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.
@cite_26: 
@cite_4: It is well known that in the standard traffic network equilibrium model with a single value of time (VOT) for all users, a so-called marginal-cost toll can drive a user equilibrium flow pattern to a system optimum. This result holds when either cost (money) or time units are used in expressing the objective function of the system optimum and the criterion for user equilibrium. This paper examines the multi-criteria or the cost-versus-time network equilibrium and system optimum problem in a network with a discrete set of VOTs for several user classes. Specifically, the following questions are investigated: Are the user-optimal flows dependent upon the unit (time or money) used in measuring the travel disutility in the presence of road pricing? Are there any uniform link tolls across all individuals (link tolls that are identical for all user classes) that can support a multi-class user equilibrium flow pattern as a system optimum when the system objective function is measured by either money or time units? What are the general properties of the valid toll set?
@cite_22: The problem of computing optimal network tolls that induce a Nash equilibrium of minimum total cost has been studied intensively in the literature, but mostly under the assumption that these tolls are unrestricted. Here we consider this problem under the more realistic assumption that the tolls have to respect some given upper bound restrictions on the arcs. The problem of taxing subnetworks optimally constitutes an important special case of this problem. We study the restricted network toll problem for both non-atomic and atomic (unweighted and weighted) players; our studies are the first that also incorporate heterogeneous players, i.e., players with different sensitivities to tolls. For non-atomic and heterogeneous players, we prove that the problem is NP-hard even for single-commodity networks and affine latency functions. We therefore focus on parallel-arc networks and give an algorithm for optimally taxing subnetworks with affine latency functions. For weighted atomic players, the problem is NP-hard already for parallel-arc networks and linear latency functions, even if players are homogeneous. In contrast, for unweighted atomic and homogeneous players, we develop an algorithm to compute optimal restricted tolls for parallel-arc networks and arbitrary (standard) latency functions. Similarly, for unweighted atomic and heterogeneous players, we derive an algorithm for optimally taxing subnetworks for parallel-arc networks and arbitrary (standard) latency functions. The key to most of our results is to derive (combinatorial) characterizations of flows that are inducible by restricted tolls. These characterizations might be of independent interest.
@cite_32: 
@cite_3: THE CAPABILITIES AND EFFICIENT OPERATION OF SEVERAL TYPES OF TRANSPORTATION SYSTEMS ARE STUDIED. FOR HIGHWAY TRAFFIC THE MAIN EMPHASIS IS ON CONGESTION AND ITS CONSEQUENCES. QUEUING MODELS ARE USED TO DETERMINE FLUCTUATIONS IN WAITING TIME FOR TWO CASES, THE FLOW OF CARS THROUGH AN INTERSECTION AND THE PASSING OF SLOWER CARS BY FASTER CARS ON A TWO-LANE HIGHWAY. THE RESULTS OF THESE STUDIES ARE INCORPORATED INTO A NON-LINEAR MODEL OF HIGHWAY TRAFFIC BEHAVIOR ON A ROAD NETWORK. THE CAPACITIES ARE EXPRESSED BY THE RELATIONSHIP BETWEEN TRAFFIC FLOWS ON ALL ROUTES AND THE COSTS ENCOUNTERED ON EACH AS A RESULT OF THESE FLOWS. HIGHWAY TRAFFIC DEMANDS ARE THE RESULT OF A GREAT MANY INDIVIDUAL DECISIONS ABOUT DESTINATION, ROUTES, AND PREFERRED SPEEDS. THE ANALOGY BETWEEN HIGHWAY TRAFFIC EQUILIBRIUM AND THE MARKET RESPONSE MODELS USED BY ECONOMISTS IS GIVEN. IF THE INDIVIDUALLY INCURRED COST OF TRANSPORTATION IS REGARDED AS THE "PRICE" IN THE TRANSPORTATION "MARKET," THE ECONOMIST'S DEMAND CURVE BECOMES APPLICABLE. FOR A SINGLE ROAD IT GIVES THE FLOW OF TRAFFIC DEMANDED FOR ANY GIVEN COST ENCOUNTERED ON THAT ROAD. GENERALLY, THE HIGHER THE COST, THE SMALLER THE FLOW. CONSIDERING A ROAD NETWORK THE DEMAND FUNCTION GIVES THE FLOWS ON EACH ROUTE IN RESPONSE TO A SET OF TRANSPORTATION COSTS ALONG THESE ROUTES. EQUILIBRIUM EXISTS IF THE FLOWS ON ALL ROADS RESULTING FROM THESE COSTS ARE THE SAME AS THE FLOWS THAT PRODUCE THESE SAME COSTS. THIS CONCEPT IS APPLIED TO A STUDY OF TRAFFIC EQUILIBRIUM AND USE OF THE ANALYSIS IN THE PREDICTION OF TRAFFIC FLOWS IS DISCUSSED. THE ANALYSIS OF DEMAND AND OF EQUILIBRIUM RECOGNIZES THAT FREEDOM OF CHOICE IS PART OF THE SERVICE RENDERED BY THE ROAD NETWORK. TOLLS FROM THE USERS OF CONGESTED ROADS AT RATES THAT WOULD MEASURE THE COST TO OTHERS CAUSED BY THE AVERAGE ROAD USER WOULD ENCOURAGE BETTER USE OF THE HIGHWAY SYSTEM. THERE ARE OBSERVATIONS ON HOW CLOSELY MAXIMUM EFFICIENCY CAN BE APPROACHED BY PROPER CHOICE OF RATES ON PRESENT TOLL ROADS AND BY OTHER WAYS OF PENALIZING ADDITIONS TO TRAFFIC CONGESTION. BECAUSE OF THE NONLINEAR RELATIONSHIP BETWEEN USE AND COST, SUCH PRICING DOES NOT NECESSARILY PRODUCE REVENUES EQUAL TO THE TOTAL COST OF OPERATING AND FINANCING THE INDIVIDUAL FACILITY. SOME OBSERVATIONS ON THE QUESTION OF CRITERIA FOR EXTENSIONS OR IMPROVEMENTS OF THE NETWORK TO RELIEVE CONGESTION ARE GIVEN. AUTHOR
@cite_34: 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.
@cite_0: We investigate how and to which extent one can exploit risk-aversion and modify the perceived cost of the players in selfish routing so that the Price of Anarchy @math is improved. We introduce small random perturbations to the edge latencies so that the expected latency does not change, but the perceived cost of the players increases due to risk-aversion. We adopt the model of @math -modifiable routing games, a variant of routing games with restricted tolls. We prove that computing the best @math -enforceable flow is @math -hard for parallel-link networks with affine latencies and two classes of heterogeneous risk-averse players. On the positive side, we show that for parallel-link networks with heterogeneous players and for series-parallel networks with homogeneous players, there exists a nicely structured @math -enforceable flow whose @math improves fast as @math increases. We show that the complexity of computing such a @math -enforceable flow is determined by the complexity of computing a Nash flow of the original game. Moreover, we prove that the @math of this flow is best possible in the worst-case, in the sense that there are instances where ii¾?the best @math -enforceable flow has the same @math , and iii¾?considering more flexible modifications does not lead to any further improvement.
@cite_19: We examine how the selfish behavior of heterogeneous users in a network can be regulated through economic disincentives, i.e., through the introduction of appropriate taxation. One wants to impose taxes on the edges so that any traffic equilibrium reached by the selfish users who are conscious of both the travel latencies and the taxes will minimize the social cost, i.e., will minimize the total latency. We generalize previous results of Cole, Dodis and Roughgarden that held for a single origin-destination pair to the multicommodity setting. Our approach, which could be of independent interest, is based on the formulation of traffic equilibria as a nonlinear complementarity problem by Aashtiani and Magnanti (1981), We extend this formulation so that each of its solutions will give us a set of taxes that forces the network users to conform, at equilibrium, to a certain prescribed routing. We use the special nature of the prescribed minimum-latency flow in order to reduce the difficult nonlinear complementarity formulation to a pair of primal-dual linear programs. LP duality is then enough to derive our results.
@cite_40: We study the negative consequences of selfish behavior in a congested network and economic means of influencing such behavior. We consider a model of selfish routing in which the latency experienced by network traffic on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route traffic on minimum-latency paths. The quality of a routing of traffic is measured by the sum of travel times (the total latency).It is well known that the outcome of selfish routing (a Nash equilibrium) does not minimize the total latency. An ancient strategy for improving the selfish solution is the principle of marginal cost pricing, which asserts that on each edge of the network, each network user on the edge should pay a tax offsetting the congestion effects caused by its presence. By pricing network edges according to this principle, the inefficiency of selfish routing can always be eradicated.This result, while fundamental, assumes a very strong homogeneity property: all network users are assumed to trade off time and money in an identical way. The guarantee also ignores both the algorithmic aspects of edge pricing and the unfortunate possibility that an efficient routing of traffic might only be achieved with exorbitant taxes. Motivated by these shortcomings, we extend this classical work on edge pricing in several different directions and prove the following results.We prove that the edges of a single-commodity network can always be priced so that an optimal routing of traffic arises as a Nash equilibrium, even for very general heterogeneous populations of network users.When there are only finitely many different types of network users and all edge latency functions are convex, we show how to compute such edge prices efficiently.We prove that an easy-to-check mathematical condition on the population of heterogeneous network users is both necessary and sufficient for the existence of edge prices that induce an optimal routing while requiring only moderate taxes.
@cite_45: We introduce a unifying model to study the impact of worst-case latency deviations in non-atomic selfish routing games. In our model, latencies are subject to (bounded) deviations which are taken into account by the players. The quality deterioration caused by such deviations is assessed by the Deviation Ratio, i.e., the worst case ratio of the cost of a Nash flow with respect to deviated latencies and the cost of a Nash flow with respect to the unaltered latencies. This notion is inspired by the Price of Risk Aversion recently studied by Nikolova and Stier-Moses [9]. Here we generalize their model and results. In particular, we derive tight bounds on the Deviation Ratio for multi-commodity instances with a common source and arbitrary non-negative and non-decreasing latency functions. These bounds exhibit a linear dependency on the size of the network (besides other parameters). In contrast, we show that for general multi-commodity networks an exponential dependency is inevitable. We also improve recent smoothness results to bound the Price of Risk Aversion.
@cite_16: 
@cite_12: Reducing traffic congestion via toll pricing has been a central topic in the operations research and transportation literature and, recently, it has been implemented in several cities all over the world. Since, in practice, it is not feasible to impose tolls on every edge of a given traffic network, we study the resulting mathematical problem of computing tolls on a predefined subset of edges of the network so as to minimize the total travel time of the induced equilibrium flow. We first present an analytical study for the special case of parallel edge networks highlighting the intrinsic complexity and nonconvexity of the resulting optimization problem. We then present algorithms for general networks for which we systematically test the solution quality for large-scale network instances. Finally, we discuss the related optimization problem of computing tolls subject to a cardinality constraint on the number of edges that have tolls. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 653, 262-285 2015
One way to restore efficiency is by means of centralized pricing. @cite_3 and Dafermos and Sparrow @cite_16 showed that marginal cost pricing induces an equilibrium flow that is optimal. Despite the effectiveness of marginal cost tolls, there are two drawbacks. First, potentially each edge in the network is tolled; an issue that is considered by @cite_26 and @cite_12 . Second, the imposed tolls can be arbitrary large; an issue that is considered by @cite_32 , @cite_0 , @cite_22 and Kleer and Sch " a fer @cite_45 . @cite_40 show that optimal tolls also exist when users are heterogeneous with respect to the tradeoff between time and money. Later this result was extended to general networks by @cite_34 , Karakostas and Kolliopoulos @cite_19 and Yang and Huang @cite_4 .