1 Upper Cumulative Independence Michael H. Birnbaum California State University, Fullerton.
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Transcript of 1 Upper Cumulative Independence Michael H. Birnbaum California 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.