Models of Choice

1. Models of Choice

2. 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

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

4. 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.

5. 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.

6. 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)

7. 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

9. Thurstone Scaling

10. 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.

11. 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.

12. A Problem with Thurstone Scaling • Works well for 2 alternatives, not more.

13. 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.

14. 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.

15. 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.

16. 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)

17. 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})

18. 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

19. 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

20. 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,

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

22. 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.

23. 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

24. Quantitative vs. Qualitative Tests

25. Quantitative vs. Qualitative Tests Prototype vs. Exemplar Theories

26. Quantitative Test Made-up #s

27. Qualitative Test <- More ‘protypical’ <- Less ‘prototypcial’

28. Qualitative Test <- Similar to A1, A3 <- Similar to A2, B6, B7 Prototype: A1>A2 Exemplar: A2>A1

29. Quantitative Test Made-up #s

30. 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.

31. 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.