415
AEA Papers and Proceedings 2022, 112: 415–420
https://doi.org/10.1257/pandp.20221090
THE BEHAVIORAL ECONOMICS OF RISKY CHOICE: NEW
PERSPECTIVES
Robustness of Rank Independence in Risky Choice
†
By
B. Douglas Bernheim, Rebecca Royer, and Charles Sprenger*
The famous Allais
(
1953
)
paradoxes chal-
lenge the validity of the independence axiom that
lies at the heart of expected utility theory
(
EUT
)
.
As originally formulated, prospect theory
(
OPT,
due to Kahneman and Tversky 1979
)
ratio-
nalizes such phenomena by allowing decision
weights to vary
nonlinearly with probabilities
but thereby introduces violations of
first-order
stochastic dominance. Cumulative prospect the-
ory
(
CPT, due to Tversky and Kahneman 1992
)
avoids this difficulty by assuming that the deci-
sion weight associated with an outcome depends
on not only its probability but also its rank.
Bernheim and Sprenger
(
2020
)
present a
novel experimental test of CPT. Specifically,
for a given
three-outcome lottery
L
=
(
X
,
Y
,
Z
;
p
,
q
, 1
−
p
−
q
)
with
Y
>
Z
, they mea-
sure discrete analogs of marginal rates of substi-
tution between
Z
and
Y
(
equalizing reductions
)
.
With
rank-dependent probability weighting, the
trade-off between
Y
and
Z
depends upon whether
X
≥
Y
>
Z
,
Y
>
X
≥
Z
, or
Y
>
Z
>
X
. As it turns
out, the percentage change in the equalizing
reduction as
X
crosses from one of these regimes
to another identifies the degree of rank depen-
dence in probability weights and, consequently,
the shape of the CPT probability weighting func-
tion,
nonparametrically. Bernheim and Sprenger
(
2020
)
present two main findings based on vari-
ation in equalizing reductions for a single set of
tasks.
FINDING 1:
The probability weights for out-
comes
Y and
Z do not vary meaningfully with
their ranks
(
comparing cases with
X
≥
Y
>
Z
to cases with
Y
>
X
≥
Z
)
.
FINDING 2:
Responses to variation in probabil-
ities with fixed ranks imply probability weights
that match standard estimates based on experi-
ments with
two-outcome lotteries.
Unlike OPT, CPT posits a form of
rank-dependent probability weighting that
makes these findings irreconcilable. Interpreted
through the lens of CPT, Finding 1 implies that
the probability weighting function must be linear
over the pertinent range. But Finding 2 implies
that the probability weighting function is highly
nonlinear over the very same range.
Since the publication of Bernheim and
Sprenger
(
2020
)
, some proponents of CPT have
raised the possibility that Finding 1 may be a
consequence of experimental procedures rather
than underlying preferences. Possible concerns
include the following:
(
i
)
comprehension of the
three-option lotteries may have been poor;
(
ii
)
the number of decision tasks may have been
overwhelming;
(
iii
)
the stakes may have been too
low;
(
iv
)
the analysis may have overlooked evi-
dence of rank dependence associated with tran-
sitions between the regimes
Y
>
X
≥
Z
and
Y
>
Z
>
X
, which shed light on
nonlinearities
in probability weighting for probabilities near 1;
and
(
v
)
the structure of the decision tasks may
have triggered a heuristic involving the “cancel-
lation” of a common outcome
(
X
)
.
1
1
In their critique of Bernheim and Sprenger
(
2020
)
,
Abdellaoui et al.
(
2020
)
raise these issues and make a num-
ber of other points.
* Bernheim: Department of Economics, Stanford
University
(
email: bernheim@stanford.edu
)
; Royer:
Department of Economics, UC San Diego.
(
email: rroyer@
ucsd.edu
)
; Sprenger: Division of Humanities and Social
Sciences, California Institute of Technology
(
email:
sprenger@caltech.edu
)
.
†
Go to https://doi.org/10.1257/pandp.20221090 to visit
the article page for additional materials and author disclo-
sure statement
(
s
)
.
MAY 2022
416
AEA PAPERS AND PROCEEDINGS
It is unlikely that the aforementioned consid-
erations account for Bernheim and Sprenger’s
(
2020
)
results. Concerns
(
i
)
–
(
iii
)
—complexity,
fatigue, and low stakes—are not specifically
entwined with rank dependence; they would
tend to suppress any nuanced feature of decision
making. But the setting is not so complex, fatigu-
ing, or inconsequential that it fails to activate
conventional probability weighting
(
Finding 2
)
.
Bernheim and Sprenger
(
2020
)
also addressed
concern
(
ii
)
(
fatigue
)
by presenting corrobo-
rating
cross-subject results based on each sub-
ject’s first task. Concern
(
iv
)
merely raises the
possibility that Finding 1 might not hold on an
unexamined portion of the probability domain;
it cannot resolve the conflict that CPT implies
between Findings 1 and 2 on the examined
portion of that domain. Finally, Bernheim and
Sprenger
(
2020
)
address concern (v
)
through a
supplemental experiment with modified tasks
that render the cancellation heuristic inappli-
cable. The supplemental experiment sacrifices
some advantages of the original but still yields
no evidence of
rank-dependent probability
weighting.
In this paper, we demonstrate that Finding 1 is
indeed robust with respect to alternative exper
-
imental procedures that address each of the five
concerns articulated above. Naturally, a compre-
hensive evaluation of CPT must consider other
evidence concerning its validity. However, other
existing tests suffer from serious confounds that
the
Bernheim-Sprenger approach avoids; see
Bernheim and Sprenger
(
2020
)
.
2
I.
Review and Extension of Methods
Regardless of whether the applicable theory is
CPT, OPT, or EUT, we can write the indifference
condition that defines the equalizing reduction
for the lottery
L
=
(
X
,
Y
,
Z
;
p
,
q
, 1
−
p
−
q
)
as
follows:
w
X
u
(
X
)
+
w
Y
u
(
Y
)
+
w
Z
u
(
Z
)
=
w
X
u
(
X
)
+
w
Y
u
(
Y
+
m
)
+
w
Z
u
(
Z
−
k
)
,
2
This approach improves significantly upon prior tests
of comonotonic and noncomonotonic independence
(
see,
e.g., Birnbaum and McIntosh 1996; Wu 1994; Wakker et al.
1994
)
by neutralizing important confounds and providing
quantitative
nonparametric measures of
nonlinearities in the
probability weighting function.
where
w
s
is the decision weight for
s
∈
{
X
,
Y
,
Z
}
. For small
m
, it follows
that
k
_
m
≈
w
Y
u
′
(
Y
)
_
w
Z
u
′
(
Z
)
. For any change from
X
′′
to
X
′
, the associated
k
′′
and
k
′
therefore satisfy
log
(
k
′
)
−
log
(
k
′′
)
≈
log
(
w
Y
′
_
w
Z
′
)
−
log
(
w
Y
′′
_
w
Z
′′
)
.
Thus, the percentage change in the equalizing
reduction
nonparametrically measures the per
-
centage change in relative decision weights
resulting from a change in
X
.
To determine whether the weights are
rank
dependent, we choose values
X
′
and
X
′′
so that the
ranks of the outcomes differ. For
X
′′
>
Y
>
Z
and
Y
>
X
′
>
Z
, CPT implies
log
(
k
′
)
−
log
(
k
′′
)
≈
log
(
π
(
q
)
_
q
)
−
log
(
π
(
p
+
q
)
−
π
(
p
)
______________
q
)
.
In other words, under the maintained hypothesis
of CPT, the percentage change in the equalizing
reduction
nonparametrically measures the per
-
centage change in the slope of the probability
weighting function between the intervals
[
0,
q
]
and
[
p
,
p
+
q
]
.
Bernheim and Sprenger
(
2020
)
found
essentially no difference in equalizing reduc-
tions between the regimes
X
′′
>
Y
>
Z
and
Y
>
X
′
>
Z
for
p
∈
{
0.1, 0.4, 0.6
}
(
with
q
=
0.3
)
, which means there is no evidence of
rank-dependent probability weighting
(
Finding
1
)
. Treating CPT as a maintained hypothesis,
they concluded that the probability weighting
function must be linear over the interval
[
0, 0.9
]
.
If CPT is valid, then one can also recover
the probability weighting function by fix-
ing ranks and varying probabilities. Defining
φ
=
(
q
_
1
−
p
−
q
)
m
_
k
and using the approxi-
mation
k
_
m
≈
w
Y
u
′
(
Y
)
_
w
Z
u
′
(
Z
)
along with the defini-
tions of
w
Y
and
w
Z
for CPT within the regime
Y
>
X
>
Z
, we see that for any change in
p
,
say from
p
′
to
p
,
log
(
φ
)
−
log
(
φ
′
)
≈
log
(
π
(
1
)
−
π
(
p
+
q
)
______________
1
−
p
−
q
)
−
log
(
π
(
1
)
−
π
(
p
′
+
q
)
______________
1
−
p
′
−
q
)
.
VOL. 112
417
ROBUSTNESS OF RANK INDEPENDENCE IN RISKY CHOICE
In other words, the percentage change in
φ
(
which is measurable
)
is a
nonparametric esti-
mate of the percentage change in the average
slope of the probability weighting function
between the intervals
[
p
′
+
q
, 1
]
and
[
p
+
q
, 1
]
.
Bernheim and Sprenger
(
2020
)
found large
differences in
φ
for
p
∈
{
0.1, 0.4, 0.6
}
(
with
q
=
0.3
)
, which means there is evidence of sub-
stantial
nonlinearities in probability weighting.
In particular, their estimates imply that the prob-
ability weighting function is highly
nonlinear
throughout the interval
[
0.4, 1
]
(
Finding 2
)
.
Treating CPT as a maintained hypothesis,
Finding 1 and Finding 2 clearly have contra-
dictory implications for the properties of the
probability weighting function over the inter
-
val
[
0.4, 0.9
]
. Bernheim and Sprenger
(
2020
)
therefore reject CPT. Their findings are instead
consistent with
nonlinear
rank-independent
probability weighting.
We extend these methods in two ways. First,
we also examine changes from
X
′
to
X
satisfying
Y
>
X
′
>
Z
and
Y
>
Z
>
X
. For CPT, we
have
log
(
k
)
−
log
(
k
′
)
≈
log
(
π
(
1
)
−
π
(
p
+
q
)
______________
1
−
p
−
q
)
−
log
(
π
(
1
−
p
)
−
π
(
q
)
______________
1
−
p
−
q
)
.
In other words, under the maintained hypothesis
of CPT, the percentage change in the equalizing
reduction between the regimes
Y
>
X
′
>
Z
and
Y
>
Z
>
X
nonparametrically measures
the percentage change in the slope of the prob-
ability weighting function between the intervals
[
q
, 1
−
p
]
and
[
p
+
q
, 1
]
. Thus, CPT can account
for invariance of equalizing reductions across all
three regimes only if the slope of the probability
weighting function is constant
(
i.e., the function
is linear
)
over the entire interval
[
0, 1
]
.
Second, we modify the method by eliciting
k
+
and
k
−
, defined as follows:
w
X
u
(
X
−
1
)
+
w
Y
u
(
Y
)
+
w
Z
u
(
Z
)
=
w
X
u
(
X
)
+
w
Y
u
(
Y
+
m
)
+
w
Z
u
(
Z
−
k
+
)
w
X
u
(
X
)
+
w
Y
u
(
Y
)
+
w
Z
u
(
Z
)
=
w
X
u
(
X
−
1
)
+
w
Y
u
(
Y
+
m
)
+
w
Z
u
(
Z
−
k
−
)
.
For small
m
, we have
0.5
(
k
+
+
k
−
)
_
m
≈
w
Y
u
′
(
Y
)
_
w
Z
u
′
(
Z
)
, so
k
and
k
M
≡
0.5
(
k
+
+
k
−
)
measure the same
theoretical object. Intuitively, substituting
k
M
for
k
immunizes our method against the cancellation
heuristic because the lotteries that define
k
+
and
k
−
involve no common outcomes. In contrast to
the supplemental experiment in Bernheim and
Sprenger
(
2020
)
, this method preserves all quan-
titative inferences concerning rank dependence.
II.
Experimental Design
Our seven conditions all measure equalizing
reductions
(
k
)
and modified equalizing reduc-
tions
(
k
+
and
k
−
)
for the probability vector
(
p
,
q
, 1
−
p
−
q
)
=
(
0.4, 0.3, 0.3
)
.
Table 1
summarizes the main features of each condition;
the online Appendix includes screenshots. We
conducted the experiment in September 2021
on the Prolific platform using Otree software
(
Chen, Schonger, and Wickens 2016
)
.
Condition 1 follows Bernheim and Sprenger
(
2020
)
in fixing
Y
=
$24 ,
Z
=
$18 , and
m
=
$5 . We used price lists to elicit equalizing
reductions and modified equalizing reductions
in random order for
X
′′
=
$31 ,
X
′
=
$22 , and
X
=
$3 . We display lottery distributions visu-
ally using the method of Lopes and Oden
(
1999
)
.
Subjects also receive training to facilitate their
comprehension of the probabilities: they draw
Table 1—Conditions for Measuring Equalizing Reductions
Elicitation
Cancellation
tasks
(
X
,
X
′
,
X
′′
)
(
Y
,
Z
)
Stakes
multiplier
m
Visual
display
Probability
training
Incentivized
Order
# of
subjects
Condition 1
Price list
Ye s
(
3, 22, 31
)
(
24, 18
)
1x
5
Ye s
Ye s
1
/
5
Random
93
Condition 2
BDM
Ye s
(
3, 22, 31
)
(
24, 18
)
1x
5
Ye s
Ye s
1
/
5
Random
109
Condition 3
BDM
Ye s
(
12, 88, 124
)
(
96, 72
)
4x
5
Ye s
Ye s
1/
20
Random
103
Condition 4
BDM
Ye s
(
12, 88, 124
)
(
96, 72
)
4x
5
Ye s
Ye s
None
Random
111
Condition 5
BDM
Ye s
(
12, 88, 124
)
(
96, 72
)
4x
20
Ye s
Ye s
1
/
20
Random
89
Condition 6
BDM
Ye s
(
48, 352, 496
)
(
384, 288
)
16x
5
Ye s
Ye s
None
Random
89
Condition 7
BDM
Ye s
(
48, 352, 496
)
(
384, 288
)
16x
80
Ye s
Ye s
None
Random
103
MAY 2022
418
AEA PAPERS AND PROCEEDINGS
from the distribution 18 times and report their
outcomes. This procedure allows the mean-
ing of the probability distribution to “sink in”
(
Heffetz 2018
)
. Thus, Condition 1 addresses
concern
(
i
)
(
comprehension of probabilities
)
through visual presentation and training, con-
cern
(
iv
)
(
limited scope
)
by encompassing
all three regimes, and concern
(
v
)
(
heuristic
cancellation
)
by eliciting modified equalizing
reductions.
Condition 2 is the same as Condition 1 except
that it employs the
titration Becker–DeGroot–
Marschak
(
BDM) mechanism of Mazar,
Koszegi, and Ariely
(
2014
)
, wherein subjects
first state a valuation, then review implications
for options just below and just above the pro-
visional point of indifference, then
(
potentially
)
revise their initial response. This procedure
improves upon the original BDM mechanism
by walking subjects through the contingent
implications of their choices. It creates the same
incentives as the corresponding price lists of
Condition 1, but subjects make only 9 decisions
rather than 585 component choices. This condi-
tion therefore addresses concern
(
ii
)
(
decision
fatigue
)
in addition to concerns
(
i
)
,
(
iv
)
, and
(
v
)
.
The remaining five conditions follow the
same procedures as Condition 2 but inflate
X
,
Y
and
Z
by a factor of 4
(
Conditions 3, 4, and 5
)
or
16
(
Conditions 6 and 7
)
. The value of
m
is
$5 in
Conditions 3, 4, and 6;
$20
in Condition 5; and
$80
in Condition 7. In other words, we either
inflate
m
by the same factor as the outcomes or
leave it fixed at
$5 . These conditions address
concern
(
iii
) (
small stakes
)
.
Conditions 1, 2, 3, and 5 involve real choices.
We paid 1 out of every 5 subjects based on one
of their choices in Conditions 1 and 2 and paid
1 out of every 20 subjects in Conditions 3 and
5. Conditions 4, 6, and 7 involve hypotheti-
cal choices. This variation provides additional
opportunities to test whether incentives induce
rank-dependent behavior.
Our procedures prevent subjects’ choices
from switching back and forth between
(
X
,
Y
,
Z
)
and
(
X
,
Y
+
m
,
Z
−
k
)
as
k
increases. This
restriction has the advantage of yielding an
unambiguous measure of
k
and
(
for Condition
1
)
of reducing each price list to a single choice,
thereby minimizing decision fatigue. A disad-
vantage is that it sacrifices a potential indicator
of poor comprehension
(
multiple switching
)
.
An alternate measure is whether the elicitations
yield boundary values. Overall, 2.3
percent
(
13.9 percent
)
of observations take on the high-
est
(
lowest
)
value. Only 4.6
percent of subjects
provide no interior values, and 80.2
percent pro-
vide 2 or fewer boundary values. Dropping these
responses does not meaningfully change our
findings.
III. Results
Table 2 and Figure 1 present our results.
Condition 1 replicates the findings of Bernheim
and Sprenger
(
2020
)
. In both cases, the esti-
mated change in decision weights between
the regimes
Z
<
X
′
<
Y
and
Z
<
Y
<
X
′′
,
log
(
k
′
)
−
log
(
k
′′
)
, is close to zero. As
before, we fail to reject the null hypothesis
of rank independence
(
i.e., equality between
k
′
and
k
′′
)
.
Condition 1 also extends the prior investi-
gation by examining tasks with
X
<
Z
<
Y
.
The differences between
k
,
k
′
, and
k
′′
are small
(
on the order of 1 to 2 percent
)
and statis-
tically insignificant, indicating the virtual
absence of rank dependence. Treating CPT
as a maintained hypothesis, one would con-
clude that the average slope of the probability
weighting function is essentially unchanged
between the intervals
[
0, 0.3
]
and
[
0.4, 0.7
]
as well as between the intervals
[
0.3, 0.6
]
and
[
0.7, 1
]
.
Results based on
k
+
and
k
−
, which are
immune to the cancellation heuristic, corrobo-
rate the
(
near
)
rank independence of probability
weights. Log differences in equalizing reduc-
tions imply that probability weights change only
slightly
(
by 1 to 4 percent
)
due to a change in
ranks. Critically, this finding does not reflect
a tendency to cancel
approximately
com-
mon outcomes—i.e., to ignore the difference
between
X
and
X
−
1 . As shown in Figure 1,
values of
k
+
are generally higher than values of
k
−
, and the difference is statistically significant
(
χ
2
=
20.94;
p
=
0.000
)
. However, the aver
-
age of
k
+
and
k
−
is statistically indistinguishable
from
k
(
χ
2
=
1.46;
p
=
0.228
)
, which sug-
gests that cancellation is unimportant.
To put the preceding findings in context,
Figure 1 also displays predicted values of
k
derived from the parameterized version of
CPT due to Tversky and Kahneman
(
1992
)
.
According to this model, equalizing reduc-
tions should exhibit discontinuous increases
VOL. 112
419
ROBUSTNESS OF RANK INDEPENDENCE IN RISKY CHOICE
Table 2—Mean Equalizing Reductions and Estimated Rank Dependence
Mean equalizing reduction
Rank dependence
Equalizing
reduction–
k
X
<
Z
<
Y
Equalizing
reduction–
k
′
Z
<
X
′
<
Y
Equalizing
reduction–
k
′′
Z
<
Y
<
X
′′
log
(
k
)
−
log
(
k
′
)
log
(
k
)
−
log
(
k
′′
)
log
(
k
′
)
−
log
(
k
′′
)
Bernheim Sprenger
(
2020
)
4.32
(
0.12
)
4.34
(
0.12
)
−
0.01
(
0.02
)
(
p
,
q
, 1
−
p
−
q
)
=
(
0.4, 0.3, 0.3
)
,
(
Y
,
Z
)
=
(
24, 18
)
Condition 1
(
price list, 1x stakes,
m
=
$5
)
4.25
(
0.29
)
4.27
(
0.28
)
4.35
(
0.26
)
−
0.00
(
0.06
)
−
0.02
(
0.05
)
−
0.02
(
0.05
)
Cancellation tasks:
0.5
(
k
+
+
k
−
)
4.05
(
0.30
)
4.21
(
0.31
)
4.29
(
0.30
)
−
0.04
(
0.06
)
−
0.06
(
0.06
)
−
0.02
(
0.07
)
Condition 2
(
BDM, 1x stakes,
m
=
$5
)
3.50
(
0.36
)
3.57
(
0.37
)
3.67
(
0.39
)
−
0.02
(
0.07
)
−
0.05
(
0.07
)
−
0.03
(
0.06
)
Cancellation tasks:
0.5
(
k
+
+
k
−
)
3.53
(
0.37
)
3.60
(
0.36
)
3.55
(
0.41
)
−
0.02
(
0.08
)
−
0.00
(
0.08
)
0.02
(
0.08
)
Condition 3
(
BDM, 4x stakes,
m
=
$5
)
5.17
(
0.35
)
5.44
(
0.39
)
4.99
(
0.33
)
−
0.05
(
0.06
)
0.04
(
0.05
)
0.09
(
0.06
)
Cancellation tasks:
0.5
(
k
+
+
k
−
)
5.19
(
0.39
)
5.34
(
0.43
)
4.93
(
0.37
)
−
0.03
(
0.08
)
0.05
(
0.07
)
0.08
(
0.07
)
Condition 4
(
BDM, 4x stakes,
m
=
$5 , hyp.
)
4.67
(
0.39
)
4.39
(
0.34
)
4.32
(
0.41
)
0.06
(
0.07
)
0.08
(
0.09
)
0.02
(
0.08
)
Cancellation tasks:
0.5
(
k
+
+
k
−
)
4.56
(
0.40
)
4.47
(
0.37
)
4.26
(
0.44
)
0.02
(
0.08
)
0.07
(
0.10
)
0.05
(
0.09
)
Condition 5 †
(
BDM, 4x stakes,
m
=
$20
)
3.42
(
0.37
)
3.52
(
0.39
)
4.07
(
0.40
)
−
0.03
(
0.10
)
−
0.17
(
0.09
)
−
0.15
(
0.07
)
Cancellation tasks:
0.5
(
k
+
+
k
−
)
3.25
(
0.38
)
3.50
(
0.42
)
4.11
(
0.41
)
−
0.07
(
0.13
)
−
0.23
(
0.11
)
−
0.16
(
0.09
)
Condition 6
(
BDM, 16x stakes,
m
=
$5 , hyp.
)
5.37
(
0.47
)
4.50
(
0.46
)
4.88
(
0.43
)
0.18
(
0.09
)
0.09
(
0.08
)
−
0.08
(
0.08
)
Cancellation tasks:
0.5
(
k
+
+
k
−
)
5.14
(
0.53
)
4.59
(
0.49
)
4.75
(
0.44
)
0.11
(
0.11
)
0.08
(
0.08
)
−
0.03
(
0.10
)
Condition 7 †
(
BDM, 16x stakes,
m
=
$80 , hyp.
)
4.85
(
0.49
)
4.55
(
0.44
)
4.44
(
0.48
)
0.06
(
0.06
)
0.09
(
0.08
)
0.03
(
0.06
)
Cancellation tasks:
0.5
(
k
+
+
k
−
)
4.88
(
0.50
)
4.60
(
0.45
)
4.37
(
0.47
)
0.06
(
0.06
)
0.11
(
0.08
)
0.05
(
0.06
)
Aggregate values †
4.47
(
0.15
)
4.34
(
0.15
)
4.39
(
0.15
)
0.03
(
0.03
)
0.02
(
0.03
)
−
0.01
(
0.03
)
Cancellation tasks:
0.5
(
k
+
+
k
−
)
4.38
(
0.16
)
4.34
(
0.15
)
4.32
(
0.16
)
0.01
(
0.03
)
0.01
(
0.03
)
0.01
(
0.03
)
Notes:
Mean values of
k
are estimated from interval regressions of experimental response on indicators for the rank of
X
, with
titration BDM data converted to equivalent price list responses.
† indicates normalized values,
k
⋅
(
5
/
m
)
, for ease of compar
-
ison across
m
values. Rows titled “Cancellation tasks” are based only on modified equalizing reductions. Standard errors, in
parentheses, are clustered at individual level and calculated using the delta method.
0
2
4
6
8
10
Equalizing reduction:
k
0
2
4
6
8
10
Equalizing reduction:
k
0
2
4
6
8
10
Equalizing reduction:
k
0
32
64
96
128
160
Equalizing reduction:
k
0
8
16
24
32
40
Equalizing reduction:
k
Condition 1
1
×
stakes,
m
=
$5
0
2
4
6
8
10
Equalizing reduction:
k
0
2
4
6
8
10
Equalizing reduction:
k
Condition 2
1
×
stakes,
m
=
$5
X
0
20
40
60
80
100
120
140
X
0
20
40
60
80
100
120
140
X
0
5
10
15
20
25
30
35
X
0
5
10
15
20
25
30
35
X
0
20
40
60
80
100
120
140
X
0
80
160
240
320
400
480
560
X
0
80
160
240
320
400
480
560
Condition 3
4
×
stakes,
m
=
$5
Condition 4
4
×
stakes,
m
=
$5, hyp
Condition 5
4
×
stakes,
m
=
$20
Condition 6
16
×
stakes,
m
=
$5, hyp
Condition 7
16
×
stakes,
m
=
$80, hyp
k
CPT predictions
k
+
k
−
Figure 1. Mean Equalizing Reductions and Standard CPT Predictions
Notes:
Black diamonds
(
with corresponding 95
percent confidence intervals
)
indicate mean values for
k
. Blue
(
red
)
circles
indicate mean values for
k
+
(
k
−
)
. Grey vertical lines mark values for outcomes
Z
and
Y
. Dashed lines indicate predicted values
of
k
in each regime based on standard CPT parametrizations, calculated at
X
,
X
′
, and
X
′′
. Dotted lines connect CPT predictions
between regimes.