First-best marginal cost toll for a traffic network with stochastic demand

https://doi.org/10.1016/j.trb.2010.04.007Get rights and content

Abstract

First-best marginal cost pricing (MCP) in traffic networks has been extensively studied with the assumption of deterministic travel demand. However, this assumption may not be realistic as a transportation network is exposed to various uncertainties. This paper investigates MCP in a traffic network under stochastic travel demand. Cases of both fixed and elastic demand are considered. In the fixed demand case, travel demand is represented as a random variable, whereas in the elastic demand case, a pre-specified random variable is introduced into the demand function. The paper also considers a set of assumptions of traveler behavior. In the first case, it is assumed that the traveler considers only the mean travel time in the route choice decision (risk-neutral behavior), and in the second, both the mean and the variance of travel time are introduced into the route choice model (risk-averse behavior). A closed-form formulation of the true marginal cost toll for the stochastic network (SN-MCP) is derived from the variational inequality conditions of the system optimum and user equilibrium assignments. The key finding is that the calculation of the SN-MCP model cannot be made by simply substituting related terms in the original MCP model by their expected values. The paper provides a general function of SN-MCP and derives the closed-form SN-MCP formulation for specific cases with lognormal and normal stochastic travel demand. Four numerical examples are explored to compare network performance under the SN-MCP and other toll regimes.

Introduction

It has long been recognized that congestion arises because the marginal social costs of road use diverge from the private costs (Pigou, 1920, Knight, 1924). Therefore, from the theoretical perspective, if a marginal cost toll is allowed to be charged on each link, then this particular first-best toll can drive the traffic flow pattern toward a social optimum (SO) under the routing principle of deterministic user equilibrium (UE). The first-best marginal cost pricing (MCP) has been extensively studies in the literature, in which all links in the network are tolled at a toll level of v dt(v)/dv, where v is the link traffic volume and t is the link travel time function (Yang and Huang, 1998). This simple and elegant formulation for traffic network models provides an important benchmark for road pricing studies. The MCP strategy is also verified to be applicable to the stochastic user equilibrium (SUE) routing principle (Yang, 1999). Maher et al. (2005) formulated a stochastic social optimum (SSO) for the probit SUE model, where the SSO aims to minimize the total expected perceived travel time of networks under the probit SUE condition. In addition to the static form of MCP, a dynamic version was analyzed by Kuwahara (2007), in which the formulation of the dynamic MCP toll is shown to be a direct extension of the static one. Yang and Huang (2004) also extended the analysis of MCP (or system optimum assignment) to the case with multi-class and multi-criteria traffic assignment.

In the abovementioned studies, travel demand is either treated as a deterministic value in the fixed demand case or modeled as a deterministic function of the average travel time (or travel cost) in the elastic demand case. Watling (2002) noted a characteristic feature of all of these models, namely, their deterministic representation of both primary input (e.g., the inter-zonal travel demand matrix and network characteristics) and output (e.g., link flows and travel times) quantity. However, such an assumption may not be plausible or realistic for traffic networks, as they are exposed to various sources of uncertainty both on the demand and supply sides. Several studies have attempted to represent such uncertainties explicitly in network modeling frameworks (stochastic networks). Some of them have focused on representing the stochastic road capacity, which can be affected by major events, such as earthquakes (Bell and Iida, 1997), or relatively minor events, such as adverse weather conditions (Lam et al., 2008). Similarly, many studies have recently begun to include stochastic travel demand (to represent the fluctuation of travel demand from day to day) into the stochastic network model (see, e.g., Clark and Watling, 2005, Shao et al., 2006, Sumalee et al., 2006, Siu and Lo, 2008). To this end, the notion of the stochastic traffic network (SN), as a framework for network modeling with stochastic inputs of travel demand and supply, has emerged. With the SN, equilibrium traffic assignment models can similarly be classified according to the behavioral assumption governing the route choice, i.e., deterministic UE and SO. For brevity, we refer to these models as SN-UE and SN-SO, respectively.

This paper aims to investigate the relationship between the SN-UE and SN-SO models to establish the first-best marginal cost toll scheme for the SN model, as previously done for the deterministic network one. The analysis includes the effect of demand uncertainty on the evaluation of first-best MCP. This problem is practically relevant. For instance, the recent change in the Electronic Road Pricing (ERP) toll adjustment scheme in Singapore involves the consideration of the 85th-percentile traffic condition (speed) to reflect the variability of traffic conditions. The paper also looks at the relationship between the original form of MCP and MCP for the stochastic network (SN-MCP) to identify an additional term caused by stochastic demand. In this paper, for the SN-UE model, we assume that no traveler can reduce his/her expected travel time (risk-neutral traveler) or his/her weighted sum of the mean and the variance of travel time (risk-averse traveler) by unilaterally changing routes. For the SN-SO model, the problem involves distributing the stochastic flows to minimize the expected total travel time in the risk-neutral case and the weighted sum of the mean and the variance of the total travel time in the risk-averse one.

The rest of this paper is organized as follows. Section 2 introduces the notations used in the paper and presents the SN-UE and SN-SO models for the inelastic (fixed) demand case. In Section 3, the relationship between the SN-UE and SN-SO models in the risk-neutral case is discussed, and the SN-MCP model for the general distribution of travel demand is derived. A closed-form SN-MCP formulation for a specific case of the SN with lognormal demand is also presented. The analysis for the risk-averse case is then presented in Section 4 together with the closed-form SN-MCP derivation under the risk-averse model. In Section 5, four numerical experiments are conducted to compare the performance of the network under different MCP principles. Finally, the conclusion and suggestions for future research are given in Section 6. The extension of the general form of the SN-MCP model to the case with elastic demand is provided in Appendix B.

Section snippets

Framework of the stochastic network model

In this section, the mathematical formulations of the SN-UE and SN-SO models for a stochastic network with demand uncertainty are presented.

Analysis of marginal cost pricing

The classical form of MCP was derived under a variety of theoretical frameworks, from the standard case of a homogenous traffic stream moving along a given uniform stretch of road to complex network-based transport model systems. As mentioned, MCP states that road users using congested roads should pay a toll equal to the difference between the marginal social cost and private cost to minimize the total system cost if the demand is fixed, or maximize the social surplus if the demand is elastic.

Model incorporating the risk-based (prone/averse) attitude

In the previous section, travelers are assumed to choose the path that will minimize their expected travel time. This assumption represents the risk-neutral case. The objective function of the SN-SO model is to minimize the expected total travel time, which is also a risk-neutral case for the system manager. Under travel time uncertainty, travelers consider both travel time variability and mean travel time (Bates et al., 2001). In this section, travelers are assumed to consider both the mean

Numerical experiments

This section presents four numerical experiments to illustrate the calculations and implications of the SN-MCP and RSN-MCP toll schemes. The equilibrium mean link flow patterns under different toll schemes are calculated, and the expected total travel times are evaluated and compared.

Conclusions

This paper investigated first-best marginal cost pricing in a stochastic traffic network in which demand uncertainty is explicitly considered. The paper derived the true first-best marginal cost toll for a general network under the assumptions of risk-neutral and risk-averse behavior of travelers in route choice decision. In the risk-neutral case, the marginal cost toll under stochastic demand involves two terms: (i) the mean link flow times the derivative of the mean link travel time with

Acknowledgements

This work was supported by the Research Grants Council of Hong Kong (PolyU 5261/07E) and National Natural Science Foundation of China (Grant No. 70831002). The authors wish to thank the two anonymous referees for their comments and suggestions, which helped to improve the quality of the paper.

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