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

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

Capacity and Price Competition in Markets with Congestion Effects

We study oligopolistic competition in service markets where firms offer a service to customers. The service quality of a firm - from the perspective of a customer - depends on the congestion and the charged price. A firm can set a price for the service offered and additionally decides on the service capacity in order to mitigate congestion. The total profit of a firm is derived from the gained revenue minus the capacity investment cost. Firms simultaneously set capacities and prices in order to maximize their profit and customers subsequently choose the services with lowest combined cost (congestion and price). For this basic model, Johari et al. (2010) derived the first existence and uniqueness results of pure Nash equilibria (PNE) assuming mild conditions on congestion functions. Their existence proof relies on Kakutani's fixed-point theorem and a key assumption for the theorem to work is that demand for service is elastic (modeled by a smooth and strictly decreasing inverse demand function). In this paper, we consider the case of perfectly inelastic demand, i.e. there is a fixed volume of customers requesting service. This scenario applies to realistic cases where customers are not willing to drop out of the market, e.g. if prices are regulated by reasonable price caps. We investigate existence, uniqueness and quality of PNE for models with inelastic demand and price caps. We show that for linear congestion cost functions, there exists a PNE. This result requires a completely new proof approach compared to previous approaches, since the best response correspondences of firms may be empty, thus standard fixed-point arguments are not directly applicable. We show that the game is C-secure (see McLennan et al. (2011)), which leads to the existence of PNE. We furthermore show that the PNE is unique, and that the efficiency compared to a social optimum is unbounded in general.Comment: A one-page abstract of this paper appeared in the proceedings of the 15th International Conference on Web and Internet Economics (WINE 2019

Nash and Wardrop equilibria in aggregative games with coupling constraints

We consider the framework of aggregative games, in which the cost function of each agent depends on his own strategy and on the average population strategy. As first contribution, we investigate the relations between the concepts of Nash and Wardrop equilibrium. By exploiting a characterization of the two equilibria as solutions of variational inequalities, we bound their distance with a decreasing function of the population size. As second contribution, we propose two decentralized algorithms that converge to such equilibria and are capable of coping with constraints coupling the strategies of different agents. Finally, we study the applications of charging of electric vehicles and of route choice on a road network.Comment: IEEE Trans. on Automatic Control (Accepted without changes). The first three authors contributed equall

Path deviations outperform approximate stability in heterogeneous congestion games

We consider non-atomic network congestion games with heterogeneous players where the latencies of the paths are subject to some bounded deviations. This model encompasses several well-studied extensions of the classical Wardrop model which incorporate, for example, risk-aversion, altruism or travel time delays. Our main goal is to analyze the worst-case deterioration in social cost of a perturbed Nash flow (i.e., for the perturbed latencies) with respect to an original Nash flow. We show that for homogeneous players perturbed Nash flows coincide with approximate Nash flows and derive tight bounds on their inefficiency. In contrast, we show that for heterogeneous populations this equivalence does not hold. We derive tight bounds on the inefficiency of both perturbed and approximate Nash flows for arbitrary player sensitivity distributions. Intuitively, our results suggest that the negative impact of path deviations (e.g., caused by risk-averse behavior or latency perturbations) is less severe than approximate stability (e.g., caused by limited responsiveness or bounded rationality). We also obtain a tight bound on the inefficiency of perturbed Nash flows for matroid congestion games and homogeneous populations if the path deviations can be decomposed into edge deviations. In particular, this provides a tight bound on the Price of Risk-Aversion for matroid congestion games

US Highway Privatization and Heterogeneous Preferences

Abstract: We assess the welfare effects of highway privatization accounting for government’s behavior in setting the sale price, firms’ strategic behavior in setting tolls in various competitive environments, and motorists’ heterogeneous preferences for speedy and reliable travel. We conclude motorists can benefit from privatization if they are able to negotiate aggressively with a private provider to obtain tolls and service that align with their varying preferences. Surprisingly, motorists are likely to be better off negotiating with a monopolist than with duopoly providers or under public-private competition. Toll regulation may be counterproductive because it would treat motorists as homogeneous. Revised June 2009.Security Breach Costs; Financial Distress; Insurance; Resource Allocation.

On the Price of Anarchy of Highly Congested Nonatomic Network Games

We consider nonatomic network games with one source and one destination. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we show that, under suitable conditions, the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case. The counterexamples occur in very simple parallel graphs.Comment: 26 pages, 6 figure