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Among those who cycle most have no regrets Michael H. Birnbaum Decision Research Center, Fullerton Outline • Family of Integrative Contrast Models • Special Cases: Regret Theory, Majority Rule (aka Most Probable Winner) • Predicted Intransitivity: Forward and Reverse Cycles • Pilot Experiment & Planned Work with Enrico Diecidue • Results: Pilot tests. Comments welcome Integrative, Interactive Contrast Models n A B (E i ) (ai ,bi ) i1 A (a1, E1;a2 , E 2 ; ;an , E n ) B (b1, E1;b2, E 2 ; ;bn , E n ) Assumptions (ai, bi ) (bi , ai ) (ai, bi ) 0 ai bi Difference Model : (ai, bi ) f [u(ai ) u(bi )] Special Cases • Majority Rule (aka Most Probable Winner) • Regret Theory • These can be represented with different functions. I will illustrate with different functions, f. Majority Rule Model 1 if u(a) u(b) f [u(a) u(b)] 0 if u(a) u(b) 1 if u(a) u(b) Regret Model f [u(a) u(b)] u(a) u(b) , u(a) u(b) 1 Predicted Intransitivity • These models violate transitivity of preference • Regret and MR cycle in opposite directions • However, both REVERSE cycle under permutation over events; i.e., “juxtaposition.” Concrete Example • • • • Urn: 33 Red, 33White, 33 Blue One marble drawn randomly Prize depends on color drawn. A = ($4, $5, $6) means win $4 if Red, win $5 if White, $6 if Blue. Majority Rule Prediction • • • • • • • A = ($4, $5, $6) B = ($5, $7, $3) C = ($9, $1, $5) AB: choose B BC: choose C CA: choose A Notation: 222 • • • • • • • A’ = ($6, $4, $5) B’ = ($5, $7, $3) C’ = ($1, $5, $9) A’B’: choose A’ B’C’: choose B’ C’A’: choose C’ Notation: 111 Regret Prediction • • • • • • • A = ($4, $5, $6) B = ($5, $7, $3) C = ($9, $1, $5) AB: choose A BC: choose B CA: choose C Notation: 111 • • • • • • • A’ = ($6, $4, $5) B’ = ($5, $7, $3) C’ = ($1, $5, $9) A’B’: choose B’ B’C’: choose C’ C’A’: choose A’ Notation: 222 Pilot Test • • • • 240 Undergraduates Tested via computers (browser) Clicked button to choose 30 choices (includes counterbalanced choices) • 10 min. task, 30 choices repeated. ABC Design Results PATTERN 111 112 121 122 211 212 221 222 TOTAL DATA PREDICTIONS One Rep not 2 Two Reps One not 2 two reps true probs 9.25 1 14.5 1.3 0.00 31.75 50.5 38.9 37.4 0.55 10.25 2.5 11.3 1.9 0.00 14.25 4.5 19.0 3.4 0.02 14.75 1.5 12.2 1.3 0.01 27.75 16 30.6 13.3 0.13 15.25 16 16.0 14.0 0.21 15.75 9 17.7 7.1 0.09 139 101 160.2 79.8 1.00 True and 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 inconsistency of response to the same choice by same person over repetitions One Choice, Two Repetitions A A B B pe 2 (1 p)(1 e)2 p(1e)e (1 p)(1 e)e p(1 e)e (1 p)(1 e)e p(1 e)2 (1 p)e 2 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: (1 e)e Estimating e Estimating p Testing if p = 0 A’B’C’ Results PATTERN 111 112 121 122 211 212 221 222 TOTAL DATA PREDICTIONS One Rep not 2 Two Reps One not 2 two reps true probs 12.75 7 19.5 5.8 0.05 31.75 71.5 43.8 55.6 0.70 13.5 6 11.3 5.9 0.06 16.25 2 19.4 2.3 0.00 11.5 2 10.1 1.9 0.01 25.25 8 25.7 7.4 0.04 10.5 8 10.2 8.8 0.10 11.5 2.5 9.7 2.5 0.03 133 107 149.8 90.2 1 ABC X A’B’C’ Analysis 111 112 121 122 211 212 221 222 111 1.0 2.0 2.0 1.8 1.3 3.3 1.8 6.8 112 3.0 59.5 2.8 5.0 3.3 22.3 3.0 4.5 121 0.5 0.8 2.3 3.5 1.5 2.0 6.3 2.8 122 1.3 3.0 2.0 1.8 2.5 1.8 3.8 2.3 211 0.5 1.5 2.0 1.3 0.5 1.5 3.8 2.5 212 1.0 12.8 0.3 1.5 3.3 10.0 2.0 2.5 221 0.8 0.8 1.3 2.8 1.8 0.8 8.8 1.8 222 2.3 2.0 0.3 1.3 2.3 2.3 2.0 1.8 ABC-A’B’C’ Analysis ABC-A'B'C' PATTERN Est. true probs 111111 0.00 112112 0.59 TAX 121121 0.04 122122 0.01 211211 0.00 212212 0.08 221221 0.16 222222 0.02 222111 0.09 MR 111222 0.02 Regret Results • Most people are transitive. • Most common pattern is 112, pattern predicted by TAX with prior parameters. • However, 2 people were perfectly consistent with MR on 24 choices. • No one fit Regret theory perfectly. Results: Continued • Among those few (est. ~10%) who cycle (intransitive), most have no regrets (i.e., they appear to satisfy MR). • Suppose 5-10% of participants are intransitive. Do we think that they indeed use a different process? Is there an artifact in the experiment? If not, can we increase the rate of intransitivity? Advice Welcome: Our Plans • We plan to test participants from the same pool was used to elicit regret function. • Assignment: Devise a theorem of integrative interactive contrast model that will lead to self-contradiction (“paradox” of regret theory). • These contrast models also imply RBI, which is refuted by our data. Summary • Regret and MR imply intransitivity whose direction can be reversed by permutation of the consequences. • Very few people are intransitive but a few do indeed appear to be consistent with MR and 2 actually show the pattern in 24 choices.