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Upper Cumulative Independence

Michael H. BirnbaumCalifornia State University,

Fullerton

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UCI is implied by CPT

• CPT, RSDU, RDU, EU satisfy UCI.• RAM and TAX violate UCI. • Violations are direct internal

contradiction in RDU, RSDU, CPT, EU.

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€

′ z > ′ x > x > y > ′ y > z > 0

′ S → ( ′ z ,1− p − q;x, p;y,q)

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

In this test, we reduce z’ in both gambles and coalesce it with x’ (in R’), and we decrease x and coalesce it with y (in S’ only).

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Lower Cumulative Independence (3-LCI)

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′ S = ( ′ z ,1− p − q;x, p;y,q) p

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

⇒

′ ′ ′ S = ( ′ x ,1− p − q;y, p + q) p

′ ′ ′ R = ( ′ x ,1− q; ′ y ,q)

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UCI implied by any model that satisfies:

• Comonotonic restricted branch independence

• Consequence monotonicity• Transitivity• Coalescing• (Proof on next page.)

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€

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

( ′ x ,1− p − q;x, p;y,q) p ( ′ x ,1− p − q; ′ x , p; ′ y ,q)

( ′ x ,1− p − q;y, p;y,q) p ( ′ x ,1− p − q;x, p;y,q)

( ′ x ,1− p − q;y, p;y,q) p ( ′ x ,1− p − q; ′ x , p; ′ y ,q)

( ′ x ,1− p − q;y, p + q) p ( ′ x ,1− q; ′ y ,q)

′ ′ ′ S p ′ ′ ′ R

Comonotonic RBI

Consequence monotonicity

Transitivity

Coalescing

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Example Test

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′ S : 10 to win $40 10 to win $44

80 to win $110

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′ R : 10 to win $10 10 to win $98

80 to win $110

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′ ′ ′ S : 20 to win $40 80 to win $98

€

′ ′ ′ R : 10 to win $10 90 to win $98

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Generic Configural Model

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

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′ S p ′ R ⇔

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⇔w3

w2

<u( ′ x ) − u(x)

u(y) − u( ′ y )

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3-2-LCI in CPT

Suppose

CPT satisfies coalescing;

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′ ′ ′ S f ′ ′ ′ R ⇔

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

⇔ w2u(y) + w3u(y) > w2u( ′ x ) + w3u( ′ y )

⇔w3

w2

>u( ′ x ) − u(y)

u(y) − u( ′ y )>

u( ′ x ) − u(x)

u(y) − u( ′ y )⇒⇐ contradiction

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′ S p ′ R ⇔w3

w2

<u( ′ x ) − u(x)

u(y) − u( ′ y )

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2 Types of Reversals:

R’S’’’: This is a violation of UCI. It refutes CPT.

S’R’’’: This reversal is consistent with LCI. (S’ made worse relative to R’.)

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RAM Weights

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w1 = a(1,3)t(1− p − q) /T

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

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

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

′ ′ ′ w 1 = a(1,2)t(1− p − q) / ′ ′ ′ T

′ ′ ′ w 2 = a(2,2)t(p + q) / ′ ′ ′ T

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

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RAM Violations• RAM violates 3-2-UCI. If t(p) is

negatively accelerated, RAM violates coalescing: coalescing branches with better consequences makes the gamble worse and coalescing the branches leading to lower consequences makes the gamble better. Even though we made S relatively worse, the coalescings made it relatively better.

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TAX Model

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w1 =t(1− p − q)[1−δ /4 −δ /4]

t(1− p − q) + t( p) + t(q)

w2 =t( p) −δt(p) /4 + δt(1− p − q) /4

t(1− p − q) + t(p) + t(q)

w3 =t(q) + δt(1− p − q) /4 + δt(p) /4

t(1− p − q) + t(p) + t(q)

′ ′ ′ w 1 =t(1− p − q) −δt(1− p − q) /3

t(1− p − q) + t( p + q)

′ ′ ′ w 2 =t( p + q) + δt(1− p − q) /3

t(1− p − q) + t( p + q)

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TAX: Violates UCI

• Special TAX model violates 3-2-UCI. Like RAM, the model violates coalescing.

• Predictions were calculated in advance of the studies, which were designed to investigate those specific predictions.

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Summary of Predictions

• EU, CPT, RSDU, RDU satisfy UCI• TAX & RAM violate UCI• CPT defends the null hypothesis

against specific predictions made by both RAM and TAX.

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Birnbaum (‘99): n = 124

No.No Choice %R

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€

′ S: 10 to win $40

10 to win $44

80 to win $110

€

′ R : 10 to win $10 10 to win $98

80 to win $110

72*

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′ ′ ′ S : 20 to win $40 80 to win $98

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′ ′ ′ R : 10 to win $10 90 to win $98

34*

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Lab Studies of UCI

• Birnbaum & Navarrete (1998): 27 tests; n = 100; (p, q) = (.25, .25), (.1, .1), (.3, .1), (.1, .3).

• Birnbaum, Patton, & Lott (1999): n = 110; (p, q) = (.2, .2).

• Birnbaum (1999): n = 124; (p, q) = (.1, .1), (.05, .05).

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Web Studies of UCI

• Birnbaum (1999): n = 1224; (p, q) = (.1, .1), (.05, .05).

• Birnbaum (2004b): 12 studies with total of n = 3440 participants; different formats for presenting gambles probabilities; (p, q) = (.1, .1), (.05, .05).

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Additional Replications A number of as unpublished

studies (as of Jan, 2005) have replicated the basic findings with a variety of different procedures in choice.

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′ S = ($ 110 , . 8 ; $ 44 , . 1 ; $ 40 , . 1 ) versus ′ R = ($ 110 ; . 8 ; $ 98 , . 1 ; $ 10 , . 1 ) , ′ ′ ′ S = ($ 98 , . 8 ; $ 40 , . 2 ) versus ′ ′ ′ R = ($ 98 , . 9 ; $ 10 , . 1 )

Choice Pattern

Choices 10 and 9

Condition

n ′ S ′ ′ ′ S ′ S ′ ′ ′ R ′ R ′ ′ ′ S ′ R ′ ′ ′ R

new tickets 141 38 11 71 21

aligned 141 36 5 74 23

unaligned 151 34 9 81 25

Negative

(reflected)200 X

277 42 157 123

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Error Analysis

• “True and Error” Model implies violations are “real” and cannot be attributed to error.

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Violations predicted by RAM & TAX, not CPT

• EU, CPT, RSDU, RDU are refuted by systematic violations of UCI.

• TAX & RAM, as fit to previous data correctly predicted the violations. Predictions published in advance of the studies.

• Violations are to CPT as the Allais paradoxes are to EU.

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To Rescue CPT:

For CPT to handle these data, make

it configural. Let < 1 for two-

branch gambles and > 1 for three-branch gambles.

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Add to the case against CPT/RDU/RSDU

• Violations of Upper Cumulative Independence are a strong refutation of CPT model as proposed.

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Next Program: UTI

• The next programs reviews tests of Upper Tail Independence (UTI).

• Violations of 3-UTI contradict any form of CPT, RSDU, RDU, including EU.

• Violations contradict Lower GDU.• They 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