of 134
On the Empirical Validity of Cumulative Prospect
Theory: Experimental Evidence of Rank-Independent
Probability Weighting
B. Douglas Bernheim
Stanford University and NBER
Charles Sprenger
UC San Diego
First Draft: December 1, 2014
This Version: January 17, 2020
Abstract
Cumulative Prospect Theory (CPT), the leading behavioral account of decisionmaking
under uncertainty, avoids the dominance violations implicit in Prospect Theory (PT) by
assuming that the probability weight applied to a given outcome depends on its ranking.
We devise a simple and direct non-parametric method for measuring the change in relative
probability weights resulting from a change in payo
ff
ranks. We find no evidence that these
weights are even modestly sensitive to ranks. Conventional calibrations of CPT preferences
imply that the percentage change in probability weights should be an order of magnitude
larger than we observe. It follows either that probability weighting is not rank-dependent,
or that the weighting function is nearly linear. Non-parametric measurement of the change
in relative probability weights resulting from changes in probabilities rules out the second
possibility. Additional tests nevertheless indicate that the dominance patterns predicted
by PT do not arise. We reconcile these findings by positing a form of complexity aversion
that generalizes the well-known certainty e
ff
ect.
JEL classification:
D81, D90
Keywords
: Prospect Theory, Cumulative Prospect Theory, Rank Dependence, Certainty Equiv-
alents.
Previous versions of this paper were titled ‘Direct Tests of Cumulative Prospect Theory.’ We are grateful
to Ted O’Donoghue, Colin Camerer, Nick Barberis, Kota Saito, seminar participants at Cornell, Caltech, MIT,
UCLA, CIDE, Tel Aviv, UC Santa Barbara, the Stanford Institute for Theoretical Economics, and five anony-
mous referees for helpful and thoughtful comments. Fulya Ersoy, Vincent Leah-Martin, Seung-Keun Martinez,
and Alex Kellogg all provided valuable research assistance.
Stanford University, Department of Economics, Landau Economics Building, 579 Serra Mall, Stanford, CA
94305; bernheim@stanford.edu.
University of California San Diego, Rady School of Management and Department of Economics, 9500 Gilman
Drive, La Jolla, CA 92093; csprenger@ucsd.edu.
Electronic copy available at: https://ssrn.com/abstract=3350196
1Introduction
Prospect Theory (PT), as formulated by Kahneman and Tversky (
1979
), provides a flexible
account of decision making under uncertainty that accommodates a wide variety of departures
from the Expected Utility (EU) paradigm. As a result, it has been enormously influential
throughout the social sciences. In contrast to the EU formulations of
von Neumann and Mor-
genstern
(
1944
),
Savage
(
1954
), and
Samuelson
(
1952
), a central premise of PT holds that
attitudes toward objective probabilities display non-linearities, with highly unlikely events re-
ceiving greater proportionate weight than nearly certain ones. This feature reconciles PT with
important behavioral puzzles such as the famous
Allais
(
1953
)paradoxes,aswellasthesimulta-
neous purchase of lottery tickets and insurance, as in
Friedman and Savage
(
1948
). Probability
weighting is also well-supported by simple and widely-replicated laboratory experiments.
1
Unfortunately, the formulation of probability weighting embedded in PT leads to conceptual
di
ffi
culties because it implies violations of first-order stochastic dominance even in relatively
simple settings. This is a serious flaw given the broad consensus that this property renders a
model of decisionmaking unappealing on both positive and normative grounds.
2
To understand
the problem, consider a lottery that pays
X
with probability
p
;forourcurrentpurpose,we
will leave other events and payo
ff
s unspecified. Now imagine a second lottery, identical to the
first, except that it splits the aforementioned event, paying
X
and
X
"
each with probability
p/
2
.
3
Given the
S
-shape of the probability weighting function, we can choose
p
so that the
1
For example, when graphing the empirical certainty equivalent,
C
,foralotterythatpays
X
with probability
p
and 0 with probability
1
p
,onetypicallyfindsaninverse
S
-shaped pattern, with
pX
exceeding
C
for moderate-
to-large values of
p
(as risk aversion would imply), but with the opposite relation for small
p
(see, e.g.,
Tversky
and Kahneman
,
1992
;
Tversky and Fox
,
1995
).
2
As noted by
Quiggin
(
1982
), “Transitivity and dominance rules command virtually unanimous assent...
even from those who sometimes violate them in practice... If a theory of decision under uncertainty is to be
consistent with any of the large body of economic theory which has already been developed... it must satisfy
these rules." (p. 325).
3
Kahneman and Tversky
(
1979
) described their theory as being concerned with lotteries that have at most
two non-zero outcomes. Hence, to apply Prospect Theory strictly in accordance with their original intent, one
would have to assume that this lottery pays zero with probability
1
p
.
Kahneman and Tversky
(
1979
)(p.288)
note that the model extends naturally to more than two non-zero outcomes, and extensions which correspond to
our three outcome formulation are provided by, for example,
Camerer and Ho
(
1994
)and
Fennema and Wakker
(
1997
).
Kahneman and Tversky
(
1979
) actually provided two formulations of Prospect Theory; we assume their
Equation 1 for ‘regular prospects.’ They implicitly invoke the same assumption when examining the Allais
1
Electronic copy available at: https://ssrn.com/abstract=3350196
total weight assigned to two events occurring with probability
p/
2
discretely exceeds the weight
assigned to a single event occurring with probability
p
. Consequently, if
X
is large and/or
"
is small, the first lottery will yield lower PT utility than the second even though it is clearly
preferrable based on first-order stochastic dominance.
4
Ultimately, “rank-dependent” probability weighting was o
ff
ered as a solution to the stochas-
tic dominance problem (
Quiggin
,
1982
), and was incorporated into a new version of PT known
as Cumulative Prospect Theory, henceforth CPT (
Tversky and Kahneman
,
1992
). To under-
stand intuitively how CPT resolves the issue, consider a lottery
L
with three possible payo
ff
s,
X>Y >Z
, occurring with probabilities
p
,
q
,and
1
p
q
.Anotherdescriptionofthe
same lottery involves cumulative probabilities: it pays
Z
with probability 1, adds
Y
Z
with
probability
p
+
q
,andthenincrementallyadds
X
Y
with probability
p
. Accordingly, within
the EU framework, one could write its expected utility as follows:
Expected Utility
=
u
(
Z
)+(
p
+
q
)(
u
(
Y
)
u
(
Z
)) +
p
(
u
(
X
)
u
(
Y
))
.
CPT involves an analogous calculation, except that a reference-dependent utility function,
u
(
·|
r
)
(where
r
is the reference point), is applied to the payo
ff
s, while a weighting function,
(
·
)
,isappliedtothecumulativeprobabilities:
5
U
(
L
)=
(1)
u
(
Z
|
r
)+
(
p
+
q
)[
u
(
Y
|
r
)
u
(
Z
|
r
)] +
(
p
)[
u
(
X
|
r
)
u
(
Y
|
r
)]
.
common consequence paradox (p. 282).
4
Kahneman and Tversky
appreciated this problematic implication of PT and attempted to address it through
an “editing” assumption: “Direct violations of dominance are prevented, in the present theory, by the assumption
that dominated alternatives are detected and eliminated prior to the evaluation of prospects" (p. 284). Most
economists have found this
adhoc
“fix” conceptually unsatisfactory, and it is rarely invoked in applications.
Kahneman and Tversky
also provided a formulation for two-outcome lotteries with either all positive or all
negative outcomes that does indeed respect dominance (see e.g., Equation 2 of
Kahneman and Tversky
,
1979
).
One can see in that formulation the roots of Cumulative Prospect Theory.
5
For simplicity, we assume here that the reference point,
r
,isbelowtheotherpayo
ff
s.
2
Electronic copy available at: https://ssrn.com/abstract=3350196
Normally this expression is rewritten in a form that attaches a weight to each outcome:
U
(
L
)=
(
p
)
u
(
X
|
r
)+[
(
p
+
q
)
(
p
)]
u
(
Y
|
r
)+[
(1)
(
p
+
q
)]
u
(
Z
|
r
)
.
(1)
Now imagine, as before, a second lottery, identical to the first, except that it splits the event
yielding the payo
ff
X
into two events paying
X
and
X
"
, each with probability
p/
2
.Inthat
case, the term
(
p/
2)
u
(
X
|
r
)+[
(
p
)
(
p/
2)]
u
(
X
"
|
r
)
replaces the term
(
p
)
u
(
X
|
r
)
.Notice
that the
total
weight assigned to the two events is still
(
p
)
,thesameasfortheoriginallottery.
Consequently, the stochastic dominance problem noted above does not arise (
Quiggin
,
1982
;
Tversky and Kahneman
,
1992
). CPT nevertheless accommodates the same assortment of EU
violations as PT. For these reasons, CPT has replaced PT as the leading behavioral model of
decisionmaking under uncertainty.
To understand the sense in which CPT involves
rank-dependent
probability weighting, con-
sider the weight applied to the event that generates the payo
ff
Y
as we change the value of
X
.
Initially
X
exceeds
Y
,andtheweighton
Y
is
(
p
+
q
)
(
p
)
.Aswereducethevalueof
X
,this
weight remains unchanged until
X
passes below
Y
, at which point it changes discontinuously
to
(
q
)
. Thus, the weight assigned to the event depends not only on probabilities, but also on
the
ranking
of the event according to the size of the payo
ff
.
The current paper devises and implements a simple and direct approach to measuring the
change in probability weights resulting from a change in payo
ff
ranks. Our method is entirely
non-parametric in the sense that it requires no maintained assumptions concerning functional
forms, either for utility and risk aversion, or for probability weighting. An essential feature of
our method is that it involves lotteries with three outcomes. To understand why the presence
of a third outcome facilitates a sharp and powerful test of the premise, consider equation (
1
).
For any small increase (
m
)inthevalueof
Y
,thereisasmall
equalizing reduction
(
k
)inthe
value of
Z
that leaves the decisionmaker indi
ff
erent. This equalizing reduction measures the
marginal rate of substitution between
Y
and
Z
,capturingrelativeprobabilityweights.
Both EU theory and PT imply that
the magnitude of the equalizing reduction is entirely
3
Electronic copy available at: https://ssrn.com/abstract=3350196
independent of the value of
X
,regardlessoffunctionalforms
. The same is true for CPT,
provided
X
remains within two ranges that we empirically examine,
X>Y
+
m
and
Y>X>Z
(as well as within the range
Z>X
). The reason is that, under CPT, the marginal rate of
substitution between
Y
and
Z
, written
MRS
YZ
,dependsonlyonthevalues,probabilities,and
ranks of
Y
and
Z
, none of which change. However, as the value of
X
crosses from one of these
ranges into the other, the ranking of
Y
changes, which causes the probability weight on
Y
to
change, while the other factors that determine
MRS
YZ
(the probabilities and values of
Y
and
Z
,andtherankof
Z
)remainfixed.Asaresult,
MRS
YZ
changes discontinuously, producing a
discontinuous change in the equalizing reduction. Critically, we show (for small changes) that
the percentage change in the equalizing reduction when
X
passes through
Y
precisely measures
the percentage change in the probability weight applied to
Y
resulting from the change in
Y
’s
rank.
Thus our strategy is to quantify the extent of rank dependence in probability weights by
eliciting equalizing reductions for
X>Y
+
m
and
X
2
(
Z, Y
)
.
Subjects in our experiment perform decision tasks that reveal their equalizing reductions
for three-outcome lotteries of the type described above. We find no evidence that probability
weights are even modestly sensitive to the ranking of outcomes. The actual percentage change
in the equalizing reductions, and hence probability weights, ranges from +3% to -3%, and in
no case can we reject the hypothesis of rank-independence. Our estimates rule out changes in
probability weights larger than 7% as ranks change with 95% confidence.
It follows from these results either that probability weighting is not rank-dependent, in
which case CPT is predicated on a false assumption, or that (contrary to other estimates) the
weighting function is nearly linear, in which case CPT does not di
ff
er from PT. To distinguish
between these possibilities, we devise and implement a non-parametric test of the hypothesis
that the probability weighting function is linear. The test exploits responses in the same tasks
to variations in probabilities, holding ranks fixed. Using this alternative source of variation, we
find evidence of substantial non-linearities. For example, responses to changes in probabilities
(with fixed ranks) imply that the average slope of the probability weighting function is roughly
4
Electronic copy available at: https://ssrn.com/abstract=3350196
19% lower on the subinterval of probabilities
[0
.
4
,
0
.
7]
than on the subinterval
[0
.
7
,
0
.
9]
,despite
the fact that – under the maintained hypothesis of rank dependence – the absence of responses to
change in ranks (with fixed probabilities) implies a constant slope for the probability weighting
function over the subinterval
[0
.
4
,
0
.
9]
.Adoptingstandardfunctionalassumptions,wethen
show that the estimated degree of curvature di
ff
ers sharply depending on whether one draws
inferences from responses to variations in payo
ff
ranks or variations in probabilities. Using
parametric models estimated based on responses to variations in probabilities with fixed ranks,
we predict the degree to which equalizing reductions should change in response to rank changes
under the assumption of rank dependence. We also perform these calculations using data
from conventional CPT elicitation tasks. In all cases, the predicted changes are an order of
magnitude greater than the observed changes, and the confidence intervals are non-overlapping.
For example, the conventional CPT calibration implies that increases in
X
that change the
rankings of
X
and
Y
in our experiment should change the equalizing reductions by -22% to -46%,
even though the actual change is negligible. Thus, the degree of curvature in the probability
weighting function implied by responses to variations in probabilities is highly inconsistent with
the constancy of equalizing reductions except under the hypothesis that probability weights are
rank-independent. Indeed, parametric estimates show that the PT formulation of probability
weighting accounts for the data on equalizing reductions more successfully than the CPT or
EU (linear) formulations.
Similar patterns are also apparent at the individual level, with a preponderance of subjects
exhibiting virtually no rank dependence for their probability weights, despite responding to
changes in probabilities in ways that imply substantial curvature of their probability weighting
functions, and hence substantial rank dependence within the CPT framework. The results are
robust with respect to a variety of alternative analytic procedures, such as using only between-
subject variation and eliminating potentially confused subjects. We also demonstrate that
our methods are robust with respect to alternative assumptions about reference points. En-
dogenizing reference points (as in
Bell
,
1985
;
Loomes and Sugden
,
1986
)changesnothingof
5
Electronic copy available at: https://ssrn.com/abstract=3350196
substance. Significantly, even with linear probability weighting, models with reference distri-
butions (
Koszegi and Rabin
,
2006
,
2007
)havesimilarpredictionsforequalizingreductions,and
hence we falsify them as well.
Our experimental design elicits equalizing reductions through choices over lotteries with
asinglecommonoutcome,
X
.Apairofearlypapersinthisarearaisedthepossibilitythat
subjects may employ a heuristic that involves the cancellation of common outcomes (
Wu
,
1994
;
Weber and Kirsner
,
1997
).
6
Under that ancillary hypothesis, our method would produce
spurious evidence of rank independence. We address this possibility by examining a similar
decision setting in which no cancellation is possible: we add
m
to
X
instead of to
Y
,and
reduce
both
Y
and
Z
by
k
. CPT rank dependence predicts discontinuities in
k
of
opposite signs
as
X
passes from
X>Y >Z
to
Y>X
0
>Z
to
Y>Z>X
00
.
7
For this modified decision task,
we again find no evidence of CPT rank dependence, clearly refuting the cancellation hypothesis
as a rationale for our results.
It is worth emphasizing that the stunning failure of CPT to account for our data is not
ameretechnicalshortcoming. Ourtestfocusesonafirst-orderimplicationofthetheory–
indeed, it isolates the critical feature that distinguishes CPT from PT. To put the matter
starkly, if equalizing reductions in three-outcome lotteries are not rank-dependent, then neither
are probability weights, and the CPT agenda is potentially on the wrong track.
What type of model should behavioral economists consider in place of CPT? One possibil-
ity is that PT is correct, and that people actually exhibit the implied violations of first-order
stochastic dominance. We test this possibility with a third experiment eliciting certainty equiv-
alents for three-outcome lotteries that pay
X
+
"
with probability
p/
2
,
X
"
with probability
p/
2
,and
Y
with probability
1
p
.Weincludethecaseof
"
=0
, which reduces to a two-outcome
6
Weber and Kirsner
(
1997
) provide evidence from certainty equivalent tasks where no cancellation is possible.
They find more support for models of rank dependence when comparing certainty equivalents for lotteries than
when comparing choices between the lotteries themselves. We thank an anonymous referee for drawing our
attention to this work and inspiring this modification. Our ‘split-event’ experiments discussed in section 5 also
explore the forces of rank dependence without the potential confound of cancellation.
7
We did not design the main portion of our investigation around these types of decision tasks because EU,
PT, and CPT all imply that the associated value of
k
should vary with
X
even when ranks do not change. This
variation complicates the task of reliably measuring the change in
k
when ranks do change.
6
Electronic copy available at: https://ssrn.com/abstract=3350196
lottery. We choose the parameters so that standard formulations of PT predict a sizable and
discontinuous drop in the certainty equivalent at
"
=0
. In contrast, CPT implies continuity.
Contrary to both predictions, we find a discontinuous
increase
in the certainty equivalent at
"
=0
. This behavior implies violations of dominance, but not the type PT predicts.
Agoodtheoryofchoiceunderuncertaintywouldthereforehavetoaccountforthreepatterns:
(1) robust evidence of probability weighting based on behavioral responses to variations in
probabilities, (2) the absence of rank dependence in equalizing reductions, and (3) the sharp
drop in certainty equivalents that results from splitting an event. EU is inconsistent with (1)
and (3). In light of (1), CPT is inconsistent with (2) and (3), and PT is inconsistent with (3).
We hypothesize that the observed behavior results from a combination of standard PT and a
form of complexity aversion: people may prefer lotteries with fewer outcomes because they are
easier to understand. One can think of the well-known
certainty e
ff
ect
as a special case of this
more general phenomenon.
Readers sympathetic to the hypothesis of rank dependence may wonder whether an en-
hanced CPT model with additional degrees of freedom might account for our data. Our analysis
rules out the possibility of achieving that objective through alternative assumptions concerning
the location of reference points, or by invoking the heuristic cancellation of common outcomes.
However, other novel hypotheses may bear investigation.
The current paper is most closely related to a handful of studies that aim to test the
axiomatic foundations of rank-dependent models (
Wu
,
1994
;
Wakker, Erev and Weber
,
1994
;
Fennema and Wakker
,
1996
;
Weber and Kirsner
,
1997
;
Birnbaum
,
2008
). Unlike our approach,
the methods used in these papers do not yield estimates of the degree to which probability
weights depend on payo
ff
ranks (non-parametric or otherwise), and the conclusions the authors
draw from them do not necessarily follow in settings with noisy choices; see Section 2.3 and
Online Appendix
B
for details.
8
8
An alternate strand of literature in psychology tests other CPT implications apart from rank dependence,
such as adherence to stochastic dominance, consistency of behavior across di
ff
erent ranges of probabilities,
separate weighting of gains and losses, and invariance to lottery description, also showing deviations (for a
broad review of these exercises see
Birnbaum
,
2008
).
7
Electronic copy available at: https://ssrn.com/abstract=3350196
Aside from the aforementioned studies, the assumption of rank-dependent probability
weighting has been the subject of surprisingly little formal scrutiny given its central role in
the leading behavioral theory of decisionmaking under uncertainty, as well as in recent appli-
cations of the theory.
9
The literature has focused instead on identifying the shapes of CPT
functions and associated parameter values based on choices involving binary lotteries (
Tversky
and Kahneman
,
1992
;
Tversky and Fox
,
1995
;
Wu and Gonzalez
,
1996
;
Gonzalez and Wu
,
1999
;
Abdellaoui
,
2000
;
Bleichrodt and Pinto
,
2000
;
Booij and van de Kuilen
,
2009
;
Booij, van Praag
and van de Kuilen
,
2010
;
Tanaka, Camerer and Nguyen
,
2010
). In cases where the experimental
tasks encompass an appropriate range of binary lotteries, one can devise and implement tests
of rank dependence, conditional on maintained assumptions about functional forms. Unfortu-
nately, an incorrect functional specification can manifest as spurious rank dependence. To our
knowledge, in cases where such data are available, no formal test of rank dependence has been
performed.
10
Defenses of rank dependence, such as the discussion in
Diecidue and Wakker
(
2001
), are instead typically based on intuitive arguments and/or point to findings concerning
the psychology of decisionmaking that arguably resonate with the premise (
Lopes
,
1984
;
Lopes
and Oden
,
1999
;
Weber
,
1994
).
9
Barseghyan, Molinari, O’Donoghue and Teitelbaum
(
2015
)investigatechoicesinvolvingarangeofinsurance
products. They demonstrate that the bracketing of risks – for example, whether people consider home and
automobile insurance together or separately – a
ff
ects the implications of probability weighting because it changes
the ranking of outcomes.
Epper and Fehr-Duda
(
2015
)examinethedatafrom
Andreoni and Sprenger
(
2012
)on
intertemporal decisionmaking under various risk conditions, which exhibits deviations from discounted expected
utility. They argue that CPT can rationalize an apparent choice anomaly if one frames two independent binary
intertemporal lotteries as a single lottery with four possible outcomes. This alternative framing delivers the
desired prediction because it alters the rankings of the four outcomes.
Barberis, Mukherjee and Wang
(
2016
)
examine historical monthly returns at the stock level for a five year window and link the CPT value of the stock’s
history to future returns, demonstrating a significant negative correlation. The interpretation for the negative
relation is that investors overvalue positively skewed, lottery-like stocks. Given 60 equi-probabable monthly
return events, PT would equally overweight all outcomes, giving no disproportionate value for skewness. CPT,
on the other hand, allows the highest ranked outcomes to receive higher proportionate weight.
Barberis et
al.
(
2016
) show that CPT substantially outperforms EU in predicting future returns. Given that that the PT
formulation (ignoring the reference point) would be collinear with the EU formulation, rank dependence would
seem critical for delivering this result.
10
As we explain in Online Appendix
A
,thedatain
Tversky and Kahneman
(
1992
)lendthemselvestosuch
tests. We show that the data from
Tversky and Kahneman
(
1992
) could be interpreted as consistent with rank
dependence. However, as noted in the Online Appendix, that finding hinges on the validity of their functional
form assumptions. We show that depending on the assumptions for the shape of utility, probability weighting
for a given chance of receiving an outcome can either appear to be rank-dependent or not.
8
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The paper proceeds as follows. Section
2
outlines the pertinent implications of CPT and
related theories. Section
3
describes our experimental design, while section
4
presents our main
results, and section 5 describes various robustness checks. Section
6
discusses implications,
including alternative theories and tests thereof. Section
7
concludes.
2TheoreticalConsiderations
Let
L
=(
{
p, q,
1
p
q
}
,
{
X, Y, Z
}
)
represent a lottery with three potential outcomes,
X, Y
,
and
Z
, played with corresponding probabilities
p
,
q
,and
1
p
q
, with
p, q
0
and
1
p
q
1
.
EU, PT, and CPT all assume that preferences over such lotteries have the following separable
form:
U
(
L
)=
w
X
u
(
X
)+
w
Y
u
(
Y
)+
w
Z
u
(
Z
)
,
where
w
i
represents the decision weight for outcome
i
.
11
Under EU and PT,
w
i
is a fixed
number that depends only on the probability of event
i
,andnotontheranksof
X
,
Y
,or
Z
.
Under CPT,
w
i
depends on the probabilities of the three events and the
ranks
of the payo
ff
s.
For PT and CPT, our notation suppresses the dependence of
u
(
·
)
on the reference point, which
for simplicity we take as fixed and assume for the moment to be less than
X, Y
,and
Z
.We
address alternative assumptions about reference dependence in section 2.2.
Our analysis employs the concept of an
equalizing reduction
,definedasthevalue
of
k
that delivers indi
ff
erence between the lottery
L
and a modified lottery
L
e
=
(
{
p, q,
1
p
q
}
,
{
X, Y
+
m, Z
k
}
)
, where
m
is a (small) fixed number. Intuitively, the
equalizing reduction approximates the marginal rate of substitution between the payo
ff
s
Y
and
Z
(
MRS
YZ
)
.
EU and PT imply that
w
Y
and
w
Z
,andtherefore
MRS
YZ
for fixed values of
Y
and
Z
,arecompletelyindependentof
X
. CPT shares this implication as long as variations in
X
do not change the payo
ff
ranks. However, as
X
crosses
Y
,therankof
Y
changes, while all
11
Our application of PT to these three-outcome lotteries corresponds to the extension of PT provided by, for
example,
Camerer and Ho
(
1994
)and
Fennema and Wakker
(
1997
).
9
Electronic copy available at: https://ssrn.com/abstract=3350196
the other factors that determine
w
Y
and
w
Z
under CPT (the probabilities of
Y
and
Z
,and
the rank of
Z
) remain fixed. To the extent rank dependence is quantitatively important, CPT
therefore implies that
MRS
YZ
changes discontinuously at
X
=
Y
.
The preceding intuition suggests an empirical strategy for evaluating the importance of rank
dependence. Assuming
m
and
k
are small, so that the payo
ff
ranks are the same for
L
and
L
e
,
then under all three theories we have
w
Y
u
(
Y
)+
w
Z
u
(
Z
)=
w
Y
u
(
Y
+
m
)+
w
Z
u
(
Z
k
)
(2)
or alternatively
k
=
Z
u
1
u
(
Z
)+
w
Y
W
Z
(
u
(
Y
)
U
(
Y
+
m
))
.
(3)
Suppose we assess
k
for various values of
X
. Under EU and PT,
w
Y
W
Z
is a fixed number (as are
Z
,
Y
,and
m
), so a graph of
k
against
X
should be a flat line. Under CPT,
w
Y
W
Z
is fixed as long
as the payo
ff
ranking is preserved, but it changes discontinously when the value of
X
passes
through the value of
Y
or
Z
. Therefore, a graph of
k
against
X
should exhibit three flat line
segments with discontinuities at
X
=
Y
and
X
=
Z
.
12
The strength of our approach is that it yields more than a
qualitative
way to gauge the
importance of rank dependence – it also provides a
quantitative
,nonparametricestimateofthe
change in relative decision weights that results from a change in payo
ff
ranks. Say we obtain
k
using the value
X
,and
k
using the value
X
.Defining
log(
a
)=log(
a
)
log(
a
)
(for the
generic variable
a
), we have:
13
Proposition 1
:Supposethereferencepointisfixed,thatdecisionweightsarefixedfora
given payo
ff
ranking, and that
u
is continuously di
ff
erentiable at
Y
and
Z
.
14
Consider any
X
and
X
distinct from
Y
and
Z
. Then
12
Technically, the discontinuities occur at
Y
and
Z
in the limit as
m
goes to zero.
13
As noted by one of our referees, it is relatively straightforward to dispense with the assumption that
u
is
di
ff
erentiable. Continuity and monotonicity of
u
(
·
)
are su
ffi
cient for the existence of positive right-derivatives,
which cancel out in the limit.
14
The continuous di
ff
erentiability requirement rules out cases in which
Y
or
Z
coincides with the reference
point. The proof extends to these cases but requires attention to some additional technical details.
10
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lim
m
!
0
log(
k
)=
log
w
Y
w
Z
Proof
:SeeAppendix.
Proposition 1 tells us that the percentage change in
k
(from
k
to
k
)
provides a quantitative
estimate of the percentage change in the relative decision weights,
w
Y
w
Z
(from
w
Y
w
Z
to
w
Y
w
Z
)resulting
from the change in
X
(from
X
to
X
)
.
To drive the implications of this point home, suppose in particular that we choose
X
and
X
such that
X>Y
+
m>Z
and
Y>X
>Z
. Then, under CPT, we have
w
Y
=
(
p
+
q
)
(
p
)
and
w
Z
=1
(
p
+
q
)
, while
w
Y
=
(
q
)
and
w
Z
=1
(
p
+
q
)
. It follows that
log
(
k
)
log
(
(
p
+
q
)
(
p
))
log
(
(
q
))
Thus, for the maintained hypothesis of CPT, in this special case the percentage change in
k
provides a quantitative estimate of the percentage change in the probability weight assigned to
payo
ff
Y
when the value of
X
passes from above
Y
to below
Y
.
15
Accordingly, the first step in our analysis is to measure changes in the relative decision
weights,
!
Y
!
Z
, associated with reversals in the ranks of
X
and
Y
.Ifwefindthatthesechanges
15
It is natural to wonder whether our central insight would apply to models in which probability weighting
functions include linear segments, so that
log
(
(
p
+
q
)
(
p
))
log
(
(
q
)) = 0
over a given range. One
prominent example is the neo-additive model of
Chateauneuf, Eichberger and Grant
(
2007
). Under the neo-
additive model with objective probabilities, decision weights for cumulative probabilities away from 0 and 1
are linear as in expected utility, but extra weight is given to the best and worst outcome in a lottery. For
Z<X
<Y
,theneo-additiveutilityis
U
=
u
(
Z
)+(1
)[
pu
(
X
)+
qu
(
Y
)+(1
p
q
)
u
(
Z
)] +
u
(
Y
)
,
where
and
represent the additional weight on the worst and best outcomes, respectively. In contrast, for
Z<Y <
X
,
U
=
u
(
Z
)+(1
)
pu
(
X
)+
qu
(
Y
)+(1
p
q
)
u
(
Z
)
+
u
(
X
)
.
Because outcome
Y
is no longer the best outcome, its weight changes discontinuously. It is straightforward to
show that
log
(
k
)
log
(1
)
q
+(1
)
q
,
so a discontinuity in equalizing reductions is predicted, and the log change in equalizing reduction again closely
approximates the change in decision weight applied to outcome
Y
.
11
Electronic copy available at: https://ssrn.com/abstract=3350196
are robustly close to zero, we can conclude that either the weights are not rank-dependent,
in which case CPT is predicated on a false assumption, or the probability weighting function
exhibits no meaningful non-linearities, in which case CPT does not di
ff
er from PT.
To be precise, suppose that for specified values of the payo
ff
s,
Y
and
Z
,andoftheproba-
bilities,
p
,
q
,and
1
p
q
,wefind
4
log
(
k
)
0
.Treatingrankdependenceasamaintained
hypothesis, we have
4
!
Z
=0
by construction, so if we find no change in
k
,wemustalso
have
4
!
Y
=[
(
p
+
q
)
(
p
)]
(
q
)=0
. Using this equation along with the assumption that
(0) = 0
(impossible events are ignored), we obtain
(
p
+
q
)
(
p
)
q
=
(
q
)
(0)
q
.
(4)
In other words,
4
log
(
k
)=0
implies that the average slope of
(
·
)
is the same over the intervals
[0
,q
]
and
[
p, p
+
q
]
.Taking
q
small, we see that, if this condition holds for all
p
2
[0
,
1
q
]
,then
(
·
)
must be linear. In our experiment, we focus on values of
p
and
q
that allow us to target the
portions of the unit interval for which previous analyses of probability weighting have found
pronounced non-linearities. Using our methods, one could obviously consider additional values
of
p
and
q
, e
ff
ectively blanketing the interval with these tests, thereby ruling out non-linearities
more comprehensively under the maintained hypothesis of rank dependence.
16
Upon determining that
4
log
(
k
)
is in fact robustly close to zero, we proceed to the second
step of our analysis, asking whether this result follows from the absence of rank dependence, or
from the absence of meaningful non-linearities in the probability weighting function. Our strat-
egy is to draw inferences about the shape of the probability weighting function from responses
in the same tasks
to variations in probabilities, holding ranks fixed. We reject rank-dependent
probability weighting, and hence CPT, if Steps 1 and 2 yield inconsistent conclusions concerning
the shape of
(
·
)
.Forexample,ifweruleoutmeaningfulnon-linearitiesinStep1(conditional
16
Apossibleobjectionisthat,inthree-outcomelotteries,itisalwaysthecasethat
p<
1
q
, which means we
cannot rule out discontinuities at
(1)
using equalizing reductions for
X>Y >Z
and
Y>X
>Z
.However,
we can easily solve that problem by also evaluating the change in
k
when
X
falls below
Z
.Althoughwedonot
perform such tests as part of our main analyses, we include them along with other robustness checks in Section
5.3.
12
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on maintaining rank dependence) but find them in Step 2, then CPT cannot account for the
pattern of equalizing reductions. However, the same pattern is consistent with PT, because the
hypothesized finding for Step 1 rules out non-linearities only if one assumes rank dependence.
For Step 2, we proceed as follows. Using the measured equalizing reductions for the same
tasks, we can compute the following quantities for multiple values of
p
and/or
q
:
(
p, q
)
q
1
p
q
m
k
and
(
p, q
)
q
1
p
q
m
k
Noting that
k
m
!
Y
!
Z
⌘⇣
u
0
(
Y
)
u
0
(
Z
)
,wehave
(
p, q
)
✓
!
Z
1
p
q
◆✓
q
!
Y
C
and
(
p, q
)
✓
!
Z
1
p
q
◆✓
q
!
Y
C,
where
C
=
u
0
(
Z
)
u
0
(
Y
)
is a constant in our experiment because we hold
Y
and
Z
fixed. If the decision
weights are proportional to the probabilities, then the bracketed terms are identically unity.
Thus, if
(
p, q
)
or
(
p, q
)
varies with
p
or
q
,theprobabilityweightingfunctioncannotbelinear.
More specifically, under CPT, we can rewrite these approximations as follows:
(
p, q
)
(1)
(
p
+
q
)
1
p
q
◆✓
q
(
p
+
q
)
(
p
)
C
and
(
p, q
)
(1)
(
p
+
q
)
1
p
q
◆✓
q
(
q
)
C.
Suppose we observe two values of
p
,callthem
p
0
>p
00
, for which
(
p
0
,q
)
6
=
(
p
00
,q
)
. Then we
can conclude that that
(
·
)
is not linear throughout the interval
[1
p
0
q,
1]
.Alternatively,
suppose we observe
(
p
0
,q
)
6
=
(
p
00
,q
)
.
Then we can conclude that
(
·
)
is not linear throughout
the interval
[min
{
1
p
0
q, p
00
}
,
1]
.Analogousstatementsholdfortheprobability
q
.
2.1 Simulated Equalizing Reductions under CPT Decisionmaking
In this section, we examine the particular lotteries studied in our experiment and show that
changes in payo
ff
ranks yield large changes in probability weights under standard parameter-
izations of CPT. We also demonstrate that the percentage change in the equalizing reduction
13
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