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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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agenda
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
Choice
  • The same choice is not always made in the “same” situation.
  • Main assumption: Choice alternatives have choice probabilities.
overview of 3 models
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
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 scaling6
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 scaling7
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 scaling10
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
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
A Problem with Thurstone Scaling
  • Works well for 2 alternatives, not more.
luce s choice theory13
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
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 axiom15
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 axiom16
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 axiom17
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 axiom18
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 axiom19
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
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 theorem21
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 & 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
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 tests25
Quantitative vs. Qualitative Tests

Prototype vs.

Exemplar

Theories

qualitative test
Qualitative Test

<- More ‘protypical’

<- Less ‘prototypcial’

qualitative test28
Qualitative Test

<- Similar to A1, A3

<- Similar to A2, B6, B7

Prototype: A1>A2

Exemplar: A2>A1

quantitative vs qualitative tests30
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
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.