1. Testing Transitivity with a True and Error Model Michael H. Birnbaum California State University, Fullerton

2. Testing Algebraic Models with Error-Filled Data Models assume or imply formal properties such as transitivity: If A > B and B > C then A > C But such properties may not hold if data contain “error.” And different people might have different “true” preferences.

3. Error Model Assumptions Each choice in an experiment has a true choice probability, p, and an error rate, e. The error rate is estimated from (and is the “reason” given for) inconsistency of response to the same choice by same person over repetitions

4. One Choice, Two Repetitions AB A B

5. Solution for e The proportion of preference reversals between repetitions allows an estimate of e. Both off-diagonal entries should be equal, and are equal to:

6. Ex: Stochastic Dominance 122 Undergrads: 59% repeated viols (BB) 28% Preference Reversals (AB or BA) Estimates: e = 0.19; p = 0.85 170 Experts: 35% show 2 violations (BB) 31% Reversals (AB or BA) Estimates: e = 0.196; p = 0.50

7. Testing Higher Properties Extending this model to properties relating 2, 3, or 4 choices: Allow a different error rate on each choice. Estimate true probability for each choice pattern. Different people can have different “true” patterns, which need not be transitive.

8. New Studies of Transitivity Work currently under way testing transitivity under same conditions as used in tests of other decision properties. Participants view choices via the WWW, click button beside the gamble they would prefer to play.

9. Some Recipes being Tested Tversky’s (1969) 5 gambles. LS: Preds of Priority Heuristic Starmer’s recipe Additive Difference Model (regret; majority rule) Birnbaum, Patton, & Lott (1999) recipe. Recipes based on Bleichrodt & Schmidt context-dependent utility models.

10. Replications of Tversky (1969) with Roman Gutierez First two studies used Tversky’s 5 gambles, but formatted with tickets instead of pie charts. Two studies with n = 417 and n = 327 with small or large prizes ($4.50 or $450) No pre-selection of participants. Participants served in other risky DM studies, prior to testing (~1 hr).

11. 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, and E preferred to A. Intransitive. Tversky (1969) reported viol of WST

12. Response Combinations Notation(A, B)(B, C)(C, A) 000ABC* 001ABA 010ACC 011ACA 100BBC 101BBA 110BCC 111BCA*

13. Weak Stochastic Transitivity

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

15. Triangle Inequality has similar problems: It is possible that everyone is transitive but WST is violated. It is possible that people are systematically intransitive and WST is satisfied. Possible that everyone is intransitive and triangle inequality is satisfied.

16. Model for Transitivity A similar expression is written for the other seven probabilities. These can in turn be expanded to predict the probabilities of showing each pattern repeatedly; i.e., up to six errors.

17. Expand and Simplify There are 8 X 8 data patterns in an experiment with 2 repetitions. However, most of these have very small frequencies. Examine probabilities of each of 8 repeated patterns. Frequencies of showing each of 8 patterns in one replicate OR the other, but NOT both. Mutually exclusive, exhaustive partition.

18. Tests of WST Percentage Choosing Column >pr Row Gamble RowABCDE A73778085 B306879 C16297478 D11162463 E13171533

19. Patterns for A, C, E PatternRep. 1Rep 2Both 00014285 001182515 01023381 0111253 10024337 101561 110301256220 11119255 Sum416 257

20. PatternBothRep 1 or 2 Not both Est pPred BothPred 1 or 2 Not both 000516.038.18.6 001156.5.0715.36.5 010129.5.004.737.2 01135.5.012.85.9 100721.5.037.826.0 10114.5.000.95.5 11022058.5.85196.667.6 111517.024.617.9 Sum2571591240.9175.1

21. Comments Results are surprisingly transitive, unlike Tversky’s data (est. 95% transitive). Of those 115 who were perfectly reliable, 93 perfectly consistent with EV (p), 8 with opposite ($), and only 1 intransitive. Differences: no pre-test, selection; Probability represented by # of tickets (100 per urn), rather than by pies. Participants have practice with variety of gambles, & choices; Tested via Computer.

22. Pie Chart Format

23. Pies: with or without Numerical probabilities 321 participants randomly assigned to same study; except probabilities displayed as pies (spinner), either with numerical probabilities displayed or without. Of 105 who were perfectly reliable, 84 were perfectly consistent with EV (prob), 13 with the opposite order ($); 1 consistent with LS.

24. Lower Standards Look at AB,BC,CD,DE choices and EA choices only: 10 of 321 participants showed this pattern; all in the pies-only condition. Although the rate is low (6% of 160), association with condition is clear! By still lower standard used by Tversky: 75% agreement with above pattern: 37 people, 27 in pies-only condition.

25. Tests of Lexicographic Semi- order and Additive Difference LS implies no integration of contrasts (additive difference model allow integration) Both LS and additive difference models imply no interactions between probability and consequences.

26. Test of Interaction RS Pies & p Pies & No p ($7.25,.95; $1.25,.05) ($4.25,.95; $3.25,.05) 1622 ($7.25,.05; $1.25,.95) ($4.25,.05; $3.25,.95) 8477

27. Among the 37 Leniently classified as Intransitive Are those 37 who are 75% consistent with the LS in the 5 choices also approx. consistent with LS in tests of Interaction? No. 26 of these have all four choices in the pattern of interaction predicted by TAX and other utility models.

28. Summary Priority Heuristic model’s predicted violations of transitivity are rare and rarely repeated when probability and prize information presented numerically. Violations of transitivity are still rare but more frequent when prob information presented only graphically. Evidence of Dimension Interaction violates PH and additive Difference models.

29. Conclusions Violations of transitivity are probably not due to intransitive strategy (LS or additive difference model), but rather to a configural assimilation of the probability values, which are then used in a numerical utility model. We are still unable to produce the higher rates of intransitivity reported by Tversky and others.

30. Transitivity Test: ADL TrialChoice%Response Pattern RepsEst. parameters 00011011pe 8 50 to win $100 50 to win $20 50 to win $60 50 to win $27 181902325230.10 3 50 to win $60 50 to win $27 50 to win $45 50 to win $34 291404439370.170.20 21 50 to win $45 50 to win $34 50 to win $100 50 to win $20 743533201720.850.12

31. Results-ADL patternRep 1Rep 2Both 000 LPH21131 001 TAX134147106 01020188 011383710 1001590 10114100 11012157 1116111 Sum260 133