1

Restricted Branch Independence

Michael H. BirnbaumCalifornia State University,

Fullerton

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RBI is Violated by CPT

• EU satisfies RBI as does SWU and PT, extended to 3-branch gambles. Cancellation

• CPT violates RBI (it MUST to explain the Allais Paradoxes)

• RAM and TAX violate RBI in the opposite direction as CPT.€

⇒ RBI

3

€

′ z > ′ x > x > y > ′ y > z > 0

S → (x, p;y, p;z,1− 2p)

R → ( ′ x , p; ′ y , p;z,1− 2p)

In this test, we move the common branch from lowest, z, to highest,

z’ consequence.

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Restricted Branch Independence (3-RBI)

€

S = (x, p,y,q;z,1− p − q) f

R = ( ′ x , p; ′ y ,q;z,1− p − q)

⇔

′ S = ( ′ z ,1− p − q;x, p, y,q) f

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

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Types of Branch Independence

• The term “restricted” is used to indicate that the number of branches and probability distribution is the same in all four gambles.

• When we further constrain z and z’ to keep the same ranks in all four gambles, it is termed “comonotonic” (restricted) branch independence.

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A Special Case

• We can make a still more restricted case of restricted branch independence, in order to test the predictions of any weakly inverse-S weighting function. Let p = q.

• This distribution has been used in most, but not all of the studies.

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Example Test

S: .80 to win $2

.10 to win $40

.10 to win $44

R: .80 to win $2

.10 to win $4

.10 to win $96

S’: .10 to win $40

.10 to win $44

.80 to win $100

R’: .10 to win $4

.10 to win $96

.80 to win $100

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Generic Configural Model

€

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

The generic model includes RDU, CPT, EU, RAM, TAX, GDU, as special cases.

€

S f R ⇔

€

⇔w2

w1

>u( ′ x ) − u(x)

u(y) − u( ′ y )

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Violation of 3-RBI

A violation will occur if S f R and

€

′ S p ′ R ⇔

€

⇔′ w 3′ w 2

<u( ′ x ) − u(x)

u(y) − u( ′ y )€

′ w 1u( ′ z ) + ′ w 2u(x) + ′ w 3u(y) < ′ w 1u( ′ z ) + ′ w 2u( ′ x ) + ′ w 3u( ′ y )

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2 Types of Violations:

€

S f R∧ ′ S p ′ R ⇔w2

w1

>u( ′ x ) − u(x)

u(y) − u( ′ y )>

′ w 3′ w 2

€

S p R∧ ′ S f ′ R ⇔w2

w1

<u( ′ x ) − u(x)

u(y) − u( ′ y )<

′ w 3′ w 2

SR’:

RS’:

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EU allows no violations

• In EU, the weights equal the probabilities; therefore

€

w2

w1

=p

p=

p

p=

′ w 3′ w 2

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RAM Weights

€

w1 = a(1,3)t(p) /T

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

w3 = a(3,3)t(1− 2p) /T

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

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RAM Violations

• RAM model violates 3-RBI.

€

w2

w1

=a(2,3)t(p)

a(1,3)t(p)≠

a(3,3)t(p)

a(2,3)t(p)=

′ w 3′ w 2

€

a(i,n) = i ⇒w2

w1

=2

1>

3

2=

w3

w2

⇒ S ′ R

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CPT/ RDU

€

w1 = W ( p) −W (0)

w2 = W (2p) −W ( p)

w3 =1−W (2p)

′ w 1 = W (1− 2p)

′ w 2 = W (1− p) −W (1− 2p)

′ w 3 =1−W (1− p)

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Inverse-S Weighting Function

0

1

0 1

Decumulative Probability

Decumulative Weight

p 2p

W(2p)

W(p)

1-p1-2p

W(1-p)

W(1-2p)

€

w1 > w2

⇒w2

w1

<1

€

′ w 3 > ′ w 2

⇒′ w 3′ w 2

>1

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CPT implies RS’ violation

• If W(P) = P, CPT reduces to EU.

• However, if W(P) is any weakly inverse-S

function, CPT implies the RS’ pattern.

• (A strongly inverse- S function is weakly inverse-S plus it crosses the identity line. If we reject weak, then we reject the strong as well.)

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CPT Analysis of Table 1, #9 and 15: RBI

0

0.5

1

1.5

2

0.5 1.0 1.5

Weighting Function Parameter, γ

, Utility Function Exponent

β

RS'

RR'

SR'

SS'

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Transfer of Attention Exchange (TAX)

• Each branch (p, x) gets weight that is a function of branch probability

• Utility is a weighted average of the utilities of the consequences on branches.

• Attention (weight) is drawn from one branch to others. In a risk-averse person, weight is transferred to branches with lower consequences.

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“Special” TAX Model

Assumptions:

€

U(G) =Au(x) + Bu(y) + Cu(z)

A + B + C

€

A = t( p) −δt(p) /4 −δt(p) /4

B = t(q) −δt(q) /4 + δt(p) /4

C = t(1− p − q) + δt(p) /4 + δt(q) /4

€

G = (x, p;y,q;z,1− p − q)

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“Prior” TAX Model

€

u(x) = x; $0 < x < $150

t( p) = pγ ; γ = 0.7

δ =1Parameters were chosen to give a rough approximation to Tversky & Kahneman (1992) data. They are used to make new predictions.

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TAX Model Weights

Each term has the same denominator; middle branch gives up what it receives when p = q.€

A = t( p) − 2δt( p) /4

B = t( p) + δt(p) /4 −δt(p) /4

C = t(1− 2p) + δt( p) /4 + δt( p) /4

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Special TAX: SR’ Violations

• Special TAX model violates 3-RBI when delta is not zero.

€

w2

w1

=t(p)

t( p) − 2δt( p) /4>

t(p) + δt(p) + δt(1− 2p)

t(p) −δt(p) + δt(1− 2p)=

′ w 3′ w 2

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Summary of Predictions

• EU, SWU, OPT satisfy RBI• CPT violates RBI: RS’

• TAX & RAM violate RBI: SR’

• Here CPT is the most flexible model, RAM and TAX make opposite prediction from that of CPT.

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Results: n = 1075

SR’ (CPT predicted RS’)

No. S R % R

9 .80 to win $2

.10 to win $40

.10 to win $44

.80 to win $2

.10 to win $4

.10 to win $96

42.4

15.10 to win $40

.10 to win $44

.80 to win $100

.10 to win $4

.10 to win $96

.80 to win $100

56.0

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Lab Studies of RBI

• Birnbaum & McIntosh (1996): 2 studies, n = 106; n = 48, p = 1/3

• Birnbaum & Chavez (1997): n = 100; 3-RBI and 4-RBI, p = .25

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

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

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

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Web Studies of RBI

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

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

• Birnbaum (2004a): 3 conditions with n = 350; p = .1. Tests combined with Allais paradox.

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Additional Replications

• SR’ pattern is significantly more frequent than RS’ pattern in judgment studies as well. (Birnbaum & Beeghley, 1997; Birnbaum & Veira, 1998; Birnbaum & Zimmermann, 1999).

• A number of as yet unpublished studies have also 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 = ($ 110 , . 8 ; $ 44 , . 1 ; $ 40 , . 1 )) . vs ′ R = ($ 110 ; . 8 ; $ 98 , . 1 ; $ 10 , . 1 ) ,Choice Pattern

Condition n S ′ S S ′ R R ′ S R ′ R

New Tickets 141 34 54 14 37

Aligned Matrix 141 28 51 13 46

Unaligned Matrix 151 28 53 14 52

Losses (reflected) 200X 2

74 104 45 174

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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 estimates that SR’ violations are

“real” and probability of RS’ is equal to zero.

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Violations predicted by RAM & TAX, not CPT

• EU, SWU, OPT are refuted in this case by systematic violations.

• Editing “cancellation” refuted.• TAX & RAM, as fit to previous data

correctly predicted the modal choices.• Violations opposite those implied by CPT

with its inverse-S W(P) function.• Fitted CPT correct when it agrees with

TAX, wrong otherwise.

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To Rescue CPT:

• CPT can handle the result of any single test, by choosing suitable parameters.

• For CPT to handle these data, let γ

> 1; i.e., an S-shaped W(P) function, contrary to previous inverse- S.

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CPT Analysis of Table 1, #9 and 15: RBI

0

0.5

1

1.5

2

0.5 1.0 1.5

Weighting Function Parameter, γ

, Utility Function Exponent

β

RS'

RR'

SR'

SS'

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Adds to the case against CPT/RDU/RSDU

• Violations of RBI as predicted by TAX and RAM but are opposite predictions of CPT.

• Maybe CPT is right but its parameters are just wrong. As we see in the next program, we can generate internal contradiction in CPT.

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Next Program: LCI

• The next programs reviews tests of Lower Cumulative Independence (LCI).

• Violations of 3-LCI contradict any form of RDU, CPT.

• They also refute EU but 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