Some New Approaches to Old Problems: Behavioral Models of Preference Michael H. Birnbaum California State University, Fullerton

Testing Algebraic Models with Error-Filled Data Algebraic models assume or imply formal properties such as stochastic dominance, coalescing, transitivity, gain-loss separability, etc. But these properties will not hold if data contain “error.”

Some Proposed Solutions Neo-Bayesian approach (Myung, Karabatsos, & Iverson. Cognitive process approach (Busemeyer) “Error” Theory (“Error Story”) approach (Thurstone, Luce) combined with algebraic models.

Variations of Error Models Thurstone, Luce: errors related to separation between subjective values. Case V: SST (scalability). Harless & Camerer: errors assumed to be equal for certain choices. Today: Allow each choice to have a different rate of error. Advantage: we desire error theory that is both descriptive and neutral.

Basic 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

One Choice, Two Repetitions AB A B

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:

Estimating e

Estimating p

Testing if p = 0

Ex: Stochastic Dominance 122 Undergrads: 59% repeated viols (BB) 28% Preference Reversals (AB or BA) Estimates: e = 0.19; p = Experts: 35% repeated violations 31% Reversals Estimates: e = 0.196; p = 0.50 Chi-Squared test reject H0: p < 0.4

Testing 2, 3, 4-Choice Properties Extending this model to properties using 2, 3, or 4 choices is straightforward. Allow a different error rate on each choice. Allow a true probability for each choice pattern.

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

Weak Stochastic Transitivity

WST Can be Violated even when Everyone is Perfectly Transitive

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.

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

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 Birnbaum, Patton, & Lott (1999) recipe. New tests: Recipes based on Schmidt changing utility models.

Priority Heuristic Brandstaetter, Gigerenzer, & Hertwig (in press) model assumes people do NOT weight or integrate information. Each decision based on one reason only. Reasons tested one at a time in fixed order.

Choices between 2-branch gambles First, consider minimal gains. If the difference exceeds 1/10 the maximal gain, choose 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.

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

Priority Heuristic Implies Violations of Transitivity Satisfies New Property: Priority Dominance. Decision based on dimension with priority cannot be overrulled by changes on other dimensions. Satisfies Independence Properties: Decision cannot be altered by any dimension that is the same in both gambles.

Fit of PH to Data Brandstaetter, et al argue that PH fits the data of Kahneman and Tversky (1979) and Tversky and Kahneman (1992) and other data better than does CPT or TAX. It also fits Tversky’s (1969) violations of transitivity.

Tversky Gambles Some Sample Data, using Tversky’s 5 gambles, but formatted with tickets instead of pie charts. Data as of May 17, 2005, n = 251. No pre-selection of participants. Participants served in other 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.

Results-ACE patternRep 1Rep 2Both sum

Test of WST

Comments Results are surprisingly transitive. Differences: no pre-test, selection; Probability represented by # of tickets (100 per urn); Participants have practice with variety of gambles, & choices; Tested via Computer.

Response Patterns Choice ( 0 = first; 1 = second) LPHLPH LHPLHP PLHPLH PHLPHL HLPHLP HPLHPL TAXTAX ($26,.1;$0)($25,.1;$20) ($100,.1;$0)($25,.1;$20) ($26,.99;$0)($25,.99;$20) ($100,.99;$0)($25,.99;$20)

Data Patterns (n = 260) Frequency BothRep 1 or 2 not bothEst. Prob

Summary True & Error model with different error rates seems a reasonable “null” hypothesis for testing transitivity and other properties. Requires data with replications so that we can use each person’s self-agreement or reversals to estimate whether response patterns are “real” or due to “error.” Priority Heuristic model’s predicted violations of transitivity do not occur and its prediction of priority dominance is violated.