Computing solutions of the multiclass network equilibrium problem with affine cost functions

Abstract

We consider a nonatomic congestion game on a 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 origindestination 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. We show that this problem is polynomially solvable when the number of vertices and the number of classes are fixed. In particular, it shows that the parallel-link case with a fixed number of classes is polynomially solvable. On a more practical side, we propose an extension of Lemke's algorithm able to solve this problem. IntroductionContext. Being able to predict the impact of a new infrastructure on the traffic in a transportation network is an old but still important objective for transport planners. In 1952, Wardrop (1952 noted that after some while the traffic arranges itself to form an equilibrium and formalized principles characterizing this equilibrium. With the terminology of game theory, the equilibrium is a Nash equilibrium for a congestion game with nonatomic users. , Beckmann et al. (1956 translated these principles as a mathematical program which turned out to be convex, opening the door to the tools from convex optimization. The currently most commonly used algorithm for such convex programs is probably the Frank-Wolfe algorithm (Frank and Wolfe, 1956), because of its simplicity and its efficiency, but many other algorithms with excellent behaviors have been proposed, designed, and experimented.One of the main assumptions used by Beckmann to derive his program is the fact that all users are equally impacted by the congestion. With the transportation terminology, it means that there is only one class. In order to improve the prediction of traffic patterns, researchers started in the 70s to study the multiclass situation where each class has its own way of being impacted by the congestion. Each class models a distinct mode of transportation, such as cars, trucks, or motorbikes. Dafermos (1972( , 1980( ) and Smith (1979 are probably the first who proposed a mathematical formulation of the equilibrium problem in the multiclass case. However, even if this problem has been the topic of many research works, an efficient algorithm for solving it is not known, except in some special cases (Florian, 1977, Harker, 1988, Mahmassani and Mouskos, 1988, Marcotte and Wynter, 2004. In particular, there are no general algorithms in the literature for solving the problem when the cost of each arc is in an affine dependence with the flow on it.Our purpose is to discuss the existence of such algorithms.Model. We are given a directed graph D = (V, A) modeling the transportation network. The set of all paths (resp. s-t paths) is denoted by P (resp. P (s,t) ). The population of players is mo...

Complexity and Parametric Computation of Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians

“…This is an interesting contrast to the PLScompleteness results for computing Nash equilibria in unsplittable congestion games using a nonconstant number of players and highly non-planar graphs [2,19]. As a byproduct, we also obtain PPAD-completeness for a related problem settling an open question from Meunier and Pradeau [41].…”

“…With the same arguments, we can also show that the computation of a multi-class Wardrop equilibrium is PPAD-complete. A multi-class Wardrop equilibrium is a multi-commodity flow that satisfies the characterization via shortest path potentials of Lemma 2 for the original cost functions l e,i instead for the marginal cost functions µ i e , i.e, x is a Wardrop equilibrium if and only if for all i ∈ [k] there is a potential vector π i with π i w − π i v = l e,i (x e ) if x i e > 0, and π i w − π i v ≤ l e,i (x e ) if x i e = 0 for all e = (v, w) ∈ E. We prove the following result settling an open question in [41]. Theorem 6.…”

“…When the cost functions of an edge are equal for all commodities, an equilibrium can be computed in polynomial time by convex programming techniques [5]. For the case of commodity-dependent affine cost functions, Meunier and Pradeau [41] very recently showed that the problem to compute a Wardrop equilibrium lies in PPAD, but left it as an open problem whether the problem is actuall PPAD-complete.…”

“…Despite considerable progress regarding the computational complexity of general equilibria in bimatrix games [10,16], Wardrop equilibria [5,41], and atomic-unsplittable congestion games [1,19], much less is known regarding the computation of equilibria in atomic splittable congestion games. For affine player-independent cost functions, Cominetti et al [12] showed that an equilibrium can be found by computing the minimum of a convex potential function, see also Huang [32] for a combinatorial algorithm for special graph topologies.…”

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

“…Indeed, coming up with proofs of existence that also guarantee rationality of solutions has been a topic of interest in the area since the very early days, way before the introduction of the relevant computational complexity classes, e.g., see [Eaves, 1976;Lemke and Howson, 1964;Lemke, 1965;Howson, 1972]. Driven by those classic results, a significant literature in computer science has attempted, and quite often has succeeded in placing the corresponding computational problems in PPAD, for several of the application domains mentioned above, including games [Sørensen, 2012;Hansen and Lund, 2018;Kintali et al, 2013;Klimm and Warode, 2020;Meunier and Pradeau, 2019], markets [Vazirani and Yannakakis, 2011;Garg and Vazirani, 2014;Garg, 2017;Garg et al, 2018], as well as the more recent domain of auto-bidding auctions [Chen et al, 2021a].…”