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This Pump Sucks: Testing Transitivity with Individual Data

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### This Pump Sucks: Testing Transitivity with Individual Data

Michael H. Birnbaum and Jeffrey P. Bahra

California State University, Fullerton

Transitivity of Preference

- If A > B and B > C then A > C.
- Satisfy it or become a money pump.
- But transitivity may not hold if data contain “error.”
- And different people might have different “true” preferences.

Tversky (1969)

- Tversky (1969) reported that selected subjects showed a pattern of intransitive data consistent with a lexicographic semi-order.
- Tversky tested Weak Stochastic Transitivity: If P(A>B) > 1/2 and P(B>C) > 1/2 then P(A>C) > 1/2.

Issues

- Iverson & Falmagne (1985) argued that Tversky’s statistical analysis was incorrect of WST.
- Tversky went on to publish transitive theories of preference (e.g., CPT).

Renewed Interest in Intransitive Preference

- New analytical methods for analysis of transitivity (Iverson, Myung, & Karabatsos; Regenwetter & Stober, et al); Error models (Sopher & Gigliotti, ‘93; Birnbaum, ‘04; others).
- Priority Heuristic (Brandstaetter, et al., 2006); stochastic difference model (González-Vallejo, 2002;similarity judgments, Leland, 1994; majority rule, Zhang, Hsee, Xiao, 2006). Renewed interest in Fishburn, as well as in Regret Theory.

Lexicographic Semi-order

- G = (x, p; y, 1 - p). F = (x’, q; y’, 1 - q).
- If y - y’ ≥ DL choose G (DL = $10)
- If y’ - y ≥ DL choose F
- If p - q ≥ DP choose G (DP = 0.1)
- If q - p ≥ DP choose F
- If x > x’ choose G; if x’ > x choose F;
- Otherwise, choose randomly.

Priority Heuristic

- “Aspiration level” is 10% of largest prize, rounded to nearest prominent number.
- Compare gambles by lowest consequences. If difference exceeds the aspiration level, choose by lowest consequence.
- If not, compare probabilities; choose by probability if difference ≥ 0.1
- Compare largest consequences; choose by largest consequences.

New Studies of Transitivity

- Work currently under way testing transitivity using same procedures as used in other decision research.
- Participants view choices via the WWW, click button beside the gamble they would prefer to play.
- Today’s talk: Single-S data.

Studies with Roman Gutierez

- Four studies used Tversky’s 5 gambles, formatted with tickets or with pie charts.
- 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.

Findings

- Results were 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; 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 conditions with 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.

Findings

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

Response to Birnbaum-Gutierrez

- Perhaps the intransitivity only develops in longer studies. Tversky used 20 replications of each choice.
- Perhaps consequences of Tversky’s gambles diminished since 1969 due to inflation. Perhaps transitivity occurs because those prizes are too small.

Birnbaum & Bahra

- Collected up to 40 choices/pair per person. (20 reps). 2 Sessions, 1.5 hrs, 1 week apart.
- Cash prizes up to $100.
- 51 participants, of whom 10 to win the prize of one of their chosen gambles.
- 3 5 x 5 Designs to test transitivity vs. Priority heuristic predictions

Notation-Two-branch Gambles

- G = (x, p; y, 1 - p); x > y ≥ 0
- L = Lower Consequence
- P = Probability to win higher prize
- H = Higher consequence

LH Design

- A = ($84, .50; $24)
- B = ($88, .50; $20)
- C = ($92, .50; $16)
- D = ($96, .50; $12)
- E = ($100, .50; $8)

LP Design

- A = ($100, .50; $24)
- B = ($100, .54; $20)
- C = ($100, .58; $16)
- D = ($100, .62; $12)
- E = ($100, .66; $8)

PH Design

- A = ($100, .50; $0)
- B = ($96, .54; $0)
- C = ($92, .58; $0)
- D = ($88, .62; $0)
- E = ($84, .66; $0)

Priority Heuristic Predictions

- LH Design: E > D > C > B > A, but A > E
- LP Design: A ~ B ~ C ~ D ~ E, but A > E
- PH Design: A > B > C > D > E but E > A

Analysis

- Each replication of each design has 20 choices; hence 1,048,576 possible data patterns (220) per rep.
- There are 1024 possible consistent patterns (Rij = 2 iff Rji = 1, all i, j).
- There are 120 (5!) possible transitive patterns.

Within-Rep Consistency

- Count the number of consistent choices in a replicate of 20 choices (10 x 2).
- If a person always chose the same button, consistency = 0.
- If a person was perfectly consistent, consistency = 10.
- Randomly choosing between 1 and 2 produces expected consistency of 5.

Within-Replicate Consistency

- The average rate of agreement was 8.63 (86% self-agreement).
- 46.4% of all replicates were scored 10; an additional 19.9% were scored 9.

Majority Data WST

- LH Design A>B>C>D>E
- LP Design A>B>C>D>E
- PH Design E>D>C>B>A
- Patterns consistent with special TAX with “prior” parameters.
- But this analysis hides individual diffs

Individual Data

- Choice proportions calculated for each individual in each design.
- These were further broken down within each person by replication.

S# 6176 C = 9.8 Rep = 20; started with this pattern, then switched to perfectly consistent with the opposite pattern for 4 replicates at the end of the first day; back to this pattern for 10 reps on day 2.

S# 684 C = 8.1 Rep = 14; an intransitive pattern opposite that predicted by priority heuristic.

S# 7663 C = 6.3 Rep = 10; an intransitive pattern consistent with priority heuristic, DP = 0.05. Few reps and low self-consistency in this case.

Data Summary

- For n = 51, there are 153 matrices. Of these, 90% were perfectly consistent with WST: P(A,B) ≥ 1/2 & P(B,C) ≥ 1/2 then P(A,C) ≥ 1/2.
- 29 people had all three arrays fitting WST; no one had all three arrays with intransitive patterns.

Within-Person Changes in Preference Pattern

- Criterion: Person must show perfect consistency (10 out of 10) to one pattern in one replication, and perfect consistency to another pattern on another replication.
- 15 Such cases were found (10%). There may be other cases where the data are less consistent.

Summary

- Recent studies fail to confirm systematic violations of transitivity predicted by priority heuristic. Adds to growing case against this descriptive model.
- Individual data are mostly transitive.
- Next Q: From individual data, can we predict, for example, from these data to other kinds of choices by same person, e. g., tests of SD?

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