Abstract We consider a network pricing game on a parallel network with congestion effects in which link owners set tolls for travel so as to maximize profit. A central authority is able to regulate this competition by means of a (uniform) price cap. The first question we want to answer is how such a cap should be designed in order to minimize the total congestion. We provide an algorithm that finds an optimal price cap for networks with affine latency functions and a full support Wardrop equilibrium. Second, we consider the induced network performance at an optimal price cap. We show that for two link networks with affine latency functions, the congestion costs at the optimal price cap are at most 8 7 times the optimal congestion costs. For more general latency functions, this bound goes up to 2 under the assumption that an uncapped Nash equilibrium exists. However, in general such an equilibrium need not exist and this can be used to show that optimal price caps can induce arbitrarily inefficient flows.
The model of @cite_44 studies competition in both tolls and capacities and finds that tolls are higher, but capacities are lower than socially desired. Other recent models of Bertrand competition in a network setting that use different ways of modelling congestion effects are Anshelevich and Sekar @cite_46 , Chawla and Niu @cite_7 , Chawla and Roughgarden @cite_24 , Papadimitriou and Valiant @cite_5 and @cite_37 .
@cite_37 @cite_7 @cite_44 @cite_24 @cite_5 @cite_46
2009182597 2084281377 2123733911 2111240209
Abstract This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum-length routing. More specifically, it is shown that in a large n-dimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is Θ(1), whereas in a Euclidean ball, the same proportion scales as Θ(1 R n−1). This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as volume2(B), whereas the same traffic load scales as volume1+1 n (B) in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel database and synthetic ... As the provision of roads has become market driven, many intriguing issues have emerged, such as the strategic interactions among private firms in determining road supply and pricing in a network and, more importantly, the resulting inefficiencies when competition substitutes for government regulation. This paper studies both toll and capacity competition among private asymmetric roads with congestion in a network with parallel links. We find that oligopolistic competition yields higher tolls but lower construction capacities than the socially optimal levels and, as a result, the level of traffic congestion does not decline. Based on some widely used assumptions, we show that the usage or the congestion level of each road controlled by each private firm, represented by the volume-capacity (v c) ratio, is independent of the other competitors' choices of capacities and tolls in the oligopolistic market. To quantify the inefficiency of the oligopolistic equilibria via higher tolls and less realized traffic demand, we establish the upper bound of the ratio of the social welfare between the competitive equilibrium and the social optimum. The inefficiency bound depends mainly on the marginal benefit of trip-makers and the realized traffic demand at equilibrium on the private roads. Exact bounds are given by considering both the worst and symmetric cases. Naturally, our results show that the distortion disappears when the oligopoly changes to perfect competition. We prove the existence of tolls to induce multicommodity, heterogeneous network users that independently choose routes minimizing their own linear function of tolls versus latency to collectively form the traffic pattern of a minimum average latency flow. This generalizes both the previous known results of the existence of tolls for multicommodity, homogeneous users (, 1956) and for single commodity, heterogeneous users (, 2003). Unlike previous proofs for single commodity users in general graphs, our proof is constructive - it does not rely on a fixed point theorem - and results in a simple polynomial-sized linear program to compute tolls when the number of different types of users is bounded by a polynomial. We show that our proof gives a complete characterization of flows that are enforceable by tolls. In particular, tolls exist to induce any traffic pattern that is the result of minimizing an arbitrary function from R sup E(G) to the reals that is nondecreasing in each of its arguments. Thus, tolls exist to induce flows with minimum average weighted latency, minimum maximum latency, and other natural objectives. We give an exponential bound on tolls that is independent of the number of network users and the number of commodities. We use this to show that multicommodity tolls also exist when users are not from discrete classes, but instead define a general function that trades off latency versus toll preference. Finally, we show that our result extends to very general frameworks. In particular, we show that tolls exist to induce the Nash equilibrium of general nonatomic congestion games to be system optimal. In particular, tolls exist even when 1) latencies depend on user type; 2) latency functions are nonseparable functions of traffic on edges; 3) the latency of a set S is an arbitrary function of the latencies of the resources contained in S. Our exponential bound on size of tolls also holds in this case; and we give an example of a congestion game that shows this is tight; it requires tolls that are exponential in the size of the game. We consider a bilevel model where the leader wants to maximize revenues from a taxation scheme, while the follower rationally reacts to those tax levels. We focus our attention on the special case of a toll-setting problem defined on a multicommodity transportation network. We show that the general problem is NP-complete, while particular instances are polynomially solvable. Numerical examples are given.