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Chapter 12 Judgment and Choice
This chapter covers the mathematical models behind the way that consumer decide and
choose. We will discuss
 The detection of sensory information
 The detection of differences between two things
 Judgments where consumers compare two things
 A model for the recognition of advertisements
 How multiple judgments are combined to make a single decision
As usual, estimation of the parameters in these models will serve as an important
theme for this chapter
Mathematical
Marketing
Slide 12.1
Judgment
and Choice
There Are Two Different Types of Judgments
 Absolute Judgment
•
•
Do I see anything?
How much do I like that?
 Comparative Judgment
•
•
Does this bagel taste better than that one?
Do I like Country Time Lemonade better than Minute
Maid?
Psychologists began investigating how people answer these sorts of questions in the 19th Century
Mathematical
Marketing
Slide 12.2
Judgment
and Choice
The Early Concept of a “Threshold”
Absolute Detection
1.0
Pr(Detect) .5
0
n
Physical measurement
Difference Detection
1.0
Pr(n Perceived > n2) .5
0
n1
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Marketing
n2
n3
Slide 12.3
Judgment
and Choice
But the Data Never Looked Like That
1.0
Pr(Detect)
.5
0
n
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Marketing
Slide 12.4
Judgment
and Choice
A Simple Model for Detection
si is the psychological impact of stimulus i
If si exceeds the threshold,
you see/hear/feel it
Pr[Detect stimulus i] = Pr[si  s0] .
We make this assumption
ei ~ N(0, 2) so that
which then implies
si ~ N(si, 2 )
We also assume
Mathematical
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si  si  ei
s0  0
Slide 12.5
Judgment
and Choice
Our Assumptions Imply That the Probability of Detection Is…
1 
2
2
p̂i 
 exp[ (si  si ) / 2 ] ds i
2 0
(Note missing left bracket in Equation 12.6 in book.)
Converting to a z-score we get
 z i2 
1 
p̂i 
 exp  dz i
2 0  s
 2 
i

(Note missing subscript i on the z in book)
Mathematical
Marketing
Slide 12.6
Judgment
and Choice
Making the Equation Simpler
 z i2 
1 
p̂i 
 exp  dz i
2 0  s
 2 
i

But since the normal distribution is symmetric about 0 we can say:
si

 z i2 
1
p̂ i 
 exp   dz
2  
 2 
 [ si / ]
Mathematical
Marketing
Slide 12.7
Judgment
and Choice
Graphical Picture of What We Just Did
Pr( s i )
2
0
Pr( z)
si
si
1
Pr(Detection)
z
 si  0
Pr( z)
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Marketing
1
0
si 
z
Slide 12.8
Judgment
and Choice
A General Rule for Pr(a > 0)
Where a Is Normally Distributed
For a ~ N[E(a), V(a)] we have
Pr [a  0] =  [E(a) /  V(a)]
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Slide 12.9
Judgment
and Choice
So Why Do Detection Probabilities Not Look Like a
Step Function?
s1 dim
s2 medium
s3 bright
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Marketing
Slide 12.10
Judgment
and Choice
Paired Comparison Data:
Pr(Row Brand > Column Brand)
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A
B
C
A
-
.6
.7
B
.4
-
.2
C
.3
.8
-
Slide 12.11
Judgment
and Choice
Assumptions of the Thurstone Model
s i  si  ei
2
ei ~ N(0,  i )
Cov(ei, ej) = ij = rij
si
Draw si
sj
Draw sj
Is si > sj?
Mathematical
Marketing
Slide 12.12
Judgment
and Choice
Deriving the E(si - sj) and V (si - sj)
pij = Pr(si > sj ) = Pr(si - sj > 0)
E(s i  s j )  E  ( si  e i )  ( s j  e j )
 si  s j
2
i ij   1 
V(s i  s j )  1  1 
2 

ij  j   1
 i2   2j  2ij
 i2   2j  2ri  j
Mathematical
Marketing
Slide 12.13
Judgment
and Choice
Predicting Choice Probabilities
For a ~ N[E(a), V(a)] we have
Pr [a  0] =  [E(a) /  V(a)]
Below si - sj plays the role of "a"

p̂ij  Pr(s i  s j )   (si  s j )
E(si  s j )
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i2   2j  2ri  j

V(si  s j )
Slide 12.14
Judgment
and Choice
Thurstone Case III

p̂ij  Pr(s i  s j )   (si  s j )
s1 = 0
i2  2j

12 = 1
s2 , s3 , , st ,  22 , 32 , ,  2t
How many unknowns are there?
How many data points are there?
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Marketing
Slide 12.15
Judgment
and Choice
Unweighted Least Squares Estimation
 
 1[Pr(s i  s j )]   1  (si  s j )
i2   2j
ẑ12  ( s1  s2 )
12   22
ẑ13  ( s1  s3 )
12   32

 
ẑ ( t 1) t  ( st 1  st )
t 1
 2t 1   2t
t
f    (z ij  ẑ ij ) 2
i 1 ji 1
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Marketing
Slide 12.16
Judgment
and Choice
Conditions Needed for Minimizing f
 f /s1   0 
 f / s   0 
2

  
   

  

f
/

s
t

  0
f /12   0 

 
2
f / 2   0 
   

  
2

f
/


0

t 
Mathematical
Marketing
Slide 12.17
Judgment
and Choice
Minimum Pearson 2
Same model:

p̂ ij   (si  s j )

i2   2j .
Different objective function
(np ij  np̂ ij ) 2
ˆ   
i j i
np̂ ij
2
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Marketing
t
t
Slide 12.18
Judgment
and Choice
Matrix Setup for Minimum Pearson 2
p  p12
p13  p( t 1) t 
pˆ   p̂12
p̂13  p̂( t 1) t 
V(pij )  V[pij  p̂ij ] 
p̂ij (1  p̂ij )
n
V(p) = V
ˆ 2  (p  pˆ ) V 1 (p  pˆ )
Mathematical
Marketing
Slide 12.19
Judgment
and Choice
Minimum Pearson 2
(np ij  np̂ ij ) 2
ˆ   
i j i
np̂ ij
2
t
t
Modified
Minimum Pearson 2
(np ij  np̂ ij ) 2
ˆ   
i j i
np ij
2
t
t
Simplifies the derivatives, and reduces the computational time required
Mathematical
Marketing
Slide 12.20
Judgment
and Choice
Definitions and Background for ML Estimation
Assume that we have two possible events A and B. The probability of A is Pr(A), and the
probability of B is Pr(B). What are the odds of two A's on two independent trials?
Pr(A) • Pr(A) = Pr(A)2
In general the Probability of p A's and q B's would be
Pr( A) p  Pr( B) q
Note these definitions and identities:
fij = npij
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Marketing
p ij 
f ij
n
p ji  1  pij 
n  f ij
n
Slide 12.21
Judgment
and Choice
ML Estimation of the Thurstone Model
According to the Model
According to the general alternative
t 1
t
f
nf
l 0    p̂ ij ij (1  p̂ ij ij )
t 1
t
f
nf
l A    p ij ij (1  p ij ij )
i 1
t 1
ji 1
i 1
t
ln(l 0 )  L0    f ij ln p̂ij  (n  f ij ) ln(1  p̂ij )
i 1 ji 1
ˆ 2  2 ln
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Marketing
t 1
ji 1
t
ln(l A )  L A    f ij ln pij  (n  f ij ) ln(1  pij )
i 1 ji 1
l0
 2[ L A  L 0 ]
lA
Slide 12.22
Judgment
and Choice
Categorical or Absolute Judgment
Love
[ ]
Like
[ ]
Dislike
[ ]
s1
1
Love
Brand 1
Brand 2
Brand 3
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Marketing
.20
.10
.05
2
Like
.30
.10
.10
s2
3
Hate
[ ]
s3
4
Dislike
.20
.60
.15
Hate
.30
.20
.70
Slide 12.23
Judgment
and Choice
Cumulated Category Probabilities
Love
Brand 1
.20
Raw
Probabilities
Cumulated
Probabilities
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Marketing
Like
.30
Dislike
.20
Hate
.30
Brand 2
.10
.10
.60
.20
Brand 3
.05
.10
.15
.70
Brand 1
.20
.50
.70
1.00
Brand 2
.10
.20
.80
1.00
Brand 3
.05
.15
.30
1.00
Slide 12.24
Judgment
and Choice
The Thresholds or Cutoffs
c0 = -
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Marketing
c1
c2
c3 (cJ-1)
c4 = +
Slide 12.25
Judgment
and Choice
A Model for Categorical Data
s i  si  ei
ei ~ N(0, 2)
p̂ij  Pr[si  c j ]  Pr[c j  si  0]
Probability that item i
is placed in category j
or less
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Marketing
Probability that the
discriminal response
to item i is less than
the upper boundary
for category j
Slide 12.26
Judgment
and Choice
The Probability of Using a Specific Category (or Less)
p̂ij  c j  si i 
Pr [a  0] =  [E(a) /  V(a)]
Below ci - sj is plays the role of "a"
Mathematical
Marketing
Slide 12.27
Judgment
and Choice
The Theory of Signal Detectability
Response
S
N
S
Hit
Miss
N
False
Alarm
Correct
Rejection
Reality
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Marketing
Slide 12.28
Judgment
and Choice