Can a Dominatrix Make My Pump Work?

Michael H. BirnbaumCSUF Decision Research Center

Acknowledgments

• Daniel Navarro-Martinez, Neil Stewart, and Christoph Ungemach contributed to this project, including research done at U. Warwick.

• Jeffrey P. Bahra assisted with data collection here at CSUF.

Outline

• Transitivity• Stochastic Dominance• Search for intransitivity predicted by an

imperfect dominance detecting editor• True and Error Model: Can estimate the

proportion of different response patterns, separate error from true violations of a property.

• Results: Very little evidence for editor.

Transitivity

• If A is preferred to B, and B is preferred to C, then A is preferred to C.

• Problem: how to separate violations due to “error” or variability of true preferences from violations due to “true” intention?

• Controversy: random utility, additive error component on utility, iid issues, and true and error models. I favor true and error model.

Stochastic Dominance

• If the probability to get prize x or more in gamble A always is greater than or equal to the corresponding probability in gamble B and sometimes strictly higher, we say gamble A dominates B by first order stochastic dominance.

• Original Prospect Theory violated stochastic dominance in cases where people satisfied dominance.

Wrong Prediction of OPT

• A = ($99, .1; $98, .1; $97, .8)• B = ($101, .2; $100, .8)

• “Stripped” Original prospect theory predicts A is preferred to B, violating dominance.

• Fishburn (1978) pointed out this type of implication with respect to Handa’s model, which like Edwards’ SWU (“SEU”) model, used decision weights.

Editing Rules of PT (1979)• Kahneman (2003): The editing rules were a pre-emptive

defense of “prospect” theory from implications thought at the time to be dubious or false. These were postulated without any evidence in 1979 paper as excuses.

• Editing rule of dominance detector: People look for dominance and satisfy it when “transparent.”

• Neither CPT nor TAX implies this type of violation of dominance. CPT (1992) implies stochastic dominance, automatically satisfies the editing rule of “combination”, and was not limited to two non-zero branches.

Violations of Stochastic Dominance

• Birnbaum (1997) proposed the following recipe, implied by RAM and TAX (configural weighted) models.

• A = ($96, .90; $14, .05; $12, .05) • B = ($96, .85; $90, .05; $12, .10)• Birnbaum & Navarrete (1998) found about

70% violations. Replicated in 42 studies with different formats, incentives, participants, etc.

Can we use Dominance to create violations of Transitivity?

Birnbaum, Patton, & Lott (1999) : Maybe people can detect dominance in simpler choices. Compare G+ and G- with G0, as follows:G+ = ($96, .9; $14, .05; $12, .05)G- = ($96, .85; $90, .05; $12, .10)G0 = ($96, .9; $12, .10)Prediction: G+ > G0, G0 > G- but G- > G+.Starmer (1999): Similar attempt.

Previous Results

• BPL (1999) results: many people violating dominance on simpler choices. Intermediate results. Concluded: maybe there is a partially accurate dominance detector used part of the time by some people. “Translucent” choices added to “transparent” and “opaque.”

• Neither BPL (1999) nor Starmer (1999) were unambiguous. At the time, the appropriate error model was lacking.

3 New Studies: Individual Data

• Each person received 40 choice problems in each “block”, randomly ordered, with problems from transitivity design separated by at least 3 trials.

• Embedded were 8 choice problems testing stochastic dominance and transitivity. 3 studies with 100 people are combined here.

• Many responses per choice problem for each person.

• Allows analysis via both gTET and iTET.

True and Error Models

• iTET: Individual person analysis.• gTET: Group analysis summarized over people.• Both models assume: Two responses to the same

choice problem by the same person in the same block of trials are based on the same “true” preferences, but may differ due to random “errors.”

• Both models violate iid: in gTET, violations mean that different people have different true preferences; iTET, violations of iid when the same person has a mixture of true preferences and changes from block to block.

Some Advantages of TE Models

• The models are testable.• They provide a framework, like ANOVA, for

statistical tests of transitivity or other properties.

• They allow estimation of the distribution of true preference patterns in a mixture.

• Some “rival” approaches are simply special cases; e.g., Thurstone, Case V.

Results-Violations of SDChoice Type

Females GS+ GS- G+ G0 G0 G- G+ G- First 7 35 60 68 Last 15 28 42 50Males First 2 29 61 59 Last 6 14 41 43

Response Patterns First Last

Pattern G F G &F G F G &F

111 13 14 4 42 29 24

*112 13 17 3 11 9 1

121 9 11 1 7 9 0

122 30 29 15 24 27 18

211 7 8 2 5 11 2

212 5 2 0 5 5 1

*221 6 5 1 2 2 2

222 17 14 4 4 8 2

Total 100 100 30 100 100 50

Response Patterns—All BlocksPattern G test F test G and F

111 1037 963 765*112 242 209 44121 212 199 33122 902 913 674211 251 257 150212 87 120 19221 38 50 6222 184 242 78

Total 2953 2953 1769

Modal Response Patterns by S

Pattern Number111 48

*112 2121 1122 35211 6212 2221 1222 3

Tests of iid

• Birnbaum (2012, 2013) tests of iid in small samples, based on random permutations of the data. A number of participants showed significant violations of iid.

• Compute preference reversals between blocks; compute correlation between gap and number of preference reversals. Median correlations were .50, .77, and .70 in 3 studies.

• Variance test also sig. for many participants. Further, there were significant decreases in violations of SD.

PPN Violations of SD in G+ vs G-

Estimation of Errors in TE model Response Pattern Parameters

Type 11 12 21 22

2

Independence p e

2

TE

GS+GS- first 92 1 6 1 5.80 0.00 0.06 5.12

GS+GS- last 82 7 8 3 4.10 0.00 0.12 3.75

G+G0 first 52 13 19 16 7.31 0.20 0.20 1.12

G+G0 last 70 14 4 12 23.77 0.13 0.11 5.25

G0G- first 26 12 15 47 19.05 0.65 0.16 0.33

G0G- last 50 13 4 33 44.10 0.39 0.11 4.52

G+G- first 19 16 19 46 6.06 0.75 0.23 0.26

G+G- last 38 18 13 31 14.47 0.44 0.19 0.80

Fit of gTET case with p(112) = 0 First Last

Pattern p G G and F p G G and F

111 0.23 14.4 6.3 0.56 36.9 23.0

112 (0) 9.6 1.1 (0) 11.4 1.5

121 0.00 11.6 1.6 0.00 9.5 1.1

122 0.56 31.4 15.1 0.32 22.0 13.3

211 0.06 7.0 2.0 0.03 7.3 1.7

212 0. 00 4.6 0.3 0.02 3.7 1.1

221 0.02 6.2 0.9 0.03 3.9 1.4

222 0.14 15.1 4.7 0.03 5.3 1.5

Total 1.00 100 30 1.00 100 44.6

Possible Problem for Prior TAX Model

• The “prior” TAX model is a special case model that assumed particular parameters and specifications for purpose of making predictions to new studies. Weight transfer = d/(n + 1).

• Prior TAX prediction is 222 (All violations), whereas most common patterns are 122 and 111. This could be fixed Post Hoc, but it might represent something deeper. [BPL (1999) attributed discrepancy to partially effective editor, but these results raise doubts.]

Two People out of 100

• Two people had data compatible with an editor that works for G+ vs. G0 and G- vs. G0 but not G+ vs. G-.

• One of these also showed evidence of changing from pattern predicted by this hypothesis to increasing conformity to SD.

• Most people had data compatible with transitive orders. Some also were consistent from start to finish (one with 78 blocks of data, perfect 122 pattern except for two “errors” out of 468 responses).

Conclusions• Results provide much clearer answer than in BPL 99. Whereas

BPL thought there might be evidence for a partially effective dominance detector working part of the time, these data show very little evidence, based on the TE analysis.

• Other tests of editing rules have also failed to find support; in fact, evidence shows systematic violation of coalescing, cancellation. Other “heuristic” or “simplifying” principles are refuted when formulated and tested as descriptive theory (JMP).

• Whereas a few people might use these on occasion, they have not yet proven successful when tested as models of the majority of participants.

Appendix

• For those who want details of the TE Models, I recommend paper in JDM:

• Birnbaum, M. H. (2013). True-and-error models violate independence and yet they are testable. Judgment and Decision Making, 8, 717-737.

• Associated Excel spreadsheets with equations for analysis of Transitivity via TE models.

• http://journal.sjdm.org/vol8.6.html