Evolution dynamic of the expressway toll-free policy impact on the mode choice in a bimodal transportation network during holidays

Abstract

The implementation of expressway toll-free policy during holidays in China has caused serious congestion and frequent accidents on expressways. Many studies have explored the policy’s macroscopic outcome and its countermeasures for policy managers, while limited attention has been paid to the influence mechanism of the policy on the individuals’ travel behavior, especially the mode choice behavior. More insight into the dynamic effects of individuals reacting to policy measure is needed. This study aims at analyzing the adaption progress of the individual’s mode choice behavior and estimating the time-varying influence of the traffic policy on the mode split. With assumptions travelers’ adapting behavior conform to the inertia and myopia principles, a Logit dynamic evolutionary model for mode choice is proposed. A unique globally stable equilibrium state for the model is derived with the strict mathematical analysis. As an application, the influence of the expressway toll-free policy on mode split is evaluated. The travel cost structure, the sensitivity of the travel distance and traffic supply, and the evolutionary dynamics of the mode split are analyzed in scenarios with and without the expressway toll-free policy. The result indicates that travel distance and network’s total supply amount remarkably affect the implementation effect of the policy.

Introduction

The rapid development of economy significantly arouses the leisure travel demand increase during public holidays. With unsynchronized construction of the multimodal traffic network, the conflict between traffic demand and supply is getting worse during holidays. Various issues, such as capacity tension of civil aviation and railway, considerably affect the traffic order and travel satisfaction. Meanwhile, with the car ownership increase during the past decades, self-driving has become an emerging way of tourism because of its flexibility, versatility and individuation. The encouragement of the self-driving tourism can get full play to the national expressway system as traffic artery and provide helpful supplementary to the national multimodal network. On this account, there are various preferential policies for expressways during holidays all over the world. For example, China has adopted the expressway toll-free policy for car users during public holidays since the National Day holiday in 2012. By reducing the travel cost of the car mode, the implementation of the expressway toll-free policy has strengthened the attractiveness of the traveling by cars on expressways, improved the national expressway system’s function as traffic artery, relieved the strain on railway and civil aviation systems, and further encouraged the self-driving tourism and stimulated the tourism economic development. The policy effect has reached its expectation to some extent. However, traffic managers’ prediction and preparation to the policy impact on the mode transfer and induced traffic volume on expressway are inadequate. Consequently, it resulted in serious congestion and frequent accidents. For instance, the Beijing’s total expressway traffic volume during the National Day holiday of 2012 increased by nearly 30% (the Special Issue of Beijing National Day Traffic Monitoring). Expressway network constantly arise to peak travel and even some of the highway paralysis.

The phenomena indicate that proper understanding of the traffic demand during holidays and the potential influence of traffic policies on the leisure travel behavior is very important to ensure the effectiveness of special traffic demand management policies and balance the traffic conflict. However, the potential effect of these policies on influencing travelers’ behavior is largely unknown. The analysis of the mode choice behavior plays an important role in evaluating the policy effectiveness. Currently, most of the previous studies on the mode choice behavior only focus on static analysis and commonly ignore the evolutionary process before reaching the static equilibrium state caused by the human bounded rationality. It would be helpful for traffic managers to do proper preparation for the implementation of a new policy and avoid unexpected congestion if they could gain clearer understanding of the time-varying influence of the traffic policy on the mode split.

In this study, we mainly focus on (1) analyzing adaption process of travelers’ mode choice behavior to the implementation of a new traffic policy, (2) practically studying the varied influence of the expressway toll-free policy on the mode split with different travel distance and service level, and (3) numerically studying the time-varying influence of the expressway toll-free policy on the mode split under a specific travel distance and supply. The remaining sections are organized as follows. In section “Literature review,” we present a comprehensive literature review on the preferential policies for expressways during holidays in countrywide and theoretical method of mode choice modeling. In section “Model description,” a Logit dynamic evolutionary model of the mode choice for a bimodal network is proposed, following the discussion of the existence of the model’s evolutionary stable strategy and the evolutionary dynamic of mode split over time. Then in section “Numerical example,” a numerical case based on the Chinese expressway toll-free policy during holidays is given to analyze the travel cost structure, test the sensitivity and identify the mode split evolutionary route in the scenarios with and without toll-free policy. Finally, the conclusions of this study are drawn in section “Conclusion.”

Literature review

Generally, preferential policies for expressways during holidays can be classified into three categories: toll free (China, Japan, South Korea and Malaysia), sub-period discount (United Kingdom, Japan and Malaysia), and fee cap (Japan). The fee cap has three patterns: travel time-based pattern, mileage-based pattern, and unlimited pattern in a day. As for the expressway toll-free policy, Akira Yanagisawa ¹ estimated the effect of freeing expressways on gasoline demand and CO₂ emissions by using a gasoline demand model. The result shows that if expressways become free of charge, gasoline demand is estimated to increase by about 7.2% (4.1 GL per year, or 9.6 Mt of CO₂ per year). K Uchida and N Sugiki ² examined the short-term effects of expressways toll-free policy that was introduced on 20% of the Japanese national expressways from 28 June 2010 to 11 March 2011. The conclusion points out that the application of the toll-free policy brought more social welfare to some parts of expressways than to all of the expressways. In China, the research on the expressway toll-free policy has become a highlight since its implementation during holidays from the National Day holiday 2012. Most relevant works focused on evaluating the outcome of the toll-free policy by statistical analysis and discussing its countermeasures—countermeasures including the economic strategy, ³ traffic flow organization, ⁴ traffic evanesce, ⁵ emergency handling of traffic accident, expressway network capacity, and improving methods. ⁶ However, limited attention has been paid to the individual’s learning process to the modified travel cost caused by the toll-free policy, and the variation of the policy’s effect with different travel distance and traffic service level.

The analysis of the mode choice behavior plays an important role in evaluating the policy effectiveness. Mode split model includes two types, the aggregated model and the disaggregate model, and have been applied and discussed extensively. However, most of the current mode split models are static.^7–10 They cannot estimate the time-varying responses of travelers’ mode choice and the transition of system mode split toward the new equilibrium state. The evolution game theory (EGT) is an emerging method of considering the effect of traffic polices on the evolution of the travel behavior and the system performance. Proposed by Mayr ¹¹ in 1970, the EGT integrates the dynamic method with the static equilibrium analysis of the game theory and is of great feasibility to model the repeated interactive choice for anonymous large groups and hence can also analyze the evolutionary process of the travel behavior in the traffic policy evaluation. Therefore, research on the application of the EGT to travel behavior becomes increasingly popular in various fields, for example, the route choice problem,^12,13 travel mode choices problem,^14,15 and rear-ending events on a congested freeway. ¹⁶ However, majority of the previous research is based on the qualitative payoff matrix that usually neglects the quantitative and empirical analysis. Moreover, they are mostly modeled by the replicator dynamic mechanism that can merely reflect the genetic and selection mechanism. The limitation of the replicator dynamic mechanism is significant in the situation that the multi-dimensional replicated dynamic model has no stable polymorphic equilibrium solution, ¹⁷ which results in its inconsistency with the diversity of population behaviors.

Model description

This article explores the impact mechanism of the expressway toll-free policy on the individual’s mode choice behavior and analyzes the evolution dynamic of the aggregate mode split. With the rapid construction of the corresponding infrastructures, expressway network and rail network have become the backbone of inter-city traffic system. In Beijing, for example, the road and railway modes account for over 90% of the total passenger volume according to the Beijing statistical yearbook from year 2000 to 2015. In order to simplify the problem, this article only concentrates on a bimodal network with expressway and railway.

Notations

The symbols in the model formulation and their representations are defined as Table 1.

Table 1.

List of notations.

Symbols of the sets
D	Set of OD pairs on the network
M	Set of all traffic modes
$P_m^d$	Set of paths connecting OD pair d in the sub-network of mode m
Lm	Set of directed links on the sub-network of mode m
Q	Set of the traveler demand
S	Set of the mode split of all OD pairs
Symbols of the indexes
d	Index of OD pairs, and $d\in D$
m	Index of traffic modes (1 = car, 2 = rail), and $m\in M$
p	Index of paths
l	Index of links
Symbols of the variables
$q^d$	Traveler demand of OD pair d
$q_m^d$	Traveler demand by mode m of OD pair d, and $q^d=\sum _{m\in M}{q}_m^d$
$s_m^d$	Share of the mode m for OD pair d, and $\sum _{m\in M}{s}_m^d=1$
$f_{m,p}^d$	Path flow on the path $p\in P_m^d$
$f_{m,l}$	Link flow on the link $l\in L_m$ , and $f_{m,l}=\sum _{d\in D}\sum _{p\in P_m^d}{f}_{m,p}^d$
$U_m^d$	Travel utility of travelers in the OD pair d who chose the mode m
$C_{m,p}^d$	Travel cost of travelers in the OD pair d who chose the path $p\in P_m^d$
$t_{m,p}^d$	Travel time of travelers in the OD pair d who chose the path $p\in P_m^d$
$g_{m,p}^d$	Fuel consumption of travelers in the OD pair d who chose the path $p\in P_m^d$ (actually only for travelers by car on the expressway)
${\tau }_{m,p}^d$	Toll charge for the car travelers and ticket price for the train travelers in the OD pair d who chose the path $p\in P_m^d$
Symbols of the parameters that need to be estimated
${\theta }_{t,m}^d$	Coefficient of the travel time component in the travel cost function of the mode m
${\theta }_{g,m}^d$	Coefficient of the fuel consumption component in the travel cost function of the mode m (actually only for travelers by car on the expressway)
${\theta }_{\tau ,m}^d$	Coefficient of the road charge or the ticket price component in the travel cost function of the mode m

Logit dynamic evolutionary model for the Mode choice

Theoretically, travelers usually want to maximize their individual random utility when making travel decisions. In this study, the individual random utility is assumed to be equal to the opposite of traffic impedance. Travel cost on each path is a random variable, and its mean value on the minimal path is addressed as the travel impedance of each mode. Therefore, we have the travel utility on each mode as in equation (1)

$$U_m^d=-E[min(C_{m,p}^d,p\in P_m^d)]+{\epsilon }_m^d$$

(1)

where ${\epsilon }_m^d$ represents travelers’ perception error on the path travel time and is usually assumed to fit a normally distribution with zero mean and constant variance. ⁹

Generally, the travel cost comprises time expense and monetary expense. For travelers by car on the expressway $(m=1)$ , it is widely acknowledged that fuel consumption and toll charge on pricing roads are both important components of the monetary expense. Thus, we combine the travel time $t_1$ , fuel consumption expense $g_1$ , and road charge ${\tau }_1$ in the car travel cost function as in equation (2)

$$\begin{array}{l}{C}_{m=1,p}^d={\theta }_{t,m=1}^d{t}_{m=1,p}^d+{\theta }_{g,m=1}^d{g}_{m=1,p}^d\\ +{\theta }_{\tau ,m=1}^d{\tau }_{m=1,p}^d,d\in D,p\in P_{m=1}^d\end{array}$$

(2)

Generally, the travel time component $t_{m=1,p}^d$ is a monotonically increasing and continuously differentiable function of mode share by car $s_{m=1}^d$ . However, the fuel consumption component $g_{m=1,p}^d$ is also a function of $s_{m=1}^d$ but not a monotonic function. In such case, the value of ${\theta }_{t,m=1}^d$ and ${\theta }_{g,m=1}^d$ should be subjected to certain constrains to ensure that the car travel cost $C_{m=1,p}^d$ is monotone increasing function of mode share by car $s_{m=1}^d$ .

For travelers by train on the rail sub-network, the travel cost comprises travel time and the ticket price. Thus, it is formulated as in equation (3)

$$\begin{array}{l}{C}_{m=2,p}^d={\theta }_{t,m=2}^d{t}_{m=2,p}^d+{\theta }_{\tau ,m=2}^d{\tau }_{m=2,p}^d,\\ d\in D,p\in P_{m=2}^d\end{array}$$

(3)

As for trains, their detailed and reliable information on the travel time and ticket price is available on their schedules. The rail mode travel cost $C_{d,2,p}$ is hence a constant which is decided by the given network. The practical analysis on the travel cost structure will be presented in section “Travel cost structure.”

In order to discuss how travelers response to the expressway toll-free policy and analyze the evolutionary trend of mode split, the dynamic evolutionary model of the mode split in a bimodal network is introduced. Due to its better rationale, the adaption of travelers’ behavior to the travel cost change caused by the implementation of a new policy will follow two assumptions: inertia and myopia. ¹⁸ Inertia means that individuals only reconsider their choices sporadically instead of continually, while Myopia indicates that travelers revise their choices according to their current behavior and utility opportunities. Denote $i\in M$ as the index of the current chosen traffic mode. Conditional switch rate ${\rho }_{\mathrm{im}}^d$ is the traveler’s choice updating procedures that describes the frequency of mode switch from current chosen mode i to mode m. It is a function of current utility $U_i^d$ and network mode split $s_m^d$ .

Let’s focus on the single OD-pair case with total demand q. The OD travel demand q is assumed to be a constant, which is decided by the population and attraction between OD pair $d\in D$ . We can deduce the expected motion of the choice updating procedures over the next dt time units, where dt is a relatively small time interval. According to the inertia principle, each traveler is equipped with a stochastic alarm clock. A ringing of a clock signals the arrival of a revision opportunity for the clock’s owner. The times between the rings of a traveler’s clock are mutually independent and subject to a rate $\lambda$ that conforms to exponential distribution. ¹⁸ Each traveler in the population accepts to receive $\lambda \mathrm{dt}$ revision opportunities during the next dt time units. If a traveler taking mode i receives a revision opportunity, he will switch to mode $m\ne i$ with probability ${\rho }_{\mathrm{im}}/\lambda$ , or remain mode i with probability $({\rho }_{\mathrm{ii}}/\lambda )=1-({\rho }_{\mathrm{im}}/\lambda )$ . Such decision is made independently of the timing of the clocks’ rings. According to the probability theory, the number of rings during time interval [0, t] follows Poisson distribution with mean $\lambda t$ . If the current mode shares are $s_1$ and $s_2$ , the number of revision opportunities received by travelers who currently taking mode i is

$$q{s}_i\lambda \mathrm{dt}$$

(4)

The probability of a traveler who gets a revision opportunity to switch to mode m is ${\rho }_{\mathrm{im}}/\lambda$ . By multiplying this probability and the number of revision opportunities, we can process the expected number of such switches during the next dt time units as

$$q{s}_i{\rho }_{\mathrm{im}}\mathrm{dt}$$

(5)

Furthermore, during the next dt time, the expected change in the number of travelers who chose mode i can be estimated by equation (6)

$$q\times d{s}_i=q\times (\sum _m{s}_m{\rho }_{\mathrm{mi}}\mathrm{dt}-\sum _m{s}_i{\rho }_{\mathrm{im}}\mathrm{dt})i,m\in M$$

(6)

We obtain a differential equation for the mode split by modifying equation (6) as in equation (7)

$${\stackrel{·}{s}}_i=\frac{d{s}_i}{\mathrm{dt}}=\sum _{m\in M}{s}_m{\rho }_{\mathrm{mi}}-s_i\sum _m{\rho }_{\mathrm{im}}$$

(7)

Equation (7) is a probability density function of the state variable $s_i$ . The mode split at a certain moment T hence can be calculated by equation (8)

$$s_i\left(T\right)=\underset{0}{\overset{T}{\int }}(\sum _{m\in M}{s}_m{\rho }_{\mathrm{mi}}-s_i\sum _m{\rho }_{\mathrm{im}})\mathrm{dt}$$

(8)

It should be noted that the process of an individual to do the iterative mode choice decision is a discrete process. In this article, such process for aggregate population is treated as a continuous evolutionary process by introducing the probability theory. In the evolutionary game theory, equation (7) is a deterministic dynamic model, namely, the mean dynamic. ¹⁸ The mean dynamic function shown in equation (7) is a continuous differential function. It is more convenient to do the equilibrium and stability analysis for the evolutionary dynamic model than the form shown in equation (8). The detailed equilibrium and stability analysis for the model are presented in the section “Network static equilibrium and dynamic evolution.”

The deduce process from equations (4) to (8) is based on the one single OD-pair case. For the general multimode and multiple OD-pair case, the mean dynamic is obtained as in equation (9)

$${\stackrel{·}{s}}_i^d=\sum _{m\in M}{s}_m^d{\rho }_{\mathrm{mi}}^d-s_i^d\sum _{m\in M}{\rho }_{\mathrm{im}}^d$$

(9)

The expressway and railway sub-networks are independent from each other and there is no over-lapping or interaction. Hence, the problem of Independence of Irrelevant Alternatives ¹⁹ can be avoided. Therefore, the mode choice behavior is assumed to conform to a binary Logit model. ²⁰ The binary Logit model obeys the individual random utility maximum principle that considers both the individual preferences and the incomplete information. Such characteristic enables the method to simulate the practical traveling decision-making behavior better than the replicated mechanism. Then, for any OD pair d, the exact conditional switch rate from the current mode i to the mode m equals the choice probability of the mode m in a binary Logit model as in equation (10)

$${\rho }_{\mathrm{im}}^d=\frac{1}{1+\exp \left[{\mu }^{-1}(U_i^d-U_m^d)\right]}$$

(10)

where $\mu >0$ is called the noise level. For any value of $\mu >0$ , each alternative receives positive probability regardless of the mode utility $U_m^d$ . For detailed discussion of the noise level $\mu$ on the switch rate ${\rho }_{\mathrm{im}}^d$ , readers can refer to Sandholm’s ¹⁸ study.

The conditional switch rates ${\rho }_{\mathrm{mi}}$ and ${\rho }_{\mathrm{im}}$ in equation (9) are substituted in equation (10). Then the generated mean dynamic is the Logit dynamic evolutionary model for the mode split on a bimodal network, as in equation (11). The corresponding mode split at a certain moment T, hence can be calculated by equation (12)

$${\stackrel{·}{s}}_i^d=\frac{d{s}_i}{\mathrm{dt}}=\frac{1}{1+\exp \left[\mu (U_m^d-U_i^d)\right]}-s_i^d$$

(11)

$$s_i^d\left(T\right)=\underset{0}{\overset{T}{\int }}(\frac{1}{1+\exp \left[\mu (U_m^d-U_i^d)\right]}-s_i^d)\mathrm{dt}$$

(12)

Network static equilibrium and dynamic evolution

In this section, we conduct the equilibrium and stability analysis for the Logit dynamic evolutionary model of the mode split in a bimodal network. Because there always exists the equality that $s_1^d+s_2^d=1$ in a bimodal network, we have $s_1^d=s_d$ and $s_2^d=1-s_d$ .

Denote the equilibrium solution of equation (11) as $S^{*}=\{s_d^{*},d\in D\}$ . There exists ${\stackrel{·}{s}}_d^{*}=0,\forall d\in D$ , that is

$${\stackrel{·}{s}}_d^{*}=\frac{1}{1+\exp \left[{\mu }^{-1}({U^{*}}_m^d-{U^{*}}_i^d)\right]}-s_d^{*}=0,\forall d\in D$$

(13)

$$\frac{1}{1+\exp \left[{\mu }^{-1}({U^{*}}_m^d-{U^{*}}_i^d)\right]}=s_d^{*}$$

(14)

Due to $1/(1+\exp [\mu (U_{d,m}-U_{d,i})])\ne 0,\forall d\in D$ . Thus, $s_d^{*}\ne 0,\forall d\in D$ . Then the equation (14) can be modified as in equation (15)

$$\exp \left[{\mu }^{-1}({U^{*}}_m^d-{U^{*}}_i^d)\right]=\frac{1}{s_d^{*}}-1$$

(15)

Then we can calculate the logarithm of both sides of equation (15) as in equation (16)

$${\mu }^{-1}({U^{*}}_m^d-{U^{*}}_i^d)=\ln (\frac{1}{s_d^{*}}-1)$$

(16)

Equation (16) can be modified as in equation (17)

$$\{\begin{array}{c}{y}_1={\mu }^{-1}(U_2^d-U_1^d),s_d\in [0,1]\hfill \\ \multicolumn{1}{c}{y_2=\ln (\frac{1}{s_d}-1),s_d\in (0,1]}\end{array}$$

(17)

Equation (17) can be solved by Graph Method. Seen from equations (1) to (3), note that the car mode utility $U_1^d$ is a function of car mode shares $S=\{s_d,\forall d\in D\}$ that increase monotonously and is continuously differentiable, while the rail mode utility $U_2^d$ is a constant. We further know that the function $y_1$ is a monotone increasing function. $y_2$ is a monotone decreasing function. Independence variables’ domain of both functions is $s_d\in (0,1]$ , and their dependence variables’ value range is both covering $(-\infty ,+\infty )$ . Thus, it is obvious that equation (17) has a unique solution $S^{*}=\{s_d^{*},d\in D\}$ as illustrated in Figure 1.

Figure 1.

Graph of the model’s equilibrium solution.

An evolution stable state (ESS) ²¹ is defined as an equilibrium state that is able to eliminate any small disturbance deviation. According to the stability analysis theory for one-dimensional dynamic systems, an equilibrium state is an ESS if the acceleration of its variation is less than zero. In order to identify the stability of the unique equilibrium solution $S^{*}=\{s_d^{*},d\in D\}$ , the acceleration of the state variation at point $S^{*}=\{s_d^{*},d\in D\}$ should be tested.

First, the acceleration of the state variation can be derived as in equation (18)

$$\frac{\partial \left({\stackrel{.}{s}}_d\right)}{\partial s_d}=-\left[1+{\mu }^{-1}\left[\frac{\partial \left(U_1^d\left(S\right)\right)}{\partial s_d}\right]\times \frac{\exp \left[{\mu }^{-1}\left(U_2^d-U_1^d\left(S\right)\right)\right]}{{\left[1+\exp \left({\mu }^{-1}\left(U_2^d-U_1^d\left(S\right)\right)\right)\right]}^2}\right]$$

(18)

According to equation (11), there holds

$$\frac{1}{1+\exp \left[{\mu }^{-1}(U_2^d-U_1^d\left({\mathbf{S}}^{*}\right))\right]}-s_d^{*}=0$$

(19)

$$\frac{1}{1+\exp \left[{\mu }^{-1}(U_2^d-U_1^d\left({\mathbf{S}}^{*}\right))\right]}=s_d^{*}$$

(20)

$$\exp \left[{\mu }^{-1}(U_2^d-U_1^d\left({\mathbf{S}}^{*}\right))\right]=\frac{1}{s_d^{*}}-1$$

(21)

Equation (18) is substituted in equation (21). Thus, we have

$$\begin{array}{l}J={\left(\frac{\partial \left({\stackrel{.}{s}}_d\right)}{\partial s_d}\right)}_{s_d^{\ast }}\\ =-\left[1+K\times s_d^{\ast }\times \left(1-s_d^{\ast }\right)\right],\mathrm{where}K={\mu }^{-1}{\left(\frac{\partial \left(U_1^d\left(S\right)\right)}{\partial s_d}\right)}_{s_d^{\ast }}>0\end{array}$$

(22)

Due to $s_d^{*}\in (0,1],\forall d\in D$ , there holds

$$-\frac{1}{s_d^{*}\times (1-s_d^{*})}\le -4<0\le K,$$

That is

$$K\times s_d^{*}\times (1-s_d^{*})\ge -1$$

On this condition, $J<0$ can be set up easily. It means the unique solution $S^{*}=\{s_d^{*},d\in D\}$ is a globally stable point. The phase diagram of evolutionary mode split model on a bimodal network defined by equation (10) is shown in Figure 2.

Figure 2.

Phase diagram of the Logit dynamic evolutionary model.

The evolutionary mode split model on a bimodal network defined by equation (10) has a unique globally stable ESS and is proved above. As the utility of car mode is a function of traffic flow on the road network, the equilibrium network flows should be calculated to gain the value of $U_1^d(S^{*})$ , and then to calculate the value of $S^{*}$ . So the equilibrium solution problem of the dynamic mode split model can be converted to the bimodal network equilibrium problem. It can be solved by a convergent cost averaging method incorporating a two-stage Monte Carlo simulation–based stochastic network loading method. The existence and uniqueness of the solution to the network equilibrium flows can be guaranteed according to Theorems 1 and 2 proposed by Cantarella. ²² For detailed description of the two methods, readers can refer to Meng and Liu’s ⁹ study.

Numerical example

As an application, the influence of the expressway toll-free policy on the mode split is evaluated. The national expressway system is the traffic artery that connects different cities and mainly provides service for the inter-city trips. Because of the limited days of a specific holiday and the low frequency of an individual’s inter-city leisure trips during a specific holiday, it is meaningless to analyze the evolutionary dynamic during a specific holiday. However, the expressway toll-free policy has implemented for 19 holidays since National Day holiday 2012, and the travelers’ adaption process to the policy can be observed from several holidays. Although individuals may change destinations in different holidays, their trip experiences in previous holidays will update their recognition to the policy and then affect their mode choice behavior as responses to the policy in the next trip. For such consideration, the following analysis in this section would focus on the reiterative travel mode choice for the inter-city leisure trips in several holidays to ensure there is plenty of time for travelers to make mode choice decisions evolutionary.

Except for the expressway toll-free policy, the mode choice evolution procedure could be influenced by several factors, for example, the economic, the traffic network’s level of service, the specific travel demand during holiday, and so on. In addition, the policy’s effect on influencing the mode choice evolution also has correlation with the other factors. However, due to the lack of the practical data, it is difficult to get rid of the other factors’ influence and only evaluate the effect of the policy. In this case, the research in this article is based on a basic assumption that other factors are constant during the evolutionary period. In the numerical example, the policy effect is evaluated by comparing the difference between evolutionary stable mode splits in two scenarios with and without the expressway toll-free policy. To make sure the comparability of these two scenarios, the same values of other factors are set for both scenarios, that is, the same economic development level, the same traffic network supply, and the fixed traffic demand. The only difference between these two scenarios is whether the expressway toll-free policy is implemented or not. Therefore, the mode choice evolution procedure in the numerical example can illustrate the impact of the policy on the mode choice behavior in the specific case. Moreover, in order to have a wider significance, the sensitivity test is done for the traffic network supply and travel distance to analyze the correlation of these two factors with the policy’s influence.

Travel cost structure

The expressway toll-free policy uses the principle of economic leverage to manage the traffic demand. It encourages travelers to travel by car on expressway by cutting their toll charge. However, increasing expressway flow will in turn affect the travel time and fuel efficiency. Practically, the proportions of different components of travel cost vary with distance. Thus, the policy effect also varies with distance. Therefore, it is necessary to better understand the policy effect on different OD pairs by analyzing how the travel cost structure varies with distance.

According to equations (1)–(3), the travel utility on road and rail sub-network is reformulated as in equations (23) and (24), respectively

$$U_1=-({\theta }_{1,t}{t}_1+{\theta }_{1,g}{g}_1+{\theta }_{1,\tau }{\tau }_1)$$

(23)

$$U_2=-({\theta }_{2,t}{t}_2+{\theta }_{2,\tau }{\tau }_2)$$

(24)

where ${\theta }_{1,t},{\theta }_{2,t}$ are defined as the traveler’s value of time to convert all un-money expense into money unit. ${\theta }_{1,g}$ is defined as the unit fuel price. The parameters’ setting is shown in Table 2, including the coefficients in different components of the travel cost and the noise level of the Logit choice probability in the dynamic model.

Table 2.

Parameters’ setting.

Parameters
${\theta }_{1,t}={\theta }_{2,t}=21.9$	The per capita disposable income of urban residents in Beijing in 2014
${\theta }_{1,g}=6.5$	The unit fuel price in 2014
${\theta }_{1,\tau }={\theta }_{2,\tau }=1$	The road charge and ticket price are originally monetary expense, and there is no need to convert them into money unit
Coefficients in components of $t_1$ and $g_1$	Refer to the practical study on the speed-flow relationship model and fuel consumption model of expressway traffic flow by W Wei27 and Jia et al.,28 which is presented in detail in equations (23) and (24)
${\gamma }_{\tau }=0.5$	Coefficients in components of ${\tau }_1$ indicate the expressway toll standard per kilometer without toll-free policy. Its value setting is based on the widely used road pricing scheme in Beijing
Coefficients in components of $t_2$ and ${\tau }_2$	Practical estimation from data of 100 city pairs
$\mu =1$	Noise level of the Logit choice probability $\mu$ , which can’t be estimated without acquired available practical data. In order to simplify the analysis, this article sets the value of the noise level as 1. For the detail discussion on the impact of the noise level on the Logit choice probability, readers can refer to Sandholm’s 22 study

Practical speed–flow relationship model and fuel consumption model for expressway traffic flow are shown in equations (25) and (26), respectively.^23,24 According to the extensive distance-based road pricing scheme in China, the toll charge of car mode is shown in equation (27)

$$\{\begin{array}{cc}\multicolumn{1}{c}{}& {\gamma }_t=1.88+4.85{(V/C)}^3\hfill \\ \multicolumn{1}{c}{}& \begin{array}{c}v=\frac{0.93\times 120}{1+{(V/C)}^{{\gamma }_t}}\mathrm{km}/\mathrm{h}\hfill \\ \multicolumn{1}{c}{t_1=\frac{l}{v}}\end{array}\hfill \end{array}$$

(25)

$$g_1=({10}^{-5}{v}^2-1.824\times {10}^{-3}v+1.34078\times {10}^{-1})\times l$$

(26)

$${\tau }_1={\gamma }_{\tau }\times l$$

(27)

where ${\gamma }_t$ is a parameter of the travel time function, $(V/C)$ is the volume–capacity ratio of expressway, v is the average travel speed on expressway, l is the travel distance (km), and ${\gamma }_{\tau }$ is the expressway toll standard per kilometer (yuan/km). According to the road pricing scheme in Beijing, ${\gamma }_{\tau }=0.5$ if the expressway is charged, ${\gamma }_{\tau }=0$ if the expressway is under toll-free policy. For the detailed estimation on the coefficients in equations (25) and (26), readers can refer to the study by W Wei ²³ and Jia et al. ²⁴

For a definite OD pair, the travel time and ticket fare on the rail link are a definite constant as shown in equation (3). For different OD pairs, the travel time and ticket fare are only associated with the distance between the OD pair. The data of 100 city pairs, including the travel distance, travel time, and the price of an economy class ticket of high-speed rails, are used to formulate the regression relationship between travel cost by train and the distance. The 100 city pairs are randomly selected and their distance range from 0 to 500 km. The linear fitting equations are shown in equations (28) and (29). The goodness of fit are, respectively, 0.9865 and 0.9225

$$t_2=0.0037l+1.2676$$

(28)

$${\tau }_2=0.4421l-1.5493$$

(29)

According to equations (24)–(27), we can generate the variation of the travel cost structure of car travelers with respect to the volume–capacity ratio of expressway (see Figure 3). It is observed that the fuel cost share keeps relatively stable between 30% and 35%. When the volume–capacity ratio of the expressway is lower than 0.8, there exists a relatively stable period for time cost share at approximately 20%, and for toll cost at approximately 47%. However, there is an enormous reduce in toll cost share from 45% to 19% when the volume–capacity ratio of expressway is >0.8. The main reason that results in such regulations is that expressway congestion improves the time cost share from 24% to 55%.

Figure 3.

Cost structure variation of car travel.

According to equations (24), (28), and (29), the variation of the cost structure of the railway with respect to the travel distance is shown in Figure 4. The trend shows the ticket price share increase from 24% when the distance is 25 km, while to 76% when the distance is 500 km.

Figure 4.

Cost structure variation of train travel.

Figure 5 shows the contour maps of the difference between travel costs of each mode $(U_2^d-U_1^d)$ with respect to the volume–capacity ratio of expressway and to the travel distance. The area that the difference is lower than zero is defined as the advantage area of the car mode, that is, traveling by car gain higher utility than by train. It can be seen from Figure 5 that under the expressway toll policy, there is only a small area (the left bottom of Figure 5, where the distance is shorter than 50 km) that has advantage for the car travelers. This area expands significantly to almost all the bottom part of Figure 5 under the expressway toll-free policy.

Figure 5.

Contour maps of travel cost comparison between car mode and rail mode.

Sensitivity test

The travel cost structure analysis indicated that the travel distance and the volume–capacity ratio play a vital role in the cooperation and competition between car and rail modes, which in turn influence the effect of toll-free policy on the mode choice behavior. When the traffic demand is fixed, the volume–capacity ratio is closely related to the supply capacity of both road and rail sub-networks. In order to evaluate how the travel distance and network supply-demand structure influence the performance of the expressway toll-free policy, following sensitivity is tested. Before the test, to simplify the problem and to concentrate on the dynamic evolutionary model for the mode split, a simplest network with two nodes, two links (one expressway link and one rail link), and one OD pair is presented as Figure 6. And we assume that the demand between the two nodes is 10,000 vehicles.

Figure 6.

The topology of the network example.

The policy effect of expressway toll-free is evaluated by the mode transfer split and measured by the deviation between stationary car mode shares of different scenarios with and without the expressway toll-free policy, that is, $\Delta s_d^{*}=s_{d,\mathrm{free}}^{*}-s_{d,\mathrm{toll}}^{*}$ . There are 20 values from 25 to 500 km selected for travel distance l, and 16 situations are set for different road and rail capacities. Figure 7 illustrates the variation of the mode transfer split with respect to the travel distance when $\mu =1$ . Each curve corresponds to a supply capacity of the traffic network. All values of these curves are above zero, which indicates that the toll-free policy has significant influence on improving the mode share of the car sub-network and relieving the stress of the rail sub-network.

Figure 7.

Relationship between travel distance and effects of toll-free policy on raising car mode share.

Among these curves, the black one represents the case of supply shortage, that is, $n_1+n_2<q$ , where $n_1$ and $n_2$ denote the road and rail capacity, respectively. The black curve keeps stable at approximately 0 for the value of $\Delta s_d^{*}$ , which illustrates that the toll-free policy has alight effect on attracting travelers from rail to car mode due to the terrible congestion.

Three red curves present the case of a balance between travel supply and demand, that is, $n_1+n_2=q$ . With the raising of distance, the red curves show a reduction trend and keep at approximately 0 for distance longer than 100 km. It indicates that, when the total supply of the traffic network is equal to the traffic demand, the toll-free policy only has slight attracting effect for short- and mid-distance travels and has no effect for long-distance travel.

Other curves stand for the case of oversupply, that is, $n_1+n_2>q$ . Different colors of curves represent different total-supply-amounts (TSA). Different line types with the same color represent different road-rail supply structure (RRSS). According to the comparison in the line colors, it is obvious that there is an increasing trend on the influence of the toll-free policy on raising car mode share along with the increase in TSA. In addition, the distance from 25 to 75 km witnesses an apparent growth in travelers’ transformation from train mode to car mode after the implementation of the toll-free policy. It is because that in this distance interval, the mode split by car is relatively high due to the apparent mode advantage, regardless of the expressway toll-free policy. Then the toll cost account for a small proportion because of the expressway congestion, which can be seen in Figure 3. The following trend shows a general decrease at the distance ranging from 75 to 500 km. In spite of expressway congestion, the reason why mode transfer split $\Delta s^{*}$ shows a decrease when the distance growing longer is that longer distance weakens the advantage of traveling by car, which can be seen in Figure 5.

In the case of oversupply, different colors of the curves in Figure 7 show an upward trend in the raising influence of the toll-free policy on the car mode share when the TSA increases. In order to intuitively summarize the relationship between traffic supply-demand structure and the policy effect, three travel distance values are selected to represent three occasions: 25 km for short-distance travel, 75 km for mid-distance travel, and 500 km for long-distance travel. Figure 8 shows the contour map of mode transfer split $\Delta s^{*}$ with respect to the network supply variation in these three occasions. Arrows at the top-right direction point to the direction that the network TSA becomes increasingly greater. Generally, the greater the network TSA, the lower the volume–capacity ratio of expressway, then more significant the effect of toll-free policy will be. Arrows at the top-left and bottom-right directions point to the direction that the RRSS is increasingly balanced. It indicates that the supply structure between RRSS has no impact on the implementation effect of the policy.

Figure 8.

Relationship between traffic supply conditions and vehicle trip attraction effects of toll-free policy.

Evolutionary dynamics

In the evolutionary dynamics, we set road capacity to 7500, rail capacity to 10,000, and travel distance to 50 km. Figure 9 presents the evolutionary trajectory of each mode share to the corresponding stationary equilibrium point with and without the expressway toll-free policy. As shown in Figure 9, regardless of the initial value, the system evolves toward a unique stable equilibrium point, which is consistent with the analysis explained in section “Network static equilibrium and dynamic evolution.” Furthermore, it is observed that without the toll-free policy, at the equilibrium point, approximately 35% of travelers choose car mode, while the rest choose the rail mode. While with the toll-free policy, travelers will reach equilibrium mode split where approximately 80% choose the car mode and the rest choose the rail mode. A significant increase in car mode share, around 45%, occurred after the implementation of the expressway toll-free policy and the objective of relieving the strain on the railway is achieved. However, attention must be paid to avoid traffic congestion on expressways.

Figure 9.

Evolutionary trajectory of mode split to corresponding stationary points in different scenarios.

Moreover, taking the evolutionary stable point (0.3477, 0.6523) in the scenario without expressway toll-free policy as the initial point in the scenario with the toll-free policy, Figure 10 indicates the evolving traces of the increase rate of the car mode share. It can be used to measure the time lag property of the policy, namely, the travelers’ response process to the toll-free policy. When the toll-free policy just put into implementation, the car travel cost is instantly reduced, which results in the transshipment of a large number of travelers from rail mode to car mode as well as the obvious raise of the car mode share. The raise of the expressway flow is accompanied by the increase in the car travel time and fuel consumption cost, which in turn gradually reduces the travelers’ transshipment from rail mode to car mode. The network mode split reaches the vicinity of the equilibrium point and keeps a stable state with slight fluctuation when the travel cost of both modes is nearly the same.

Figure 10.

The evolving traces of car mode shares increase rate.

Conclusion

This article dealt with the evolutionary process of the travelers’ mode choice behavior when a new traffic demand management policy is implemented. This study is oriented on two basic assumptions: (1) travelers’ mode choice conforms to the random utility maximum principle and (2) travelers’ mode choice adapting behavior conforms to the inertia and myopia principles, a Logit dynamic evolutionary model for the mode choice is proposed by this study. A unique globally stable equilibrium mode split for Logit dynamic evolutionary model is derived from the strict mathematical analysis. In the numerical example study, the influence of expressway toll-free policy on mode split is evaluated on a bimodal transportation network with expressway and railway modes. Two asymmetric nonlinear travel cost functions are formulated to better adapt to the practical situation, which synthetically consider the travel time, the fuel consumption, and the toll charges for car travelers. Based on the travel cost functions, the travel cost structure, the sensitivity of the travel distance, and traffic supply, the evolutionary route of the mode split is analyzed and compared in scenarios with and without expressway toll-free policy. The conclusions are as follows:

1. The expressway toll-free policy can enhance the mode share of the car sub-network and relieve the strain on the rail sub-network. However, the policy’s effect on raising the car mode share is significantly influenced by the travel distance and the network’s total supply amount (TSA) of both modes, while it is hardly impacted by the RRSS.

2. Generally, the greater the TSA, the more significant the policy’s effect on raising the car mode share. The policy’s effect is most significant on an oversupply network than on a supply-demand balance or a supply shortage one.

3. The travel distance has a nonlinear impact on the policy’s enhancing effect. In the oversupply network, the policy’s effect on raising the car mode share first strengthens and then weakens with the increase in the travel distance. The policy’s effect reached up to the maximum when the travel distance is nearly 75 km.

4. In the supply-demand balance network, the policy has a slight enhancing effect. With the increase in the travel distance, the mode transfer split first reduces when travel distance is shorter than 100 km and then graduates away. In the supply shortage network, the policy does not have obvious impact on the traffic mode split regardless of the travel distance.

Conclusions drawn from this article can provide useful guidance on the road pricing management. First, a distance-based preferential policy for expressways can be more considered than the existed indifference toll-free policy, which can ensure the policy’s enhancing effect on the car mode share and somehow take the expressway operate companies’ interests into account. Second, because of the bounded rationality, the effect of a new policy has a time lag property. Traffic managers should pay more attention to the evolution of travelers’ adaption process and try to make adequate preparation for the congestion caused by the preferential policy at the beginning when a new preferential policy is implemented.

The article is based on fixed travel demand. Further development can look into induced traffic demand and multimodal network, which would make it more complicated to investigate the network equilibrium problem.