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Upper Cumulative Independence. Michael H. Birnbaum California State University, Fullerton. 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. - PowerPoint PPT Presentation
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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
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′ ′ ′ 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
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′ 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.