Daganzo, C. The uniqueness of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transportation Science 19 1 , Traffic congestion pricing methodologies and technologies. Dupuis, P.

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## Traffic Congestion Pricing: Methodologies and Equity Implications

Transportation Science 15 1 , Horn, R. Matrix Analysis, 2nd Edition. Iryo, T. Equivalent optimization problem for finding equilibrium in the bottleneck model with departure time choices. In: Heydecker, B. Elsevier, pp.

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Kuwahara, M. Equilibrium queueing patterns at a two-tandem bottleneck during the morning peak. Transportation Science 24 3 , Lindsey, R. Existence, uniqueness, and trip cost function properties of user equilibrium in the bottleneck model with multiple user classes. Transportation Science 38 3 , Monderer, D. Potential games. Games and Economic Behavior 14 1 , Mossay, P. On spatial equilibria in a social interaction model.

Journal of Economic Theory 6 , Mun, S. Flextime, traffic congestion and urban productivity. Journal of Transport Economics and Policy 40 3 , Peer, S. Equilibrium at a bottleneck when long-run and short-run scheduling preferences diverge. Transportation Research Part B: Methodological 57, Rosenthal, R. A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory 2 1 , Sandholm, W.

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A simultaneous equilibrium model of work start time and departure time choices with bottleneck congestion. Simsek, A. Generalized Poincare-Hopf theorem for compact nonsmooth regions. Mathematics of Operations Research 32 1 , However, the data originating from these systems cannot adequately address critical aspects considered in the design of discriminatory road pricing schemes see Section 3. In contrast, active sensors can identify particular classes and trajectories of vehicles in a network, thus allowing the appropriate toll imposition for the usage of some facility [].

First advanced developments related to automatic debiting and toll col- lection systems based on microwave technology were considered to facilitate user payments at toll highway entrances or urban cordon toll points. Over the past few years, there have been many advances in the field of road pricing technologies see [38, 41] , which can offer a range of new options and support the monitoring of user responses [, , ], while reducing the cost of adoption. Other technologies refer to area-wide communication systems, e. Also, combinations of these technology options e.

All these advanced systems may encounter a complete electronic billing system, e. The criteria for selecting the appropriate road pricing technology encom- pass a number of issues related to the pricing strategy, like the location requirements, the timeframe of congestion charging or point toll imposition, and the type of roads and vehicles charged [, ]. Particularly, in view of the potential benefits that accrue from the synergy of road pricing systems with other network traffic information and control man- agement systems see Section 3.

A key requirement to handle these challenges is the collaboration of all stakeholders, including governments, regulators, automotive industries, public transport authorities, toll operators and different user groups, in order to share costs and create the conditions for the establishment of a mass market.

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This set of assumptions has allowed the development of complex as well as application-oriented design schemes which can be practically implemented and evaluated over a certain planning horizon. As it is shown graphically in Fig. The optimal charge yields a net welfare benefit shown in the patterned area by internalizing the congestion externality, as it reduces the traffic from the user-equilibrium flow level, que , to the system optimal one, qso.

Given that the maximum traffic level, qmax , implied by a possible capacity bound, is held fixed, an increase rightward shift of the demand would result in a higher congestion charge as well as a larger welfare benefit. The study of marginal-cost pricing MCP schemes typically ignores de- ployment costs and, particularly, other externalities, such as those on urban economic activity and environment [].

It essentially corresponds to the short-run optimal price, since, as mentioned before, it does not generally consider the optimal service level usually expressed in terms of capacity provision. This is because urban road capacity often does not have constant returns to scale, due to the high costs of capacity addition in dense urban areas. The MCP charges can be theoretically obtained through achieving a system optimum SO travel cost minimizing flow pattern that satisfies the tolled user-equilibrium conditions.

In a narrow economic sense, first-best pricing refers to the unconstrained maximization of the social welfare or surplus total benefits minus total costs and is often used as a benchmark solution, since it involves some strong rather unrealistic assumptions, like the ubiquity of user charges and perfect information []. Some of these game-theoretic modeling structures resemble the ones adopted to address pricing problems in other types of networks, like those of telecommunications [3] see also Section 4.

In particular, noncooperative game theoretic models of network traffic are gaining an increasing importance in both large-scale communication e. In such models, data streams or single packets equivalent to road vehicles managed by selfish agents players trying to minimize their individual latencies are charged with the aim to reduce the price of anarchy. This price measures the degradation of network efficiency due to uncoordinated behavior, through a comparison between the latency experienced in a worst case Nash equilibrium decentralized user-optimization problem and the latency experienced in a centralized system-optimization problem [, ].

The function G represents the constraint set of the decision vector, which may involve a range of different requirements or restrictions see below. At the lower-level problem, f represents the response or reaction function of users with respect to tolls or charges and possibly other control measures, towards achieving an equilibrium flow distribution q p , subject to a set of constraints denoted by g. The road user costs include the travel cost from the point of entry to the point of exit of the tolled area plus the toll rate on the entry or the charged links.

A more realistic extension of this principle refers to the stochastic user-equilibrium SUE , based on which no traveler could decrease the perceived travel cost by unilaterally changing the route [81]. The latter model allows representing variations in the perceived generalized travel cost and uncertainty in the route choice behavior of users.

## Mathematical and Economic Theory of Road Pricing

Smith et al. Several types of optimization problem formulations with alternative objec- tive functions and combinations of constraints have been proposed and tested. These studies have shown the similarity of the resulting equilibrium flows between the unconstrained tolled user problem and the side constrained one. This outcome has important implications for meeting certain policy requirements, addressing deployment issues and promoting the political support and social acceptability of road pricing.

More specifically, constrained optimization problem formulations aim- ing to achieve a SO solution also characterized as first-best, in a broader sense , with either fixed [36, ] or elastic [, ] demand, refer to the minimal revenue toll design, which seeks to minimize total toll revenues without MINSYS or with MINREV the possibility for subsidies negative tolls on some links also see [20, 70, , , , ], toll design which minimize the number of toll booths MINTB and hence deployment costs the cheapest one , and toll design that minimizes the maximum toll on any one link MINMAX.

Extensions of the MINREV design scheme include the incorporation of stochasticity in the SO flow pattern under logit and probit assignment methods []. Formulations of second-best road pricing design problems typically include constraints on imposing tolls only on a sub-area or subset of links.

Such problems have been investigated with regard to the optimal toll level, number and location of tolled links, trip duration, delay or distance traveled in the charged area and number of crossings, i. Specifically, [] considered the optimal congestion toll levels for a given subset of tollable links under deterministic user-optimal conditions, while the logit-based SUE model has been used for finding the optimal link-toll pattern that would induce the lowest total travel cost [64], possibly combined with minimal toll revenues [].

May and Milne [] employed an elastic deterministic user-optimal traffic assignment procedure to show that delay- based charging outperforms a number of other road charging schemes. Santos [] suggested the use of a second, outer cordon jointly with an inner cordon surrounding the centre of a town to enhance the social welfare gains, in comparison with a single inner cordon. Mun et al. Ho et al. Zhang and Yang [] used a combination of a genetic algorithm GA and a grid search method to simultaneously determine optimal toll levels and locations in single- and multi-layered cordon pricing schemes in road networks.

Also, [] examined the optimal toll level and location problem through a GA approach, in comparison to derivative-based methods [], in order to address the strong assumption of the perfect convergence of the lower-level problem to user-equilibrium conditions when dealing with realistic large-scale networks. A similar approach was employed in [] and [] to indicate the social welfare gains of an optimal single cordon scheme, compared to the best judgmentally designed cordon, subject to constraints on desired level of revenue and equity impact.

Several more elaborate travel demand and network supply models have also been used to gain more insight in the evaluation of different aspects of second-best congestion tolling schemes.