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Testing Critical Properties of Models of Risky Decision Making

Testing Critical Properties of Models of Risky Decision Making. Michael H. Birnbaum Fullerton, California, USA Sept. 13, 2007 Luxembourg. Outline. I will discuss critical properties that test between nonnested theories such as CPT and TAX.

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Testing Critical Properties of Models of Risky Decision Making

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  1. Testing Critical Properties of Models of Risky Decision Making Michael H. Birnbaum Fullerton, California, USA Sept. 13, 2007 Luxembourg

  2. Outline • I will discuss critical properties that test between nonnested theories such as CPT and TAX. • I will also discuss properties that distinguish Lexicographic Semiorders and family of transitive integrative models (including CPT and TAX).

  3. Cumulative Prospect Theory/ Rank-Dependent Utility (RDU)

  4. “Prior” TAX Model Assumptions:

  5. TAX Parameters For 0 < x < $150 u(x) = x Gives a decent approximation. Risk aversion produced by d. d = 1 .

  6. TAX Model

  7. TAX and CPT nearly identical for binary (two-branch) gambles • CE (x, p; y) is an inverse-S function of p according to both TAX and CPT, given their typical parameters. • Therefore, there is no point trying to distinguish these models with binary gambles.

  8. Non-nested Models

  9. CPT and TAX nearly identical inside the prob. simplex

  10. Testing CPT TAX:Violations of: • Coalescing • Stochastic Dominance • Lower Cum. Independence • Upper Cumulative Independence • Upper Tail Independence • Gain-Loss Separability

  11. Testing TAX Model CPT: Violations of: • 4-Distribution Independence (RS’) • 3-Lower Distribution Independence • 3-2 Lower Distribution Independence • 3-Upper Distribution Independence (RS’) • Res. Branch Indep (RS’)

  12. Stochastic Dominance • A test between CPT and TAX: G = (x, p; y, q; z) vs. F = (x, p – s; y’, s; z) Note that this recipe uses 4 distinct consequences: x > y’ > y > z > 0; outside the probability simplex defined on three consequences. CPT  choose G, TAX  choose F Test if violations due to “error.”

  13. Violations of Stochastic Dominance 122 Undergrads: 59% two violations (BB) 28% Pref Reversals (AB or BA) Estimates: e = 0.19; p = 0.85 170 Experts: 35% repeat violations 31% Reversals Estimates: e = 0.20; p = 0.50 Chi-Squared test reject H0: p violations < 0.4

  14. Pie Charts

  15. Aligned Table: Coalesced

  16. Summary: 23 Studies of SD, 8653 participants • Large effects of splitting vs. coalescing of branches • Small effects of education, gender, study of decision science • Very small effects of probability format, request to justify choice. • Miniscule effects of event framing (framed vs unframed)

  17. Lower Cumulative Independence R:39%S:61% .90 to win $3 .90 to win $3 .05 to win $12 .05 to win $48 .05 to win $96 .05 to win $52 R'':69%S'':31% .95 to win $12 .90 to win $12 .05 to win $96 .10 to win $52

  18. Upper Cumulative Independence R':72%S':28% .10 to win $10 .10 to win $40 .10 to win $98 .10 to win $44 .80 to win $110 .80 to win $110 R''':34%S''':66% .10 to win $10 .20 to win $40 .90 to win $98 .80 to win $98

  19. Summary: UCI & LCI 22 studies with 33 Variations of the Choices, 6543 Participants, & a variety of display formats and procedures. Significant Violations found in all studies.

  20. S’: .1 to win $40 .1 to win $44 .8 to win $100 S: .8 to win $2 .1 to win $40 .1 to win $44 R’: .1 to win $10 .1 to win $98 .8 to win $100 R: .8 to win $2 .1 to win $10 .1 to win $98 Restricted Branch Indep.

  21. S’: .10 to win $40 .10 to win $44 .80 to win $100 S2’: .45 to win $40 .45 to win $44 .10 to win $100 R’: .10 to win $4 .10 to win $96 .80 to win $100 R2’: .45 to win $4 .45 to win $96 .10 to win $100 3-Upper Distribution Ind.

  22. S’: .80 to win $2 .10 to win $40 .10 to win $44 S2’: .10 to win $2 .45 to win $40 .45 to win $44 R’: .80 to win $2 .10 to win $4 .10 to win $96 R2’: .10 to win $2 .45 to win $4 .45 to win $96 3-Lower Distribution Ind.

  23. Gain-Loss Separability

  24. Notation

  25. Wu and Markle Result

  26. Birnbaum & Bahra--% F

  27. Allais Paradox Dissection

  28. Summary: Prospect Theories not Descriptive • Violations of Coalescing • Violations of Stochastic Dominance • Violations of Gain-Loss Separability • Dissection of Allais Paradoxes: viols of coalescing and restricted branch independence; RBI violations opposite of Allais paradox.

  29. Summary-2

  30. Summary-3

  31. Results: CPT makes wrong predictions for all 12 tests • Can CPT be saved by using different formats for presentation? More than a dozen formats have been tested. • Violations of coalescing, stochastic dominance, lower and upper cumulative independence replicated with 14 different formats and thousands of participants.

  32. Lexicographic Semiorders • Renewed interest in issue of transitivity. • Priority heuristic of Brandstaetter, Gigerenzer & Hertwig is a variant of LS, plus some additional features. • In this class of models, people do not integrate information or have interactions such as the probability x prize interaction in family of integrative, transitive models (EU, CPT, TAX, and others)

  33. LPH LS: G = (x, p; y) F = (x’, q; y’) • If (y –y’ > D) choose G • Else if (y ’- y > D) choose F • Else if (p – q > d) choose G • Else if (q – p > d) choose F • Else if (x – x’ > 0) choose G • Else if (x’ – x > 0) choose F • Else choose randomly

  34. Family of LS • In two-branch gambles, G = (x, p; y), there are three dimensions: L = lowest outcome (y), P = probability (p), and H = highest outcome (x). • There are 6 orders in which one might consider the dimensions: LPH, LHP, PLH, PHL, HPL, HLP. • In addition, there are two threshold parameters (for the first two dimensions).

  35. Non-Nested Models Allais Paradoxes Lexicographic Semiorders TAX, CPT Integration Intransitivity Interaction Transitive Priority Dominance

  36. New Tests of Independence • Dimension Interaction: Decision should be independent of any dimension that has the same value in both alternatives. • Dimension Integration: indecisive differences cannot add up to be decisive. • Priority Dominance: if a difference is decisive, no effect of other dimensions.

  37. Taxonomy of choice models

  38. Testing Algebraic Models with Error-Filled Data • Models assume or imply formal properties such as interactive independence. • But these properties may not hold if data contain “error.” • Different people might have different “true” preference orders, and different items might produce different amounts of error.

  39. Error Model Assumptions • Each choice pattern in an experiment has a true probability, p, and each choice has an error rate, e. • The error rate is estimated from inconsistency of response to the same choice by same person over repetitions.

  40. Priority Heuristic Implies • Violations of Transitivity • Satisfies Interactive Independence: Decision cannot be altered by any dimension that is the same in both gambles. • No Dimension Integration: 4-choice property. • Priority Dominance. Decision based on dimension with priority cannot be overruled by changes on other dimensions. 6-choice.

  41. Dimension Interaction

  42. Family of LS • 6 Orders: LPH, LHP, PLH, PHL, HPL, HLP. • There are 3 ranges for each of two parameters, making 9 combinations of parameter ranges. • There are 6 X 9 = 54 LS models. • But all models predict SS, RR, or ??.

  43. Results: Interaction n = 153

  44. Analysis of Interaction • Estimated probabilities: • P(SS) = 0 (prior PH) • P(SR) = 0.75 (prior TAX) • P(RS) = 0 • P(RR) = 0.25 • Priority Heuristic: Predicts SS

  45. Probability Mixture Model • Suppose each person uses a LS on any trial, but randomly switches from one order to another and one set of parameters to another. • But any mixture of LS is a mix of SS, RR, and ??. So no LS mixture model explains SR or RS.

  46. Dimension Integration Study with Adam LaCroix • Difference produced by one dimension cannot be overcome by integrating nondecisive differences on 2 dimensions. • We can examine all six LS Rules for each experiment X 9 parameter combinations. • Each experiment manipulates 2 factors. • A 2 x 2 test yields a 4-choice property.

  47. Integration Resp. Patterns

  48. 54 LS Models • Predict SSSS, SRSR, SSRR, or RRRR. • TAX predicts SSSR—two improvements to R can combine to shift preference. • Mixture model of LS does not predict SSSR pattern.

  49. Choice Percentages

  50. Test of Dim. Integration • Data form a 16 X 16 array of response patterns to four choice problems with 2 replicates. • Data are partitioned into 16 patterns that are repeated in both replicates and frequency of each pattern in one or the other replicate but not both.

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