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Lower Cumulative Independence

Michael H. BirnbaumCalifornia State University,

Fullerton

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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)

€

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

€

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

€

S f R ⇔

€

⇔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;

€

′ ′ S p ′ ′ R ⇔

€

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:

€

S f R ⇒

w2

w1

=q

p>

u( ′ x ) − u(x)

u(y) − u( ′ y )⇒ ′ ′ S f ′ ′ R

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RAM Weights

€

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

€

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.

mbirnbaum@fullerton.edu