Effect of Pre-Trip Information in a Traffic Network with Stochastic Travel Conditions: Role of Risk Attitude

Abstract

Empirical studies have suggested that travelers’ risk attitudes affect their choice behavior when travel conditions are stochastic. By considering the travelers’ risk attitudes, we extend the classical two-route model, in which road capacities vary due to such shocks as bad weather, accidents, and special events. Two information regimes have been investigated. In the zero-information regime, we postulate that travelers acquire the variability in route travel time based on past experiences and choose the route to minimize the travel time budget. In the full-information regime, travelers have pre-trip information of the road capacities and thus choose the route to minimize the travel time. User equilibrium states of the two regimes have been analyzed, based on the canonical BPR travel time function with power coefficient $p$. In the special case $p=1$, the closed form solutions have been derived. Three cases and eleven subcases have been classified concerning the dependence of expected total travel times on the risk attitude in the zero-information regime. In the general condition $p>0$, although we are not able to derive the closed form solutions, we proved that the results are qualitatively unchanged. We have studied the benefit gains/losses by shifting from the zero-information to the full-information regime. The circumstance under which pre-trip information is beneficial has been identified. A numerical analysis is conducted to further illustrate the theoretical findings.

1. Introduction

Transportation system dynamics are fundamentally shaped by traffic demand and travelers’ choice behavior, encompassing transportation mode selection, departure scheduling, and route planning. This intrinsic relationship underscores the critical importance of understanding commuter behavioral mechanisms for effective system management and optimization. Among numerous behavioral determinants, the role of information provision has emerged as a pivotal research focus in transportation studies.

Historically, traffic information dissemination relied on conventional media channels, including print publications, broadcast systems, and dynamic signage. The technological evolution has witnessed the proliferation of Advanced Traveler Information Systems (ATISs) [1,2,3,4,5,6,7], integrating multimodal data streams through digital platforms such as navigation applications, mobile devices, and personalized travel assistants [8,9,10,11]. Such real-time data provisions empower travelers to optimize their mobility strategies across multiple dimensions—from parking location selection to itinerary customization—aimed at minimizing individual travel disutility [12,13,14,15,16,17,18,19]. While empirical evidence suggests potential societal benefits through enhanced network efficiency [20,21,22,23], contrasting findings reveal an “information paradox”, where excessive information provision paradoxically degrades system performance. This counterintuitive phenomenon manifests through three primary mechanisms: concentration, overreaction, and oversaturation [13]. Notably, Emmerink et al. demonstrated how amplified congestion levels could emerge from disproportionate user reactions to traffic alerts [24]. Subsequent studies by Verhoef et al. and Emmerink et al. further identified scenarios where expanded informational access precipitated network-wide performance deterioration [25,26]. Acemoglu et al. found that the informational Braess’s Paradox can occur in a network congestion game, where commuters with access to information about alternative routes paradoxically experience worse outcomes than those lacking such information [27].

In particular, Lindsey et al. conducted seminal research on the classical “two-route network”, analyzing pre-trip information efficacy under stochastic travel conditions [28]. Their model compares two information regimes: zero-information and full-information regimes. In the zero-information regime, travelers only know the unconditional probability distribution of traffic conditions on the two routes, while in the other regime, they are provided the road conditions before choosing a route. Travelers are assumed to be risk-neutral with respect to travel times, so that they prefer the route with a lower expected travel cost in their model. They found that information can have a negative impact on social welfare when travel cost functions on the two routes are quadratic. By contrast, when capacities vary independently and travel cost functions are linear, pre-trip information is always welfare-improving. Complementary experimental work by Rapoport et al. validated these theoretical predictions through observed behavioral patterns [6].

However, emerging empirical evidence challenges the risk-neutrality assumption, highlighting the significant role of risk perception in shaping travelers’ route selection strategies [29,30]. In response to these behavioral observations, transportation researchers have developed sophisticated equilibrium models to analyze risk-adaptive behaviors in network performance scenarios [31,32,33,34,35]. Lo et al. introduced an innovative travel time budget (TTB) framework that incorporates historical variability data into commuters’ decision-making processes, formulating a multi-class equilibrium model to simulate the route choice behaviors of risk-averse travelers [36,37]. Parallel developments include Yin et al.’s expected disutility paradigm that quantifies risk-taking behavior [38]. Recent theoretical advancements by Tan et al. have systematically compared the Pareto efficiency of various reliability-based equilibrium states, employing statistical descriptors of the central tendency and dispersion for network performance assessment [39]. Li studied heterogeneous risk preferences through a revealed preference analysis. They found differences in the willingness to pay for time savings and reduced travel time variability [40].

As mentioned above, most scholars have employed equilibrium analysis and empirical research methods to analyze risk-adaptive behaviors in network performance scenarios. However, there is still a lack of detailed theoretical analysis on the impact of pre-trip information and travelers’ risk attitudes on travel choice behaviors.

In this study, we extend the model proposed by Lindsey et al. to consider travelers’ risk attitudes. We postulate that travelers acquire the variability in route travel time based on past experiences and choose the route to minimize the travel time budget in the zero-information regime. We study the welfare gains or losses by shifting from the zero-information to the full-information regime. Using the canonical BPR travel time function with power coefficient p, user equilibrium states of the two regimes are analyzed. In the special case p = 1, the closed form solutions have been derived. Three cases and eleven subcases have been classified concerning the dependence of expected total travel times on the risk attitude in the zero-information regime. In the general condition p > 0, the travel time function is not linear. Although we are not able to derive the closed form solutions, we proved that the results are qualitatively unchanged. We have studied the benefit gains/losses by shifting from the zero-information to the full-information regime. The circumstance under which pre-trip information is beneficial has been identified. Finally, a numerical analysis is conducted to further illustrate the theoretical findings.

The rest of this paper is organized as follows. Section 2 gives a brief introduction to the model. Section 3 derives the closed form user equilibrium solutions in the special case p = 1 and discusses the general condition p > 0. Section 4 studies welfare gains or losses from information. The numerical examples are presented in Section 5 to further illustrate the theoretical findings. The conclusion is given in Section 6.

2. The Model

The notations used in the paper are listed in

Table 1. Notational glossary.

Variable	Description
$F$	full-information regime
$Z$	zero-information regime
$i$	index of route
$B$	bad condition
$G$	good condition
$s$	possible condition
$s^{\prime }$	possible condition on route 2
$S$	Set of conditions, $S=\left\{G,B\right\}$
$a_i$	free-flow travel time on route i
$b_{is}$	stochastic congestion coefficient on route i when it is in state s
$Q$	total traffic demand
$Q_i$	traffic demand on route i
$Q_{is{s}^{\prime }}^F$	traffic demand on route i when route 1 is in state s and route 2 is in state $s^{\prime }$ in the full-information regime
$T_{is}$	travel time function on route i when it is in state s
$E\left[\right]$	expectations operator
$T_i$	travel time on route i
$TT$	total travel time on the network
${TT}_i$	total travel time on route i
${TT}_{s{s}^{\prime }}$	total travel time on the network when route 1 is in state s and route 2 is in state $s^{\prime }$
$E\left[T_i^Z\right]$	the expected travel time on route i in the zero-information regime
$E\left[{TT}^Z\right]$	the expected total travel time in the zero-information regime
$E\left[{TT}^F\right]$	the expected total travel time in the full-information regime
$B_i^Z$	travel time budget on route i in the zero-information regime
$var\left(T_i^Z\right)$	the variance of $T_i$ in the zero-information regime
$\lambda$	risk-preference coefficient of travelers
$\pi$	degradation probability of capacity ($0\le \pi \le 1$)
${\lambda }_h$	threshold of $\lambda$, h = 0, 1, 2, …, 8 in the special case $p=1$
${\lambda }_h^{\prime }$	threshold of $\lambda$, h = 0, 1, 2, …, 8 in the general condition $p>0$
$\varphi ,\varpi ,\phi$	three different thresholds of $a_2-a_1$
$G^{ZF}$	welfare gains (losses) from pre-trip information
${\epsilon }_i$	${\epsilon }_i=\pi b_{iB}+\left(1-\pi \right)b_{iG}$
$k_i$	$k_i={\epsilon }_i+\lambda \left(b_{iB}-b_{iG}\right)\sqrt{\pi \left(1-\pi \right)}$
$[Q_1^Z{]}^{\#}$	${[Q_1^Z{]}^{\#}=\left[Q_1^Z\right]}_{\lambda \to \pm \infty }$
$E_{\#}$	$E_{\#}=E\left[{TT}^z\right](Q_1^Z=[Q_1^Z{]}^{\#})$
$E_0$	$E_0=E\left[{TT}^z\right](Q_1^Z=0)$
$E_Q$	$E_Q=E\left[{TT}^z\right]\left(Q_1^Z=Q\right)$

.

Consider the classical “two-route network” model as shown in Figure 1, in which a single origin (O) is connected to a single destination (D) by two routes. The traffic demand is fixed at $Q$. Assuming constant demand in a commuting corridor is justified, as we consider a long-run equilibrium, and the volume of daily commuters generally exhibits minimal fluctuation over a long period. Travel conditions on each route vary from day to day because of bad weather, accidents, roadwork, or other shocks. The state of the route, denoted as $s$, is random and drawn from the set $S=\left\{G,B\right\}$, where $G$ and $B$ represent the good (normal) and bad (incident) conditions, respectively. The traffic demand on route $i$ is denoted as $Q_i$. The travel time on route $i$ when it is in state $s$ is assumed to follow the canonical U.S. Bureau of Public Roads form [41],

$$T_{is}\left(Q_i\right)=a_i+b_{is}{Q}_i^p,p\ge 0,i=\mathrm{1,2},\forall s.$$

(1)

Let $a_i$ denote the free-flow travel time on route $i$. Parameter $b_{is}$ represents the stochastic congestion coefficient on route $i$, and $p$ is the power coefficient. For simplicity, we do not consider the situation that $a_i$ can also change in a stochastic way.

Following existing studies, we adopt the following assumptions in our model:

Assumption 1. 

Travel conditions on each route are either good ($G$) or bad ($B$), with $b_{iB}>b_{iG}$.

In contrast to the deterministic bottleneck model, we consider a bottleneck capacity by two distinct operational states: good ($G$) and bad ($B$), varying from day to day due to random factors (e.g., traffic incidents, construction activities), with probabilities $\pi$ for the bad state and $1-\pi$ for the good state. The assumed binary capacity model is practical. For instance, a two-lane road operates at full capacity without accidents but loses half its capacity if a lane is blocked by an accident.

Assumption 2. 

Without loss of generality, assume $b_{1B}/b_{1G}<b_{2B}/b_{2G}$.

Assumption 3. 

The congestion coefficients $b_{1s}$ and $b_{2s}$ are statistically independent.

Th ratio $b_{iB}/b_{iG}$ quantifies the congestion elasticity, indicating the proportional increase in congestion due to incidents relative to baseline conditions. Assumption 2 indicates that route 1 exhibits less significant congestion growth compared to route 2 under the same incident severity. This higher reliability may stem from road designs, such as emergency shoulders or parallel arterials, which reduce $b_{1B}$ by limiting bottleneck formations. Assumption 3 guarantees that special incidents occur stochastically on each route. When a pair of routes are geographically separated and disruptions like accidents occur on one of the two routes, the capacity of the other route can remain the designed capacity. Such a separation allows parameters $b_{1s}$ and $b_{2s}$ to exhibit statistical independence.

Route capacity degradations can cause route travel time variability. Travelers do not know their exact a priori travel time without pre-trip information. As Lo et al. proposed, in the presence of travel condition variations, to hedge against uncertainty, travelers generally add or deduct a travel time margin to the expected trip time to form their travel time budget [3]. The travel time budget depends on the risk attitudes of the travelers and also on the trip purposes. For trips that have a high penalty for lateness, such as trips to job interviews or trips to the airport to catch a flight, one can expect that travelers can be risk-averse and reserve a larger travel time budget. The travel time budget is defined as

$$\left[TravelTimeBudget\right]=\left[ExpectedTravelTime\right]+\left[TravelTimeMargin\right].$$

(2)

Mathematically, the travel time budget on route $i$, $B_i$, can be expressed as

$$B_i=E\left[T_i\right]+\lambda \sqrt{Var\left(T_i\right)},$$

(3)

where $T_i$ is the random variable of travel time on route $i$; $E\left[T_i\right]$ and $Var\left(T_i\right)$ are the mean and variance of $T_i$, respectively; and $\lambda$ is the risk-preference coefficient. The risk-averse travelers with $\lambda >0$ will reserve a relatively large travel time budget; the risk-preferred travelers with $\lambda <0$ will reserve a relatively small travel time budget. For $\lambda =0$, travelers are risk-neutral.

3. User Equilibrium

In this section, two information regimes are considered: zero-information ($Z$) and full-information ($F$) regimes. Travelers are assumed to adopt pure strategies and follow the user equilibrium (UE) principle so that they choose a route deterministically. Three scenarios (A–C) and seven situations (i–vii) are studied in this section.

3.1. User Equilibrium in the Zero-Information Regime

In the zero-information regime, travelers only know the unconditional probability distribution of conditions on the route (i.e., the probability of a bad condition ($B$) occurring is $\pi$, and the probability of a good condition ($G$) occurring is $1-\pi$), but do not know the specific congestion state of the two routes on each day. Travelers are assumed to prefer the route with lower travel time budget. Let $Q_i^Z$ denote the traffic demand on route $i$ in the zero-information regime. The expected travel time on route $i$ can be written as

$$E\left[T_i^Z\right]=\pi T_{iB}\left(Q_i^Z\right)+\left(1-\pi \right)T_{iG}\left(Q_i^Z\right)=a_i+{\epsilon }_i{\left(Q_i^Z\right)}^p,$$

(4)

in which ${\epsilon }_i=\pi b_{iB}+\left(1-\pi \right)b_{iG}$. Then, the expected total travel time on the network can be calculated

$$E\left[T{T}^Z\right]=Q_1^{Z}E\left[T_1^Z\right]+Q_2^{Z}E\left[T_2^Z\right]={\epsilon }_1{\left(Q_1^Z\right)}^{p+1}+{\epsilon }_2{\left(Q-Q_1^Z\right)}^{p+1}+\left(a_1-a_2\right)Q_1^Z+a_{2}Q.$$

(5)

Similarly, the expectation of ${\left(T_i^Z\right)}^2$ can be written as

$$\begin{array}{cc}& E\left[{\left(T_i^Z\right)}^2\right]=\pi [T_{iB}\left(Q_i^Z\right){]}^2+\left(1-\pi \right)[T_{iG}\left(Q_i^Z\right){]}^2\\ \hfill =& \pi {\left[a_i+b_{iB}{\left(Q_i^Z\right)}^p\right]}^2+\left(1-\pi \right){\left[a_i+b_{iG}{\left(Q_i^Z\right)}^p\right]}^2.\hfill \end{array}$$

(6)

Thus, the variance in travel time can be obtained,

$$Var\left(T_i^Z\right)=E\left[{\left(T_i^Z\right)}^2\right]-[E\left(T_i^Z\right){]}^2=\pi \left(1-\pi \right){\left(b_{iB}-b_{iG}\right)}^2{\left(Q_i^Z\right)}^{2p}.$$

(7)

The travel time budget of route $i$, $B_i^Z$ is a function of $Q_i^Z$, and can be written as

$$B_i^Z\left(Q_i^Z\right)=E\left[T_i^Z\right]+\lambda \sqrt{Var\left(T_i^Z\right)}=a_i+k_i{\left(Q_i^Z\right)}^p,$$

(8)

in which $k_i={\epsilon }_i+\lambda \left(b_{iB}-b_{iG}\right)\sqrt{\pi \left(1-\pi \right)}$.

Let $f\left(\lambda ,Q_1^Z\right)=B_1^Z\left(Q_1^Z\right)-B_2^Z\left(Q_1^Z\right)$ denote a composite function of $\lambda$ and $Q_1^Z$; then, the following can be obtained:

$$\begin{gathered} f\left(\lambda ,Q_1^Z\right)=a_1-a_2+\left[{\epsilon }_1+\lambda \left(b_{1B}-b_{1G}\right)\sqrt{\pi \left(1-\pi \right)}\right]{\left(Q_1^Z\right)}^p \\ -\left[{\epsilon }_2+\lambda \left(b_{2B}-b_{2G}\right)\sqrt{\pi \left(1-\pi \right)}\right]{\left(Q-Q_1^Z\right)}^p. \end{gathered}$$

(9)

The user equilibrium (UE) is realized when the travel time budget on the two routes is equal, i.e.,

$$f\left(\lambda ,Q_1^Z\right)=0.$$

(10)

3.1.1. Three Scenarios Under $p=1$

Given $p=1$, one can see that the function of $B_i^Z$ versus $Q_i^Z$ is a linear function. Equation (8) can be simplified into

$$B_1^Z\left(Q_1^Z\right)=a_1+k_1{Q}_1^Z,\mathrm{a}\mathrm{n}\mathrm{d}$$

(11)

$$B_2^Z\left(Q_1^Z\right)=a_2+k_2{Q}_2^Z=a_2+k_{2}Q-k_2{Q}_1^Z.$$

(12)

Then, the equation $f\left(\lambda ,Q_1^Z\right)=0$ has a unique solution, unless $k_1=-k_2$. In this case, the solution can be denoted as

$$Q_1^Z={\overline{Q}}_1^Z={\displaystyle \frac{a_2-a_1+{\epsilon }_{2}Q+\lambda \left(b_{2B}-b_{2G}\right)Q\sqrt{\pi \left(1-\pi \right)}}{{\epsilon }_1+{\epsilon }_2+\lambda \left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right)\sqrt{\pi \left(1-\pi \right)}}}.$$

(13)

Note that from Equation (13), one can easily derive that the solution ${\overline{Q}}_1^Z$ is a constant when $\lambda \to \pm \infty$, no matter which situation it is. We denote it as $[Q_1^Z{]}^{\#}$,

$${\overline{Q}}_1^Z=[Q_1^Z{]}^{\#}=\left(b_{2B}-b_{2G}\right)Q/\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right),$$

(14)

and it is obvious that $0<[Q_1^Z{]}^{\#}<Q$.

As a result, there are three scenarios under $p=1$, i.e., scenario A–C, which are classified based on the relationship between $k_1$ and $k_2$. Under each scenario, there are different possible UE patterns, denoted as pattern ij, where i = A, B, C and j = 1, 2 …, which are classified due to the possibilities of $\lambda$.

• Scenario A: $k_1=-k_2$

We can derive that scenario A exists when

$$\lambda ={\lambda }_0=-\left({\epsilon }_1+{\epsilon }_2\right)/\left[\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right)\sqrt{\pi \left(1-\pi \right)}\right]$$

(15)

by solving $k_1=-k_2$. In this scenario, functions $B_1^Z\left(Q_1^Z\right)$ and $B_2^Z\left(Q_1^Z\right)$ are parallel (or identical) to each other. One has $k_1=\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)/\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right)$ by substituting $\lambda ={\lambda }_0$ into $k_1$. One can easily derive that $k_1>0,$ since $b_{1B}/b_{1G}<b_{2B}/b_{2G}$ (Assumption 2). As a result, in this scenario, there are three possible UE patterns, as shown in Figure 2 and

Table 2. Three possible UE patterns in scenario A.

Pattern	Existence Condition	Main Characteristics	UE State	Example
A1	$a_1=a_2+k_{2}Q$	$B_1^Z\left(Q_1^Z\right)$$\mathrm{and}{B}_2^Z\left(Q_1^Z\right)$ are identical	$0\le Q_1^Z\le Q$	Figure 2a
A2	$a_1>a_2+k_{2}Q$	$B_1^Z\left(Q_1^Z\right)$$\mathrm{is}\mathrm{above}{B}_2^Z\left(Q_1^Z\right)$	$Q_1^Z=0$	Figure 2b
A3	$a_1<a_2+k_{2}Q$	$B_1^Z\left(Q_1^Z\right)$$\mathrm{is}\mathrm{below}{B}_2^Z\left(Q_1^Z\right)$	$Q_1^Z=Q$	Figure 2c

.

2.

Scenario B: $k_1{\ne -k}_2$ and $k_1{k}_2\le 0$

By solving $k_1{\ne -k}_2$ and $k_1{k}_2\le 0$, we can derive that scenario B exists when $\lambda \in \left[{\lambda }_1\right.\left.,{\lambda }_0\right)\cup \left({\lambda }_0\right.\left.,{\lambda }_2\right]$. Here, ${\lambda }_1$ and ${\lambda }_2$ can be obtained by solving $k_1=0$ and $k_2=0$, separately,

$$\left\{\begin{array}{c}{\lambda }_1=-{\epsilon }_1/\left[\left(b_{1B}-b_{1G}\right)\sqrt{\pi \left(1-\pi \right)}\right]\\ {\lambda }_2=-{\epsilon }_2/\left[\left(b_{2B}-b_{2G}\right)\sqrt{\pi \left(1-\pi \right)}\right].\end{array}\right.$$

(16)

In this scenario, the equation $f\left(\lambda ,Q_1^Z\right)=0$ has a unique solution, $Q_1^Z={\overline{Q}}_1^Z$, as shown in Equation (13). Substituting $\lambda ={\lambda }_1$ into $k_1$, one has $k_1=0$. Since $k_1$ is an increasing function of $\lambda$, one has $k_1\ge 0$ in scenario B. Accordingly, $k_2\le 0$. Therefore, the slopes of functions $B_1^Z\left(Q_1^Z\right)$ and $B_2^Z\left(Q_1^Z\right)$ have the same sign. As a result, in this scenario, there are six possible patterns, as shown in Figure 3 and

Table 3. Six possible UE patterns in scenario B.

Pattern	Existence Condition		UE State	Example
B1	$k_1<-k_2$ $\left({\lambda }_1\le \lambda <{\lambda }_0\right)$	$0<{\overline{Q}}_1^Z<Q$	$Q_1^Z=0$ $\mathrm{or}{Q}_1^Z=Q$	Figure 3a
B2		${\overline{Q}}_1^Z\ge Q$	$Q_1^Z=0$	Figure 3b
B3		${\overline{Q}}_1^Z\le 0$	$Q_1^Z=Q$	Figure 3c
B4	$k_1>-k_2$ $\left({\lambda }_0<\lambda \le {\lambda }_2\right)$	0 < $Q_1^Z$ < Q	$Q_1^Z$ = $Q_1^Z$	Figure 3d
B5		${\overline{Q}}_1^Z\ge Q$	$Q_1^Z=Q$	Figure 3e
B6		${\overline{Q}}_1^Z\le 0$	$Q_1^Z=0$	Figure 3f

.

3.

Scenario C: $k_1\ne -k_2$ and $k_1{k}_2>0$

Obviously, scenario C exists when $\lambda <{\lambda }_1$ or $\lambda >{\lambda }_2$. In this scenario, the equation $f\left(\lambda ,Q_1^Z\right)=0$ has a unique solution, as shown in Equation (13), and the slopes of functions $B_1^Z\left(Q_1^Z\right)$ and $B_2^Z\left(Q_1^Z\right)$ have a different sign. As a result, in this scenario, there are six possible UE patterns, as shown in Figure 4 and

Table 4. Six possible UE patterns in scenario C.

Pattern	Existence Condition		UE State	Example
C1	$k_1<0,k_2<0$ $\left(\lambda <{\lambda }_1\right)$	$0<{\overline{Q}}_1^Z<Q$	$Q_1^Z=0$ $\mathrm{or}{Q}_1^Z=Q$	Figure 4a
C2		${\overline{Q}}_1^Z\ge Q$	$Q_1^Z=0$	Figure 4b
C3		${\overline{Q}}_1^Z\le 0$	$Q_1^Z=Q$	Figure 4c
C4	$k_1>0,k_2>0$ $\left(\lambda >{\lambda }_2\right)$	$0<{\overline{Q}}_1^Z<Q$	$Q_1^Z={\overline{Q}}_1^Z$	Figure 4d
C5		${\overline{Q}}_1^Z\ge Q$	$Q_1^Z=Q$	Figure 4e
C6		${\overline{Q}}_1^Z\le 0$	$Q_1^Z=0$	Figure 4f

.

3.1.2. Closed Form UE Solutions Under $p=1$

In this section, we first discuss the relation between $Q_1^Z$ and $\lambda$ in the zero-information regime based on the above patterns in the three scenarios. By setting ${\overline{Q}}_1^Z=0$ and ${\overline{Q}}_1^Z=Q$, respectively, we obtain two thresholds:

$$\left\{\begin{array}{c}{\lambda }_3=\left(a_1-a_2-{\epsilon }_{2}Q\right)/\left[\left(b_{2B}-b_{2G}\right)Q\sqrt{\pi \left(1-\pi \right)}\right]\\ {\lambda }_4=\left(a_2-a_1-{\epsilon }_{1}Q\right)/\left[\left(b_{1B}-b_{1G}\right)Q\sqrt{\pi \left(1-\pi \right)}\right].\end{array}\right.$$

(17)

Note that ${\lambda }_3>{\lambda }_4$ when $a_2-a_1<\varphi$, and ${\lambda }_3<{\lambda }_4$ when $a_2-a_1>\varphi$. Here,

$$\varphi =\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)Q/\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right),$$

(18)

by solving ${\lambda }_3={\lambda }_4$. Due to Assumption 2, it is obvious that $\varphi >0$. Moreover, the other two thresholds of $a_2-a_1$ can be obtained by solving ${\lambda }_2={\lambda }_4$ and ${\lambda }_1={\lambda }_3$, respectively,

$$\left\{\begin{array}{c}\phi =\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)Q/\left(b_{2B}-b_{2G}\right)\\ \varpi =\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)Q/\left(b_{1B}-b_{1G}\right).\end{array}\right.$$

(19)

In the special case $a_2-a_1=\varphi$, ${\overline{Q}}_1^Z$ in Equation (13) is a constant no matter what the risk attitude $\lambda$ is (see Figure 5c). We denote the constant ${\overline{Q}}_1^Z$ as $[Q_1^Z{]}^{\prime }$,

$${\overline{Q}}_1^Z=[Q_1^Z{]}^{\prime }=\left(b_{2B}-b_{2G}\right)Q/\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right).$$

(20)

If $a_2-a_1\ne \varphi$, the function (13) of ${\overline{Q}}_1^Z$ versus $\lambda$ is a hyperbolic curve. When $a_2-a_1<\varphi$, the function is an increasing one for the two branches (see Figure 5a,b); when $a_2-a_1>\varphi$, the function is a decreasing one for the two branches (see Figure 5d–g). Note that the two branches are separated by $\lambda ={\lambda }_0$. From Equation (13), ${\overline{Q}}_1^Z$ can be less than 0 or larger than $Q$; however, $0\le Q_1^Z\le Q$ should be met.

According to the above analysis, under UE, seven situations are given based on the relationship of $a_1$ and $a_2$, and different scenarios may belong to one of the situations.

1.

Situations (i) and (ii)

When $a_2-a_1<\varphi$, situations (i) and (ii) can exist. In the two situations, when $\lambda ={\lambda }_0$, pattern A2 exists, in which all travelers choose route 2 under UE.

• Situation (i) corresponds to ${\lambda }_4<{\lambda }_1$ and ${\lambda }_2<{\lambda }_3$ (see Figure 5a). In this situation, patterns C1, C2, B2, B6, C6, and C4 exist when the risk attitude $\lambda$ belongs to $\left(-\infty ,{\lambda }_4\right)$, $\left({\lambda }_4,{\lambda }_1\right)$, $\left({\lambda }_1,{\lambda }_0\right)$, $\left({\lambda }_0,{\lambda }_2\right)$, $\left({\lambda }_2,{\lambda }_3\right)$, and $\left({\lambda }_3,+\infty \right)$, respectively.
• Situation (ii) corresponds to ${\lambda }_1<{\lambda }_4$ and ${\lambda }_3<{\lambda }_2$ (see Figure 5b). In this situation, patterns C1, B1, B2, B6, B4, and C4 exist when $\lambda$ belongs to $\left(-\infty ,{\lambda }_1\right)$, $\left({\lambda }_1,{\lambda }_4\right)$, $\left({\lambda }_4,{\lambda }_0\right)$, $\left({\lambda }_0,{\lambda }_3\right)$, $\left({\lambda }_3,{\lambda }_2\right)$, and $\left({\lambda }_2,+\infty \right)$, respectively.
• Note that when $a_2-a_1<\varphi$, ${\lambda }_2<{\lambda }_3$, and ${\lambda }_1<{\lambda }_4$ cannot be not met simultaneously, ${\lambda }_2>{\lambda }_3$ and ${\lambda }_1>{\lambda }_4$ cannot be not met simultaneously, either.

2.

Situation (iii)

When $a_2-a_1=\varphi$, one has situation (iii). In this situation, when $\lambda ={\lambda }_0$, pattern A1 exists, in which $Q_1^Z$ is not uniquely determined under UE, which can be any value between 0 and $Q$ ($0\le Q_1^Z\le Q$). Moreover, patterns C1, B1, B4, and C4 exist when $\lambda$ belongs to $\left(-\infty ,{\lambda }_1\right)$, $\left({\lambda }_1,{\lambda }_0\right)$, $\left({\lambda }_0,{\lambda }_2\right)$, and $\left({\lambda }_2,+\infty \right)$, respectively (see Figure 5c).

3.

Situations (iv)–(vii)

When $a_2-a_1>\varphi$, situations (iv), (v), or (vi), and (vii) can exist. In the four situations, when $\lambda ={\lambda }_0$, pattern A3 exists, in which all travelers choose route 1 under UE.

• Situation (iv) corresponds to ${\lambda }_1<{\lambda }_3$ and ${\lambda }_4<{\lambda }_2$ (see Figure 5d). In this situation, patterns C1, B1, B3, B5, B4, and C4 exist when $\lambda$ belongs to $\left(-\infty ,{\lambda }_1\right)$, $\left({\lambda }_1,{\lambda }_3\right)$, $\left({\lambda }_3,{\lambda }_0\right)$, $\left({\lambda }_0,{\lambda }_4\right)$, $\left({\lambda }_4,{\lambda }_2\right)$, and $\left({\lambda }_2,+\infty \right)$, respectively.
• Situation (v) corresponds to ${\lambda }_1<{\lambda }_3$ and ${\lambda }_2<{\lambda }_4$ (see Figure 5e). In this situation, patterns C1, B1, B3, B5, C5, and C4 exist when $\lambda$ belongs to $\left(-\infty ,{\lambda }_1\right)$, $\left({\lambda }_1,{\lambda }_3\right)$, $\left({\lambda }_3,{\lambda }_0\right)$, $\left({\lambda }_0,{\lambda }_2\right)$, $\left({\lambda }_2,{\lambda }_4\right)$, and $\left({\lambda }_4,+\infty \right)$, respectively.
• Situation (vi) corresponds to ${\lambda }_3<{\lambda }_1$ and ${\lambda }_4<{\lambda }_2$ (see Figure 5f). In this situation, patterns C1, C3, B3, B5, B4, and C4 exist when $\lambda$ belongs to $\left(-\infty ,{\lambda }_3\right)$, $\left({\lambda }_3,{\lambda }_1\right)$, $\left({\lambda }_1,{\lambda }_0\right)$, $\left({\lambda }_0,{\lambda }_4\right)$, $\left({\lambda }_4,{\lambda }_2\right)$, and $\left({\lambda }_2,+\infty \right)$, respectively.
• Situation (vii) corresponds to ${\lambda }_3<{\lambda }_1$ and ${\lambda }_2<{\lambda }_4$ (see Figure 5g). In this situation, patterns C1, C3, B3, B5, C5, and C4 exist when $\lambda$ belongs to $\left(-\infty ,{\lambda }_3\right)$, $\left({\lambda }_3,{\lambda }_1\right)$, $\left({\lambda }_1,{\lambda }_0\right)$, $\left({\lambda }_0,{\lambda }_2\right)$, $\left({\lambda }_2,{\lambda }_4\right)$, and $\left({\lambda }_4,+\infty \right)$, respectively.

The boundary conditions for the seven possible situations are shown in

Table 5. Boundary conditions for seven possible situations in the zero-information regime under $p=1$.

Situation	Existence Condition	Example
(i)	${\lambda }_4<{\lambda }_1$ and ${\lambda }_2<{\lambda }_3$ $\Rightarrow a_2-a_1<0$	Figure 5a
(ii)	${\lambda }_1<{\lambda }_4$ and ${\lambda }_3<{\lambda }_2$ $\Rightarrow 0<a_2-a_1<\varphi$	Figure 5b
(iii)	${\lambda }_1<{\lambda }_3={\lambda }_4<{\lambda }_2$ $\Rightarrow a_2-a_1=\varphi$	Figure 5c
(iv)	${\lambda }_1<{\lambda }_3$ and ${\lambda }_4<{\lambda }_2$ $\Rightarrow \varphi <a_2-a_1<min\left\{\phi ,\varpi \right\}$	Figure 5d
(v)	${\lambda }_1<{\lambda }_3$ and ${\lambda }_2<{\lambda }_4$ $\Rightarrow \phi <a_2-a_1<\varpi$	Figure 5e
(vi)	${\lambda }_3<{\lambda }_1$ and ${\lambda }_4<{\lambda }_2$ $\Rightarrow \varpi <a_2-a_1<\phi$	Figure 5f
(vii)	${\lambda }_3<{\lambda }_1$ and ${\lambda }_2<{\lambda }_4$ $\Rightarrow a_2-a_1>max\left\{\phi ,\varpi \right\}$	Figure 5g

. One can see that the existence conditions of situation (vi) and (v) are mutually exclusive. In other words, with other parameters fixed, when $a_2-a_1$ increases from $-\infty$ to +$\infty$, while situations (i)–(iv) and (vii) can always be observed, either situation (v) or situation (vi) can emerge.

If we only concern the relation between $Q_1^Z$ and $\lambda$, and do not care for the specific scenario, then the seven situations can be attributed to three cases.

Case 1 includes situation (i) and (ii), in which $Q_1^Z=0$ or $Q_1^Z=Q$ when $\lambda <{\lambda }_4$; $Q_1^Z=0$ when ${\lambda }_4<\lambda <{\lambda }_3$; and $Q_1^Z={\overline{Q}}_1^Z$ when $\lambda >{\lambda }_3$.

Case 2 includes situation (iii), in which $Q_1^Z=0$ or $Q_1^Z=Q$ when $\lambda <{\lambda }_0$; ${0\le Q}_1^Z\le Q$ when $\lambda ={\lambda }_0$; and $Q_1^Z=[Q_1^Z{]}^{\prime }$ when $\lambda >{\lambda }_0$.

Case 3 includes situation (iv)-(vii), in which $Q_1^Z=0$ or $Q_1^Z=Q$ when $\lambda <{\lambda }_3$; $Q_1^Z=Q$ when ${\lambda }_3<\lambda <{\lambda }_4$; and $Q_1^Z={\overline{Q}}_1^Z$ when $\lambda >{\lambda }_4$.

Now we investigate the relation between $E\left[T{T}^Z\right]$ and $\lambda$ based on the three cases. Note that from Equation (5), the expected total travel time, $E\left[T{T}^Z\right]$, is a quadratic function of $Q_1^Z$ under $p=1$, and reaches a minimum value when

$$Q_1^Z=[Q_1^Z{]}^{\ast }=\left(a_2-a_1+2{\epsilon }_{2}Q\right)/\left(2{\epsilon }_1+2{\epsilon }_2\right),$$

(21)

which corresponds to

$${\lambda }_5={\displaystyle \frac{\left(a_2-a_1\right)\left({\epsilon }_1+{\epsilon }_2\right)}{[\left(a_2-a_1\right)\left(b_{1B}+b_{2B}-b_{1G}-b_{2G}\right)-2\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)Q]\sqrt{\pi \left(1-\pi \right)}}}.$$

(22)

Obviously, $[Q_1^Z{]}^{\ast }$ may be less than 0 or greater than $Q$; however, $0\le Q_1^Z\le Q$ should be met. Substituting $Q_1^Z=0$, $Q_1^Z=Q$ and $Q_1^Z=[Q_1^Z{]}^{\#}$ into Equation (5), respectively, one has three critical values of expected total travel time,

$$\left\{\begin{array}{l}{E}_0=E\left[{TT}^z\right]\left(Q_1^Z=0\right)=a_{2}Q+{\epsilon }_2{Q}^2\\ E_Q=E\left[{TT}^z\right]\left(Q_1^Z=Q\right)=a_{1}Q+{\epsilon }_1{Q}^2\\ E_{\#}=E[{TT}^z](Q_1^Z=[Q_1^Z{]}^{\#})={\displaystyle \frac{\left[a_1\left(b_{2B}-b_{2G}\right)+\left(b_{1B}-b_{1G}\right)\left(a_2+{\epsilon }_{2}Q\right)\right]Q}{b_{1B}-b_{1G}+b_{2B}-b_{2G}}}+{\displaystyle \frac{\left(b_{2B}-b_{2G}\right)\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)Q^2}{{\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right)}^2}}.\end{array}\right.$$

(23)

By setting $E_0=E_Q$, one has

$$a_2-a_1=\psi =\left({\epsilon }_1-{\epsilon }_2\right)Q.$$

(24)

Moreover, based on the three critical values, we have the following thresholds of risk coefficient $\lambda$,

$$\left\{\begin{array}{l}{\lambda }_6={\displaystyle \frac{{\epsilon }_2\left({\epsilon }_1+{\epsilon }_2\right)Q}{\left[\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)Q-\left(a_2-a_1+{\epsilon }_{2}Q\right)\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right)\right]\sqrt{\pi \left(1-\pi \right)}}}\\ {\lambda }_7={\displaystyle \frac{{\epsilon }_1\left({\epsilon }_1+{\epsilon }_2\right)Q}{\left[\left(a_2-a_1-{\epsilon }_{1}Q\right)\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right)-\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)Q\right]\sqrt{\pi \left(1-\pi \right)}}}\\ {\lambda }_8={\displaystyle \frac{\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)\left({\epsilon }_1+{\epsilon }_2\right)Q}{\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right)\left[\left(a_2-a_1\right)\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right)-2\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)Q\right]\sqrt{\pi \left(1-\pi \right)}}},\end{array}\right.$$

(25)

by solving $E\left[{TT}^z\right]=E_0$, $E\left[{TT}^z\right]=E_Q$, and $E\left[{TT}^z\right]=E_{\#}$, respectively.

As a result, the three cases can be classified into eleven subcases (see Figure 6, Figure 7 and Figure 8 and

Table 6. Boundary conditions for 11 subcases in the zero-information regime under $p=1$.

Case	Subcase	Existence Condition	Example
1	1a	${E_0<E}_{\#}<E_Q$$\mathrm{and}{\lambda }_3>{\lambda }_5$  $\Rightarrow a_2-a_1<-2{\epsilon }_{2}Q$	Figure 6a
	1b	${E_0<E}_{\#}<E_Q$$\mathrm{and}{\lambda }_3<{\lambda }_5$  $\Rightarrow -2{\epsilon }_{2}Q<a_2-a_1<\varphi -{\epsilon }_{2}Q$	Figure 6b
	1c	$E_{\#}<E_0<E_Q$$\mathrm{and}{\lambda }_3<{\lambda }_5$  $\Rightarrow \varphi -{\epsilon }_{2}Q<a_2-a_1<min\left\{\varphi ,\psi \right\}$	Figure 6c
	1d	$E_{\#}<E_Q<E_0$$\mathrm{and}{\lambda }_3<{\lambda }_5$  $\Rightarrow \psi <a_2-a_1<\varphi$	Figure 6d
2		$E_{\#}<min\left\{E_Q,E_0\right\}$$\mathrm{and}{\lambda }_0={\lambda }_3={\lambda }_4$  $\Rightarrow$  $a_2-a_1=\varphi$	Figure 7
3	3a	$E_{\#}<E_0<E_Q$$\mathrm{and}{\lambda }_5<{\lambda }_3$  $\Rightarrow \varphi <a_2-a_1<min\left\{2\varphi ,\psi \right\}$	Figure 8a
	3b	$E_{\#}<E_Q<E_0$$\mathrm{and}{\lambda }_5<{\lambda }_3$  $\Rightarrow max\left\{\varphi ,\psi \right\}<a_2-a_1<2\varphi$	Figure 8b
	3c	$E_{\#}<E_0<E_Q$$\mathrm{and}{\lambda }_5>{\lambda }_4$  $\Rightarrow$  $2\varphi <a_2-a_1<\psi$	Figure 8c
	3d	$E_{\#}<E_Q<E_0$$\mathrm{and}{\lambda }_5>{\lambda }_4$  $\Rightarrow max\left\{2\varphi ,\psi \right\}<a_2-a_1<\varphi +{\epsilon }_{1}Q$	Figure 8d
	3e	$E_Q<E_{\#}<E_0$ and ${\lambda }_5>{\lambda }_4$ $\Rightarrow \varphi +{\epsilon }_{1}Q<a_2-a_1<2{\epsilon }_{1}Q$	Figure 8e
	3f	$E_Q<E_{\#}<E_0$ and ${\lambda }_5<{\lambda }_4$ $\Rightarrow$ $a_2-a_1>2{\epsilon }_{1}Q$	Figure 8f

).

• Case 1 in Figure 6a–d: When $a_2-a_1<\varphi$, case 1 exists. One can easily prove that $E_{\#}<E_Q$ and ${\lambda }_3>{\lambda }_4$ are always met in case 1. Case 1 can be further classified into cases 1a–1d.

In case 1a, ${\lambda }_3>{\lambda }_5$ and $E_0<E_{\#}<E_Q$. In this subcase, $E\left[T{T}^Z\right]$ increases with $\lambda$ when $\lambda \in \left({\lambda }_3,+\infty \right)$; there are two possible UE states when $\lambda \in \left(-\infty ,{\lambda }_4\right)$, and $E\left[T{T}^Z\right]$ is a constant when $\lambda \in ({\lambda }_4,{\lambda }_3)$ (see Figure 6a).

In case 1b–1d, ${\lambda }_3<{\lambda }_5$. The three subcases correspond to $E_0<E_{\#}<E_Q$, $E_{\#}<E_0<E_Q$, and $E_{\#}<E_Q<E_0$, respectively. In the three subcases, $E\left[T{T}^Z\right]$ increases with $\lambda$ when $\lambda \in \left({\lambda }_5,+\infty \right)$ and decreases with $\lambda$ when $\lambda \in \left({\lambda }_3,{\lambda }_5\right)$; there are two possible UE states when $\lambda \in \left(-\infty ,{\lambda }_4\right)$, and $E\left[T{T}^Z\right]$ is a constant when $\lambda \in ({\lambda }_4,{\lambda }_3)$; $E\left[T{T}^Z\right]$ reaches the minimum value when $\lambda ={\lambda }_5$ (see Figure 6b–d).

• Case 2 in Figure 7: When $a_2-a_1=\varphi$, case 2 exists. One can easily prove that $E_{\#}<min\left\{E_0,E_Q\right\}$ and ${\lambda }_0={\lambda }_3={\lambda }_4$ are always met in case 2. In this case, there are two possible UE states when $\lambda \in \left(-\infty ,{\lambda }_0\right)$; the expected total travel time $E\left[T{T}^Z\right]$ is not uniquely determined when $\lambda ={\lambda }_0$, which can be any value between $E_{\ast }$ and $max\left\{E_0,E_Q\right\}$. Here,

  $$E_{\ast }=E\left[{TT}^z\right](Q_1^Z=\left[Q_1^Z{]}^{\ast }\right),$$

  (26)

  which denotes the minimum value of the expected total travel time. Note that $0\le [Q_1^Z{]}^{\ast }\le Q$ is met due to $E_{\#}<min\left\{E_0,E_Q\right\}$. Finally, $E\left[T{T}^Z\right]$ is a constant no matter what the risk attitude is when $\lambda \in \left({\lambda }_0,+\infty \right)$, which can be calculated,

  $$E\left[{TT}^z\right]=\left[\left(b_{2B}-b_{2G}\right)\left(a_1+\varphi \right)+\left(b_{1B}-b_{1G}\right)\left(a_2+{\epsilon }_{2}Q\right)\right]Q/\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right).$$

  (27)
• Case 3 in Figure 8a–f: When $a_2-a_1>\varphi$, case 3 exists. One can easily prove that $E_{\#}<E_0$ and ${\lambda }_3<{\lambda }_4$ are always met in case 3. Case 3 can be further classified into cases 3a–3f.

In case 3a and case 3b, ${\lambda }_5<{\lambda }_3$. The two subcases correspond to $E_{\#}<E_0<E_Q$ and $E_{\#}<E_Q<E_0$, respectively (see Figure 8a,b). In the two subcases, $E\left[T{T}^Z\right]$ decreases with $\lambda$ when $\lambda \in \left({\lambda }_4,+\infty \right)$; there are two possible UE states when $\lambda \in \left(-\infty ,{\lambda }_3\right)$, and $E\left[T{T}^Z\right]$ is a constant when $\lambda \in \left({\lambda }_3,{\lambda }_4\right)$.

In case 3c–3e, ${\lambda }_5>{\lambda }_4$. The three subcases correspond to $E_{\#}<E_0<E_Q$, $E_{\#}<E_Q<E_0$, and $E_Q<E_{\#}<E_0$, respectively (see Figure 8c–e). In the three subcases, $E\left[T{T}^Z\right]$ decreases with $\lambda$ when $\lambda \in \left({\lambda }_4,{\lambda }_5\right)$, and increases with $\lambda$ when $\lambda \in \left({\lambda }_5,+\infty \right)$; there are two possible UE states when $\lambda \in \left(-\infty ,{\lambda }_3\right)$, and $E\left[T{T}^Z\right]$ is a constant when $\lambda \in \left({\lambda }_3,{\lambda }_4\right)$; $E\left[T{T}^Z\right]$ reaches the minimum value when $\lambda ={\lambda }_5$.

In case 3f, ${\lambda }_3<{\lambda }_5<{\lambda }_4$ and $E_Q<E_{\#}<E_0$ (see Figure 8f). $E\left[T{T}^Z\right]$ increases with $\lambda$ when $\left({\lambda }_4,+\mathrm{\infty }\right)$; there are two possible UE states when $\lambda \in \left(-\infty ,{\lambda }_3\right)$, and $E\left[T{T}^Z\right]$ is a constant when $\lambda \in \left({\lambda }_3,{\lambda }_4\right)$.

Actually, with other parameters fixed, not all 11 subcases can be observed when $a_2-a_1$ increases from $-\infty$ to +$\infty$. For example, when $\psi <\varphi$, case 3a and case 3c do not exist; when $\varphi <\psi <2\varphi$, case 1d and case 3c do not exist; and when $\psi >2\varphi$, case 1d and case 3b do not exist (see Table 6). This will be further illustrated in numerical examples in Section 5.

3.1.3. General Results Under $p>0$

In the zero-information regime, we have derived the closed form solutions under $p=1$ in the above subsection. However, the equation $f\left(\lambda ,Q_1^Z\right)=0$ does not have closed form solutions for arbitrary $p$ ($p>0$ and $p\ne 1$) if the free-flow travel times are different (i.e., $a_1\ne a_2$). Nevertheless, we can prove that the results are qualitatively unchanged.

For arbitrary $p$ ($p>0$), denote the solution of equation $f\left(\lambda ,Q_1^Z\right)=0$ for a given $\lambda$ as $Q_1^Z={\widehat{Q}}_1^Z$, and assume that the slopes of functions $B_1^Z\left(Q_1^Z\right)$ and $B_2^Z\left(Q_1^Z\right)$ under the solution are

$$\left\{\begin{array}{l}{k}_1^{\prime }=p{k}_1{\left({\widehat{Q}}_1^Z\right)}^{p-1}\\ k_2^{\prime }=-p{k}_2{\left(Q-{\widehat{Q}}_1^Z\right)}^{p-1}.\end{array}\right.$$

(28)

It is obvious that the state $Q_1^Z={\widehat{Q}}_1^Z$ is not a UE state when ${\widehat{Q}}_1^Z<0$ or ${\widehat{Q}}_1^Z>Q$, because $0\le Q_1^Z\le Q$ should be met. Based on the definition of the UE principle, we have the following property.

Property 1. 

Assume that the state $Q_1^Z={\widehat{Q}}_1^Z$ satisfies $0\le {\widehat{Q}}_1^Z\le Q$. Then the following is obtained:

(a) 

when $k_1^{\prime }=k_2^{\prime }$, if the functions $B_1^Z\left(Q_1^Z\right)$ and $B_2^Z\left(Q_1^Z\right)$ are identical, the state is a UE state; otherwise, the two functions are a tangent at $Q_1^Z={\widehat{Q}}_1^Z$, which is not a UE state, unless $Q_1^Z=0$ (if $B_1^Z\left(Q_1^Z\right)$ is above $B_2^Z\left(Q_1^Z\right)$) or $Q_1^Z=Q$ (if $B_1^Z\left(Q_1^Z\right)$ is below $B_2^Z\left(Q_1^Z\right)$);

(b) 

when $k_1^{\prime }<k_2^{\prime }$, the state is not a UE state;

(c) 

when $k_1^{\prime }>k_2^{\prime }$, the state is a UE state.

By substituting $Q_1^Z=0$ and $Q_1^Z=Q$ into equation $f\left(\lambda ,Q_1^Z\right)=0$, respectively, one has

$$\left\{\begin{array}{l}{\lambda }_3^{\prime }=\left(a_1-a_2-{\epsilon }_2{Q}^p\right)/\left[\left(b_{2B}-b_{2G}\right)Q^p\sqrt{\pi \left(1-\pi \right)}\right]\\ {\lambda }_4^{\prime }=\left(a_2-a_1-{\epsilon }_1{Q}^p\right)/\left[\left(b_{1B}-b_{1G}\right)Q^p\sqrt{\pi \left(1-\pi \right)}\right].\end{array}\right.$$

(29)

By solving ${\lambda }_3^{\prime }={\lambda }_4^{\prime }$, one has

$$a_2-a_1={\varphi }^{\prime }=\left(b_{1G}{b}_{2B}-b_{1B}{b}_{2G}\right)Q^p/\left(b_{1B}-b_{1G}+b_{2B}-b_{2G}\right).$$

(30)

As a result, we have the following propositions.

Proposition 1. 

When $\lambda \to +\infty$, the traffic demand $Q_1^Z$ under UE is a constant no matter what $a_2-a_1$ is.

Proof. 

Substituting $\lambda \to +\infty$ into the UE condition $f\left(\lambda ,Q_1^Z\right)=0$, one can easily derive that the solution is

$$Q_1^Z=[Q_1^Z{]}^{\#\#}=Q/\left(1+\sqrt[p]{\left(b_{1B}-b_{1G}\right)/\left(b_{2B}-b_{2G}\right)}\right),$$

(31)

which is a constant independent of $a_2-a_1$. When $\lambda \to +\infty$, one has $k_1^{\prime }>0$. Moreover, $k_2^{\prime }<0$ is always met, provided $0<Q_1^Z<Q$. Thus, ${\displaystyle \frac{\partial f\left(\lambda ,Q_1^Z\right)}{\partial Q_1^Z}}=k_1^{\prime }-k_2^{\prime }>0$. Thereby, the solution is unique. According to Property 1(c), the state $Q_1^Z=[Q_1^Z{]}^{\#\#}$ is a unique UE state. □

Proposition 2. 

(a) 

when $a_2-a_1<{\varphi }^{\prime }$, there are two possible UE states ($Q_1^Z=0$ or $Q_1^Z=Q$) when $\lambda <{\lambda }_4^{\prime }$. Otherwise, a unique UE state exists: $Q_1^Z=0$ when ${\lambda }_4^{\prime }<\lambda <{\lambda }_3^{\prime }$; $0\le Q_1^Z<[Q_1^Z{]}^{\#\#}$ and $Q_1^Z$ increases with $\lambda$ when $\lambda >{\lambda }_3^{\prime }$;

(b) 

when $a_2-a_1={\varphi }^{\prime }$, there are two possible UE states ($Q_1^Z=0$ or $Q_1^Z=Q$) when $\lambda <{\lambda }_3^{\prime }$; the UE traffic demand $Q_1^Z$ is not uniquely determined, which can be any value between 0 and $Q$ ($0\le Q_1^Z\le Q$) when $\lambda ={\lambda }_3^{\prime }$; the UE traffic demand is a constant $Q_1^Z=[Q_1^Z{]}^{\#\#}$ when $\lambda >{\lambda }_3^{\prime }$;

(c) 

when $a_2-a_1>{\varphi }^{\prime }$, there are two possible UE states ($Q_1^Z=0$ or $Q_1^Z=Q$) when $\lambda <{\lambda }_3^{\prime }$. Otherwise, a unique UE state exists: $Q_1^Z=Q$ when ${\lambda }_3^{\prime }<\lambda <{\lambda }_4^{\prime }$; $[Q_1^Z{]}^{\#\#}<Q_1^Z\le Q$; and $Q_1^Z$ decreases with $\lambda$ when $\lambda >{\lambda }_4^{\prime }$.

Proof. 

The proof of the proposition can be found in Appendix A1. □

Proposition 3. 

The function of $E\left[T{T}^Z\right]$ versus $\lambda$ has at most one extreme point.

Proof. 

The proof of the proposition can be found in Appendix A2. □

Proposition 2 is consistent with the three cases of the relation between $Q_1^Z$ and $\lambda$ under $p=1$. The result of Proposition 3 is consistent with that of the 11 subcases under $p=1$, in which there is no extreme point in cases 1a, 2, 3a, 3b and 3f, and one extreme point exists at ${\lambda =\lambda }_5$ in other cases. Given the three propositions, one can conclude that the results under the general condition ($p>0$) are qualitatively unchanged as that under $p=1$.

3.2. User Equilibrium in the Full-Information Regime

In the full-information regime, travelers can be informed about the accurate travel conditions before choosing a route and thus choose the route to minimize the travel time. Given Assumption 3, there are four possible combinations for the system: $GG$, $BG$, $GB$, and $BB$, where the first letter refers to the condition of route 1, and the second refers to the condition of route 2 as shown in Figure 9. Accordingly, there are four possible UE usage levels of route $i$: $Q_{iGG}^F$, $Q_{iGB}^F$, $Q_{iBG}^F$, and $Q_{iBB}^F$. The expected total travel time on the network $E\left[T{T}^F\right]$ can be calculated by

$$E\left[T{T}^F\right]={\left(1-\pi \right)}^2{TT}_{GG}^F+\left(1-\pi \right)\pi {TT}_{GB}^F+\pi \left(1-\pi \right){TT}_{BG}^F+{\pi }^2{TT}_{BB}^F,$$

(32)

in which ${TT}_{s{s}^{\prime }}^F$ denotes the total travel time on the network when route 1 is in state $s$ and route 2 is in state $s^{\prime }$, $\forall s,s^{\prime }\in S=\left\{G,B\right\}$,

$${TT}_{s{s}^{\prime }}^F=Q_{1s{s}^{\prime }}^F{T}_{1s}\left(Q_{1s{s}^{\prime }}^F\right)+Q_{2s{s}^{\prime }}^F{T}_{2s^{\prime }}\left(Q_{2s{s}^{\prime }}^F\right)$$

(33)

With full information, the UE division of traffic between routes is realized when the travel time is equal, i.e., $g\left(Q_{1s{s}^{\prime }}^F\right)=0$; here,

$$g\left(Q_{1s{s}^{\prime }}^F\right)=T_{1s}\left(Q_{1s{s}^{\prime }}^F\right)-T_{2s^{\prime }}\left(Q_{2s{s}^{\prime }}^F\right)=a_1-a_2+b_{1s}{\left(Q_{1s{s}^{\prime }}^F\right)}^p-b_{2s^{\prime }}{\left(Q-Q_{1s{s}^{\prime }}^F\right)}^p.$$

(34)

Given $p=1$, substituting Equation (1) into $g\left(Q_{1s{s}^{\prime }}^F\right)=0$, the UE traffic demand on route 1 can be obtained,

$$\left\{\begin{array}{l}{Q}_{1GG}^F=max\left\{0,min\left\{{\displaystyle \frac{a_2-a_1+b_{2G}Q}{b_{1G}+b_{2G}}},Q\right\}\right\}\\ Q_{1GB}^F=max\left\{0,min\left\{{\displaystyle \frac{a_2-a_1+b_{2B}Q}{b_{1G}+b_{2B}}},Q\right\}\right\}\\ Q_{1BG}^F=max\left\{0,min\left\{{\displaystyle \frac{a_2-a_1+b_{2G}Q}{b_{1B}+b_{2G}}},Q\right\}\right\}\\ Q_{1BB}^F=max\left\{0,min\left\{{\displaystyle \frac{a_2-a_1+b_{2B}Q}{b_{1B}+b_{2B}}},Q\right\}\right\}.\end{array}\right.$$

(35)

and one has $Q_{2s{s}^{\prime }}^F=Q-Q_{1s{s}^{\prime }}^F,\forall s,s^{\prime }\in S$.

In the full-information regime, for arbitrary $p$ ($p>0$), denote the solution of $g\left(Q_{1s{s}^{\prime }}^F\right)=0$ as ${Q_{1s{s}^{\prime }}^F=\widehat{Q}}_{1s{s}^{\prime }}^F$; then, we have the following proposition.

Proposition 4. 

In the full-information regime, there is a unique UE state for each combination $s{s}^{\prime }$, and

(a) 

when $a_2-a_1\le -b_{2s^{\prime }}{Q}^p$, all travelers choose route 2 under UE (i.e., $Q_{1s{s}^{\prime }}^F=0$);

(b) 

when $-b_{2s^{\prime }}{Q}^p<a_2-a_1<b_{1s}{Q}^p$, the UE state is ${Q_{1s{s}^{\prime }}^F=\widehat{Q}}_{1s{s}^{\prime }}^F$ ($0<{\widehat{Q}}_{1s{s}^{\prime }}^F<Q$);

(c) 

when $a_2-a_1\ge b_{1s}{Q}^p$, all travelers choose route 1 under UE (i.e., $Q_{1s{s}^{\prime }}^F=Q$).

Proof. 

The proof of the proposition can be found in Appendix A3. □

4. Welfare Gains or Losses from Information

In this section, we study the welfare gains/losses from pre-trip information. We compare the expected total travel time between the zero-information and full-information regimes. The welfare gains/losses under equilibrium by shifting from the zero-information to full-information regimes are

$$G^{ZF}=E\left[T{T}^Z\right]-E\left[T{T}^F\right].$$

(36)

Welfare improvement is defined as $G^{ZF}\ge 0$. $G^{ZF}<0$ corresponds to the information paradox.

Proposition 5. 

For the general condition ($p>0$), full information is always welfare-improving under the UE state $Q_1^Z=0$ or $Q_1^Z=Q$.

Proof. 

The proof of the proposition can be found in Appendix A4. □

Proposition 5 indicates that providing pre-trip information is of vital importance to avoid the overuse of one route that arises from uncertainty. In the zero-information regime, if most travelers are risk-averse, they may all end up choosing the same route. This is an outcome we obviously wish to avoid. This homogeneity occurs because risk-averse travelers tend to be more cautious, and they allocate more time to travel and prefer stable, predictable routes over accident-prone, unreliable ones. To mitigate this inefficiency, it is prudent to provide accurate pre-trip information via intelligent transportation systems, diverting a portion of travelers to less-used routes. Such strategic guidance can significantly enhance the transportation network’s operational efficiency and improve overall travel quality. Based on proposition 5, we have the following corollaries.

Corollary 1. 

In case 1a and case 3f, full information is always welfare-improving.

Proof. 

In case 1a, the minimum of $E\left[T{T}^Z\right]$ is achieved when $Q_1^Z=0$. In case 3f, the minimum of $E\left[T{T}^Z\right]$ is achieved when $Q_1^Z=Q$. According to proposition 5, one has $G^{ZF}\ge 0$. □

By denoting ${\lambda }_6^{\prime }$ and ${\lambda }_7^{\prime }$ as the two critical values of $\lambda$ by solving $E\left[T{T}^Z\right]=E_0$ and $E\left[T{T}^Z\right]=E_Q$ under $p>0$, respectively, we have the following corollary.

Corollary 2. 

Full information is welfare-improving, if one of the following cases occurs in the zero-information regime:

(a) 

when $\lambda <{\lambda }_3^{\prime }$ in either case 1c or case 2;

(b) 

when $\lambda <{\lambda }_4^{\prime }$ in either case 3b or case 3d;

(c) 

when $\lambda <{\lambda }_6^{\prime }$ in either case 3a or case 3c;

(d) 

when $\lambda <{\lambda }_7^{\prime }$ in case 1d;

(e) 

when $\lambda <{\lambda }_3^{\prime }$ or $\lambda >{\lambda }_6^{\prime }$ in case 1b;

(f) 

when $\lambda <{\lambda }_4^{\prime }$ or $\lambda >{\lambda }_7^{\prime }$ in case 3e.

Proof. 

In the above-mentioned range of $\lambda$, $E\left[T{T}^Z\right]$ is not smaller than either $E_0$ or $E_Q$. According to Proposition 5, one has $G^{ZF}\ge 0$. □

Since risk attitudes affect travelers’ route choice behavior and therefore welfare gain, next, we compare the welfare gain of risk-neutral travelers with that of risk-preferred ones and risk-averse ones. Since there is no closed form solution in the general condition ($p>0$), we focus on the special condition ($p=1$).

Proposition 6. 

In case 1a and case 3f, risk-neutral travelers obtain no more welfare gains than risk-preferred ones or risk-averse ones.

Proof. 

In case 1a, from $a_2-a_1<-2{\epsilon }_{2}Q$, one can derive ${\lambda }_3>0$. In case 3f, from $a_2-a_1>2{\epsilon }_{1}Q$, one can derive ${\lambda }_4>0$. Since $E\left[T{T}^Z\right]$ reaches the minimum when $\lambda <{\lambda }_3$ in case 1a and when $\lambda <{\lambda }_4$ in case 3f, risk-neutral travelers can obtain a minimum welfare gain. □

Proposition 7. 

In case 2, risk-neutral travelers obtain no more welfare gains than risk-preferred ones with risk attitude $\lambda <{\lambda }_0$.

Proof. 

In case 2, $E_{\#}={E\left[T{T}^Z\right]}_{\lambda =0}<min\left\{E_0,E_Q\right\}$ (see Figure 7). Thus, risk-neutral travelers obtain no more welfare gains than risk-preferred ones with risk attitude $\lambda <{\lambda }_0$. □

Proposition 8. 

(a) 

In case 3a and case 3c, risk-neutral travelers obtain no more welfare gains than risk-preferred ones with risk attitude $\lambda <{\lambda }_6$;

(b) 

In case 1d, risk-neutral travelers obtain no more welfare gains than risk-preferred ones with risk attitude $\lambda <{\lambda }_7$.

Proof. 

In case 3a and case 3c, from $\varphi <a_2{-a}_1<\psi$, one can derive ${\lambda }_6<0$. Thus, one can easily prove that $E_0={E\left[T{T}^Z\right]}_{\lambda ={\lambda }_6}>{E\left[T{T}^Z\right]}_{\lambda =0}$ is always met in the two cases. Therefore, risk-neutral travelers obtain no more welfare gains than risk-preferred ones with risk attitude $\lambda <{\lambda }_6$ (see Figure 8a,c).

In case 1d, from $\psi <a_2{-a}_1<\varphi$, one can derive ${\lambda }_7<0$. Thus, one can easily prove that $E_Q={E\left[T{T}^Z\right]}_{\lambda ={\lambda }_7}>{E\left[T{T}^Z\right]}_{\lambda =0}$ is always met in this case. Therefore, risk-neutral travelers obtain no more welfare gains than risk-preferred ones with risk attitude $\lambda <{\lambda }_7$ (see Figure 6d). □

Proposition 9. 

(a) 

In case 1b, risk-neutral travelers obtain no more welfare gains than risk-averse ones with risk attitude $\lambda >{\lambda }_6$;

(b) 

In case 3e, risk-neutral travelers obtain no more welfare gains than risk-averse ones with risk attitude $\lambda >{\lambda }_7$.

Proof. 

In case 1b, from $-2{\epsilon }_{2}Q<a_2{-a}_1<\varphi -{\epsilon }_{2}Q$, one can derive ${\lambda }_6>0$. Thus, one can easily prove that $E_0={E\left[T{T}^Z\right]}_{\lambda ={\lambda }_6}>{E\left[T{T}^Z\right]}_{\lambda =0}$ is always met in this case. Therefore, risk-neutral travelers obtain no more welfare gains than risk-averse ones with risk attitude $\lambda >{\lambda }_6$ (see Figure 6b).

In case 3e, from $\varphi +{\epsilon }_{1}Q<a_2{-a}_1<2{\epsilon }_{1}Q$, one can derive ${\lambda }_7>0$. Thus, one can easily prove that $E_Q={E\left[T{T}^Z\right]}_{\lambda ={\lambda }_7}>{E\left[T{T}^Z\right]}_{\lambda =0}$ is always met in this case. Therefore, risk-neutral travelers obtain no more welfare gains than risk-averse ones with risk attitude $\lambda >{\lambda }_7$ (see Figure 8e). □

Proposition 10. 

(a) 

In case 3c and case 3d, risk-neutral travelers obtain more welfare gains than risk-averse ones with risk attitude $\lambda >{\lambda }_8$;

(b) 

In case 3e, risk-neutral travelers obtain more welfare gains than risk-averse ones with risk attitude ${\lambda }_4<\lambda <{\lambda }_7$.

Proof. 

In case 3c and case 3d, from $a_2{-a}_1>2\varphi$, one can derive ${\lambda }_8>0$. Thus, one can easily prove that $E_{\#}={E\left[T{T}^Z\right]}_{\lambda ={\lambda }_8}<{E\left[T{T}^Z\right]}_{\lambda =0}$ is always met in these two cases (see Figure 8c,d). Therefore, risk-neutral travelers obtain more welfare gains than risk-averse ones with risk attitude $\lambda >{\lambda }_8$.

In case 3e, from $\varphi +{\epsilon }_{1}Q<a_2{-a}_1<2{\epsilon }_{1}Q$, one can derive ${\lambda }_4>0$. Thus, one can easily prove that $E_Q={E\left[T{T}^Z\right]}_{\lambda ={\lambda }_4}={E\left[T{T}^Z\right]}_{\lambda ={\lambda }_7}<{E\left[T{T}^Z\right]}_{\lambda =0}$ is always met in this case (see Figure 8e). Therefore, risk-neutral travelers obtain more welfare gains than risk-averse ones with risk attitude ${\lambda }_4<\lambda <{\lambda }_7$. □

Propositions 6–10 indicate that risk-neutral travelers sometimes obtain less (or larger) welfare gains. The assumption that travelers are risk-neutral may underestimate or overestimate the benefit gains or losses of the information. As demonstrated in Propositions 6–9, risk-preferred and risk-averse travelers may exhibit a stronger preference for choosing specific routes compared to risk-neutral travelers. This route selection behavior could increase the total system travel cost due to congestion on preferred routes. Such scenarios are more likely to occur when a route exhibits higher elasticity in its capacity response to demand fluctuations. Under these circumstances, providing pre-trip information proves more advantageous for risk-preferred and risk-averse travelers. Conversely, when two routes maintain comparable and stable capacities, the situation described in Proposition 10 becomes more prevalent. In this case, providing pre-trip information yields greater benefits if all travelers are risk-neutral.

5. Numerical Analysis

In this section, we present the results of the numerical analysis to illustrate further the effects of pre-trip information on system efficiency under varying risk-preference coefficients.

5.1. The Effect of Pre-Trip Information in the General Condition

Firstly, we consider UE states $Q_1^Z=0$ and $Q_1^Z=Q$ in the general condition. Figure 10 shows examples of welfare gains by shifting from zero information to full information in the two UE states. One can see that providing pre-trip information is always welfare-improving ($G^{ZF}\ge 0$). Moreover, we can see that providing pre-trip information can gain more benefits when $a_2-a_1$ increases/decreases for $Q_1^Z=0$/$Q_1^Z=Q$. This is in consistent with Proposition 5. In real-world traffic scenarios, we often encounter situations where two alternative routes between an origin and destination (OD) offer similar travel times. One route is a longer-distance highway with higher speeds, while the other is a shorter urban arterial road featuring more traffic lights and lower speeds. Without traffic information, most travelers may prefer the highway for its perceived smoother flow. However, increased highway traffic could lead to significant travel time delays, which should be prevented. A judicious solution involves providing pre-trip information to divert a portion of travelers to urban arterial roads, thereby effectively minimizing total system travel costs.

Figure 11 shows benefit gains/losses from pre-trip information under different $\lambda$. For the specific parameters, case 3a and case 3c do not exist when $a_2-a_1$ increases from $-\infty$ to +$\infty$. In Figure 11a–d,f–i, there are two UE states (the welfare gains $G_0$ or $G_Q$ corresponding to UE state $Q_1^Z=0$ or $Q_1^Z=Q$ in the zero-information regime) when $\lambda <{\lambda }_4^{\prime }$ ($\lambda <{\lambda }_3^{\prime }$); the welfare gains $G_0$ correspond to UE state $Q_1^Z=0$ when ${\lambda }_4^{\prime }<\lambda <{\lambda }_3^{\prime }$ (${\lambda }_3^{\prime }<\lambda <{\lambda }_4^{\prime }$); and a unique UE state exists when $\lambda >{\lambda }_3^{\prime }$ ($\lambda >{\lambda }_4^{\prime }$). This is in consistent with Proposition 2. Furthermore, the minimum value is observed at $\lambda ={\lambda }_5^{\prime }$ in cases 1b, 1c, 1d, 3d, and 3e. This is in consistent with Proposition 3.

Moreover, from Figure 11a,i, one can see that pre-trip information is always beneficial ($G^{ZF}\ge 0$) in case 1a and 3f. This is in consistent with Corollary 1. Providing pre-trip information is welfare-improving when $\lambda <{\lambda }_3^{\prime }$ in case 1c and case 2; when $\lambda <{\lambda }_4^{\prime }$ in case 3b and case 3d; when $\lambda <{\lambda }_7^{\prime }$ in case 1d; when $\lambda <{\lambda }_3^{\prime }$ or $\lambda >{\lambda }_6^{\prime }$ in case 1b; and when $\lambda <{\lambda }_4^{\prime }$ or $\lambda >{\lambda }_7^{\prime }$ in case 3e. This is in consistent with Corollary 2. Under other circumstances, providing pre-trip information may have either positive or negative impact on improving the welfare gains, which depends on the parameters.

5.2. The Effect of Pre-Trip Information in the Special Case $p=1$

The Figure 12 plots show the benefit gains from pre-trip information under different $\lambda$ in the special case $p=1$. For the specific parameters, (i) case 1d and case 3b do not exist when $a_2-a_1$ increases from $-\infty$ to +$\infty$; and (ii) providing pre-trip information is always welfare-improving.

From Figure 12a,h, one can see that risk-neutral travelers ($\lambda =0$) obtain no more benefit gains than risk-preferred ones ($\lambda <0$) or risk-averse ones ($\lambda >0$) in case 1a or 3f. This is in consistent with Proposition 6. Moreover, risk-neutral travelers obtain no more welfare gains than risk-preferred ones with risk attitude (i) $\lambda <{\lambda }_3$ in case 2; (ii) $\lambda <{\lambda }_6$ in case 3a and case 3c, or risk-averse ones with risk attitude (iii) $\lambda >{\lambda }_6$ in case 1b; (iv) $\lambda >{\lambda }_7$ in case 3e. This is consistent with Proposition 7–9. Nevertheless, risk-neutral travelers obtain more welfare gains than risk-averse ones with risk attitude (v) $\lambda >{\lambda }_8$ in case 3c and case 3d; (vi) ${\lambda }_4<\lambda <{\lambda }_7$ in case 3e. This is consistent with Proposition 10.

5.3. The Effect of Pre-Trip Information on Different Routes

Figure 13 shows two examples of benefit gains/losses from pre-trip information on different routes. In example 1 (Figure 13a), given $a_1=a_2$, full information is always welfare-improving on the network ($G^{ZF}\ge 0$) (see Figure 13a). While providing pre-trip information is welfare-improving on both routes when $-0.3<\lambda <0.15$, information is welfare-improving on route 2 (route 1) and welfare-reducing on route 1 (route 2) when $\lambda <-0.3$ ($\lambda >0.15$). In example 2, the free-flow travel costs are significantly larger on route 2 ($a_2=5$) than those on route 1 ($a_1=0$). Unfortunately, a counterintuitive information paradox occurs when travelers are risk-averse with the risk-preference coefficient $\lambda >2.14$ (see Figure 13b). In this example, providing information exerts adverse effects. This result suggests that we should not blindly provide pre-trip information to travelers. Instead, it is crucial to comprehensively consider factors such as free-flow travel costs, the reliability of different routes, and travelers’ risk attitudes when deciding whether to disseminate information.

6. Conclusions and Discussion

In this study, we extended the classical two-route network model proposed by Lindsey et al. [28] and considered travelers’ risk attitudes. The travel time variability due to stochastic network link capacity variations has been taken into account. We postulated that travelers acquire the variability in route travel time based on past experiences and factor such variability into their route choice consideration in the form of a “travel time budget”. Two information regimes, zero-information and full-information regimes, were considered. Commuters only know the unconditional probability distribution of conditions on the two routes in the zero-information regime, while they are provided the road conditions before choosing a route in the full-information regime.

User equilibrium states of the two regimes have been analyzed. In the special case $p=1$, the closed form solutions have been derived. Three cases and eleven subcases have been classified concerning the dependence of expected total travel times on the risk attitude in the zero-information regime. In the general condition $p>0$, although we are not able to derive the closed form solutions, we proved that the results are qualitatively unchanged.

We have studied the benefit gains/losses by shifting from the zero-information to the full-information regime. The circumstance under which pre-trip information is beneficial has been identified. We also demonstrated that risk-averse (risk-preferred) travelers might obtain more/less welfare than risk-neutral ones. Thus, the assumption that travelers are risk-neutral may underestimate or overestimate the benefit gains/losses of the information. Finally, a numerical analysis was conducted to further illustrate the benefits of information under the general conditions, which also demonstrated that while the network welfare improves/reduces, welfare on one of the routes may reduce/improve. Counterintuitively, the information paradox is more likely to occur when risk-averse travelers confront routes with substantial disparities in free-flow travel costs. Therefore, it is crucial to comprehensively consider factors such as free-flow travel costs, the reliability of different routes, and travelers’ risk attitudes when formulating traffic information dissemination strategies.

Our study has several limitations that could provide opportunities for future research. With pre-trip information, travelers might adjust their decisions from multiple aspects, e.g., travel route, departure time, travel mode, and destination. Our model only considers route choices based on the classical “two-route network”. Thus, the first extension would be to incorporate simultaneous multi-dimensional decision-making. Analyzing how travelers prioritize goals in multi-OD complex networks could deepen our understanding of pre-trip information’s impact on transportation efficiency. Second, our model features two polar information regimes. Full information is an idealization that is practically unattainable. Information on travel conditions may be imperfect and collected with delays, which should be considered in further work. Third, we assume a constant demand in commuting corridors. Yet, in emerging residential areas, dynamic demand fluctuations may significantly influence travel behavior, necessitating further exploration. Fourth, the model simplifies route states to binary conditions (good/bad). Extending this to multi-state stochastic scenarios would clarify how risk attitudes and information interact under realistic variability. Finally, we treat travelers as homogeneous. Future work should examine heterogeneous populations with diverse risk preferences to better reflect real-world decision dynamics.