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Upper Distribution Independence

Michael H. BirnbaumCalifornia State University,

Fullerton

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UDI is Violated CPT

• CPT violates UDI but EU and RAM satisfy it.

• TAX violates UDI in the opposite direction as CPT.

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€

′ z > ′ x > x > y > ′ y > 0

S → ( ′ z ,1− 2p;x, p;y, p)

R → ( ′ z ,1− 2p; ′ x , p; ′ y , p)The upper branch consequence, z’, has different probabilities in the two choices.

€

′ p > p⇒ 1− 2 ′ p <1− 2p

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Upper Distribution Independence (3-UDI)

€

′ S = ( ′ z ,1− 2p;x, p;y, p) f

′ R = ( ′ z ,1− 2p; ′ x , p; ′ y , p)

⇔

S ′ 2 = ( ′ z ,1− 2 ′ p ;x, ′ p ;y, ′ p ) f

R ′ 2 = ( ′ z ,1− 2 ′ p ; ′ x , ′ p ; ′ y , ′ p )

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Example Test

S’: .10 to win $40

.10 to win $44

.80 to win $100

R’: .10 to win $4

.10 to win $96

.80 to win $100

S2’: .45 to win $40

.45 to win $44

.10 to win $100

R2’: .45 to win $4

.45 to win $96

.10 to win $100

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Generic Configural Model

€

U(G) = w1u( ′ z ) + w2u(x) + w3u(y)

where

€

u( ′ z ) > u(x) > u(y) > 0

CPT, RAM, and TAX disagree on

€

w1,w2,w3

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Generic Configural Model

€

w1u( ′ z ) + w2u(x) + w3u(y) > w1u( ′ z ) + w2u( ′ x ) + w3u( ′ y )

The generic model includes RDU, CPT, EU, RAM, TAX, GDU, as special cases.

€

′ S f ′ R ⇔

€

⇔w3

w2

>u( ′ x ) − u(x)

u(y) − u( ′ y )

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Violation of 3-UDI

€

′ w 1u( ′ z ) + ′ w 2u(x) + ′ w 3u(y) < ′ w 1u( ′ z ) + ′ w 2u( ′ x ) + ′ w 3u( ′ y )

A violation will occur if S’ f R’ and

€

S ′ 2 p R ′ 2 ⇔

€

⇔′ w 3′ w 2

<u( ′ x ) − u(x)

u(y) − u( ′ y )

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2 Types of Violations:

€

′ S f ′ R ∧S ′ 2 p R ′ 2 ⇔w3

w2

>u( ′ x ) − u(x)

u(y) − u( ′ y )>

′ w 3′ w 2

€

′ S p ′ R ∧S ′ 2 f R ′ 2 ⇔w3

w2

<u( ′ x ) − u(x)

u(y) − u( ′ y )<

′ w 3′ w 2

S’R2’:

R’S2’:

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EU allows no violations

• In EU, the weights are equal to the probabilities; therefore

€

w3

w2

=p

p=

′ p ′ p =

′ w 3′ w 2

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RAM Weights

€

w1 = a(1,3)t(1− 2p) /T

w2 = a(2,3)t(p) /T

w3 = a(3,3)t(p) /T

T = a(1,3)t(1− 2 p) + a(2,3)t( p) + a(3,3)t( p)

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RAM allows no Violations

• RAM model with any parameters satisfies 3-UDI.

€

w3

w2

=a(3,3)t(p)

a(2,3)t(p)=

a(3,3)t( ′ p )

a(2,3)t( ′ p )=

′ w 3′ w 2

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Cumulative Prospect Theory/ RDU

€

w1 = W (1− 2p)

w2 = W (1− p) −W (1− 2p)

w3 =1−W (1− p)

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CPT implies violations

• If W(P) = P, CPT reduces to EU.• From previous data, we can

calculate where to expect violations and predict which type of violation should be observed.

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CPT Analysis of 3-UDI Choices 15 & 18

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4Weighting Function Parameter, γ

, Exponent of Utility Function

β

2R'S '

2S'R '

2S'S '

2R'R '

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CPT implies S’R2’ Violations

• When γ = 1, CPT reduces to EU.• Given the inverse-S weighting function,

the fitted CPT model implies S’R2’ pattern.

• If γ > 1, S-Shaped, but the model can handle the opposite pattern.

• A series of tests can be devised to provide overlapping combinations of parameters.

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TAX Model

Each term has the same denominator; however, unlike the case of LDI, here the middle branch can gain more weight than it gives up.

€

w1 =t(1− 2p) − 2δt(1− 2 p) /4

t(1− 2 p) + t(p) + t( p)

w2 =t( p) + δt(1− 2p) /4 −δt( p) /4

t(1− 2p) + t( p) + t(p)

w3 =t( p) + δt(1− 2p) /4 + δt( p) /4

t(1− 2p) + t( p) + t(p)

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Special TAX: R’S2’ Violations

• Special TAX model violates 3-UDI. • Here the ratio depends on p.

€

w3

w2

=t( p) + δt(1− 2p) /4 + δ t(p) /4

t( p) + δt(1− 2p) /4 −δ t(p) /4<

′ w 3′ w 2

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Summary of Predictions

• RAM, & EU satisfy 3-UDI• CPT violates 3-UDI: S’R2’

• TAX violates 3-UDI: R’S2’

• Here CPT is the most flexible model, RAM defends the null hypothesis, TAX makes opposite prediction from that of CPT.

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Results n = 1075

€

′ S

€

′ R 1075

.10 to win $40

.10 to win $44

.80 to win $100

.10 to win $4

.10 to win $96

.80 to win $100

56

.45 to win $40

.45 to win $44

.10 to win $100

.45 to win $4

.45 to win $96

.10 to win $100

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R’S2’

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Results: n = 503

20 white to win $28

20 blue to win $30

60 red to win $100

20 yellow to win $4

20 green to win $96

60 black to win $100

70.8*

45 white to win $28

45 purple to win $30

10 blue to win $100

45 black to win $4

45 green to win $96

10 red to win $100

59.1*

R’R2’ (CPT predicted S’R2’ )

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Summary: Observed Violations fit TAX, not CPT

• RAM and EU are refuted in this case by systematic violations.

• TAX model, fit to previous data correctly predicted the modal choices.

• Violations opposite those implied by CPT with its inverse-S W(P) function.

• Fitted CPT was correct when it agreed with TAX, wrong otherwise.

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To Rescue CPT:

• CPT can handle the result of any single test, by choosing suitable parameters.

• For CPT to handle these data, let γ

> 1; i.e., an S-shaped W(P) function, contrary to previous inverse- S.

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CPT Analysis of 3-UDI Choices 15 & 18

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4Weighting Function Parameter, γ

, Exponent of Utility Function

β

2R'S '

2S'R '

2S'S '

2R'R '

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Adds to the case against CPT/RDU/RSDU

• Violations of 3-UDI favor TAX over RAM and are opposite predictions of CPT.

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Preview of Next Program

• The next programs reviews tests of Restricted Branch Independence (RBI).

• It turns out the violations of 3-RBI are opposite the predictions of CPT with inverse-S function.

• They refute EU but are consistent with RAM and TAX.

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For More Information:

http://psych.fullerton.edu/mbirnbaum/

Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers.

mbirnbaum@fullerton.edu