Direct Tests of Cumulative Prospect Theory
∗
B. Douglas Bernheim
†
Stanford University and NBER
Charles Sprenger
‡
UC San Diego
First Draft: December 1, 2014
This Version: January 15, 2019
Abstract
Cumulative Prospect Theory (CPT), the leading behavioral account of decisionmaking
under uncertainty, assumes that the probability weight applied to a given outcome depends
on its ranking. This assumption is needed to avoid the violations of dominance implied
by Prospect Theory (PT). We devise a simple and direct non-parametric method for
measuring the change in relative probability weights resulting from a change in payoff
ranks. We find no evidence that these weights are even modestly sensitive to ranks. The
estimated changes in relative weights range from +3% to -3%, and in no case can we
reject the hypothesis of rank-independence. Our estimates rule out changes in relative
probability weights larger than a few percent as ranks change with 95% confidence. In
contrast, conventional calibrations of CPT preferences for the same subjects imply that
probability weights should change by 20% to 40%. Models with reference distributions
(notably Koszegi and Rabin, 2006) have similar implications, and hence we falsify them as
well. 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 effect.
JEL classification:
D81, D90
Keywords
: Prospect Theory, Cumulative Prospect Theory, Rank Dependence, Certainty Equiv-
alents.
∗
We are grateful to Ted O’Donoghue, Colin Camerer, Nick Barberis, Kota Saito, and seminar participants
at Cornell, Caltech, and UC Santa Barbara for helpful and thoughtful discussions. 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.
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1 Introduction
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, as well as the simulta-
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
difficulties 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
; for our current purpose, we
will leave other events and payoffs unspecified. Now imagine a second lottery, identical to the
first, except that it splits the aforementioned event, paying
X
and
X
−
ε
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
, for a lottery that pays
X
with probability
p
and 0 with probability
1
−
p
, one typically finds an inverse
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
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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
ε
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 offered as a solution to the stochas-
tic dominance problem (Quiggin, 1982; Schmeidler, 1989), and was incorporated into a new
version of PT known as Cumulative Prospect Theory, henceforth CPT (Tversky and Kahne-
man, 1992). To understand intuitively how CPT resolves the issue, consider a lottery
L
with
three possible payoffs,
X > Y > Z
, occurring with probabilities
p
,
q
, and
1
−
p
−
q
. Another
description of the same lottery involves cumulative probabilities: it pays
Z
with probability 1,
adds
Y
−
Z
with probability
p
+
q
, and then incrementally adds
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 payoffs, while a weighting function,
π
(
·
)
, is applied to the cumulative probabilities:
U
(
L
) =
π
(1)
u
(
Z
|
r
) +
π
(
p
+
q
)[
u
(
Y
|
r
)
−
u
(
Z
|
r
)] +
π
(
p
)[
u
(
X
|
r
)
−
u
(
Y
|
r
)]
.
Normally this expression is rewritten in a form that attaches a weight to each outcome:
U
(
L
) =
π
(
p
)
u
(
X
|
r
) + [
π
(
p
+
q
)
−
π
(
p
)]
u
(
Y
|
r
) + [
π
(1)
−
π
(
p
+
q
)]
u
(
Z
|
r
)
.
(1)
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
ad hoc
“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.
2
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Now imagine, as before, a second lottery, identical to the first, except that it splits the event
yielding the payoff
X
into two events paying
X
and
X
−
ε
, each with probability
p/
2
. In that
case, the term
π
(
p/
2)
u
(
X
|
r
) + [
π
(
p
)
−
π
(
p/
2)]
u
(
X
−
ε
|
r
)
replaces the term
π
(
p
)
u
(
X
|
r
)
. Notice
that the
total
weight assigned to the two events is still
π
(
p
)
, the same as for the original lottery.
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 payoff
X
as we change its value.
Initially
X
exceeds
Y
, and its weight is
π
(
p
)
. As we reduce the value of
X
, the weight remains
unchanged until
X
passes below
Y
, at which point it changes discontinuously to
π
(
p
+
q
)
−
π
(
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 payoff.
The current paper devises and implements a simple and direct approach to measuring the
change in probability weights resulting from a change in payoff 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
) in the value of
Y
, there is a small
equalizing reduction
(
k
) in the
value of
Z
that leaves the decisionmaker indifferent. This equalizing reduction measures the
marginal rate of substitution between
Y
and
Z
, capturing relative probability weights.
Both EU theory and PT imply that
the magnitude of the equalizing reduction is entirely
independent of the value of
X
, regardless of functional forms
. The same is true for CPT,
provided
X
remains within one of the following three ranges:
X > Y
+
m
,
Y > X > Z
, or
Z
−
k > X
. However, as the value of
X
crosses from one of these ranges into another, the
3
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ranking of the payoffs changes, which causes the probability weights to change. This change
in probability weight alters the marginal rate of substitution between
Y
and
Z
, and thus the
equalizing reduction. Critically, we show (for small changes) that
the percentage change in the
equalizing reduction precisely measures the percentage change in the probability weights applied
to the outcomes
. Thus our strategy is to quantify the extent of rank-dependence in probability
weights by eliciting equalizing reductions for
X > Y
+
m
and
X
∈
(
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.
In light of these findings, it is important to confirm that there is nothing unusual about our
subjects, and in particular that they exhibit the standard choice patterns usually associated
with CPT. Accordingly, following previous studies (Tversky and Kahneman, 1992; Tversky and
Fox, 1995), we also elicit subjects’ certainty equivalents for a collection of binary lotteries,
which we use to derive their CPT parameters. This method reproduces standard findings
regarding probability weights: subjects apparently attach disproportionately high weight to
low probabilities and disproportionately low weight to high probabilities, so the
π
(
·
)
curve has
the standard inverse
S
-shape. Moreover, our estimates of the curvature parameters correspond
closely to those reported in the prior literature.
Many economists have adopted these types of CPT calibrations for the purpose of studying
applied problems; see, for example, the discussion of asset pricing in Barberis, Mukherjee and
Wang (2016). Such work proceeds from the assumption that valuations of binary lotteries
reveal the values of "deep" CPT preference parameters that are stable across a wide range of
contexts. It is therefore of interest to treat these calibrations as benchmarks, and to compare
their implications for rank-induced changes in relative probability weights to the estimates
4
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obtained through our methods. According to the calibrations, 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. Critically, this contrast does not reflect some
inherent difference between the standard Prospect Theory elicitation tasks and our equalizing
reduction tasks. It is also possible to estimate the implied curvature of the probability weighting
function directly from the latter design using responses to variations in probabilities across
tasks. Precisely the same implications follow: the implied degree of curvature in the probability
weighting function is highly inconsistent with the constancy of the equalizing reduction except
under the hypothesis that probability weights are rank-independent.
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.
Endogenizing reference points (as in Bell, 1985; Loomes and Sugden, 1986) changes nothing of
substance. Models with reference distributions (Koszegi and Rabin, 2006, 2007) have similar
predictions for equalizing reductions, and hence we falsify them as well.
Our experimental design elicits equalizing reductions through choices over lotteries with
a single common outcome,
X
. A pair of early papers in this area raised the possibility that
subjects may employ a heuristic that involves the cancellation of common outcomes (Wu,
1994; Weber and Kirsner, 1997).
5
Under that ancillary hypothesis, our method would produce
spurious evidence of rank-independence. We address this possibility by examining a similar
5
Weber and Kirsner (1997) provide evidence from certainty equivalents tasks where no cancelation 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 cancelation.
5
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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
′
> Z
to
Y > Z > X
′′
.
6
For this modified decision task,
we again find no evidence of CPT rank dependence, clearly refuting the cancelation hypothesis
as a rationale for our results.
It is worth emphasizing that the stunning failure of CPT to account for our data is not
a mere technical shortcoming. Our test focuses on a first-order implication of the theory –
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 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
+
ε
with probability
p/
2
,
X
−
ε
with probability
p/
2
, and
Y
with probability
1
−
p
. We include the case of
ε
= 0
, which reduces to a two-outcome
lottery. We choose the parameters so that standard formulations of PT predict a sizable and
discontinuous drop in the certainty equivalent at
ε
= 0
. In contrast, CPT implies continuity.
Contrary to both predictions, we find a discontinuous
increase
in the certainty equivalent at
ε
= 0
. This behavior implies violations of dominance, but not the type PT predicts.
A good theory of choice under uncertainty would therefore have to account for three patterns:
(1) the inverse
S
-shaped certainty equivalent profile, (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), while 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
6
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
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with fewer outcomes because they are easier to understand. One can think of the well-known
certainty effect
as a special case of this more general phenomenon.
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 payoff 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
Appendix B for details.
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 applica-
tions of the theory.
7
Fehr-Duda and Epper’s (2012) recent review of the literature acknowledges
this point.
8
The literature has focused instead on identifying the shapes of CPT functions and
associated parameter values based on choices involving binary lotteries (Tversky and Kahne-
man, 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
7
Barseghyan, Molinari, O’Donoghue and Teitelbaum (2015) investigate choices involving a range of insurance
products. They demonstrate that the bracketing of risks – for example, whether people consider home and
automobile insurance together or separately – affects the implications of probability weighting because it changes
the ranking of outcomes. Epper and Fehr-Duda (2015) examine the data from 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 et al. (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.
8
They state “It is our impression that this feature of rank-dependent utility has often not been properly
understood. For example, an inverse S-shaped probability weighting function does not imply that all small
probabilities are overweighted. Whether a small probability is overweighted or underweighted depends on the
rank of the outcome to which it is attached" (p. 571).
7
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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. Unfortunately, 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.
9
An additional strand of literature in psychology tests further implications, such as adherence
to stochastic dominance and invariance to lottery description, also showing deviations (for dis-
cussion, see, e.g., Birnbaum, 2008).
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).
The paper proceeds as follows. Section 2 outlines the pertinent implications of CPT and
related theories. Section 3 elaborates our experimental design, while section 4 presents our main
results and robustness checks. Section 5 discusses implications, including alternative theories
and tests thereof. Section 6 concludes.
2 Theoretical Considerations
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
.
EUT, PT, and CPT all assume that preferences over such lotteries have the following separable
9
As we explain in Appendix A, the data in Tversky and Kahneman (1992) lend themselves to such tests. We
show that the data from Tversky and Kahneman (1992) could be interpreted as consistent with rank dependence.
However, as noted in the 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.
10
One notable phenomenon discussed by Birnbaum (2008) is a sensitivity of lottery valuations to descrip-
tion of events. Describing two lotteries as
(
{
0
.
85
,
0
.
10
,
0
.
05
}
;
{
100
,
50
,
50
}
)
and
(
{
0
.
85
,
0
.
10
,
0
.
05
}
;
{
100
,
100
,
7
}
)
leads to qualitatively different hypothetical binary choice patterns than
(
{
0
.
85
,
0
.
15
}
;
{
100
,
50
}
)
and
(
{
0
.
95
,
0
.
05
}
;
{
100
,
7
}
)
. This failure of ‘coalescing’ is part of a number of violations reviewed by Birnbaum
(2008) and is clearly at odds with CPT. Our split-event design in section 5 carries some similarity to this work
as we do vary the presentation of lotteries between subjects and then split events. For example we elicit certainty
equivalents both for
(
{
0
.
3
,
0
.
3
,
0
.
4
}
;
{
30
,
30
,
20
}
)
and
(
{
0
.
6
,
0
.
4
}
;
{
30
,
20
}
)
. In our incentivized decisions, we do
not see the failure of coalescing noted by Birnbaum (2008) for hypothetical choice.
8
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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 EUT and PT,
w
i
is a fixed
number that depends only on the probability of event
i
, and not on the values of
X
,
Y
, or
Z
. Under CPT,
w
i
depends on the probabilities of the three events and the
ordering
of the
payoffs. For PT and CPT, our notation suppresses the dependence of
u
on the reference point,
which for simplicity we take as fixed and assume to be less than
X, Y
, and
Z
. We address
more sophisticated forms of reference dependence in section 2.2.
Our analysis employs the concept of an
equalizing reduction
, defined as the value
of
k
that delivers indifference 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 payoffs
Y
and
Z
(
MRS
Y Z
)
.
EUT and PT imply that
MRS
Y Z
is completely independent of
X
. CPT shares
this implication as long as variations in
X
do not change the payoff ranks. However, if the
ranks change, CPT implies that
MRS
Y Z
will change discontinuously.
The preceding intuition suggests a sharp qualitative test of EUT, PT, and CPT. Assuming
m
and
k
are small, so that the payoff 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 EUT 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 payoff ranking is preserved, but it changes discontinously when the value of
X
passes
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).
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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
test of the underlying
theories – it also provides a
quantitative
, nonparametric estimate of the change in relative
decision weights that results from a change in payoff 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
: Suppose the reference point is fixed, that decision weights are fixed for a
given payoff ranking, and that
u
is continuously differentiable at
Y
and
Z
.
14
Consider any
X
and
X
distinct from
Y
and
Z
. Then
lim
m
→
0
∆ log(
k
) = ∆ log
(
w
Y
w
Z
)
Proof
: See Appendix E.1.
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
=
π
(
p
+
q
)
−
π
(
p
)
and
w
Z
= 1
−
π
(
p
+
q
)
, while
w
Y
=
π
(
q
)
and
w
Z
= 1
−
π
(
p
+
q
)
. It follows that
∆
log
(
k
)
≈
log
(
π
(
p
+
q
)
−
π
(
p
))
−
log
(
π
(
q
))
Thus, for the maintained hypothesis of CPT, in this special case the percentage change in
k
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
differentiable. Continuity and monotonicity of
u
are sufficient for the existence of positive right-derivatives,
which cancel out in the limit.
14
The continuous differentiability 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.
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provides a quantitative estimate of the percentage change in the probability weight assigned to
payoff
Y
when the value of
X
passes from above
Y
to below
Y
.
15
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 payoff ranks yield large changes in probability weights under standard parameter-
izations of CPT. We also demonstrate that the percentage change in the equalizing reduction
approximates the percentage change in the probability weights to a high degree of accuracy
even when
m
represents a discrete payoff increment of non-trivial magnitude.
We focus on the parametric specification used in the original formulation of CPT (Tversky
and Kahneman, 1992),
16
which posited a probability weighting function,
π
(
p
) =
p
γ
/
(
p
γ
+ (1
−
p
)
γ
)
1
/γ
, a reference point of
r
= 0
, and a utility function
u
(
x
) =
x
α
for
x > r
= 0
. The
parameters identified by Tversky and Kahneman (1992) were
γ
= 0
.
61
and
α
= 0
.
88
.
Consider the lottery,
L
, with
{
X
,Y,Z
}
=
{
$30
,
$24
,
$18
}
and
{
p,q,
1
−
p
−
q
}
=
{
0
.
4
,
0
.
3
,
0
.
3
}
. Increase
Y
by
m
= $5
, from
$24
to
$29
. For the parameters
γ
= 0
.
61
and
α
= 0
.
88
, the equalizing reduction is
k
= 1
.
67
.
17
Now consider the lottery
L
′
with
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
(
π
(
p
+
q
)
−
π
(
p
))
−
log
(
π
(
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 utiility, but extra weight is given to the best and worst outcome in a lottery. For
Z <
X < Y
, the neo-additive utility is
U
=
γu
(
Z
) + (1
−
γ
−
λ
)
[
pu
(
X
) +
qu
(
Y
) + (1
−
p
−
q
)
u
(
Z
)
]
+
λu
(
Y
)
,
where
γ
and
λ
represent the additional weight on the worst and best outcomes, respectively. In contrast, for
Z < Y < X
,
U
=
γu
(
Z
) + (1
−
γ
−
λ
) [
pu
(
X
) +
qu
(
Y
) + (1
−
p
−
q
)
u
(
Z
)] +
λu
(
X
)
.
Because outcome
Y
is no longer the best outcome, its weight changes discontinuously. It is straightforward to
show that
∆
log
(
k
)
≈
log
(
(1
−
γ
−
λ
)
q
λ
+ (1
−
γ
−
λ
)
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
.
16
Tversky and Fox (1995) and Gonzalez and Wu (1999) employ a similar two parameter
π
(
p
)
function. See
Prelec (1998) for alternative
S
-shaped specifications.
17
Note that
Y
and
Z
are received with equal probability, so that a risk neutral decisionmaker would exhibit
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{
X,Y,Z
}
=
{
$23
,
$24
,
$18
}
and
{
p,q,
1
−
p
−
q
}
=
{
0
.
4
,
0
.
3
,
0
.
3
}
. For the same CPT pa-
rameters as above, the equalizing reduction for
m
= 5
is
k
= $3
.
22
.
18
Thus, a standard
parameterization of CPT implies a sharp discontinuity in equalizing reductions: moving from
X < Y
to
X
> Y
+
m
cuts the equalizing reduction roughly in half. The log difference in
equalizing reductions,
∆
log
(
k
) =
−
0
.
66
, closely approximates the change in probability weight
associated with the outcome
Y
, as
log
(
π
(0
.
7)
−
π
(0
.
3))
−
log
(
π
(0
.
3)) =
−
0
.
66
as well. In-
deed, the approximation remains quite close even when the utility function has much greater
curvature. For example, with
α
= 0
.
5
,
∆
log
(
k
) =
−
0
.
65
, and for
α
= 0
.
25
,
∆
log
(
k
) =
−
0
.
64
.
In Table 1, we provide additional simulations with the same values of
X,Y,Z
, and
m
as
above, but using three different values of
γ
, 0.4, 0.61, 0.8, as well as three different probability
vectors,
{
p,q,
1
−
p
−
q
}
=
{
0
.
6
,
0
.
3
,
0
.
1
}
,
{
0
.
4
,
0
.
3
,
0
.
3
}
, and
{
0
.
1
,
0
.
3
,
0
.
6
}
.
19
For the CPT
parameter values of Tversky and Kahneman (1992), the probability weight on payoff
Y
changes
by 29 to 66 percent as
X
passes through
Y
. Even with more modest curvature of the probability
weighting function (
γ
= 0
.
8
), the change in probability weight remains sizable. The effect is
largest for the third probability vector. In that case, shifting
X
from below
Y
to above
Y
changes the decision weight on
Y
by removing the relatively large weight associated with
the first 10% lump of probability and adding the relatively small weight associated with the
fourth 10% lump of probability. Critically, in all cases, the percentage change in the equalizing
reduction approximates the percentage change in the probability weight associated with payoff
Y
to a high degree of accuracy.
an equalizing reduction of
k
= $5
.
18
One again, note that a risk neutral decisionmaker would exhibit an equalizing reduction of
k
= $5
.
19
To demonstrate the dependence of discontinuities in equalizing reduction on the extent of probability weight-
ing, we hold
α
fixed at
0
.
88
throughout.
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Table 1: Cumulative Prospect Theory Simulated Equalizing Reductions
γ
= 0
.
4
γ
= 0
.
61
γ
= 0
.
8
{
p,q,
1
−
p
−
q
}
k k
∆
log
(
k
)
k k
∆
log
(
k
)
k k
∆
log
(
k
)
∆
log
(
w
Y
w
Z
)
=
log
(
π
(
p
+
q
)
−
π
(
p
)
π
(
q
)
)
∆
log
(
w
Y
w
Z
)
=
log
(
π
(
p
+
q
)
−
π
(
p
)
π
(
q
)
)
∆
log
(
w
Y
w
Z
)
=
log
(
π
(
p
+
q
)
−
π
(
p
)
π
(
q
)
)
{
0
.
6
,
0
.
3
,
0
.
1
}
1.97 1.33
-0.39
5.17 3.88
-0.29
9.21 7.84
-0.16
-0.39
-0.29
-0.17
{
0
.
4
,
0
.
3
,
0
.
3
}
1.61 0.53
-1.12
3.22 1.67
-0.66
4.29 3.13
-0.31
-1.12
-0.66
-0.32
{
0
.
1
,
0
.
3
,
0
.
6
}
1.45 0.40
-1.30
2.39 1.39
-0.55
2.60 2.08
-0.22
-1.30
-0.55
-0.22
Notes
: Dollar values for equalizing reductions in
Z
for increase in
Y
to
Y
+
m
.
k
calculated with
{
X
,Y,Z
}
=
{
$30
,
$24
,
$18
}
,
m
= $5
.
k
calculated with
{
X,Y,Z
}
=
{
$23
,
$24
,
$18
}
,
m
= $5
. CPT calculations with
u
(
x
) =
x
α
,α
= 0
.
88
; and
π
(
p
) =
p
γ
/
(
p
γ
+ (1
−
p
)
γ
)
1
/γ
with
γ
varying by column.
2.2 Reference Point Formulation and Alternative Models of Refer-
ence Dependence
Throughout the previous discussion, we assumed that the reference point is fixed and below all
potential payoffs. While this assumption is a reasonable starting point, one naturally wonders
whether our conclusions are robust with respect to other possibilities.
First consider the possibility that the reference point is exogenous but falls either (i) above
all payoffs, or (ii) between the lottery’s payoffs, which it segregates into gains and losses. Case
(ii) may seem particularly concerning because CPT applies probability weighting to gains and
losses separately. Notice, however, that Proposition 1 subsumes these possibilities because it
is proved for a specification with general decision weights. Because CPT still implies that the
weights change when the value of
X
passes through
Y
, precisely the same implications follow.
20
Notably, additional discontinuities in equalizing reductions emerge (for similar reasons) at the
point where the reference point passes the outcomes
Y
and
Z
(see Appendix C.1). Thus the
equalizing reduction approach offers not only a novel test of rank dependence, but could also
be used to test the hypothesis that gains and losses relative to the reference point are weighted
20
An additional complication arises for non-infinitessimal values of
m
, in that an increase from
Y
to
Y
+
m
could cross the reference point, or cause
Z
−
k
to cross the reference point. As shown in Appendix C.1, the
implications of CPT are nevertheless unchanged.
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separately.
Next consider the possibility that the reference point depends on the lottery’s payoffs, as in
Bell (1985) and Loomes and Sugden (1986). We will use
r
to denote a generic reference point
and
R
(
X,Y,Z
)
to denote the reference point for a lottery that yields payoffs
{
X,Y,Z
}
with
probabilities
{
p,q,
1
−
p
−
q
}
. (The reference point may also depend on the probabilities, but
we hold them constant, and consequently suppress those arguments for notational simplicity.)
Here we will focus on cases in which the reference point
r
coincides with neither
Y
nor
Z
.
Proposition 2
: Suppose decision weights are fixed for a given payoff ranking, that
u
(
x,r
)
is
continuously differentiable in neighborhoods of
(
Y,R
(
Y,Y,Z
))
and
(
Z,R
(
Y,Y,Z
))
, and that
R
is continuously differentiable in a neighborhood of
(
Y,Y,Z
)
. Consider any sequence
(
X
n
,
̄
X
n
)
→
(
Y,Y
)
such that
X
n
> Y >
̄
X
n
> Z
. Then
lim
n
→∞
lim
m
→
0
∆ log(
k
) = ∆ log
(
w
Y
w
Z
)
Proof
: See Appendix E.1.
Thus, even with an endogenous reference point, the discontinuity in the
X
-
k
schedule at
X
=
Y
still measures the percentage change in the relative decision weights on
Y
and
Z
. That
said, if the reference point depends on the payoff
X
, then the
X
-
k
schedule may no longer
be flat within intervals with fixed decision weights, a possibility that could in principle make
the size of any discontinuity more difficult to measure. However, as we will see, there is no
indication that this potential issue materializes in practice. On the contrary, the flatness of the
empirical
X
-
k
schedule eliminates any complications arising from the potential endogeneity of
the reference point.
With an endogenous reference point, an additional discontinuity in the equalizing reduction
emerges where
X
cross the reference point, even in the absence of probability weighting (see
Appendix C.2). Thus our approach offers a novel test of theories with endogenous reference
points, such as Disappointment Aversion.
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A final possibility worth considering is that subjects have reference point distributions, as in
Koszegi and Rabin (2006, 2007). We show in Appendix C.3 that this preference formulation also
yields discontinuities in the equalizing reduction as
X
passes through
Y
and
Z
, even without
probability weighting.
21
Furthermore, when we apply a calibrated model to our experimental
tasks, we find that the implied discontinuities are substantial, and that their signs are opposite
those implied by the CPT calibrations. Thus our approach also offers a novel and discerning
test of the Koszegi and Rabin (2006, 2007) framework.
2.3 Relation to Existing Tests of Cumulative Prospect Theory
As mentioned in the introduction, our work is most closely related to a handful of studies that
aim to test the axiomatic foundations of rank-dependent models (Wu, 1994; Wakker et al., 1994;
Fennema and Wakker, 1996; Weber and Kirsner, 1997; Birnbaum, 2008). A defining feature
of those models is that they assume the independence axiom holds on a limited domain. In
particular, we say that two lotteries are
comonotonic
if they induce the same payoff ranking over
states of nature. Under EUT, PT, and CPT, if two comonotonic lotteries yield the same payoff,
x
j
, in some state
j
, then a change in
x
j
that leaves the ranking intact should have no effect
on preferences between the lotteries. This property reflects an axiom known as
Comonotonic
Independence
(CI), which EUT, PT, and CPT all satisfy. Naturally, one can also ask whether
preferences between the lotteries are invariant with respect to changes in
x
j
that alter the
payoff ranking. This type of invariance follows from a property known as
Non-Comonotonic
Independence
(NCI), which EUT and PT satisfy, but CPT does not. Thus, evidence validating
both CI and NCI would point to either EUT or PT, and evidence favoring CI while challenging
NCI would point to CPT.
When interpreting laboratory evidence concerning conformance with choice axioms, it is
important to allow for the possibility that observed choices are somewhat noisy. As a result,
21
Masatlioglu and Raymond (2016); Barseghyan et al. (2015) note a tight connection between rank dependent
theories and the Koszegi and Rabin (2006, 2007) model. This work presages the results outlined in Appendix
C.3.
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even if a theory captures the essence of decisionmaking, one would expect to observe violations
of the axioms that characterize it. What then can one conclude from the frequency of violations?
Existing tests of rank dependence involve comparisons between the prevalence of violations for
different axioms. For instance, Wakker et al. (1994) attribute the differential between the
frequency of NCI violations and CI violations to rank dependence. Because they find little
difference in these frequencies, they conclude that rank dependence is not supported.
These types of frequency comparisons raise two difficulties, both stemming from the fact
that the results are difficult to interpret without a parametric model of noisy choice. First, the
premise of the approach – that violation frequencies are necessarily higher for invalid axioms
– is flawed. For reasonable models of noisy choice, noise-induced violations of choice axioms
are more likely to occur when the parameters of the tasks place the decisionmaker closer to
the point of indifference. Existing approaches provide no way to ensure that the “distance to
indifference” is held constant when comparing CI and NCI violations. It is therefore easy to
construct examples in which a “noisy” CPT decisionmaker violates CI just as frequently, or even
more frequently, than NCI. Second, even if one could control for “distance to indifference,” this
approach offers no basis for judging whether a given discrepancy between the frequencies of CI
and NCI violations is large or small relative to the implications of a reasonably parameterized
CPT model. For any given degree of rank dependence, one can construct simple examples
(with constant “distance to indifference”) in which the differential between violation frequencies
falls anywhere between zero and unity. See Appendix B for a description of the aforementioned
examples.
Because rank dependence is characterized by the restriction of the independence axiom to
comonotonic lotteries (Wakker et al., 1994), any valid test of the hypothesis is necessarily related
to the existing studies, and ours is no exception. However, our use of equalizing reductions
has no counterpart in the existing literature. Instead of counting violatons of CI and NCI,
we measure equalizing reductions separately for each lottery, and then compare them across
lotteries. Certainly, the constancy of the equalizing reduction over values of
X
that preserve
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