# Research on Decision-Making Behavior of Discretionary Lane-Changing Based on Cumulative Prospect Theory

Authors: Xueqin Long; Liancai Zhang; Shanshan Liu; Jianjun Wang
Journal: Journal of Advanced Transportation
 (2020)

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## Abstract

In this paper, the decision-making model of discretionary lane-changing is established using cumulative prospect theory (CPT). Through analyzing the vehicles’ dynamic running states, safety spacing calculating approaches for discretionary lane-changing and lane-keeping have been put forward firstly. Then, based on CPT, a lane-changing decision model with accelerating space as its utility is proposed by estimating the difference between actual spacings and the safety spacings for discretionary lane-changing as well as lane-keeping. In order to calculate the utility of discretionary lane-changing, dynamic reference points and a parameter representing driver’s risk preference are introduced into the model. With the real data collected from an urban expressway, the distribution of discretionary lane-changing duration is analyzed, and the model parameters are also calibrated. Furthermore, the applicability of the model is evaluated by comparing with the actual observation and random unity model. Finally, the sensitivity analysis of the model is carried out, that is, assessing the influence degree of each variable on the decision result. The study reveals that the CPT-based model can describe discretionary lane-changing behavior more accurately, which consider drivers’ risk-aversion during decision-making.


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## Body

## 1. Introduction

Lane-changing is a common traffic phenomenon. It is difficult for drivers to predict the lane-changing behavior of surrounding vehicles especially the discretionary lane-changing, which usually takes only a few seconds from intention generation to operation completion. Discretionary lane-changing usually interferes with traffic flow and may cause traffic jams or even traffic accidents. The purpose of this paper is to propose a method to describe the discretionary lane-changing behavior and predict whether the driver will change lanes.As one of the most common traffic behaviors, lane-changing behavior has been paid extensive attention, and a series of achievements have been obtained in recent decades. The earliest study on lane-changing model dates back to 1986 when Gipps established a rule-based lane-changing model in the presence of obstacles from the perspective of gap acceptance threshold [1]. Later, many scholars developed some widely used models based on Gipps’ model, such as microscopic traffic simulator (MITSIM), simulation of intelligent transport systems (SITRAS), and corridor simulation (CORSIM) [2–4]. Presently, cellular automata, game theory, and discrete selection model are principally used in lane-changing behavior study. However, the current studies either ignore the driver factors or assume the driver is perfectly rational. Lots of experiments have confirmed that the results generated by the models based on perfect rationality are quite different from the actual observations.Simon pointed out that people are boundedly rational in the decision-making process because of limited perception and computing ability [5]. They are more likely to choose the satisfactory scheme rather than the optimal one calculated based on perfect rationality. Therefore, it is necessary to consider the driver’s bounded rationality in the study of discretionary lane-changing decision-making.In order to accurately describe the relationship between the driver’s lane-changing intention and the surrounding traffic conditions as well as personal risk preference, the cumulative prospect theory (CPT) under the framework of bounded rationality has been introduced into the paper. Considering the unpredictability of lane-changing duration, this article explores that the minimum safety spacing varies with the dynamic lane-changing duration and describes how the accelerating space of objective vehicle varies with the adjacent vehicles’ running states. On this basis, a decision-making model of discretionary lane-changing based on dynamic reference points is established. This study considers the limitation of driver’s perception and the difference in attitudes towards losses and gains. It breaks through the limitation of perfect rationality assumption and can describe the driver’s actual lane-changing process more accurately.The rest of this paper is organized as follows. Section2 provides a literature review on lane-changing behavior as well as the application of CPT in transportation. Section 3 analyzes the involved vehicles’ dynamic running states during lane-changing process and the minimum safety spacing for lane-changing between the subjective vehicle and the leading and following vehicles in the destination lane. Section 4 analyzes the minimum safety spacing for lane-keeping between the subjective vehicle and the leading vehicle in the originating lane. Based on Sections 3 and 4, a lane-changing decision-making model with accelerating space as its utility is established in Section 5. Combining with the real collected data, Section 6 analyzes the distribution of discretionary lane-changing duration and calibrates drivers’ risk preference coefficients. Further, driver’s lane-changing behavior is predicted using actual samples, and the sensitivity of the lane-changing decision-making model is analyzed. Finally, Section 7 provides conclusions and suggestions for future studies.


## 2. Literature Review

### 2.1. Lane-Changing Behavior

The study of cellular automata in lane-changing behavior began in the 1990s. In 1997, based on the single-lane cellular automata model (Nasch model) proposed by Nagel and Schreckenberg [6], Chowdhury et al. established the two-lane cellular automata model (STCA) by introducing lane-changing rules [7]. STCA model can well describe lane-changing process with simple evolution rules. Using cellular automata model, the asymmetric lane-changing behavior between the fast lane and the slow lane and effects of lane-changing rules on multilane highway traffic were discussed, respectively [8, 9].Some researchers adopted game theory to study lane-changing behavior. Kesting et al. first introduced game theory into lane-changing study and established a game model considering acceleration as the payoff of lane-changing [10]. Based on automated driving systems, Wang et al. put forward an approach to generate optimal lane-changing strategies while obeying safety and comfort requirements, including strategic overtaking, cooperative merging, and selecting a safe gap [11]. Also, game theory was used to analyze the influence of vehicle aggregation situations and driver’s aggressiveness on the lane-changing behavior [12, 13].Probabilistic methods were also employed to study lane-changing behavior. Li et al. proposed a lane-changing intention recognition algorithm combining with the hidden Markov model and Bayesian filtering techniques, which has been proved to have high recognition accuracy [14]. Lee et al. established an exponential probability model to study the relationship between lane-changing probability and relative gap as well as relative velocity [15]. In addition, some researchers took driver’s risk perception and uncertainties of surrounding vehicles into consideration while building probabilistic models [16–18].The lane-changing model based on utility selection was firstly proposed by Ahmed et al. [19]. The model takes individual vehicle as the research object and generates lane-changing demand by evaluating the utility (satisfaction degree) of driving on each lane. On this basis, Toledo et al. further integrated mandatory and discretionary lane-changing behavior into a utility model [20]. Later, through real vehicle experiments, Sun and Elefteriadou considered driver factors and established utility-based lane-changing models in various scenarios [21].The next generation simulation program (NGSIM), cosponsored by the U.S. Department of Transportation and the Federal Highway Administration, greatly promotes the research on highway lane-changing behavior by reducing the difficulty of obtaining actual data. Talebpour et al. constructed a lane-changing simulation framework, which was verified by NGSIM data to have higher accuracy than the basic gap acceptance model [22]. By using NGSIM data, Woo et al. and Vechione et al. evaluated the performance of a dynamic potential energy model on lane-changing behavior prediction and the effects of decision variables on discretionary lane-changing as well as mandatory lane-changing, respectively [23, 24].In recent years, the great popularity of artificial intelligence has led to some lane-changing behavior studies based on it. The neural network model can predict lane-changing behavior more accurately than the multinomial logit model [25]. It was adopted to explore the relationship between the driver’s lane-changing intention and lane-changing acceptance [26] and to analyze traveling heading angle as well as acceleration during lane-changing [27]. Recently, Balal et al. built a binary decision model of discretionary lane-changing based on fuzzy inference system to predict driver’s lane-changing intention and the choice of the timing of lane-changing [28].In the existing studies about lane-changing behavior, most of the lane-changing models were built based on the assumption of perfect rationality, which assume that drivers can accurately perceive the utility of lane-changing or lane-keeping and make the choice with maximum utility. However, studies have shown that experimental results based on perfect rationality deviate from the actual traffic behavior. Because while facing risk in decision-making, people cannot make accurate quantitative analysis objectively, but rely on subjective perception [29]. Therefore, a method that can reflect the driver’s perception error and risk attitude is needed to describe the driver’s decision-making behavior more accurately.


### 2.2. The Application of CPT in Transportation Research

Simon came up with the concept of bounded rationality, which has been widely accepted [5]. It is believed that when faced with decision-making, people’s perception, judgment, and prediction abilities are limited. Based on this basic idea, Tversky and Kahneman proposed the prospect theory and developed it into CPT [29, 30].CPT was mostly used to investigate travel behavior such as route choice and travel mode choice. It has not been applied to the study of driving behavior until recent years. Hamdar et al. conducted a series of studies on car-following behavior based on CPT [31–34]. In 2008, Hamdar et al. established a car-following model that captures risk-taking behavior under uncertainty and reflects drivers’ cognition of driving conditions of surrounding vehicles. In this study, gains and losses are assumed as the increases and decreases of velocity, respectively. In the subsequent studies, Hamdar et al. improved the previous car-following model through correlating subjective utilities with acceleration and deceleration and probed the effects of stochastic acceleration and lane-changing behavior on travel time reliability. In addition, the acceleration model was extended to explore the driver’s perception of dynamical driving environment (i.e., different weather conditions and road geometry configurations) and the way to execute acceleration maneuvers accordingly. Chow et al. used CPT to investigate driver’s choice behavior between high-occupancy-vehicle lanes and mixed-use lanes and calibrated CPT parameters with the genetic algorithm [35]. Taking the lane choice behavior between general purpose lanes and managed lanes as research objects, Huang et al. compared the difference between the performances of CPT and expected utility theory (EUT) in travel time estimation [36] and explored the influence of gender, age, and income on the probability weighting for risky travel times [37].At present, CPT has been rarely applied to driving behavior study, especially for lane-changing behavior. Several scholars used CPT to analyze the risky travel time differences of lanes with different characteristics and functions (such as management lane and general lane) and discussed the driver’s choice behavior of lanes with diverse attributes. However, these studies in essence belong to the category of route choice research and fail to reflect the relationship between the stochastic lane-changing behavior among common lanes with the same attributes and the surrounding traffic conditions.Through applying CPT, a free lane-changing decision-making model based on dynamic reference point is established in this paper. The measurement criteria of collision risk and gains are unified by introducing safety spacings, which eliminates the uncertainty of collision risk in previous studies. In addition, the driver risk factor is introduced to assess the influence of surrounding vehicles’ dynamic running states on discretionary lane-changing, which enriches the lane-changing decision-making studies under the framework of bounded rationality. CPT can well describe the driver group’s perception error and risk attitude, but it also has a limitation to reflect the characteristic differences among individuals.


## 2.1. Lane-Changing Behavior

The study of cellular automata in lane-changing behavior began in the 1990s. In 1997, based on the single-lane cellular automata model (Nasch model) proposed by Nagel and Schreckenberg [6], Chowdhury et al. established the two-lane cellular automata model (STCA) by introducing lane-changing rules [7]. STCA model can well describe lane-changing process with simple evolution rules. Using cellular automata model, the asymmetric lane-changing behavior between the fast lane and the slow lane and effects of lane-changing rules on multilane highway traffic were discussed, respectively [8, 9].Some researchers adopted game theory to study lane-changing behavior. Kesting et al. first introduced game theory into lane-changing study and established a game model considering acceleration as the payoff of lane-changing [10]. Based on automated driving systems, Wang et al. put forward an approach to generate optimal lane-changing strategies while obeying safety and comfort requirements, including strategic overtaking, cooperative merging, and selecting a safe gap [11]. Also, game theory was used to analyze the influence of vehicle aggregation situations and driver’s aggressiveness on the lane-changing behavior [12, 13].Probabilistic methods were also employed to study lane-changing behavior. Li et al. proposed a lane-changing intention recognition algorithm combining with the hidden Markov model and Bayesian filtering techniques, which has been proved to have high recognition accuracy [14]. Lee et al. established an exponential probability model to study the relationship between lane-changing probability and relative gap as well as relative velocity [15]. In addition, some researchers took driver’s risk perception and uncertainties of surrounding vehicles into consideration while building probabilistic models [16–18].The lane-changing model based on utility selection was firstly proposed by Ahmed et al. [19]. The model takes individual vehicle as the research object and generates lane-changing demand by evaluating the utility (satisfaction degree) of driving on each lane. On this basis, Toledo et al. further integrated mandatory and discretionary lane-changing behavior into a utility model [20]. Later, through real vehicle experiments, Sun and Elefteriadou considered driver factors and established utility-based lane-changing models in various scenarios [21].The next generation simulation program (NGSIM), cosponsored by the U.S. Department of Transportation and the Federal Highway Administration, greatly promotes the research on highway lane-changing behavior by reducing the difficulty of obtaining actual data. Talebpour et al. constructed a lane-changing simulation framework, which was verified by NGSIM data to have higher accuracy than the basic gap acceptance model [22]. By using NGSIM data, Woo et al. and Vechione et al. evaluated the performance of a dynamic potential energy model on lane-changing behavior prediction and the effects of decision variables on discretionary lane-changing as well as mandatory lane-changing, respectively [23, 24].In recent years, the great popularity of artificial intelligence has led to some lane-changing behavior studies based on it. The neural network model can predict lane-changing behavior more accurately than the multinomial logit model [25]. It was adopted to explore the relationship between the driver’s lane-changing intention and lane-changing acceptance [26] and to analyze traveling heading angle as well as acceleration during lane-changing [27]. Recently, Balal et al. built a binary decision model of discretionary lane-changing based on fuzzy inference system to predict driver’s lane-changing intention and the choice of the timing of lane-changing [28].In the existing studies about lane-changing behavior, most of the lane-changing models were built based on the assumption of perfect rationality, which assume that drivers can accurately perceive the utility of lane-changing or lane-keeping and make the choice with maximum utility. However, studies have shown that experimental results based on perfect rationality deviate from the actual traffic behavior. Because while facing risk in decision-making, people cannot make accurate quantitative analysis objectively, but rely on subjective perception [29]. Therefore, a method that can reflect the driver’s perception error and risk attitude is needed to describe the driver’s decision-making behavior more accurately.


## 2.2. The Application of CPT in Transportation Research

Simon came up with the concept of bounded rationality, which has been widely accepted [5]. It is believed that when faced with decision-making, people’s perception, judgment, and prediction abilities are limited. Based on this basic idea, Tversky and Kahneman proposed the prospect theory and developed it into CPT [29, 30].CPT was mostly used to investigate travel behavior such as route choice and travel mode choice. It has not been applied to the study of driving behavior until recent years. Hamdar et al. conducted a series of studies on car-following behavior based on CPT [31–34]. In 2008, Hamdar et al. established a car-following model that captures risk-taking behavior under uncertainty and reflects drivers’ cognition of driving conditions of surrounding vehicles. In this study, gains and losses are assumed as the increases and decreases of velocity, respectively. In the subsequent studies, Hamdar et al. improved the previous car-following model through correlating subjective utilities with acceleration and deceleration and probed the effects of stochastic acceleration and lane-changing behavior on travel time reliability. In addition, the acceleration model was extended to explore the driver’s perception of dynamical driving environment (i.e., different weather conditions and road geometry configurations) and the way to execute acceleration maneuvers accordingly. Chow et al. used CPT to investigate driver’s choice behavior between high-occupancy-vehicle lanes and mixed-use lanes and calibrated CPT parameters with the genetic algorithm [35]. Taking the lane choice behavior between general purpose lanes and managed lanes as research objects, Huang et al. compared the difference between the performances of CPT and expected utility theory (EUT) in travel time estimation [36] and explored the influence of gender, age, and income on the probability weighting for risky travel times [37].At present, CPT has been rarely applied to driving behavior study, especially for lane-changing behavior. Several scholars used CPT to analyze the risky travel time differences of lanes with different characteristics and functions (such as management lane and general lane) and discussed the driver’s choice behavior of lanes with diverse attributes. However, these studies in essence belong to the category of route choice research and fail to reflect the relationship between the stochastic lane-changing behavior among common lanes with the same attributes and the surrounding traffic conditions.Through applying CPT, a free lane-changing decision-making model based on dynamic reference point is established in this paper. The measurement criteria of collision risk and gains are unified by introducing safety spacings, which eliminates the uncertainty of collision risk in previous studies. In addition, the driver risk factor is introduced to assess the influence of surrounding vehicles’ dynamic running states on discretionary lane-changing, which enriches the lane-changing decision-making studies under the framework of bounded rationality. CPT can well describe the driver group’s perception error and risk attitude, but it also has a limitation to reflect the characteristic differences among individuals.


## 3. Minimum Safety Spacing for Discretionary Lane-Changing

### 3.1. Discretionary Lane-Changing Process

Now, presume a scenario of discretionary lane-changing, as shown in Figure1. Vehicle M (the merging vehicle) is going to drive from the originating to the destination lane. Vehicle Ld is the leading vehicle in the destination lane, and vehicle Fd is the following vehicle in the destination lane. While vehicle Lo is the leading vehicle in originating lane and vehicle Fo is the following vehicle in originating lane. Vehicle M needs to drive into the gap between vehicle Ld and vehicle Fd with a certain longitudinal and lateral acceleration in order to implement to lane-changing process. Driving velocities of vehicle Ld and Fd and the longitudinal spacings between vehicle M and Ld, M, and Fd are mainly considered to evaluate the potential risks of lane-changing for vehicle M. Vehicle M can easily control the distance from vehicle Lo, while the distance from vehicle Fo is mainly adjusted by vehicle Fo. Then, we can consider that there is no need for vehicle M to notice vehicle Fo during lane-changing. Therefore, the minimum longitudinal safety spacings between vehicle M and Ld, M and Fd are mainly analyzed while evaluating the risk of lane-changing.Figure 1
Pre-lane-changing configuration showing position of the involved vehicles.t = 0 is defined as the moment when vehicle M starts to accelerate laterally, that is, the moment when the lane-changing starts. t = T is the moment when vehicle M stopped laterally accelerating after entering the destination lane, that is, the end time of the lane-changing. And, T is the lane-changing duration.Shladover et al. proposed that the trajectory of the merging vehicle during the lane-changing can be described by a sinusoidal function [38]. The lateral acceleration of vehicle M is defined as alatt, which can be expressed by(1)alatt=2πDT2×sin2πtT,t∈0,T.By integrating equation (1) two times, the lateral displacement yt of vehicle M can be obtained as follows:(2)yt=−D2π×sin2πtT+DtT,t∈0,T,where D is the lateral displacement of vehicle M when the lane-changing is completed. During the lane-changing, vehicle M needs to move from the centerline of the originating lane to the centerline of the destination lane. Therefore, D can be considered as the width of a complete lane.The continuous process of lane-changing is divided into two phases. Phase I is from the momentt = 0 to the moment t=tC when the central point of vehicle M coincides with the lane boundary line. At the end of phase I, the lateral displacement of vehicle M reaches D/2, and then we can get t=T/2 according to equation (2). Phase II starts from the moment t=tC to the moment t = T. The process of discretionary lane-changing is shown in Figure 2:Figure 2
Discretionary lane-changing process.The vehicles’ following relationship will change during the process of lane-changing. In phase I, vehicleM follows vehicle Lo and vehicle Ld at the same time. The driver of vehicle M will pay attention to the leading vehicle Lo in the originating lane and the leading vehicle Ld in the destination lane simultaneously. So, the longitudinal acceleration of vehicle M is affected by both of vehicle Lo and Ld. Vehicle Fd shifts its attention from vehicle Ld to vehicle M when perceives the lane-changing intention of vehicle M. And, the leading vehicle of vehicle Fd changes from vehicle Ld to vehicle M. So, the acceleration of vehicle Fd is assumed to be only affected by vehicle M.In phase II, vehicleM has already entered the destination lane and has been driving following vehicle Ld. In this stage, the acceleration of vehicle M is only affected by vehicle Ld. At the same time, vehicle Fd continually drives following vehicle M, and the acceleration of vehicle Fd is still only affected by vehicle M.VehicleLo and vehicle Ld are both leading vehicles and will not be affected by the following vehicle M. Considering that the entire lane-changing process generally lasts for only a few seconds, for the convenience of study, this paper considers that vehicle Lo and vehicle Ld moves at the constant velocity vLo0 and vLd0, respectively. That is, the longitudinal acceleration of vehicle Lo and vehicle Ld are both 0.


### 3.2. Full Velocity Difference Model

In previous studies on driving behavior, the mainstream car-following models only considered the factor of spacing, but ignored the influence of relative velocity on acceleration. To make up for this defect, Jiang et al. [39] proposed the full velocity difference model (FVDM), which can exactly explain the phenomenon that when the velocity of the leading vehicle is greater than the following vehicle, the following vehicle will not slow down, even if the spacing between them is very small. FVDM can be expressed as follows:(3)ai,i+1t=κVΔxi,i+1t−vi+1t+εΔvi,i+1t,VΔxi,i+1t=V1+V2tanhC1Δxi,i+1t−li−C2,where ai,i+1t is the acceleration of the following vehicle i + 1 behind the leading vehicle i at the moment t, κVΔxi,i+1t−vi+1t represents the influence of spacing on the acceleration of following vehicle, εΔvi,i+1t represents the influence of relative velocity on the acceleration of following vehicle, Δxi,i+1t=xit−xi+1t is the spacing between the following vehicle i + 1 and the leading vehicle i at the moment t, Δvi,i+1t=vit−vi+1t is the relative velocity of vehicle i + 1 and vehicle i at the moment t, and vi+1t is the velocity of vehicle i + 1 at the moment t. li is the length of vehicle i; κ and ε are driver’s response sensitivity coefficients. V⋅ is optional velocity function. C1 represents the jam density. C2 is the adjustment factor. V1 and V2 are the parameters related to velocity.In this paper, FVDM is applied to lane-changing scene to study the acceleration changes of vehicleM and vehicle Fd.


### 3.3. Longitude Acceleration of VehicleM

The driver of vehicleM has a limited perceptive sensitivity and needs a reaction time Δt. It is assumed that from t = 0, the driver updates the perception of the involved vehicles’ running status every Δt. The driver adjusts the longitudinal acceleration according to the latest perceived driving status and drives with the latest updated longitudinal acceleration for Δt until the next time when longitudinal acceleration updates. The natural numbers N1 and N2 are assumed to satisfy the following relationships:(4)N1Δt≤tC<N1+1Δt,N2Δt≤T<N2+1Δt.


#### 3.3.1. Lane-Changing Phase I

In phase I, vehicleM drives following both the vehicle Lo and Ld at the same time. When FVDM is extended to the double-anticipative car-following situation, aM1i⋅Δt, the longitudinal acceleration of vehicle M at the moment t = i⋅Δt, can be calculated by(5)aM1i⋅Δt=μi⋅Δt⋅aLo,Mi⋅Δt+1−μi⋅Δt⋅aLd,Mi⋅Δt,i=0,1,2,…,N1.μi⋅Δt is the influence degree coefficient of vehicle Lo on the longitudinal acceleration of vehicle M. While 1−μi⋅Δt is the influence degree coefficient of vehicle Ld on the longitudinal acceleration of vehicle M. In phase I, with lateral displacement of vehicle M increasing, the driver’s attention to the vehicle Lo gradually decreases, while the attention to the vehicle Ld gradually increases. Therefore, during the lane-changing, μi⋅Δt gradually decreases from 1 to 0, and 1−μi⋅Δt gradually increases from 0 to 1 accordingly. In phase I, the lateral displacement of vehicle M gradually increases from 0 toD/2, and the relationship between μi⋅Δt and the lateral displacement yi⋅Δt of vehicle M is as follows:(6)μi⋅Δt=1−2yi⋅ΔtD,i=0,1,2,…,N1.Combining equations (2) and (6), it can be rewritten as(7)μi⋅Δt=1πsin2πi⋅ΔtT−2i⋅ΔtT+1,i=0,1,2,…,N1.According to FVDM, to calculateaM1i⋅Δt, variates vMi⋅Δt, ΔxLd,Mi⋅Δt, and ΔxLo,Mi⋅Δt are necessary, which can be obtained as follows:(8)vMi⋅Δt=vM0+∑j=0i−1aM1j⋅Δt×Δt,i=0,1,2,…,N1,(9)ΔxLd,Mi⋅Δt=ΔxLd,M0+∑j=0i−1∫0Δt∫0φ−aM1j⋅Δtdτdφ+vLd0−vMj⋅Δt×Δt,i=0,1,2,…,N1,(10)ΔxLo,Mi⋅Δt=ΔxLo,M0+∑j=0i−1∫0Δt∫0φ−aM1j⋅Δtdτdφ+vLo0−vMj⋅Δt×Δt,i=0,1,2,…,N1,where vM0 is the initial longitudinal velocity of vehicle M. ΔxLo,Mi⋅Δt and ΔxLd,Mi⋅Δt are the initial space headways between vehicle M and Lo, vehicle M and Ld, respectively.


#### 3.3.2. Lane-Changing Phase II

At the beginning of phase II,t=tC, vehicle M has already entered the destination lane and has been driving following vehicle Ld. Every time interval Δt, vehicle M adjusts the longitudinal acceleration according to the relative velocity and the longitudinal spacing between vehicle Ld and itself. In phase II, aM2i⋅Δt, the longitudinal acceleration of vehicle M at the moment i⋅Δt, can be obtained as follows:(11)aM2i⋅Δt=aLd,Mi⋅Δt,i=N1+1,N1+2,…,N2.Similarly, to calculateaM2i⋅Δt, variates vMi⋅Δt and ΔxLd,Mi⋅Δt are necessary, which can be obtained as follows:(12)vMi⋅Δt=vMN1Δt+∑j=N1i−1aM2j⋅Δt×Δt,i=N1+1,N1+2,…,N2,(13)ΔxLd,Mi⋅Δt=ΔxLd,MN1Δt+∑j=N1i−1∫0Δt∫0φ−aM2j⋅Δtdτdφ+vLd0−vMj⋅Δt×Δt,i=N1+1,N1+2,…,N2.


### 3.4. Longitude Acceleration of VehicleFd

In the same way, vehicleFd updates its acceleration following the similar principle as vehicle M does. That is, considering the same reaction time Δt of vehicle Fd, the longitudinal acceleration is updated every time interval Δt. The acceleration of the vehicle Fd is only influenced by vehicle M. Thus, during the lane-changing process, the longitudinal acceleration aFdi⋅Δt of vehicle Fd can be calculated by(14)aFdi⋅Δt=aM,Fdi⋅Δt,i=0,1,2,…,N2.According to FVDM, to calculateaFdi⋅Δt, variates vFdi⋅Δt and ΔxM,Fdi⋅Δt are necessary, which can be obtained as follows:(15)vFdi⋅Δt=vFd0+∑j=0i−1aFdj⋅Δt×Δt,i=0,1,2,…,N2,(16)ΔxM,Fdi⋅Δt=ΔxM,Fd0+∑j=0i−1∫0Δt∫0φaM1j⋅Δt−aFdj⋅Δtdτdφ+vMj⋅Δt−vFdj⋅Δt×Δt,i=1,2,…,N1,ΔxM,FdN1Δt+∑j=N1i−1∫0Δt∫0φaM2j⋅Δt−aFdj⋅Δtdτdφ+vMj⋅Δt−vFdj⋅Δt×Δt,i=N1+1,N1+2,…,N2,where vFd0 is the initial longitudinal velocity of vehicle Fd. ΔxM,Fd0 is the initial longitudinal space headway between vehicle M and Fd.


### 3.5. Minimum Safety Spacing

Jula et al. proposed that vehicleM should consider the safety spacing between each related vehicle and itself during the lane-changing [40]. In order to prevent from colliding, it is only necessary to ensure that both SrLd,Mt and SrM,Fdt are more than 0 at any moment within the interval tC,T. That is, SrLd,Mt and SrM,Fdt should satisfy the following equations:(17)SrLd,Mt=SrLd,M0+∫0t∫0φaLdτ−aMτdτdφ+vLd0−vM0×t>0,∀t∈tC,T,(18)SrM,Fdt=SrM,Fd0+∫0t∫0φaMτ−aFdτdτdφ+vM0−vFd0×t>0,∀t∈tC,T,where aLdτ, aMτ, and aFdτ are the accelerations of the vehicle Ld, M, and Fd during the lane-changing process, respectively. vLd0, vM0, and vFd0 are the initial velocities of vehicles Ld, M, and Fd, respectively. SrLd,M0 is the initial longitudinal spacing between the vehicle M and Ld. SrM,Fd0 is the initial longitudinal spacing between vehicles Fd and M.Under the condition that equations (17) and (18) are satisfied, the minimum values of SrLd,M0 and SrM,Fd0 are, respectively, denoted as the initial minimum safety spacings that vehicle M needs to keep with vehicle Ld and vehicle Fd before lane-changing, which can be marked as MSSLd,M and MSSM,Fd, and they can be calculated according to the following equations:(19)MSSLd,M=max∫0t∫0φaMτ−aLdτdτdφ+vM0−vLd0×t,∀t∈tC,T,(20)MSSM,Fd=max∫0t∫0φaFdτ−aMτdτdφ+vFd0−vM0×t,∀t∈tC,T.Therefore, to calculateMSSLd,M and MSSM,Fd, the longitudinal accelerations of vehicles Ld, M, and Fd should be obtained first. According to Sections 3.3 and 3.4, equations (19) and (20) can be rewritten as(21)MSSLd,M=max∑j=0N1−1∫0Δt∫0φaM1j⋅Δtdτdφ+vMjΔt−vLd0×Δt+∑j=N1i−1∫0Δt∫0φaM2j⋅Δtdτdφ+vMj⋅Δt−vLd0×Δt,i=N1+1,N1+2,…,N2,(22)MSSM,Fd=max∑j=0N1−1∫0Δt∫0φaFdj⋅Δt−aM1j⋅Δtdτdφ+vFdj⋅Δt−vMj⋅Δt×Δt+∑j=N1i−1∫0Δt∫0φaFdj⋅Δt−aM2j⋅Δtdτdφ+vFdj⋅Δt−vMj⋅Δt×Δt,i=N1+1,N1+2,…,N2.According to the above analysis, when the initial states of vehicleM, Ld, Lo, and Fd are fixed, MSSLd,M and MSSM,Fd are determined by T (that is, the lane-changing duration) and vary with T.


## 3.1. Discretionary Lane-Changing Process

Now, presume a scenario of discretionary lane-changing, as shown in Figure1. Vehicle M (the merging vehicle) is going to drive from the originating to the destination lane. Vehicle Ld is the leading vehicle in the destination lane, and vehicle Fd is the following vehicle in the destination lane. While vehicle Lo is the leading vehicle in originating lane and vehicle Fo is the following vehicle in originating lane. Vehicle M needs to drive into the gap between vehicle Ld and vehicle Fd with a certain longitudinal and lateral acceleration in order to implement to lane-changing process. Driving velocities of vehicle Ld and Fd and the longitudinal spacings between vehicle M and Ld, M, and Fd are mainly considered to evaluate the potential risks of lane-changing for vehicle M. Vehicle M can easily control the distance from vehicle Lo, while the distance from vehicle Fo is mainly adjusted by vehicle Fo. Then, we can consider that there is no need for vehicle M to notice vehicle Fo during lane-changing. Therefore, the minimum longitudinal safety spacings between vehicle M and Ld, M and Fd are mainly analyzed while evaluating the risk of lane-changing.Figure 1
Pre-lane-changing configuration showing position of the involved vehicles.t = 0 is defined as the moment when vehicle M starts to accelerate laterally, that is, the moment when the lane-changing starts. t = T is the moment when vehicle M stopped laterally accelerating after entering the destination lane, that is, the end time of the lane-changing. And, T is the lane-changing duration.Shladover et al. proposed that the trajectory of the merging vehicle during the lane-changing can be described by a sinusoidal function [38]. The lateral acceleration of vehicle M is defined as alatt, which can be expressed by(1)alatt=2πDT2×sin2πtT,t∈0,T.By integrating equation (1) two times, the lateral displacement yt of vehicle M can be obtained as follows:(2)yt=−D2π×sin2πtT+DtT,t∈0,T,where D is the lateral displacement of vehicle M when the lane-changing is completed. During the lane-changing, vehicle M needs to move from the centerline of the originating lane to the centerline of the destination lane. Therefore, D can be considered as the width of a complete lane.The continuous process of lane-changing is divided into two phases. Phase I is from the momentt = 0 to the moment t=tC when the central point of vehicle M coincides with the lane boundary line. At the end of phase I, the lateral displacement of vehicle M reaches D/2, and then we can get t=T/2 according to equation (2). Phase II starts from the moment t=tC to the moment t = T. The process of discretionary lane-changing is shown in Figure 2:Figure 2
Discretionary lane-changing process.The vehicles’ following relationship will change during the process of lane-changing. In phase I, vehicleM follows vehicle Lo and vehicle Ld at the same time. The driver of vehicle M will pay attention to the leading vehicle Lo in the originating lane and the leading vehicle Ld in the destination lane simultaneously. So, the longitudinal acceleration of vehicle M is affected by both of vehicle Lo and Ld. Vehicle Fd shifts its attention from vehicle Ld to vehicle M when perceives the lane-changing intention of vehicle M. And, the leading vehicle of vehicle Fd changes from vehicle Ld to vehicle M. So, the acceleration of vehicle Fd is assumed to be only affected by vehicle M.In phase II, vehicleM has already entered the destination lane and has been driving following vehicle Ld. In this stage, the acceleration of vehicle M is only affected by vehicle Ld. At the same time, vehicle Fd continually drives following vehicle M, and the acceleration of vehicle Fd is still only affected by vehicle M.VehicleLo and vehicle Ld are both leading vehicles and will not be affected by the following vehicle M. Considering that the entire lane-changing process generally lasts for only a few seconds, for the convenience of study, this paper considers that vehicle Lo and vehicle Ld moves at the constant velocity vLo0 and vLd0, respectively. That is, the longitudinal acceleration of vehicle Lo and vehicle Ld are both 0.


## 3.2. Full Velocity Difference Model

In previous studies on driving behavior, the mainstream car-following models only considered the factor of spacing, but ignored the influence of relative velocity on acceleration. To make up for this defect, Jiang et al. [39] proposed the full velocity difference model (FVDM), which can exactly explain the phenomenon that when the velocity of the leading vehicle is greater than the following vehicle, the following vehicle will not slow down, even if the spacing between them is very small. FVDM can be expressed as follows:(3)ai,i+1t=κVΔxi,i+1t−vi+1t+εΔvi,i+1t,VΔxi,i+1t=V1+V2tanhC1Δxi,i+1t−li−C2,where ai,i+1t is the acceleration of the following vehicle i + 1 behind the leading vehicle i at the moment t, κVΔxi,i+1t−vi+1t represents the influence of spacing on the acceleration of following vehicle, εΔvi,i+1t represents the influence of relative velocity on the acceleration of following vehicle, Δxi,i+1t=xit−xi+1t is the spacing between the following vehicle i + 1 and the leading vehicle i at the moment t, Δvi,i+1t=vit−vi+1t is the relative velocity of vehicle i + 1 and vehicle i at the moment t, and vi+1t is the velocity of vehicle i + 1 at the moment t. li is the length of vehicle i; κ and ε are driver’s response sensitivity coefficients. V⋅ is optional velocity function. C1 represents the jam density. C2 is the adjustment factor. V1 and V2 are the parameters related to velocity.In this paper, FVDM is applied to lane-changing scene to study the acceleration changes of vehicleM and vehicle Fd.


## 3.3. Longitude Acceleration of VehicleM

The driver of vehicleM has a limited perceptive sensitivity and needs a reaction time Δt. It is assumed that from t = 0, the driver updates the perception of the involved vehicles’ running status every Δt. The driver adjusts the longitudinal acceleration according to the latest perceived driving status and drives with the latest updated longitudinal acceleration for Δt until the next time when longitudinal acceleration updates. The natural numbers N1 and N2 are assumed to satisfy the following relationships:(4)N1Δt≤tC<N1+1Δt,N2Δt≤T<N2+1Δt.


### 3.3.1. Lane-Changing Phase I

In phase I, vehicleM drives following both the vehicle Lo and Ld at the same time. When FVDM is extended to the double-anticipative car-following situation, aM1i⋅Δt, the longitudinal acceleration of vehicle M at the moment t = i⋅Δt, can be calculated by(5)aM1i⋅Δt=μi⋅Δt⋅aLo,Mi⋅Δt+1−μi⋅Δt⋅aLd,Mi⋅Δt,i=0,1,2,…,N1.μi⋅Δt is the influence degree coefficient of vehicle Lo on the longitudinal acceleration of vehicle M. While 1−μi⋅Δt is the influence degree coefficient of vehicle Ld on the longitudinal acceleration of vehicle M. In phase I, with lateral displacement of vehicle M increasing, the driver’s attention to the vehicle Lo gradually decreases, while the attention to the vehicle Ld gradually increases. Therefore, during the lane-changing, μi⋅Δt gradually decreases from 1 to 0, and 1−μi⋅Δt gradually increases from 0 to 1 accordingly. In phase I, the lateral displacement of vehicle M gradually increases from 0 toD/2, and the relationship between μi⋅Δt and the lateral displacement yi⋅Δt of vehicle M is as follows:(6)μi⋅Δt=1−2yi⋅ΔtD,i=0,1,2,…,N1.Combining equations (2) and (6), it can be rewritten as(7)μi⋅Δt=1πsin2πi⋅ΔtT−2i⋅ΔtT+1,i=0,1,2,…,N1.According to FVDM, to calculateaM1i⋅Δt, variates vMi⋅Δt, ΔxLd,Mi⋅Δt, and ΔxLo,Mi⋅Δt are necessary, which can be obtained as follows:(8)vMi⋅Δt=vM0+∑j=0i−1aM1j⋅Δt×Δt,i=0,1,2,…,N1,(9)ΔxLd,Mi⋅Δt=ΔxLd,M0+∑j=0i−1∫0Δt∫0φ−aM1j⋅Δtdτdφ+vLd0−vMj⋅Δt×Δt,i=0,1,2,…,N1,(10)ΔxLo,Mi⋅Δt=ΔxLo,M0+∑j=0i−1∫0Δt∫0φ−aM1j⋅Δtdτdφ+vLo0−vMj⋅Δt×Δt,i=0,1,2,…,N1,where vM0 is the initial longitudinal velocity of vehicle M. ΔxLo,Mi⋅Δt and ΔxLd,Mi⋅Δt are the initial space headways between vehicle M and Lo, vehicle M and Ld, respectively.


### 3.3.2. Lane-Changing Phase II

At the beginning of phase II,t=tC, vehicle M has already entered the destination lane and has been driving following vehicle Ld. Every time interval Δt, vehicle M adjusts the longitudinal acceleration according to the relative velocity and the longitudinal spacing between vehicle Ld and itself. In phase II, aM2i⋅Δt, the longitudinal acceleration of vehicle M at the moment i⋅Δt, can be obtained as follows:(11)aM2i⋅Δt=aLd,Mi⋅Δt,i=N1+1,N1+2,…,N2.Similarly, to calculateaM2i⋅Δt, variates vMi⋅Δt and ΔxLd,Mi⋅Δt are necessary, which can be obtained as follows:(12)vMi⋅Δt=vMN1Δt+∑j=N1i−1aM2j⋅Δt×Δt,i=N1+1,N1+2,…,N2,(13)ΔxLd,Mi⋅Δt=ΔxLd,MN1Δt+∑j=N1i−1∫0Δt∫0φ−aM2j⋅Δtdτdφ+vLd0−vMj⋅Δt×Δt,i=N1+1,N1+2,…,N2.


## 3.3.1. Lane-Changing Phase I

In phase I, vehicleM drives following both the vehicle Lo and Ld at the same time. When FVDM is extended to the double-anticipative car-following situation, aM1i⋅Δt, the longitudinal acceleration of vehicle M at the moment t = i⋅Δt, can be calculated by(5)aM1i⋅Δt=μi⋅Δt⋅aLo,Mi⋅Δt+1−μi⋅Δt⋅aLd,Mi⋅Δt,i=0,1,2,…,N1.μi⋅Δt is the influence degree coefficient of vehicle Lo on the longitudinal acceleration of vehicle M. While 1−μi⋅Δt is the influence degree coefficient of vehicle Ld on the longitudinal acceleration of vehicle M. In phase I, with lateral displacement of vehicle M increasing, the driver’s attention to the vehicle Lo gradually decreases, while the attention to the vehicle Ld gradually increases. Therefore, during the lane-changing, μi⋅Δt gradually decreases from 1 to 0, and 1−μi⋅Δt gradually increases from 0 to 1 accordingly. In phase I, the lateral displacement of vehicle M gradually increases from 0 toD/2, and the relationship between μi⋅Δt and the lateral displacement yi⋅Δt of vehicle M is as follows:(6)μi⋅Δt=1−2yi⋅ΔtD,i=0,1,2,…,N1.Combining equations (2) and (6), it can be rewritten as(7)μi⋅Δt=1πsin2πi⋅ΔtT−2i⋅ΔtT+1,i=0,1,2,…,N1.According to FVDM, to calculateaM1i⋅Δt, variates vMi⋅Δt, ΔxLd,Mi⋅Δt, and ΔxLo,Mi⋅Δt are necessary, which can be obtained as follows:(8)vMi⋅Δt=vM0+∑j=0i−1aM1j⋅Δt×Δt,i=0,1,2,…,N1,(9)ΔxLd,Mi⋅Δt=ΔxLd,M0+∑j=0i−1∫0Δt∫0φ−aM1j⋅Δtdτdφ+vLd0−vMj⋅Δt×Δt,i=0,1,2,…,N1,(10)ΔxLo,Mi⋅Δt=ΔxLo,M0+∑j=0i−1∫0Δt∫0φ−aM1j⋅Δtdτdφ+vLo0−vMj⋅Δt×Δt,i=0,1,2,…,N1,where vM0 is the initial longitudinal velocity of vehicle M. ΔxLo,Mi⋅Δt and ΔxLd,Mi⋅Δt are the initial space headways between vehicle M and Lo, vehicle M and Ld, respectively.


## 3.3.2. Lane-Changing Phase II

At the beginning of phase II,t=tC, vehicle M has already entered the destination lane and has been driving following vehicle Ld. Every time interval Δt, vehicle M adjusts the longitudinal acceleration according to the relative velocity and the longitudinal spacing between vehicle Ld and itself. In phase II, aM2i⋅Δt, the longitudinal acceleration of vehicle M at the moment i⋅Δt, can be obtained as follows:(11)aM2i⋅Δt=aLd,Mi⋅Δt,i=N1+1,N1+2,…,N2.Similarly, to calculateaM2i⋅Δt, variates vMi⋅Δt and ΔxLd,Mi⋅Δt are necessary, which can be obtained as follows:(12)vMi⋅Δt=vMN1Δt+∑j=N1i−1aM2j⋅Δt×Δt,i=N1+1,N1+2,…,N2,(13)ΔxLd,Mi⋅Δt=ΔxLd,MN1Δt+∑j=N1i−1∫0Δt∫0φ−aM2j⋅Δtdτdφ+vLd0−vMj⋅Δt×Δt,i=N1+1,N1+2,…,N2.


## 3.4. Longitude Acceleration of VehicleFd

In the same way, vehicleFd updates its acceleration following the similar principle as vehicle M does. That is, considering the same reaction time Δt of vehicle Fd, the longitudinal acceleration is updated every time interval Δt. The acceleration of the vehicle Fd is only influenced by vehicle M. Thus, during the lane-changing process, the longitudinal acceleration aFdi⋅Δt of vehicle Fd can be calculated by(14)aFdi⋅Δt=aM,Fdi⋅Δt,i=0,1,2,…,N2.According to FVDM, to calculateaFdi⋅Δt, variates vFdi⋅Δt and ΔxM,Fdi⋅Δt are necessary, which can be obtained as follows:(15)vFdi⋅Δt=vFd0+∑j=0i−1aFdj⋅Δt×Δt,i=0,1,2,…,N2,(16)ΔxM,Fdi⋅Δt=ΔxM,Fd0+∑j=0i−1∫0Δt∫0φaM1j⋅Δt−aFdj⋅Δtdτdφ+vMj⋅Δt−vFdj⋅Δt×Δt,i=1,2,…,N1,ΔxM,FdN1Δt+∑j=N1i−1∫0Δt∫0φaM2j⋅Δt−aFdj⋅Δtdτdφ+vMj⋅Δt−vFdj⋅Δt×Δt,i=N1+1,N1+2,…,N2,where vFd0 is the initial longitudinal velocity of vehicle Fd. ΔxM,Fd0 is the initial longitudinal space headway between vehicle M and Fd.


## 3.5. Minimum Safety Spacing

Jula et al. proposed that vehicleM should consider the safety spacing between each related vehicle and itself during the lane-changing [40]. In order to prevent from colliding, it is only necessary to ensure that both SrLd,Mt and SrM,Fdt are more than 0 at any moment within the interval tC,T. That is, SrLd,Mt and SrM,Fdt should satisfy the following equations:(17)SrLd,Mt=SrLd,M0+∫0t∫0φaLdτ−aMτdτdφ+vLd0−vM0×t>0,∀t∈tC,T,(18)SrM,Fdt=SrM,Fd0+∫0t∫0φaMτ−aFdτdτdφ+vM0−vFd0×t>0,∀t∈tC,T,where aLdτ, aMτ, and aFdτ are the accelerations of the vehicle Ld, M, and Fd during the lane-changing process, respectively. vLd0, vM0, and vFd0 are the initial velocities of vehicles Ld, M, and Fd, respectively. SrLd,M0 is the initial longitudinal spacing between the vehicle M and Ld. SrM,Fd0 is the initial longitudinal spacing between vehicles Fd and M.Under the condition that equations (17) and (18) are satisfied, the minimum values of SrLd,M0 and SrM,Fd0 are, respectively, denoted as the initial minimum safety spacings that vehicle M needs to keep with vehicle Ld and vehicle Fd before lane-changing, which can be marked as MSSLd,M and MSSM,Fd, and they can be calculated according to the following equations:(19)MSSLd,M=max∫0t∫0φaMτ−aLdτdτdφ+vM0−vLd0×t,∀t∈tC,T,(20)MSSM,Fd=max∫0t∫0φaFdτ−aMτdτdφ+vFd0−vM0×t,∀t∈tC,T.Therefore, to calculateMSSLd,M and MSSM,Fd, the longitudinal accelerations of vehicles Ld, M, and Fd should be obtained first. According to Sections 3.3 and 3.4, equations (19) and (20) can be rewritten as(21)MSSLd,M=max∑j=0N1−1∫0Δt∫0φaM1j⋅Δtdτdφ+vMjΔt−vLd0×Δt+∑j=N1i−1∫0Δt∫0φaM2j⋅Δtdτdφ+vMj⋅Δt−vLd0×Δt,i=N1+1,N1+2,…,N2,(22)MSSM,Fd=max∑j=0N1−1∫0Δt∫0φaFdj⋅Δt−aM1j⋅Δtdτdφ+vFdj⋅Δt−vMj⋅Δt×Δt+∑j=N1i−1∫0Δt∫0φaFdj⋅Δt−aM2j⋅Δtdτdφ+vFdj⋅Δt−vMj⋅Δt×Δt,i=N1+1,N1+2,…,N2.According to the above analysis, when the initial states of vehicleM, Ld, Lo, and Fd are fixed, MSSLd,M and MSSM,Fd are determined by T (that is, the lane-changing duration) and vary with T.


## 4. Minimum Safety Spacing for Lane-Keeping

If vehicleM does not make the decision of lane-changing, but continues to keep driving following vehicle Lo in the originating lane, there is also a minimum safety spacing for vehicles M and Lo. If the initial spacing between vehicles M and Lo is bigger than the minimum safety spacing for lane-keeping, vehicle M has surplus space for accelerating; if not, vehicle M will decelerate to ensure safety and finally keep a safety spacing with vehicle Lo. Such a scenario is assumed: vehicle Lo suddenly brakes until completely stopping, and the maximum braking deceleration of vehicle Lo is as the same as vehicle M. In order to avoid the collision, vehicles M and Lo need to keep a minimum safety spacing MSSLo,M between themselves, which can be described as follows:(23)MSSLo,M=vM0td+vM20−vLo202amax,where amax is the maximum braking deceleration generally valued as 6 m/s2∼8 m/s2 [41]. Because different types of vehicle have different power performances, it is impossible to determine the specific maximum braking deceleration of each vehicle. Therefore, amax in this paper takes 7 m/s2, the empirical value of maximum braking deceleration of small cars on the dry pavement. td is the braking delay time of vehicle M, that is, the time interval from the moment vehicle Lo starts braking to the moment vehicle M starts braking. It is related to driver characteristics and vehicle performance and is generally valued at 1.2 s∼2.0 s in other similar studies [41]. It is hard to observe the braking delay time of each vehicle, so td in this paper takes 1.5 s, the empirical value under the same environment in congeneric studies.


## 5. Modeling of Discretionary Lane-Changing Decision-Making Behavior

### 5.1. Functions of CPT

CPT believes that the decision-maker is boundedly rational [30]. The decision-maker processes and judges the utility relying on the value function and the probability weight function. CPT considers the decision-maker’s characteristics of bounded rationality as follows: (1) decision-makers make judgments and choices by predicting the potential gains and losses of different options. The gains and losses depend on reference points. (2) Different people have different sensitivity of gains and losses. When faced with gains, people tend to be risk-averse. When faced with losses, people tend to be risk-seeking. (3) While making decisions, people tend to overestimate the small probability event and underestimate the large probability event.The above characteristics can be described by the value functionΦxi and the probability weight function wpi as follows:(24)Φxi=xi−x0α,xi−x0≥0,−λ−xi+x0β,xi−x0<0,(25)wpi=piγpiγ+1−piγ1/γ,where x0 is the reference point and xi is the possible utility of an option. When xi≥x0, xi represents a gain, and when xi<x0, xi represents a loss. pi is the probability corresponding to xi. α, β, λ, and γ are related parameters. Among them, α and β represent decision-makers’ attitude to risk. The values of α and β are between 0 and 1. And, the larger α and β are, the more inclined the decision-maker is to pursue risk. λ represents the sensitivity of the decision-maker to the losses. γ reflects the deviation degree between the probability perceived by the decision-maker and the actual probability.The shapes ofΦxi and wpi are shown in Figure 3.Figure 3
(a) Value function. (b) Probability weighing function.
(a)
(b)While applying CPT, the possible utilities are sorted from small to large and divided inton + 1 gains (the utilities equal to or larger than the reference point) and m losses (the utilities less than the reference point). The possible utilities are expressed as x−m,…,x1,x0,…,xn, and the corresponding probabilities are p−m,…,p−1,p0,…,pn. Therefore, the cumulative prospect value CPVx,p is calculated as follows:(26)π+pi=wpi+…+pn−wpi+1+…+pn,0≤i≤n,(27)π−pi=wp−m+…+p−i−wp−m+…+p−i−1,−m≤−i<0,(28)CPVx,p=∑i=0nΦxiπ+pi+∑i=−m−1Φxiπ−pi.The decision-maker will choose the option with the largest prospect cumulative value.


### 5.2. Model Establishment

The main basis for the driver to make decisions is the accelerating space obtained from lane-changing or lane-keeping. The accelerating space depends on the longitudinal spacings between the objective vehicle and related vehicles. Regardless of lane-changing or lane-keeping, if the longitudinal spacing is bigger than the corresponding minimum safety spacing, the driver will obtain a gain; otherwise, the driver will obtain a loss. Therefore, whether to change lanes becomes a risk decision-making behavior, which meets the applicable conditions of CPT.From the above analysis, it is known that whether the accelerating space can be obtained is the criterion for judging gains or losses. The utilities of the lane-changing and the lane-keeping are defined as the corresponding initial longitudinal spacings between vehicleM and the related vehicles. And, the reference points are the dynamic minimum safety spacings corresponding to the lane-changing duration.


#### 5.2.1. The Duration of Discretionary Lane-Changing

In the actual driving situation, drivers cannot accurately perceive the duration of discretionary lane-changing that has not yet happened. They can only predict it based on the previous experience of lane-changing in the same driving environment. Therefore, the duration of discretionary lane-changing is not a definite value, but subjects to a probability distribution. It is assumed that the duration of discretionary lane-changing follows a normal distribution:T∼N(μ0σ2). The significant distribution range of lane-changing duration is determined under a confidence of 99.8%, assuming that it is [T1, T2]. The minimum perceived time interval is denoted as Δt, and the range of lane-changing duration [T1, T2] is divided into n + m + 1 subintervals. Since Δt is small, the mean value T¯i=1/2T1+i⋅Δt+T1+i+1⋅Δt of each subinterval T1+i⋅Δt,T1+i+1⋅Δt (i from 0 to n + m) is used to represent any value within the subinterval. Then, the duration of discretionary lane-changing T has been discretized, and there are n + m + 1 values in total. The probability of each optional value of T can be expressed by the following probability distribution function:(29)PT=T¯i=PT1+i⋅Δt≤T<T1+i+1⋅Δt=∫T1+i⋅ΔtT1+i+1⋅Δt1σ2πe−t−μ02/2σ2dt,i=0,1,2,…,n+m.


#### 5.2.2. The Cumulative Prospect Value of Lane-Changing

At the initial moment of lane-changing, there is a longitudinal spacing and corresponding minimum safety spacing between vehiclesM and Ld, M and Fd, so the utility of lane-changing consists of front-utility and rear-utility. Front-utility is defined as the difference between the initial front longitudinal spacing SrLd,M0 and the front minimum safety spacing MSSLd,M. Rear-utility is defined as the difference between the initial rear longitudinal spacing SrM,Fd0 and the rear minimum safety spacing MSSM,Fd. The corresponding front-subjective utility ΦLd,M and rear-subjective utility ΦM,Fd can be calculated by(30)ΦLd,M=SrLd,M0−MSSLd,Mα,SrLd,M0−MSSLd,M≥0,−λ−SrLd,M0+MSSLd,Mβ,SrLd,M0−MSSLd,M<0,(31)ΦM,Fd=SrM,Fd0−MSSM,Fdα,SrM,Fd0−MSSM,Fd≥0,−λ−SrM,Fd0+MSSM,Fdβ,SrM,Fd0−MSSM,Fd<0.The total subjective utility of lane-changing is composed of front-subjective utility and rear-subjective utility and is calculated by(32)ΦLd,M,Fd=θΦLd,M+1−θΦM,Fd,where the value of the parameter θ is between 0 and 1, reflecting the driver’s attention degree on front-subjective utility. The larger value of θ indicates that when the accelerating space (front-utility) provided by lane-changing is larger, the driver shows a stronger tendency to change lanes for pursuing higher velocity. Therefore, θ represents the risk attitude of drivers. Larger θ denotes that, the driver is less sensitive to the risk of collision with vehicle Fd during lane-changing, which reflects the driver’s risk-seeking tendency when faced with gains. Smaller θ denotes that, the driver pays more attention to safety rather than velocity benefit which reflects the driver’s risk-aversion tendency when faced with gains.MSS
L
d
,
M and MSSLd,M are related to the lane-changing duration T, so the reference points of front-utility and rear-utility are dynamic reference points changing with T.According to previous analysis, the lane-changing durationT may take n + m + 1 possible values. For each possible value T¯i, the driver may obtain a gain or a loss. Each front-subjective utility and rear-subjective utility of lane-changing has the same probability as the corresponding lane-changing duration. That is, the probability of front-subjective utility and rear-subjective utility can be expressed as follows:(33)PΦLd,MT¯i=PT=T¯i,i=0,1,2,…,n+m,(34)PΦM,FdT¯i=PT=T¯i,i=0,1,2,…,n+m.The total subjective utility of lane-changing and the corresponding probability whenT takes different values can be calculated. The total subjective utility of lane-changing can be divided into m negative values and n + 1 nonnegative values from small to large: Φ−m,…,Φ−1,Φ0,…,Φn. And, the corresponding probabilities are p−m,…,p−1,p0,…,pn. Based on equations (26)∼(28), the cumulative prospect value of lane-changing can be calculated by(35)CPVC=∑i=0nΦiπ+pi+∑i=−m−1Φiπ−pi.


#### 5.2.3. The Cumulative Prospect Value of Lane-Keeping

If vehicleM decides to keep following vehicle Lo in the originating lane, the utility of lane-keeping is the difference between the initial spacing SrLo,M0, and the minimum safety spacing for lane-keeping MSSLo,M. Then, the subjective utility of lane-keeping can be calculated by(36)ΦLo,M=SrLo,M0−MSSLo,Mα,SrLo,M0−MSSLo,M≥0,−λ−SrLo,M0+MSSLo,Mβ,SrLo,M0−MSSLo,M<0.MSSLo,M is determined by the initial velocity of vehicle M and Lo. Therefore, the subjective utility of lane-keeping can be regarded as a certain value with probability 1, and the cumulative prospect value of lane-keeping CPVK is equal to the subjective utility, that is,(37)CPVK=ΦLo,M.


## 5.1. Functions of CPT

CPT believes that the decision-maker is boundedly rational [30]. The decision-maker processes and judges the utility relying on the value function and the probability weight function. CPT considers the decision-maker’s characteristics of bounded rationality as follows: (1) decision-makers make judgments and choices by predicting the potential gains and losses of different options. The gains and losses depend on reference points. (2) Different people have different sensitivity of gains and losses. When faced with gains, people tend to be risk-averse. When faced with losses, people tend to be risk-seeking. (3) While making decisions, people tend to overestimate the small probability event and underestimate the large probability event.The above characteristics can be described by the value functionΦxi and the probability weight function wpi as follows:(24)Φxi=xi−x0α,xi−x0≥0,−λ−xi+x0β,xi−x0<0,(25)wpi=piγpiγ+1−piγ1/γ,where x0 is the reference point and xi is the possible utility of an option. When xi≥x0, xi represents a gain, and when xi<x0, xi represents a loss. pi is the probability corresponding to xi. α, β, λ, and γ are related parameters. Among them, α and β represent decision-makers’ attitude to risk. The values of α and β are between 0 and 1. And, the larger α and β are, the more inclined the decision-maker is to pursue risk. λ represents the sensitivity of the decision-maker to the losses. γ reflects the deviation degree between the probability perceived by the decision-maker and the actual probability.The shapes ofΦxi and wpi are shown in Figure 3.Figure 3
(a) Value function. (b) Probability weighing function.
(a)
(b)While applying CPT, the possible utilities are sorted from small to large and divided inton + 1 gains (the utilities equal to or larger than the reference point) and m losses (the utilities less than the reference point). The possible utilities are expressed as x−m,…,x1,x0,…,xn, and the corresponding probabilities are p−m,…,p−1,p0,…,pn. Therefore, the cumulative prospect value CPVx,p is calculated as follows:(26)π+pi=wpi+…+pn−wpi+1+…+pn,0≤i≤n,(27)π−pi=wp−m+…+p−i−wp−m+…+p−i−1,−m≤−i<0,(28)CPVx,p=∑i=0nΦxiπ+pi+∑i=−m−1Φxiπ−pi.The decision-maker will choose the option with the largest prospect cumulative value.


## 5.2. Model Establishment

The main basis for the driver to make decisions is the accelerating space obtained from lane-changing or lane-keeping. The accelerating space depends on the longitudinal spacings between the objective vehicle and related vehicles. Regardless of lane-changing or lane-keeping, if the longitudinal spacing is bigger than the corresponding minimum safety spacing, the driver will obtain a gain; otherwise, the driver will obtain a loss. Therefore, whether to change lanes becomes a risk decision-making behavior, which meets the applicable conditions of CPT.From the above analysis, it is known that whether the accelerating space can be obtained is the criterion for judging gains or losses. The utilities of the lane-changing and the lane-keeping are defined as the corresponding initial longitudinal spacings between vehicleM and the related vehicles. And, the reference points are the dynamic minimum safety spacings corresponding to the lane-changing duration.


### 5.2.1. The Duration of Discretionary Lane-Changing

In the actual driving situation, drivers cannot accurately perceive the duration of discretionary lane-changing that has not yet happened. They can only predict it based on the previous experience of lane-changing in the same driving environment. Therefore, the duration of discretionary lane-changing is not a definite value, but subjects to a probability distribution. It is assumed that the duration of discretionary lane-changing follows a normal distribution:T∼N(μ0σ2). The significant distribution range of lane-changing duration is determined under a confidence of 99.8%, assuming that it is [T1, T2]. The minimum perceived time interval is denoted as Δt, and the range of lane-changing duration [T1, T2] is divided into n + m + 1 subintervals. Since Δt is small, the mean value T¯i=1/2T1+i⋅Δt+T1+i+1⋅Δt of each subinterval T1+i⋅Δt,T1+i+1⋅Δt (i from 0 to n + m) is used to represent any value within the subinterval. Then, the duration of discretionary lane-changing T has been discretized, and there are n + m + 1 values in total. The probability of each optional value of T can be expressed by the following probability distribution function:(29)PT=T¯i=PT1+i⋅Δt≤T<T1+i+1⋅Δt=∫T1+i⋅ΔtT1+i+1⋅Δt1σ2πe−t−μ02/2σ2dt,i=0,1,2,…,n+m.


### 5.2.2. The Cumulative Prospect Value of Lane-Changing

At the initial moment of lane-changing, there is a longitudinal spacing and corresponding minimum safety spacing between vehiclesM and Ld, M and Fd, so the utility of lane-changing consists of front-utility and rear-utility. Front-utility is defined as the difference between the initial front longitudinal spacing SrLd,M0 and the front minimum safety spacing MSSLd,M. Rear-utility is defined as the difference between the initial rear longitudinal spacing SrM,Fd0 and the rear minimum safety spacing MSSM,Fd. The corresponding front-subjective utility ΦLd,M and rear-subjective utility ΦM,Fd can be calculated by(30)ΦLd,M=SrLd,M0−MSSLd,Mα,SrLd,M0−MSSLd,M≥0,−λ−SrLd,M0+MSSLd,Mβ,SrLd,M0−MSSLd,M<0,(31)ΦM,Fd=SrM,Fd0−MSSM,Fdα,SrM,Fd0−MSSM,Fd≥0,−λ−SrM,Fd0+MSSM,Fdβ,SrM,Fd0−MSSM,Fd<0.The total subjective utility of lane-changing is composed of front-subjective utility and rear-subjective utility and is calculated by(32)ΦLd,M,Fd=θΦLd,M+1−θΦM,Fd,where the value of the parameter θ is between 0 and 1, reflecting the driver’s attention degree on front-subjective utility. The larger value of θ indicates that when the accelerating space (front-utility) provided by lane-changing is larger, the driver shows a stronger tendency to change lanes for pursuing higher velocity. Therefore, θ represents the risk attitude of drivers. Larger θ denotes that, the driver is less sensitive to the risk of collision with vehicle Fd during lane-changing, which reflects the driver’s risk-seeking tendency when faced with gains. Smaller θ denotes that, the driver pays more attention to safety rather than velocity benefit which reflects the driver’s risk-aversion tendency when faced with gains.MSS
L
d
,
M and MSSLd,M are related to the lane-changing duration T, so the reference points of front-utility and rear-utility are dynamic reference points changing with T.According to previous analysis, the lane-changing durationT may take n + m + 1 possible values. For each possible value T¯i, the driver may obtain a gain or a loss. Each front-subjective utility and rear-subjective utility of lane-changing has the same probability as the corresponding lane-changing duration. That is, the probability of front-subjective utility and rear-subjective utility can be expressed as follows:(33)PΦLd,MT¯i=PT=T¯i,i=0,1,2,…,n+m,(34)PΦM,FdT¯i=PT=T¯i,i=0,1,2,…,n+m.The total subjective utility of lane-changing and the corresponding probability whenT takes different values can be calculated. The total subjective utility of lane-changing can be divided into m negative values and n + 1 nonnegative values from small to large: Φ−m,…,Φ−1,Φ0,…,Φn. And, the corresponding probabilities are p−m,…,p−1,p0,…,pn. Based on equations (26)∼(28), the cumulative prospect value of lane-changing can be calculated by(35)CPVC=∑i=0nΦiπ+pi+∑i=−m−1Φiπ−pi.


### 5.2.3. The Cumulative Prospect Value of Lane-Keeping

If vehicleM decides to keep following vehicle Lo in the originating lane, the utility of lane-keeping is the difference between the initial spacing SrLo,M0, and the minimum safety spacing for lane-keeping MSSLo,M. Then, the subjective utility of lane-keeping can be calculated by(36)ΦLo,M=SrLo,M0−MSSLo,Mα,SrLo,M0−MSSLo,M≥0,−λ−SrLo,M0+MSSLo,Mβ,SrLo,M0−MSSLo,M<0.MSSLo,M is determined by the initial velocity of vehicle M and Lo. Therefore, the subjective utility of lane-keeping can be regarded as a certain value with probability 1, and the cumulative prospect value of lane-keeping CPVK is equal to the subjective utility, that is,(37)CPVK=ΦLo,M.


## 5.2.1. The Duration of Discretionary Lane-Changing

In the actual driving situation, drivers cannot accurately perceive the duration of discretionary lane-changing that has not yet happened. They can only predict it based on the previous experience of lane-changing in the same driving environment. Therefore, the duration of discretionary lane-changing is not a definite value, but subjects to a probability distribution. It is assumed that the duration of discretionary lane-changing follows a normal distribution:T∼N(μ0σ2). The significant distribution range of lane-changing duration is determined under a confidence of 99.8%, assuming that it is [T1, T2]. The minimum perceived time interval is denoted as Δt, and the range of lane-changing duration [T1, T2] is divided into n + m + 1 subintervals. Since Δt is small, the mean value T¯i=1/2T1+i⋅Δt+T1+i+1⋅Δt of each subinterval T1+i⋅Δt,T1+i+1⋅Δt (i from 0 to n + m) is used to represent any value within the subinterval. Then, the duration of discretionary lane-changing T has been discretized, and there are n + m + 1 values in total. The probability of each optional value of T can be expressed by the following probability distribution function:(29)PT=T¯i=PT1+i⋅Δt≤T<T1+i+1⋅Δt=∫T1+i⋅ΔtT1+i+1⋅Δt1σ2πe−t−μ02/2σ2dt,i=0,1,2,…,n+m.


## 5.2.2. The Cumulative Prospect Value of Lane-Changing

At the initial moment of lane-changing, there is a longitudinal spacing and corresponding minimum safety spacing between vehiclesM and Ld, M and Fd, so the utility of lane-changing consists of front-utility and rear-utility. Front-utility is defined as the difference between the initial front longitudinal spacing SrLd,M0 and the front minimum safety spacing MSSLd,M. Rear-utility is defined as the difference between the initial rear longitudinal spacing SrM,Fd0 and the rear minimum safety spacing MSSM,Fd. The corresponding front-subjective utility ΦLd,M and rear-subjective utility ΦM,Fd can be calculated by(30)ΦLd,M=SrLd,M0−MSSLd,Mα,SrLd,M0−MSSLd,M≥0,−λ−SrLd,M0+MSSLd,Mβ,SrLd,M0−MSSLd,M<0,(31)ΦM,Fd=SrM,Fd0−MSSM,Fdα,SrM,Fd0−MSSM,Fd≥0,−λ−SrM,Fd0+MSSM,Fdβ,SrM,Fd0−MSSM,Fd<0.The total subjective utility of lane-changing is composed of front-subjective utility and rear-subjective utility and is calculated by(32)ΦLd,M,Fd=θΦLd,M+1−θΦM,Fd,where the value of the parameter θ is between 0 and 1, reflecting the driver’s attention degree on front-subjective utility. The larger value of θ indicates that when the accelerating space (front-utility) provided by lane-changing is larger, the driver shows a stronger tendency to change lanes for pursuing higher velocity. Therefore, θ represents the risk attitude of drivers. Larger θ denotes that, the driver is less sensitive to the risk of collision with vehicle Fd during lane-changing, which reflects the driver’s risk-seeking tendency when faced with gains. Smaller θ denotes that, the driver pays more attention to safety rather than velocity benefit which reflects the driver’s risk-aversion tendency when faced with gains.MSS
L
d
,
M and MSSLd,M are related to the lane-changing duration T, so the reference points of front-utility and rear-utility are dynamic reference points changing with T.According to previous analysis, the lane-changing durationT may take n + m + 1 possible values. For each possible value T¯i, the driver may obtain a gain or a loss. Each front-subjective utility and rear-subjective utility of lane-changing has the same probability as the corresponding lane-changing duration. That is, the probability of front-subjective utility and rear-subjective utility can be expressed as follows:(33)PΦLd,MT¯i=PT=T¯i,i=0,1,2,…,n+m,(34)PΦM,FdT¯i=PT=T¯i,i=0,1,2,…,n+m.The total subjective utility of lane-changing and the corresponding probability whenT takes different values can be calculated. The total subjective utility of lane-changing can be divided into m negative values and n + 1 nonnegative values from small to large: Φ−m,…,Φ−1,Φ0,…,Φn. And, the corresponding probabilities are p−m,…,p−1,p0,…,pn. Based on equations (26)∼(28), the cumulative prospect value of lane-changing can be calculated by(35)CPVC=∑i=0nΦiπ+pi+∑i=−m−1Φiπ−pi.


## 5.2.3. The Cumulative Prospect Value of Lane-Keeping

If vehicleM decides to keep following vehicle Lo in the originating lane, the utility of lane-keeping is the difference between the initial spacing SrLo,M0, and the minimum safety spacing for lane-keeping MSSLo,M. Then, the subjective utility of lane-keeping can be calculated by(36)ΦLo,M=SrLo,M0−MSSLo,Mα,SrLo,M0−MSSLo,M≥0,−λ−SrLo,M0+MSSLo,Mβ,SrLo,M0−MSSLo,M<0.MSSLo,M is determined by the initial velocity of vehicle M and Lo. Therefore, the subjective utility of lane-keeping can be regarded as a certain value with probability 1, and the cumulative prospect value of lane-keeping CPVK is equal to the subjective utility, that is,(37)CPVK=ΦLo,M.


## 6. Numerical Analysis and Model Assessment

### 6.1. Data Collection

In order to obtain discretionary lane-changing data effectively, a two-way six-lane urban expressway, the southern section of the second ring road in Xi’an, Shaanxi province, China, was selected as the survey site. The selected road section with no large vehicles entering is about 1300 m in length. The driving behavior in the east-to-west way was observed, where the middle lane and the inside lane are both straight-going lanes and the outside lane is the straight-right-lane. In order to exclude the mandatory lane-changing behavior, the vehicles that move from the middle lane to the inside lane or move from the outside lane to the middle lane are recorded as discretionary lane-changing vehicles. Video shooting was carried out on the pedestrian overpass above the selected road section for several times with good weather and high visibility in December 2018. And, the valid camera shooting range is about 150 m long.After the video recording, Simi Motion, a video image processing software was used to analyze the video frame by frame (sampling frequency is 30 fps) to track the vehicle’s location at every moment. The first-order Butterworth filter and symmetric exponential moving average method (SEMA) were, respectively, used to filter velocity noise and random errors of lateral position. The vehicles in the middle lane or the outside lane were considered target vehicles. And, left-lane-changing vehicles were recorded as lane-changing samples while non-lane-changing vehicles were recorded as lane-keeping samples. Each sample recorded the velocity and the position of the target vehicle, the front vehicle, the left front vehicle, and the left rear vehicle. In addition, the lane-changing sample also contained the duration of lane-changing of the target vehicle. Finally, 940 samples were obtained, including 401 discretionary lane-changing samples and 539 lane-keeping samples. Descriptive statistics of lane-changing samples and lane-keeping samples are shown in Table1.Table 1
Descriptive statistics of samples.
Variable
Lane-changing
Lane-keeping
Mean
S.D.
Mean
S.D.
v
M
10.50
1.11
10.69
1.45
v
L
o
10.04
1.38
11.13
1.62
v
L
d
11.38
1.42
11.45
1.80
v
F
d
11.70
1.83
11.99
1.93
Δ
x
L
o
,
M
16.88
4.78
21.81
5.91
Δ
x
L
d
,
M
43.39
12.06
28.33
10.10
Δ
x
M
,
F
d
19.67
6.70
12.89
4.86


### 6.2. Distribution of Discretionary Lane-Changing Duration

401 observed samples of discretionary lane-changing are used to analyze the lane-changing duration. One-sample Kolmogorov–Smirnov test indicates that the data obey normal distribution:T ∼ N (5.48, 0.75), as shown in Table 2 and Figure 4.Table 2
One-sample Kolmogorov–Smirnov test.
Number
401
Normal parameters
Mean
5.477
Std. Deviation
0.748
Most extreme differences
Absolute
0.034
Positive
0.027
Negative
−0.034
Kolmogorov–smirnov Z
0.673
Asymp. sig. (2-tailed)
0.755Figure 4
The distribution of discretionary lane-changing duration.According to statistics, the 99.8% confidence interval of lane-changing duration is [μ0 − 3.01σ, μ0 + 3.01σ], that is, T∈3.22,7.73. Assuming Δt = 0.5 s, interval [3.22, 7.73] can be divided into 9 subintervals by Δt. The mean value of each subinterval is used to represent any value within the subinterval, and then the optional values of lane-changing duration are 3.47 s, 3.97 s, 4.47 s, 4.97 s, 5.47 s, 5.97 s, 6.47 s, 6.97 s, and 7.47 s, respectively. The probability of each lane-changing can be calculated by equation (27). Furthermore, the probability of each subjective utility of lane-changing is the same as the probability of corresponding lane-changing duration according to equations (33) and (34). Then, the cumulative prospect value of lane-changing and lane-keeping can be calculated.


### 6.3. Parameters Calibration

#### 6.3.1. Parameters Calibration of FVDM

This part will calibrate the parameters of FVDM according to the data extracted from the video. The optional velocity functionVΔxi,i+1t in FVDM represents the optimal velocity of the vehicle i + 1 when the space headway between the leading vehicle i and the following vehicle i + 1 is Δxi,i+1t. When the traffic density reaches the maximum, that is, the jam density C1, the average space headway reaches the minimum. In this case, vehicles can hardly move, so the optimal velocity of vehicle i + 1 is 0, that is,(38)VΔxjam=V1+V2tanhC1Δxjam−li−C2=0.WhenΔxi,i+1t⟶∝, vehicle i + 1 can run at free velocity, so the optimized speed is equal to free velocity vfree, that is,(39)limΔx⟶∝VΔx=V1+V2=vfree.In this study, the observation objects are small cars, andli takes 5 m. According to the observed data, the average minimum space headway Δxjam=7.69m, so C1=1/Δxjam=0.13m−1. The speed limit of the survey section is 70 km/h (19.44 m/s); therefore, the free speed vfree = 19.44 m/s. According to equations (38) and (39), V1 and V2 can be expressed by the equation with C2. Therefore, the next step is to calibrate κ, ε, and C2 in FVDM.In this part, the least square method was adopted. The relevant data of vehicleM and Fd in 401 lane-changing samples is used to estimate the parameters, and the root-mean-squared error (RMSE) is shown as follows:(40)RMSEκ,ε,C2=1802∑i=1401vi,MTi−vi,MrealTi2+vi,FdTi−vi,FdrealTi2,where vi,MTi and vi,FdTi, respectively, represent the calculated velocity of vehicle M and vehicle Fd at the end of lane-changing Ti in sample i. vi,MrealTi and vi,FdrealTi, respectively, represent the actual velocity of vehicle M and vehicle Fd at the end of lane-changing in sample i. When RMSEκ,ε,C2 reaches the minimum, the corresponding parameter values are shown in Table 3.Table 3
Estimated value and empirical value of the parameters in FVDM.
Parameters
Estimated value
Empirical value [39]
κ
0.62 s−1
0.41 s−1
ε
0.44 s−1
0.5 s−1
V
1
8.55 m/s
6.75 m/s
V
2
10.89 m/s
7.91 m/s
C
1
0.13 m−1
0.13 m−1
C
2
1.42
1.57


#### 6.3.2. Parameters Calibration of CPT

To calibrate the parameters of CTP, 940 samples are randomly divided into two data sets. Data set I contains 174 lane-changing samples and 255 lane-keeping samples; data set II contains 227 lane-changing samples and 284 lane-keeping samples. Samples in data set I were used to calibrate parameters, while samples in data set II were used to evaluate the model accuracy.The lane-changing samples in data set I were numbered from 1 to 174, and the lane-keeping samples were numbered from 175 to 429. And, the RMSE between the results predicted by model and drivers’ actual decision-making is shown in equations (41) and (42), and the least square method was used to estimate the parameters.(41)RMSEα,β,λ,γ,θ=1429∑i=1174Ri−12+∑i=175429Ri2,(42)Ri=1,CPVC,i>CPVK,i,0,CPVC,i≤CPVK,i,where CPVC,i and CPVK,i represents the cumulative prospect value of lane-changing and lane-keeping for the sample numbered i, respectively.WhenRMSEα,β,λ,γ,θ reaches the minimum, the corresponding parameter values are obtained as shown in Table 4.Table 4
Estimated value and empirical value of the parameters in CPT.
Parameters
Estimated value
Empirical value [30]
α
0.58
0.88
β
0.83
0.88
λ
1.94
2.25
γ
Gain
0.71
0.61
Loss
0.86
0.69
θ
0.68
—α
<
β represents that decision-makers are more sensitive to losses than gains, and λ>1 indicates that decision-makers tend to be loss-averse. γ<1 represents that decision-makers tend to overestimate small probabilities and underestimate large ones. θ>0.5 shows that drivers pay more attention to the front longitudinal spacing than the rear longitudinal spacing during lane-changing.


### 6.4. Model Evaluation

#### 6.4.1. Prediction Based on CPT

Samples in data set II were used to evaluate the accuracy of the calibrated model. Firstly, the cumulative prospect values of both lane-changing and lane-keeping of all samples in data set II are calculated. Then, the lane-changing samples and lane-keeping samples in data set II are numbered, respectively, in the ascending order according to the cumulative prospect value of lane-changing from small to large. Cumulative prospect values of lane-changing samples (no. 1–227) and lane-keeping samples (no. 228–511) are shown in Figure5.Figure 5
The distribution of cumulative prospect values of lane-changing and lane-keeping (a) for lane-changing samples and (b) for lane-keeping samples.
(a)
(b)As can be seen from Figure5(a), most of the cumulative prospect value of lane-changing is located above those of lane-keeping, that is, most of the lane-changing samples in data set II are predicted to change lanes, and only a few of them are predicted to keep the lane, which is consistent with most of the actual travel behavior. In Figure 5(b), the cumulative prospect values of lane-keeping for most lane-keeping samples are above those of lane-changing, showing that the results predicted by the calibrated model are mostly consistent with drivers’ actual behaviors. In addition, it is found that most of the prospect values of lane-keeping for lane-changing samples do not exceed 2 in Figure 5(a). It indicates that there exists a sufficient accelerating space for vehicle M in the current lane when the cumulative prospect value of lane-keeping reaches a certain level. This makes it unnecessary for vehicle M to change lanes, even if the cumulative prospect value of lane-changing is much larger than that of lane-keeping (as shown in the upper right area of Figure 5(b)).As can be seen intuitively from Figure5, the results generated from the model do not deviate too much from the drivers’ actual decision-makings. Numerical comparison of predicted results and observed samples is shown in Table 5.Table 5
Estimated results in data set I and data set II.
Samples
Lane-changing
Lane-keeping
Estimated number
Observed number
Matching rate (%)
Estimated number
Observed number
Matching rate (%)
Data set I
156
174
89.96
224
255
87.84
Data set II
200
227
88.11
241
284
84.86As shown in Table5, the matching rates are over 84%, indicating that the model has a high accuracy in predicting drivers’ decision-making behaviors. Whether in data set I or data set II, the matching rate on lane-keeping samples is lower than that on lane-changing samples. According to the decision-making model, when the cumulative prospect value of lane-changing is equal to or slightly greater than that of lane-keeping, drivers will choose to change lanes. However, in actual driving situation, drivers may not change lanes for pursuing a relatively small gain because there is a certain operation cost of lane-changing. These may explain why the matching rate on lane-keeping samples is lower than that on lane-changing samples.


#### 6.4.2. Comparison with Random Utility Model

In order to compare the prediction performance of CPT-based model with that of perfect rationality-based model, classic random utility model (RUM) is also considered. And, the utility functions based on RUM are shown as follows:(43)UK=VK+ξK=α1+β1SrLo,M0−MSSLo,M+ξK,(44)UC=VC+ξC=∑i=0n+mPT=T¯iβ2SrLd,M0−MSSLd,MT¯i+β3SrM,Fd0−MSSM,FdT¯i+ξC,where VK and VC, respectively, represent the observable utility of lane-keeping and lane-changing; ξK and ξC are the random error of lane-keeping and lane-changing, respectively; α1, β1, β2, and β3 are coefficients, which are calibrated as −0.122, 0.146, 0.097, and 0.049, respectively. Assuming that ξK and ξC are independent and obey Gumbel distribution, the probability of lane-changing is shown as follows:(45)PC=PUC>UK=11+eVK−VC.Data set II was used to evaluate the prediction performance of RUM. Take the lane-changing probability calculated through RUM of each sample as the classification threshold, which determines the prediction error. And, the prediction result of RUM in the case with the smallest prediction error was compared with that of CPT, as shown in Table6.Table 6
Prediction confusion matrix of data set II based on CPT and RUM.
Estimated number via CPT
Estimated number via RUM
Lane-changing
Lane-keeping
Lane-changing
Lane-keeping
Observed number
Lane-changing
200
27
188
39
Lane-keeping
43
241
72
212Take lane-changing as positive alternative and lane-keeping as negative alternative. Then, the true positive rate (TPR) = 200/227 = 0.8811 and the false positive rate (FPR) = 43/284 = 0.1514 of CPT-based model. Similarly, the TPR and FPR of RUM are 0.8282 and 0.2535, respectively. TPR of CPT based model is slightly larger than that of RUM, while FPR of CPT-based model is significantly smaller than that of RUM. This indicates that compared with CPT, RUM is more likely to overestimate the possibility of lane-changing. The receiver operating characteristic (ROC) curves of CPT and RUM based on data set II are shown in Figure6.Figure 6
Receiver operating characteristic curves.The area under curve (AUC) of CPT-based model is 0.865, bigger than AUC of RUM, 0.772. Table6 and Figure 6 indicate that CPT-based model performs better on prediction than RUM. It proves that the bounded rationality hypothesis of CPT is more consistent with the driver’s actual decision-making behavior.


### 6.5. Sensitivity Analysis

In order to study the relationship between the driving states of involved vehicles and drivers’ decision-makings, some decision variables’ effects on the cumulative prospect values are analyzed.


#### 6.5.1. Effects ofvM, ΔxLo,M, and ΔxLo,M on the Cumulative Prospect Value of Lane-Keeping

Influence ofΔvLo,M and ΔxLo,M when vM = 10 m/s and influence of vM and ΔxLo,Mwhen ΔvLo,M = 3 m/s are analyzed, respectively, as shown in Figure 7.Figure 7
Contour plots of the cumulative prospect value of lane-keeping (a) as function ofΔvLo,M and ΔxLo,M for vM = 10 m/s and (b) as function of vMand ΔxLo,M for ΔvLo,M = 3 m/s.
(a)
(b)The series of curves in Figure7 are contour lines, where each point on the same curve represents the equal cumulative prospect value of lane-keeping. Obviously, Figure 7(a) shows that the cumulative prospect value of lane-keeping increases significantly with the increase of ΔvLo,M and ΔxLo,M. In addition, the farther away from the contour line valued 0, the sparser it is, indicating that the driver’s sensitivity to gains or losses decreases gradually with the increase of gains or losses. This reflects the actual driving situation: when ΔvLo,M or ΔxLo,M is large enough, vehicle M can obtain sufficient accelerating space, and its intention to follow vehicle Lo will not significantly become stronger with the increase of ΔvLo,M or ΔxLo,M.Figure7(b) demonstrates that the prospect value of lane-keeping increases with ΔxLo,M increasing, which is consistent with the conclusion shown in Figure 7(a). However, when ΔvLo,M and ΔxLo,M are fixed, the prospect value of the lane-keeping presents a downward trend with the increase of vM. Because a greater vM will lead to larger minimum safety spacing for lane-keeping, which will bring about less gains, the relatively flat contour lines in Figure 7(b) indicates that the prospect value of the lane-keeping is not significantly affected by vM.


#### 6.5.2. Effects ofΔxLd,M, ΔxM,Fd, ΔvLd,M, and ΔvM,Fd on the Cumulative Prospect Value of Lane-Changing

Influence ofΔxLd,M and ΔxM,Fd on cumulative prospect value of lane-changing when vM = vLd = vLo = vFd = 10 m/s and ΔxLo,M = 20 m and influence of ΔvLd,M and ΔvM,Fd when vM = 10 m/s and ΔxLo,M = ΔxLd,M = ΔxM,Fd = 20 m are analyzed, respectively, as shown in Figure 8.Figure 8
Contour plots of the cumulative prospect value of lane-changing (a) as function ofΔxLd,M and ΔxM,Fd for vM = vLd = vLo = vFd = 10 m/s and (b) as function of ΔvLd,M and ΔvM,Fd for ΔxLo,M = ΔxLd,M = ΔxM,Fd = 20 m.
(a)
(b)In Figure8(a), the cumulative prospect value of lane-changing increases with ΔxLd,M and ΔxM,Fd increasing, and ΔxLd,M has a significant impact on it. In addition, it can be found that the left area and lower area in Figure 8(a) are darker in color, indicating that the cumulative prospect value of lane-changing is much smaller than that of other areas. Especially in the left area of Figure 8(a), steep contour lines indicate that when ΔxLd,M is very small, the gain of lane-changing keeps almost unchanged or even present as a loss even if ΔvM,Fd increases continuously. Further, the sensitivity of cumulative prospect value of lane-changing to ΔxM,Fd decreases sharply with the increase of ΔxM,Fd. Because the intention of vehicle M to change lanes will not substantially become stronger with the increase of ΔxM,Fd when the minimum safety space for lane-changing is met.In Figure8(b), it can be observed that the cumulative prospect value of lane-changing increases with ΔvLd,M and ΔvM,Fd increasing. And, the steep contour lines show that the cumulative prospect value of lane-changing is very sensitive to ΔvLd,M. That means, the greater the vLd is, the greater the increment of velocity that vehicle M will obtain after lane-changing and the stronger the intention of vehicle M to change lanes. However, ΔvM,Fd has little effect on the intention of vehicle M to change lanes. Therefore, the relative velocity between vehicle M and vehicle Ld is one of the decisive factors for the driver’s decision-making.


## 6.1. Data Collection

In order to obtain discretionary lane-changing data effectively, a two-way six-lane urban expressway, the southern section of the second ring road in Xi’an, Shaanxi province, China, was selected as the survey site. The selected road section with no large vehicles entering is about 1300 m in length. The driving behavior in the east-to-west way was observed, where the middle lane and the inside lane are both straight-going lanes and the outside lane is the straight-right-lane. In order to exclude the mandatory lane-changing behavior, the vehicles that move from the middle lane to the inside lane or move from the outside lane to the middle lane are recorded as discretionary lane-changing vehicles. Video shooting was carried out on the pedestrian overpass above the selected road section for several times with good weather and high visibility in December 2018. And, the valid camera shooting range is about 150 m long.After the video recording, Simi Motion, a video image processing software was used to analyze the video frame by frame (sampling frequency is 30 fps) to track the vehicle’s location at every moment. The first-order Butterworth filter and symmetric exponential moving average method (SEMA) were, respectively, used to filter velocity noise and random errors of lateral position. The vehicles in the middle lane or the outside lane were considered target vehicles. And, left-lane-changing vehicles were recorded as lane-changing samples while non-lane-changing vehicles were recorded as lane-keeping samples. Each sample recorded the velocity and the position of the target vehicle, the front vehicle, the left front vehicle, and the left rear vehicle. In addition, the lane-changing sample also contained the duration of lane-changing of the target vehicle. Finally, 940 samples were obtained, including 401 discretionary lane-changing samples and 539 lane-keeping samples. Descriptive statistics of lane-changing samples and lane-keeping samples are shown in Table1.Table 1
Descriptive statistics of samples.
Variable
Lane-changing
Lane-keeping
Mean
S.D.
Mean
S.D.
v
M
10.50
1.11
10.69
1.45
v
L
o
10.04
1.38
11.13
1.62
v
L
d
11.38
1.42
11.45
1.80
v
F
d
11.70
1.83
11.99
1.93
Δ
x
L
o
,
M
16.88
4.78
21.81
5.91
Δ
x
L
d
,
M
43.39
12.06
28.33
10.10
Δ
x
M
,
F
d
19.67
6.70
12.89
4.86


## 6.2. Distribution of Discretionary Lane-Changing Duration

401 observed samples of discretionary lane-changing are used to analyze the lane-changing duration. One-sample Kolmogorov–Smirnov test indicates that the data obey normal distribution:T ∼ N (5.48, 0.75), as shown in Table 2 and Figure 4.Table 2
One-sample Kolmogorov–Smirnov test.
Number
401
Normal parameters
Mean
5.477
Std. Deviation
0.748
Most extreme differences
Absolute
0.034
Positive
0.027
Negative
−0.034
Kolmogorov–smirnov Z
0.673
Asymp. sig. (2-tailed)
0.755Figure 4
The distribution of discretionary lane-changing duration.According to statistics, the 99.8% confidence interval of lane-changing duration is [μ0 − 3.01σ, μ0 + 3.01σ], that is, T∈3.22,7.73. Assuming Δt = 0.5 s, interval [3.22, 7.73] can be divided into 9 subintervals by Δt. The mean value of each subinterval is used to represent any value within the subinterval, and then the optional values of lane-changing duration are 3.47 s, 3.97 s, 4.47 s, 4.97 s, 5.47 s, 5.97 s, 6.47 s, 6.97 s, and 7.47 s, respectively. The probability of each lane-changing can be calculated by equation (27). Furthermore, the probability of each subjective utility of lane-changing is the same as the probability of corresponding lane-changing duration according to equations (33) and (34). Then, the cumulative prospect value of lane-changing and lane-keeping can be calculated.


## 6.3. Parameters Calibration

### 6.3.1. Parameters Calibration of FVDM

This part will calibrate the parameters of FVDM according to the data extracted from the video. The optional velocity functionVΔxi,i+1t in FVDM represents the optimal velocity of the vehicle i + 1 when the space headway between the leading vehicle i and the following vehicle i + 1 is Δxi,i+1t. When the traffic density reaches the maximum, that is, the jam density C1, the average space headway reaches the minimum. In this case, vehicles can hardly move, so the optimal velocity of vehicle i + 1 is 0, that is,(38)VΔxjam=V1+V2tanhC1Δxjam−li−C2=0.WhenΔxi,i+1t⟶∝, vehicle i + 1 can run at free velocity, so the optimized speed is equal to free velocity vfree, that is,(39)limΔx⟶∝VΔx=V1+V2=vfree.In this study, the observation objects are small cars, andli takes 5 m. According to the observed data, the average minimum space headway Δxjam=7.69m, so C1=1/Δxjam=0.13m−1. The speed limit of the survey section is 70 km/h (19.44 m/s); therefore, the free speed vfree = 19.44 m/s. According to equations (38) and (39), V1 and V2 can be expressed by the equation with C2. Therefore, the next step is to calibrate κ, ε, and C2 in FVDM.In this part, the least square method was adopted. The relevant data of vehicleM and Fd in 401 lane-changing samples is used to estimate the parameters, and the root-mean-squared error (RMSE) is shown as follows:(40)RMSEκ,ε,C2=1802∑i=1401vi,MTi−vi,MrealTi2+vi,FdTi−vi,FdrealTi2,where vi,MTi and vi,FdTi, respectively, represent the calculated velocity of vehicle M and vehicle Fd at the end of lane-changing Ti in sample i. vi,MrealTi and vi,FdrealTi, respectively, represent the actual velocity of vehicle M and vehicle Fd at the end of lane-changing in sample i. When RMSEκ,ε,C2 reaches the minimum, the corresponding parameter values are shown in Table 3.Table 3
Estimated value and empirical value of the parameters in FVDM.
Parameters
Estimated value
Empirical value [39]
κ
0.62 s−1
0.41 s−1
ε
0.44 s−1
0.5 s−1
V
1
8.55 m/s
6.75 m/s
V
2
10.89 m/s
7.91 m/s
C
1
0.13 m−1
0.13 m−1
C
2
1.42
1.57


### 6.3.2. Parameters Calibration of CPT

To calibrate the parameters of CTP, 940 samples are randomly divided into two data sets. Data set I contains 174 lane-changing samples and 255 lane-keeping samples; data set II contains 227 lane-changing samples and 284 lane-keeping samples. Samples in data set I were used to calibrate parameters, while samples in data set II were used to evaluate the model accuracy.The lane-changing samples in data set I were numbered from 1 to 174, and the lane-keeping samples were numbered from 175 to 429. And, the RMSE between the results predicted by model and drivers’ actual decision-making is shown in equations (41) and (42), and the least square method was used to estimate the parameters.(41)RMSEα,β,λ,γ,θ=1429∑i=1174Ri−12+∑i=175429Ri2,(42)Ri=1,CPVC,i>CPVK,i,0,CPVC,i≤CPVK,i,where CPVC,i and CPVK,i represents the cumulative prospect value of lane-changing and lane-keeping for the sample numbered i, respectively.WhenRMSEα,β,λ,γ,θ reaches the minimum, the corresponding parameter values are obtained as shown in Table 4.Table 4
Estimated value and empirical value of the parameters in CPT.
Parameters
Estimated value
Empirical value [30]
α
0.58
0.88
β
0.83
0.88
λ
1.94
2.25
γ
Gain
0.71
0.61
Loss
0.86
0.69
θ
0.68
—α
<
β represents that decision-makers are more sensitive to losses than gains, and λ>1 indicates that decision-makers tend to be loss-averse. γ<1 represents that decision-makers tend to overestimate small probabilities and underestimate large ones. θ>0.5 shows that drivers pay more attention to the front longitudinal spacing than the rear longitudinal spacing during lane-changing.


## 6.3.1. Parameters Calibration of FVDM

This part will calibrate the parameters of FVDM according to the data extracted from the video. The optional velocity functionVΔxi,i+1t in FVDM represents the optimal velocity of the vehicle i + 1 when the space headway between the leading vehicle i and the following vehicle i + 1 is Δxi,i+1t. When the traffic density reaches the maximum, that is, the jam density C1, the average space headway reaches the minimum. In this case, vehicles can hardly move, so the optimal velocity of vehicle i + 1 is 0, that is,(38)VΔxjam=V1+V2tanhC1Δxjam−li−C2=0.WhenΔxi,i+1t⟶∝, vehicle i + 1 can run at free velocity, so the optimized speed is equal to free velocity vfree, that is,(39)limΔx⟶∝VΔx=V1+V2=vfree.In this study, the observation objects are small cars, andli takes 5 m. According to the observed data, the average minimum space headway Δxjam=7.69m, so C1=1/Δxjam=0.13m−1. The speed limit of the survey section is 70 km/h (19.44 m/s); therefore, the free speed vfree = 19.44 m/s. According to equations (38) and (39), V1 and V2 can be expressed by the equation with C2. Therefore, the next step is to calibrate κ, ε, and C2 in FVDM.In this part, the least square method was adopted. The relevant data of vehicleM and Fd in 401 lane-changing samples is used to estimate the parameters, and the root-mean-squared error (RMSE) is shown as follows:(40)RMSEκ,ε,C2=1802∑i=1401vi,MTi−vi,MrealTi2+vi,FdTi−vi,FdrealTi2,where vi,MTi and vi,FdTi, respectively, represent the calculated velocity of vehicle M and vehicle Fd at the end of lane-changing Ti in sample i. vi,MrealTi and vi,FdrealTi, respectively, represent the actual velocity of vehicle M and vehicle Fd at the end of lane-changing in sample i. When RMSEκ,ε,C2 reaches the minimum, the corresponding parameter values are shown in Table 3.Table 3
Estimated value and empirical value of the parameters in FVDM.
Parameters
Estimated value
Empirical value [39]
κ
0.62 s−1
0.41 s−1
ε
0.44 s−1
0.5 s−1
V
1
8.55 m/s
6.75 m/s
V
2
10.89 m/s
7.91 m/s
C
1
0.13 m−1
0.13 m−1
C
2
1.42
1.57


## 6.3.2. Parameters Calibration of CPT

To calibrate the parameters of CTP, 940 samples are randomly divided into two data sets. Data set I contains 174 lane-changing samples and 255 lane-keeping samples; data set II contains 227 lane-changing samples and 284 lane-keeping samples. Samples in data set I were used to calibrate parameters, while samples in data set II were used to evaluate the model accuracy.The lane-changing samples in data set I were numbered from 1 to 174, and the lane-keeping samples were numbered from 175 to 429. And, the RMSE between the results predicted by model and drivers’ actual decision-making is shown in equations (41) and (42), and the least square method was used to estimate the parameters.(41)RMSEα,β,λ,γ,θ=1429∑i=1174Ri−12+∑i=175429Ri2,(42)Ri=1,CPVC,i>CPVK,i,0,CPVC,i≤CPVK,i,where CPVC,i and CPVK,i represents the cumulative prospect value of lane-changing and lane-keeping for the sample numbered i, respectively.WhenRMSEα,β,λ,γ,θ reaches the minimum, the corresponding parameter values are obtained as shown in Table 4.Table 4
Estimated value and empirical value of the parameters in CPT.
Parameters
Estimated value
Empirical value [30]
α
0.58
0.88
β
0.83
0.88
λ
1.94
2.25
γ
Gain
0.71
0.61
Loss
0.86
0.69
θ
0.68
—α
<
β represents that decision-makers are more sensitive to losses than gains, and λ>1 indicates that decision-makers tend to be loss-averse. γ<1 represents that decision-makers tend to overestimate small probabilities and underestimate large ones. θ>0.5 shows that drivers pay more attention to the front longitudinal spacing than the rear longitudinal spacing during lane-changing.


## 6.4. Model Evaluation

### 6.4.1. Prediction Based on CPT

Samples in data set II were used to evaluate the accuracy of the calibrated model. Firstly, the cumulative prospect values of both lane-changing and lane-keeping of all samples in data set II are calculated. Then, the lane-changing samples and lane-keeping samples in data set II are numbered, respectively, in the ascending order according to the cumulative prospect value of lane-changing from small to large. Cumulative prospect values of lane-changing samples (no. 1–227) and lane-keeping samples (no. 228–511) are shown in Figure5.Figure 5
The distribution of cumulative prospect values of lane-changing and lane-keeping (a) for lane-changing samples and (b) for lane-keeping samples.
(a)
(b)As can be seen from Figure5(a), most of the cumulative prospect value of lane-changing is located above those of lane-keeping, that is, most of the lane-changing samples in data set II are predicted to change lanes, and only a few of them are predicted to keep the lane, which is consistent with most of the actual travel behavior. In Figure 5(b), the cumulative prospect values of lane-keeping for most lane-keeping samples are above those of lane-changing, showing that the results predicted by the calibrated model are mostly consistent with drivers’ actual behaviors. In addition, it is found that most of the prospect values of lane-keeping for lane-changing samples do not exceed 2 in Figure 5(a). It indicates that there exists a sufficient accelerating space for vehicle M in the current lane when the cumulative prospect value of lane-keeping reaches a certain level. This makes it unnecessary for vehicle M to change lanes, even if the cumulative prospect value of lane-changing is much larger than that of lane-keeping (as shown in the upper right area of Figure 5(b)).As can be seen intuitively from Figure5, the results generated from the model do not deviate too much from the drivers’ actual decision-makings. Numerical comparison of predicted results and observed samples is shown in Table 5.Table 5
Estimated results in data set I and data set II.
Samples
Lane-changing
Lane-keeping
Estimated number
Observed number
Matching rate (%)
Estimated number
Observed number
Matching rate (%)
Data set I
156
174
89.96
224
255
87.84
Data set II
200
227
88.11
241
284
84.86As shown in Table5, the matching rates are over 84%, indicating that the model has a high accuracy in predicting drivers’ decision-making behaviors. Whether in data set I or data set II, the matching rate on lane-keeping samples is lower than that on lane-changing samples. According to the decision-making model, when the cumulative prospect value of lane-changing is equal to or slightly greater than that of lane-keeping, drivers will choose to change lanes. However, in actual driving situation, drivers may not change lanes for pursuing a relatively small gain because there is a certain operation cost of lane-changing. These may explain why the matching rate on lane-keeping samples is lower than that on lane-changing samples.


### 6.4.2. Comparison with Random Utility Model

In order to compare the prediction performance of CPT-based model with that of perfect rationality-based model, classic random utility model (RUM) is also considered. And, the utility functions based on RUM are shown as follows:(43)UK=VK+ξK=α1+β1SrLo,M0−MSSLo,M+ξK,(44)UC=VC+ξC=∑i=0n+mPT=T¯iβ2SrLd,M0−MSSLd,MT¯i+β3SrM,Fd0−MSSM,FdT¯i+ξC,where VK and VC, respectively, represent the observable utility of lane-keeping and lane-changing; ξK and ξC are the random error of lane-keeping and lane-changing, respectively; α1, β1, β2, and β3 are coefficients, which are calibrated as −0.122, 0.146, 0.097, and 0.049, respectively. Assuming that ξK and ξC are independent and obey Gumbel distribution, the probability of lane-changing is shown as follows:(45)PC=PUC>UK=11+eVK−VC.Data set II was used to evaluate the prediction performance of RUM. Take the lane-changing probability calculated through RUM of each sample as the classification threshold, which determines the prediction error. And, the prediction result of RUM in the case with the smallest prediction error was compared with that of CPT, as shown in Table6.Table 6
Prediction confusion matrix of data set II based on CPT and RUM.
Estimated number via CPT
Estimated number via RUM
Lane-changing
Lane-keeping
Lane-changing
Lane-keeping
Observed number
Lane-changing
200
27
188
39
Lane-keeping
43
241
72
212Take lane-changing as positive alternative and lane-keeping as negative alternative. Then, the true positive rate (TPR) = 200/227 = 0.8811 and the false positive rate (FPR) = 43/284 = 0.1514 of CPT-based model. Similarly, the TPR and FPR of RUM are 0.8282 and 0.2535, respectively. TPR of CPT based model is slightly larger than that of RUM, while FPR of CPT-based model is significantly smaller than that of RUM. This indicates that compared with CPT, RUM is more likely to overestimate the possibility of lane-changing. The receiver operating characteristic (ROC) curves of CPT and RUM based on data set II are shown in Figure6.Figure 6
Receiver operating characteristic curves.The area under curve (AUC) of CPT-based model is 0.865, bigger than AUC of RUM, 0.772. Table6 and Figure 6 indicate that CPT-based model performs better on prediction than RUM. It proves that the bounded rationality hypothesis of CPT is more consistent with the driver’s actual decision-making behavior.


## 6.4.1. Prediction Based on CPT

Samples in data set II were used to evaluate the accuracy of the calibrated model. Firstly, the cumulative prospect values of both lane-changing and lane-keeping of all samples in data set II are calculated. Then, the lane-changing samples and lane-keeping samples in data set II are numbered, respectively, in the ascending order according to the cumulative prospect value of lane-changing from small to large. Cumulative prospect values of lane-changing samples (no. 1–227) and lane-keeping samples (no. 228–511) are shown in Figure5.Figure 5
The distribution of cumulative prospect values of lane-changing and lane-keeping (a) for lane-changing samples and (b) for lane-keeping samples.
(a)
(b)As can be seen from Figure5(a), most of the cumulative prospect value of lane-changing is located above those of lane-keeping, that is, most of the lane-changing samples in data set II are predicted to change lanes, and only a few of them are predicted to keep the lane, which is consistent with most of the actual travel behavior. In Figure 5(b), the cumulative prospect values of lane-keeping for most lane-keeping samples are above those of lane-changing, showing that the results predicted by the calibrated model are mostly consistent with drivers’ actual behaviors. In addition, it is found that most of the prospect values of lane-keeping for lane-changing samples do not exceed 2 in Figure 5(a). It indicates that there exists a sufficient accelerating space for vehicle M in the current lane when the cumulative prospect value of lane-keeping reaches a certain level. This makes it unnecessary for vehicle M to change lanes, even if the cumulative prospect value of lane-changing is much larger than that of lane-keeping (as shown in the upper right area of Figure 5(b)).As can be seen intuitively from Figure5, the results generated from the model do not deviate too much from the drivers’ actual decision-makings. Numerical comparison of predicted results and observed samples is shown in Table 5.Table 5
Estimated results in data set I and data set II.
Samples
Lane-changing
Lane-keeping
Estimated number
Observed number
Matching rate (%)
Estimated number
Observed number
Matching rate (%)
Data set I
156
174
89.96
224
255
87.84
Data set II
200
227
88.11
241
284
84.86As shown in Table5, the matching rates are over 84%, indicating that the model has a high accuracy in predicting drivers’ decision-making behaviors. Whether in data set I or data set II, the matching rate on lane-keeping samples is lower than that on lane-changing samples. According to the decision-making model, when the cumulative prospect value of lane-changing is equal to or slightly greater than that of lane-keeping, drivers will choose to change lanes. However, in actual driving situation, drivers may not change lanes for pursuing a relatively small gain because there is a certain operation cost of lane-changing. These may explain why the matching rate on lane-keeping samples is lower than that on lane-changing samples.


## 6.4.2. Comparison with Random Utility Model

In order to compare the prediction performance of CPT-based model with that of perfect rationality-based model, classic random utility model (RUM) is also considered. And, the utility functions based on RUM are shown as follows:(43)UK=VK+ξK=α1+β1SrLo,M0−MSSLo,M+ξK,(44)UC=VC+ξC=∑i=0n+mPT=T¯iβ2SrLd,M0−MSSLd,MT¯i+β3SrM,Fd0−MSSM,FdT¯i+ξC,where VK and VC, respectively, represent the observable utility of lane-keeping and lane-changing; ξK and ξC are the random error of lane-keeping and lane-changing, respectively; α1, β1, β2, and β3 are coefficients, which are calibrated as −0.122, 0.146, 0.097, and 0.049, respectively. Assuming that ξK and ξC are independent and obey Gumbel distribution, the probability of lane-changing is shown as follows:(45)PC=PUC>UK=11+eVK−VC.Data set II was used to evaluate the prediction performance of RUM. Take the lane-changing probability calculated through RUM of each sample as the classification threshold, which determines the prediction error. And, the prediction result of RUM in the case with the smallest prediction error was compared with that of CPT, as shown in Table6.Table 6
Prediction confusion matrix of data set II based on CPT and RUM.
Estimated number via CPT
Estimated number via RUM
Lane-changing
Lane-keeping
Lane-changing
Lane-keeping
Observed number
Lane-changing
200
27
188
39
Lane-keeping
43
241
72
212Take lane-changing as positive alternative and lane-keeping as negative alternative. Then, the true positive rate (TPR) = 200/227 = 0.8811 and the false positive rate (FPR) = 43/284 = 0.1514 of CPT-based model. Similarly, the TPR and FPR of RUM are 0.8282 and 0.2535, respectively. TPR of CPT based model is slightly larger than that of RUM, while FPR of CPT-based model is significantly smaller than that of RUM. This indicates that compared with CPT, RUM is more likely to overestimate the possibility of lane-changing. The receiver operating characteristic (ROC) curves of CPT and RUM based on data set II are shown in Figure6.Figure 6
Receiver operating characteristic curves.The area under curve (AUC) of CPT-based model is 0.865, bigger than AUC of RUM, 0.772. Table6 and Figure 6 indicate that CPT-based model performs better on prediction than RUM. It proves that the bounded rationality hypothesis of CPT is more consistent with the driver’s actual decision-making behavior.


## 6.5. Sensitivity Analysis

In order to study the relationship between the driving states of involved vehicles and drivers’ decision-makings, some decision variables’ effects on the cumulative prospect values are analyzed.


### 6.5.1. Effects ofvM, ΔxLo,M, and ΔxLo,M on the Cumulative Prospect Value of Lane-Keeping

Influence ofΔvLo,M and ΔxLo,M when vM = 10 m/s and influence of vM and ΔxLo,Mwhen ΔvLo,M = 3 m/s are analyzed, respectively, as shown in Figure 7.Figure 7
Contour plots of the cumulative prospect value of lane-keeping (a) as function ofΔvLo,M and ΔxLo,M for vM = 10 m/s and (b) as function of vMand ΔxLo,M for ΔvLo,M = 3 m/s.
(a)
(b)The series of curves in Figure7 are contour lines, where each point on the same curve represents the equal cumulative prospect value of lane-keeping. Obviously, Figure 7(a) shows that the cumulative prospect value of lane-keeping increases significantly with the increase of ΔvLo,M and ΔxLo,M. In addition, the farther away from the contour line valued 0, the sparser it is, indicating that the driver’s sensitivity to gains or losses decreases gradually with the increase of gains or losses. This reflects the actual driving situation: when ΔvLo,M or ΔxLo,M is large enough, vehicle M can obtain sufficient accelerating space, and its intention to follow vehicle Lo will not significantly become stronger with the increase of ΔvLo,M or ΔxLo,M.Figure7(b) demonstrates that the prospect value of lane-keeping increases with ΔxLo,M increasing, which is consistent with the conclusion shown in Figure 7(a). However, when ΔvLo,M and ΔxLo,M are fixed, the prospect value of the lane-keeping presents a downward trend with the increase of vM. Because a greater vM will lead to larger minimum safety spacing for lane-keeping, which will bring about less gains, the relatively flat contour lines in Figure 7(b) indicates that the prospect value of the lane-keeping is not significantly affected by vM.


### 6.5.2. Effects ofΔxLd,M, ΔxM,Fd, ΔvLd,M, and ΔvM,Fd on the Cumulative Prospect Value of Lane-Changing

Influence ofΔxLd,M and ΔxM,Fd on cumulative prospect value of lane-changing when vM = vLd = vLo = vFd = 10 m/s and ΔxLo,M = 20 m and influence of ΔvLd,M and ΔvM,Fd when vM = 10 m/s and ΔxLo,M = ΔxLd,M = ΔxM,Fd = 20 m are analyzed, respectively, as shown in Figure 8.Figure 8
Contour plots of the cumulative prospect value of lane-changing (a) as function ofΔxLd,M and ΔxM,Fd for vM = vLd = vLo = vFd = 10 m/s and (b) as function of ΔvLd,M and ΔvM,Fd for ΔxLo,M = ΔxLd,M = ΔxM,Fd = 20 m.
(a)
(b)In Figure8(a), the cumulative prospect value of lane-changing increases with ΔxLd,M and ΔxM,Fd increasing, and ΔxLd,M has a significant impact on it. In addition, it can be found that the left area and lower area in Figure 8(a) are darker in color, indicating that the cumulative prospect value of lane-changing is much smaller than that of other areas. Especially in the left area of Figure 8(a), steep contour lines indicate that when ΔxLd,M is very small, the gain of lane-changing keeps almost unchanged or even present as a loss even if ΔvM,Fd increases continuously. Further, the sensitivity of cumulative prospect value of lane-changing to ΔxM,Fd decreases sharply with the increase of ΔxM,Fd. Because the intention of vehicle M to change lanes will not substantially become stronger with the increase of ΔxM,Fd when the minimum safety space for lane-changing is met.In Figure8(b), it can be observed that the cumulative prospect value of lane-changing increases with ΔvLd,M and ΔvM,Fd increasing. And, the steep contour lines show that the cumulative prospect value of lane-changing is very sensitive to ΔvLd,M. That means, the greater the vLd is, the greater the increment of velocity that vehicle M will obtain after lane-changing and the stronger the intention of vehicle M to change lanes. However, ΔvM,Fd has little effect on the intention of vehicle M to change lanes. Therefore, the relative velocity between vehicle M and vehicle Ld is one of the decisive factors for the driver’s decision-making.


## 6.5.1. Effects ofvM, ΔxLo,M, and ΔxLo,M on the Cumulative Prospect Value of Lane-Keeping

Influence ofΔvLo,M and ΔxLo,M when vM = 10 m/s and influence of vM and ΔxLo,Mwhen ΔvLo,M = 3 m/s are analyzed, respectively, as shown in Figure 7.Figure 7
Contour plots of the cumulative prospect value of lane-keeping (a) as function ofΔvLo,M and ΔxLo,M for vM = 10 m/s and (b) as function of vMand ΔxLo,M for ΔvLo,M = 3 m/s.
(a)
(b)The series of curves in Figure7 are contour lines, where each point on the same curve represents the equal cumulative prospect value of lane-keeping. Obviously, Figure 7(a) shows that the cumulative prospect value of lane-keeping increases significantly with the increase of ΔvLo,M and ΔxLo,M. In addition, the farther away from the contour line valued 0, the sparser it is, indicating that the driver’s sensitivity to gains or losses decreases gradually with the increase of gains or losses. This reflects the actual driving situation: when ΔvLo,M or ΔxLo,M is large enough, vehicle M can obtain sufficient accelerating space, and its intention to follow vehicle Lo will not significantly become stronger with the increase of ΔvLo,M or ΔxLo,M.Figure7(b) demonstrates that the prospect value of lane-keeping increases with ΔxLo,M increasing, which is consistent with the conclusion shown in Figure 7(a). However, when ΔvLo,M and ΔxLo,M are fixed, the prospect value of the lane-keeping presents a downward trend with the increase of vM. Because a greater vM will lead to larger minimum safety spacing for lane-keeping, which will bring about less gains, the relatively flat contour lines in Figure 7(b) indicates that the prospect value of the lane-keeping is not significantly affected by vM.


## 6.5.2. Effects ofΔxLd,M, ΔxM,Fd, ΔvLd,M, and ΔvM,Fd on the Cumulative Prospect Value of Lane-Changing

Influence ofΔxLd,M and ΔxM,Fd on cumulative prospect value of lane-changing when vM = vLd = vLo = vFd = 10 m/s and ΔxLo,M = 20 m and influence of ΔvLd,M and ΔvM,Fd when vM = 10 m/s and ΔxLo,M = ΔxLd,M = ΔxM,Fd = 20 m are analyzed, respectively, as shown in Figure 8.Figure 8
Contour plots of the cumulative prospect value of lane-changing (a) as function ofΔxLd,M and ΔxM,Fd for vM = vLd = vLo = vFd = 10 m/s and (b) as function of ΔvLd,M and ΔvM,Fd for ΔxLo,M = ΔxLd,M = ΔxM,Fd = 20 m.
(a)
(b)In Figure8(a), the cumulative prospect value of lane-changing increases with ΔxLd,M and ΔxM,Fd increasing, and ΔxLd,M has a significant impact on it. In addition, it can be found that the left area and lower area in Figure 8(a) are darker in color, indicating that the cumulative prospect value of lane-changing is much smaller than that of other areas. Especially in the left area of Figure 8(a), steep contour lines indicate that when ΔxLd,M is very small, the gain of lane-changing keeps almost unchanged or even present as a loss even if ΔvM,Fd increases continuously. Further, the sensitivity of cumulative prospect value of lane-changing to ΔxM,Fd decreases sharply with the increase of ΔxM,Fd. Because the intention of vehicle M to change lanes will not substantially become stronger with the increase of ΔxM,Fd when the minimum safety space for lane-changing is met.In Figure8(b), it can be observed that the cumulative prospect value of lane-changing increases with ΔvLd,M and ΔvM,Fd increasing. And, the steep contour lines show that the cumulative prospect value of lane-changing is very sensitive to ΔvLd,M. That means, the greater the vLd is, the greater the increment of velocity that vehicle M will obtain after lane-changing and the stronger the intention of vehicle M to change lanes. However, ΔvM,Fd has little effect on the intention of vehicle M to change lanes. Therefore, the relative velocity between vehicle M and vehicle Ld is one of the decisive factors for the driver’s decision-making.


## 7. Conclusion

Based on previous studies, this paper studies the driver’s decision-making behavior of discretionary lane-changing under the framework of bounded rationality and achieves the following results:(1)
The dynamic car-following relationship between the involved vehicles during discretionary lane-changing is analyzed. A method to calculate the acceleration during lane-changing is proposed through applying a car-following model, thus the minimum safety spacing for lane-changing is derived.(2)
Considering the unpredictability of lane-changing duration, the minimum safety spacing for lane-changing is put forward as the dynamic reference point for decision-making, and the decision-making model of lane-changing with the accelerating space as its utility is established based on CPT. The risk attitude parameterθ is introduced into the model to represent the driver’s attention degree to velocity and safety during the process of lane-changing.(3)
940 samples were collected and randomly divided into two data sets. Data set I was used to calibrate parameters, and data set II was used to evaluate the accuracy of the model. The result indicates that CPT-based model performs better in decision-making behavior prediction than RUM. It is also found that when the cumulative prospect value of lane-keeping reaches a certain level, vehicleM will not make lane-changing even if the cumulative prospect value of lane-changing is much larger than that of lane-keeping, which verifies the driver’s bounded rationality.(4)
The effects of the driving states of involved vehicles on the cumulative prospect value of lane-keeping and lane-changing are analyzed, respectively. It is found that the cumulative prospect value of lane-keeping is sensitive toΔvLo,M and ΔxLo,M, but not sensitive to vM. While the cumulative prospect value of lane-changing is sensitive to the changes of ΔxLd,M and ΔvLd,M, but not sensitive to ΔvM,Fd. Moreover, the sensitivity of the cumulative prospect value of lane-changing to ΔxM,Fd decreases sharply with the increase of ΔxM,Fd.The current lane-changing decision-making models are mostly established in the framework of perfect rationality, which cannot accurately reflect the limitation of driver’s perception and the difference in attitudes towards losses and gains. This study considers the characteristics of the driver’s bounded rationality and better simulates the driver’s actual decision-making behavior, which will enrich the research on microscopic traffic behavior. But there are also some shortcomings as follows:(1)
The driver’s risk attitude parameterθ introduced into the study reflects the overall attributes to risk of the driver group, but fail to well present the attitude’s differences between individual drivers.(2)
The operation cost of lane-changing was not considered in the model.The above shortcomings lead to a certain deviation between the prediction results and the observation results, which will be improved in further studies.



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*Source: 1291342-2020-01-09.xml*