Algorithms for the Multiclass Network Equilibrium Problem

International audienceWe consider a nonatomic congestion game on a connected graph, with several classes of players. Each player wants to go from its origin vertex to its destination vertex at the minimum cost and all players of a given class share the same characteristics: cost functions on each arc, and origin-destination pair. In the transportation terminology, each class models a distinct mode of transportation, such as cars, trucks, or motorbikes for example.A strategy profile is a mapping determining for each player a route from its origin to its destination. A strategy profile is a (pure) Nash equilibrium if no player would benefit by changing its route. This game enters in the category of nonatomic congestion games with player-specific cost functions, see Milchtaich (Congestion games with player-specific payoff functions, Games Econom. Behavior, 13:111-124, 1996). The problem of finding a Nash equilibrium for such a game is called the Multiclass Network Equilibrium Problem. Under some mild conditions, it is known that a Nash equilibrium exists, but the computation of an equilibrium in the multiclass case is an open problem for general functions. We consider the specific case where the cost functions are affine and stricly increasing. In this case, we write the Multiclass Network Equilibrium Problem as a linear complementarity problem. Our contribution is twofold: - We prove the existence of a polynomial algorithm solving the problem when the graph consists in parallel arcs. The main idea of the algorithm relies on properties of hyperplane arrangements.- We build a pivoting algorithm, adapted from Lemke (Bimatrix equilibrium points and equilibrium programming, Management Science, 11:681-689, 1965), solving the problem for general graphs. To our knowledge, it is the first algorithm solving this problem. We prove its efficiency through computational experiments. Moreover, this algorithm provides the first constructive proof of the existence of an equilibrium for this problem.</p

A Lemke-like algorithm for the Multiclass Network Equilibrium Problem

14 pagesWe consider a nonatomic congestion game on a connected graph, with several classes of players. Each player wants to go from its origin vertex to its destination vertex at the minimum cost and all players of a given class share the same characteristics: cost functions on each arc, and origin-destination pair. Under some mild conditions, it is known that a Nash equilibrium exists, but the computation of an equilibrium in the multiclass case is an open problem for general functions. We consider the specific case where the cost functions are affine and propose an extension of Lemke's algorithm able to solve this problem. At the same time, it provides a constructive proof of the existence of an equilibrium in this case

Analysis and treatment of errors due to high velocity gradients in particle image velocimetry

International audienceThis paper deals with errors occurring in 2D cross-correlation PIV algorithms (with window shifting), when high velocity gradients are present. A first bias error is due to the difference between the Lagrangian displacement of a particle and the real velocity. This error is calculated theoretically as a function of the velocity gradients, and is shown to reach values up to 1 pixel if only one window is translated. However, it becomes negligible when both windows are shifted in a symmetric way. A second error source is linked to the image pattern deformation, which decreases the height of the correlation peaks. In order to reduce this effect, the windows are deformed according to the velocity gradients in an iterative process. The problem of finding a sufficiently reliable starting point for the iteration is solved by applying a Gaussian filter to the images for the first correlation. Tests of a PIV algorithm based on these techniques are performed, showing their efficiency, and allowing the determination of an optimum time separation between images for a given velocity field

Uniqueness of equilibrium on rings

We consider congestion games on networks with nonatomic users and user-specific costs. We are interested in the uniqueness property defined by Milchtaich [Milchtaich, I. 2005. Topological conditions for uniqueness of equilibrium in networks. Math. Oper. Res. 30 225-244] as the uniqueness of equilibrium flows for all assignments of strictly increasing cost functions. He settled the case with two-terminal networks. In the present work we characterize completely bidirectional rings for which the uniqueness property holds. The main result is that it holds precisely for nine networks and those obtained from them by elementary operations. For other bidirectional rings, we exhibit affine cost functions yielding to two distinct equilibrium flows. Related results are also proven

Bounding the price of anarchy for games with player-specific cost functions

9 pagesWe study the efficiency of equilibria in atomic splittable congestion games on networks. We consider the general case where players are not affected in the same way by the congestion. Extending a result by Cominetti, Correa, and Stier-Moses (The impact of oligopolistic competition in networks, Oper. Res., 57, 1421--1437 (2009)), we prove a general bound on the price of anarchy for games with player-specific cost functions. This bound generalizes some of their results, especially the bound they obtain for the affine case. However our bound still requires some dependence between the cost functions of the players. In the general case, we prove that the price of anarchy is unbounded, by exhibiting an example with affine cost functions and only two players

Lyapunov functions, Identities and the Cauchy problem for the Hele-Shaw equation

This article is devoted to the study of the Hele-Shaw equation. We introduce an approach inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the Dirichlet to Neumann operator and exploiting various cancellations, we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface. This allows to quasi-linearize the equations in a very simple way. By combining these exact identities with convexity inequalities, we prove the existence of hidden Lyapunov functions of different natures. We also deduce from these identities and previous works on the water wave problem a simple proof of the well-posedness of the Cauchy problem. The analysis contains two side results of independent interest. Firstly, we give a principle to derive estimates for the modulus of continuity of a PDE under general assumptions on the flow. Secondly we prove a convexity inequality for the Dirichlet to Neumann operator.Comment: We have added some reference

Capacity Investment under Demand Uncertainty: The Role of Imports in the U.S. Cement Industry

Demand uncertainty is thought to in uence irreversible capacity decisions. Suppose local demand can be sourced from domestic (rigid) production or from (fl exible) imports. This paper shows that the optimal domestic capacity is either increasing or decreasing with demand uncertainty depending on the relative level of the costs of domestic production and imports. This relationship is tested with data on the U.S. cement industry, where, because cement is costly to transport over land, the diff erence in marginal cost between domestic production and imports varies across local U.S. markets. Industry data for 1999 to 2010 are consistent with the predictions of the model. The introduction of two technologies to the production set one rigid and one exible is crucial in understanding the relationship between capacity choice and uncertainty in this industry because there is no relationship at the aggregated U.S. data. The analysis presented in the paper reveals that the relationship is negative for coastal districts, and signi cantly more positive in landlocked districts