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Lower Distribution Independence

Michael H. BirnbaumCalifornia State University,

Fullerton

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Testing Among Models of Risky Decision Making

• Previous studies tested properties implied by CPT.

• Violations of Stochastic Dominance, Coalescing, Upper and Lower Cumulative Independence, Upper Tail Independence refute CPT.

• “Unfair” to test CPT this way?

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Test Predicted Effects

• Instead of testing implied invariance of CPT

• Test predicted violations of EU• Put RAM and TAX in position of

defending the null hypothesis against violations predicted by CPT

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LDI is Violated by CPT

• LDI is implied by EU.• CPT violates LDI but RAM and

Special TAX models satisfy it.• In this test, RAM and TAX defend

the null hypothesis against predictions of specific violations made by CPT.

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Cumulative Prospect Theory/ Rank-Dependent

Utility (RDU)

€

CPU(G ) = [W ( pj )− W ( pj )j =1

i −1

∑j =1

i

∑i =1

n

∑ ]u(xi )

Probability Weighting Function, W(P)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Decumulative Probability

Decumulative Weight

CPT Value (Utility) Function

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

Objective Cash Value

Subjective Value

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RAM Model

€

x1 > x2 > K > xi > K > xn > 0

€

RAMU(G ) =

a( i,n)t( pi )u(xi )i =1

n

∑

a( i,n)t( pi )i =1

n

∑

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RAM Model Parameters

Probability Weighting Function, t(p)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Objective Probability, p

€

a(1,n) = 1; a(2,n) = 2;K ; a( i,n) = i;K ; a(n ,n) = n

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RAM implies inverse-SCertainty Equivalents of

($100, p; $0)

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Probability to Win $100

Certainty Equivalent

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Special TAX Model

€

TAX (G ) =

[t( pi )+δ

n +1t( pj )

j =1

i −1

∑ −δ

n +1t( pi )

j =i+1

n

∑ ]u(xi )i =1

n

∑

t( pi )i =1

n

∑

€

G = (x1,p1;x2 ,p2;K ;xn ,pn )

€

x1 > x2 >K > xn > 0;t(p) = pγ ;γ = 0.7;δ =1

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TAX Model

• The weight of a branch depends on the branch’s probability.

• Each branch gains weight from branches with higher consequences.

• Each branch gives up weight to branches with lower consequences.

• Predictions nearly identical to those of CPT and RAM for binary gambles.

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€

′ x > x > y > ′ y > z > 0

S → (x, p;y, p;z,1− 2p)

R → ( ′ x , p; ′ y , p;z,1− 2p)The lower branch, z, has different probabilities in the two choices.

€

′ p > p⇒ 1− 2 ′ p <1− 2p

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Lower Distribution Independence (3-LDI)

€

S = ( x , p ; y , p ; z , 1 − 2 p ) f

R = ( ′ x , p ; ′ y , p ; z , 1 − 2 p )

⇔

S 2 = ( x , ′ p ; y , ′ p ; z , 1 − 2 ′ p ) f

R 2 = ( ′ x , ′ p ; ′ y , ′ p ; z , 1 − 2 ′ p )

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Example Test

S: .60 to win $2

.20 to win $56

.20 to win $58

R: .60 to win $2

.20 to win $4

.20 to win $96

S2: .10 to win $2

.45 to win $56

.45 to win $58

R2: .10 to win $2

.45 to win $4

.45 to win $96

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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, RAM, TAX, GDU, & others as special cases.

€

S f R ⇔

€

⇔w2

w1

>u( ′ x )− u(x)

u(y)− u( ′ y )

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Violation of 3-LDI

€

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

A violation will occur if S f R and

€

S2 p R2 ⇔

€

⇔′ w 2′ w 1

<u( ′ x ) − u(x)

u(y) − u( ′ y )

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2 Types of Violations:

€

S f R∧S2 p R2 ⇔w2

w1

>u( ′ x ) − u(x)

u(y) − u( ′ y )>

′ w 2′ w 1

€

S p R∧S2 f R2 ⇔w2

w1

<u( ′ x ) − u(x)

u(y) − u( ′ y )<

′ w 2′ w 1

SR2:

RS2:

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EU allows no violations

• In EU, the weights are the probabilities; therefore

€

w2

w1

=p

p=

′ p ′ p =

′ w 2′ w 1

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CPT implies violations

• If W(P) = P, CPT reduces to EU; however, when W(P) is nonlinear, CPT violates LDI systematically.

• From previous data, we can calculate where to expect violations and predict which type of violation should be observed.

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CPT Model of Tversky and Kahneman (1992)

€

W (P) =Pγ

[Piγ +(1− Pi )

γ ]1γ

€

γ=0.61

u(x) = x β

β = 0.88

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CPT Analysis of Example 1: 3-LDI

0

1

2

0.5 1 1.5

Weighting Function Parameter, γ

, Utility Function Exponent

β 2RR

2RS

2SR

2SS

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CPT implies RS2 Violations

• When γ = 1, CPT reduces to EU.• Given the inverse-S weighting function,

the fitted CPT model implies RS2 pattern.

• If γ > 1, however, the model can handle the opposite pattern.

• A series of tests can be devised to provide overlapping combinations of parameters.

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RAM allows no Violations

• RAM model with any parameters satisfies 3-LDI.

€

w2

w1

=a(2,3)t(p)

a(1,3)t(p)=

a(2,3)t( ′ p )

a(1,3)t( ′ p )=

′ w 2′ w 1

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Special TAX: No Violations• The Special TAX model, with one

configural parameter, allows no violations of 3-LDI.

• The middle branch gains as much weight as it gives up for any p.

€

w2

w1

=t( p)+(δ t( p)

4 )− (δ t( p)4 )

t( p)− (24)δ t( p)

=t( p)

t( p)[1− 2δ4]

=′ w 2′ w 1

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Summary of Predictions

• RAM, TAX, & EU satisfy 3-LDI• CPT violates 3-LDI

• Fitted CPT implies RS2 pattern of violation

• Here CPT is the most flexible model, RAM and TAX defend the null hypothesis.

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Web-Based Studies• Two from a Series of Studies tests:

classical and new paradoxes in decision making.

• People come on-line via WWW (some tested in lab for comparison).

• Choose between gambles; 1 person per month (about 1% of participants) wins the prize of one of their chosen gambles. 20 or 22 choices.

• Data arrive 24-7; sample sizes are large; results are clear.

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Results n = 503Choice % RNo.

S R n = 503

6 60 blue to win $2

20 red to win $56

20 white to win $58

60 green to win $2

20 black to win $4

20 purple to win $96

23.6*

12 10 black to win $2

45 green to win $56

45 purple to win $58

10 white to win $2

45 red to win $4

45 blue to win $96

18.7*

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Results: n = 1075

S R n = 1075

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

12 .10 to win $2

.45 to win $40

.45 to win $44

.10 to win $2

.45 to win $4

.45 to win $96

30.2

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3-2 Lower Distribution Independence

• In this property, the probability of the branch with the lowest consequence goes to zero and the branch is removed.

• CPT again predicts violations• Special TAX and RAM again satisfy

the property

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Test of 3-2 LDI; n = 1075S R 1075

.04 to win $2

.48 to win $40

.48 to win $44

.04 to win $2

.48 to win $4

.48 to win $96

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.50 to win $40

.50 to win $44

.50 to win $4

.50 to win $96

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Fitted CPT predicts RS2 Pattern

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Summary: Predicted Violations of CPT failed to

Materialize• TAX model, fit to previous data

correctly predicted the modal choices.

• RAM makes the same predictions in this case.

• Fitted CPT was correct when it agreed with TAX, wrong otherwise except 1 case in 12.

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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, the values of β must be much smaller

or γ much larger than those reported in the literature.

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CPT Analysis of Example 1: 3-LDI

0

1

2

0.5 1 1.5

Weighting Function Parameter, γ

, Utility Function Exponent

β 2RR

2RS

2SR

2SS

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RAM and TAX have been found more accurate than

CPT in other tests• Are they simply more flexible? No.• In the tests of 3-LDI and 3-2-LDI,

CPT is the most flexible model.• Why then has there been a

growing consensus for CPT? I suspect lack of familiarity with the results of studies like these.

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Case against CPT/RDU

• Violations of Stochastic Dominance• Violations of Coalescing (Event-

Splitting)• Violations of 3-Upper Tail Independence• Violations of Lower Cumulative

Independence• Violations of Upper Cumulative

Independence

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More Evidence against CPT/RDU/RSDU

• Violations of Gain-Loss separability.• Violations of Restricted Branch

Independence are opposite predictions of fitted CPT.

• Violations of 4-distribution independence, 3-UDI favor TAX over RAM and opposite predictions of CPT.

• Failure of predicted violations of 3-LDI and 3-2 LDI to materialize.

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Preview of Next Program

• The next programs reviews tests of Upper Distribution Independence, assuming the viewer has seen this program.

• EU and RAM predict no violations, CPT and TAX predict opposite patterns. Data agree with 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