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Testing Transitivity with a True and Error Model. Michael H. Birnbaum California State University, Fullerton. 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
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Testing Transitivity with a True and Error Model Michael H. Birnbaum California State University, Fullerton
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.
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
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:
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
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.
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.
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.
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).
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; $0, 0.54) Priority Heurisitc Predicts: A preferred to C; C preferred to E, and E preferred to A. Intransitive. Tversky (1969) reported viol of WST
WST Can be Violated even when Everyone is Perfectly Transitive
Triangle Inequality has similar problems: • It is possible that everyone is transitive but WST is violated. • It is possible that people are systematically intransitive and WST is satisfied. • Possible that everyone is intransitive and triangle inequality is satisfied.
Model for Transitivity A similar expression is written for the other seven probabilities. These can in turn be expanded to predict the probabilities of showing each pattern repeatedly; i.e., up to six errors.
Expand and Simplify • There are 8 X 8 data patterns in an experiment with 2 repetitions. • However, most of these have very small frequencies. • Examine probabilities of each of 8 repeated patterns. • Frequencies of showing each of 8 patterns in one replicate OR the other, but NOT both. Mutually exclusive, exhaustive partition.
Comments • Results are surprisingly transitive, unlike Tversky’s data (est. 95% transitive). • Of those 115 who were perfectly reliable, 93 perfectly consistent with EV (p), 8 with opposite ($), 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.
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.
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.
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.
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.
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.
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.
