NETWORK EQUILIBRIUM UNDER CUMULATIVE PROSPECT THEORY WITH ENDOGENOUS STOCHASTIC DEMAND & SUPPLY

Dr. Agachai SumaleeHong Kong Polytechnic University

Based on the paper Sumalee et al (2009) presented at ISTTT

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Two-link Network

TT on route 1

TT on route 2

How do drivers choose between routes with uncertain travel times?

Origin Destination

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Approaches

Mean Travel Time ⇒ Route 1Route 1

Route 2

Prob Late Arrival ⇒ Route 2

Mean + stdDevMean + Variance95th Percentile of TT distribution…

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Empirical Evidence leads to PT

• Choice modellers have shown that decisions under uncertainty do not conform to maximising the expected benefit.

• The empirical evidence suggests two major modifications to ‘Expected Utility Maximisation’:

• the carriers of value are gains/losses relative to a reference point

• the value of each outcome is multiplied by a decision weight, not by an additive probability.

This generalisation of EUmax is Prospect Theory

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Context

• A traffic network has many sources of variability in demand and supply: results in Travel Time Variability.

• Drivers are aware that travel times are uncertain and include this TTV in their decision making.

• Senbil, M., & Kitamura, R., (2004). Reference points in commuter departure time choice: a prospect theoretic test of alternative decision frames. Journal of Intelligent Transportation Systems 8, 19–31

• Jou, R.C., Kitamura, R., Weng, M.C., Chen, C.C., (2008). Dynamic commuter departure time choice under uncertainty. Transportation Research Part A 42(5), 774-783.

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Empirical evidence

Senbil and Kitamura (2004)

7/280 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

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probability

Decision Weight Transform

Gain WeightsLoss Weights

Prospect Theory Transformations

The fundamental embodiment (and impact) of PT is through the following two transformations:

Value Function g(.) Probability Weighting Function w(.)

Reference Point

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Reference Dependence

Empirical studies show that people’s preferences over finaloutcomes depend on the reference point from which they arejudged:Kahneman & Tversky (1979) for decision under risk

Kahneman et al (1990) for choice among commodity bundles

Loewenstein & Prelec (1992) for inter-temporal choice

Bateman et al (1997) for contingent valuation

Dolan & Robinson (2001) for welfare theory

Bleichrodt & Pinto (2002) for multi-attribute utility

Senbil&Kitamura (2004) for departure time choice

Jou, Kitamura et al (2008) for departure time choice

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Travel Time → Utility

If we have a distribution of path costs

It is convenient to consider the utility ( )( )~ ,k k kC N c σx

( )k dest k dest k kU U C U c ε= − = − −x

Travel Time Distribution

Utility Distribution

LossesGains Losses Gains

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Applying the CPT Transforms

Weighting FunctionMaps Probabilities

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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probability

Decision Weight Transform

Gain WeightsLoss Weights

Value Function Maps Outcome Utilities

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CPT Transformations

Utility: Cumulative Distribution

Losses Gains

12/28Losses Gains

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Cumulative Prospect Value

( )( ) ( )0u

Udw F ug u du

du−∞

+ ⋅∫( )( ) ( )

0

1 U

u

dw F ucpv g u du

du

∞ −= − ⋅∫

CPV for continuous distribution of outcomes

FU (u) = Path Utility CDF (the outcome distribution)g(.) = Value Functionw(.) = Weight Functionu0 = Reference Point

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Two-link Network

How do drivers choose between routes with uncertain travel times?

Origin Destination

Choose between alternatives according to CPV

CPV = 0.5

CPV = -1.0

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Making Variability Endogenous

• Part B Paper Stochastic link travel times with exogenously defined distribution

Reference point (and other CPT parameters) assumed

Compute equilibrium: show dependence on RefPt and other parameters

• Extension here – to endogenize variability Stochastic OD demand…

…split into path flows via ratio of means f/q hence stochastic path flows

Gives rise to stochastic link flows (with correlations)

Independently stochastic link capacities

Can compute resulting stochastic travel times (with correlations)

CPT applied to route choice

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Stochastic Demand

Lognormal Demand Q ~ LN(μ,σ) with mean q

Path Flows Lognormal Distributed Fk ~ LN(μk,σk)

Conservation condition on means:

Gives stochastic, correlated link flows.k

kq f= ∑

Split Q into path flows Fk = (fk/q)Q

Where fk is mean flow on path k

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Stochastic Supply

Link capacities are assumed to be independent random variables

01

an

aa a

a

XT t bC

= +

With Ca ~ LN(μca,σca) and independent of Xa

We therefore have non-trivial covariance matrices for link flows and link travel times.

Path costs arise from standard link-additive model.

BUT CPV is not link-additive!

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Equilibrium Condition

With cpvk(x) the flow dependent CPV on route k we have that (demand feasible) f* is a CPV-UE if and only if:

( ) ( )* * 0T

F− ≤ ∀ ∈cpv f f f f

( ) min G∈Ωf

f

( ) ( ) ( )( )max T TG∈Ω

= ⋅ − ⋅g

f cpv f g cpv f f

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Equilibrium assignment algorithm

• Generate path set. Find initial feasible mean path flow f

• Compute travel time distribution for each path.

• Compute prospect value for each path & find ‘best’ path

• Assign mean OD flow to best path; gives auxiliary flow g

• Step from f toward g (as in MSA)

• Test convergence

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Nguyen and Dupuis Network

1 12

5 6 4 7 8

9 10 11 2

13 3

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8 6 11 10

14 12

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4

19

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Destination

Destination

Origin

Orig¥Dest 2 31 1000 (0.2) 800 (0.25)4 1500 (0.2) 1000 (0.25)

OD demand (CoV)

Origin

Link Capacities CoV = 0.1Reference Points = 100 (initially)

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Iteration k

Mea

n flo

w o

n pa

th 1

Reference point = 100Reference point = 160Reference point = 200

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5 6 4 7 8

9 10 11 2

13 3

1

2

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5 3 9 7

8 6 11 10

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16 13

4

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Destination

Destination

Origin

Origin

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1 12

5 6 4 7 8

9 10 11 2

13 3

1

2

18 17

5 3 9 7

8 6 11 10

14 12

16 13

4

19

15

Destination

Destination

Origin

Origin

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Summary

Cumulative Prospect Theory as paradigm for route choice with stochastic travel times.

Continuous formulation of cumulative prospect theory presented.Equilibrium condition defined. Potential future studies: Solve for equilibrium efficiently and consider larger networks Path based criterion so needs thought

Network inference (estimation) from the observed link/path data Investigate impact of assuming CPV over E[Umax] or ‘reliability’ terms How is UE recovered from CPV in the limit?

Define reference point via a day-to-day learning process? Initial reference point updated by experience, new alternatives added to choice set

as reference point changes? Does this converge? To a sensible solution? Is this process stable?