Testing Lexicographic Semi-Order Models: Generalizing the Priority Heuristic

1. Testing Lexicographic Semi-Order Models: Generalizing the Priority Heuristic Michael H. Birnbaum California State University, Fullerton

2. Outline • Priority Heuristic for risky decisions • New Critical Tests: Allow each person to have a different LS with different parameters • Results for three tests: Interaction, Integration, and Transitivity. • Discussion and questions

3. Priority Heuristic • Brandstätter, Gigerenzer, & Hertwig • Consider one dimension at a time. • If that “reason” is decisive, other reasons not considered. No integration; no interactions. • Very different from utility models. • Very similar to CPT.

4. Priority Heuristic • Brandstätter, et al (2006) model assumes people do NOT weight or integrate information. • Each decision based on one dimension only. • Only 4 dimensions considered. • Order fixed: L, P(L), H, P(H).

5. 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 best highest consequence.

6. Priority Heuristic examples

7. Example: Allais Paradox

8. 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

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

10. 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.

11. Modifications: • People act as if they compute ratio of EV and choose higher EV when ratio > 2. • People act as if they can detect stochastic dominance. • Although these help, they do not improve model to more than 50% accuracy. • Today: Suppose different people have different LS with different parameters.

12. 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).

13. 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.

14. Taxonomy of choice models

15. 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.

16. 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.

17. 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.

18. Dimension Interaction

19. 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 ??.

20. Results: Interaction n = 153

21. 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

22. 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.

23. 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.

24. Integration Resp. Patterns

25. 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.

26. Choice Percentages

27. 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.

28. Data Patterns (n = 260)

29. 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.

30. 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.

31. Replication of Tversky (‘69) with Roman Gutierez • 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).

32. 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

33. Tests of WST (Exp 1)

34. Response Combinations

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

36. Results-ACE

37. 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.

38. 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.