2
UDI is Violated CPT
• CPT violates UDI but EU and RAM satisfy it.
• TAX violates UDI in the opposite direction as CPT.
3
€
′ z > ′ x > x > y > ′ y > 0
S → ( ′ z ,1− 2p;x, p;y, p)
R → ( ′ z ,1− 2p; ′ x , p; ′ y , p)The upper branch consequence, z’, has different probabilities in the two choices.
€
′ p > p⇒ 1− 2 ′ p <1− 2p
4
Upper Distribution Independence (3-UDI)
€
′ S = ( ′ z ,1− 2p;x, p;y, p) f
′ R = ( ′ z ,1− 2p; ′ x , p; ′ y , p)
⇔
S ′ 2 = ( ′ z ,1− 2 ′ p ;x, ′ p ;y, ′ p ) f
R ′ 2 = ( ′ z ,1− 2 ′ p ; ′ x , ′ p ; ′ y , ′ p )
5
Example Test
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
S2’: .45 to win $40
.45 to win $44
.10 to win $100
R2’: .45 to win $4
.45 to win $96
.10 to win $100
6
Generic Configural Model
€
U(G) = w1u( ′ z ) + w2u(x) + w3u(y)
where
€
u( ′ z ) > u(x) > u(y) > 0
CPT, RAM, and TAX disagree on
€
w1,w2,w3
7
Generic Configural Model
€
w1u( ′ z ) + w2u(x) + w3u(y) > w1u( ′ z ) + w2u( ′ x ) + w3u( ′ y )
The generic model includes RDU, CPT, EU, RAM, TAX, GDU, as special cases.
€
′ S f ′ R ⇔
€
⇔w3
w2
>u( ′ x ) − u(x)
u(y) − u( ′ y )
8
Violation of 3-UDI
€
′ w 1u( ′ z ) + ′ w 2u(x) + ′ w 3u(y) < ′ w 1u( ′ z ) + ′ w 2u( ′ x ) + ′ w 3u( ′ y )
A violation will occur if S’ f R’ and
€
S ′ 2 p R ′ 2 ⇔
€
⇔′ w 3′ w 2
<u( ′ x ) − u(x)
u(y) − u( ′ y )
9
2 Types of Violations:
€
′ S f ′ R ∧S ′ 2 p R ′ 2 ⇔w3
w2
>u( ′ x ) − u(x)
u(y) − u( ′ y )>
′ w 3′ w 2
€
′ S p ′ R ∧S ′ 2 f R ′ 2 ⇔w3
w2
<u( ′ x ) − u(x)
u(y) − u( ′ y )<
′ w 3′ w 2
S’R2’:
R’S2’:
10
EU allows no violations
• In EU, the weights are equal to the probabilities; therefore
€
w3
w2
=p
p=
′ p ′ p =
′ w 3′ w 2
11
RAM Weights
€
w1 = a(1,3)t(1− 2p) /T
w2 = a(2,3)t(p) /T
w3 = a(3,3)t(p) /T
T = a(1,3)t(1− 2 p) + a(2,3)t( p) + a(3,3)t( p)
12
RAM allows no Violations
• RAM model with any parameters satisfies 3-UDI.
€
w3
w2
=a(3,3)t(p)
a(2,3)t(p)=
a(3,3)t( ′ p )
a(2,3)t( ′ p )=
′ w 3′ w 2
14
CPT implies violations
• If W(P) = P, CPT reduces to EU.• From previous data, we can
calculate where to expect violations and predict which type of violation should be observed.
15
CPT Analysis of 3-UDI Choices 15 & 18
0.4
0.6
0.8
1
0.6 0.8 1 1.2 1.4Weighting Function Parameter, γ
, Exponent of Utility Function
β
2R'S '
2S'R '
2S'S '
2R'R '
16
CPT implies S’R2’ Violations
• When γ = 1, CPT reduces to EU.• Given the inverse-S weighting function,
the fitted CPT model implies S’R2’ pattern.
• If γ > 1, S-Shaped, but the model can handle the opposite pattern.
• A series of tests can be devised to provide overlapping combinations of parameters.
17
TAX Model
Each term has the same denominator; however, unlike the case of LDI, here the middle branch can gain more weight than it gives up.
€
w1 =t(1− 2p) − 2δt(1− 2 p) /4
t(1− 2 p) + t(p) + t( p)
w2 =t( p) + δt(1− 2p) /4 −δt( p) /4
t(1− 2p) + t( p) + t(p)
w3 =t( p) + δt(1− 2p) /4 + δt( p) /4
t(1− 2p) + t( p) + t(p)
18
Special TAX: R’S2’ Violations
• Special TAX model violates 3-UDI. • Here the ratio depends on p.
€
w3
w2
=t( p) + δt(1− 2p) /4 + δ t(p) /4
t( p) + δt(1− 2p) /4 −δ t(p) /4<
′ w 3′ w 2
19
Summary of Predictions
• RAM, & EU satisfy 3-UDI• CPT violates 3-UDI: S’R2’
• TAX violates 3-UDI: R’S2’
• Here CPT is the most flexible model, RAM defends the null hypothesis, TAX makes opposite prediction from that of CPT.
20
Results n = 1075
€
′ S
€
′ R 1075
.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
.45 to win $40
.45 to win $44
.10 to win $100
.45 to win $4
.45 to win $96
.10 to win $100
33
R’S2’
21
Results: n = 503
20 white to win $28
20 blue to win $30
60 red to win $100
20 yellow to win $4
20 green to win $96
60 black to win $100
70.8*
45 white to win $28
45 purple to win $30
10 blue to win $100
45 black to win $4
45 green to win $96
10 red to win $100
59.1*
R’R2’ (CPT predicted S’R2’ )
22
Summary: Observed Violations fit TAX, not CPT
• RAM and EU are refuted in this case by systematic violations.
• TAX model, fit to previous data correctly predicted the modal choices.
• Violations opposite those implied by CPT with its inverse-S W(P) function.
• Fitted CPT was correct when it agreed with TAX, wrong otherwise.
23
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.
24
CPT Analysis of 3-UDI Choices 15 & 18
0.4
0.6
0.8
1
0.6 0.8 1 1.2 1.4Weighting Function Parameter, γ
, Exponent of Utility Function
β
2R'S '
2S'R '
2S'S '
2R'R '
25
Adds to the case against CPT/RDU/RSDU
• Violations of 3-UDI favor TAX over RAM and are opposite predictions of CPT.
26
Preview of Next Program
• The next programs reviews tests of Restricted Branch Independence (RBI).
• It turns out the violations of 3-RBI are opposite the predictions of CPT with inverse-S function.
• They refute EU but are consistent with RAM and TAX.
27
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.
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