lower cumulative independence
DESCRIPTION
Lower Cumulative Independence. Michael H. Birnbaum California State University, Fullerton. LCI is implied by CPT. EU satisfies LCI CPT, RSDU, RDU satisfy LCI. RAM and TAX violate LCI. Violations are direct internal contradiction in RDU, RSDU, CPT, EU. - PowerPoint PPT PresentationTRANSCRIPT
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LCI is implied by CPT
• EU satisfies LCI• CPT, RSDU, RDU satisfy LCI.• RAM and TAX violate LCI. • Violations are direct internal
contradiction in RDU, RSDU, CPT, EU.
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′ x > x > y > ′ y > z > 0
S → (x, p;y,q;z,1− p − q)
R → ( ′ x , p; ′ y ,q;z,1− p − q)
In this test, we increase z in both gambles and coalesce it with y’ (in R), and we increase y and coalesce it with x (in S only). So we have improved S relative to R.
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Lower Cumulative Independence (3-2-LCI)
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S = (x, p;y,q;z,1− p − q) f
R = ( ′ x , p; ′ y ,q;z,1− p − q)
⇒
′ ′ S = (x, p + q; ′ y ,1− p − q) f
′ ′ R = ( ′ x , p; ′ y ,1− p)
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LCI implied by any model that satisfies:
• Consequence monotonicity• Transitivity• Coalescing• Comonotonic restricted branch
independence
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Example Test of 3-2-LCI
S: 90 to win $3
05 to win $48
05 to win $52
R: 90 to win $3
05 to win $12
05 to win $96
S”: 90 to win $12
10 to win $52
R”: 95 to win $12
05 to win $96
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Generic Configural Model
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w1u(x) + w2u(y) + w3u(z) > w1u( ′ x ) + w2u( ′ y ) + w3u(z)
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S f R ⇔
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⇔w2
w1
>u( ′ x ) − u(x)
u(y) − u( ′ y )
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CPT Cannot Handle Violation of 3-2-LCI
Suppose
CPT satisfies coalescing;
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′ ′ S p ′ ′ R ⇔
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w1u(x) + w2u(x) + w3u( ′ y ) < w1u( ′ x ) + w2u( ′ y ) + w3u( ′ y )
⇔ w1u(x) + w2u(x) < w1u( ′ x ) + w2u( ′ y )
⇔w2
w1
<u( ′ x ) − u(x)
u(x) − u( ′ y )<
u( ′ x ) − u(x)
u(y) − u( ′ y )⇒⇐ contradiction
€
S f R ⇔w2
w1
>u( ′ x ) − u(x)
u(y) − u( ′ y )
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2 Types of Reversals:
SR”: This is a violation of LCI. If systematic, it refutes CPT.
RS”: This reversal is perfectlyconsistent with LCI. (We Improved S” relative to R”.)
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EU allows no violations
EU implies LCI:
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S f R ⇒
w2
w1
=q
p>
u( ′ x ) − u(x)
u(y) − u( ′ y )⇒ ′ ′ S f ′ ′ R
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RAM Weights
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w1 = a(1,3)t(p) /T
w2 = a(2,3)t(q) /T
w3 = a(3,3)t(1− p − q) /T
T = a(1,3)t(p) + a(2,3)t(q) + a(3,3)t(1− p − q)
′ ′ w 1 = a(1,2)t(p + q) / ′ ′ T
′ ′ w 2 = a(2,2)t(1− p − q) / ′ ′ T
′ ′ T = a(1,2)t(p + q) + a(2,2)t(1− p − q)
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RAM Violations• RAM violates 3-LCI. Even when the
a(i,n) are all equal, 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 improved S, we made R seem better.
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TAX Model
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w1 =t( p) −δt(p) /4 −δt(p) /4
t( p) + t(q) + t(1− p − q)
w2 =t(q) −δt(q) /4 + δt(p) /4
t( p) + t(q) + t(1− p − q)
w3 =t(1− 2p) + δt( p) /4 + δt( p) /4
t( p) + t(q) + t(1− p − q)
′ ′ w 1 =t( p + q) −δt(p + q) /3
t(p + q) + t(1− p − q)
′ ′ w 2 =t(1− p − q) + δt(p + q) /3
t(p + q) + t(1− p − q)
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TAX: Violates LCI
• Special TAX model violates 3-LCI. Like RAM, the model violates coalescing.
• Predictions were calculated in advance of the studies, which were designed to test those specific predictions.
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TAX Model Violates LCIAccording to parameters estimated from previous data, TAX implies a violation of LCI in the example used. The prediction is fairly robust with respect to the parameters, and holds for > .36 and < 1.06. As shown later, violations are observed here, showing that at least some participants have parameters in this region. By adjusting the consequences, we can test other combinations of parameters.
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Summary of Predictions
• EU, CPT satisfy LCI• TAX & RAM violate LCI• Here CPT defends the null
hypothesis against a specific prediction made by both RAM and TAX. Predictions were made in advance and studies designed to test them.
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Birnbaum (‘99), n = 124
No Safe Risky %Risky
17 S: 90 to win $3
05 to win $48
05 to win $52
R: 90 to win $3
05 to win $12
05 to win $96
39*
20 S”: 90 to win $12
10 to win $52
R”: 95 to win $12
05 to win $96
69*
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Lab Studies of LCI
• Birnbaum & Navarrete (1998): 27 tests; n = 100; (p, q) = (.25, .25), (.1, .1), (.3, .1), (.1, .3).
• Birnbaum, Patton, & Lott (1999): n = 110; (p, q) = (.2, .2).
• Birnbaum (1999): n = 124; (p, q) = (.1, .1), (.05, .05).
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Web Studies of LCI
• Birnbaum (1999): n = 1224; (p, q) = (.1, .1), (.05, .05).
• Birnbaum (2004b): 12 studies with total of n = 3440 participants; different formats for presenting gambles probabilities; (p, q) = (.1, .1), (.05, .05).
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Additional Replications A number of as yet unpublished
studies have replicated the basic findings with a variety of different procedures in choice.
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S = ($ 44 , . 1 ; $ 40 , . 1 ; $ 2 , . 8 ) . vs R = ($ 98 , . 1 ; $ 10 , . 1 ; $ 2 , . 8 )
′ ′ S = ($ 44 , . 2 ; $ 10 , . 8 ) vs. ′ ′ R = ($ 98 ; . 1 ; $ 10 , . 9 )
Choice Pattern
Condition n S ′ ′ S S ′ ′ R R ′ ′ S R ′ ′ R
new tickets 141 20 67 11 39
aligned 141 32 48 12 49
unaligned 151 29 52 13 55
Negative
(reflected)
200 X2
82 96 54 165
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Error Analysis
• We can fit “true and error” model to data with replications to separate “real” violations from those attributable to “error”.
• Model implies violations are “real” and cannot be attributed to error.
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Violations predicted by RAM & TAX, not CPT
• EU and CPT are refuted by systematic violations of LCI.
• TAX & RAM, as fit to previous data correctly predicted the modal choices. Predictions published in advance of the studies.
• Violations of LCI are to CPT as the Allais paradoxes are to EU.
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To Rescue CPT:
• CPT cannot handle the results unless it becomes a configural model.
• For CPT to handle these data, allow different W(P) functions depending on the number of branches in the gambles.
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 Lower Cumulative Independence are a strong refutation of CPT model as proposed.
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Next Program: UCI
• The next programs reviews tests of Upper Cumulative Independence (UCI).
• Violations of 3-2-UCI contradict any form of RDU, CPT, including EU.
• They are consistent with (and were predicted by) 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.