1. Testing Heuristic Models of Risky Decision Making Michael H. Birnbaum California State University, Fullerton

2. Outline Priority Heuristic New Critical Tests: Allow each person to have a different LS with different parameters Four tests: Interaction, Integration, Transitivity, & Priority Dominance.

3. Priority Heuristic Brandstätter, et al (2006) model assumes people do NOT weight or integrate information. Examine dimensions in order Only 4 dimensions considered. Order fixed: L, P(L), H, P(H).

4. PH for 2-branch gambles First: minimal gains. If the difference exceeds 1/10 the (rounded) maximal gain, choose by best minimal gain. If minimal gains not decisive, consider probability; if difference exceeds 1/10, choose best probability. Otherwise, choose gamble with the better highest consequence.

5. Priority Heuristic examples A:.5 to win $100.5 to win $0 B: $40 for sure Reason: lowest consequence. C:.02 to win $100.98 to win $0 Reason: highest consequence. D: $4 for sure

6. PH Reproduces Some Data… Predicts 100% of modal choices in Kahneman & Tversky, 1979. Predicts 85% of choices in Erev, et al. (1992) Predicts 73% of Mellers, et al. (1992) data

7. …But not all Data Birnbaum & Navarrete (1998): 43% Birnbaum (1999): 25% Birnbaum (2004): 23% Birnbaum & Gutierrez (in press): 30%

8. Problems No attention to middle branch, contrary to results in Birnbaum (1999) Fails to predict stochastic dominance in cases where people satisfy it in Birnbaum (1999). Fails to predict violations when 70% violate stochastic dominance. Not accurate when EVs differ. No individual differences and no free parameters. Different data sets have different parameters. Delta >.12 & Delta <.04.

9. Modification: Suppose different people have different LS with different parameters.

10. 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).

11. 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.

12. 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*

13. 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.

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

15. 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 ??.

16. Results: Interaction n = 153 RiskySafe% Safe Est. p ($95,.1;$5)($55,.1;$20)71%.76 ($95,.99;$5)($55,.99;$20)17%.04

17. 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

18. 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.

19. 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.

20. 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

21. 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.

22. 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

23. 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.

24. 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

25. 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.

26. New Studies of Transitivity LS models violate transitivity: A > B and B > C implies A > C. Birnbaum & Gutierrez 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.

27. Three of Tversky’s (1969) Gambles A = ($5.00, 0.29; $0, 0.71) C = ($4.50, 0.38; $0, 0.62) E = ($4.00, 0.46; $0, 0.54) Priority Heurisitc Predicts: A > C; C > E, but E > A. Intransitive. TAX (prior): E > C > A

28. Tests of WST (Exp 1)

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

30. Summary Priority Heuristic’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.