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Information Provision and Congestion Pricing in a Risky
Two-Route Network with Heterogeneous Travelers
Yang Liu*1,2 and Zhenyu Yang2
1Department of Civil and Environmental Engineering, National University of
Singapore, Singapore
2Department of Industrial Systems Engineering and Management, National University
of Singapore, Singapore
March 13, 2021
Abstract
Travel information and congestion pricing are two major approaches to relieving conges-
tions. This paper examines travelers’ travel strategies and route choices in a road network with
a risky route and a safe route, where travel information and congestion pricing are jointly im-
plemented. We derive the equilibrium of the risk-averse travelers with heterogeneous Values
of Time (VOTs) under various information regimes. We find that information regarding the
risky road conditions would not necessarily improve the expected system travel time. When
there is no toll, free information surprisingly makes congestion worse. When information is
not free, congestion may be reduced if both toll and information are properly priced. We
also examine the distributional effects of information provision across travelers with different
VOTs. Travelers with low VOTs who do not use information are not affected by the price
of information. Travelers with intermediate VOTs always benefit from information provision
regardless of its price. Finally, travelers with high VOTs are better off when the information
price exceeds a threshold.
Keywords: User equilibrium, Information provision, Risk aversion, Congestion pricing
1 Introduction
Over the past decades, various forms of sensing, computational, and wireless communication
technologies have been developed and applied in transportation systems. Travel information can
be collected, processed, and disseminated in real-time. The availability and accuracy of travel
information may affect travelers’ choices of travel modes, departure times, and routes. As a user-
friendly instrument, information may guide travelers to make better decisions against recurrent
(Yang, 1998) and non-recurrent (Mahmassani and Jayakrishnan, 1991) congestion. A substantial
*Corresponding author, Phone: 65-6516-2334; Email: iseliuy@nus.edu.sg
1
number of models have been developed that describe travelers’ decision-making behavior infor-
mation provided through ATIS (e.g., Mahmassani and Jayakrishnan (1991); Ben-Akiva and De
Palma (1991); Arnott et al. (1991); Yang (1998); de Palma and Picard (2006); de Palma et al. (2012);
Liu and Liu (2018); Liu et al (2018); Jiang et al. (2020)). However, information may not always
benefit system performance if travelers’ responses to the information are not taken into account.
Many studies in literature investigate the “information paradox” problem in which information
provision could be welfare-reducing (e.g., Lindsey et al. (2014); Acemoglu et al. (2016)). For
example, the shortest path suggested by an ATIS may become very congested if many travelers
follow the guidance and use this path.
Given the limited road capacities, as a complement to information provision, congestion pric-
ing has been extensively studied and applied because it can effectively manage travel demand
and reduce congestion. However, congestion pricing appears to bring a direct loss to those who
do not value their travel time saving high enough to justify the paid toll or those who are “tolled
off” the road and have to use undesirable alternatives. By relaxing the value of time (VOT)
parameter from a constant to a random variable with an arbitrary probability distribution, Dial
(1996) demonstrates that the traffic assignment model with VOTs is more general than the con-
ventional models. The welfare effects of congestion pricing on travelers with the heterogeneous
VOT have been examined, and the design of a Pareto-improving pricing scheme has been dis-
cussed in the existing literature (e.g., Kockelman and Kalmanje (2005); Liu et al. (2009); Nie and
Liu (2010); Liu and Nie (2011); Wu et al. (2011); Liu and Nie (2017)). Many studies (e.g., Liu
et al. (2009); Nie and Liu (2010); Liu and Nie (2011); Van den Berg and Verhoef (2011); Liu and
Nie (2017)) show that whether a traveler would gain or lose from congestion pricing schemes de-
pends on the VOT distribution of the population. Thus, it is important to consider heterogeneous
travel preferences in policy decision-making.
This paper studies a realistic situation where information provision and congestion pricing
are jointly implemented in a road network. From the transportation planning perspective, road
and communication infrastructure may be developed and designed for facilitating both ATIS and
congestion pricing systems (Yang, 1999). For example, the next-generation Electronic Road Pric-
ing (ERP) system based on Global Navigation Satellite System will be implemented within a few
years in Singapore, which may provide more accurate and flexible road information in addition
to the existing congestion pricing scheme. We first investigate how the joint implementation of
the two instruments affects travelers’ route choices and travel strategy at equilibrium when in-
formation is provided for free or at a price. Then we explicitly examine the effectiveness of two
instruments in relieving congestion and improving travelers’ welfare. Last, we investigate the
design of the toll scheme and information price so as to relieve congestion and improve travelers’
welfare.
Limited studies consider the joint application of information and congestion pricing (e.g.
Verhoef et al., 1996; de Palma and Lindsey, 1998; Emmerink et al., 1996; Yang, 1999; Zhang and
Verhoef, 2006; Enrique Fern´andez et al., 2009; Gardner et al., 2011; Gubins et al., 2012; Chen
et al., 2015; Rambha et al., 2018). Some studies consider the situation that all the travelers are
provided with information (Verhoef et al., 1996; de Palma and Lindsey, 1998; Gardner et al.,
2011). Verhoef et al. (1996) study a similar setting in which information and congestion pricing
are jointly implemented in a stochastic two-route network. Beyond their consideration of elastic
2
demand and stochasticity for both routes, Verhoef et al. (1996) allow the toll to be fluctuating with
the link state. In addition to important modeling differences, their work has a different purpose.
We ask how tolling and information provision diversify travelers’ information acquisition and
subsequent route choices when allowing heterogeneity in value of time. In contrast, Verhoef et al.
(1996) look into the route split and modal split effects of five different information and pricing
instruments, in which information is disseminated to either all travelers or no travelers. de Palma
and Lindsey (1998) focus on the impact of information qualities in a one-link network, and thus,
they do not study route choices. Gardner et al. (2011) formulate a stochastic congestion pricing
problem in general networks. They allow the network manager to decide link tolls before or after
the network realization. They consider homogenous travelers such that either all travelers or no
travelers receive information on the realization of supply and treat the demand realization as
common knowledge. Another stream of the literature assumes that not all travelers are provided
with information. Enrique Fern´andez et al. (2009) extend the model in Verhoef et al. (1996) by
considering the market penetration of information is exogenously determined. Some studies
consider the information penetration is endogenously determined. For example, Yang (1998)
assumes that the information penetration depends on the information price and the equilibrium
costs of travelers with and without information. Emmerink et al. (1996) show that subsidy for
information provision is not required to maximize welfare when the first-best congestion pricing
is imposed. Using the same model, Zhang and Verhoef (2006) examine the case with a monopolist
supplier of information. The papers with endogenous market penetration focus on the impact of
information on travel demand. Therefore, most of them do not consider the route choices, except
for Gubins et al. (2012), which captures the route choices, while the travel time on each route is
assumed flow-independent.
Our study focuses on the joint implementation of information provision and congestion pric-
ing in a road network with a risky route and a safe route. The risky route is congested, and its
travel time functions vary from day to day according to a known probability. The travel time
of the safe route is constant. We examine travelers’ travel strategies (among always using the
safe route, always using the risky route, and using the information to make route choices) and
subsequent route choices. We consider three information regimes, i.e.,no information,free informa-
tion, and costly information regimes. Under the no information regime, no travelers are aware of
the current traffic conditions. They make route choices according to the knowledge gained from
the daily commuting experience. Under the free information regime, all travelers have free access
to real-time information about travel conditions. Under the costly information regime, only some
travelers that purchased information before departure have access to the traffic information. The
information penetration is endogenously determined by the information price and the toll. We
derive the equilibrium of the risk-averse travelers with heterogeneous Values of Time (VOTs)
under various information regimes. The heterogeneity of VOTs has been rarely discussed in the
information literature. Nevertheless, we consider travelers’ heterogeneity because it helps to bet-
ter evaluate the effectiveness of information and congestion pricing. It also allows examining the
distributional effects of the two instruments across travelers with different VOTs.
Three results stand out. First, we derive the equilibrium of the risk-averse travelers with
heterogeneous VOTs under three information regimes and prove that equilibrium is unique. We
found that, under the free information regime, travelers with intermediate VOTs always choose to
3
use information, and those with high (low) VOTs always choose the risky (safe) route regardless
of information state. However, under the costly information regime where information is provided
at a price, the travelers who use information in equilibrium could vary accordingly with the in-
formation price and the toll. We derive six cases of equilibrium and show that either a traveler
with high VOT or intermediate VOT purchases information. Second, we found that information
would not necessarily improve the expected system travel time. When there is no toll, free infor-
mation surprisingly worsens congestion. Under the costly information regime, congestion may be
reduced by properly designing toll and information price. Third, we also examine the distribu-
tional effects of information provision across travelers with different VOTs. Travelers with low
VOTs, who do not use the information and always choose the safe route, are not affected by both
free information and costly information, while travelers with intermediate VOTs always benefit from
using information. However, travelers with high VOTs may be worse off under free information
and costly information regimes. Their welfare will increase when the information price is higher
than a threshold.
The paper is organized as follows. Section 2 introduces assumptions and model settings.
Section 3 derives the equilibrium and discusses the impact of free information on social and indi-
vidual welfare, while in Section 4, the costly information regime is investigated. Finally, Section 5
presents numerical results. Section 6 concludes our findings.
2 The Model
This section presents a model, which is a variant of the model in de Palma et al. (2012). de Palma
et al. (2012) study the information provision problem considering heterogeneity in risk-aversion.
They use a general utility function to describe travelers’ risk-averse behaviors. Travelers are
assumed to have the same VOTs. By contrast, our model is more general in that traveler differs
in their VOTs but have the same degree of risk aversion. Beyond the difference between the
behavioral assumptions, our model uses a different network, which will be explained later. In
our study, both information provision and congestion pricing are implemented in order to reduce
congestion and improve travelers’ welfare. Risk-averse travelers choose their routes and travel
strategies in a stochastic and congested road network where travel time is uncertain.
A summary table of the notations may be found in Table 1.
We consider a road network with two parallel routes between an origin and a destination,
as shown in Figure 1(a). The travel demand dbetween the origin and the destination is fixed.
Travelers must choose between a risky route, denoted by R, and a safe route, denoted by S. The
travel time on routes Rand Scan be described as:
TS(f) = tS,
P(TR(f) = t−(f)) = p,
P(TR(f) = t+(f)) = 1−p.
(1a)
(1b)
(1c)
Equation (1a) implies that the travel time on Route S,TS(f), is deterministic and equal to a
constant tS. Route Srepresents a local arterial connecting the origin and the destination. The
capacity of Route Sis sufficiently large, and therefore there is no congestion. Equations (1b) and
(1c) imply that the travel time on Route R, TR(f), depends on the traffic flow fon Route Rand the
4
Table 1: Notations
Model Parameters
d: total demand between an origin and a destination;
p: probability of a good day, p∈(0, 1);
f: traffic flow on Route R;
g: travel strategy g,g∈ {Ar,As,Ai};
q: information regime q,q∈ {0, f,c};
nq
g: equilibrium number of travelers choosing Strategy gunder regime q;
tS: deterministic travel time on Route S;
TR(f): stochastic travel time on Route Rwhen the traffic flow is f;
t+(f): travel time on Route Ron bad days when the traffic flow is f;
t−(f): travel time on Route Ron good days when the traffic flow is f;
mRtoll on Route R;
θ: risk aversion parameter;
F(β): Cumulative Distribution Function of VOT β;
βu/βl: the maximum and the minimum VOT among all travelers;
π: information price
state of nature. There are two states of nature: a good state and a bad state. A binary probability
distribution of the state of nature is adopted. The probability of a good state is p∈(0, 1). For
brevity, the two states will hereafter be called a good day and a bad day. Congestion effects are
considered on Route Ron both good days and bad days. Specifically, the travel time on Route
R, denoted by t−(f)on good days and denoted by t+(f)on bad days, is a strictly increasing
function of the traffic flow f. By contrast, de Palma et al. (2012) also consider a single Origin-
Destination pair connected by a risky route Rand a safe route S. In their model, Route Ris not
congested on good days, and otherwise congested. Route Sis congested on both days.
We made the following Assumption 1, for guaranteeing the existence of an interior equilib-
rium solution when tolls are not applied. Therefore, by imposing Assumption 1, we focus on the
cases where both routes are used by travelers.
Assumption 1. The travel times on Route R, i.e., t−(f)on good days and t+(f)on bad days, are
continuous and strictly increasing functions of the traffic flow f . The following inequalities hold:
t−(0) = t+(0)<tS,
t+(f)>t−(f),∀f∈(0, d],
t+(d)>t−(d)>tS.
(2a)
(2b)
(2c)
Assumption 1 is illustrated in Figure 1(b). First, Inequality (2a) assumes that Route Ris faster
than Route Son both good days and bad days when the traffic flow f=0. Second, Inequality
(2b) requires the travel time of Route Ron bad days is always longer than that on good days
when the traffic flow f>0. Finally, if all the travelers choose Route R, Route Ris slower than
Route S, regardless of the state of nature (Inequality (2c)).
The heterogeneous travel preferences in terms of VOTs are considered here. Travelers differ
in their degree of risk aversion in de Palma et al. (2012), whereas they differ in their VOTs in
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Figure 1: A network with a risky route and a safe route
this paper. Consequently, travelers here sort themselves between routes on the basis of their
VOTs rather than their degrees of risk aversion. The VOTs in the population follow a continuous
distribution. Let F(β)denote the cumulative distribution function (CDF) of the VOT β.F(·)is
continuous and strictly increasing defined on [βl,βu], where βland βuare the minimum and
maximum VOTs among all travelers, i.e., F(βu) = 1 and F(βl) = 0. βl≥0 and βu∈[βl,∞).
Thus, any individual can be identified according to the VOT βor a ranking in the population
k=d×(1−F(β)). Conversely, for any given k,β(k) = F−1(1−k/d)identifies a unique β. A
toll is imposed on Route Rto relieve congestion. The toll on Route Ris denoted by mR≥0, and
Route Sis not tolled. If Route Ris faster than Route Sand the toll on Route Ris positive, then a
risk-neutral traveler with a high VOT would choose Route R.
Now, we consider three distinct information regimes under which information is provided
in different manners. The information regimes have been discussed for travelers with different
risk-averse degrees without congestion pricing in de Palma et al. (2012):
(a) No information (0): the probability pof the state is known, but the realized state of nature is
unknown before departure.
(b) Free information (F): free and accurate information about the realized state of nature is pro-
vided to all travelers before departure.
(c) Costly information (C): accurate information about the realized state of nature is provided to
travelers who are willing to purchase the information at a price π. The information price
πcan be considered as the out-of-pocket cost for receiving information services, including
costs of the necessary communication devices, information subscription fees, and so on.
Under each of the three information regimes, travelers choose among three alternative travel
strategies:
(a) Always choosing Route R(travel strategy Ar): travelers do not utilize the information and
always choose Route Rregardless of states of nature.
6
(b) Always choosing Route S(travel strategy As): travelers do not utilize the information and
always choose Route Sregardless of states of nature.
(c) Choosing Route Ror Saccording to the realized state (travel strategy Ai): travelers utilize
the information at a price π≥0 and make route choices according to realized states.
To incorporate the impacts of uncertainties of travel time into the travelers’ decision-making,
we adopt a prevailing method in the transportation literature, the travel time budget (TTB),
which combines the mean travel time with the cost of risk (measured by the standard deviation
of travel time) (e.g., Uchida and Iida, 1993; Shao et al., 2006; Lo et al., 2006; Nikolova and Stier-
Moses, 2014). The TTB of Route Sis equal to tS. For Route Rwith a flow f, its TTB is specified
in the following formulation:
TT BR(f) = ETR(f) + θ
qVar(TR(f)),
where θis the risk aversion parameter.
The travel cost under risk (TCUR) measured by TTB is defined below, incorporating both
travel time and out-of-pocket cost.
Definition 1. Travelers prefer a travel strategy g with minimal Travel Cost Under Risk (TCUR). The
TCUR of a traveler with the VOT βwho chooses a travel strategy g is given below:
cg(β)≡E(βTg+Mg) + θ
qVar(βTg+Mg),∀g∈ {As,Ar,Ai}
where Tgis a random variable of travel time by choosing Strategy g, Mgis a random variable of out-of-
pocket cost including the toll and the information price by choosing Strategy g.
Since the state of nature is assumed to follow a binary probability distribution, travelers with
the VOT βchoosing any Strategy gwould experience a total cost c−
g(β) = βT−
g+M−
gon good
days and c+
g(β) = βT+
g+M+
gon bad days, where T−
gand T+
gdenote the travel times of the
chosen routes by strategy gon good and bad days, respectively. And M−
gand M+
gdenote the
out-of-pocket costs of the chosen route on good and bad days, respectively. Thus, the TCUR of
Strategy gis reformulated as
cg(β) = pc−
g(β) + (1−p)c+
g(β) + θ
qp(1−p)|c+
g(β)−c−
g(β)|(3)
For a bounded VOT distribution, when mRis large enough, corner solutions may emerge in
which all travelers choose Route S. Since we focus on interior solutions where both routes are
used, the following assumption is imposed.
Assumption 2. 0≤mR<βu(tS−t−
r(0)).
The following Assumption 3 ensures the decision made according to the TCUR is consistent
with the property of the statewise stochastic dominance, which is a special case of the canonical
first-order stochastic dominance (FSD) (see Quiggin (1990); Wu and Nie (2011)). For notational
brevity, we use c−
gand c+
ginstead of c−
g(β)and c−
g(β). According to the statewise stochastic
dominance property, when c−
g<c+
g, the TCUR should decrease with the probability pof good
days. Also, it should increase with the travel cost on good days (c−
g) or the travel cost on bad
days (c+
g). As proved in Appendix A, when c−
g<c+
g, it requires the following:
7
Assumption 3. The risk-aversion parameter θand the probability p of the good state satisfy:
θ2/(1+θ2)<p.(4a)
3 No information and free information
In this section, we first derive the equilibria under the no information regime and the free informa-
tion regime. Then, we examine the effectiveness of information in improving system performance
under the free information regime.
In equilibrium, no traveler can lower its TCUR by unilaterally changing its travel strategy.
Denote numbers of travelers choosing travel strategy Ar,As, and Ai by nr,ns, and ni, respectively.
The numbers of travelers choosing each travel strategy (i.e., strategy splits), route choices, and
travel costs under risk under the three information regimes are summarized in Table 2. Under
the no information regime, travelers make route choices without knowing the realized state of
nature before departures, and they choose between travel strategies Ar and As. They know the
probability distribution of the state of nature through their daily commuting experiences.
Table 2: Travel Strategies, Strategy Splits, Route Choices, and Travel Costs Under Risk
Regime Strategy Splits Route Choice Travel Cost Under Risk
+ -
No information Ar n0
rR R mR+βTT BR(n0
r)
As n0
sS S βtS
Free information
Ar nF
rR R mR+βMS(t−(nF
r+nF
i),t+(nF
r))
As nF
sS S βtS
Ai nF
iS R MS(mR+βt−(nF
r+nF
i),βtS)
Costly information
Ar nC
rR R mR+βMS(t−(nC
r+nC
i),t+(nC
r))
As nC
sS S βtS
Ai nC
iS R MS(mR+βt−(nC
r+nC
i),βtS) + π
Based on Assumptions 1-3, the following lemma is obtained. For simplicity, we introduce a
function MS(x1,x2)≡px1+ (1−p)x2+θ
pp(1−p)|x2−x1|for any positive real numbers x1
and x2.MS(x1,x2)can be understood as TCUR of a strategy with travel costs x1on good days
and x2on bad days. That is, we can write cg(β)in Equation 3 as MS(c−
g,c+
g).
Lemma 1. Given Assumptions 1-3, when mR>0, the equilibrium traffic flows on Route R on good days
f+and on bad days f −are such that (1) t+(f+)>t−(f−)and (2) tS>MS(t−(f−),t+(f+)) under
the no information, free information, and costly information regimes.
Proof. See Appendix B.
Lemma 1 concludes that, in equilibrium, the travel time t+(f+)of Route Ron bad days is
always longer than the travel time t−(f−)on good days. Therefore, travelers who adopt the
travel Strategy Ai will choose Route Ron good days and Route Son bad days. Lemma 1 also
indicates that the travel time budget MS(t−(f−),t+(f+)) of Route Ris always lower than that
on Route S. Thus, Route Routperforms Route Sin terms of travel time budget.
8
3.1 No information
In the no information regime, no travelers know the state and actual travel costs on each route.
In this setting, travelers choose between routes so that TCURs are equal at the equilibrium. The
following proposition describes the traffic distribution at the equilibrium under the no information
regime.
Proposition 1. Under Assumptions 1-3, there exists a unique interior equilibrium traffic flow f 0∈(0, d)
on Route R under the no information regime. That is,
(a) When mR=0, everyone is indifferent between strategies As and Ar at equilibrium. The traffic flow
on Route R, f 0, can be solved using the equation TTBR(f0) = tS. f 0is monotonically decreasing
with respect to θ, and it is monotonically increasing with respect to p.
(b) When mR>0, there exists a unique VOT threshold β0
sr such that a traveler with β>β0
sr selects
travel strategy Ar, a traveler with β<β0
sr selects travel strategy As. Variables β0
sr and f 0can be
solved using the equations
(f0=d−dF(β0
sr),
mR+β0
sr TTBR(f0) = β0
srtS.
(5a)
(5b)
(c) When mR>0, the VOT threshold β0
sr is monotonically increasing with respect to θ, or mR, and it
is monotonically decreasing with respect to p.
Proof. See Appendix C
Proposition 1 implies, with no toll, travelers are indifferent to strategies As and Ar (As ∼Ar).
When mR>0, travelers with VOTs higher than β0
sr prefer strategy Ar to As (Ar ≻As), and
the rest prefer strategy As to Ar (Ar ≺As). The traffic operators can manage the traffic flow
distribution by adjusting tolls under the no information regime. As the toll mRincreases, the VOT
threshold β0
sr will increase, and thereby more travelers will be tolled off the risky Route Rand
switch to Route S.
3.2 Free information
We now turn to characterize equilibrium under the free information regime, in which all the
travelers are informed of the state of nature before departures. Under the free information regime,
the travel time on Route Ris deterministic and is known as a function of its traffic flow. The
following proposition describes the equilibrium under the free information regime.
Proposition 2. Under Assumptions 1-3, there exists a unique interior equilibrium under the free infor-
mation regime such that the flows of Route R on good and bad days are f−
F∈(f0,d)and f +
F∈(0, f0),
respectively.
(a) When mR=0, all travelers are indifferent to strategies As, Ar, and Ai; Traffic flows f −
Fand f +
Fcan
be solved using equations t−(f−
F) = tSand t+(f+
F) = tS, respectively, and they are independent of
θand p.
9
(b) When mR>0, there exist unique VOT thresholds βF
si and βF
ri such that a traveler with β<βF
si
selects strategy As, a traveler with β∈(βF
si,βF
ri )selects strategy Ai, and a traveler with β>βF
ri
selects strategy Ar. Variables f −
F,f+
F,βF
si and βF
ri can be solved using the following equations:
f−
F=d−d·F(βF
si),
mR+β0
srt−(f−
F) = βF
sitS,
f+
F=d−d·F(βF
ri ),
mR+β0
srt+(f+
F) = βF
ri tS.
(6a)
(6b)
(6c)
(6d)
(c) When mR>0, the VOT thresholds βF
si and βF
ri are monotonically increasing with respect to mR,
and is independent of θand p.
Proof. See Appendix D
Proposition 2 reveals travelers’ travel strategies and flow patterns when free information is
applied with and without congestion pricing. Figure 2 illustrates the travel strategies across
travelers with heterogeneous VOTs under the no information and free information regimes when
congestion pricing is implemented. The equilibrium flow of Route Rincreases on good days and
decreases on bad days comparing to that under the no information regime. (a) and (b) address that
the equilibrium is unique. (a) shows that when free information is provided without congestion
pricing, travelers will be indifferent to strategies As,Ar, and Ai. (b) shows that when free infor-
mation is provided with congestion pricing, travelers will choose different strategies according to
their VOTs. In the presence of a toll, travelers with VOTs less than βF
si choose strategy As, and
travelers with VOTs higher than βF
ri choose strategy Ar. Travelers with VOTs between βF
si and
βF
ri choose strategy Ai, and they pick Route Ron good days and Route Son bad days. On good
days, the indifferent traveler to strategies Ar and Ai would also be indifferent to both routes; on
bad days, the indifferent traveler to strategies As and Ai would be indifferent to both routes. (c)
concludes the VOTs of the indifferent travelers are increasing in the toll.
Different from the equilibrium under the no information regime, which is affected by the risk
aversion parameter θ, the equilibrium under the free information regime is not affected by θ.
Because the free information eliminates the uncertainty on Route R, the equilibrium under the free
information regime is reduced to the equilibrium with risk-neutral travelers (θ=0). Therefore,
the risk-aversion level will not affect the traffic flow distribution between the two routes under
the free information regime. If travelers are more risk-averse, the traffic flow of Route Runder the
no information regime will be lower, while the traffic flow of Route Runder the free information
regime will remain the same. It results in an information paradox in the following Proposition
3. Proposition 3 reveals that, surprisingly, providing free information worsens congestion when a
toll is not imposed.
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Proposition 3. Under Assumptions 1-3, when the toll mRon Route R is equal to zero,
1. the expected system travel time GFunder the free information regime is higher than the expected
system travel time G0under the no information regime: f 0ETR(f0) + (d−f0)tS<dtS;
2. free information reduce the travel time variability for travelers choosing route R in the no information
regime, and does not change the travel time variability for travelers choosing route S in the no
information regime.
Proof. 1. Under the free information regime, t+(f+
F) = tSand t−(f−
F) = tSby Proposition 2,
and the expected system travel time is GF=dtS.
Under the no information regime, we have TTBR(f0) = tSby Proposition 1, and the expected
system travel time is
G0=f0ETR(f0) + (d−f0)tS.
When θ>0, we have
TT BR(f0) = ETR(f0) + θ
qVar(TR(f0)) >ETR(f0).
Therefore, we have
G0≡f0ETR(f0) + (d−f0)tS
<f0TT BR(f0) + (d−f0)tS
=dtS.
(7)
Therefore, we have GF≡dtS>G0.
2. Under the no information and free information regimes, travel times on both routes are listed
in the following table:
Row Regime Route Good days Bad days
1 no information R t−(f0)t+(f0)
2 no information S tStS
3 free information R/S tStS
11
That travel time variability decreases for the group of travelers choosing route Runder the
no information regime follows from rows 1 and 3 and t+(f0)>t−(f0). That travel time
variability does not change and is equal to zero for the group of travelers choosing route S
under the no information regime follows from rows 2 and 3.
Proposition 3 implies that when there is no toll, providing free information can worsen traffic
congestion, i.e., the expected system travel time increases after introducing free information. That
is because, in the absence of toll, the congestion externality on Route Ris higher than that on
Route Sunder the no information regime. The congestion externality on Route Ris offset by
the risk-averse travel behavior to a certain extent under the no information regime. However,
providing free information eliminates uncertainty in travel time, which results in the traffic flow
of Route Runder the free information regime becomes higher than that under the no information
regime. Therefore, the congestion externality on Route Ris further increased by free information
compared to the equilibrium with no information. That explains why free information worsens
traffic congestion.
When congestion pricing is implemented (mR>0), free information may increase or decrease
the expected system travel time, which will be demonstrated in the numerical experiments. For
example, our numerical experiments show that when the toll is properly designed, joint im-
plementation of free information and congestion pricing could probably alleviate congestion and
ensure that everyone is not worse off by information provision. Our model exhibits different
results compared to the model in de Palma et al. (2012). Although both models show that the
equilibrium usage of Route Rincreases on good days and decreases on bad days after the in-
troduction of free information, the impact of free information on system travel time is different. In
de Palma et al. (2012), free information reduces expected travel time for all travelers. However, our
results show that free information provision may not necessarily reduce congestion. It is mainly
due to that de Palma et al. (2012) assume that travel time on Route R, is constant on good days,
and it is not flow-dependent. Therefore, free information will unambiguously decrease the ex-
pected travel time on Route R. In our model, travel time on Route Ris an increasing function
of its traffic flow in both states of nature. Free information may attract too many travelers to use
Route Ron good days, which increases the congestion externality. Hence, free information may
increase traffic congestion.
Other metrics, such as the total travel time budget cost, can also be used to measure the system
performance. We examine several alternative metrics numerically in Section 5.2. Here, we focus
on the expected system travel time for the following reasons. First, despite the assumption of
risk-averse travelers in this study, a traffic management authority tends to be risk-neutral. A risk-
neutral authority may emphasize reducing the average congestion, which directly affects fuel
consumption and carbon emissions. Second, the cost associated with travelers’ time variability
also depends on travelers’ risk aversion degree. If the traffic management authority aims to
reduce both congestion and the cost associated with risk, the congestion may be worsen when
travelers’ risk aversion degree is sufficiently large. Third, the goal of congestion minimization
allows this study to be extended in the future. For example, we can consider a Pareto-improving
pricing scheme by adding constraints to the congestion minimizing problem, which ensures
everyone is better off from our instruments in addition to congestion minimization.
12
Now, we examine the distributional effects of free information on individual welfare across
travelers with heterogeneous VOTs. We use the compensating variation (CV), as defined in
Definition 2, to measure the welfare change of an individual traveler caused by free or costly
information.
Definition 2. Under Assumption 1, if the travel demand is fixed, then the compensating variation CV q(β)
of a traveler with VOT βunder information regime q,∀q∈ {F,C}, corresponds to the gap between TCURs
under the no information regime and information regime q. CVq(β)is given by
CVq(β) = c0(β)−cq(β),
where c0(β)and cq(β)denote TCURs of strategies g0and gqselected by the traveler with VOT βunder
the no information regime and regime q, respectively.
According to the strategies chosen under the free information and no information regimes, trav-
elers can be classified into four groups, denoted by SS, SI, RI, and RR, as shown in Figure 2.
Travelers in Group SS (β<βF
si) choose strategy As under both regimes. Travelers in Group SI
(βF
si <β<β0
sr) choose strategy As under the no information regime and choose strategy Ai under
the free information regime. Travelers in Group RI (β0
sr <β<βF
ri ) choose strategy Ar under the
no information regime and Ai under the free information regime. Travelers in Group RR (β>βF
ri )
choose strategy Ar under both regimes. Let TTB0
Rand TT BF
Rdenote the TTBs of strategy Ar
under the no information and free information regimes, respectively, and TTBF
idenote the TTB of
strategy Ai under the free information regime. The CVs for travelers in each group are specified
in Proposition 4.
Proposition 4. Under Assumptions 1-3, under the free information regime,
(a) given a toll mR=0, CVF(β) = 0holds for all travelers.
(b) Given a toll mR>0,
1. for Group SS, CVF(β) = 0;
2. for Group SI, CVF(β)>0and is monotonically increasing with respect to β.
3. for Groups RI, when TTB0
R>TT BF
i, CVF(β)>0and is monotonically increasing with respect to
β; when TTB0
R=TT BF
i, CVF(β)>0and is constant; when TTB0
R∈(TT BF
R,TT BF
i], CVF(β)>
0and is monotonically decreasing with respect to β; when TTB0
R<TT BF
R, CVF(β0
rs)>0and
CVF(βF
ri )<0, and CV F(β)is monotonically decreasing with respect to β.
4. for Group RR, when TTB0
R>TT BF
R, CVF(β)>0and is monotonically increasing with respect to
β; when TT B0
R=TT BF
R, CVF(β) = 0; when TTB0
R<TT BF
R, CVF(β)<0and is monotonically
increasing with respect to β;
Proof. See Appendix E.
Proposition 4 reveals the distributional effects of free information provision across travelers
with different VOTs. When the toll is not implemented (mR=0), individual welfare is not
influenced by the free information provision. According to Proposition 1 and 2, travelers are
indifferent to all alternative travel strategies under both regimes regardless of their VOTs. Thus,
13
their TCURs remain the same as the TCUR of choosing travel strategy As, which is a constant
under both regimes.
Proposition 4 also shows when mR>0, there exists a trade-off between the out-of-pocket cost
and travel time cost when travelers make route choices, and hence the equilibrium depends on
the VOT distribution. Property (1) in Proposition 4 shows travelers with low VOTs (i.e., Group
SS) , who choose As under both the free information and no information regimes, break-even, and
free information does not affect individual welfare. Property (2) shows travelers with intermediate
VOT (i.e., Group SI), who choose As under the no information regime and Ai in the free information
regime, benefit from free information, and the benefit increases with the VOT. However, travelers
with high VOTs (i.e., Groups RI and RR), who choose Ar under the no information regime could
incur a loss from free information. Properties (3) - (4) present the conditions under which groups
RR and RI benefit or lose from the free information. For Group R I, if the TTB of Ai is higher than
Ar, CV is increasing in the VOT, and all group members’ CVs are positive; Otherwise, CV is
decreasing in the VOT, and some travelers’ CV may be negative. For Group RR, when the TTB of
Ar under the no information regime is higher than that in the free information, all group members
benefit fromfree information provision. Otherwise, free information makes the travelers in Group
RR worse off.
As we have shown, heterogeneity in the VOT is a critical determinant of travel decisions when
congestion pricing is imposed. The heterogeneity in our model facilitates the evaluation of the
distributional welfare effect across travelers with different VOTs. If we ignore the heterogeneity
and assume an identical VOT among the population, all the travelers will be indifferent to the
alternative travel strategies, and their TCURs will remain the same under the no information and
free information regimes. Therefore, the introduction of free information makes everyone break
even. However, when heterogeneous VOTs are considered, we prove that travelers may benefit,
lose, or break even, depending on their VOTs.
4 Costly information
Now, we consider the costly information regime under which traffic information on the state of
nature is provided by a private sector at a price π. Travelers may decide whether they are willing
to purchase information, and they will choose the travel strategies among Ar,As, and Ai. The
costly information regime is much more complicated than the free information regime. We will show
that there are six cases of equilibrium under the costly information regime, which depend on the
information price and the toll. Unlike the free information regime under which the travelers with
intermediate VOTs always choose to use information (i.e., travel strategy Ai), and those with
high VOTs always choose to use Route R(i.e., strategy Ar), we will show the travel strategies of
travelers with heterogeneous VOTs may vary with the toll and the information price under the
costly information regime.
Given the information price πand the traffic condition (ns,nr,ni), the properties of the
preference ranking between strategies Ar,As, and Ai are described in the following lemmas.
Lemma 2. Under Assumptions 1-3,
14
(a) there exists a unique threshold ˇ
βsr such that
Ar ≻As ⇔β>ˇ
βsr(mR,ns,nr,ni),where
ˇ
βsr =(mR
tS−MS(t−(nr+ni),t+(nr) ) ,if tS>MS(t−(nr+ni),t+(nr));
∞,otherwise;
(b) for 0<ˇ
βsr <∞,ˇ
βsr is independent of π, increasing in mR, nr, niand decreasing in ns.
Proof. See Appendix F.
Lemma 2 reveals the properties of the preference ranking between strategies Ar and As given
the traffic flow f−=nr+niand f+=nron Route R. According to Lemma 1, tS>MS(t−(nr+
ni),t+(nr)) must hold at equilibrium. Thus, a traveler with a high VOT (i.e., β>ˇ
βsr) would
prefer Ar to As at equilibrium.
Lemma 3. Under Assumptions 1-3,
(a) there exists a unique threshold ˇ
βsi such that
Ai ≻As ⇔β>ˇ
βsi (mR,π,ns,nr,ni),where
ˇ
βsi(mR,π,ns,nr,ni) =
π+(p−θ
√p(1−p))mR
tS−MS(t−(nr+ni),tS),if tS>MS(t−(nr+ni),tS);
∞,otherwise ;
(b) for ˇ
βsi <∞,ˇ
βsi is increasing in nrand ni, decreasing in ns, and increasing in π.
Proof. See Appendix G.
Lemma 3 addresses the properties of the preference ranking between strategies As and Ai
given the traffic flow f−=nr+niand f+=nron Route R. According to Lemma 1 and
Assumption 3, tS>MS(t−(nr+ni),t+(nr)) must hold at equilibrium. Thus, a traveler with a
high VOT (i.e., β>ˇ
βsi) would prefer Ai to As at equilibrium.
Lemma 4. Under Assumptions 1-3, let ˜
π= (1−p+θ
pp(1−p))mR.
(a) When π<˜
π, there exists a unique threshold ˇ
βri (mR,π,ns,nr,ni)such that
Ar ≻Ai ⇔β>ˇ
βri (mR,π,ns,nr,ni),where
ˇ
βri (mR,π,ns,nr,ni) = (˜
π−π
MS(t−(nr+ni),tS)−MS(t−(nr+ni),t+(nr)) ,if t+(nr)<tS;
∞,otherwise ;
for ˇ
βri <∞,ˇ
βri is decreasing in π, increasing in mRand nr; otherwise, ˇ
βri =∞;
15
(b) when π>˜
π, there exists a unique threshold ˇ
βri (mR,π,ns,nr,ni)such that
Ar ≻Ai ⇔β<ˇ
βri (mR,π,ns,nr,ni),where
ˇ
βri (mR,π,ns,nr,ni) = (π−˜
π
MS(t−(nr+ni),t+(nr))−MS(t−(nr+ni),tS),if t+(nr)>tS;
∞,otherwise ;
for ˇ
βri <∞,ˇ
βri is increasing in π, decreasing in mRand nr; otherwise, ˇ
βri =∞;
(c) when π=˜
π, individual preference rankings for the information regimes are
Ai ≻Ar,∀β,if t+(nr)>tS;
Ai ≺Ar,∀β,if t+(nr)<tS;
Ai ∼Ar,∀β,if t+(nr) = tS.
Proof. See Appendix H.
Lemma 4 addresses the properties of the preference ranking between strategies Ar and Ai
given the traffic flow f−=nr+niand f+=nron Route R. The relative magnitude of the travel
time and out-of-pocket costs of Strategies Ar and Ai are both undetermined. When strategy Ar
dominates Ai in terms of travel time, and strategy Ai dominates Ar in terms of the out-of-pocket
cost, travelers with β>ˇ
βri would prefer strategy Ar; when strategy Ai dominates Ar in terms
of travel time and strategy Ar dominates Ai in terms of the out-of-pocket cost, travelers with
β>ˇ
βri would prefer strategy Ai; When a strategy Ai or Ar dominates the other one in terms of
both travel time and cost, no one chooses the dominated strategy Ai or Ar.
Lemmas 2- 4 lead to the following lemma.
Lemma 5. Under Assumptions 1-3, for any (ns,nr,ni),there exists a unique VOT threshold
ˇ
βsr(mR,ns,nr,ni)and two functions ˇ
βsi(mR,π,ns,nr,ni)and ˇ
βri (mR,π,ns,nr,ni)such that
(a) when π<˜
π, a traveler selects strategy As if β<ˇ
βsi(mR,π,ns,nr,ni)and β<ˇ
βsr(mR,ns,nr,ni);
selects Ai if β<ˇ
βri (mR,π,ns,nr,ni)and β>ˇ
βsi (mR,π,ns,nr,ni); selects Ar if β>
ˇ
βri (mR,π,ns,nr,ni)and β>ˇ
βsi (mR,π,ns,nr,ni);βsi is increasing in πwhen βsi <∞;βr i
is decreasing in πwhen βri <∞;
(b) when π>˜
π, a traveler selects strategy As if β<ˇ
βsi(mR,π,ns,nr,ni)and β<ˇ
βsr(mR,ns,nr,ni);
selects Ai if β>ˇ
βri (mR,π,ns,nr,ni)and β>ˇ
βsi (mR,π,ns,nr,ni); selects Ar if β<
ˇ
βri (mR,π,ns,nr,ni)and β>ˇ
βsi(mR,π,ns,nr,ni);βsi and βri are both increasing in πwhen
βsi <∞and βri <∞;
(c) When tS>MS(t−(nr+ni),t+(nr)) and tS>MS(t−(nr+ni),tS), there exists a unique infor-
mation price threshold ˇ
πsuch that ni=0if and only if π≥ˇ
π.
If t+(nr)<tS,
ˇ
π<˜
πand ˇ
βsr(mR,ns,nr,ni) = ˇ
βsi(mR,ˇ
π,ns,nr,ni) = ˇ
βri (mR,ˇ
π,ns,nr,ni);
if t+(nr)>tS,
ˇ
π>˜
πand ˇ
βri (mR,ˇ
π,ns,nr,ni) = βu;
16
if t+(nr) = tS,
ˇ
π=˜
πand ˇ
βsr(mR,ns,nr,ni) = ˇ
βsi(mR,ˇ
π,ns,nr,ni).
Lemma 5 characterizers individual traveler strategies choices with costly information, given
traffic conditions. Lemma 5 implies that when the information price is low, the decisions of
travel strategies require a comparison between strategies As and Ai for travelers with relatively
low VOTs, and a comparison between Ai and Ar for travelers with relatively high VOTs. When
the information price is high, the decisions of travel strategies require a comparison between
strategies As and Ar for travelers with relatively low VOTs, and a comparison between Ai and
Ar for travelers with relatively high VOTs.
As illustrated in Figure 3, depending on the toll and the information price, there are six cases
of equilibrium patterns. The travel strategies for travelers with heterogeneous VOTs in Cases 1-6
are presented in Figures 4(a)- 4(f). First, in Case 5, when the information price is sufficiently high
(π>¯
π), nobody will purchase information. Consider a toll threshold ¯
mRat which t+(n0
r) = tS
under the no information regime. We present the results in the following proposition.
,QIRUPDWLRQSULFH ߨ
݉ோ
&DVH
&DVH
&DVH
&DVH
&DVH
݉ ൌ ݉ഥோ
&DVH
Figure 3: Six cases of equilibrium patterns under the costly information regime
Proposition 5. Under Assumptions 1-3, under the costly information regime, there exists an information
price threshold ¯
π>0such that when π≥¯
π(Case 5), the equilibrium is the same as that under the no
information regime. And,
¯
π=
β0
sr(tS−MS(t−(f0),tS)) −(p−θpp(1−p))mR<˜
π,if mR>¯
mR;
βu(TT BR(f0)−MS(t−(f0),tS)) + ˜
π>˜
π,if mR<¯
mR;
˜
π,if mR=¯
mR;
Proof. See Appendix I.
17
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Figure 4: Travel strategies for travelers with heterogeneous VOTs under the costly information
regime
Proposition 5 implies that the equilibrium under the costly information regime will be reduced
to that with no information if the information price is higher than a unique threshold. That is,
when the information price is higher than the threshold, travelers with high VOTs will choose
strategy Ar, travelers with low VOTs will choose strategy As, and no one will choose to purchase
and use information. Therefore, the equilibrium solution is the same as that under no information
regime. Particularly, if the toll mRis lower than the toll threshold ¯
mR, the threshold ¯
πfor
information price is higher than ˜
π, and vice versa. With the expression of ¯
πas described in
Proposition 5, the information price threshold ¯
πis increasing in θ. That is, no one will use
information when information is provided to a less risk-averse population at a price higher than
¯
π. However, if the information is provided to a population with a higher risk aversion degree at
the same price, some travelers in the population may use information.
Next, we will examine the equilibrium Cases 2, 3, 4, and 6 in Figure 3, which occur when the
information price πis lower than ¯
π, and the toll mRis lower than ¯
mR.
Proposition 6. Under Assumptions 1-3, there exists a unique equilibrium under the costly information
regime when mR<¯
mRand π<¯
π. And,
18
(a) when π∈(0, ˜
π)and mR∈(0, ¯
mR)(Case 2), there exist unique VOT thresholds βC
ri and βC
si such
that a traveler with β<βC
si selects strategy As, a traveler with β∈(βC
si,βC
ri )selects strategy Ai
and a traveler with β>βC
si selects strategy Ar;
(b) when π∈(˜
π,¯
π)and mR∈(0, ¯
mR)(Case 3), there exist unique VOT thresholds βC
ri =βC
si such
that a traveler with β<βC
ri selects strategy As, a traveler with β∈(βC
sr,βC
ri )selects strategy Ar,
and a traveler with β>βC
ri selects strategy Ai;
(c) when π=˜
πand mR∈(0, ¯
mR)(Case 4), there exist unique VOT thresholds βC
si =βC
sr such that a
traveler with β<βC
sr selects strategy As, and a traveler with β>βC
sr is indifferent to strategies Ar
and Ai;
(d) when π∈(˜
π,¯
π)and mR=0(Case 6), there exist unique VOT thresholds βC
si =βC
ri such that
a traveler with β<βC
si is indifferent to strategies Ar and As, and a traveler with β>βC
si selects
strategy Ai;
(e) nC
sand nC
rare increasing in π. nC
sis increasing in mR, and nC
ris independent of mR.
Proof. See Appendix J.
Proposition 6 describes the equilibrium patterns in Cases 2, 3, 4 and 6. Travelers are classified
into three groups, i.e., the group with low, intermediate, and high VOTs. Proposition 6(a) shows
that when the information price is low (Case 2), travelers with low (high) VOTs will choose
strategy As (Ar), and travelers with intermediate VOTs will purchase and use the information.
It is similar to the travel strategies chosen under the free information regime (see Figures 2 and
4(a).) On the contrary, Proposition 6(b) shows that in Case 3, travelers with high VOTs would
purchase information, while travelers with low (intermediate) VOTs would choose strategy As
(Ar). Also, Proposition 6(c) shows that in Case 4 where information price π=˜
π, travelers with
low VOTs would choose strategy As, but others would be indifferent to the travel strategies Ar
and Ai; Proposition 6(d) shows that in Case 6, where the toll mR=0, travelers with high VOTs
would choose strategy Ai, but others would be indifferent to the travel strategies Ar and As.
Note that the information price threshold ˜
πthat separates Case 2 and 3 is increasing in θ. That
is, given a fixed information price π∈(˜
π,¯
π), when the toll is low (mR<¯
mR)and travelers are
highly risk-averse, travelers with intermediate VOTs tend to switch from strategy Ar to Ai, and
travelers with high VOTs tend to switch from strategy Ai to Ar.
Next, we show that Case 1 in Figure 3 occurs when the information price is lower than the
threshold ¯
π, and the toll is not lower than ¯
mR.
Proposition 7. Under Assumptions 1-3, there exists a unique equilibrium solution under the costly
information regime when mR≥¯
mRand π<¯
π(case 1). That is, there exist unique thresholds βC
si and
βC
ri such that travelers with β<βC
si select strategy As, travelers with β∈(βC
si,βC
ri )select strategy Ai and
travelers with β>βC
si select strategy Ar. nC
sand nC
rare increasing in π. nC
sis increasing in mRand nC
r
is independent of mR.
Proof. By Proposition 5, ¯
π<˜
π. Similarly, the equilibrium is unique, and the proof is the same as
that for Case 2 in the proof of Proposition 6.
19
Proposition 7 describes the equilibrium Case 1, which occurs when the toll is high, and the
information price is lower than the threshold. In this case, travelers with low (high) VOTs would
always use the safe (risky) route, while travelers with intermediate VOTs would purchase the
information. As shown in Figure 2 and Figure 4(a), the travel strategies across VOT groups in
Cases 1-2 yield similar patterns as that under the free information regime.
Next, we benchmark the costly information regime against the no information regime to examine
individuals’ welfare change using CV defined in Definition 2. Similarly, according to travel
strategy choices under the no information regime and the costly information regime, as shown in
Figures 4(a)-4(e), travelers are grouped into SS, SI, SR, RI, RR.
Proposition 8. Under Assumptions 1-3,
(a) when 0<mR<¯
mRand π∈(˜
π,¯
π)(Case 3), (1) there are four groups SS, SR, RR and R I when
nC
s+nC
r>n0
s; (2) there are four groups SS, SR, SI and R I when nC
s+nC
r<n0
s. For Group SS,
CVC(β) = 0. For the other three groups, CVC(β)>0and CVC(β)is monotonically increasing
with respect to β;
(b) when 0<mR<¯
mRand π=˜
π(Case 4), there are five groups SS, SR, SI, RR, and RI. For
Group SS, CVC(β) = 0. For the other three groups, CVC(β)>0and CVC(β)is monotonically
increasing with respect to β;
(c) when π≥¯
π(Case 5), there are two groups SS and RR. And CVC(β) = 0,∀β∈[βl,βu];
(d) when mR=0and π∈(0, ¯
π)(Case 6), there are five groups SS, SR, S I, RR and R I. For Groups
SI and R I, CVC(β)>0and CVC(β)is monotonically increasing with respect to β. For the other
four groups, CVC(β) = 0.
Proof. See Appendix K.
Proposition 8 reveals that, in Cases 3-6, costly information will not reduce any traveler’s wel-
fare. Proposition 8 (a)-(b) shows that in Cases 3-4, when the toll is relatively low and the in-
formation price is moderate, except that the welfare of group SS remains unchanged, the other
three groups all benefit from costly information. It is worth noting that the benefit from costly in-
formation increases with an individual’s VOT, and the travelers in Groups SR and RR also benefit
even though they do not purchase and use information. Proposition 8(c) shows that when the
information price is sufficiently high (Case 5), nobody will be affected by the costly information
because the information is not used due to its high price. Proposition 8(d) shows that in Case 6,
when there is no toll and the information price is moderate, all groups, except Groups SI and R I,
remain unchanged. Groups SI and R I, which choose strategy Ar in the costly information regime,
benefit from costly information.
Proposition 9. Under Assumptions 1-3, when the information price π<min {˜
π,¯
π}, i.e., Cases 1 and
2, there are four groups SS, SI, RI, and RR.
(a) For Group SS, CVC(β) = 0.
(b) For Group SI, CVC(β)>0and CVC(β)is monotonically increasing with respect to β.
20
(c) When MS(t−(f−
C),tS)<TT BR(f0), for Groups RI and RR, CVC(β)>0and CV F(β)is mono-
tonically increasing with respect to β;
(d) When MS(t−(f−
C),tS)>TT BR(f0)>MS(t−(f−
C),t+(f+
C)), for Groups R I and RR, CVC(β)>
0. CVC(β)is monotonically decreasing with respect to βfor Group RI and is monotonically in-
creasing with respect to βfor Group RR.
(e) When MS(t−(f−
C),tS)>MS(t−(f−
C),t+(f+
C)) >TTBR(f0), for Groups RI and RR, CVC(β)
is monotonically decreasing with respect to β. CVC(β)<0for Group RR. CV C(β)>0for the
traveler with the VOT equal to β0
sr. CVC(β)<0for the traveler with the VOT equal to βC
ri .
Proof. See Appendix L
Proposition 9 shows that travelers with high VOTs may be worse off from costly information
in some cases, while the welfare of travelers with low VOTs remains unchanged or increase. It
is intuitive to see the welfare effects of Groups SS and SI. Property (a) shows that Group SS
would not be affected by the information because they always choose Route Swith deterministic
and constant travel time. The reason why Group SS breaks even from information provision
is that travel demand is fixed in this study. It is worthwhile to mention that the welfare of
travelers in Group SS will be affected if travel demand is elastic here. For example, if information
provision lowers the perceived travel costs for most users, then the overall demand will be raised.
Therefore, travelers with lower travel benefits may balk from making a trip and be hurt from
information provision.
Property (b) shows that Group SI would switch from strategy As to Ai because they must
benefit from costly information. Furthermore, the benefit increases with the VOT. However, it is
less straightforward to determine whether Groups RI and RR benefit or lose from costly informa-
tion. Specifically, it depends on three variables, including the TTB of Route Rwith no information,
the TTB of Route Rwith costly information, and the expected utility of travel time by utilization
information. Property (c) shows that when the TTB of Route Rwith no information is the highest
among the three, the CV is positive and increasing with the VOT for both Groups RI and RR.
Property (d) shows that when the TTB of Route Rwith no information is between the other two
variables, the CV is positive for both Groups RI and RR, but the monotonicity is different from
that in (c). For Group RI, the CV is decreasing in the VOT. For Group RR, the CV is increasing
in the VOT. Property (e) reveals that when the TTB of Route Rwith no information is the lowest
among the three, the CV is decreasing in the VOT for Groups RR and R I. The welfare of Group
RR is reduced. There exists a VOT threshold such that travelers with a relatively lower VOT in
Group RI would benefit from information, and travelers with a relatively higher VOT in Group
RI would incur a loss.
Combined with the result in Proposition 8 that costly information will not reduce any traveler ’s
welfare in Cases 3-6, we can immediately conclude that travelers with intermediate VOTs tend
to gain from costly information provision in all cases. This result is similar to that in Emmerink
et al. (1996), who consider a setting with one link and individuals who differ in their willingness
to pay (WTP) for a trip. They show that individuals with intermediate WTPs gain the most
from the information. There are two kinds of information benefits: internal benefits that accrue
to individuals who receive information and alter their travel decisions accordingly, and external
21
benefits that arise from the reactions of other individuals to information (Emmerink et al., 1996).
In Cases 1 and 2, by saying travelers with intermediate VOTs, we are referring to all travelers in
Group SI and some travelers in RI with VOTs lower than a threshold. The internal benefits of
Groups SI and RI must be positive because they prefer strategy Ai to either As or Ar under the
costly information regime. For Group S I, the expected travel time of strategy As is not affected by
information provision, which results in zero external benefits. Thus, Group SI always benefits
from information. For Group RI, the traffic flow of Route Ron good (bad) days is increased
(decreased) by information provision. Thus, the expected travel time of strategy Ar could be
increased or decreased by information provision. The external benefit or cost depends on the
VOT and is proportional to the gain or loss in the expected travel time. If a traveler’s VOT in
Group RI is relatively low, then her possible external cost would not exceed the internal benefit.
Thus, some travelers in Group RI with relatively low VOTs would benefit from the information
even though they could incur a higher travel time cost.
5 Numerical experiment
We conduct numerical experiments in this section to illustrate our results. We also further discuss
the design of toll and information price here. The following BPR functions describe the travel
time function on Route R:(t−(f) = t0(1+ ( f/c1)b),
t+(f) = t0(1+ ( f/c2)b),(8)
where t0is the free-flow travel time on Route R, c1,c2are capacities on good days and bad days,
respectively. For the base case, the risk aversion parameter is θ=0.2, the probability of good
days is p=0.8, and the travel demand is d=10000. Other parameters values are tS=25
minutes, t0=20 minutes, c1=10000 vehicles per hour, c2=3000 vehicles per hour and b=2.
βfollows a truncated log-normal probability distribution between 0 and 50 ($/hour) with mean
$21/hour and the standard deviation $10.5/hour. The probability density function of the VOT
distribution is shown in Figure 5. The parameters of the VOT distribution are adopted from a
study of commuters on State Route 91 located in Orange County, CA (Lam and Small, 2001).
22
0 5 10 15 20 25 30 35 40 45 50
0
0.01
0.02
0.03
0.04
0.05
0.06
Figure 5: Comparison between the log-normal distribution used in the base-case and the piece-
wise linear distribution with which some travelers are worse off
5.1 Base-case results
Consider two operators: a road operator and an information provider who make independent
decisions and do not cooperate. The road operator levies a toll on the risky and congested route.
The information provider sets the information price. Given the information price and toll, Figure
6 illustrates how the congestion (i.e., the expected system travel time) varies with the information
price πand the toll mR. The six equilibrium Cases 1-6 are demonstrated. The toll threshold ¯
mR
is $1.62. Note that when the toll mR=0, information provision is separately applied without
congestion pricing. When the information price is sufficiently high (Case 5), i.e., π>¯
π, nobody
will purchase information, and hence it is reduced to the equilibrium under the no information
regime.
We identify four Scenarios A-Dwith the separate implementation of information or conges-
tion pricing and another four typical Scenarios E-Hwith the joint implementation, as specified
in Table 3. Gubins et al. (2012) study the effect of the ownership regimes of the information
provider and the road operator on system performance. They model the effect of joint imple-
mentation on the elastic travel demand and assume travel time is not flow-dependent. We study
the problem in a congested network and consider heterogeneous risk-averse travelers in terms
of the VOT. We compare different pricing strategies and the system performance under different
scenarios with public or private road operators and information providers, as shown in Figure 7
and Table 3. The competition between the road operator and the information provider admits a
Bertrand equilibrium, in which each firm assumes that the price of service offered by the other
firm is fixed. Each of them chooses an optimal price in the competition simultaneously. We do
not consider cooperation here.
In scenarios A,E, and G, the public road operator levies a road toll that would minimize the
system congestion. Given an information price π∗, the public road operator will set the toll equal
23
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
0.5
1
1.5
2
2.5
2.42
2.43
2.44
2.45
2.46
2.47
2.48
2.49
105
Case 6
,QIRUPDWLRQSULFHѝ
7ROOP5
Figure 6: The expected system travel time given different combinations of information prices and
tolls
Table 3: Scenarios A-Hregarding the roles of the information provider and the road operator
Scenarios Information provider Road operator
ANo information Congestion-minimizing
BNo information Revenue-maximizing
CCongestion-minimizing No toll
DRevenue-maximizing No toll
ECongestion-minimizing Congestion-minimizing
FCongestion-minimizing Revenue-maximizing
GRevenue-maximizing Congestion-minimizing
HRevenue-maximizing Revenue-maximizing
to the solution to the mathematical program:
minimize
mR
p(t−(f−)f−+tS(d−f−)) + (1−p)(t+(f+)f−+tS(d−f+))
subject to (f−,f+)∈Eq(mR,π∗)
where Eq(mR,π∗)represents the set of equilibrium link flows given the information price π∗and
the toll mR.
In scenarios B,F, and H, the private road operator levies a road toll that would maximize the
toll revenue. Given an information price π∗, a private road operator will set the toll equal to the
solution to the mathematical program:
maximize
mR
mR(p f −+ (1−p)f+)
subject to (f−,f+)∈Eq(mR,π∗)
24
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%
&
'
+
(
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Figure 7: Optimal information prices and optimal tolls with a private or a public operator
The solid blue line in Figure 7 presents the optimal toll to minimize the expected system travel
time from the perspective of a public operator given different information prices π. The dotted
blue line in Figure 7 presents the optimal toll to maximize the toll revenue from the perspective
of a private road operator given different information prices π. We found that the private road
operator always set a higher (optimal) toll than the public road operator. When the information
price πis lower than a threshold, the optimal tolls for the public and private road operators are
both monotone decreasing in π, and the number of travelers purchasing information is positive;
otherwise, the optimal tolls for the public and private road operators are constant at which no
one purchases information under the costly information regime. Kinks can be found on these lines
where the lines intersect with case boundaries. The reason for these kinks is that the equilibrium
patterns in two adjacent cases are different. Thus, there could be more than one local optimum in
the optimization problem of minimizing system congestion or maximizing revenue. Therefore,
these optimal information prices or tolls are piecewise functions of tolls or information prices.
In scenarios C,E, and F, the public information provider set the information price to a value
that would minimize the system congestion. Given a toll m∗
R, the public information provider
will set the information price equal to the solution to the mathematical program:
minimize
πp(t−(f−)f−+tS(d−f−)) + (1−p)(t+(f+)f−+tS(d−f+))
subject to (f−,f+)∈Eq(m∗
R,π)
In scenarios D,Gand H, the private information providers set the information price to a
value that maximizes the information revenue. Given a fixed toll m∗
R, the private information
provider will set the information price equal to the solution to the mathematical program:
maximize
ππ(f−+f+)
subject to (f−,f+)∈Eq(m∗
R,π)
25
The solid red line in Figure 7 presents the optimal information prices to minimize the ex-
pected system travel time from the perspective of the public information provider. The dotted
red line in Figure 7 presents the optimal information prices to maximize the revenue from the
perspective of the private information provider. We found that when the toll is higher than $0.2,
the optimal information price, which maximizes the information provider’s revenue, is always
greater than the optimal price that minimizes the congestion. When the toll is less than $0.2, the
optimal information price, which minimizes congestion, would be higher than the optimal price
that maximizes the information revenue.
This result is counter-intuitive because revenue-maximizing or profit-maximizing prices are
usually higher than welfare-maximizing prices. A policymaker in the public sector usually sets
the price equal to the social marginal cost to maximize social welfare. For example, Gubins et al.
(2012) study the joint implementation of road pricing and traffic information in a road network
of multiple parallel routes with elastic demand. In their setting, travel times do not depend on
the flow, and therefore there are no externality effects in the network. That is, the social marginal
cost of information provision is zero. They show that when the toll is zero, the information
price should be zero to maximize social welfare and be positive to maximize the profit of the
information provider.
Compared to the model in Gubins et al. (2012), our model considers the congestion effect
here, and the information provider aims to minimize the expected system travel time instead
of maximizing the social welfare. Now we explain why the information price to minimize the
system travel time is high when mRis low in our model. First, the information price should be
set equal to the social marginal cost if a welfare-maximizing information provider is considered.
As observed in Figure 6, the expected system travel time is unimodal in πwhen mRis small
enough. And the minimal expected system travel time Ttotal is achieved at an intermediate
value of π. Thus, the social marginal cost of information provision ∂Ttotal
∂nc
iis positive when πis
zero. Apparently, the information price does not equal the marginal social cost when the price
equals zero. Thus, in our setting, we can conclude that the information price should be set to
a positive value to maximize social welfare. Second, we show that the congestion-minimizing
price is higher than the welfare-maximizing price. By minimizing the system travel time, we do
not consider the factor of social benefits (total willingness to pay) here, which is normally used
in calculating social welfare. Thus, we choose an information price minimizing the social cost
(system travel time) regardless of the social benefits. The welfare-maximizing price is lower than
the chosen price because a low price would normally increase the social benefits.
We start with situations where information provision or congestion pricing is implemented
separately. First, Scenario Arepresents the optimal toll set by the public operator, which mini-
mizes the congestion under the no information regime. The expected system travel time and toll
revenue at different values of tolls when congestion pricing is separately implemented without
information provision, are shown in Figure 8(a). The minimal expected system travel time is
244465 minutes by setting the optimal toll mR=$1.39 dollars. Scenario Brepresents the op-
timal toll, mR=$1.87, set by a private road operator, which maximizes the toll revenue (i.e.,
$2478.21). Second, we consider that information is provided without congestion pricing. The
expected system travel time and toll revenue at different values of information price when infor-
mation provision is separately implemented without congestion pricing are shown in Figure 8(b).
26
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Figure 8: The separate implementation of information provision or congestion pricing
Scenario Cpresents the optimal information price π=$0.89 of a public information provider so
that minimal congestion is obtained (i.e., 245837). Scenario Dpresents the optimal information
price π=$0.79 of the private information provider, which can receive the maximum information
revenue (i.e., $1192).
Next, we consider the situation where both information provision and congestion pricing are
implemented. In Scenario E, both the information provider and the road operator are from the
public sector, and therefore they both aim to mitigate congestions. In this scenario, the global
minimal congestion is achieved at mR=$1.35 and π=0. Scenario Frepresents the equilibrium
decisions of the public information provider who aims to minimize congestion and the private
road operator who aims to maximize toll revenue. In this scenario, mR=1.63 and π=0.
Scenario Grepresents the equilibrium decisions of the public road operator who aims to reduce
congestions, while the private information provider aims to maximize information revenue. And,
mR=0.9 and π=0.44. In Scenario H, both the private road operator and the private information
provider aim to maximize their own revenues. Their equilibrium decisions are mR=1.51 and
π=0.24.
Summary statistics for the equilibria in the aforementioned scenarios with the base-case pa-
rameters are listed in columns of Table 4.
Table 4: Flow patterns and system performance of information provision and congestion pricing
Scenarios Initial A B C D E F G H
mR0 1.39 1.87 0 0 1.35 1.63 0.9 1.51
π- - - 0.89 0.79 0 0 0.44 0.24
nC
s7446 8340 8675 6416 6283 7139 7573 7295 7913
nC
r2554 1660 1325 2249 2199 1064 965 1677 1298
nC
i0 0 0 1335 1519 1797 1462 1028 788
t+(f+
C)34.49 26.13 23.90 31.24 30.74 22.51 22.07 26.25 23.75
t−(f−
C)21.30 20.55 20.35 22.57 22.76 21.64 21.18 21.46 20.87
TT BR25.00 22.11 21.35 25.00 25.00 21.88 21.43 22.80 21.68
Expected
system time 247299 244465 244781 245837 245874 241774 242014 242766 242782
Toll revenue 0 2304.67 2478.21 0 0 3371.74 3472.26 2249.21 2918.54
Information
revenue 0 0 0 1191.79 1204.21 0 0 455.18 191.55
27
Figures 9(a)-9(f) reveal the distributional welfare effects of information provision across the
travelers with different VOTs in Scenarios A-Hby comparing with the welfare before and after
the information provision. No traveler is worse off by information provision in both scenarios.
Figure 9(e) shows that, as indicated by Propositions 8 and 9, in Scenario G(Case 3), Group SS is
not affected by the information. The other three groups benefit from the information provision,
and the benefit is monotonically increasing with respect to β. Figures 9(a) and 9(b) show that,
in Scenarios C,D(Case 6), Groups SI and R I benefit from the information provision and the
benefit is monotonically increasing with respect to β. The other four groups are not affected by
the information. Figures 9(c), 9(d), and 9(e) show that, in Scenarios E,G(Case 2), and F(Case 1),
it still holds that no one is worse off by information. However, the benefit is not monotonically
increasing with respect to βamong the travelers who benefit from information. We will also
show that when the VOTs conform to some particular distributions, some travelers could incur a
loss from information provision in the following context.
RI RU SI RR RU SS or SR or RS RU RU
(a) Scenario C
RU RU RI RU SI RR RU SS or SR or RS
(b) Scenario D
RU RU RU RU
RR RI SI SS
(c) Scenario E
RU RU RU RU
RR RI SI SS
(d) Scenario F
RU RU RU RU
RI RR SR SS
(e) Scenario G
RU RU RU RU
RR RI SI SS
(f) Scenario H
Figure 9: Individual welfare with and without information in Scenarios C-H
Figures 10(a) and 10(b) reveal the distributional welfare effects of the two instruments (infor-
mation provision and congestion pricing) across the travelers with different VOTs in Scenarios
A−Hby comparing the welfare before and after the implementation. For travelers with high
VOTs, their TCURs are reduced by policies in all referred scenarios. With a higher rank in the
population, the VOT is lower according to our definition of VOT distribution in Section 2. As
shown in Figures 10(a) and 10(b), the welfare of individual travelers with low VOTs (i.e., high
ranks k) is not affected by the two instruments in all scenarios. Generally, the welfare gains in-
crease with the VOT in all scenarios. The travelers with intermediate or high VOTs are all better
28
off from the implementation of congestion pricing or/and information provision.
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Figure 10: The distributional welfare effect across heterogeneous VOTs
5.2 Social welfare
In the previous section, we evaluate the system performance with the expected system travel
time. Recall that we measure an individual’s welfare change with CV. CV is determined by
the expected travel cost, VOT, risk aversion degree, and travel cost variability. In comparison,
we exclude the out-of-pocket costs comprising toll and information costs in the calculation of
social welfare, assuming that toll revenues and the information cost remain internal to society by
transfer payments (Sharon et al., 2017). Thus, we include only the cost associated with the travel
time when evaluating the social welfare for our instruments.
Now, we examine the optimal social welfare achieved by our instruments and consider the
impact of the value of time distribution and travel time variability. First, we evaluate the social
welfare with the expected system travel time cost (ESTTC), which is defined as the weighted
sum of each traveler’s travel time according to the value of time. If T(β)is the stochastic travel
time for a traveler with a VOT of β, and ET(β)is the expected travel time for a traveler with a
VOT of β, then the expected system travel time cost is defined as dRβu
βlβET(β)dF(β). Second,
we also calculate the aggregate travel time budget cost (ATTBC), which considers the travel time
variability in addition to the VOT distribution. If ¯
t(β) = ET(β) + θ
pVar(T(β)) is the travel time
budget, then ATTBC is defined as dRβu
βlβ¯
t(β)dF(β).
29
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Information price
0.5
1
1.5
2
2.5
Toll mR
ATTBC-minimizing information prices
ESTTC-minimizing information prices
Congestion-minimizing information prices
(a) The optimal information prices for three metrics when
the toll is fixed
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Information price
0.5
1
1.5
2
2.5
Toll mR
ATTBC-minimizing tolls
ESTTC-minimizing tolls
Congestion-minimizing tolls
(b) The optimal tolls for three metrics when the informa-
tion price is fixed
Figure 11: Comparison between the optimal information prices or tolls when the expected system
travel time, the expected system travel time cost, and the aggregate travel time budget cost are
used as metrics, respectively
Figure 11(a) shows that the optimal information price minimizing the ESTTC is not lower than
the congestion-minimizing information price. This result is intuitive in Case 3 where travelers
with high VOTs use information. Note that travelers following strategy Ai choose route Ron
good days and Son bad days. With a higher information price, fewer travelers use Route R
on good days, and therefore the travel time on Route Ris reduced. Since travel time on Route
Sis constant, the expected travel time of strategy Ai becomes lower as well. Since travelers
following strategy Ai are subject to higher VOTs, a reduction in their travel time tends to reduce
the ESTTC even when travelers with lower VOTs may incur higher travel times. Figure 11(a) also
shows that the congestion-minimizing information price is higher than the ATTBC-minimizing
information price when mRis lower than 0.5. This is also intuitive because a higher information
price would decrease the number of travelers using information, which would eventually increase
the variability of travel time on Route R.
Figure 11(b) shows that the ESTTC-minimizing toll is higher than the congestion-minimizing
toll. It is a natural result since a traveler on Route Rhas a higher VOT compared to a traveler on
Route S, and the usage of Route Ris decreasing in mR. Increasing the toll will make low-VOT
travelers worse off and high VOT travelers better off. Figure 11(b) also shows that ATTBC-
minimizing toll is higher than the ESTTC-minimizing toll. Since only Route Ris subject to travel
time variability, it is reasonable to increase the ESTTC-minimizing toll to toll off more travelers
from route Rand reduce the travel time variability of Route R.
5.3 Sensitivity analysis
The sensitivity analysis is conducted by varying θand pand holding fixed other parameters
at their base-case values. Figure 12(a) shows that the reduction of expected system travel time
30
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(b) Congestion reduction varies with the probability p
of good days
Figure 12: Sensitivity analysis of θand pregarding travel-time minimization
by the joint implementation (Scenario E) is always higher than that by separate implementation
regardless of the value of θ. When θ<0.4, the reduction by separately implementing congestion
pricing is higher than separately implementing information provision. When θ>1.2, separately
implementing congestion pricing would not reduce the expected system travel time. The possible
failure of the separate implementation of congestion pricing happens because the risk aversion
of travelers serves as the role of externality in route choice, which eventually benefits the system.
When θis small, travelers’ risk-averse behavior has already offset some congestion externality
on Route R. However, this offset is not enough to drive the equilibrium to system optimum, i.e.,
the traffic flow of Route Ris still larger than its system optimal flow. Thus, a small amount of
toll on Route Rcould toll off some travelers from this route, which eventually drives the flow
distribution towards the system optimal flow. In contrast, information can reduce the congestion
externality of Route Ron bad days, but it will also increase the congestion externality of Route
Ron good days. Thus, the congestion reduction by information provision without congestion
pricing is not as effective as that by congestion pricing without information. When θis high,
travelers tend to abandon Route Runder the no information regime, which results in a higher
marginal congestion cost on Route Scompared to Route R. Under this circumstance, information
provision can reduce uncertainty and increase the usage of Route R, which eventually reduces
congestion. Meanwhile, when θis sufficiently high, the impact of risk aversion is larger than
the necessary externality. Therefore, congestion pricing without information could not help to
improve the system performance because a positive toll mRon Route Rwill further increase the
externality perceived by travelers. Meanwhile, system performance would be improved if it were
possible to impose a toll on Route S.
Figure 12(b) shows that the reduction of expected system travel time by the joint implementa-
tion (Scenario H) is always higher than that by separate implementation regardless of the value
of p. Meanwhile, the reduction by separately implementing congestion pricing is higher than
separately implementing information provision. The reduction by implementing information
provision separately reaches a maximum for some p∈[0.5, 0.7]. In the limits as p→0 (p→1),
under the no information regime, Route R’s average travel time is close to the travel time on bad
(good) days, respectively, and the standard deviation of travel time is close to zero. The equi-
31
librium flow on Route Ron bad (good) days is close to that under the no information regime,
and information provision can not decrease (increase) the usage of Route R(S) significantly on
bad (good) days. Information provision would have a significant impact on the flow distribution
when it is a good (bad) day, which only occurs with a small probability. Thus, the reduction in
the expected system travel time flow from information provision would be negligible.
Figure 13 shows that some travelers can be worse off from the information when we consider
a special VOT distribution. Here we assume that the VOTs in the population follow the piecewise
linear distribution as illustrated in Figure 5 instead of the log-normal distribution in the base case.
In particular, the PDF of the piecewise linear distribution is as follows
f(β) =
2.93 ×10−2, if β∈[0, 27.67);
1.87 ×10−2, if β∈[27.67, 31.15);
1.17 ×10−3, if β∈[31.15, 46.53);
3.09 ×10−2, if β∈[46.53, 50].
The other parameters are consistent with the base case. Figure 13 displays individuals’ welfare
change when the toll mRis $2 and the information is free. We carefully design the piecewise linear
distribution as follows. First, with the piecewise linear distribution, the number of travelers with
VOTs lower than the threshold β0
sr is the same as that with the log-normal distribution. Therefore,
the flow of Route Rwithout information is the same as that with the log-normal distribution.
Second, the piecewise linear distribution preserves the same number of travelers with VOTs
lower than the threshold βC
si as that with the log-normal distribution. That is, the flow of route R
on good days with information is the same as that with the log-normal distribution. Third, the
piecewise linear distribution features a larger portion of travelers with high VOTs and a smaller
portion of travelers with intermediate VOTs. In contrast, the log-normal distribution in the base-
case features a smaller portion of travelers with high VOTs and a larger portion of travelers with
intermediate VOTs. According to Proposition 2, travelers with intermediate VOTs choose Ai with
information, and travelers with higher VOTs prefer strategy Ar to save travel time cost. With the
piecewise linear distribution, the number of travelers from Group SI is much higher than that
from Group RI. That is, on good days, many travelers from Group SI switch from Route Sto
Route Rafter information provision and increase the travel time of route R. Meanwhile, on bad
days, only a few travelers from Group RI switch from Route Rto Route Sand the reduction
of travel time on route Ris negligible. According to Assumption 3, increasing the travel cost of
Route Ron good days while holding the travel cost on bad days fixed would result in a higher
TCUR of Route R. Thus, with the tailored piecewise linear distribution, travelers in Group RR
are made worse off by information provision.
6 Conclusions
This paper examines the impact of joint implementation of travel information provision and
congestion pricing on route choices, welfare effects, and system congestion. The heterogeneity
of travelers’ VOTs facilitates better evaluating the two instruments not only in terms of efficiency
but also in terms of fairness. In our model, risk-averse travelers choose between a safe route
32
RU RU RU RU
RR RI SI SS
Figure 13: The individual welfare change with the piecewise linear distribution when some
travelers are worse off from the information provision
and a risky route. Three information regimes, including no information,free information, and costly
information, are examined.
Our results demonstrate that the travelers who are willing to use information to choose their
routes could vary accordingly with the information price and toll. We derive six equilibrium
cases under the costly information regime where information can be received at a price. We fur-
ther investigate the congestion at different information prices and tolls. We prove that providing
free information without toll would surprisingly increase the congestion. Under the costly infor-
mation regime, congestion may be reduced by properly designing the toll and the information
price. Note that the expected system travel time may increase even with a joint implementation
of information and congestion pricing if the toll and the information price are not cautiously
designed. We further investigate the distributional welfare effects of the information provision
across travelers with different VOTs. We find that travelers with low VOTs who always use
the safe route will not be affected by both free information and costly information, while travelers
with intermediate VOTs will always benefit from information. However, we find that travelers
with high VOTs that choose the risky route under the no information regime are the potential
losers from the information provision. Through extensive numerical experiments, we show that
the joint implementation of information provision and congestion pricing can achieve a higher
reduction in terms of congestion compared to the separate implementation of information or
congestion pricing. When travelers’ risk-aversion degree is low, congestion pricing without in-
formation will reduce congestion more effectively compared to information provision without
congestion pricing.
There are some limitations of the proposed model. First of all, we assume that the current
toll on Route Rdoes not change with the state of nature. It would be interesting to extend
33
the model to include state-dependent tolls, but it introduces complications in deriving equilibria
cases because of the additional parameters. Second, we use a simple network to analyze the dis-
tributional effects of travelers’ routing strategies. Although this network is analytically tractable,
the results may not always hold for general networks. One can also expand the current two-route
network to complex networks with multiple links and states and examine the distributional and
welfare effects. One can consider multiple states of nature to make the model more practicable
(Lindsey et al., 2014). Also, the “all or nothing” assumption under which travelers choose be-
tween using perfect information and not using any information can be relaxed by considering
rational inattentive travelers (Jiang et al., 2020). Furthermore, the current model focuses on the
impacts of pre-trip information, which can be expanded to study en-route information provision
by assuming sequential route choice.
Acknowledgment
The work was supported by the Singapore Ministry of Education Academic Research Fund Tier
2 (MOE2017-T2-2-128).
A Property of TCUR under Assumption 3
When c−
g<c+
g, we have cg(β) = pc−
g+ (1−p)c+
g+θ
pp(1−p)(c+
g−c−
g). According to the
statewise stochastic dominance property, the following conditions should be met.
First, cg(β)should decrease with p, i.e.,
∂cg(β)
∂p=c−
g−c+
g+θ(c+
g−c−
g)1−2p
2
pp(1−p)
<0.
Hence, ∂cg(β)
∂p>0 holds when 1
2(1−1
√1+θ2)<p.
Second, cg(β)should increase with c+
g, i.e.,
∂cg(β)
∂c+
g
=1−p+θ
qp(1−p)>0.
Hence ∂cg(β)
∂c+
g
>0 holds for all p∈(0, 1)and θ>0.
Third, cg(β)should increase with c−
g, i.e.,
∂cg(β)
∂c−
g
=p−θ
qp(1−p)>0.
Hence, ∂cg(β)
∂c−
g
>0 holds when θ2/(1+θ2)<p.
Note that θ2/(1+θ2)>1
2(1−1
√1+θ2)always holds. Hence, when c−
g<c+
g,cg(β)would
decrease with p, and increase with c−
gor c+
gprovided that θ2/(1+θ2)<p.
34
B Proof of Lemma 1
(a) Under the no information regime, f−=f+=f0⇒t−(f−)<t+(f+).
Under the free information and costly information regime, we proceed by contradiction.
Assume that t−(f−)≥t+(f+). Then, traffic flows on Route Rmust satisfy
f+=nr+niand f−=nr.
Since nr≥0, we have
f+≥f−⇔t−(f−)<t+(f+),
which is a contradiction.
(b) Once again, we proceed by contradiction.
Assume that tS≤MS(t−(f−),t+(f+)). Since mR>0, we have
cAs(β)<cAr (β)⇔Ar ≻As,∀β.
Thus, nr=0, f+=0 and t+(f+) = t+(0).
According to Assumption 1, we have t+(0)<tS.
According to Assumption 3, we have MS(t−(f−),t+(f+)) <t+(f+). According to Part (a), we
have t−(f−)<t+(f+). Therefore, we have
MS(t−(f−),t+(f+)) <t+(f+) = t+(0)<tS,
which contradicts.
C Proof of Proposition 1
(a) mR=0.
When mR=0, the TCURs of Ar and As are TTBR(f)and tS, respectively. In equilibrium,
if TTBR(f0)>tS,f0=0 and therefore TTBR(0)<tS, which contradicts by Assumption 1.
If TT BR(f0)<tS,f0=dand therefore TTBR(d)>tS, which contradicts by Assumption 1.
Thus, TTBR(f0) = tSmust hold in equilibrium.
Define Ψ(f) = TT BR(f)−tS. Since Ψ(f)is a continuous and strictly increasing function
of fby Assumption 1, we have Ψ(0) = TT BR(0)−tS<0 and Ψ(d) = TTBR(d)−tS>0.
Therefore, there exists a unique f0∈(0, d)such that
Ψ(f0) = TTBR(f0)−tS=0.
By totally differentiating the above equation, one obtains:
∂f
∂θ =−∂TTBR(f)
∂θ /∂TTBR(f)
∂f<0.
∂f
∂p=−∂TT BR(f)
∂p/∂TT BR(f)
∂f>0.
Therefore, f0is monotonically decreasing with respect to θ, and it is monotonically increas-
ing with respect to p.
35
(b) mR>0.
Note that TTBR(f)<tSmust hold in equilibrium by Lemma 1. Consider f0
0such that
TT BR(f0
0) = tS. Since TTBR(f)is monotone increasing in f, we have TTBR(f)<tSif and
only if f<f0
0.
Define ¯
Ψ(β,f) = mR+β(TTBR(f)−tS). Hence when f<f0
0,¯
Ψ(β,f)decreases with βand
there always exists some βsr =mR/(tS−TT BR(f)) such that ¯
Ψ(β,f) = 0. Thus, given the
flow f, if β=βsr , we have ¯
Ψ(β,f) = 0 and Ar ∼As; if β>βsr, we have ¯
Ψ(β,f)<0 and
Ar ≻As; if β<βsr, we have ¯
Ψ(β,f)>0 and Ar ≺As.
Therefore, the number of travelers preferring Ar is
nr=d−d·F(βsr).
Because the CDF F(β)is continuous and strictly increasing for β∈[βl,βu], Equation 5a
defines a decreasing relationship between βand f, with f(βl) = dand f(βu) = 0.
Because TT BR(f)is continuous and strictly increasing for f, Equation 5b defines an in-
creasing relationship between βand fwhen f≤f0
0, with
β(0) = mR
tS−TT BR(0)<βuand lim
fրf0
0
β(f) = +∞.
The two curves therefore cross exactly once, which defines β0
sr and f0<f0
0.
(c) According to Lemma 1, we have tS>TT BR(f0). According to Assumption 1, we have
∂TTBR(f0)
∂f0>0. Because the CDF F(β)is continuous and strictly increasing for β∈[βl,βu],
we have ∂f0
∂β0
rs
=−∂F(βrs )
∂β0
rs
<0.
Because of Assumption 3, we have
∂TT BR(f0)
∂θ >0 and ∂TTBR(f0)
∂p<0.
By totally differentiating the above equation system (5a)-(5b), one obtains:
dmR+dβ0
rs (TTBR(f0) + β0
rs
∂TT BR(f0)
∂f0
∂f0
∂β0
rs −tS) + dθ∂TTBR(f0)
∂θ +d p ∂TTBR(f0)
∂p=0
Set dθ=dp =0 and divide both sides by dmR, and we have
∂β0
rs
∂mR
=1
tS−TT BR(f0)−β0
rs
∂TTBR(f0)
∂f0
∂f0
∂β0
rs
>0.
Set dmR=dp =0 and divide both sides by dθ, and we have
∂β0
rs
∂θ =β0
rs
TTBR(f0)
θ
tS−TT BR(f0)−∂TTBR(f0)
∂f0
∂f0
∂β0
rs
>0,
36
Set dθ=dmR=0 and divide both sides by dp, and we have
∂β0
rs
∂p=β0
rs
TTBR(f0)
p
tS−TT BR(f0)−∂TTBR(f0)
∂f0
∂f0
∂β0
rs
<0.
This completes the proof.
D Proof of Proposition 2
(a) mR=0.
Define Ψ1(f) = t−(f)−tSand Ψ2(f) = t+(f)−tS. By Assumption 1, Ψ1(f)and Ψ2(f)
are both continuous and strictly increasing functions of f. According to Assumption 3, we
have
Ψ1(f0) = t−(f0)−tS<TTBR(f0)−tS=0,
Ψ1(d) = t−(f)(d)−tS>0,
Ψ2(0) = t+(f)−tS<0,
Ψ2(f0) = t+(f)( f0)−tS>TTBR(f0)−tS=0.
Hence, there exist a unique f−
F∈(f0,d)such that
Ψ1(f0) = t−(f−
F)−tS=0
and a unique f+
F∈(0, f0)such that
Ψ2(f0) = t+(f+
F)−tS=0
Thus, it is immediate to conclude that f−
Fand f+
Fare independent of θand p.
(b) mR>0.
Let f−
F0and f+
F0denote the flow of Route Ron good days and bad days when mR=0,
respectively. On good days, according to Assumption 1, given any f<f−
F0, it is immediate
to conclude that there exists a VOT threshold
βsi =mR
tS−t−(f)
such that a traveler with β>βsi would prefer Route Rand the number of travelers prefer-
ring Route Ris
f−=d−d·F(βsi).
On bad days, given any f<f+
F0, it is immediate to conclude that there exists a VOT
threshold
βri =mR
tS−t+(f)
such that a traveler with β>βri would prefer Route Rand the number of travelers prefer-
ring Route Ris
f−=d−d·F(βri ).
37
Because the CDF F(β)is continuous and strictly increasing for β∈[βl,βu], Equation (6a)
defines a decreasing relationship between βand f, with f(βl) = dand f(βu) = 0.
Because t−(f)is continuous and strictly increasing for f, Equation (6b)defines an increasing
relationship between βand fwhen f≤f−
F0, with
lim
fրf−
F0
βsi(f) = +∞and βsi (f0) = mR
tS−t−(f0)<mR
tS−TT BR(f0)=F−1(1−f0
d).
Therefore, the two curves in 6(a) and 6(b) therefore cross exactly once, which defines βF
si
and f−
F∈(f0,f−
F0).
Similar to Equation (6a), Equation (6c) also defines a decreasing relationship between βand
f, with f(βl) = dand f(βu) = 0. Because t+(f)is continuous and strictly increasing for f,
Equation (6d) defines an increasing relationship between βand fwhen f≤f+
F0, with
βri (0)<βuand βri (f0) = mR
tS−t+(f0)>mR
tS−TT BR(f0)=F−1(1−f0
d).
Therefore, the two curves in (6c) and (6d) also cross exactly once, which defines βF
ri and
f+
F<f0.
(c) βF
si and βF
ri are independent of θand paccording to the above equation system. By totally
differentiating the above equation system, one obtains:
∂βF
si
∂mR
=1
tS−t−(f−
F)−βF
si
∂t−(f−
F)
∂f−
F
∂f−
F
∂βF
si
>0,
∂βF
ri
∂mR
=1
tS−t+(f+
F)−β0
ri
∂t+(f+
F)
∂f+
F
∂f+
F
∂βF
ri
>0.
This completes the proof.
E Proof of Proposition 4
Part (a). mR=0.
Under the no information regime, we have c0(β) = βtS,∀β.
Under the free information regime, we have t+(f+
F) = t−(f−
F) = tS. Therefore,
cF(β) = MS(βtS,βtS) = βtS,∀β.
Hence we have
CVF(β) = c0(β)−cF(β) = 0, ∀β.
Part (b). mR>0.
1. For Group SS (k∈(f−
F,d)), CVF(β) = βtS−βtS=0;
38
2. For Group S I (k∈(f0,f−
F)), we have
Ai ≻As ⇔βtS>MS(mR+βt−(f−
F),βtS).
Therefore, we have
CVF(β) = βtS−MS(mR+βt−(f−
F),βtS)>0.
According to Proposition 2, tS>t−(f−
F). With Assumption 3, tS>MS(t−(f−
F),tS). There-
fore,
CVF(β) = β(tS−TTBF
i)−(p−θ
qp(1−p))mR
is monotonically increasing with respect to β;
3. For Group R I (k∈(f+
F,f0)), we have
CVF(β) = mR+βTTB0
R−MS(mR+βt−(f−
F),βtS)
=β(TT B0
R−TT BF
i) + [1−p+θ
qp(1−p)]mR
Therefore, we have
∂CVF(β)
∂β =TTB0
R−TT BF
i,
of which the sign is determined by the relative magnitude of TTB0
Rand TT BF
i. For β=β0
rs,
Ar ∼As under the no information regime and Ai ∼As under the free information regime.
Thus,
β0
rs tS=β0
rs TTB0
R+mR
and
β0
rs tS>MS(mR+β0
rs t−(f−
F),β0
rs tS)
must hold. Hence, we have
CVF(β0
rs ) = β0
rstS−MS(mR+β0
rs t−(f−
F),β0
rs tS)>0.
For β=βF
ri,Ar ∼Ai under the free information regime. Thus,
MS(mR+β0
rs t−(f−
F),β0
rstS) = mR+β0
rs TTBF
R
must hold. Hence, we have
CVF(βF
ri ) = βF
ri (TTB0
R−TT BF
R),
of which the sign is determined by the relative magnitude of TTB0
Rand TT BF
R.
4. For Group RR (k∈(0, f+
F)), we have
CVF(β) = βTTB0
R−βTT BF
R.
Therefore, we have
∂CVF(β)
∂β =TTB0
R−TT BF
R.
Hence, signs of CV F(β)and ∂CVF(β)/∂β are both determined by the relative magnitude of TTB0
R
and TT BF
R.
39
F Proof of Lemma 2
The preference ranking between Ar and As is described by the condition:
Ar RAs ⇔βMS(t−(nr+ni),t+(nr)) + mR⋚βtS.
If tS>MS(t−(nr+ni),t+(nr)),Ar ≻Ai ⇔β>βsr =mR
tS−MS(t−(nr+ni),t+(nr) ) .
If tS≤MS(t−(nr+ni),t+(nr)),Ar ≻Ai ⇔βsr >∞.
G Proof of Lemma 3
The preference ranking between As and Ai is described by the condition:
Ai ≻As ⇔βtS>βMS(t−(nr+ni),tS) + (p−θ
qp(1−p))mR+π.
If tS>MS(t−(nr+ni),tS),Ai ≻As ⇔β>βsi =(p−θ
√p(1−p))mR+π
tS−MS(t−(nr+ni),tS).
If tS≤MS(t−(nr+ni),tS),Ai ≺As,∀β.
H Proof of Lemma 4
The preference ranking between Ar and Ai is described by the condition:
Ar ∼Ai ⇔
βri MS(t−(nr+ni),t+(nr)) + mR<βr i MS(t−(nr+ni),tS) + ( p−θ
qp(1−p))mR+π.
(a) π<˜
π.
If t+(nr)≥tS,Ai ≻Ar,∀β. If t+(nr)<tS, we have
Ar ≻Ai ⇔β>βri =(1−p+θ
pp(1−p))mR−π
MS(t−(nr+ni),tS)−MS(t−(nr+ni),t+(nr)) >0.
∂βri
∂π =−1
MS(t−(nr+ni),tS)−MS(t−(nr+ni),t+(nr)) <0,
∂βri
∂mR
=(1−p+θ
pp(1−p))
MS(t−(nr+ni),tS)−MS(t−(nr+ni),t+(nr)) >0,
Let ∆MS =MS(t−(nr+ni),tS)−MS(t−(nr+ni),t+(nr)), then we have
∂βri
∂nr
=(1−p+θ
pp(1−p))mR−π
−∆MS2
∂∆MS
∂nr
,
where ∂∆MS
∂nr
=−(1−p+θ
qp(1−p)) ∂t+(nr)
∂nr
<0.
Hence, we have ∂βri
∂nr
>0.
40
(b) π>˜
π.
If t+(nr)≤tS,Ar ≻Ai,∀β. If t+(nr)>tS, we have
Ar ≻Ai ⇔β<βri =(1−p+θ
pp(1−p))mR−π
MS(t−(nr+ni),tS)−MS(t−(nr+ni),t+(nr)) .
∂βri
∂π =1
MS(t−(nr+ni),t+(nr)) −MS(t−(nr+ni),tS)>0,
∂βri
∂mR
=−(1−p+θ
pp(1−p))
MS(t−(nr+ni),t+(nr)) −MS(t−(nr+ni),tS)<0,
∂βri
∂nr
=π−˜
π
−∆MS2
∂∆MS
∂nr
,
where ∂∆MS
∂nr
= (1−p+θ
qp(1−p)) ∂t+(nr)
∂nr
>0.
Hence, we have ∂βri
∂nr
<0.
(c) π= (1−p+θ
pp(1−p))mR.
If t+(nr)>tS,As ≻Ai,∀β; if t+(nr)<tS,As ≺Ai,∀β; if t+(nr) = tS,As ∼Ai,∀β;
I Proof of Proposition 5
By Lemma 5, there exists a unique information price threshold ˇ
πsuch that ni≥0 if and only
if π>ˇ
π. That is, for any (ns,nr,ni), if π=ˇ
π, then the number of travelers choosing Ar is
nr=d−d·F(βsr). For π=¯
π, the equilibrium is such that nC
i=0. Therefore, when π=¯
π, the
equilibrium flow nC
rand βC
sr can be solved using the equations:
nC
r=d−d·F(βC
sr),
βC
sr =mR
tS−TT BR(nC
r).
(9a)
(9b)
Because the CDF F(β)is continuous and strictly increasing for β∈[βl,βu], Equation (9a)
defines a decreasing relationship between βand nr, with nr(βl) = dand nr(βu) = 0.
Because TT BR(nr)is continuous and strictly increasing for nr, Equation (9b) defines an in-
creasing relationship between βand nrwhen nr≤f0
0, with
β(0) = mR
tS−TT BR(0)<βuand lim
nrրf0
0
β(nr) = +∞.
Hence, the system of equations in (9a)-(9b) has a unique solution nC
r=f0and βC
rs =β0
rs.
(1) mR>¯
mR, i.e., t+(f0)<tS.
If ¯
π>˜
πand t+(f0)<tS, then we have
ˇ
βri (mR,¯
π,ns,nr,ni) = βu
41
according to Lemma 5. However, if t+(f0)<tSand ¯
π>˜
π, then we have ˇ
βri (mR,¯
π,ns,nr,ni) =
∞by Lemma 3, which is a contradiction.
If ¯
π<˜
πand t+(f0)<tS, according to Lemma 5, then we have
ˇ
βsr(mR,ns,nr,ni) = ˇ
βsi(mR,¯
π,ns,nr,ni) = ˇ
βri (mR,¯
π,ns,nr,ni),
⇒β0
rs =ˇ
βri (mR,¯
π,ns,nr,ni),
⇒¯
π=β0
sr(tS−MS(t−(f0),tS)) −(p−θqp(1−p))mR.
If ¯
π=˜
πand t+(f0)<tS, then we have βsi =βsr according to Lemma 5. However, we have
βsr =mR
tS−MS(t−(nr+ni),t+(nr)) <mR
tS−MS(t−(nr+ni),tS)=βsi,
which is a contradiction. Therefore, ¯
π<˜
πmust hold when m>¯
mR.
(2) mR<¯
mR, i.e., t+(f0)>tS.
If ¯
π>˜
πand t+(f0)>tS, according to Lemma 5, then we have
ˇ
βri (mR,¯
π,ns,nr,ni) = βu.
⇒¯
π=βu(TT BR(f0)−MS(t−(f0),tS)) + ˜
π,
If ¯
π<˜
πand t+(f0)>tS, according to Lemma 5, we have
ˇ
βsr(mR,ns,nr,ni) = ˇ
βsi(mR,¯
π,ns,nr,ni) = ˇ
βri (mR,¯
π,ns,nr,ni).
However, we have ˇ
βri (mR,¯
π,ns,nr,ni) = ∞by Lemma 3, which is a contradiction.
If ¯
π=˜
πand t+(f0)>tS, then Ar ≻Ai,∀β. Therefore, nC
r=0 and t+(nC
r)<tS, which
contradicts the condition t+(f0)>tS. Therefore, ¯
π>˜
πmust hold when mR<¯
mR.
(3) mR=¯
mR, i.e., t+(f0) = tS.
If ¯
π<˜
πand t+(f0) = tS, according to Lemma 5, then we have Ar ≺Ai,∀β. Therefore, nr=0
and t+(nr)<tSwhich contradicts the condition t+(f0) = tS.
If ¯
π>˜
πand t+(f0) = tS, then we have
ˇ
βri (mR,¯
π,ns,nr,ni) = βu.
However, ˇ
βri (mR,ˇ
π2,ns,nr,ni) = ∞by Lemma 3, which is a contradiction.
If ¯
π=˜
π, according to Lemma 5, then we have
ˇ
π=˜
πand ˇ
βsr(mR,ns,nr,ni) = ˇ
βsi (mR,ˇ
π,ns,nr,ni).
Therefore, ¯
π=˜
πmust hold when mR=¯
mR.
42
J Proof of Proposition 6
(a) π∈(0, ˜
π)and mR∈(0, ¯
mR).
According to Lemma 5, a traveler selects strategy As if β<βsi and β<βsr ; selects Ai if
β<βri and β>βsi ; selects Ar if β>βri and β>βsi.
When t+(nr)<tS, we have
βsr =mR
tS−MS(t−(nr+ni),t+(nr)) <π+ (p−θ
pp(1−p))mR
tS−MS(t−(nr+ni),tS)=βsi.
Thus, the number of travelers choosing As is ns=d·F(βsi). The number of travelers
choosing Ar is nr=d−d·F(βri ). Variables nC
s,nC
r,βC
ri and βC
si can be solved using the
equations:
nC
s=d·F(βC
si),
βC
si =π+ (p−θ
pp(1−p))mR
tS−MS(t−(d−nC
s),tS),
nC
r=d·(1−F(βC
ri )),
βC
ri =˜
π−π
MS(t−(d−nC
s),tS)−MS(t−(d−nC
s),t+(nC
r)) .
(10a)
(10b)
(10c)
(10d)
Because the CDF F(β)is continuous and strictly increasing for β∈[βl,βu], Equations
(10a)defines an increasing relationship between βsi and ns, with nr(βl) = 0 and nr(βu) = d.
Recall that we denote the flow of Route Ron good days and bad days when mR=0 as f−
F0
and f+
F0, respectively. Because t−(f)is continuous and strictly increasing for f, Equation
(10b) defines a decreasing relationship between βand ns, with lim
nsրd−f−
F0
βsi(ns) = +∞and
βsi(n0
s) = π+ (p−θ
pp(1−p))mR
tS−MS(t−(d−n0
s),tS)
<mR
tS−TT BR(d−n0
s)
=F−1(n0
s
d).
Therefore, the two curves in Equation (10a) and Equation (10b) cross exactly once, which
defines βC
si and nC
s∈(d−f−
F0,n0
s).
Because the CDF F(β)is continuous and strictly increasing for β∈[βl,βu], Equations (10c)
defines a decreasing relationship between βri and nr, with
nr(βl) = dand nr(βu) = 0.
Because t+(f)is continuous and strictly increasing for f, Equation (10d) defines an increas-
ing relationship between βri and nr, with
βri (n′
r) = ˜
π−π
MS(t−(d−nC
s),tS)−t−(d−nC
s)<mR
tS−t−(0)<βu
43
and
lim
nrրf+
F0
βri (nr) = +∞,
where n′
ris a constant such that t+(n′
r) = t−(d−nC
s). Therefore, the two curves in Equation
(10c) and Equation (10d) cross exactly once, which defines βC
ri and nC
r∈(n′
r,f+
F0).
When t+(nr)≥tS, we have βsr =∞and nr=0. Therefore, t+(nr) = t+(0)<tS, which is a
contraction.
(b) π∈(˜
π,¯
π)and mR∈(0, ¯
mR)(Case 3).
According to Lemma 5, a traveler selects strategy As if β<βsi and β<βsr ; selects Ai if
β>βri and β>βsi ; selects Ar if β<βri and β>βsi;βsi and βri are both increasing in π
when βsi <∞and βri <∞.
When t+(nr)≤tS, we have βri =∞and ni=0 for π∈(˜
π,¯
π), which contradicts Proposi-
tion 5.
When t+(nr)>tS, we have βri =π−˜
π
MS(t−(nr+ni),t+(nr))−MS(t−(nr+ni),tS). Thus, the number of
travelers selecting strategy Ai is ni=dF(d−ni). The number of travelers choosing As is
ns=d·F(βsi). Variables nC
s,nC
r,βC
sr and βC
ri can be solved using the equations:
nC
s=d·F(βC
sr),
βC
sr =mR
tS−MS(t−(d−nC
s),t+(nC
r)) ,
nC
r+nC
s=d·F(βC
ri ),
βC
ri =π−˜
π
MS(t−(d−nC
s),t+(nC
r)) −MS(t−(d−nC
s),tS)) .
(11a)
(11b)
(11c)
(11d)
Because the CDF F(β)is continuous and strictly increasing for β∈[βl,βu], and t−(f)
is continuous and strictly increasing for f, Equations (11a) and (11b) define a decreasing
relationship between nrand ns, with
ns(f+
F0) = d·F(mR
tS−MS(t−(d−ns),tS))<d·F(π+ ( p−θ
pp(1−p))mR
tS−MS(t−(d−n0
s),tS)) = n0
s
and
ns(f0) = ns(0).
Because the CDF F(β)is continuous and strictly increasing for β∈[βl,βu], and t+(f)
is continuous and strictly increasing for f, Equations (11c)and (11d) define an increasing
relationship between nrand ns, with
lim
nr→f+
F0
ns(nr) = ∞and ns(f0) = dF(βC
ri )−f0<d(βu)−f0=n0
s.
Therefore, the two curves in Equations (11a)- (11d) cross exactly once, which defines nC
r∈
(f+
F0,f0)and nC
s∈(0, n0
s).
44
(c) π=˜
πand mR∈(0, ¯
mR)(Case 4).
If t+(nr)<tS, according to Lemma 5, then Ai ≺Ar,∀β. Thus, ni=0 which contradicts
Proposition 5.
If t+(nr)>tS, according to Lemma 5, then Ar ≺Ai,∀β. Thus, nr=0 and t+(nr)<tS
which contradicts the condition t+(nr)>tS.
If t+(nr) = tS, then Ai ∼Ar,∀β. The number of travels selecting As is ns=d·F(βsi).
Therefore, the variable nscan be solved using Equations (10a) and (10b), of which the
uniqueness has been proved. The variable nrcan be solved by using the equation t+(nr) =
tS.
(d) π∈(˜
π,¯
π)and mR=0 (Case 6).
If tS>MS(t−(nr+ni),t+(nr)), according to Lemma ??, then βsr =0 and ns=0. Thus,
t−(nr+ni) = t−(d)>tS, which contradicts Proposition 5.
If tS=MS(t−(nr+ni),t+(nr)), according to Lemma ??, then As ∼Ar,∀β. Thus, the
number of travelers choosing Ai is ni=d−d·F(βri). Variables nC
i,nC
r, and βC
ri can be
solved using the equations:
ni=d−d·F(βC
ri ),
βC
ri =π
MS(t−(nC
i+nC
r),t+(nC
r)) −MS(t−(nC
i+nC
r),tS)) ,
tS=MS(t−(nr+ni),t+(nr)).
(12a)
(12b)
(12c)
Because the CDF F(β)is continuous and strictly increasing for β∈[βl,βu], Equations (12a)
and (12b) define an increasing relationship between nrand ni, with
lim
nr→f+
F
ni(nr) = d−d·F(βu) = 0 and ni(f0)>d−d·F(βu) = 0.
Equation (12c) defines a decreasing relationship between nrand ni, with
lim
nr→f+
F
ni(nr)>0 and ni(f0) = 0.
Therefore, the two curves in Equations (12b) - (12c) cross exactly once, which defines nC
r∈
(f+
F0,f0)and nC
i.
(e) By totally differentiating the equation system (10a) - (10d), one obtains:
∂nC
s
∂π =
∂F(βC
si)
∂βC
si
d
tS−MS(t−(d−nC
s),t+(nC
r))
1−d∂F(βC
si )
∂βC
si
π+(p−θ
√p(1−p))mR
(tS−MS(t−(d−nC
s),tS))2
∂MS(t−(d−nC
s),t+(nC
r))
∂nC
s
>0,
∂nC
r
∂π =
∂F(βC
ri)
∂βC
ri
d
MS(t−(d−nC
s),tS)−MS(t−(d−nC
s),t+(nC
r))
1+d∂F(βC
ri )
∂βC
ri
˜
π−π
(MS(t−(d−nC
s),tS)−MS(t−(d−nC
s),t+(nC
r)))2
∂MS(t−(d−nC
s),t+(nC
r))
∂nC
r
>0,
∂nC
s
∂mr
=
∂F(βC
si)
∂βC
si
d(p−θ
√p(1−p))
tS−MS(t−(d−nC
s),t+(nC
r))
1−d∂F(βC
si)
∂βC
si
π+(p−θ
√p(1−p))mR
(tS−MS(t−(d−nC
s),tS))2
∂MS(t−(d−nC
s),t+(nC
r))
∂nC
s
>0,
45
∂nC
r
∂mr
=0.
By totally differentiating the above equation system (11a) - (11d), one obtains:
∂nC
r
∂π =
1+d∂F(βC
ri)
∂βC
ri
π−˜
π
(MS(t−(d−nC
s),t+(nC
r))−MS(t−(d−nC
s),tS))2
∂MS(t−(d−nC
s),t+(nC
r))
∂nr
d∂F(βC
ri )
∂βC
ri
1
MS(t−(d−nC
s),t+(nC
r))−MS(t−(d−nC
s),tS)
>0
∂nC
s
∂π =∂nC
r
∂π
1−d∂F(βC
sr)
∂βC
sr
mR
(tS−MS(t−(d−nC
s),t+(nC
r)))2
∂MS(t−(d−nC
s),t+(nC
r))
∂ns
d∂F(βC
sr )
∂βC
sr
mR
(tS−MS(t−(d−n0
s),t+(nC
r))2
∂MS(t−(d−nC
s),t+(nC
r))
∂ns
>0,
∂nC
s
∂mr
=
1−d∂F(βC
ri )
∂βC
ri
mR
(tS−MS(t−(d−nC
s),t+(nC
r)))2
∂MS(t−(d−nC
s),t+(nC
r))
∂ns
d∂F(βC
sr)
∂βC
sr
1
tS−MS(t−(d−nC
s),t+(nC
r))
>0,
∂nC
r
∂mr
=0.
For any (π,mR)that falls into Cases 4 and 6, the equilibrium solutions are continuous at
such points as described in (c) and (d). Thus, nC
sand nC
rare increasing in π.nC
sis increasing
in mRand nC
ris independent of mRwhen mR<¯
mRand π<¯
π.
K Proof of Proposition 8
(a) Case 3.
(1) When nC
s+nC
r>n0
s, there are four Groups SS,SR,RR, and R I:
For Group SS (β<βC
sr),
CVC(β) = βtS−βtS=0.
For Group SR (βC
sr <β<β0
sr),
CVC(β) = βtS−mR−βMS(t−(nC
r+nC
i),t+(nC
r)).
Because Ar ≻As in the costly information regime for Group SR,
βtS>mR+βMS(t−(nC
r+nC
i),t+(nC
r)).
Therefore, CVC(β)>0. Note that CV(βC
sr) = 0. By continuity, tS>MS(t−(nC
r+
nC
i),t+(nC
r)) must hold and therefore CVC(β)is monotonically increasing in β.
For Group RR (β0
sr <β<βC
ri ),
CVC(β) = βTTBR(f0)−βMS(t−(nC
r+nC
i),t+(nC
r)).
Note that CV C(β0
sr)>0. By continuity, TTBR(f0)>MS(t−(nC
r+nC
i),t+(nC
r)) must hold.
Therefore, CVC(β)is increasing in β.
For Group RI (β>βC
ri ),
CVC(β) = βTTBR(f0)−MS(mR+βt−(nC
r+nC
i),βtS)−π
=β(TT BR(f0)−MS(t−(f−
C),tS)) + ˜
π−π
46
Note that CVC(βC
ri )>0 and ˜
π−π<0 in Case 3. By continuity, MS(TTBR(f0)>
MS(t−(f−
C),tS)) must hold. Therefore, CVC(β)is increasing in β.
(2) When nC
s+nC
r<n0
s, travelers are classified into Groups SS,SR,SI,RI:
For Group SS ,
CVC(β) = βtS−βtS=0.
For Group SR,
CVC(β) = βtS−mR−βMS(t−(nC
r+nC
i),t+(nC
r)).
Since Ar ≻As in the costly information regime for Group SR,
βtS>mR+βMS(t−(nC
r+nC
i),t+(nC
r))
and hence CVC(β)>0. Note that mR>0. By continuity, tS>MS(t−(nC
r+nC
i),t+(nC
r))
must hold if CVC(β)>0. Therefore, CVC(β)is monotonically increasing in β.
For Group SI,
CVC(β) = βtS−MS(mR+βt−(nC
r+nC
i),βtS)−π.(13)
Since Ai ≻As in the costly information regime for Group S I,
βtS>MS(mR+βt−(nC
r+nC
i),βtS) + π
and hence CVC(β)>0. We can rewrite Equation (13) as follows:
CVC(β) = β(tS−MS(t−(f−
C),tS)) −π−(p−θ
qp(1−p))mR.
According to Assumption 3, p>θ
pp(1−p)holds. tS−MS(t−(f−
C),tS)>0 must also
hold if CVC(β)>0. Therefore, CV C(β)is monotonically increasing with respect to β.
For Group R I, similar to the proof when nC
s+nC
r>n0
s,CVC(β)>0 and it is increasing in
βwithin Group RI.
(b) Case 4. The proof is similar to Case 3.
(c) Case 5. In Case 5, the costly information regime is reduced to the no information regime.
Thus, CVC(β) = 0, ∀β∈[βl,βu]
(d) Case 6. The proof is similar to Case 3.
L Proof of Proposition 9
First, we identify all possible situations regarding the relative magnitude of MS(t−(f−
C),tS),
TT BR(f0)and MS(t−(f−
C),t+(f+
C)). If β=βC
ri, then Ar ∼Ai under the costly information regime.
Particularly, we have
MS(mR+βC
ri t−(f−
C),βC
ri tS)
=βC
ri MS(t−(f−
C),tS) + π−˜
π
=βC
ri MS(t−(f−
C),t+(f+
C)).
47
Note that here π<˜
π. Hence if π<min {˜
π,¯
π}, then MS(t−(f−
C),tS)>MS(t−(f−
C),t+(f+
C))
must hold. Thus, there are three possible situations regarding the relative magnitude of MS(t−(f−
C),tS),
TT BR(f0)and MS(t−(f−
C),t+(f+
C)).
According to Proposition 7, there are four Groups SS,SI,RI, and RR in Cases 1 and 2.
(a) For Group SS (β<βC
si),
CVC(β) = βtS−βtS=0.
(b) For Group SI (βC
si <β<β0
sr),
CVC(β) = βtS−MS(mR+βt−(f−
C),βtS)−π.(14)
Because Ai ≻As in the costly information regime for Group SI, we have
βtS>MS(mR+βt−(f−
C)),βtS) + π.
Thus, CVC(β)>0. In addition, we can rewrite (14) as
CVC(β) = β(tS−MS(t−(f−
C),tS)) −π−(p−θ
qp(1−p))mR>0.
According to Assumption 3, p>θ
pp(1−p)holds. Therefore,
β(tS−MS(t−(f−
C),tS)) >p−θ
qp(1−p)>0
and CVC(β)is monotonically increasing with respect to β.
(c) For Group RI (β0
sr <β<βC
ri),
CVC(β) = βTTBR(f0)−MS(mR+βt−(f−
C),βtS)−π
=β(TT BR(f0)−MS(t−(f−
C),tS)) + ˜
π−π.(15)
For Group RR (β>βC
ri ),
CVC(β) = β(TTBR(f0)−MS(t−(f−
C),t+(f+
C))).(16)
According to Equations (15) and (16), if MS(t−(f−
C),tS)<TT BR(f0), then CVC(β)>0 for
Groups RI and RR and CVF(β)is monotonically increasing with respect to β;
(d) According to Equations (15) and (16), if MS(t−(f−
C),tS)>TT BR(f0)>MS(t−(f−
C),t+(f+
C)),
then we have
CVC(β) = β(TTBR(f0)−MS(t−(f−
C),tS)) + ˜
π−π
with CVC(βC
ri )>0 for Group R I, and
CVC(β) = β(TTBR(f0)−MS(t−(f−
C),t+(f+
C))) >0,
for Group RR. Since TTBR(f0)<MS(t−(f−
C),tS),CVC(β)is monotonically decreasing
with respect to βfor Group R I. Since TTBR(f0)>MS(t−(f−
C),t+(f+
C)),CV C(β)is mono-
tonically increasing with respect to βfor Group RR.CVC(β)<0 for Group RR.
48
(e) According to Equations (15) and (16), if MS(t−(f−
C),tS)>MS(t−(f−
C),t+(f+
C)) >TTBR(f0),
then we have
CVC(β) = β(TTBR(f0)−MS(t−(f−
C),tS)) + ˜
π−π,
with CVC(βC
ri )<0 for Group R I, and
CVC(β) = β(TTBR(f0)−MS(t−(f−
C),t+(f+
C))) <0,
for Group RR. Since TTBR(f0)<MS(t−(f−
C),t+(f+
C)) <MS(t−(f−
C),tS),CVC(β)is mono-
tonically decreasing with respect to βfor Groups RI and RR.
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