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Testing Transitivity (and other Properties) Using a True and Error Model

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Testing Transitivity (and other Properties) Using a True and Error Model

Michael H. Birnbaum

- 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.”

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

- 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

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

122 Undergrads: 59% repeated viols (BB)

28% Preference Reversals (AB or BA)

Estimates: e = 0.19; p = 0.85

170 Experts: 35% repeated violations

31% Reversals

Estimates: e = 0.196; p = 0.50

Chi-Squared test reject H0: p < 0.4

- 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.

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.

- A = ($15, 0.2; $0, 0.8)
- B = ($8; 0.3; $0, 0.7)
- C = ($8, 0.15; $7.75; 0.15; $0, .7)
- Starmer predicted intransitivity from Prospect Theory and the dominance detection (editing) mechanism.

Full model is underdetermined. One error

Fixed to zero; but other errors not equal.

Most people recognized dominance.

- 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.

- 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.

- Tversky’s (1969) 5 gambles.
- LS: Preds of Priority Heuristic
- Starmer’s recipe
- Additive Difference Model
- Birnbaum, Patton, & Lott (1999) recipe.

- Some Sample Data, using Tversky’s 5 gambles, but formatted with tickets instead of pie charts.
- Data as of May 5, 2005, n = 123.
- No pre-selection of participants.
- Participants served in other studies, prior to testing (~1 hr).

- A = ($5.00, 0.29; $0, 0.79)
- C = ($4.50, 0.38; $0, 0.62)
- E = ($4.00, 0.46; $0, 0.54)

- Preliminary results were surprisingly transitive.
- Difference: no pre-test, selection
- Probability represented by # of tickets (100 per urn)
- Participants have practice with variety of gambles, & choices.
- Tested via Computer

- Same Structure as Transitivity
- Property implied by CPT, RSDU
- Property violated by TAX.
- Loss Aversion: people do not like fair bets to win or lose.
- CPT: Loss Aversion due to utility function for gains and losses.

- Wu & Markle (2004) found evidence of violation of GLS. Modified CPT.
- Birnbaum & Bahra (2005) also find evidence of violation of GLS, violations of modified CPT as well.
- TAX: In mixed gambles, losses get greater weight. Data do not require kink in the utility function at zero.

- 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.”