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  

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

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

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. Restricted Branch Indep. 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

21. 3-Upper Distribution Ind. 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

22. 3-Lower Distribution Ind. 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

23. Gain-Loss Separability

24. Notation

25. Wu and Markle Result

26. Birnbaum & Bahra--% F

27. Allais Paradox Dissection Restricted Branch Independence CoalescingSatisfiedViolated SatisfiedEU, PT*,CPT*CPT ViolatedPTTAX

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 PropertyCPTRAMTAX LCINo ViolsViols UCINo ViolsViols UTINo ViolsR ’ S1Viols LDIRS2 ViolsNo Viols 3-2 LDIRS2 ViolsNo Viols

30. Summary-3 PropertyCPTRAMTAX 4-DI RS ’ Viols No Viols SR ’ Viols UDIS ’ R2 ’ Viols No ViolsR ’ S2 ’ Viols RBI RS ’ ViolsSR ’ Viols

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’ >  ) choose G Else if (y ’- y >  ) choose F Else if (p – q >  ) choose G Else if (q – p >  ) 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 TAX, CPT Lexicographic Semiorders Semiorders Intransitivity Allais Paradoxes Priority Dominance Integration Interaction Transitive

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 TransitiveIntransitive Interactive & Integrative EU, CPT, TAX Regret, Majority Rule Non-interactive & Integrative Additive, CWA Additive Diffs, SD Not interactive or integrative 1-dim.LS, PH*

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 RiskySafeTAXLPHHPL ($95,.1;$5)($55,.1;$20)SSR ($95,.99;$5)($55,.99;$20)RSR

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 RiskySafe% Safe Est. p ($95,.1;$5)($55,.1;$20)71%.76 ($95,.99;$5)($55,.99;$20)17%.04

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 Choice Risky= 0 Safe = 1 LPHLPH LPHLPH LPHLPH HPLHPL HPLHPL HPLHPL TAXTAX ($51,.5;$0)($50,.5;$50) 1101101 ($51,.5;$40)($50,.5;$50) 1001101 ($80,.5;$0)($50,.5;$50) 1100101 ($80,.5;$40)($50,.5;$50) 1000100

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 Risky Safe % safe ($51,.5;$0)($50,.5;$50)93 ($51,.5;$40)($50,.5;$50)82 ($80,.5;$0)($50,.5;$50)79 ($80,.5;$40)($50,.5;$50)19

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.

51. Data Patterns (n = 260) PatternFrequency BothRep 1Rep 2Est. Prob 0000 1160.03 0001 1160.01 0010 0630.02 0011 0000 0100 0340.01 0101 0110 0110 * 0200 0111 0100 1000 01340 1001 0010 1010 * 426140.02 1011 0760 1100 PHL, HLP,HPL * 620360.04 1101 0640 1110 TAX 981491320.80 1111 LPH, LHP, PLH * 924430.06

52. Results: Dimension Integration Data strongly violate independence property of LS family Data are consistent instead with dimension integration. Two small, indecisive effects can combine to reverse preferences. Replicated with all pairs of 2 dims.

53. New Studies of Transitivity LS models violate transitivity: A > B and B > C implies A > C. Birnbaum & Gutierrez (2007) tested transitivity using Tversky’s gambles, but using typical methods for display of choices. Also used pie displays with and without numerical information about probability. Similar results with both procedures.

54. Replication of Tversky (‘69) with Roman Gutierrez Two studies used Tversky’s 5 gambles, formatted with tickets instead of pie charts. Two conditions used pies. Exp 1, n = 251. No pre-selection of participants. Participants served in other studies, prior to testing (~1 hr).

55. Three of Tversky’s (1969) Gambles A = ($5.00, 0.29; $0, 0.79) C = ($4.50, 0.38; $0, 0.62) E = ($4.00, 0.46; $0, 0.54) Priority Heurisitc Predicts: A preferred to C; C preferred to E, But E preferred to A. Intransitive. TAX (prior): E > C > A

56. Tests of WST (Exp 1)

57. Response Combinations Notation(A, C)(C, E)(E, A) 000ACE* PH 001ACA 010AEE 011AEA 100CCE 101CCA 110CEETAX 111CEA*

58. WST Can be Violated even when Everyone is Perfectly Transitive

59. Results-ACE patternRep 1Rep 2Both 000 (PH)10215 00111139 01014231 011710 10016194 101431 110 (TAX)176154133 11113173 sum251 156

60. Comments Results were surprisingly transitive, unlike Tversky’s data. Differences: no pre-test, selection; Probability represented by # of tickets (100 per urn); similar results with pies. Participants have practice with variety of gambles, & choices; Tested via Computer.

61. Summary Priority Heuristic model’s predicted violations of transitivity are rare. Dimension Interaction violates any member of LS models including PH. Dimension Integration violates any LS model including PH. Data violate mixture model of LS. Evidence of Interaction and Integration compatible with models like EU, CPT, TAX.