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Distribution Independence

Michael H. BirnbaumCalifornia State University,

Fullerton

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4-DI is violated by CPT

If W(P) is nonlinear, we should be able to predict violations of 4-DI from CPT.

• RAM satisfies 4-DI• TAX violates 4-DI in the

opposite way as CPT with its inverse-S weighting function.

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€

′ z > ′ x > x > y > ′ y > z > 0

S → ( ′ z ,r;x, p;y, p;z,1− 2p − r)

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

We manipulate r in both gambles, r’ > r. This changes where the two equally probable branches fall with respect to the gamble’s distribution.

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4-Distribution Independence (4-DI)

€

S = ( ′ z ,r;x, p;y, p;z,1− 2p − r) f

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

⇔

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

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

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Example Test of 4-DIS: 59 to win $3

20 to win $45

20 to win $49

01 to win $109

R: 59 to win $3

20 to win $11

20 to win $97

01 to win $109

S’: 01 to win $3

20 to win $45

20 to win $49

59 to win $109

R’: 01 to win $3

20 to win $11

20 to win $97

59 to win $109

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Generic Configural Model

€

w1u( ′ z ) + w2u(x) + w3u(y) + w4u(z) > w1u( ′ z ) + w2u( ′ x ) + w3u( ′ y ) + w4u(z)

€

S f R ⇔

€

⇔w3

w2

>u( ′ x ) − u(x)

u(y) − u( ′ y )

There will be no violation if this ratio is independent of r

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CPT Analysis of S vs. R

Choice between S and R, r small

0

1

0 1

Decumulative Probability, P

Decumulative Weight, W(P)

r p p 1 - 2p - r

w2 > w3

w1

w2

w3

w4

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CPT Analysis of S’ vs. R’Choice between S' and R', r' large

0

1

0 1

Decumulative Probability, P

Decumulative Weight, W(P)

r' p p 1 - 2p - r

w2 < w3

w1

w2

w3

w4

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Violation of 4-DI in CPT

€

w2 > w3 ⇒w3

w2

<1∩ ′ w 2 < ′ w 3 ⇒ 1<′ w 3′ w 2

R ′ S : S p R∧ ′ S f ′ R ⇔w3

w2

<u( ′ x ) − u(x)

u(y) − u( ′ y )<

′ w 3′ w 2

If W(P) has inverse-S shape, the ratios

depend on r. CPT implies RS’.

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RAM Weights

€

w1 = a(1,4)t(r) /T

w2 = a(2,4)t(p) /T

w3 = a(3,4)t(p) /T

w4 = a(4,4)t(1− 2 p − r) /T

T = a(1,4)t(r) + a(2,4)t(p) + a(3,4)t(p) +

+a(4,4)t(1− 2p − r)

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RAM Satisfies 4-DI• RAM satisfies 4-DI because the

ratio of weights is independent of r.

€

w3

w2

=a(3,4)t(p)

a(2,4)t(p)=

′ w 3′ w 2

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TAX Model

€

w1 =t(r) − 3δt(r) /5

t(r) + t(p) + t( p) + t(1− 2 p − r)

w2 =t( p) −δt(p) /5 −δt(p) /5 + δt(r) /5

t(r) + t(p) + t( p) + t(1− 2 p − r)

w3 =t( p) −δt(p) /5 + δt(p) /5 + δt(r) /5

t(r) + t(p) + t( p) + t(1− 2 p − r)

w4 =t(1− 2p − r) + δt( p) /5 + δt( p) /5 + δt(r) /5

t(r) + t( p) + t(p) + t(1− 2p − r)

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TAX Model Implies SR’• TAX violates 4-DI in the opposite

pattern as CPT with inverse-S.• Weight ratios:

• This implies the SR’ pattern.

€

w3

w2

=t( p) −δt(p) /5 + δt(p) /5 + δt(r) /5

t( p) −δt(p) /5 −δt(p) /5 + δt(r) /5>

′ w 3′ w 2

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Summary of Predictions

• EU and RAM satisfy 4-DI.• CPT as fit to previous data violates

4-DI with RS’ pattern. • TAX as fit to previous data predicts

the SR’ pattern of violations.

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Study of 4-DI• Birnbaum, M. H., & Chavez, A. (1997).

Tests of Theories of Decision Making: Violations of Branch Independence and Distribution Independence. Organizational Behavior and Human Decision Processes, 71, 161-194.

• 100 participants, 12 tests with (r, r’) = (.01, .59) and (.05, .55).

• Study also tested RBI and other properties. Significantly more SR’ than RS’ violations.

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Example TestS: 59 to win $3

20 to win $45

20 to win $49

01 to win $109

R: 59 to win $3

20 to win $11

20 to win $97

01 to win $109

S’: 01 to win $3

20 to win $45

20 to win $49

59 to win $109

R’: 01 to win $3

20 to win $11

20 to win $97

59 to win $109

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Results for this Example

Choice Pattern

SS’ SR’ RS’ RR’

43 23* 6 28

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Violations predicted by TAX, not CPT

• EU and RAM are refuted by systematic violations of 4-DI.

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

• CPT, with its inverse-S weighting function predicted opposite pattern.

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To Rescue CPT:

• CPT can handle the results if it uses an S-shaped rather than an inverse-S shaped weighting function.

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Summary

Property CPT RAM TAX

4-DI RS’Viols No Viols SR’ Viols

UDI S’R2’

ViolsNo Viols R’S2’

Viols

RBI RS’ Viols SR’ Viols SR’ Viols

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Summary-Continued

Property CPT RAM TAX

LCI No Viols Viols Viols

UCI No Viols Viols Viols

UTI No Viols R’S1Viols R’S1Viols

LDI RS2 Viols No Viols No Viols

3-2 LDI RS2 Viols No Viols No Viols

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Summary-Continued

• CPT violates RBI, 4-DI, and UDI, but the results show the opposite pattern. It violates 3-LDI and 3-2 LDI, but violations not found. CPT satisfies LCI, UCI, and UTI, but there are systematic violations.

• TAX correctly predicts all 8 results; RAM correct in 6 cases where it agrees with TAX; RAM disproved by violations of 4-DI and UDI.

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End of Series on Tests of Independence

• This presentation concludes the series on Lower and Upper Cumulative Independence, Lower and Upper Distribution Independence, Upper Tail Independence, Restricted Branch Independence, and 4-Distribution Independence.

• If you have not yet viewed them, the series of programs on Stochastic Dominance Violations and Allais Paradoxes will also be of interest, as will the separate programs on various models of decision making.

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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