectors at rest point and complete information equilibrium can be different. We present some conditions for complete learning and illustrate situations when such an outcome is not guaranteed

Sensitivity analysis of generalized variational inequalities

AbstractDafermos studied the sensitivity properties of the solutions of a variational inequality with regard to continuity and differentiability of such solutions with respect to a parameter λ. In the present paper we extend this analysis for a generalized variational inequality of the type introduced by Noor of which the variational inequality of Dafermos is a particular case. Our results are such that they automatically extend the regularity properties of solutions with respect to a parameter λ when the variational inequality is treated on a Hilbert space

A heuristic methodology to tackle the Braess Paradox detecting problem tailored for real road networks

Adding a new road to help traffic flow in a congested urban network may at first appear to be a good idea. The Braess Paradox (BP) says, adding new capacity may actually worsen traffic flow. BP does not only call for extra vigilance in expanding a network, it also highlights a question: Does BP exist in existing networks? Literature reveals that BP is rife in real world. This study proposes a methodology to find a set of roads in a real network, whose closure improve traffic flow. It is called the Braess Paradox Detection (BPD) problem. Literature proves that the BPD problem is highly intractable especially in real networks and no efficient method has been introduced. We developed a heuristic methodology based on a Genetic Algorithm to tackle BPD problem. First, a set of likely Braess-tainted roads is identified by simply testing their closure (one-by-one). Secondly, a seraph algorithm is devised to run over the Braess-tainted roads to find a combination whose closure improves traffic flow. In our methodology, the extent of road closure is limited to some certain level to preserve connectivity of the network. The efficiency and applicability of the methodology are demonstrated using the benchmark Hagstrom–Abrams network, and on a network of city of Winnipeg in Canada

Mechanisms that Govern how the Price of Anarchy varies with Travel Demand

Selfish routing, represented by the User-Equilibrium (UE) model, is known to be inefficient when compared to the System Optimum (SO) model. However, there is currently little understanding of how the magnitude of this inefficiency, which can be measured by the Price of Anarchy (PoA), varies across different structures of demand and supply. Such understanding would be useful for both transport policy and network design, as it could help to identify circumstances in which policy interventions that are designed to induce more efficient use of a traffic network, are worth their costs of implementation. This paper identifies four mechanisms that govern how the PoA varies with travel demand in traffic networks with separable and strictly increasing cost functions. For each OD movement, these are expansions and contractions in the sets of routes that are of minimum cost under UE and minimum marginal total cost under SO. The effects of these mechanisms on the PoA are established via a combination of theoretical proofs and conjectures supported by numerical evidence. In addition, for the special case of traffic networks with BPR-like cost functions having common power, it is proven that there is a systematic relationship between link flows under UE and SO, and hence between the levels of demand at which expansions and contractions occur. For this case, numerical evidence also suggests that the PoA has power law decay for large demand

Application of robust and inverse optimization in transportation

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 79-82).We study the use of inverse and robust optimization to address two problems in transportation: finding the travel times and designing a transportation network. We assume that users choose the route selfishly and the flow will eventually reach an equilibrium state (User Equilibrium). The first part of the thesis demonstrates how inverse and robust optimization can be used to find the actual travel times given a stable flow on the network and some noisy information on travel times from different users. We model the users' perception of travel times using three different sets and solve the robust inverse problem for all of them. We also extend the idea to find parametric functional forms for travel times given historical data. Our numerical results illustrate the significant improvement obtained by our models over a simple fitting model. The second part of the thesis considers the network design problem under demand uncertainty. We show that for affine travel time functions, the deterministic problem can be formulated as a mixed integer programming problem with quadratic objective and linear constraints. For the robust network design problem, we propose a decomposition scheme: breaking a tri-level programming problem into two smaller problems and re-iterating until a good solution is obtained. To deal with the expensive computation required by large networks, we also propose a heuristic robust simulated annealing approach. The heuristic algorithm is computationally tractable and provides some encouragingly results in our simulations.by Thai Dung Nguyen.S.M