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.”

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This Pump Sucks: Testing Transitivity with Individual Data

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This pump sucks testing transitivity with individual data

This Pump Sucks: Testing Transitivity with Individual Data

Michael H. Birnbaum and Jeffrey P. Bahra

California State University, Fullerton


Transitivity of preference

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)

  • 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

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

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

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

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

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

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

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

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.


Pie chart format

Pie Chart Format


Pies with or without numerical probabilities

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.


Findings1

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

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

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

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

LH Design

  • A = ($84, .50; $24)

  • B = ($88, .50; $20)

  • C = ($92, .50; $16)

  • D = ($96, .50; $12)

  • E = ($100, .50; $8)


Lp design

LP Design

  • A = ($100, .50; $24)

  • B = ($100, .54; $20)

  • C = ($100, .58; $16)

  • D = ($100, .62; $12)

  • E = ($100, .66; $8)


Ph design

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

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


One rep 2 choices pair

One Rep = 2 choices/pair


Analysis

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

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.


Intransitive and consistent

Intransitive and Consistent


Within replicate consistency

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.


Lh design overall proportions choosing second gamble

LH Design: Overall Proportions Choosing Second Gamble


Lp design overall proportions choosing second gamble

LP Design: Overall Proportions Choosing Second Gamble


Ph design overall proportions choosing second gamble

PH Design: Overall Proportions Choosing Second Gamble


Majority data wst

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

Individual Data

  • Choice proportions calculated for each individual in each design.

  • These were further broken down within each person by replication.


S 8328 c 9 6 rep 20

S# 8328 C = 9.6 Rep = 20


S 8328 c 9 8 rep 20

S# 8328 C = 9.8 Rep = 20


S 8328 c 9 9 rep 20

S# 8328 C = 9.9 Rep = 20


This pump sucks testing transitivity with individual data

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


This pump sucks testing transitivity with individual data

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

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.


Summary of wst individuals

Summary of WST Individuals


29 people with 3 perfectly wst patterns

29 People with 3 Perfectly WST Patterns


Within person changes in preference pattern

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

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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