Models of Choice

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Models of Choice
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Agenda

• 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

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Choice

• The same choice is not always made in the same
  situation.
• Main assumption Choice alternatives have choice
  probabilities.

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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.

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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.

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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 gt xj) P(xk - xj gt 0)

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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

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Thurstone Scaling
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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.

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Luces 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.

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A Problem with Thurstone Scaling

• Works well for 2 alternatives, not more.

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Luces 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.

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Luces 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) gt P(sausage)
• T2 ice cream, sausages, sauerkraut
• P(sausage) gt P(ice cream)
• Need a psychological theory.

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Luces 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.

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Luces Choice Axiom

• PT(S) is the probability of choosing any element
  of S given a choice from T.
• Pchicken, beef, pork, veggies(chicken, pork)

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Luces 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)

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Luces 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
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Luces 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)
• Why? If x is dominated by any element in T, it is
  dominated by all elements. Causes division
  problems.

T
S
X
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Luces Choice Theorem

• Theorem There exists a positive real-valued
  function v on T, which is unique up to
  multiplication by a positive constant, such that
  for every S ? T,

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Luces Choice Theorem

• Proof Define v(x) kPT(x), for k gt 0. Then, by
  the choice axiom (proof of uniqueness left to
  reader),

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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.

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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(10A, 10) (4 - 3)/(4 - 3 3 - 3) 1
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Quantitative vs. Qualitative Tests
Dimensions Dimensions Dimensions 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
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Quantitative vs. Qualitative Tests
Dimensions Dimensions Dimensions 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
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Quantitative Test
P(Correct) P(Correct) 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
GOF .0119 .0103
Made-up s
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Qualitative Test
Dimensions Dimensions Dimensions 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
lt- More protypical
lt- Less prototypcial
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Qualitative Test
Dimensions Dimensions Dimensions 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
lt- Similar to A1, A3
lt- Similar to A2, B6, B7
Prototype A1gtA2 Exemplar A2gtA1
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Quantitative Test
P(Correct) P(Correct) 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
GOF .0119 .0103
Made-up s
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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.

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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.