A prospect theory model of route choice with context dependent reference points

A prospect theory model of route choice with context
dependent reference points
International Association for Travel Behaviour Research (IATBR) 2018
Santa Barbara, CA, USA
Pablo Guarda1,2 Felipe González1 Juan Carlos Muñoz1
1Department of Transportation Engineering and Logistics (DTEL)
Pontifical Catholic University of Chile (PUC)
2Department of Experimental Psychology
University College London (UCL)
July 17, 2018

Introduction Context
Why focussing on prospect theory?
1 Prominent descriptive theory for modelling decisions under risk in
cognitive sciences
2 Compact mathematical model to represent psychological processes
inherent to human bounded rationality, which is built upon the
classical normative utilitarian models of human decision-making
3 Widely applied for modelling monetary decisions but rarely for
time-related decisions. There is an ongoing debate on the capabilities
of PT to represent travellers’ decision-making
Pablo Guarda (UCL and PUC) IATBR 2018 July 17, 2018 1 / 23

Introduction Study Overview
Study overview
Motivation
1 Time variability is a natural form of prospects, therefore, Prospect
Theory is a useful framework
Research questions
1 What are the drivers - according to PT - of the typical risk-averse
behaviour exhibited by travellers in time-related decisions?
2 Can PT increase the goodness of ﬁt obtained with standard
speciﬁcations of route choice models?

Introduction Study Overview
Study overview
Motivation
1 Time variability is a natural form of prospects, therefore, Prospect
Theory is a useful framework
Research questions
1 What are the drivers - according to PT - of the typical risk-averse
behaviour exhibited by travellers in time-related decisions?
2 Can PT increase the goodness of ﬁt obtained with standard
speciﬁcations of route choice models?
Main objectives
1 Formulate an empirically tractable PT model of route choice with
context-dependent reference points
2 Jointly estimate reference points, curvatures and slopes of the PT
model in a multi-attribute decision problem

Theoretical framework Model speciﬁcation
Impact of time variability according to PT models (1)
PT value function
u(t) =
λ+(r − t)α+
if t < r (gains)
−λ−(t − r)α−
if t ≥ r (losses)
where:
t : time outcome
r : reference point
λ+
, λ−
: Slopes of the value function in the gains and losses domains
α+
, α−
: Curvatures of the value function in the gains’ and losses’ domain

Impact of time variability according to PT models (2)
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

Subjective value of two-outcomes prospects in a single
attribute decision
Case 1: Mixed prospect
Uk (t) = w+
k (p+
k ) u+
k (t+
k ) + w−
k (p−
k ) u−
k (t−
k )
Case 2: Non-mixed prospect framed in the gains’ domain
Uk (t) = w+
k (pk,1) u+
k (tk,1) + w+
k (pk,2) u+
k (tk,2)
Case 3: Non-mixed prospect framed in the losses’ domain
Uk (t) = w−
k (pk,1) u−
k (tk,1) + w−
k (pk,2) u−
k (tk,2)

Subjective value of two-outcomes prospects in a single
attribute decision
Case 1: Mixed prospect
Uk (t) = w+
k (p+
k ) u+
k (t+
k ) + w−
k (p−
k ) u−
k (t−
k )
Case 2: Non-mixed prospect framed in the gains’ domain
Uk (t) = w+
k (pk,1) u+
k (tk,1) + w+
k (pk,2) u+
k (tk,2)
Case 3: Non-mixed prospect framed in the losses’ domain
Uk (t) = w−
k (pk,1) u−
k (tk,1) + w−
k (pk,2) u−
k (tk,2)
Subjective value of alternative j
Uj (t, p) =
∀k∈K
Uk (tk , pk |θk , αk , γk , rk )

Context-dependent reference point
Deﬁnition
Context-dependent reference points are a convex combination of the
time outcomes of the available alternatives

Deﬁnition
In theory, for each choice scenario s, a reference point rs could be
estimated for each time attribute:
rs =
js oj
ρj,otj,o
js oj
ρj,o

Definition
In theory, for each choice scenario s, a reference point rs could be
estimated for each time attribute:
rs =
js oj
ρj,otj,o
js oj
ρj,o
In practice, due to identifiability issues, we made the following
simplification:
¯rs = ρ
js oj
tj,o
Os
= ρ ¯ts

Method Task and materials
Method
1 Materials
Virtual environment programmed in PyQt
2 Cognitive task
Participants 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 and the level of variability
of the time attributes
3 Experimental conditions (between-subjects)
A priori condition: Prospects showing the probabilities of occurrence of
the time outcomes in each route
A posteriori condition: Tables showing the waiting and in-vehicle times
during a two-day period 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 Standard route choice model (SRUM)
Standard route choice model (SRUM)
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)
• Average in-vehicle time (vj ∈ 1, 2, 4, 6, 7, 9)
• 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
• Between-subjects conditions: A priori vs. A posteriori

Results Standard route choice model (SRUM)
Logit estimation results
Table 5: Binary logit model (BL) estimation results. Disaggregation by between-
subjects condition (a priori / a posteriori)
Between-Subject Conditions
Variable (t-test) A priori A posteriori Both
Average Waiting Time (θµ
w) −1.559 (−6.6) −1.949 (−6.2) −1.724 (−9.2)
Average In-Vehicle Time (θµ
v ) −1.285 (−6.3) −1.589 (−5.9) −1.412 (−8.7)
Variability Waiting Time (θσ
w) −0.346 (−5.0) −0.311 (−4.4) −0.326 (−6.6)
Variability In-Vehicle Time (θσ
v ) −0.228 (−3.4) −0.417 (−5.6) −0.317 (−6.4)
Ratio Average Waiting/Travel (γµ
w,v) 1.21***
1.23***
1.22***
Ratio Variability Waiting/Travel (γσ
w,v) 1.52*
0.75 1.03
Akaike Information Criteria (AIC) 578 539 1118
Log-likelihood -285 -266 -555
Observations 504 504 1,008

Results CPT
CPT model
Utility function
Ujn =
∀k∈K
Ujk(tjk, pjk|θk, αk, γk, rk) + ejn, ejn iid EV (0, µ)
Assumptions about parameters
1 Equal sensitivity across domains
α+
k = α−
k
2 Equal probability weighting across domains
γ+
k = γ−
k → wk(p) = 0.5

Results CPT
Models speciﬁcation
Table: CPT models speciﬁcation
Model Reference points Curvature
M1 Absolute (rk = ˆrk) No (αk = 1)
M2 Absolute (rk = ˆrk) Yes (αk = 1)
M3 Relative (rk,s = ˆρk ¯tk,s) No (αk = 1)
M4 Relative (rk,s = ˆρk ¯tk,s) Yes (αk = 1)

Results CPT
Goodness of ﬁt
Table: Comparison of log-likelihood (LL) and Akaike Information Criteria (AIC) in
CPT and SRUM models
Model Parameters LL ∆ LL ∆ AIC
SRUM 4 -554.8 0 0
M1 6 -553.8 -1.0 0.0
M2 8 -551.2 3.6 0.8
M3 6 -546.6 8.2 -12.4
M4 8 -542.7 12.1 -16.2

Results CPT
Value functions using absolute reference points (rk = ˆrk)
M1 (αk = 1)
0 2 4 6 8 10
−10−505
Outcome [t]
SubjectiveUtility
M2 (αk = 1)
0 2 4 6 8 10
−10−505
Outcome [t]
SubjectiveUtility

Results CPT
Value functions using relative reference points
(rk,s = ˆρk ¯tk,s)
M3 (αk = 1)
0 2 4 6 8 10
−10−505
Outcome [t]
SubjectiveUtility
M4 (αk = 1)
0 2 4 6 8 10
−10−505
Outcome [t]
SubjectiveUtility

Results CPT vs SRUM
Loss aversion
Table: Ratio of the slopes of the value functions in the losses and gains domains
Model SRUM M1 M2 M3 M4
In-vehicle time ratio 1 1 0.99 1.24 1.29
Waiting time ratio 1 1.44 2.05 1.94 1.78

Conclusions CPT vs SRUM model
Conclusions: CPT vs SRUM model
Goodness of fit
• The CPT models (M1 and M2) with absolute reference points
were not statistically different than the SRUM model
• The CPT models (M3 and M4) with relative reference points had
a higher statistical fit than the SRUM model (∆AIC > 10)

Conclusions: CPT vs SRUM model
Goodness of fit
• The CPT models (M1 and M2) with absolute reference points
were not statistically different than the SRUM model
• The CPT models (M3 and M4) with relative reference points had
a higher statistical fit than the SRUM model (∆AIC > 10)
Context-dependence in reference point
• Setting the reference point as the mean of the distribution of time
outcomes is a plausible assumption for in-vehicle times
(ρv = [1.00−1.04]) but not for waiting times (ρw = [0.77−0.94])
• The specifications using context-dependent reference points
increase parameters’ stability

Conclusions: CPT parameters
Loss aversion
• Consistent with PT, the models’ estimation results confirmed the
presence of loss aversion for both time attributes
• The only exception was model M2 where the ratio of the slopes
was not significantly different than 1 for in-vehicle times

Conclusions: CPT parameters
Loss aversion
• Consistent with PT, the models’ estimation results confirmed the
presence of loss aversion for both time attributes
• The only exception was model M2 where the ratio of the slopes
was not significantly different than 1 for in-vehicle times
Curvatures
• Consistent with PT, the curvatures for both time attributes were
between 0 and 1 (diminishing sensitivity)
• When reference points are a convex-combination of the prospects’
outcomes, the curvatures mediates risk-averse and risk-seeking
behaviour.

Further Research Further research
Further research
Estimation procedure
• The parameters obtained via maximum log-likelihood estimation
(MLE) are sensitive to the starting points (local optimums).
• Compares the parameters’ estimates obtained via MLE and
Bayesian estimation

Further research
Estimation procedure
• The parameters obtained via maximum log-likelihood estimation
(MLE) are sensitive to the starting points (local optimums).
• Compares the parameters’ estimates obtained via MLE and
Bayesian estimation
Marginal rates of substitution (MRS)
• The SRUM model, by construction, assumes that the MRS are
constant.
• The CPT model allows a richer representation of the MRS
between (i) the mean of the attributes, (ii) the variality of the
attributes, and the mean and variability of each attribute (risk
premiums)

Thanks for your attention

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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