Travelers’ Bi-Attribute Decision Making on the Risky Mode Choice with Flow-Dependent Salience Theory

: The purpose of this paper is to provide new insights into travelers’ bi-attribute (travel time and travel cost) risky mode choice behavior with one risky option (i.e., the highway) and one non-risky option (i.e., the transit) from the long-term planning perspective. In the classical Wardropian User Equilibrium principle, travelers make their choice decisions only based on the mean travel times, which might be an unrealistic behavioral assumption. In this paper, an alternative approach is proposed to partially remedy this unrealistic behavioral assumption with ﬂow-dependent salience theory, based on which we study travelers’ context-dependent bi-attribute mode choice behavior, focusing on the effect of travelers’ salience characteristic. Travelers’ attention is drawn to the bi-attribute salient travel utility, and then the objective probability of each state for the risky world is distorted in favor of this bi-attribute salient travel utility. A long-term bi-attribute salient user equilibrium will be achieved when no traveler can improve their bi-attribute salient travel utility by unilaterally changing the choice decisions. Conditions for the existence and uniqueness of the bi-attribute salient user equilibrium are presented, and based on the equilibrium results, we analyze travelers’ risk attitudes in this bi-attribute risky choice problem. Finally, numerical examples are conducted to examine the sensitivity of equilibrium solutions to the input parameters, which are cost difference and salience bias.


Introduction
One of the powerful principles widely used in the traditional four-step transportation planning model is the Wardrop's user equilibrium model, also known as Wardrop's first principle proposed in [1], which can be stated as: No traveler can decrease his (her) travel time by unilaterally changing his (her) choice decisions. In this paper, we use this principle to study travelers' mode choice behavior as [2].
There are several unrealistic assumptions underlying this principle. One of these is that travel times on the traffic network are deterministic. However, several studies show that uncertainty is inevitable, which could come from the demand side, and/or the supply side (e.g., [3,4]). Numerous studies that consider the uncertainty on the traffic network have been conducted. For example, the authors in [5] classified the travelers into three categories on the stochastic traffic network, which are risk-neutral, risk-seeking and risk-averse. The authors in [6] incorporated the random evolution of traffic states in the cell-based multi-class dynamic traffic assignment. The authors in [7] studied a risk-neutral congestion pricing problem and formulated it as a stochastic programming problem. The authors in [8] proposed a general approach to incorporate stochastic variations into the macroscopic traffic model. The authors in [9] studied a simultaneous route and departure time choice problem with stochastic travel times, where the demand is fixed and the link capacity is stochastic. The work of [10] classified uncertainty during the decision making reference [29]. Finally, decision makers' preference reversals can be captured by the salience theory, but not the prospect theory, which might be helpful to study travelers' preference reversals, e.g., [30]. Other differences between these two theories can be found in [31].
Prospect theory and its extension, the cumulative prospect theory [32], have been widely used in the studies on transportation and traffic (e.g., [28,29,33,34]). However, to the best of our knowledge, only a few scholars discuss the effect of travelers' salience characteristic on the policy design and implementation, e.g., [35,36] studied the effect of travelers' salience characteristic on the pricing policy, which does not attract much attention in transportation and traffic studies.
Considering that the research on the effect of travelers' salience characteristic is still in its infancy, we focus on a stylized situation in this paper, where we consider travelers' mode choice between two options (one risky option, i.e., the highway, and one non-risky option, i.e., the transit) in the risky world with two state, and travel cost here refers to the toll for the highway and fare for the transit. The motivation on this two-option choice analysis is that travelers usually consider two modes in practice (e.g., [37,38]), while the motivation for the two-state world assumption is the study in [39], where the authors point out two flaws about the original salience theory. The first one is that, for some ranges of probabilities, certainty equivalent is not defined, and the second one is that monotonicity is violated by the model. We conjecture that these two flaws also exist in the flow-dependent salience theory, an extension on the original salience theory, which is the foundation of our analysis. Although we focus on the stylized situation, we make a thorough analysis on it and obtain some novel results. The other contributions are summarized as follows.
We propose a bi-attribute salient travel utility model with the continuous salience ranking (also known as endogenous salience ranking), compared to the conventional discrete ranking method proposed in [18]. Furthermore, we develop a bi-attribute salient user equilibrium model based on this choice model and prove its solution existence and uniqueness. Moreover, we analyze travelers' risk attitudes in the bi-attribute mode choice problem based on the equilibrium results. Finally, we conduct the numerical examples to investigate the sensitivity of equilibrium solutions to the input parameters, which are cost difference and salience bias, to shed light on travelers' bi-attribute salient behavior. Our findings provide insights into travelers' behavioral studies related to the travel cost, which can further provide implications for the policy design and implementation, especially on the congestion tolling and transit fare design.
The remainder of the paper is organized as follows. In Section 2, we present the basic definitions and notations used in this paper. In Section 3, we propose the bi-attribute salient travel utility model after introducing the flow-dependent salience theory. In Section 4, we analyze the solution existence and uniqueness of the bi-attribute salient user equilibrium, and based on this, we discuss travelers' risk attitudes on the bi-attribute risky mode choice. In Section 5, we conduct the numerical examples to show the performance of the proposed model via the sensitivity analysis, followed by the conclusions and major findings in Section 7.

Definitions and Notations
We assume N travelers go from the common origin, e.g., the suburb area, to the common destination, e.g., the core area, and study their bi-attribute (travel time and travel cost) risky mode choice behavior with one risky option, i.e., the highway, denoted by R, and one non-risky option, i.e., the transit, denoted by NR. Furthermore, we investigate its long-run effect and assume there is no other mode options, e.g., staying at home. As aforementioned, travelers mainly focus on these two travel modes in practice. There are two states of the world, say good state and bad state, and the set of states is denoted by S. We use p(p ∈ (0, 1)) to denote the probability of bad state and then 1−p is the probability of good state. When p= 0 or p= 1, the risky option degenerates into a non-risky option, which will not be discussed furthermore. On the non-risky option NR, the travel utility functions are the same for both states, denoted by u NR , while the travel utility function on the risky option depends on the state, denoted by u + in the good state, and u − in the bad state.
The nonlinear travel utility functions shown in Equation (1) are used in our study, which are in the form of the difference between the utility value (due to the arrival at the destination) U and the generalized cost (weighted sum of travel time and travel cost) between the origin and the destination. Moreover, we assume that travelers could specify a suitable value U to make the utility functions always positive as [28] and [40].
where n R denotes the traffic flows on the risky option, and then travel flows on the nonrisky option is N − n R . Travel costs on these two options, risky and non-risky option, are τ 1 and τ 2 , respectively, and β > 0 denotes value of travel cost, which converts the travel cost into travel time. At the same time, t − (n R ) and t NR (N − n R ) are defined as travel time functions on the risky option and the non-risky option, respectively. In this paper, the travel time functions t − (n R ) and t NR (N − n R ) are considered to be a continuous strictly increasing function of the traffic volume on the corresponding option, and then we have In addition, it is assumed that t − (N) > t NR (N), i.e., travel time of the risky option in bad state is larger than that of the non-risky option when the flow is N, and the free flow travel times t 0 on these two options are identical. Therefore, the bad state is the most undesirable. Finally, the utility in the good state of the risky option is a constant, which is normalized as U − (t 0 + βτ 1 ). Assumptions made here could simplify our discussions in this paper, but our method can be extended accordingly if some assumptions are relaxed, and nevertheless, the insights and implications obtained according to our results will not be changed. The relationship between these travel time functions is shown in Figure 1.

Bi-Attribute Salient Travel Utility Model Based on Flow-Dependent Salience Theory
In this section, we introduce the flow-dependent salience theory first, and then propose the bi-attribute salient route utility model based on this theory.

Flow-Dependent Salience Theory
Following the principle of expected utility theory, the bi-attribute expected travel utility on the risky option is (1 − p)u + (n R ) + pu − (n R ), while the bi-attribute travel utility on the non-risky option is u NR (N − n R ). However, decision makers' mind might focus on whatever is odd, unusual or different, which is the essential meaning of salience ( [41]). Therefore, travelers could over-weight the route's most salient states in S, and assign a subjective probability to each state s (good or bad state), which depends on the objective probability of the state s and its salience. To formally define the flow-dependent salience theory, let u max s and u min s denote the largest and smallest utilities for each state s, respectively, and u −i s denote the utility of route j, j = s Definition 1. The salience of state s for option i, i = R, NR, is a continuous and bounded functionσ u i s (n R ), u −i s (n R ) for a given flow n R that satisfies the following two conditions: 1.
Ordering. If for states, s ∈ S we have that u min 2. Diminishing sensitivity. If u i s (n R ) > 0 f or i = R, NR, then for any ε > 0, In the original salience theory ( [18]), there is another property for the salience function, called reflection, which can handle negative utility. However, this property is not needed in our study, because all the values of utility functions are positive. To illustrate Definition (1), we use salience function It can be verified that this salience function satisfies both properties used in Definition (1). For a given value of n R , the ordering property shows that the difference between the utility u i s (n R ) of option u −i s (n R ) of other option increases, and thus the salience of the state s will increase, which is captured by u i s (n R ) − u −i s (n R ) . For a given value of n R , the diminishing sensitivity property shows that if a state's utility becomes larger, and thus the salience will decrease, which is captured by u i s (n R ) − u −i s (n R ). Moreover, we see that the salience function (4) also satisfies the symmetry property, because we discuss the mode choice between two options. Particularly, given states s, s ∈ S, state s is said to be more salient than s for option . We see that the travel utilities and the salience function are both flow-dependent, where the flow denotes the traffic flow on the risky option, and thus, we call it flowdependent salience theory.

Bi-Attribute Salient Travel Utility Model
Based on the flow-dependent salience theory, we propose the bi-attribute salient travel utility model as follows.

Definition 2.
According to the flow-dependent salience theory, travelers evaluate the bi-attribute salient travel utility on the risky option according to the formulation while travelers evaluate the bi-attribute salient travel utility on the non-risky option according to the formulation where δ 1 = δ −σ 1 (u + (n R ),u NR (N−n R )) , and δ 2 = δ −σ 2 (u − (n R ),u NR (N−n R )) . According to Equation (4), we have From Definition (2), we see that the bi-attribute salient travel utility on the risky option can be written as (1 − p )u + (n R ) + p u − (n R ), where p = δ 2 p δ 1 (1−p)+δ 2 p , and the biattribute salient travel utility on the non-risky option remains the same, regardless of the composition of the choice set. Compared to the formulation for the bi-attribute expected utility theory, i.e., (1 − p)u + (n R ) + pu − (n R ), we see that the only difference is the change of the objective probability, which is called distorted probability for bad state and good state based on flow-dependent salience theory.
Parameter δ(δ ∈ (0, 1]) indicates travelers' susceptibility to the salience, called salience bias. δ = 1 means rational travelers, and thus there is no distortion on the objective probabilities of the states. In the following study, when δ < 1, the travelers are called salient travelers. Smaller value of δ means stronger salience bias, and vice versa. In the special situation where δ → 0 , the salient travelers will only pay their attention on the most bi-attribute salient travel utility.
Based on the bi-attribute salient route utility presented in Definition (2), we obtain the following proposition about the mode preference, which is basis of our equilibrium analysis. Proposition 1. For a given flow n R , a salient traveler will choose the risky option if and only if the following inequality satisfied.
Proof. For a given flow variable n R , based on the bi-attribute salient travel utility, a salient traveler prefers the risky option to the non-risky option if and only if U R (n R ) > U NR (n R ) i.e., Equation (9) can be obtained by Equation (10) after some rearrangement, which completes the proof.

Bi-Attribute Salient User Equilibrium Analysis
In this section, we study the long-run effect of bi-attribute salient travel utility, and propose the bi-attribute salient user equilibrium (BaSUE), which can be stated as: No traveler can improve their bi-attribute salient travel utility by unilaterally changing their options. Furthermore, we compare the results of the BaSUE with those of bi-attribute expected user equilibrium (BaEUE), which can be stated as: No traveler can improve their bi-attribute expected travel utility by unilaterally changing their options.
Based on Proposition (1), we see that at the BaSUE, the following equation is satisfied.
Furthermore, we define the following flow-dependent mode preference function based on Equation (11), which is the foundation of our equilibrium analysis. Definition 3. The flow-dependent mode preference function, denoted by V R→NR (n R )is defined as (3), we see that when V R→NR (n R ) > 0, the salient travelers prefer the risky option to the non-risky option, when V R→NR (n R ) < 0, the salient travelers prefer the non-risky option to the risky option, and when V R→NR (n R ) = 0, the BaSUE is reached.

Formal Results on the BaSUE
Next, we present the formal analysis on the BaSUE, and before that, following the principle of expected utility theory, we present the analysis on the BaEUE as a benchmark case for comparison. We always use n Re to denote the equilibrium flow on the risky option.
Proof. The proof can be completed based on the following logic. A corner BaEUE is obtained when either of the following two conditions is satisfied.

1.
When n R = 0, the bi-attribute expected travel utility of the risky option is still not greater than the travel utility of the non-risky option, i.e., In this situation, we obtain that n Re = 0 is a corner BaEUE. Substituting the travel utility functions into this inequality and rearranging it, we obtain When n R = N, the bi-attribute expected travel utility of the risky option is still not less than the travel utility of the non-risky option, i.e., (1 − p)u + (N) + pu − (N) ≤ u NR (0). In this situation, we obtain that n Re = N is a corner BaEUE. Substituting the travel utility functions into this inequality and rearranging it, we obtain (2) is that when the travel cost of one option is too large (or too small) compared to that of the other option, travelers' mode choice decisions cannot be changed, even though increase in traffic flow causes the increase in travel time. That is, when the travel cost of a certain option is too large, no matter how the traffic flow is distributed, travelers will choose the option with a small travel cost, and vice versa. According to this proposition, we only discuss the situations where

The intuitive interpretation on Proposition
Surprisingly, result in Proposition (2) is also a sound prerequisite for the analysis on BaSUE due to the inherent relationship between the expected utility theory and salience theory as discussed in the part of flow-dependent salience theory, which will be elaborated in the following discussions. We divide the into two cases, and the motivation for this division is the utility value when the flow on each option is N.
. In this case, the corresponding relationship between different travel utility functions is shown in Figure 2 schematically. We see that when the flow on each option is N, The relationship u + (N) = u NR (N) when τ 1 = τ 2 is not shown explicitly, and one can modify Figure 2 for this straightforwardly. Moreover, relationship u + (N) = u NR (N) can also be incorporated into Case 2 without loss of generality. 2.
In this case, the corresponding relationship between different travel utility functions is shown in Figure 3 schematically. We see that when the flow on each option is N,  From Figures 2 and 3, we see that when n R ∈ [0, n t ] (here, n t denotes the temporary value of n R , and is obtained by solving the equation . Substituting these two inequalities into the flow-dependent mode preference function and with δ 1 and δ 2 being positive, we have i.e., the salient travelers always prefer the risky option to the non-risky option. Therefore, we obtain that when n R ∈ [0, n t ], there is no BaSUE. The aforementioned discussions are true for both cases, while for Case 2, we can obtain more results.
When n R ∈ [n t , N], N] (here, n t also denotes the temporary value of n R , and is obtained by solving the equation Substituting these two inequalities into the flow-dependent mode preference function and with δ 1 and δ 2 being positive, we have i.e., the salient travelers always prefer the non-risky option to the risky option. Therefore, we obtain that when n R ∈ [n t , N], there is no BaSUE. Therefore, in the following discussions, we only consider interval (n t , N] for Case 1, and interval (n t , n t ) for Case 2.
Next we discuss the BaSUE in these two cases separately based on the following technical lemma. Lemma 1. The flow-dependent mode preference function V R→NR(n R ) is a strictly decreasing function of the flow n R in the considered interval (n t , N] or (n t , n t ).
Proof. We prove this lemma is true for interval (n t , n t ), and the proof for interval (n t , N] can be completed similarly, which is omitted for brevity. It is obvious that in the interval (n t , n t ), To prove the strict monotonicity of the flow-dependent mode preference function, the following steps are followed. When n R (n t , n t )σ Figure 3, we see that u + (n R ) − u NR (N − n R ) > 0,u − (n R ) − u NR (N − n R ) < 0 when n R ∈ (n t , n t ), which can also be verified with the travel utility functions. There- 1) implies ln δ < 0 and we have δ 1 > 0 and δ 2 > 0.
Combining all these results, we can show that dV R→NR(n R ) dn R < 0, ∀n R ∈ (n t , n t ).

Case 1
In this case, is satisfied, which corresponds to Figure 2, and we present the formal results on the equilibrium in this section.
The right-hand side of this inequality is the value when n R = N with the objective probability, which motivates us to discuss the value when n R = N with the distorted probability.
Proof. We complete the first part of the proof in two steps. When Therefore, the salient travelers always prefer the risky option. That is, there is no BaSUE when , there is a unique BaSUE between n t and N. It can be verified that Following the similar steps used in the proof of Lemma (1), we obtain N). Combining the results that V R→NR (n t ) > 0 and V R→NR (N) ≤ 0, we can show that there is a unique value of n R ∈ (n t , N] that satisfies V R→NR (n Re ) = 0, which completes the first part of the proof.
Finally, δ 2 p δ 1 (1−p)+δ 2 p < p means that the cost difference, which ensures that there is still no corner equilibrium, becomes smaller.

Proposition 4.
If δ 1 = δ 2 when n R = N, there is a unique BaSUE between n t and N. In particular, the cost difference, which ensures that there is still no corner equilibrium, remains the same.
Proof. If δ 1 = δ 2 when n R = N, the BaSUE becomes the BaEUE. Follow the aforementioned discussions, we have N). Combining the results that V R→NR (n t ) > 0 and V R→NR (N) ≤ 0, we complete the first part of the proof.
Finally, δ 2 p δ 1 (1−p)+δ 2 p = p means that the cost difference, which ensures that there is still no corner equilibrium, remains the same. Proposition 5. If δ 1 < δ 2 when n R = N, there is a unique BaSUE between n t and N. In particular, we can increase the cost difference, and meanwhile, guarantee that there is still no corner equilibrium.
Proof. If δ 1 < δ 2 when n R = N, we have Follow the aforementioned discussions, we have N). Combining the results that V R→NR (n t ) > 0 and V R→NR (N) ≤ 0, we complete the first part of the proof.
Finally, δ 2 p δ 1 (1−p)+δ 2 p > p means that we can increase the cost difference, and meanwhile, guarantee that there is still no corner equilibrium.
All the results in the above three propositions about the corner equilibrium can be verified with the logic discussed in Proposition (2), by replacing the objective probability p with the distorted probability p .

Case 2
In this case, − t NR (N) − t 0 < β(τ 2 − τ 1 ) < 0 is satisfied, which corresponds to Figure 3, and we present the formal results on the equilibrium in this section. In this case, there is no objective probability p on left-hand side of the inequality, which is different from that in Case 1.

Proposition 6.
There is a unique BaSUE n Re between n t and n t . (1), we obtain that there is a unique value of n R ∈ (n t , n t ) hat satisfies V R→NR (n Re ) = 0, which completes the proof.

Travelers' Risk Attitude on the Bi-Attribute Risky Mode Choice
In this section, we present the results on travelers' attitude based on the BaSUE.
Proof. At the BaSUE, if no state is salient, we have δ 1 = δ 2 , and thus Equation (13) becomes Therefore, we obtain the BaEUE. The distorted probability of good state will become larger if it is salient, which will increase the left-hand-side value of Equation (16). Therefore, more travelers will choose the risky option, i.e., they are risk-seeking. Similarly, the distorted probability of the bad state will become larger if it is salient, which will decrease the left-hand-side value of Equation (16). Therefore, more travelers will choose the non-risky option, i.e., they are risk-averse.
In practice, maybe only one state is salient for a specific situation, as shown in the following Figure 4. Next, we propose the flow-dependent salience ranking analysis to present more detailed results based on Theorem (1), and obtain the following two propositions. Result in Proposition (7) is motivated by the discussion on the BaSUE in Case 1, where the relationship between δ 1 and δ 2 is a key prerequisite when n R = N. Here, we discuss the conditions for this relationship.
With the sufficient conditions, we obtain the following proposition.

2.
σ 1 (N) = σ 2 (N)is satisfied, and then when n R ∈ (n t , N), σ 1 > σ 2 , i.e., good state is salient and travelers are risk-seeking, and when n R = N, σ 1 = σ 2 , i.e., no state is salient and travelers are risk-neutral. 3. σ 1 (N) > σ 2 (N)is satisfied, and then whenn R ∈ (n t , N], σ 1 > σ 2 , i.e., good state is salient and travelers are risk-seeking. Proof. According to the aforementioned discussions in Lemma (1), σ 1 (n R ) and σ 2 (n R ) are both continuous functions of n R . dσ 1 dn R < 0, ∀n R ∈ (n t , N], i.e., σ 1 is a strictly decreasing function of n R when n R ∈ (n t , N], and dσ 2 dn R > 0, ∀n R ∈ (n t , N], i.e.,σ 2 is a strictly increasing function of n R when n R ∈ (n t , N] We have discussed the relationship between σ 1 and σ 2 when n R = N. With the strict monotonicity properties, we need to figure out the relationship between σ 1 and σ 2 when n R = n t . Recalling that n t is obtained by solving the equation u − (n R ) − u NR (N − n R ) = 0, we have σ 2 (n t ) = 0 and σ 1 (n t ) > 0 by Equations (7) and (8). Therefore, combining all the results, we obtain the following three possible situations.
Next, we talk about another case, i.e., Case 2.
Proof. It is obvious that in the interval (n t , n t ), σ 1 (n R ) and σ 2 (n R ) are both continuous func- , when n R ∈ (n t , n t ). According to Lemma (1), dσ 1 dn R < 0, ∀n R ∈ (n t , n t ),i.e.,σ 1 is a strictly decreasing function of n R when n R ∈ (n t , n t ), and dσ 2 dn R > 0, ∀n R ∈ (n t , n t ), i.e., σ 2 is a strictly increasing function of n R when n R ∈ (n t , n t ). Moreover, when n R = n t , we have σ 2 (n t ) = 0 and σ 1 (n t ) > 0; when n R = n t , σ 2 (n t ) = 0 and σ 1 (n t ) = 0, recalling the calculation of n t . Therefore, there is a value between n t and n t , defined as n * R , that σ 1 = σ 2 , as schematically shown in Figure 5. Furthermore, when n R ∈ (n t , n * R ), σ 1 > σ 2 ; when n R ∈ (n * R , n t ), σ 2 > σ 1 . Combining with the results in Theorem (1), we complete the proof.

Numerical Examples
In this section, we conduct the numerical examples to show the performance of the proposed method, focusing on the sensitivity of equilibrium solution to the input parameters, the cost difference and the salience bias. We assume that there are N = 3000 travelers in the system, and use the following quadratic travel utility functions.

Case 1
In this case,       Before we discuss the main insights of the results, we present four common aspects in all the testings. (1) In our testing, the sufficient condition σ 1 (N) < σ 2 (N) is satisfied, i.e., the relationship between σ 1 and σ 2 are shown in the upper-left figure of Figure 4. One can modify the travel utility func-tions to test other two situations; (2) In all the following figures, we claim that the cross point (denoted by n e R ) of the two curves, the curve for expected flow following the principle of expected utility theory and the curve for BaSUE flow following the principle of flow-dependent salience theory, is a special situation, where travelers are risk-neutral according to Theorem (1). The condition, that the solution to the flow-dependent mode choice function and the solution to the equation σ 1 (n R ) < σ 2 (n R ) coincide, needs to be satisfied, which can be verified by the related definitions. The above condition can also be used to figure out the P of the bad state when travelers are riskneutral. (3) When p = 0 or p = 1,we obtain the degeneration values for the equilibrium solution. In both cases, when p = 0, the solution is obtained by solving the equation u + (n R ) = u NR (N − n R ), and when p = 1, the solution is obtained by solving the equation u − (n R ) = u NR (N − n R ), given the specific values of τ 1 and τ 2 . (4) In all the testings, we can always find the unique BaSUE solution, which demonstrates Proposition (3).
From these tables and figures, we see the following insights.
(1) Given a specific objective probability (e.g., p = 0.2), when the cost difference (cost difference here means τ 2 − τ 1 ) become larger and larger, traffic flow on the risky option will become larger and larger, and vice versa. The intuitive explanation for this is that the increase of travel cost on non-risky option will make travelers choose the risky option. (2) When the probability of bad state is not large (e.g., p = 0.2), travelers are risk-averse, and when the probability of bad state is relatively large (e.g., p = 0.7), travelers are risk-seeking. We also see the cross point and its corresponding p, where travelers are risk-neutral as discussed before. For p ∈ (0, p), travelers are risk averse, and for p ∈ ( p, 1), travelers are risk-seeking. These results seem to be non-intuitive, and it seems that if the travelers know that the risky option is in bad state (in good state) with high probability, they should choose the non-risky (risky) option, i.e., they should be risk-averse (risk-seeking). However, travelers' salience characteristic means they put their attention on the unusual aspect. That is, if the risky option is in bad state (in good state) with high probability, considering the low (high) flow on the risky option (see , the good state (the bad state) is the unusual aspect according to our travel utility functions. Moreover, we can combine the results in Proposition (7) to present more interpretation on these phenomena. When the probability of bad state is not large, the equilibrium solution n Re is larger than n c R , and thus, we know bad state is salient according to Proposition (7). In contrast, we know good state is salient when the probability of bad state is large. Therefore, we see these phenomena in the testings. All the above discussions also demonstrate the validity of Theorem (1).
(3) When the cost difference become larger and larger, the extent of travelers' risk-averse attitude becomes stronger almost, except when p is very small, e.g., p = 0.02. This is because larger cost difference will make travelers choose the risky option, which further makes the difference between travel utility of risky option in good state and the travel utility of non-risky option become smaller, considering the high flow on the risky option (see . Therefore, the bad state is more salient, which leads to the stronger extent of travelers' risk-averse attitude. However, there is almost no difference between the risk-neutral equilibrium flow, i.e., the expected flow, and the risk-seeking equilibrium flow, i.e., the BaSUE flow when p is relatively large. That is, travelers' risk-seeking attitude following the flow-dependent salience theory has almost no effect on the equilibrium flow, compared to the results of expected utility theory. This is because the flow on the risky option is not large (see Figures 6-9), which make the difference between the utilities on risky and non-risky option small. (4) With the increase of cost difference, we clearly see the corner equilibrium following the principle of expected utility theory. Meanwhile, larger cost difference means more situations with corner equilibrium. However, there is no corner equilibrium for BaSUE even though we increase the cost difference, which demonstrate Proposition (5).
Next, we fix the travel cost on the risky option (τ 1 = 20) and non-risky option (τ 1 = 25), and change the values of salience bias δ. We choose four representative values for δ, which are 0.1, 0.3, 0.5 and 0.7. δ = 0.1 implies that travelers have stronger salience bias, while δ = 0.7 implies the salience bias is weaker. The final equilibrium results are shown in Figures 10-13. From these figures, we see the following insights. (1) With the increase of the value for δ, the extent of travelers' risk-averse attitude and risk-seeking attitude become weaker, and vice versa. (2) The value of δ has no effect on the cross point, where travelers are risk-neutral, as discussed before. (3) We clearly see the values for p, where there exists the corner equilibrium for expected flow, and the value of δ has no effect on this. (4) Even though travelers have very strong salience bias, e.g., δ = 0.1, the effect of travelers' risk-seeking attitude following the flow-dependent salience theory on the equilibrium results can be ignored, compared to the results of expected utility theory. We can present similar explanations as the above testing.

Case 2
In this case, − t NR (N) − t 0 < β(τ 2 − τ 1 ) < 0 is satisfied. The travel cost on the non-risky option is 20, while the cost on the risky option is increased from 21 to 30 with the step size being 3. That is, we choose four representative values for travel cost on risky option. The salience bias δ is also 0.3. The final equilibrium results are shown in Tables 5-8  (here we also     From these tables and figures, we see the following insights. (1) Given a specific value of objective probability (e.g., p = 0.2), when the cost difference (cost difference here means τ 1 − τ 2 ) becomes larger and larger, traffic flow on the risky option will become smaller and smaller, and vice versa. The intuitive explanation for this is that the increase of travel cost on risky option will make travelers choose the non-risky option. (2) We also see travelers' risk-seeking and risk-averse attitudes for different objective probability ps, and can give similar explanations as those in Case 1. Also, the risk-neutral attitude is a special condition.
(3) When the cost difference become larger and larger, the extent of travelers' risk-averse attitude almost remains the same, even though it becomes weaker. Increase in the cost difference will make more travelers choose non-risky option, and the difference between the utility of the risky option in good state and the utility of the non-risky option become larger. Meanwhile, increase of the value of τ 1 makes this difference become smaller. Therefore, this difference remains almost the same, and we see this phenomenon on travelers' risk-averse attitude. Similarly, there is almost no difference between the expected flow, and the BaSUE flow when travelers' are risk-seeking, and we can present similar explanation as those in Case 1. (4) In this test, there is no corner equilibrium for both principles, and one can modify the travel utility functions and travel costs to show the corner equilibrium.
Next, we fix the travel cost on the risky option (τ 1 = 25) and non-risky option (τ 2 = 20), and change the values of salience bias δ. We also choose four different value for δ, which are 0.

Conclusions
In this paper, we discussed travelers' bi-attribute (travel time and travel cost) risky mode choice behavior with one risky option (e.g., the highway) and one non-risky option (e.g., the transit), and provided new insights to this kind of behavior. We proposed to use the flow-dependent salience theory to support this research aim, and developed the biattribute salient travel utility model. Furthermore, we examined the long-run effect of this model and developed the bi-attribute salient user equilibrium model, which can partially remedy the unrealistic behavioral assumptions used in the classical Wardropian user equilibrium principle. We theoretically proved the existence and unique of the equilibrium, and based on the equilibrium results, we analyzed the travelers' risk attitudes on the bi-attribute mode choice problem. Finally, we run the numerical examples to show the sensitivity of the equilibrium results to the input parameters, which are cost difference and salience bias.
The novelty of our paper is to study the effect of travelers' salience characteristic on the bi-attribute mode choice behavior. Salience here is context-dependent, and travelers' attention is drawn to the bi-attribute salient travel utility. The main findings of this paper are summarized as follows. (1) Travelers' attitude can be risk-averse, risk seeking or risk-neutral in the bi-attribute mode choice problem. If the objective probability of bad state p is relatively small, travelers are risk-averse, and if p is relatively large, travelers are risk-seeking. Only in the special situation, where the solution to the flow-dependent mode choice function and the solution to the equation σ 1 (n R ) = σ 2 (n R ) coincide, travelers are risk-neutral. This condition can also be used to figure out the p of the bad state when travelers are risk-neutral, which can further be used to figure out the ps for risk-averse attitude and the ps for risk-seeking attitude. (2) The extent of travelers' risk-averse and risk-seeking attitudes can be affected by their salience bias and the cost difference, while their risk-neutral attitude is only affected by the cost difference. (3) If the travel cost on the risky option is smaller than that on the non-risky option, we see that the extent of travelers' risk-averse attitude becomes stronger with the increase of the cost difference, while if the travel cost on the non-risky is smaller, we see that the extent of travelers' risk-averse attitude almost remains the same with the change of cost difference. In both cases, we find that the effect of travelers' risk-seeking attitude can be ignored, which is identical to the effect of their risk-neutral attitude almost, even though travelers' salience bias is very strong. (4) The cost difference is closely related to the corner equilibrium, and travelers' salience characteristic can affect the occurrence of the corner equilibrium. (5) The relationship between the salience functions can be affected by the travel cost, and we identify the sufficient conditions for this (see the sufficient condition (17)).
Our findings can provide implications for the policy design and implementation, especially on the congestion tolling and transit fare design, which are summarized as follows.
(1) The salience characteristic can cause non-intuitive risk attitudes for travelers, which needs to be considered in the policy design and implementation. (2) The policy makers can use the congestion tolling and transit fare design (focusing on the cost difference) to change travelers' risk attitudes, and further change the distribution of the traffic flow. Especially, the change on travelers' risk-averse attitude is effective. (3) If the policy makers focus on the corner equilibrium, the effect of travelers' salience characteristic is notable, which can change the effect of cost difference. (4) The policy makers can also use the congestion tolling and transit fare design to change the relationship between the salience functions, focusing on our sufficient conditions.
There are several directions which merit further study. Firstly, in this paper, we only focus on the effect of travel cost on the equilibrium, and, in the future, we will discuss the optimal congestion tolling and transit fare design problem considering the salience characteristic, where the results of our paper can serve as the basis. Secondly, there is perception error for travelers, and in further research we will discuss the effect of perceived objective probability. Thirdly, our method can also be used for the two-route equilibrium analysis, but an extension to the multiple options is needed for the analysis on the general traffic network. Finally, a behavioral experiment can be conducted to estimate the key parameters in our method, i.e., the salience bias, as discussed in the introduction. Acknowledgments: Special thanks go to the reviewers for their constructive suggestions.

Conflicts of Interest:
The authors declare no conflict of interest.

Notations
The following notations are used in this manuscript: N The number of travelers R The risky option NR The non-risky option S The set of states of the world p The probability of bad state u NR The travel utility functions on NR for both states u + The travel utility function on the risky option in good state u − The travel utility function on the risky option in bad state n R The traffic flows on the risky option Travelers' susceptibility to the salience n t The temporary value of n R