Who are these People Who Violate Stochastic

Dominance, Anyway? What, if anything, are they

thinking?

Michael H. BirnbaumCalifornia State University,

Fullerton

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Violations of Stochastic Dominance

• CPT/RSDU/RDU/EU and other models of decision-making imply satisfaction of first order stochastic dominance.

• RAM/TAX/GDU/PRT and others violate it in specified (but rare) situations.

• RAM/TAX correctly led to a method for constructing choices that show 70% violations in undergraduates.

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Recruiting Highly Educated Participants

• Would highly educated participants show the same violations of CPT as undergraduates do?

• Web study recruited members of Societies for Math-Psych and Judgment and Decision Making (professors).

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Web Studies A and B

• Convenient: no mailing costs, printing, lab assts, or data entry.

• Recruited via email, n = 2085.• At the time, the method was novel,

and there was great interest. Data arrived very quickly. Two studies, 20 choices.

• 1% of participants get one of their chosen gambles; $ prizes.

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DemographicsCharacteristic A-1224 B-737 Lab-124

Age ≤ 22 years 20 22 91

Age > 40 years 20 24 0

College Grad 60 47 0

Doctorates 11 3 0

Read on DM 31 19 13

Female 56 61 73

Viols of SD 52 59 68

Viols of CM 7 8 11

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Results-Average % ViolsSex Edu G– f G+

A B Lab

GS– f GS+

A B Lab

Sample Size, n

A B Lab

F < 16 60 66 70 10 11 12 318 248 91

F 16 62 57 10 8 206 148 0

F 17-19 45 58 7 13 108 41 0

F 20 42 54 2 9 54 11 0

M < 16 53 56 63 6 8 11 163 141 33

M 16 43 56 6 10 195 98 0

M 17-19 36 56 2 3 88 32 0

M 20 38 57 6 7 80 14 0

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286 Doctorates A & follow-up studies

JDM SEX ViolsSD

ViolsCM

Sample

n

No F 51 3 52

No M 42 5 63

Yes F 56 7 74

Yes M 44 7 97

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Summary: Correlations

• Rate of Violations is correlated with education, gender.

• Lower Division Undergrads at CSUF: ~ 70% violations

• College Graduates: ~ 60% Violations

• Doctorates: ~ 50% Violations

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Questions

• Are these violations diminished or amplified in mixed gambles?

• What would happen if participants got additional practice?

• Are there individual differences, with education fixed?

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Mixed Gambles

• 92 undergraduates made 250 choices between gambles with losses and mixed consequences.

• Same recipe with consequences reduced by c, where c = $25, $50, or $100. 4 problems with 2 reps.

• 24 tests of SD (3 X 4 X 2) per person. 24 tests of CM.

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Results (n = 92 X 4 X 2)

90 to win $715 to lose $115 to lose $13

85 to win $715 to win $6510 to lose $13

79%

90 to win $465 to lose $365 to lose $38

85 to win $465 to win $4010 to lose $38

75%

90 to lose $45 to lose $865 to lose $88

85 to lose $45 to lose $1010 to lose $88

75%

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Ind. Differences (parens show preds:no Ind. diffs)

• 0 had 0-4 violations (0)• 6 had 5-9 violations (0)• 9 had 10-14 violations (3.5)• 34 had 15-19 violations (61)• 43 had 20-24 violations (27.5) incl.

12 with 24 Violations (0)• Split-half correlation: r = .78

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Individual Analysis

• We can reject the hypothesis that there are 50% or fewer violations in 57 of 92 (62%) by individual tests. Only 11 people had 50% or fewer violations.

• Conclusions: There appear to be syst. individual differences; effects of practice (24 tests/250 trials) minimal; rates of violation are similar in mixed and loss-only gambles to results with positive consequences.

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What do they say they are thinking?

• 268 Undergrads in lab, after completing 250 choices, were asked to make 6 more and explain why they made each of those 6 choices. One was a test of SD.

• Reasons: Categorized as: money, probability, prob & money, tradeoff, branch contrast, dominance, miscl.

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87 reasons cited money

• “Greater sums of money.”• “You could win more this way.”• “You win more money.”• “More money.”• “I’d win more money.”

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65 cited probability

• “There are more of the marbles that are worth more money so as making the odds a little more enticing.”

• “All around I have a better chance.”• “Better chances of winning.”• “Better odds in my favor.”• “.9 to win 96.”• “I could win 90 with twice the chance.”

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42: Probability & Money

• “There is a greater liklihood of me winning more money.”

• “There is a greater chance that I would get more money.”

• “More chance to win higher pay.”• “I chose J because the probability

was higher to win more money.”

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17 (leniently) classified as using Dominance

• “I can win 96 over 90 with the same number of marbles.”

• “I is statistically better.”• “I is better all around-higher chance of

winning $96, and if I don’t the other two options are priced same or better than $12.”

• “It is obvious that J is much better for there’s a better chance and a higher percentage of winning more money.”

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15 Cited Tradeoffs• “The amount of marbles to the amount

of money.”• “Even thought (sic) its 5 percent more

I’ll end up with 12 I’d take the risk for the 90.”

• “I would choose this one because the odds are about even to win the $96 on both so then I look at the next one and it is 5% to win $90 instead of $14. That extra 5% doesn’t weigh out the odds in my opinion.”

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11: branch contrasts• “You can win more with the green

marbles and the same with the others, so go for the higher one.”

• “high chance of getting $96 with both bets, but J has the opportunity to also get $90 instead of just 12 or 14, so that sounds better. Both are a win-win situation.”

• “Because there is a high chance of me to win $90 and the other choice is only $14.”

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Reason/Decision RelationMoney 79 8

Probability 43 22

Prob & Money 40 2

Dominance 4 13

Tradeoffs 13 2

Branch Contrast 10 1

Other reasons 22 9

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Summary• 79% violations. Apparently the request

to give reasons did not reduce incidence of violations of SD.

• Not easy to understand the students’ reasons; hard to predict from the reason to the decision.

• Ben Franklin: Man is a reasonable animal; he can always find a reason to justify what he is inclined to do.

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Final Words• Although there are some systematic differences

in the rates of violation, all groups show more violations in the coalesced than split form. Evidence of individual differences within demographic group, but high rates of violation persist in fairly long experiments and when people have to explain their choices.

• Can we teach people to satisfy SD? If so, what else would this special training do to a person’s choices for tests of other properties?

• 7 tutorials available on LCI, UCI, 3-LDI, UDI, RBI, 4-DI, UTI. All tests favor TAX over CPT. See also: Allais Dissection.

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For More Information:

http://psych.fullerton.edu/mbirnbaum/

Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers.

mbirnbaum@fullerton.edu