presidential address: a program of web-based research on decision making
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Presidential Address: A Program of Web-Based Research on Decision Making. Michael H. Birnbaum SCiP, St. Louis, MO November 18, 2010. Preliminary Remarks. I am grateful for the honor to be president of SCiP and this opportunity to speak. - PowerPoint PPT PresentationTRANSCRIPT
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Presidential Address: A Program of Web-Based Research on Decision
Making
Michael H. BirnbaumSCiP, St. Louis, MONovember 18, 2010
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Preliminary Remarks• I am grateful for the honor to be president of SCiP and this opportunity to speak.
• You are invited to attend the Edwards Bayesian Research Conference in Fullerton in January 2011. Look for details by email.
• You, your students, and colleagues are invited to apply to the NSF-funded training sessions in Web-based DRMS research. Look for details by email. Next session follows Edwards Bayesian Meetings.
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Preliminary Remarks:A Personal History of
Computing in Psychology
Michael H. BirnbaumDecision Research Center
California State UniversityFullerton, CA
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Lab Research in the 1960s
Michael at the Keypunch-- ~ 1965-66 in Parducci Lab in Franz Hall, UCLA
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The “monster” in the basement --Parducci
Lab
•
•This machine’s relays and rheostats controlled equipment in the next room. Programmed by paper tape reader, knobs and dials, it recorded Ss’ responses on computer cards.
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We had to use Mainframe Computers to Analyze data--One Run a Day…
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We all Awaited the Personal Computer…Predicted for 2004
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Difficulties of Lab Research
• Limited “seating” in lab, limited hours
• Reconfigure equipment for each study
• Lab assistant needed who has skill to trouble-shoot problems with equipment
• Experimenter must be present• Experimenter bias effects...
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1970: The Lab Computer --Programmable in BASIC--
Expensive, But it still lacked real power
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1980s: Then came affordable personal
computers
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One Computer per Participant
• Still limited to facilities in a lab
• Networks made it more efficient to assemble data from different participants and to mediate interactions between participants.
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1989-90: The WWW…1995: Browsers & HTML2
• One could now: • …Test large numbers of participants
• …Test people at remote locations• …Recruit people with special characteristics
• …Run the same study in the lab and via the Web for comparison
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My First Web Studies
• I wanted to compare Highly Educated Participants with Undergraduates.
• Critical Tests refuted then-popular theories of Decision Making
• Recruit PhDs who are members of SJDM and SMP and studied DM.
• Recruit and Test participants via the WWW
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Critical Tests are Theorems of One Model that are Violated by
Another Model• This approach has advantages over tests or comparisons of fit.
• It is not the same as “axiom testing.”
• Use model-fitting to rival model to predict where to find violations of theorems deduced from model tested.
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Outline
• I will discuss critical properties that test between nonnested theories: CPT and TAX.
• Lexicographic Semiorders vs. family of transitive, integrative models (including CPT and TAX).
• Integrative Contrast Models (e.g., Regret, Majority Rule) vs. transitive, integrative models.
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Cumulative Prospect Theory/ Rank-Dependent
Utility (RDU)
Probability Weighting Function, W(P)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Decumulative Probability
Decu
mu
lati
ve W
eig
ht
CPT Value (Utility) Function
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140
Objective Cash Value
Su
bje
ctiv
e V
alu
e
CPU(G ) [W ( pj ) W ( pj )j1
i 1
j1
i
i1
n
]u(xi )
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TAX Model
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“Prior” TAX Model
Assumptions:
U(G) Au(x) Bu(y) Cu(z)
A B C
At(p) t(p) /4 t(p) /4
Bt(q) t(q) /4 t(p) /4
C t(1 p q) t(p) /4 t(q) /4
G(x, p;y,q;z,1 p q)
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TAX Parameters
Probability transformation, t(p)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Probability
Tra
nsf
orm
ed P
rob
abil
ity
For 0 < x < $150u(x) = x Gives a decentapproximation.Risk aversion produced by
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TAX and CPT nearly identical for binary (two-
branch) gambles• CE (x, p; y) is an inverse-S function of p according to both TAX and CPT, given their typical parameters.
• Therefore, there is little point trying to distinguish these models with binary gambles.
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Non-nested Models
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CPT and TAX nearly identical inside the prob.
simplex
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Testing CPT
• Coalescing• Stochastic Dominance
• Lower Cum. Independence
• Upper Cumulative Independence
• Upper Tail Independence
• Gain-Loss Separability
TAX:Violations of:
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Testing TAX Model
• 4-Distribution Independence (RS’)
• 3-Lower Distribution Independence
• 3-2 Lower Distribution Independence
• 3-Upper Distribution Independence (RS’)
• Res. Branch Indep (RS’)
CPT: Violations of:
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Stochastic Dominance
• A test between CPT and TAX:G = (x, p; y, q; z) vs. F = (x, p – s; y’, s; z)
Note that this recipe uses 4 distinct consequences: x > y’ > y > z > 0; outside the probability simplex defined on three consequences.
CPT choose G, TAX choose FTest if violations due to “error.”
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Error Model Assumptions
• Each choice pattern in an experiment has a true probability, p, and each choice has an error rate, e.
• The error rate is estimated from inconsistency of response to the same choice by same person over repetitions.
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Violations of Stochastic Dominance
122 Undergrads: 59% TWO violations (BB) 28% Pref Reversals (AB or BA) Estimates: e = 0.19; p = 0.85170 Experts: 35% repeated violations 31% Reversals Estimates: e = 0.20; p = 0.50
A: 5 tickets to win $12 5 tickets to win $14 90 tickets to win $96
B: 10 tickets to win $12 5 tickets to win $90 85 tickets to win $96
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42 Studies of Stochastic Dominance, n = 12,152
• Large effects of splitting vs. coalescing of branches
• Small effects of education, gender, study of decision science
• Very small effects of 15 probability formats and request to justify choices.
• Miniscule effects of event framing (framed vs unframed)
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Summary: Prospect Theories not Descriptive• Violations of Coalescing, Stochastic Dominance, Gain-Loss Separability, and 9 other critical tests.
• Summary in my 2008 Psychological Review paper.
• JavaScript Web pages contain calculators--predictions
• Archives show Exp materials and data
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Lexicographic Semiorders
• Intransitive Preference.• Priority heuristic of Brandstaetter, Gigerenzer & Hertwig is a variant of LS, plus some additional features.
• In this class of models, people do not integrate information or have interactions such as the probability X prize interaction in family of integrative models (CPT, TAX, GDU, EU and others)
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LPH LS: G = (x, p; y) F = (x’, q; y’)
• If (y –y’ > ) choose G• Else if (y ’- y > ) choose F• Else if (p – q > ) choose G• Else if (q – p > ) choose F• Else if (x – x’ > 0) choose G• Else if (x’ – x > 0) choose F• Else choose randomly
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Family of LS
• In two-branch gambles, G = (x, p; y), there are three attributes: L = lowest outcome (y), P = probability (p), and H = highest outcome (x).
• There are 6 orders in which one might consider the attributes: LPH, LHP, PLH, PHL, HPL, HLP.
• In addition, there are two threshold parameters (for the first two attributes).
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Testing Lexicographic Semiorder Models
Allais Paradoxes
Violations ofTransitivity
Violations ofPriority DominanceIntegrative IndependenceInteractive Independence
EU CPTTAX
LS
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New Tests of Independence
• Dimension Interaction: Decision should be independent of any dimension that has the same value in both alternatives.
• Dimension Integration: indecisive differences cannot add up to be decisive.
• Priority Dominance: if a difference is decisive, no effect of other dimensions.
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Taxonomy of choice modelsTransitive
Intransitive
Interactive & Integrative
EU, CPT, TAX
Regret, Majority Rule
Non-interactive & Integrative
Additive,CWA
Additive Diffs, SDM
Not interactive or integrative
1-dim. LS, PH*
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Dimension Interaction
Risky Safe TAX LPH HPL
($95,.1;$5)
($55,.1;$20) S S R
($95,.99;$5)
($55,.99;$20)
R S R
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Family of LS
• 6 Orders: LPH, LHP, PLH, PHL, HPL, HLP.
• There are 3 ranges for each of two parameters, making 9 combinations of parameter ranges.
• There are 6 X 9 = 54 LS models.• But all models predict SS, RR, or ??.
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Results: Interaction n = 153
Risky Safe % Safe
Est. p
($95,.1;$5) ($55,.1;$20)
71% .76
($95,.99;$5)
($55,.99;$20)
17% .04
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Analysis of Interaction
• Estimated probabilities:• P(SS) = 0 (prior PH)• P(SR) = 0.75 (prior TAX)• P(RS) = 0• P(RR) = 0.25• Priority Heuristic: Predicts SS
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Results: Dimension Integration
• Data strongly violate independence property of LS family
• Data are consistent instead with dimension integration. Two small, indecisive effects can combine to reverse preferences.
• Observed with all pairs of 2 dims.
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New Studies of Transitivity
• LS models violate transitivity: A > B and B > C implies A > C.
• Birnbaum & Gutierrez (2007) tested transitivity using Tversky’s gambles, using typical methods for display of choices.
• Text displays and pie charts with and without numerical probabilities. Similar results with all 3 procedures.
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Replication of Tversky (‘69) with Roman
Gutierrez• 3 Studies used Tversky’s 5 gambles, formatted with tickets and pie charts.
• Exp 1, n = 251, tested via computers.
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Three of Tversky’s (1969) Gambles
• A = ($5.00, 0.29; $0)• C = ($4.50, 0.38; $0)• E = ($4.00, 0.46; $0)Priority Heurisitc Predicts: A preferred to C; C preferred to E,
But E preferred to A. Intransitive.
TAX (prior): E > C > A
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Response CombinationsNotation (A,
C)(C, E)
(E, A)
000 A C E * PH
001 A C A
010 A E E
011 A E A
100 C C E
101 C C A
110 C E E TAX
111 C E A *
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Results-ACEpattern Rep 1 Rep 2 Both
000 (PH) 10 21 5
001 11 13 9
010 14 23 1
011 7 1 0
100 16 19 4
101 4 3 1
110 (TAX)
176 154 133
111 13 17 3
sum 251 251 156
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Summary
• Priority Heuristic model’s predicted violations of transitivity are rare.
• Dimension Interaction violates any member of LS models including PH.
• Dimension Integration violates any LS model including PH.
• Evidence of Interaction and Integration compatible with models like EU, CPT, TAX.
• Birnbaum, J. Mathematical Psych. 2010.
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Integrative Contrast Models
• Family of Integrative Contrast Models
• Special Cases: Regret Theory, Majority Rule (aka Most Probable Winner)
• Predicted Intransitivity: Forward and Reverse Cycles
• Research with Enrico Diecidue
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Integrative, Interactive Contrast Models
AB (E i)(ai,bi)i1
n
A(a1,E1;a2,E2;;an,E n )
B(b1,E1;b2,E 2;;bn,E n )
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Assumptions
(ai, bi) (bi, ai)
(ai, bi) 0 ai biDifference Model :
(ai, bi) f [u(ai) u(bi)]
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Special Cases
• Majority Rule (aka Most Probable Winner)
• Regret Theory • Other models arise with different functions, f.
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Regret Aversion
[a, c] [a, b][b, c], u(a) u(b) u(c)
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Regret Model
f [u(a) u(b)]u(a) u(b) , u(a) u(b)
f [u(a) u(b)] u(a) u(b) , u(b) u(a)
1
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Majority Rule Model
f [u(a) u(b)]1 if u(a) u(b)
0 if u(a) u(b) 1 if u(a) u(b)
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Predicted Intransitivity
• These models violate transitivity of preference
• Regret and MR cycle in opposite directions
• However, both REVERSE cycle under permutation over events; i.e., “juxtaposition.”
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Concrete Example
• Urn: 33 Red, 33White, 33 Blue• One marble drawn randomly• Prize depends on color drawn.• A = ($4, $5, $6) means win $400 if Red, win $500 if White, $600 if Blue. (Study used values x 100).
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Majority Rule Prediction
• A = ($4, $5, $6)• B = ($5, $7, $3)• C = ($9, $1, $5)• AB: choose B• BC: choose C• CA: choose A• Notation: 222
• A’ = ($6, $4, $5)
• B’ = ($5, $7, $3)
• C’ = ($1, $5, $9)
• A’B’: choose A’• B’C’: choose B’• C’A’: choose C’• Notation: 111
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Regret Prediction
• A = ($4, $5, $6)• B = ($5, $7, $3)• C = ($9, $1, $5)• AB: choose A• BC: choose B• CA: choose C• Notation: 111
• A’ = ($6, $4, $5)
• B’ = ($5, $7, $3)
• C’ = ($1, $5, $9)
• A’B’: choose B’• B’C’: choose C’• C’A’: choose A’• Notation: 222
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Non-Nested Models
TAX, CPT, TAX, CPT, GDU, etc.GDU, etc.
Integrative Integrative Contrast ModelsContrast Models
Intransitivity
Allais Paradoxes
ViolationsOf RBI
Transitive
Recycling
RestrictedBranch Independence
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Study
• 240 Undergraduates• Tested via computers (browser)• Clicked button to choose• 30 choices (includes counterbalanced choices)
• 10 min. filler task, 30 choices repeated.
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Recycling Predictions of Regret and Majority Rule
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ABC-A’B’C’ Analysis
ABC-A'B'C'PATTERN Est. true probs
111111 0.00112112 0.59 TAX121121 0.04122122 0.01211211 0.00212212 0.08221221 0.16222222 0.02222111 0.09 MR111222 0.02 Regret
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Results
• Most people are transitive.• Most common pattern is 112, pattern predicted by TAX with prior parameters.
• However, 2 people were perfectly consistent with MR on 24 choices (incl. Recycling pattern).
• No one fit Regret theory perfectly.
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Results: Continued
• Systematic Violations of RBI, refuting this class as descriptive of majority.
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Testing Individuals• Recent Work: Lab Studies: large amounts of data from a few people, but still use WWW as network.
• Question: Are some people consistent with different models? For example, do some satisfy CPT when tested in each person? Recall CPT implies stochastic dominance and predicts a different pattern of violation of RBI.
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2 Studies with Jeff Bahra
• 158 choices. Participants came back one week later, received the same choices, up to 20 repetitions per choice.
• 43 and 59 individuals, each provided up to 3160 responses.
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RBI
• Restricted Branch Independence• 3 equally likely events: slips in urn.• (x, y, z) := prizes x, y, or z, x < y < z
• RBI:(x, y, z) f (x', y', z)
(x, y, z') f (x', y', z')
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TAX, RDU,& CPT Violate RBI
• 0 < z < x' < x < y < y' < z'• (x, y) is “Safe”, S• (x', y') is “Risky”, R• (z, x, y) f (z, x', y') wLu(z) + wMu(x) + wHu(y) >
wLu(z) + wMu(x') + wHu(y')
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TAX implies SR' Violations
Inverse-S & CPT => RS’
wLwM
< u(y') - u(y)u(x) - u(x')
< wMwH
SR': S = (x, y, z) R = (x', y', z) AND
S' = (x, y, z') R' = (x', y', z')
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RBI for 10 peoplesubj ID SS' SR' RS' RR' SUM
1 83 163 7 0 0 170 2 91 21 80 0 18 119 3 203 0 0 0 170 170 4 244 0 36 0 65 101 5 259 1 4 0 216 221 6 278 0 27 0 91 118 7 327 38 48 1 15 102 8 364 26 86 0 6 118 9 410 6 76 1 19 102
10 590 0 3 0 211 214
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Stochastic Dominance 10 people
no subj GG FG total pGG revsls est_p est_ e 1 83 16 3 19 0.84 0.16 1.00 0.08 2 91 8 6 15 0.53 0.40 1.00 0.27 3 203 12 5 18 0.67 0.28 0.96 0.17 4 244 0 4 11 0.00 0.36 0.00 0.24 5 259 14 6 20 0.70 0.30 1.00 0.17 6 278 7 4 15 0.47 0.27 0.65 0.16 7 327 7 6 13 0.54 0.46 1.00 0.29 8 364 13 1 14 0.93 0.07 1.00 0.04 9 410 9 4 13 0.69 0.31 1.00 0.18
10 590 8 11 20 0.40 0.55 1.00 0.39
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Observed Violations of RBI and SD
0.00
0.20
0.40
0.60
0.80
1.00
-60 -40 -20 0 20 40 60 80 100
RBI violations (SR' - RS') as percentage
Pro
port
ion
of
Vio
lati
on
s o
f S
D (
3.1
an
d 3
.3)
CPT EU
TAX
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Summary
• Search for individuals who agree with CPT--NOT one individual found.
• One person showed RS' pattern but had 94% violations of SD in coalesced form (30 out of 32 trials).
• The others violated CPT or fit EU.
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Conclusions
• Evidence most consistent with Integrative, Interactive, and transitive models
• CPT not descriptive• LS not descriptive of majority• Regret, MR not descriptive, but some show MR viols of transitivity
• Thanks for your attention!