Transcript Models of Choice

Models of Choice
Agenda
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Administrivia
– Readings
– Programming
– Auditing
– Late HW
– Saturated
– HW 1
Models of Choice
– Thurstonian scaling
– Luce choice theory
– Restle choice theory
Quantitative vs. qualitative tests of models.
Rumelhart & Greeno (1971)
Conditioning…
Next assignment
Choice
• The same choice is not always made in
the “same” situation.
• Main assumption: Choice alternatives
have choice probabilities.
Overview of 3 Models
• Thurstone & Luce
– Responses have an associated ‘strength’.
– Choice probability results from the
strengths of the choice alternatives.
• Restle
– The factors in the probability of a choice
cannot be combined into a simple strength,
but must be assessed individually.
Thurstone Scaling
• Assumptions
– The strongest of a set of alternatives will
be selected.
– All alternatives gives rise to a probabilistic
distribution (discriminal dispersions) of
strengths.
Thurstone Scaling
• Let xj denote the discriminal process
produced by stimulus j.
• The probability that Object k is preferred
to Stimulus j is given by
– P(xk > xj) = P(xk - xj > 0)
Thurstone Scaling
• Assume xj & xk are normally distributed
with means j & k, variances j & k,
and correlation rjk.
• Then the distribution of xk - xj is normal
with
– mean k - j
– variance j2 + k2 - 2 rjkjk = jk2
P(x k  x j )
 P(x k  x j  0)



0

N(k   j , 2jk )
Thurstone Scaling
k   j
zkj 
 jk
P(x k  x j )
 P(x k  x j  0)


 N(
k
  j , )
2
jk
0


 N(0,1)
z kj

k   j  zkj   jk
Thurstone Scaling
• Special cases:
– Case III: r = 0
• If n stimuli, n means, n variances, 2n
parameters.
– Case V: r = 0, j2 = k2
• If n stimuli, n means, n parameters.
Luce’s Choice Theory
• Classical strength theory explains
variability in choices by assuming that
response strengths oscillate.
• Luce assumed that response strengths
are constant, but that there is variability
in the process of choosing.
– The probability of each response is
proportional to the strength of that
response.
A Problem with Thurstone
Scaling
• Works well for 2 alternatives, not more.
Luce’s Choice Theory
• For Thurstone with 3 or more alternatives, it
can be difficult to predict how often B will be
selected over A. The probabilities of choice
may depend on what other alternatives are
available.
• Luce is based on the assumption that the
relative frequency of choices of B over C
should not change with the mere availability
of other choices.
Luce’s Choice Axiom
• Mathematical probability theory cannot
extend from one set of alternatives to
another. For example, it might be possible
for:
– T1 = {ice cream, sausages}
• P(ice cream) > P(sausage)
– T2 = {ice cream, sausages, sauerkraut}
• P(sausage) > P(ice cream)
• Need a psychological theory.
Luce’s Choice Axiom
• Assumption: The relative probabilities of
any two alternatives would remain
unchanged as other alternatives are
introduced.
– Menu: 20% choose beef, 30% choose
chicken.
– New menu with only beef & chicken: 40%
choose beef, 60% choose chicken.
Luce’s Choice Axiom
• PT(S) is the probability of choosing any
element of S given a choice from T.
– P{chicken, beef, pork, veggies}(chicken, pork)
Luce’s Choice Axiom
• Let T be a finite subset of U such that,
for every S  T, Ps is defined, Then:
– (i) If P(x, y)  0, 1 for all x, y  T, then for R
 S  T, PT(R) = PS(R) PT(S)
– (ii) If P(x, y) = 0 for some x, y in T, then for
every S  T, PT(S) = PT-{x}(S-{x})
Luce’s Choice Axiom
T
(i) If P(x, y)  0,
1 for all x, y 
T, then for R 
S  T, PT(R) =
PS(R) PT(S)
S
R
Luce’s Choice Axiom
•(ii) If P(x, y) = 0 for
some x, y in T, then
for every S  T, PT(S)
= PT-{x}(S-{x})
T
•Why? If x is
dominated by any
element in T, it is
dominated by all
elements. Causes
division problems.
S
X
Luce’s Choice Theorem
• Theorem: There exists a positive realvalued function v on T, which is unique
up to multiplication by a positive
constant, such that for every S  T,
v(x)
PS (x) 
v(y)
y S
Luce’s Choice Theorem
• Proof: Define v(x) = kPT(x), for k > 0.
Then, by the choice axiom (proof of
uniqueness left to reader),
PT (x)
PT (S)
kPT (x)

 kPT (x)
PS (x) 
y s

v(x)
 v(y)
y s
Thurstone & Luce
• Thurstone's Case V model becomes
equivalent to the Choice Axiom if its
discriminal processes are assumed to
be independent double exponential
random variables
– This is true for 2 and 3 choice situations.
– For 2 choice situations, other discriminal
processes will work.
Restle
• A choice between 2 complex and
overlapping choices depends not on
their common elements, but on their
differential elements.
– $10 + an apple
– $10
XXX X
XXX
P($10+A, $10) = (4 - 3)/(4 - 3 + 3 - 3) = 1
Quantitative vs. Qualitative
Tests
Dimensions
Stimulus
Legs
Eye
Head
Body
A1
1
1
1
0
A2
1
0
1
0
A3
1
0
1
1
A4
1
1
0
1
A5
0
1
1
1
B1
1
1
0
0
B2
0
1
1
0
B3
0
0
0
1
B4
0
0
0
0
Quantitative vs. Qualitative
Tests
Dimensions
Stimulus
Legs
Eye
Head
Body
A1
1
1
1
0
A2
1
0
1
0
A3
1
0
1
1
A4
1
1
0
1
A5
0
1
1
1
B1
1
1
0
0
B2
0
1
1
0
B3
0
0
0
1
B4
0
0
0
0
Prototype vs.
Exemplar
Theories
Quantitative Test
P(Correct)
Stimulus
Data
Prototype
Exemplar
A1
.58
.65
.60
A2
.66
.60
65
A3
.58
.61
.61
A4
.71
.74
.78
A5
.45
.45
.40
B1
.41
.42
.40
B2
.47
.46
.45
B3
.59
.60
.60
B4
.65
.61
.63
.0119
.0103
GOF
Made-up #s
Qualitative Test
Dimensions
Stimulus
Legs
Eye
Head
Body
A1
1
1
1
0
A2
1
0
1
0
A3
1
0
1
1
A4
1
1
0
1
A5
0
1
1
1
B1
1
1
0
0
B2
0
1
1
0
B3
0
0
0
1
B4
0
0
0
0
<- More ‘protypical’
<- Less ‘prototypcial’
Qualitative Test
Dimensions
Stimulus
Legs
Eye
Head
Body
A1
1
1
1
0
A2
1
0
1
0
A3
1
0
1
1
A4
1
1
0
1
A5
0
1
1
1
B1
1
1
0
0
B2
0
1
1
0
B3
0
0
0
1
B4
0
0
0
0
<- Similar to A1, A3
<- Similar to A2, B6, B7
Prototype: A1>A2
Exemplar: A2>A1
Quantitative Test
P(Correct)
Stimulus
Data
Prototype
Exemplar
A1
.58
.65
.60
A2
.66
.60
65
A3
.58
.61
.61
A4
.71
.74
.78
A5
.45
.45
.40
B1
.41
.42
.40
B2
.47
.46
.45
B3
.59
.60
.60
B4
.65
.61
.63
.0119
.0103
GOF
Made-up #s
Quantitative vs. Qualitative
Tests
• You ALWAYS have to figure out how to
split up your data.
– Batchelder & Riefer, 1980 used E1, E2, etc
instead of raw outputs.
– Rumelhart & Greeno, 1971 looked at
particular triples.
Caveat
• Qualitative tests are much more
compelling and, if used properly, telling,
but
– qualitative tests can be viewed as
specialized quantitative tests, i.e., on a
subset of the data.
– “qualitative” tests often rely on quantitative
comparisons.