Using prospect theory to investigate cross national and gender differences in the (dis)like for waiting and travelling in public transport

Using prospect theory to investigate cross-national and
gender differences in the (dis)like for waiting and
travelling in public transport
Interdisciplinary Choice Workshop (ICW), Santiago, Chile
Pablo Guarda1,2 and Juan Carlos Muñoz1
1Department of Transportation Engineering and Logistics (DTEL)
Pontifical Catholic University of Chile (PUC)
2Department of Experimental Psychology
University College London (UCL)
August 10, 2018

Introduction Context
How do travellers make route choices?
Decision attributes
• Monetary cost
• Physical eﬀort (e.g. while walking)
• Time spent (e.g. while walking, waiting or travelling)
• Service reliability (e.g. associated to time variability)
• ...
Pablo Guarda (UCL and PUC) ICW 2018 August 10, 2018 1 / 28

How do travellers make route choices?
Decision attributes
• Monetary cost
• Physical eﬀort (e.g. while walking)
• Time spent (e.g. while walking, waiting or travelling)
• Service reliability (e.g. associated to time variability)
• ...
Decision-making models for risky choice
• Mean variance-approach (Markowitz, 1952)
• Prospect theory (Kahneman & Tversky, 1979)
• Priority heuristic (Brandst¨atter et al., 2006)

Why should choice modellers be interested on using
prospect theory (PT) to study time-related decisions?
1 Individuals typically make decisions in complex environments with
multiple probabilistic outcomes

2 PT is one of the most inﬂuential descriptive account of decision
making under risk and uncertainty

3 PT presents a compact mathematical representation that preserves
the concept of utility maximization used in traditional economic
models of decision-making, while also accounting for features of
human bounded rationality

3 PT presents a compact mathematical representation that preserves
the concept of utility maximization used in traditional economic
models of decision-making, while also accounting for features of
human bounded rationality
4 PT has been widely applied and with relative success for modelling
decisions with a single monetary attribute. In contrast, PT has
been few applied to study time-related decisions

Psychological principles integrated in PT
• Reference dependence
• Diminishing sensitivity
• Loss aversion (losses loom larger than gains)

Introduction Study Overview
Study overview
Research questions
1 Can PT properly represent travellers’ trade-oﬀ between waiting
and in-vehicle times?
2 Does the heterogeneity in travellers’ characteristics (gender and
city of residence) mediates these trade-oﬀs?

Introduction Study Overview
Study overview
Research questions
1 Can PT properly represent travellers’ trade-oﬀ between waiting
and in-vehicle times?
2 Does the heterogeneity in travellers’ characteristics (gender and
city of residence) mediates these trade-oﬀs?
Main objectives
1 Analytically derive themarginal rate of substitution (MRS)
between the average and variability of two attributes using PT
2 Jointly estimate the components of the PT value function in a
two-attribute decision problem
3 Compute and compare the MRSs across genders and cities
using PT and mean-variance (MV) models

Theoretical framework Mean-variance (MV) model
Marginal rate of substitutions between the average (ρµ
k1,k2)
and variability (ρσ
k1,k2) of two attributes
Utility function consistent with the mean-variance approach
(Jackson & Jucker, 1981)
Uj =
∀k∈K
θµ
k
¯tkj +
∀k∈K
θσ
k σkj

k1,k2)
Uj =
∀k∈K
θµ
k
¯tkj +
∀k∈K
θσ
k σkj
MRS between the average values of two attributes tk1, tk2
ρµ
k1,k2 =
∂U/∂tk1
∂U/∂tk2
=
θµ
k1
θµ
k2

k1,k2)
Uj =
∀k∈K
θµ
k
¯tkj +
∀k∈K
θσ
k σkj
MRS between the average values of two attributes tk1, tk2
ρµ
k1,k2 =
∂U/∂tk1
∂U/∂tk2
=
θµ
k1
θµ
k2
MRS between the variability of two attributes tk1, tk2
ρσ
k1,k2 =
∂U/∂σk1
∂U/∂σk2
=
θσ
k1
θσ
k2

ρµ
w,v reported in the literature for Santiago and London
Study Methoda Sample Variabilityb Mode City MRS
Guarda (2017) SP 504 Bus Santiago [1.13, 1.3]
SP 504 Bus London [0.92,1.14]
Batarce et al. (2015) SP 3,380 Bus Metro Santiago [1.08,2.33]
RP 28,961 Metro Santiago [0.94,1.56]
SP RP 32,341 Bus Metro Santiago [1.01,2.26]
Basso Silva (2014) MA - Bus Santiago 1.93
MA - Bus London 2.5
Raveau et al. (2014) c RP 28,961 Metro Santiago [1.46,4.48]
RP 17,073 Metro London [0.61,1.93]
Raveau et al. (2011) RP 28,961 Metro Santiago [1.07,1.41]
Guo Wilson (2011) RP 25,036 Metro London [0.64,1.14]
Yáñez et al. (2009) RP 1,290 d Santiago [1.37,2.75]
a
Stated Preferences (SP), Revealed Preferences (RP), Meta-analysis (MA).
a
Yes (); No (). The study computes the MRS of variability in waiting time for variability in in-vehicle time.
b
Raveau et al. (2014) presents MRSs for morning peak trips with restrictive purpose (e.g. going to work).
c
Yáñez et al. (2009) computed MRSs using data from 10 transportation modes (e.g. car-driver, bus/Metro, bus)

Theoretical framework Prospect theory (PT) model
PT value function for time-related decisions
Two-part power value function (inverse S-shaped for time outcomes)
u(t) =
θ+(r − t)α+
if t r (gains)
−θ−(t − r)α−
if t r (losses)
where:
t : time outcome
r : reference point
θ+
, θ−
: Slopes of the value function in the gains and losses domains
α+
, α−
: Curvatures of the value function in the gains’ and losses’ domain

PT probability weighting function
w(p) =



pγ+
(pγ+
+ (1 − p)γ+
)1/γ+ (gains)
pγ−
(pγ−
+ (1 − p)γ−
)1/γ−
(losses)
where:
w(·) : Decision weight
p : outcome probability
γ+
, γ−
: elevation of the probability weighting function in the gains’ and losses’ domain

MRS between the average of two attributes (ρµ
w±,v±)
Case i: Loss waiting Loss travelling
ρµ
w−,v− =
∂U(w−
)/∂w−
∂U(v−)/∂v−
=
−λ−
w α−
w (w−
− rw )α−
w −1
−λ−
v α−
v (v− − rv )α−
v −1

w±,v±)
ρµ
w−,v− =
∂U(w−
)/∂w−
∂U(v−)/∂v−
=
−λ−
w α−
w (w−
− rw )α−
w −1
−λ−
v α−
v (v− − rv )α−
v −1
Case ii: Gain waiting Loss travelling
ρµ
w+,v− =
−∂U(w+
)/∂w+
∂U(v−)/∂v−
=
−λ+
w α+
w (rw − w+
)α+
w −1
−λ−
v α−
v (v− − rv )α−
v −1

w±,v±)
ρµ
w−,v− =
∂U(w−
)/∂w−
∂U(v−)/∂v−
=
−λ−
w α−
w (w−
− rw )α−
w −1
−λ−
v α−
v (v− − rv )α−
v −1
ρµ
w+,v− =
−∂U(w+
)/∂w+
∂U(v−)/∂v−
=
−λ+
w α+
w (rw − w+
)α+
w −1
−λ−
v α−
v (v− − rv )α−
v −1
Case iii: Loss waiting Gain travelling
ρµ
w−,v+ =
∂U(w−
)/∂w−
−∂U(v+)/∂v+
=
−λ−
w α−
w (w−
− rw )α−
w −1
−λ+
v α+
v (rv − v+)α+
v −1

w±,v±)
ρµ
w−,v− =
∂U(w−
)/∂w−
∂U(v−)/∂v−
=
−λ−
w α−
w (w−
− rw )α−
w −1
−λ−
v α−
v (v− − rv )α−
v −1
ρµ
w+,v− =
−∂U(w+
)/∂w+
∂U(v−)/∂v−
=
−λ+
w α+
w (rw − w+
)α+
w −1
−λ−
v α−
v (v− − rv )α−
v −1
Case iii: Loss waiting Gain travelling
ρµ
w−,v+ =
∂U(w−
)/∂w−
−∂U(v+)/∂v+
=
−λ−
w α−
w (w−
− rw )α−
w −1
−λ+
v α+
v (rv − v+)α+
v −1
Case iv: Gain waiting Gain travelling
ρµ
w+,v+ =
−∂U(w+
)/∂w+
−∂U(v+)/∂v+
=
−λ+
w α+
w (rw − w+
)α+
w −1
−λ+
v α+
v (rv − v+)α+
v −1

MRS between the variability of two attributes (ρσ
w±w±,v±v±)
• In decision scenarios where a single attribute has time variability, the
prospect’s value (or alternative) can be computed in 3 ways:
(1) mixed-prospect; (2) non-mixed prospect in the gains’ domain; (3)
non-mixed prospect in the losses’ domain .

w±w±,v±v±)
• Thus, the MRS between the variability of two attributes can be computed in
9 possible ways (3x3). The MRS also depends on the level of variability of
each attribute (∆w, ∆v)

w±w±,v±v±)
• Let’s consider the case where the reference point of each attribute (rw , rv ) is
located between the prospects’ outcomes (tk = t∗
k ± ∆tk ) and the level of
variability in both attributes is the same (∆t = ∆w = ∆v):

w±w±,v±v±)
• Let’s consider the case where the reference point of each attribute (rw , rv ) is
located between the prospects’ outcomes (tk = t∗
k ± ∆tk ) and the level of
variability in both attributes is the same (∆t = ∆w = ∆v):
ρσ
w+w−,v+v− =
∂U(w+
, w−
)/∂∆w
∂U(v+, v−)/∂∆v
=
α+
w π+
w λ+
w (rw − w+
+ ∆t)α+
w −1
− α−
w π−
w λ−
w (w−
+ ∆t − rw )α−
w −1
α+
v π+
v λ+
v (rv − v+ + ∆t)α+
v −1 − α−
v π−
v λ−
v (v− + ∆t − rv )α−
v −1

Method Task and materials
Method
1 Materials
Virtual environment programmed in PyQt

Method
1 Materials
2 Cognitive task
Participants (72) were asked to make a choice between two bus routes
in 14 decision scenarios.
Each decision scenario presented two bus routes with diﬀerent waiting
and in-vehicle times
The scenarios manipulated the average value (8) and the level of
variability (6) of the time attributes

Method
1 Materials
2 Cognitive task
Participants (72) were asked to make a choice between two bus routes
in 14 decision scenarios.
Each decision scenario presented two bus routes with diﬀerent waiting
and in-vehicle times
The scenarios manipulated the average value (8) and the level of
variability (6) of the time attributes
3 Experimental conditions (between-subjects)
A priori condition: Prospects showing the probabilities (100% or 50%)
of occurrence of the time outcomes in each route
A posteriori condition: Tables showing the waiting and in-vehicle times
for two bus trips in each route

A priori condition (prospects)

A posteriori condition (tables)

Method Participants
Participants
City: Santiago, Chile
Place: Computer Lab (Engineering), PUC
Date: June 2017
Participants: 36 university students
Average Session: 35 minutes
City: London, UK
Place: CogSys Lab (Psychology), UCL
Date: July 2017
Participants: 36 university students
Average Session: 40 mins*

Results Mean-variance model (MV)
Mean-variance model (MV) of route choice
Utility function
Ujn = θw ¯wj + θv ¯vj + θσ
w σwj + θσ
v σvj + ejn, ejn iid EV (0, µ)
Predictors
• Average waiting time (wj ∈ 1, 2, 3, 4, 6, 7, 8, 9, ¯w ≈ 4)
• Average in-vehicle time (vj ∈ 1, 2, 4, 6, 7, 9, ¯w ≈ 6)
• Standard deviation of waiting time (±2, ±4, wmin = 0, wmax = 8)
• Standard deviation of in-vehicle time (±2, ±4,vmin = 2,vmax = 10)
Levels of analysis
• Gender (Male/Female) and City (Santiago/London)

Results PT model of route choice
Prospect theory (PT) model of route choice
Utility function
Ujn =
∀k∈K
Ujk(tjk, pjk|θk, αk, γk, rk) + ejn, ejn iid EV (0, µ)
Parameters
1 Reference points: rk
2 Slopes: λ−
k , λ+
k
3 Curvatures: γ−
k , γ+
k
Assumptions
1 Equal sensitivity across domains
α+
k = α−
k
2 Equal probability weighting across domains
γ+
k = γ−
k → wk(p) = 0.5

Sample description
Data City Gender Individuals Choices
S-M Santiago (S) Male (M) 20 280
S-F Santiago (S) Female (F) 14 196
L-M London (L) Male (M) 14 196
L-F London (L) Female (F) 22 308
All S L M F 70 980

Goodness of ﬁt
Table: Comparison of log-likelihood (LL) and Akaike Information Criteria (AIC)
between PT and MV models
MV PT PT - MV
Data Pars LL AIC Pars LL AIC ∆ LL ∆ AIC
S-M 4 -133.6 275.2 8 -131.4 278.9 2.2 3.7
S-F 4 -86.4 180.7 8 -86.0 188.0 0.4 7.3
L-M 4 -116.6 241.2 8 -117.2 250.4 -0.6 9.2
L-F 4 -178.3 364.6 8 -177.8 371.7 0.5 7.1
All 4 -554.8 1117.6 8 -551.2 1118.4 3.6 0.8

Value functions Uk(·) between cities
Santiago, Chile
0 2 4 6 8 10
−20−1001020
Time outcome [min]
SubjectiveUtility
London, UK
0 2 4 6 8 10
−20−1001020
Time outcome [min]
SubjectiveUtility
In−vehicle time Waiting time

Curvatures αk and reference points rk
Reference points
3.0 3.5 4.0 4.5 5.0
7.07.58.08.59.0
Reference point waiting (rw)
Referencepointtravelling(rt)
q
Curvatures
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0 Curvature waiting (αw)
Curvaturetravelling(αt)
q
qL−M L−F S−M S−F

Loss aversion
λ−
k
λ+
k
0 1 2 3 4 5
012345
Gain parameter (λ+
)
Lossparameter(λ−
)
q
q
q qIn−vehicle time Waiting time
qL−M L−F S−M S−F

Figure: Marginal rate of substitution between waiting and in-vehicle times for a
given amount of in-vehicle time (rv ≈ 8, rw ≈ 4)
0 2 4 6 8 10
012345
Waiting time [min]
Marginalrateofsubstition
In−vehicle time [min]
4 (++)

0 2 4 6 8 10
012345
Waiting time [min]
4 (++)
6 (+)

0 2 4 6 8 10
012345
Waiting time [min]
4 (++)
6 (+)
8 (−)

0 2 4 6 8 10
012345
Waiting time [min]
4 (++)
6 (+)
8 (−)
10 (−−)

Marginal rates of substitution ρµ
w,v between genders and
cities
Between cities
0 2 4 6 8 10
012345
Waiting Time
MRS
London Santiago
Between genders
0 2 4 6 8 10
0123456
Waiting Time
MRS
Males Females

Marginal rates of substitution ρµ
w,v between genders by city
London between genders
0 2 4 6 8 10
01234567
Waiting Time
MRS
Males Females
Santiago between genders
0 2 4 6 8 10
0123456
Waiting Time
MRS
Males Females

Marginal rates of substitution ρσ
w,v between genders by city
London, UK
Level of variability [min]
1 2 3 4
−2−1012
Males Females
Santiago, Chile
Level of variability [min]
1 2 3 4
−202468
Males Females

Conclusions PT vs MV
Main results: PT vs MV
Goodness of fit
• The difference in AIC between the models was lower than 10 for
the five choice samples (S-M;S-F;L-M;L-F;All) used for estimation,
so there is partial to substantial support for the PT model.
• The larger is the sample size, the better is the goodness of fit of
the PT model compared with the MV model

Main results: PT vs MV
Goodness of fit
• The difference in AIC between the models was lower than 10 for
the five choice samples (S-M;S-F;L-M;L-F;All) used for estimation,
so there is partial to substantial support for the PT model.
• The larger is the sample size, the better is the goodness of fit of
the PT model compared with the MV model
Analytical properties
• Although the PT model does not significantly improve the
goodness of fit obtained with the MV model, it has much richer
analytical properties to study time-related decisions under risk.
• The MV model, by construction, assumes that the MRSs are
constant. The PT model provides a richer representation of the
MRS between the mean and variability of the attributes

Main results: marginal rate of substitution (MRS)
MRS between the average of waiting and in-vehicle times (ρµ
w,v )
• ρµ
w,v reaches its peak at values closer to the reference point of
waiting time (rw )
• The higher is the in-vehicle time, lower is ρµ
w,v . However, ρµ
w,v
begins to increase when in-vehicle times are above rv .
• ρµ
w,v are higher for females than males and higher for Santiago
than London

Main results: marginal rate of substitution (MRS)
MRS between the average of waiting and in-vehicle times (ρµ
w,v )
• ρµ
w,v reaches its peak at values closer to the reference point of
waiting time (rw )
• The higher is the in-vehicle time, lower is ρµ
w,v . However, ρµ
w,v
begins to increase when in-vehicle times are above rv .
• ρµ
w,v are higher for females than males and higher for Santiago
than London
MRS between the variability of waiting and in-vehicle times (ρσ
w,v )
• The higher is the level of variability in both time attributes, the
greater is ρσ
w,v
• ρσ
w,v is higher for females than males in London but no diﬀerences
were found between genders in Santiago.

Acknowledgements
Acknowledgements
This research was beneﬁted from the support of:
• Becas Chile Masters Scholarship Program from the Chilean National
Commission for Scientiﬁc and Technological Research (CONICYT)
• Bus Rapid Transit Centre of Excellence, funded by the Volvo Research
and Educational Foundations (VREF)

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