Reconsidering Risk Aversion
Mark Fontana1
Daniel Benjamin1
1 University
Miles Kimball2
of Southern California
2 University
of Michigan
January 10, 2016
We are grateful to NIH/NIA (R21-AG037741) for financial support, and
thank Mike Gideon for helpful early conversations.
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
1
Motivation
Coefficient of relative risk aversion is a central parameter
Practical motivation of measuring it for calibrating long-term
retirement savings (personal investing or fund contribution defaults)
Focus here on normative issue of measuring risk aversion to give
individuals advice, not positive issue of describing behavior
Problem: how you ask the question (frame) partially determines the
answer, e.g. short-run v long-run risky choices
Our approach motivated by the philosophical tradition of deliberative
thinking and logical reconciliation between contradictions
Develop procedure to lead people through inconsistencies in their own
decision making and provide opportunities to revise choices
Goals: 1) identify deliberate violations of ”normative” axioms from
mistakes, 2) reduce range of uncertainty about risk aversion
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
2
Motivation: Welfare Consequences of Getting it Wrong
Continuous time 35 year time frame model, safe and risky asset with
stochastic return (Merton (1969), except with Kreps-Porteus
preferences that separate the EIS from relative risk aversion)
What are the welfare consequences of being γ but optimizing like
you’re γ̂, in terms of wealth (asymmetry! extra risk more costly; we’re
dealing with 1 v 2)
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
3
Related Literature: Framing
Problem: how you ask about risk aversion colors the answer
Benartzi and Thaler 1999: myopic loss aversion, people choose to
hold more stocks if given long-term rather than one-year rates of
return (verified this with our piloting)
Druckman 2001: Asian disease problem, use ”both” frame as baseline
McNeil et al. 1988: medical decisions; suggest asking positive frame,
negative frame, and combination of positive and negative frame as
sort of sensitivity analysis
See also: Beshears et al. 2008, Chong and Druckman 2007,
Druckman and Nelson 2003
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
4
Our Setup and Approach
Choices among 5 investment plans framed 7 different ways, lotteries
involve ”amount you have to spend every year during retirement,
from age 65 on”
Confront people with inconsistencies and placebos between frames
(call this a ”step”) and intransitivities within frames, allowing them
to reason through their preferences to decide what they really want
Break down the problem into axiomatic ”baby steps” so that we can
isolate specific axiomatic violations as distinct from mistakes
Assume we’re moving from original ”untutored” preferences toward
”reasoned” preferences
Do not want to be paternalistic, but deferentially light-handed
While our focus is on risk preferences, this approach could be adopted
for any sort of preferences
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
5
Motivating Questions
Which EU axioms do people initially endorse, and where are they
most likely to correct where they’ve said they’ve made a mistake?
Toward which frames do people revise?
How do estimates of risk aversion and incidence of decision errors
vary by frame? How do these change between untutored and
reasoned preferences?
How does cognition relate to untutored preferences, reasoned
preferences, and the frequency and type of decision errors we
uncover? Are reasoned preferences less correlated with cognition and
more homogeneous across individuals than untutored preferences?
What about other covariates? e.g. gender, order of frames, etc
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
6
Overview
6 rounds of data collected at Cornell’s LEEDR and BSL
n = 601, almost all undergraduates
Brought subset of people back people in for a followup (2-4 weeks
later) that repeated the main body of the experiment (311 invited,
264 showed)
Sessions are scheduled for 2 hours each, but mean completion time is
68 minutes (not including initial introductions and interactions which
last about 10 minutes)
Paid $40 for 2 hours
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
7
Survey Overview
1
Pre-test
2
Training
Main Body: untutored and reasoned preference elicitation
3
Inconsistencies
Intransitivities
4
Demographics and cognition measures
Subset of participants brought back for a second ”wave” to repeat
pre-test and main body
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
8
Survey Overview
1
Pre-test
2
Training
Main Body: untutored and reasoned preference elicitation
3
Inconsistencies
Intransitivities
4
Demographics and cognition measures
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
9
Untutored Preferences
Simplest frame shown (1 q): ”complete contingent action plan”
6 other frames derived from this one
Two-period investment horizon with binary lotteries, over 30+ years
5 possible plans: A, BCE, BCF, BDE, BDF (safest to riskiest)
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
10
Design Choices
Training section walks people through how to understand the figures
Many rounds of pilot testing to make sure training and figures were
clear and understood
Lotteries over yearly income during retirement to measure risk
aversion over consumption and reduce cognitive burden
”Conservative” to highlight guaranteed amount
Spinners and dotted lines to highlight uncertainty (again explained in
training section), only use 50%-50% and 50%-25%-25% probabilities
(unlike much of the literature!)
Include ages to incorporate different investment horizons, our focus is
on retirement savings
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
11
Monetary Levels
Gambles over amount each year during retirement from age 65 on
Monetary amounts for an individual always drawn from one of six sets
(always the same set for an individual despite slides being drawn from
different monetary levels)
Each monetary level is associated with a coefficient of relative risk
aversion that should make individuals indifferent between certain
gambles: 1.576, 2.958, 4.865, 7.184, 12.113, 17.967
Random half get all amounts cut in half
1
2
3
4
5
6
52K
64K
74K
81K
88K
92K
72K
80K
86K
90K
94K
96K
100K 100K 100K 100K 100K 100K
108K 120K 129K 135K 141K 144K
150K 150K 150K 150K 150K 150K
225K 225K 225K 225K 225K 225K
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
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The Frames
Nodewise Action Choice Frames
Single action in isolation (2 q’s)
Single action with backdrop (2 q’s)
Two contingent actions with backdrop (1 q)
Complete contingent action plan (1 q)
Pairwise Strategy Choice Frames
Pairwise choices between complete strategies (10 q’s)
Pairwise choices between compound lotteries (10 q’s)
Pairwise choices between reduced simple lotteries (10 q’s)
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
13
Single Action in Isolation
2 questions
Choice between C v D and E v F, chosen individually without context
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
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Single Action with Backdrop
2 questions
Choice between C v D and E v F, chosen individually
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
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Two Contingent Actions with Backdrop
1 question
Choice between C v D and E v F, simultaneously
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
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Complete Contingent Action Plan
1 question
Choice between A v B, C v D, and E v F, simultaneously
Basis of 3 other Nodewise Action Choice Frames
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
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Pairwise Choices Between Complete Strategies
10 questions
Choices between all pairwise complete action plans, for example
between BDE and BDF below
Basis of 2 other Pairwise Strategy Choice Frames
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
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Pairwise Choices Between Compound Lotteries
10 questions
Strip backdrop, do not reduce compound lotteries, for example
between BDF and BDE
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
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Pairwise Choices Between Reduced Simple Lotteries
10 questions
Reduce compound lotteries, for example between BDE and BDF
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
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Survey Overview
1
Pre-test
2
Training
Main Body: untutored and reasoned preference elicitation
3
Inconsistencies
Intransitivities
4
Demographics and cognition measures
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
21
Reasoned Preferences and Inconsistencies
Participants asked whether they think two choices that are
inconsistent between the most similar frames (”steps”) should be the
same; given the opportunity to revise
Vast majority who revise say they originally made a mistake or learned
something about their preferences in answering questions; those who
don’t say the situations are different enough to merit different answers
Also present ”placebo inconsistencies” i.e. given the opportunity to
change consistent choices (intermixed with the regular inconsistencies,
with identical text and questions), but people rarely change these (less
than 2%) compared to greater than 40% revise actual inconsistencies
Everyone receives some placebos
Two rounds of inconsistency resolutions and two rounds of
intransitivity checks, call the resulting preferences ”reasoned”
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
22
Steps and Frames
Step/Axiom
Irrelevance of Background Counterfactuals
Simple Actions =
State-Contingent
Actions
Irrelevance of Counterfactual Choices
Fusion + Shift from
Nodewise to Pairwise
Frame 1
Single Action in Isolation
Single Action with
Backdrop
Frame 2
Single Action with
Backdrop
Two Contingent Actions with Backdrop
Two Contingent Actions with Backdrop
Complete Contingent
Action Plan
Complete Strategies =
Implied Lotteries
Pairwise Choices between Complete Strategies
Pairwise Choices Between Compound Lotteries
Complete Contingent
Action Plan
Pairwise Choices between Complete Strategies
Pairwise Choices Between Compound Lotteries
Pairwise Choices Between Reduced Simple
Lotteries
Reduction of Compound Lotteries
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
23
Irrelevance of Background Counterfactuals
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
24
Simple Actions = State-Contingent Actions
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
25
Irrelevance of Counterfactual Choices
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
26
Fusion + Shift from Nodewise to Pairwise
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
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Complete Strategies = Implied Lotteries
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
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Reduction of Compound Lotteries
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
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Percentage Any Inconsistency
Table: Percentage of respondents with any inconsistency among all wave 1
participants (N = 601). Note participants with any missing responses are omitted.
Variable
Wave 1 Untutored
Wave 1 Reasoned
Mean
0.942
0.677
Std. Dev.
0.233
0.468
N
573
573
Table: Percentage of respondents with any inconsistency among participants
invited to two waves (N = 311, with 30 potential inconsistencies). Participants
with missing responses are omitted.
Variable
Wave 1 Untutored
Wave 1 Reasoned
Wave 2 Untutored
Wave 2 Reasoned
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Mean
0.955
0.75
0.867
0.671
Std. Dev.
0.207
0.434
0.34
0.471
Reconsidering Risk Aversion
N
268
268
249
249
January 10, 2016
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Average Inconsistency Rates, Wave 1
Table: Average inconsistency rates by axiom = (total inconsistencies) / (total
potential inconsistencies). P-values from two-sided tests for differences in
proportions. N denotes number of participants whose responses are used for a
given row’s statistics. Participants with missing responses for a given pair of
frames are omitted.
Step/Axiom
Irrelevance of Background Counterfactuals
Simple Actions = StateContingent Actions
Irrelevance of Counterfactual
Choices
Fusion + Shift from Nodewise to
Pairwise
Complete Strategies = Implied
Lotteries
Reduction of Compound Lotteries
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Total
2
Untutored
0.123
Reasoned
0.057
P-Value
<0.0005
N
601
2
0.122
0.084
0.002
598
2
0.109
0.116
0.600
580
4
0.232
0.122
<0.0005
581
10
0.197
0.083
<0.0005
595
10
0.232
0.084
<0.0005
595
Reconsidering Risk Aversion
January 10, 2016
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Reactions To Our Procedure, Wave 1
Emotion
Enjoyment
Annoyance
Stress
Frustration
1 (least)
11.8%
4.6%
23.8%
15.2%
Mark Fontana, Daniel Benjamin, Miles Kimball ()
2
18.1%
17.0%
22.2%
21.1%
3
25.4%
24.2%
20.3%
22.2%
4
32.5%
23.8%
19.5%
24.6%
Reconsidering Risk Aversion
5
9.0%
17.8%
10.2%
10.2%
6 (most)
3.2%
12.6%
3.8%
6.6%
January 10, 2016
32
How Respondents Update Inconsistencies, All Data
Table: Percentage of the time that subjects updated toward each frame, did not
update, or swapped their choices for normal inconsistency checks.
Step/Axiom
Irrelevance of Background Counterfactuals
Simple
Actions
=
StateContingent Actions
Irrelevance
of
Counterfactual
Choices
Fusion + Shift from Nodewise to
Pairwise
Complete Strategies = Implied
Lotteries
Reduction of Compound Lotteries
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Toward
Frame 1
23%
Toward
Frame 2
24%
No Update
49%
Swap
Choices
3%
197
20%
12%
62%
5%
196
7%
3%
88%
0%
246
21%
17%
55%
6%
449
21%
25%
48%
4%
1504
27%
20%
45%
5%
2005
Reconsidering Risk Aversion
Total
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33
How Respondents Update Placebos, All Data
Table: Percentage of the time that subjects updated each frame, did not update,
or updated both their choices for placebo inconsistency checks.
Axiom
Irrelevance of Background Counterfactuals
Simple
Actions
=
StateContingent Actions
Irrelevance
of
Counterfactual
Choices
Fusion + Shift from Nodewise to
Pairwise
Complete Strategies = Implied
Lotteries
Reduction of Compound Lotteries
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Update
Frame 1
0%
Update
Frame 2
0%
No
Update
100.0%
Update
Both
0%
506
0%
0%
100.0%
0%
449
0%
0.58%
99.42%
0%
172
1.13%
3.10%
95.21%
0.56%
355
0.57%
0.85%
97.82%
0.76%
2107
1.08%
0.44%
97.55%
0.93%
2249
Reconsidering Risk Aversion
Total
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34
Average Inconsistency Rates, Waves 1 + 2
Table: Average inconsistency rates. 264 of 311 invited participants returned.
Step/Axiom
Irrelevance
of
Background Counterfactuals
Simple Actions =
State-Contingent
Actions
Irrelevance
of
Counterfactual
Choices
Fusion + Shift
from Nodewise to
Pairwise
Complete Strategies = Implied
Lotteries
Reduction of Compound Lotteries
U1
R1
P-value
U1-R1
0.123 0.062 0.094 0.042 <0.0005
P-value
U1-U2
0.115
P-value
U2-R2
0.0009
N1
0.129 0.092 0.069 0.071 0.046
0.0009
0.903
283 262
0.136 0.158 0.101 0.137 0.304
0.086
0.080
273 252
0.235 0.141 0.180 0.115 <0.0005
0.002
<0.0005
275 256
0.220 0.098 0.152 0.089 <0.0005
<0.0005
<0.0005
280 261
0.260 0.100 0.190 0.097 <0.0005
<0.0005
<0.0005
278 262
Mark Fontana, Daniel Benjamin, Miles Kimball ()
U2
R2
Reconsidering Risk Aversion
N2
284 262
January 10, 2016
35
Why Not Revise Inconsistent Choices, Wave 1 + 2
Step/Axiom
Irrelevance
of
Background Counterfactuals
Simple Actions =
State-Contingent
Actions
Irrelevance
of
Counterfactual
Choices
Fusion + Shift
from Nodewise to
Pairwise
Complete Strategies = Implied
Lotteries
Reduction of Compound Lotteries
Total
Different
situations
54.6%
IDK
Confused
Other
Total
21.6%
Expt’er
Demand
3.1%
9.3%
9.3%
2.1%
97
73.2%
17.9%
2.4%
1.6%
4.1%
0.8%
123
55.0%
25.8%
1.7%
9.2%
3.3%
5.0%
120
62.7%
20.5%
2.0%
7.2%
4.0%
3.6%
249
55.9%
24.3%
3.7%
7.3%
3.2%
5.7%
725
54.5%
27.4%
3.0%
5.1%
4.1%
5.9%
920
24.8%
3.0%
6.3%
4.0%
5.1%
2234
56.9%
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Indiff.
Reconsidering Risk Aversion
January 10, 2016
36
Why Revised Inconsistent Choices, Wave 1 + 2
Step/Axiom
Mistake
Learned
Indiff.
IDK
Confused
Other
11.7%
Expt’er
Demand
1.1%
Irrelevance
of
Background Counterfactuals
Simple Actions =
State-Contingent
Actions
Fusion + Shift
from Nodewise to
Pairwise
Complete Strategies = Implied
Lotteries
Reduction of Compound Lotteries
Total
47.9%
35.1%
1.1%
1.1%
2.1%
36.9%
36.9%
16.9%
0.0%
4.6%
1.5%
3.1%
47.1%
31.0%
13.2%
1.1%
4.6%
2.3%
0.6%
45.7%
34.7%
10.9%
1.0%
4.7%
1.3%
1.8%
45.1%
36.4%
12.0%
1.1%
2.3%
1.4%
1.6%
45.4%
35.3%
11.8%
1.0%
3.4%
1.4%
1.7%
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
37
Why Not Revise Placebo, Wave 1 + 2
Axiom
Irrelevance
of
Background Counterfactuals
Simple Actions =
State-Contingent
Actions
Irrelevance
of
Counterfactual
Choices
Fusion + Shift
from Nodewise to
Pairwise
Complete Strategies = Implied
Lotteries
Reduction of Compound Lotteries
Total
Same Situations
90.2%
IDK
Confused
Other
Total
7.6%
Expt’er
Demand
0.2%
0.8%
0.6%
0.6%
490
85.2%
11.1%
0.2%
1.9%
0.9%
0.7%
432
85.5%
9.3%
0.0%
0.0%
1.7%
3.5%
172
91.4%
7.1%
0.0%
0.6%
0.6%
0.3%
338
88.9%
7.0%
0.5%
1.2%
1.1%
1.3%
2063
90.6%
6.7%
0.4%
0.8%
1.0%
0.6%
2191
7.3%
0.4%
1.0%
1.0%
0.9%
5686
89.4%
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Indiff.
Reconsidering Risk Aversion
January 10, 2016
38
Motivating Questions
Which EU axioms do people initially endorse, and where are
they most likely to correct where they’ve said they’ve made a
mistake? Toward which frames do people revise?
For any given axiom, most people endorse consistency, but there are
always recalcitrant people; relatively few are completely consistent
across all axioms; would continuing the procedure reap further gains?
People rarely move from consistency to inconsistency
Initially endorse axioms involving nodewise action choices
Initially do not endorse Shift Between Nodewise and Pairwise
(Delta-Epsilon) and Reduction of Compound Lotteries (Eta-Theta)
Most correction for Reduction of Compound Lotteries, toward their
choices in the non-reduced lotteries; least correction between
Irrelevance of Counterfactual Choices (Gamma-Delta)
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
39
Survey Overview
1
Pre-test
2
Training
Main Body: untutored and reasoned preference elicitation
3
Inconsistencies
Intransitivities
4
Demographics and cognition measures
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
40
Intransitivities, Wave 1 + 2
For pairwise strategy choice frames, allow intransitivity resolution
27.8% ”Cannot rank” (indifferent or don’t know what they prefer)
Wave
Wave
Wave
Wave
Wave
Wave
Wave
Wave
Wave
Wave
1,
1,
1,
1,
1,
2,
2,
2,
2,
2,
Stage of Survey
untutored
after 1st inconsistency check
after 1st intransitivity check
after 2nd inconsistency check
after 2nd intransitivity check
untutored
after 1st inconsistency check
after 1st intransitivity check
after 2nd inconsistency check
after 2nd intransitivity check
Intransitivities
1.158
1.043
0.205
0.424
0.137
0.8
0.754
0.265
0.377
0.177
N
278
278
278
278
278
260
260
260
260
260
Table: Average Total Intransitivities, Waves 1 + 2. Only includes participants
having answered all pairwise questions.
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
41
Quantifying Risk Aversion with MLE: Motivating Questions
How do estimates of risk aversion and incidence of decision
errors vary by frame? How do these change between untutored
and reasoned preferences
How do cognitive traits relate to untutored preferences, reasoned
preferences, and the frequency and type of decision errors we
uncover? Are reasoned preferences less correlated with cognition and
more homogeneous across individuals than untutored preferences are?
How does untutored risk aversion vary with gender? How about
reasoned risk aversion?
Will reasoned preferences depend on the order in which individuals
reason through their violations of normative axioms?
Mark Fontana, Daniel Benjamin, Miles Kimball ()
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January 10, 2016
42
Quantifying Risk Aversion with MLE
Assume EU with CRRA utility function with (directly unobserved) risk
aversion parameter γift , for individual i, frame f , and snapshot t
Because of individual response error, iftq , we can only observe ηiftq ,
where q denotes each of the possible 10 pairs of choices
xift = ln(γift )
(1)
ηiftq = xift + iftq
(2)
iftq ∼ N(0, σ2ftq )
(3)
xift ∼ N(µft , σx2ft )
(4)
Estimate: (µft , σxft , σftq )
Note σftq is within-frame decision error
Mark Fontana, Daniel Benjamin, Miles Kimball ()
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43
Certainty Equivalents
Define CE k as the log certainty equivalent of a gamble conditional on
some known coefficient of relative risk aversion γ
Function of participant’s assigned monetary level
Mark Fontana, Daniel Benjamin, Miles Kimball ()
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Certainty Equivalents
Mark Fontana, Daniel Benjamin, Miles Kimball ()
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January 10, 2016
45
Certainty Equivalents
CE A = ln[ML3 ]
CE BCE = ln ((1 − γ)(
(5)
1
1 ML2 1−γ
1 ML5 1−γ 1−γ
+
))
2 1−γ
2 1−γ
1
1 ML2 1−γ
1 ML4 1−γ
1 ML6 1−γ 1−γ
CE BCF = ln ((1 − γ)(
+
+
))
2 1−γ
4 1−γ
4 1−γ
1
1 ML1 1−γ
1 ML4 1−γ
1 ML5 1−γ 1−γ
CE BDE = ln ((1 − γ)(
+
+
))
4 1−γ
4 1−γ
2 1−γ
1
1 ML4 1−γ
1 ML6 1−γ 1−γ
1 ML1 1−γ
+
+
))
CE BDF = ln ((1 − γ)(
4 1−γ
2 1−γ
4 1−γ
Mark Fontana, Daniel Benjamin, Miles Kimball ()
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January 10, 2016
(6)
(7)
(8)
(9)
46
Multinomial Logit: Pairwise Frames
Respondent chooses one in each of 10 pairs
Choose q1 over q2 (for the pair defined by q) when the former
certainty equivalent is greater than the latter (after including error on
each term so a gamble is evaluated as CE k + k , where response
errors are distributed normal with zero mean and σ )
!
Z ∞
n
−(x−µ)2 Y
X
CE
CE
1
q1
q2
√ e 2σx2
f
,
max
ln
dx
(10)
µ,σx ,σ
σ
σ
−∞ σx 2π
q
i=1
f(
CE q1 CE q2
,
)=
σ
σ
Mark Fontana, Daniel Benjamin, Miles Kimball ()
1
1+e
Reconsidering Risk Aversion
CE q2 −CE q1
σ
(11)
January 10, 2016
47
Pairwise Frames (SE’s not CI’s!)
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
48
Pairwise Frames
Limited convergence in mean risk aversion
Reduced lotteries associated with less variation in risk aversion
measure
Reduced lotteries associated with less error
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
49
Multinomial Logit: Nodewise Frames
Respondent chooses either among 4 or 5 plans
CE k is the chosen option’s conditional log certainty equivalent
CE −k is the vector of other options’ conditional log certainty
equivalents
!
Z ∞
n
−(x−µ)2
X
CE
CE
1
k
−k
√ e 2σx2 f
,
dx
max
ln
µ,σx ,σ
σ
σ
σ
2π
−∞ x
(12)
i=1
f(
CE k CE −k
1
,
) = P CE m −CE k
σ
σ
σ
me
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
(13)
January 10, 2016
50
Notes on Complete Contingent Action Plan
Complete-1 considers both the 1st and 2nd choice information (fully
contingency)
Complete-4 considers combination of:
1
2
Those for whom A was their first choice, use only second choice (of 4)
Those form whom A was not their first choice, use their first choice
among non-A responses (again, of 4)
Note: Complete-4 takes into account the fluctuating number of
people who chose A snapshot by snapshot
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
51
Nodewise Frames
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
52
Nodewise Frames
Need to toss out ”Two Backdrop” to see anything
Note: diagnostics show likelihood function breaks down
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
53
Nodewise Frames sans ”Two Backdrop”
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
54
Nodewise Frames sans ”Two Backdrop”
Try to toss out two more sets of things:
”Complete-1” and ”Complete-4”
Wave 2 snapshots (6-10)
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
55
Nodewise Frames sans ”Two Backdrop” and ”Complete”
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
56
Nodewise Frames sans ”Two Backdrop” and ”Complete”
”Isolation” and ”Backdrop” indistinguishable!
Consistent with raw data and consistency
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
57
Nodewise Frames sans ”Two Backdrop” and Wave 2
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
58
Nodewise Frames sans ”Two Backdrop” and Wave 2
Most frames indistinguishable
”Complete-1” is more difficult (higher error response variance)
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
59
Motivating Questions: Risk Aversion and Decision Errors
How do estimates of risk aversion and incidence of decision
errors vary by frame? How do these change between untutored
and reasoned preferences?
Reduced pairwise strategy choices associated with greater risk
aversion among pairwise frames
Risk aversion mildly converges among pairwise frames
Decision errors more prevalent in non-reduced pairwise frames
Error response variance declines, most for non-reduced pairwise frames
Nodewise frames mostly indistinguishable where not underpowered;
need to ask follow-up questions to elicit rank
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
60
Conclusions
People generally revise toward consistency (also reduced decision error
variance), but not ubiquitously across all potential inconsistencies
Demonstrated method that makes progress in reducing the
uncertainty about normative risk aversion
Incomplete convergence in risk aversion among frames; no single
measure for investing or policy defaults yet
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
61
Moving Forward
More heavy-handed reconciliation procedure (to facilitate even more
updating toward consistency)
Direct elicitation of second-, third-, and fourth-favorite choices for
nodewise frames (to allow for greater comparability across frames and
boost the statistical power of our MLE procedure)
Bring subjects back into the lab for third or even fourth waves (to
test the extent to which our procedure “sticks” over the course of
more than a few weeks
Collect data from more nationally representative sample (given
concerns about external validity)
Shorter survey that approximates the above; make into an app!
actually have people use this
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
62
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
63
Alpha and Beta Inconsistency Matrices
Untutored preferences:
aa
a Beta CE
Alpha aa
a
a
CE
CF
DE
DF
rsums
57
13
8
6
84
CF
DE
DF
csums
12
116
4
31
163
3
2
27
14
46
4
14
21
269
308
76
145
60
320
601
CF
DE
DF
csums
4
143
1
18
166
2
1
34
3
40
5
7
7
302
321
73
155
48
325
601
Reasoned preferences:
aa
a Beta CE
Alpha aa
a
a
CE
CF
DE
DF
rsums
Mark Fontana, Daniel Benjamin, Miles Kimball ()
62
4
6
2
74
Reconsidering Risk Aversion
January 10, 2016
1
Beta and Gamma Inconsistency Matrices
Untutored preferences:
aa
aGamma CE
Beta aa
a
a
CE
CF
DE
DF
rsums
54
11
3
4
72
CF
DE
DF
csums
17
130
6
22
175
4
3
29
21
57
8
18
8
260
294
83
162
46
307
598
CF
DE
DF
csums
8
135
5
16
164
6
2
27
11
46
3
18
4
292
317
73
164
40
321
598
Reasoned preferences:
aa
aGamma CE
Beta aa
a
a
CE
CF
DE
DF
rsums
Mark Fontana, Daniel Benjamin, Miles Kimball ()
56
9
4
2
71
Reconsidering Risk Aversion
January 10, 2016
2
Gamma and Delta Inconsistency Matrices
Untutored preferences:
aa
aaDelta CE
Gamma
aa
a
CE
CF
DE
DF
rsums
35
12
2
5
54
CF
DE
DF
csums
13
126
7
18
164
7
4
29
9
49
2
14
16
243
275
57
156
54
275
542
CF
DE
DF
csums
13
114
8
19
154
5
4
23
11
43
4
18
4
261
287
57
146
41
298
542
Reasoned preferences:
aa
aaDelta CE
Gamma
aa
a
CE
CF
DE
DF
rsums
Mark Fontana, Daniel Benjamin, Miles Kimball ()
35
10
6
7
58
Reconsidering Risk Aversion
January 10, 2016
3
Delta and Epsilon Inconsistency Matrices
Considers epsilon plans with perfect 4-0 win records
Untutored preferences:
aa
aEpsilon A
Delta aa
a
a
A
BCE
BCF
BDE
BDF
rsums
27
0
0
0
1
28
BCE
BCF
BDE
BDF
csums
7
5
3
2
3
20
15
8
65
4
39
131
4
2
8
6
10
30
15
5
32
13
173
238
68
20
108
25
226
447
BCE
BCF
BDE
BDF
csums
4
6
4
1
0
15
9
9
76
1
16
111
1
4
6
8
3
22
8
4
24
8
220
264
53
23
112
19
240
447
Reasoned preferences:
aa
aEpsilon A
Delta aa
a
a
A
BCE
BCF
BDE
BDF
rsums
Mark Fontana, Daniel Benjamin, Miles Kimball ()
31
0
2
1
1
35
Reconsidering Risk Aversion
January 10, 2016
4
Epsilon and Eta Inconsistency Matrices
Considers eta and theta plans with perfect 4-0 win records
Untutored preferences:
aa
a
Eta
Epsilonaa
A
BCE
BCF
BDE
BDF
csums
A
BCE
BCF
BDE
BDF
rsums
22
4
2
0
0
28
0
8
0
0
0
8
3
2
75
1
28
109
1
2
1
5
13
22
1
1
30
7
176
215
27
17
108
13
217
382
Eta
Epsilonaa
A
BCE
BCF
BDE
BDF
csums
A
BCE
BCF
BDE
BDF
rsums
28
2
0
0
0
30
1
7
1
0
2
11
1
1
74
2
15
93
0
0
0
9
10
19
2
2
14
2
209
229
32
12
89
13
236
382
a
a
Reasoned preferences:
aa
a
a
a
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
5
Eta and Theta Inconsistency Matrices
Considers eta and theta plans with perfect 4-0 win records
Untutored preferences:
aa
a Theta A
Eta aa
a
a
A
BCE
BCF
BDE
BDF
rsums
24
1
1
0
5
31
BCE
BCF
BDE
BDF
csums
0
5
11
0
5
21
1
2
35
0
21
59
0
0
9
17
41
67
0
1
32
11
138
182
25
9
88
28
210
360
BCE
BCF
BDE
BDF
csums
0
9
0
1
3
13
0
0
60
0
10
70
1
0
3
23
12
39
0
2
11
3
195
211
28
11
74
27
220
360
Reasoned preferences:
aa
a Theta A
Eta aa
a
a
A
BCE
BCF
BDE
BDF
rsums
Mark Fontana, Daniel Benjamin, Miles Kimball ()
27
0
0
0
0
27
Reconsidering Risk Aversion
January 10, 2016
6
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
7
Riskier Choice
Axiom
Irrelevance of Background
Counterfactuals
Simple Actions = StateContingent Actions
Irrelevance of Counterfactual Choices
Fusion + Shift from Nodewise to Pairwise
Complete Strategies = Implied Lotteries
Reduction of Compound
Lotteries
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Frame 1
Riskier
.4
Frame 2
Riskier
.59
197
.52
.47
196
.50
.50
246
.38
.61
449
.47
.52
1504
.51
.48
2005
Reconsidering Risk Aversion
Total
January 10, 2016
1
Frame 1 Riskier
Axiom
Irrelevance of Background
Counterfactuals
Simple Actions = StateContingent Actions
Irrelevance of Counterfactual Choices
Fusion + Shift from Nodewise to Pairwise
Complete Strategies = Implied Lotteries
Reduction of Compound
Lotteries
Toward
Frame 1
.33
Toward
Frame 2
.22
No
Update
.41
Swap
Choices
.02
Total
.22
.07
.66
.02
103
.05
.01
.92
0
120
.21
.17
.55
.04
174
.23
.22
.48
.05
715
.31
.17
.45
.05
1023
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
80
2
Frame 2 Riskier
Axiom
Irrelevance of Background
Counterfactuals
Simple Actions = StateContingent Actions
Irrelevance of Counterfactual Choices
Fusion + Shift from Nodewise to Pairwise
Complete Strategies = Implied Lotteries
Reduction of Compound
Lotteries
Toward
Frame 1
.16
Toward
Frame 2
.25
No
Update
.54
Swap
Choices
.03
Total
.18
.17
.56
.07
93
.09
.05
.84
0
126
.2
.17
.54
.06
275
.19
.28
.47
.04
789
.23
.24
.46
.05
982
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
117
3
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
4
Flow of Inconsistency Checks
In one question you chose X over Y but in another question you chose
Y over X. Do you think the two situations are different enough that it
makes sense to have different choices, or should they be the same?
(And show pictures of the two choices)
It makes sense to have the same choice in both questions.
It makes sense to have different choices.
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
1
Flow of Inconsistency Checks
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
2
Flow of Inconsistency Checks
In one question you chose X over Y but in another question you chose
Y over X. Do you think the two situations are different enough that it
makes sense to have different choices, or should they be the same?
(And show pictures of the two filled out choices)
It makes sense to have the same choice in both questions.
It makes sense to have different choices.
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
3
It makes sense to have different choices...
Why do you want to make different choices in these two situations?
The two situations are different enough that I want different choices
Some of the options are equally good to me, so it doesn’t matter which
one I choose
I chose how I thought the experimenters wanted me to chose
I don’t know which options I prefer
I don’t know or am confused
Other
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
4
It makes sense to have different choices...
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
5
Flow of Inconsistency Checks
In one question you chose X over Y but in another question you chose
Y over X. Do you think the two situations are different enough that it
makes sense to have different choices, or should they be the same?
(And show pictures of the two choices)
It makes sense to have the same choice in both questions.
It makes sense to have different choices.
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
6
Flow of Inconsistency Checks
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
7
It makes sense to have the same choice...
Please look at your choices from before. Which better represents
what you want to do in both, X or Y?
Choice of X (with image showing filled out choice)
Choice of Y (with image showing filled out choice)
I changed my mind: I realized that it does make sense to have different
choices in these two situations. I would like to change *both* of my
choices.
I changed my mind: I realized that it does make sense to have different
choices in these two situations. I would like to keep my current choices.
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
8
It makes sense to have the same choices...
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
9
Verify new preferences...
Is this what you wanted your choices to be changed to? If so, click
next. If not, click back and change your choices to what you want.
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
10
Verify prefer BDF > BDE...
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
11
Why new preferences...
Why did you want to change your choices as you did?
I made a mistake when I first chose
Answering all of these questions made me change what I want
Some of the options are equally good to me, so it doesn’t matter which
one I choose
I chose how I thought the experimenters wanted me to chose
I don’t know which options I prefer
I don’t know or am confused
Other
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
12
Why prefer BDF > BDE...
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
13
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
14
Flow of Placebo Inconsistencies
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
1
It makes sense to have the same choices...
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
2
Flow of Placebo Inconsistencies
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
3
It makes sense to have different choices...
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
4
Verify I would like to change *both* my choices...
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
5
Why swap...
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
6
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
7
Duration Histograms
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
1
Monetary Levels
e.g. γ = 1.576
100k fixed, 150k fixed, 225k fixed
72k = given γ, dollar amount such that indifferent between 100k for
sure and 50-50 chance of 150k and X-k, assuming CRRA and EU
52k = 72k squared / 100
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
2
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
3
Intransitivity Example
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
1
Followup Why ”Cannot Rank”
Recall 27.8% of the time participants said they could not resolve an
intransitivity (”Cannot rank”).
Among 311 participants invited for two waves, followed up with those
who said ”cannot rank” by asking why?
Of 186 instances:
17.74%: ”Optionals all equally good”
48.92%: ”I don’t know what I prefer”
22.58%: ”Too hard to rank”
10.75%: ”Other” (Allowed free-response, most are complaining about
being tired or that the task was too hard.)
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
2
Intransitivities, Wave 1
At each of the 5 ”snapshots” over the course of the survey
Note: ”Reduced” slightly more intransitivities than ”Complete” and
”Compound” to begin with, but by the end of the survey,
convergence among the frames in terms of intransitivity.
Table: Average total intransitivities, Wave 1. Only includes participants having
answered all pairwise questions.
Stage of Survey
Untutored
After 1st inconsistency check
After 1st intransitivity check
After 2nd inconsistency check
After 2nd intransitivity check
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Intransitivities
1.067
0.874
0.229
0.36
0.167
Reconsidering Risk Aversion
N
593
593
594
594
594
January 10, 2016
3
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
4
Power Issues and Diagnostic Tests
Test how well-powered analyses are:
Manually vary one parameter while optimizing over the other two
Plot the log likelihood as a function of the manually varied parameter
Use critical values from the likelihood ratio test to draw horizontal lines
representing confidence intervals at appropriate vertical distances from
global optimum
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
1
Power Issues and Diagnostic Tests
Conclusions (see appendix slides):
Pairwise frames using all available data: well-powered across all
snapshots using all available data
Nodewise frames using all available data: snapshots 1-5 are marginally
well-powered, while snapshots 6-10 are underpowered
Can ”stack” nodewise frames: snapshots 1-5 are now well-powered,
while snapshots 6-10 are still underpowered; however, this is cheating!
need to account for ”within person” correlation
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
2
Diagnostic Tests: Pairwise Frames, Epsilon, Snapshot 1
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
3
Diagnostic Tests: Pairwise Frames, Eta, Snapshot 1
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
4
Diagnostic Tests: Pairwise Frames, Theta, Snapshot 1
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
5
Diagnostic Tests: Nodewise Frames, alpha, Mu
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
6
Diagnostic Tests: Nodewise Frames, alpha, SigX
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
7
Diagnostic Tests: Nodewise Frames, alpha, SigEps
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
8
Diagnostic Tests: Nodewise Frames, beta, Mu
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
9
Diagnostic Tests: Nodewise Frames, beta, SigX
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
10
Diagnostic Tests: Nodewise Frames, beta, SigEps
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
11
Diagnostic Tests: Nodewise Frames, gamma, Mu
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
12
Diagnostic Tests: Nodewise Frames, gamma, SigX
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
13
Diagnostic Tests: Nodewise Frames, gamma, SigEps
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
14
Diagnostic Tests: Nodewise Frames, delta1, Mu
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
15
Diagnostic Tests: Nodewise Frames, delta1, SigX
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
16
Diagnostic Tests: Nodewise Frames, delta1, SigEps
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
17
Diagnostic Tests: Nodewise Frames, delta4, Mu
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
18
Diagnostic Tests: Nodewise Frames, delta4, SigX
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
19
Diagnostic Tests: Nodewise Frames, delta4, SigEps
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
20
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
21
Survey Overview
1
Pre-test
2
Training
Main Body: untutored and reasoned preference elicitation
3
Inconsistencies
Intransitivities
4
Demographics and cognition measures
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
1
Overview: Pre-test
4 rounds of 3 questions each, yields 4 measures of risk aversion
Rounds vary the extent of upside of the risky choice
Apply Kimball, Sahm, Shapiro procedure to impute single measure
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
2
Pre-test Results
Subjects with higher cognition are more risk tolerant
Cognition measured by probabilistic sophistication battery (coming
HRS), number series battery (HRS), self-described logical adjectives
battery, number of statistics and economics classes taken, SAT math
score, cognitive reflex task (Frederick 2005), and need for cognition
battery (Cacioppo et al. 1984)
Female subjects are more risk averse
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
3
Pre-test Randomization
For subset of participants, randomized when the pre-test questions
appear
Control: beginning of survey
Treatment: end of the survey
Decline in log risk aversion associated with treatment in both waves 1
and 2
If we think of risk aversion as a sort of cognitive bias, our procedure
(at least temporarily) alleviates said bias
T-test of means:
Wave 1: diff = 0.16, diff > 0: p = 0.0038
Wave 2: diff = 0.20, diff > 0: p = 0.0144
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
4
Pre-test Randomization – Histograms
1
0
1
Density
0
0
.5
.5
Density
1
1
1.5
1.5
0
0
2
4
6
0
2
4
6
0
2
imputed_log_gamma
Graphs by calibration_random
Mark Fontana, Daniel Benjamin, Miles Kimball ()
4
6
0
2
4
6
imputed_log_gamma_2
Graphs by calibration_random_2
Reconsidering Risk Aversion
January 10, 2016
5
Pre-test Across Waves
Participants asked the same set of pre-test questions in each wave
(separated by a few weeks)
Between waves 1 and 2: no difference in risk aversion (p = 0.5350)
We really don’t need persistence to get a cleaner measure of risk
aversion
Cognitive measures explain less variation in risk aversion in wave 2
than in wave 1 (separated by a few weeks)
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
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0
0
20
50
40
Frequency
60
Frequency
100
150
80
200
100
250
Pre-test Across Waves – Histograms
0
1
2
3
imputed_log_gamma
Mark Fontana, Daniel Benjamin, Miles Kimball ()
4
5
0
Reconsidering Risk Aversion
1
2
3
imputed_log_gamma_2
4
5
January 10, 2016
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Pre-test Risk Aversion Wave 1 v Wave 2
(1)
imputed log gamma
-0.0216
(0.0334)
(2)
imputed log gamma 2
-0.0106
(0.0395)
numeracy
-0.0239
(0.0209)
0.00696
(0.0247)
logical
-0.00506
(0.00327)
-0.00493
(0.00387)
classes
-0.0186
(0.0131)
-0.0291
(0.0154)
-0.000263∗
(0.000114)
-0.000219
(0.000135)
probsoph
SAT math
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
8
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
9
Quantifying Risk Aversion with MLE: Motivating Questions
How do estimates of risk aversion and incidence of decision errors
vary by frame? How do these change between untutored and
reasoned preferences
How do cognitive traits relate to untutored preferences,
reasoned preferences, and the frequency and type of decision
errors we uncover? Are reasoned preferences less correlated
with cognition and more homogeneous across individuals than
untutored preferences are?
How does untutored risk aversion vary with gender? How about
reasoned risk aversion?
Will reasoned preferences depend on the order in which
individuals reason through their violations of normative axioms?
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
1
Motivating Questions: Gender
How does untutored risk aversion vary with gender? How about
reasoned risk aversion?
Women more risk tolerant and higher prevalence of decision errors
Women have wider risk aversion distribution (higher st. dev.)
No convergence between genders after revising
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
2
Pairwise by Gender
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
3
Measuring Cognition
Principal component analysis combining:
Probabilistic sophistication battery
Number series battery
Number of stats classes taken
Number of econ classes taken
SAT math score
PC1: 33.4% total variation, all positive factor loadings
PC2: 25.7% total varaition, all positive loadings except classes
Focus only on PC1 split by median
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
4
Pairwise by Cognition
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
5
Motivating Questions: Cognition
How do cognitive traits relate to untutored preferences,
reasoned preferences, and the frequency and type of decision
errors we uncover? Are reasoned preferences less correlated
with cognition and more homogeneous across individuals than
untutored preferences are?
High cognition associated with greater risk tolerance, fewer errors
Risk aversion does not converge across low and high cognitive ppl
Decision error variance does not converge across low and high
cognitive individuals either
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
6
Epsilon by Frame Order
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
7
Eta by Frame Order
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
8
Theta by Frame Order
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
9
Motivation Questions: Frame Order
Will reasoned preferences depend on the order in which
individuals reason through their violations of normative axioms?
Few systematic differences
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
10
Pairwise by Survey Frustration
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
11
Pairwise by Cut By Half Randomization
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
12
Pairwise by Figure Orientation Randomization
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
13
Pairwise by Training Randomization
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
14
Pairwise, Duration Robustness Checks
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
15
Pairwise
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
16
Robustness to Survey Duration
Main MLE results robust to dropping bottom quintile of survey duration
(i.e. fastest survey takers); less so to dropping the top quintile of survey
duration (i.e. slowest survey takers)
Those who move most quickly through the survey are most risk tolerant;
aren’t answering randomly
Those who take the most time have highest error response variance;
people who take longest are struggling with questions
Conclusion: no need to throw out anyone based on duration
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
17
Mark Fontana, Daniel Benjamin, Miles Kimball ()
Reconsidering Risk Aversion
January 10, 2016
18