Restricted Branch
Independence
Michael H. Birnbaum
California State University,
Fullerton
1
RBI is Violated by CPT
• EU satisfies RBI as does SWU and PT,
extended to 3-branch gambles.
Cancellation
 RBI
• CPT violates RBI (it MUST to explain
the Allais Paradoxes)
• RAM and TAX violate RBI in the
as CPT.
opposite direction
2
z x  x  y  y  z  0
S  (x, p;y, p;z,1 2 p)
R  ( x , p; y , p;z,1 2 p)
In this test, we move the common
branch from lowest, z, to highest, z’
consequence.
3
Restricted Branch
Independence (3-RBI)
S  (x, p, y,q;z,1 p  q)
R  ( x , p; y ,q;z,1 p  q)

S  ( z,1 p  q;x, p, y,q)
R ( z,1 p  q; x , p; y ,q)
4
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.
5
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.
6
Example Test
S:
.80 to win $2
.10 to win $40
.10 to win $44
S’: .10 to win $40
.10 to win $44
.80 to win $100
R:
.80 to win $2
.10 to win $4
.10 to win $96
R’: .10 to win $4
.10 to win $96
.80 to win $100
7
Generic Configural Model
The generic model includes RDU, CPT, EU,
RAM, TAX, GDU, as special cases.
S
R
w1u(x)  w2u(y)  w3u(z)  w1u( x)  w2u( y)  w3u(z)
w2 u( x )  u(x)


w1 u(y)  u( y )
8
Violation of 3-RBI
A violation will occur if S f R and
S R
w1u(z)  w
u(z)  w
)  w
)
2u(x)  w
3u(y)  w1
2u( x
3u( y
w
u( x )  u(x)
3


w
u(y)  u( y )
2
9
2 Types of Violations:
SR’:
S
w2 u( x )  u(x) w
R  S R

 3
w1 u(y)  u( y ) w
2
’
RS :
S
w2 u( x )  u(x) w
R  S R

 3
w1 u(y)  u( y ) w
2
10
EU allows no violations
• In EU, the weights equal the
probabilities; therefore
w2 p p w
3
  
w1 p p w
2
11
RAM Weights
w1  a(1,3)t( p) /T
w 2  a(2,3)t( p) /T
w 3  a(3,3)t(1 2 p) /T
T  a(1,3)t( p)  a(2,3)t( p)  a(3,3)t(1 2 p)
12
RAM Violations
• RAM model violates 3-RBI.
w2 a(2,3)t( p) a(3,3)t( p) w


 3
w1 a(1,3)t( p) a(2,3)t( p) w
2
w2 2 3 w3
a(i,n)  i 
  
 SR
w1 1 2 w2
13
CPT/ RDU
w1  W ( p)  W (0)
w 2  W (2 p)  W ( p)
w 3  1 W (2 p)
w1 W (1 2 p)
w 2  W (1 p)  W (1 2 p)
w 3  1 W (1 p)
14
w1  w 2
w2

1
w1
1
Decumulative Weight
w 3  w 2
w 3

1
w 2
Inverse-S Weighting Function
W(1-p)
W(1-2p)
W(2p)
W(p)
0
0
p
2p
1-2p
1-p
1
Decumulative Probability
15
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.)
16
2
Utility Function Exponent,

CPT Analysis of Table 1, #9 and 15: RBI
1.5
RR'
1
SR'
RS'
0.5
SS'
0
0.5
1.0
Weighting Function Parameter,
1.5

17
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.
18
“Special” TAX Model
Assumptions:
G  (x, p;y,q;z,1 p  q)
Au(x)  Bu(y)  Cu(z)
U(G) 
A BC

A  t( p)  t( p) /4  t( p) /4
B  t(q)  t(q) /4  t( p) /4
C  t(1 p  q)  t( p) /4  t(q) /4
19
“Prior” TAX Model
u(x)  x; $0  x  $150

t( p)  p ;   0.7
 1
Parameters were chosen to give a rough
approximation to Tversky & Kahneman
(1992) data. They are used to make new
predictions.
20
TAX Model Weights
A  t( p)  2t( p) /4
B  t( p)  t( p) /4  t( p) /4
C  t(1 2 p)  t( p) /4  t( p) / 4

Each term has the same denominator; middle
branch gives up what it receives when p = q.
21
Special TAX:
’
SR
Violations
• Special TAX model violates 3-RBI when
delta is not zero.
w2
t( p)
t( p)  t( p)  t(1 2p) w


 3
w1 t( p)  2t( p) /4 t( p)  t( p)  t(1 2p) w
2
22
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.
23
Results: n = 1075
No.
9
15
S
R
.80 to win $2
.80 to win $2
.10 to win $40
.10 to win $4
.10 to win $44
.10 to win $96
.10 to win $40
.10 to win $4
.10 to win $44
.10 to win $96
.80 to win $100
.80 to win $100
%R
4 2.4
5 6.0
SR’ (CPT predicted RS’)
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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.
25
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.
27
vs. R  ($98,.1;$10,.1;$2,.8)
S  ($44,.1;$40,.1;$2,.8)
S  ($110,.8;$44,.1;$40,.1)) vs. R ($110;.8;$98,.1;$10,.1) ,
Cho ice Pattern
Cond iti on
n
SS 
SR 
RS 
RR 
New Tic kets
141
34
54
14
37
Ali gned Matrix
141
28
51
13
46
Una li gned Matrix
151
28
53
14
52
Losses (reflected)
200
X 2
74
104
45
1 74
28
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.
29
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.
30
To Rescue CPT:
• CPT can handle the result of any single
test, by choosing suitable parameters.
• For CPT to handle these data, let  > 1;
i.e., an S-shaped W(P) function,
contrary to previous inverse- S.
31
2
Utility Function Exponent,

CPT Analysis of Table 1, #9 and 15: RBI
1.5
RR'
1
SR'
RS'
0.5
SS'
0
0.5
1.0
Weighting Function Parameter,
1.5

32
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.
33
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.
34
For More Information:
[email protected]
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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