Models of Choice

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# Models of Choice - PowerPoint PPT Presentation

Models of Choice. 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. Choice.

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### Models of Choice

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

T

S

X

Luce’s 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,
Luce’s Choice Theorem
• Proof: Define v(x) = kPT(x), for k > 0. Then, by the choice axiom (proof of uniqueness left to reader),
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

Prototype vs.

Exemplar

Theories

Qualitative Test

<- More ‘protypical’

<- Less ‘prototypcial’

Qualitative Test

<- Similar to A1, A3

<- Similar to A2, B6, B7

Prototype: A1>A2

Exemplar: A2>A1

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.