Direct Tests of Cumulative Prospect Theory - Harvard … · Direct Tests of Cumulative Prospect Theory ... In contrast to the EU formulations of von Neumann and Mor ... consider a

Direct Tests of Cumulative Prospect Theory⇤

B. Douglas Bernheim †

Stanford University and NBER

Charles Sprenger‡

UC San Diego

First Draft: December 1, 2014This Version: November 8, 2016

Abstract

Cumulative Prospect Theory (CPT), the leading behavioral account of decision makingunder uncertainty, assumes that the probability weight applied to a given outcome dependson its ranking. This assumption is needed to avoid the violations of dominance implied byProspect Theory (PT). We devise a simple test involving three-outcome lotteries, basedon the implication that compensating adjustments to the payo↵s in two states shoulddepend on their rankings compared with payo↵s in a third state. In an experiment, we areunable to find any support for the assumption of rank dependence. We separately elicitprobability weighting functions for the same subjects through conventional techniquesinvolving binary lotteries. These functions imply that changes in payo↵ rank shouldchange the compensating adjustments in our three-outcome lotteries by 20-40%, yet wecan rule out any change larger than 7% at standard confidence levels. Additional testsnevertheless indicate that the dominance patterns predicted by PT do not arise. Wereconcile these findings by positing a form of complexity aversion that generalizes thewell-known certainty e↵ect.

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 participantsat Cornell, Caltech, and Gary’s Conference (UC Santa Barbara) for helpful and thoughtful discussions. FulyaErsoy, 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, CA94305; [email protected].

‡University of California San Diego, Rady School of Management and Department of Economics, 9500 GilmanDrive, La Jolla, CA 92093; [email protected].

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 utility

is non-linear in proabilities, with highly unlikely events receiving 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 simultaneous 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�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 decision making 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 payo↵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

total weight assigned to two events occurring with probability p/2 discretely exceeds the weight

1For example, when graphing the empirical certainty equivalent, C, for a lottery that pays X with probabilityp 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., Tverskyand Kahneman, 1992; Tversky and Fox, 1995).

2As 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 beconsistent with any of the large body of economic theory which has already been developed... it must satisfythese rules.” (p. 325).

3Kahneman and Tversky (1979) described their theory as being concerned with lotteries that have at mosttwo non-zero outcomes. Hence, to apply Prospect Theory strictly in accordance with their original intent, onewould have to assume that this lottery pays zero with probability 1�p. Kahneman and Tversky (1979) actuallyprovided two formulations of Prospect Theory; we we assume their Equation 1 for ‘regular prospects.’ Theyimplicitly invoke the same assumption when examining the Allais common consequence paradox (p. 282).

1

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↵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↵s,

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 payo↵s, while a weighting function,

⇡(·), is applied to the cumulative probabilities:

U(L) = ⇡(1)u(Z|r) + ⇡(p+ q)[u(Y |r)� u(Z|r)] + ⇡(p)[u(X|r)� u(Y |r)].

Normally this 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)

4 Kahneman and Tversky appreciated this problematic implication of PT and attempted to address itthrough an “editing” assumption: “Direct violations of dominance are prevented, in the present theory, bythe 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 invokedin applications. Kahneman and Tversky also provided a formulation for two-outcome lotteries with either allpositive or all negative outcomes that does indeed respect dominance (see e.g., Equation 2 of Kahneman andTversky, 1979). One can see in that formulation the roots of Cumulative Prospect Theory.

2

Now imagine, as before, a second lottery, identical to the first, except that it splits the event

yielding the payo↵ X into two events paying X and X � ", each with probability p/2. In that

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), 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

decision making 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↵ X as we change its value.

Initially X exceeds Y , and its weight is ⇡(p). As we reduce the value of X, the weight remains

unchanged until X passes below Y , at which point it changes discontinuously to ⇡(p+q)�⇡(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↵.

Given the central role the assumption of rank-dependent probability weighting plays in the

leading behavioral theory of decision-making under uncertainty, as well as in recent applications

of the theory,5 it has been the object of surprisingly little formal scrutiny. The prior literature’s

failure to explore and test the implications of rank dependence systematically is acknowledged

5 Barseghyan, Molinari, O’Donoghue and Teitelbaum (2015) investigate choices involving a range of insur-ance products. They demonstrate that the bracketing of risks – for example, whether people consider homeand automobile insurance together or separately – a↵ects the implications of probability weighting because itchanges the ranking of outcomes. Epper and Fehr-Duda (Forthcoming) examine the data from Andreoni andSprenger (2012) on intertemporal decision-making under various risk conditions, which exhibits deviations fromdiscounted expected utility. They argue that CPT can rationalize an apparent choice anomaly if one framestwo independent binary intertemporal lotteries as a single lottery with four possible outcomes. This alternativeframing delivers the desired prediction because it alters the rankings of the four outcomes. Barberis, Mukherjeeand Wang (Forthcoming) examine historical monthly returns at the stock level for a five year window and linkthe CPT value of the stock’s history to future returns, demonstrating a significant negative correlation. Theinterpretation for the negative relation is that investors overvalue positively skewed, lottery-like stocks. Given 60equi-probabable monthly return events, PT would equally overweight all outcomes, giving no disproportionatevalue for skewness. CPT, on the other hand, allows the highest ranked outcomes to receive higher proportionateweight. Barberis et al. (Forthcoming) 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.

3

in the review of Fehr-Duda and Epper (2012).6 The literature has focused instead on identi-

fying 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. Unfortunately, an incorrect functional specification can manifest as

spurious rank dependence. To our knowledge, in cases where such data are available, no for-

mal test of rank dependence has been performed.7 The literature does include a small number

of attempts to test the axiomatic foundations of rank-dependent models (Wu, 1994; Wakker,

Erev and Weber, 1994; Fennema and Wakker, 1996), and the findings have been on the whole

unfavorable.8 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 decision making that arguably resonate with the premise (Lopes, 1984; Lopes

and Oden, 1999; Weber, 1994).

The current paper contributes to the literature by devising and implementing a simple,

direct, and robust test of rank-dependent probability weighting that requires no maintained

6They state “It is our impression that this feature of rank-dependent utility has often not been properlyunderstood. For example, an inverse S-shaped probability weighting function does not imply that all smallprobabilities are overweighted. Whether a small probability is overweighted or underweighted depends on therank of the outcome to which it is attached” (p. 571).

7As we explain in Appendix A, the data in Tversky and Kahneman (1992) lend themselves to such tests.Reanalyzing results from Tversky and Kahneman (1992), we find some support for rank dependence. However,as noted in the text, that finding hinges on the validity of their functional form assumptions. We show thatdepending on the assumptions for the shape of utility, probability weighting for a given chance of receiving anoutcome can either appear to be rank dependent or not.

8Wu (1994) tests the concept of ‘ordinal independence,’ which means that when two lotteries have commontails, one should be able to replace both tails without changing how anyone ranks the lotteries. The authorsobserve significant violations, which they explain by assuming that decision makers practice a form of editingprior to maximizing the CPT objective function. Wakker et al. (1994) and Fennema and Wakker (1996) providesimilar tests of comonotonic independence for both risk and ambiguity, which show more severe violations thanone would expect assuming noisy rank-dependent decision-making. They also find that a rank-dependentmodel does not meaningfully improve upon EU in terms of explanatory power. More recently, in the domain ofambiguity, L’Haridon and Placido (2010) investigate the closely related concept of ‘tail separability’, and alsodemonstrate significant violations.

4

assumptions concerning functional forms, either for utility and risk aversion, or for probability

weighting. An essential feature of our test is that it involves lotteries with three outcomes

rather than two. 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 decision maker

indi↵erent. Both EU theory and PT imply that the magnitude of the equalizing reduction is

entirely indepedent 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

ranking of the payo↵s changes, which causes the probability weights to change, thus altering

the equalizing reduction. In particular, if we increase X from a value just below Y to a value

just above Y + m, the weight on u(Y |r) changes from ⇡(q) to [⇡(p + q) � ⇡(p)] , while the

weight on u(Z|r) is una↵ected. Provided q is not too large, the shape of the standard CPT

probability weighting function implies [⇡(p + q) � ⇡(p)] < ⇡(q). Accordingly, the equalizing

reduction should decrease discontinuously when X crosses this boundary, with the magnitude

of the change reflecting the quantitative importance of rank dependence. Our strategy is to

test the weaker implication that the equalizing reduction should be smaller for X > Y + m

than for X 2 (Z, Y ).

Subjects in our experiment perform decision tasks that reveal their equalizing reductions for

three-outcome lotteries of the type described in the previous paragraph. Following previous

studies (Tversky and Kahneman, 1992; Tversky and Fox, 1995), we also elicit their certainty

equivalents for a collection of binary lotteries, which we use to derive their CPT parameters.

The results are striking. Using the data from the tasks involving binary lotteries, we reproduce

standard findings regarding probability weights: subjects apparently attach disproportionately

high weight to low probabilities and disproportionately low weight to high probabilities, so

the ⇡(·) curve has the standard inverse S-shape. Moreover, our estimates of the curvature

parameters correspond closely to those reported in the prior literature. Given this finding,

5

CPT yields dramatic predictions for our three-outcome lotteries: an increase in X that changes

the ranking of X and Y should change the equalizing reductions by –20% to –37%. Contrary to

this implication, we find no evidence that equalizing reductions are even modestly sensitive to

the ranking of outcomes. The actual change in the equalizing reduction ranges from +2.87%

to -2.74%, and in no case can we reject the hypothesis of rank-independence. However, in all

cases we can reject the hypothesis that the equalizing reduction falls by more than 7%, and

the confidence intervals for actual and predicted changes are always non-overlapping. These

patterns are also apparent at the individual level, with a preponderance of subjects exhibiting

virtually no rank dependence for their equalizing reductions, despite manifesting preferences

over binary lotteries that would imply 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

consider alternative formulations of CPT, and assess related models that posit expectations-

based reference dependence (Koszegi and Rabin, 2006, 2007; Bell, 1985; Loomes and Sugden,

1986). These models have similar predictions for equalizing reductions, and hence we reject

them as well.

It is worth emphasizing that this 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 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 by eliciting certainty equivalents for three out-

come lotteries that pay X + " with probability p/2, X � " with probability p/2, and Y with

probability 1 � p. We include the case of " = 0, which reduces to a two-outcome lottery. We

choose the parameters so that standard formulations of PT predict a sizable and discontinuous

6

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.

A good theory of choice under uncertainty would therefore have to account for three pat-

terns: (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 split-

ting 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

with fewer outcomes because they are easier to understand. One can think of the well-known

certainty e↵ect as a special case of this more general phenomenon.

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

We focus on the implications of Cumulative Prospect Theory (Tversky and Kahneman, 1992)

(CPT) for three outcome lotteries. Let L = ({p, q, 1�p�q}, {X, Y, Z}) represent a lottery with

three potential outcomes, X, Y, Z, with X > Y > Z > r, paid with corresponding probabilities

p, q, 1� p� q; 0 p, q, 1� p� q 1. The CPT representation is

U(L) = ⇡(p)u(X|r) + [⇡(p+ q)� ⇡(p)]u(Y |r) + [1� ⇡(p+ q)]u(Z|r).

We first introduce the notion of an ‘equalizing reduction.’ The equalizing reduction for a given

lottery L = ({p, q, 1 � p � q}, {X, Y, Z}) is the value k that delivers indi↵erence between L

and Le = ({p, q, 1 � p � q}, {X, Y + m,Z � k}). That is, k is the equalizing reduction to

7

Z that exactly compensates the decisionmaker for increasing Y by a given m. The intuition

for why the equalizing reduction may be of interest in the analysis of CPT decisionmaking is

clear: the CPT rank-dependent weighting of q depends upon whether the outcomes are ranked

X > Y > Z > r or Y > X 0 > Z > r. As such, the equalizing reduction will depend on whether

Y is the second highest or the highest ranked outcome. In contrast, Expected Utility Theory

(EUT) and PT imply that the equalizing reduction should be entirely independent of X.9

2.1 Equalizing Reductions: X > Y > Z

We first consider equalizing reductions for a lottery such as L, above, with X > Y > Z > r.

The equalizing reduction for increasing Y to Y +m < X (i.e., without changing the ranks) is

the value

k = Z � u�1

✓u(Z|r)�

⇡(p+ q)� ⇡(p)

1� ⇡(p+ q)

�[u(Y +m|r)� u(Y |r)]

◆. (2)

The notation k represents the equalizing reduction when Y and Y +m lie below X.10 Note that

k depends upon the weighted probability ratioh⇡(p+q)�⇡(p)1�⇡(p+q)

i. For an expected utility decision

maker this ratio is q1�p�q .

11 Under PT, the ratio would be ⇡(q)1�⇡(p)�⇡(q) . For all three theories, we

have @k/@X = 0, such that equalizing reductions are constant with respect to the outcome X,

provided X > Y, Y +m.

9An alternative approach would be to assess certainty equivalents for the lottery with outcomes X, Y, andZ, varying X as in our method. Under CPT (but not PT or EUT), the certainty equivalent would changediscontinuously when X crosses Y. An important di↵erence is that the certainty equivalent would also changewith X for a fixed ranking of X and Y, whereas with our method the equivalent reduction remains constant.Consequently, with the alternative method, one would have to disentangle rank dependence from the e↵ects ofrisk aversion. Our approach avoids that potential confound.

10 Solve for k as k satisfying

⇡(p)u(X|r) + [⇡(p + q) � ⇡(p)]u(Y |r) + [1 � ⇡(p + q)]u(Z|r) =

⇡(p)u(X|r) + [⇡(p + q) � ⇡(p)]u(Y + m|r) + [1 � ⇡(p + q)]u(Z � k|r).

11For an expected utility decisionmaker, k satisfies the equation

1 � p� q[u(Z) � u(Z � k)] = q[u(Y + m) � u(Y )].

8

2.2 Equalizing Reductions: Y > X 0 > Z

We now consider equalizing reductions for a lottery, L0 , with Y > X 0 > Z > r and the same

probabilities as before. The only di↵erence between lotteries L and L0 is that X is reduced to

X 0. The values of Y and Z remain unchanged, but Y now becomes the highest ranked outcome.

Given the new configuration of ranks, the equalizing reduction for increasing Y to Y +m is

k = Z � u�1

✓u(Z|r)�

⇡(q)

1� ⇡(p+ q)

�[u(Y +m|r)� u(Y |r)]

◆. (3)

The notation k represents the equalizing reduction when Y and Y +m lie above X 0.12 Note that

k now depends upon the weighted probability ratioh

⇡(q)1�⇡(p+q)

i. Once again, the ratio would

be q1�p�q under EUT and ⇡(q)

1�⇡(p)�⇡(q) under PT. Under all three theories, @k/@X 0 = 0, so that

equalizing reductions are constant with respect to the outcome X 0, provided X 0 < Y, Y +m.

The critical insight from examining (2) and (3) is that under CPT, k and k will generally

di↵er. This is because the relevant ratios,h

⇡(q)1�⇡(p+q)

iand

h⇡(p+q)�⇡(p)1�⇡(p+q)

i, will as a general matter

coincide only when ⇡(·) is linear.13 Hence, equalizing reductions will depend on whether the

ranks are arrayed as X > Y > Z or Y > X 0 > Z. This insight, in combination with the

observations that @k/@X = 0 and @k/@X 0 = 0, implies a distinctive relationship between k

and X: the equalizing reduction is constant as long as the fixed ranking remains, but shifts

discontinuously whenX passes across Y .14 Our strategy is to test the corresponding implication

that the equalizing reduction should be systematically di↵erent for X > Y + m than for

X 2 (Z, Y ). Critically, both EUT and PT imply that k should be invariant with respect to X.

In the next sub-section we provide a sense for the magnitude of the discontinuity implied by

CPT under standard parameterizations.

12 Solve for k as

⇡(q)u(Y |r) + [⇡(p + q) � ⇡(q)]u(X 0|r) + [1 � ⇡(p + q)]u(Z|r) =

⇡(q)u(Y + m|r) + [⇡(p + q) � ⇡(q)]u(X 0|r) + [1 � ⇡(p + q)]u(Z � k|r).

13With linearity, we have ⇡(q) = ⇡(p + q) � ⇡(p) .14Technically, the discontinuity occurs at Y in the limit as m goes to zero.

9

2.3 Simulated Equalizing Reductions under CPT Decisionmaking

In this section we derive the equalizing reductions, k and k, for given values of X,X 0, Y, Z and

m, and probabilities, p, q, and 1 � p � q. We focus on the parametric specification used in

the original formulation of CPT (Tversky and Kahneman, 1992),15 which posited probability

weighting function, ⇡(p) = p�/(p� + (1 � p)�)1/�, a reference point of r = 0, and a utility

function u(x) = x↵ for x > r = 0. The parameters identified by Tversky and Kahneman (1992)

were � = 0.61 and ↵ = 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 � = 0.61 and

↵ = 0.88, the equalizing reduction is k = 1.67. The CPT decision maker is indi↵erent between

the original lottery, L, and the perturbed lottery, Le, in which Y is $5 higher and Z is $1.67

lower.16

Now consider the lottery L0 with {X 0, Y, Z} = {$23, $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 same CPT parameters as

above, the equalizing reduction is k = $3.22.17 The CPT decisionmaker is indi↵erent between

the original lottery, L0, and the perturbed lottery, L0e, in which Y is $5 higher and Z is $3.22

lower.

Thus, a standard parameterization of CPT implies a sharp discontinuity in equalizing re-

ductions: moving from X 0 < Y to X > Y +m cuts the equalizing reduction roughly in half. In

Table 1, we provide further simulations with the above values of X, Y, Z, and m and di↵ering

parameter levels for �.18 Table 1 exhibits the results of analogous calculations for three di↵erent

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}. For the

CPT parameter values of Tversky and Kahneman (1992), substantial di↵erences between k and

15Tversky and Fox (1995) and Gonzalez and Wu (1999) employ a similar two parameter ⇡(p) function. SeePrelec (1998) for alternative S -shaped specifications.

16Note that Y and Z are received with equal probability, so that a risk neutral decisionmaker would exhibitan equalizing reduction of k = $5.

17One again, note that a risk neutral decisionmaker would exhibit an equalizing reduction of k = $5.18To demonstrate the dependence of discontinuities in equalizing reduction on the extent of probability weight-

ing, we hold fixed, ↵ = 0.88 throughout.

10

k are predicted in all cases. The table also includes results for other values of the probability

weighting parameter. Even with more modest curvature of the probability weighting function

(� = 0.8), the discontinuities remain sizable. The final column of Table 1 shows that the dis-

continuities disappear when the probability weighting function becomes linear (� = 1), which

of course corresponds to the case of expected utility.19

Table 1: Cumulative Prospect Theory Simulated Equalizing Reductions

� = 0.4 � = 0.61 � = 0.8 � = 1

{p, q, 1� p� q} k k k - k k k k- k k k k - k k k k - k(% Change) (% Change) (% Change) (% Change)

{0.6, 0.3, 0.1} 1.97 1.33 -0.63 5.17 3.88 -1.29 9.21 7.84 -1.37 13.42 13.42 0(-32%) (-25%) (-15%) (0%)

{0.4, 0.3, 0.3} 1.61 0.53 -1.09 3.22 1.67 -1.56 4.29 3.13 -1.16 4.69 4.69 0(-68%) (-48%) (-27%) (0%)

{0.1, 0.3, 0.6} 1.45 0.40 -1.06 2.39 1.39 -1.00 2.60 2.08 -0.51 2.37 2.37 0(-73%) (-42%) (-20%) (0%)

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 0

, Y, Z} = {$23, $24, $18}, m = $5. CPT calculations withu(x) = x

↵,↵ = 0.88; and ⇡(p) = p

�/(p� + (1 � p)�)1/� with � varying by column.

2.4 Reference Point Formulation and Alternative Models of Refer-

ence Dependence

Throughout the previous discussion, we implicitly assumed that the reference point, r, lay below

all potential outcomes. Though this is a natural starting point, one may imagine situations

where the reference point falls between payo↵s, which it segregates into gains and losses. CPT

treats gain and loss probabilities di↵erently, and consequently could in principle have somewhat

di↵erent implications in these settings. It is particularly important to consider the sensitivity

of the implied discontinuity to formulations that endogenize the reference point, such as Bell

(1985), Loomes and Sugden (1986), and Koszegi and Rabin (2006, 2007). In Appendices B

19Because the popularized functional form of Tversky and Kahneman (1992) features both convex and concaveregions of probability weighting, non-monotonicities in equalizing reductions with respect to the weightingparameter � can exist. One such non-monotonicity is observed for k in {0.1, 0.3, 0.6}.

11

and C, we consider these possibilities in detail, demonstrating that alternative formulations of

static reference points as well as models of expectations-based reference-dependence all exhibit

substantial sensitivity of equivalent reductions to the ranking of outcomes.20

3 Design

Our experimental design follows closely the theoretical discussion of section 2. Conditional on

various probability vectors, {p, q, 1 � p � q}, we test for di↵erences in equalizing reductions

between lotteries with ranks X > Y > Z and those with ranks Y > X 0 > Z. Subjects

also complete a battery of certainty equivalent tasks involving binary lotteries; these tasks

are commonly used to derive risk-preference parameters within the CPT framework. This

strategy allows us to compare the observed equalizing reductions to predicted values based on

elicited risk preferences for the same individuals. We divide our discussion of design into three

subsections. First we describe the elicitation of equalizing reductions; second we detail the

conventional elicitation of CPT preference parameters; third, we discuss other design details

including task orders and payment procedures.

3.1 Elicitation of Equalizing Reduction

We elicited equalizing reductions using the method of price lists. In each task, subjects made a

series of decisions between ‘Option A’ and ‘Option B’, both three outcome lotteries. Option A

was fixed throughout the task as either a lottery withX > Y > Z or a lottery with Y > X 0 > Z.

Option B was constructed by adding $5 to Y and reducing Z by $k. The value of k varied

throughout the task. The point at which an individual switched from choosing Option A to

choosing Option B places tight bounds on the equalizing reduction, either k or k. Panels A and

20A subtle feature this analysis involves the distinction between the models of Koszegi and Rabin (2006, 2007)and those of Bell (1985); Loomes and Sugden (1986). Because these models feature di↵erent formulations ofexpectations-based reference points, their predictions di↵er. Whereas Koszegi and Rabin (2006, 2007) predictchanges in equivalent reductions resulting from changes in ranks, the models of Bell (1985) and Loomes andSugden (1986) predict discontinuities as X crosses the induced lottery’s certainty equivalent. See Appendix Cfor further detail.

12

Figure 1: equalizing Reduction Tasks

Panel A : X > Y > Z Panel B : Y > X 0 > Z

k k

TASK 8On this page you will make a series of decisions between two uncertain options. Option A will be a 40

in 100 chance of receiving $30, a 30 in 100 chance of receiving $24 and 30 in 100 chance of receiving $18.Initially Option B will be a 40 in 100 chance of receiving $30, a 30 in 100 chance of receiving $29 and 30 in100 chance of receiving $18. As you proceed, Option B will change. The lowest amount you receive with30 in 100 chance will decrease. For each row, decide whether you prefer Option A or Option B.

Option A or Option B40 in 100 30 in 100 30 in 100 40 in 100 30 in 100 30 in 100Chance Chance Chance Chance Chance Chance

1) $30 $24 $18 2 or $30 $29 $18.00 22) $30 $24 $18 2 or $30 $29 $17.75 23) $30 $24 $18 2 or $30 $29 $17.50 24) $30 $24 $18 2 or $30 $29 $17.00 25) $30 $24 $18 2 or $30 $29 $16.75 26) $30 $24 $18 2 or $30 $29 $16.50 27) $30 $24 $18 2 or $30 $29 $16.25 28) $30 $24 $18 2 or $30 $29 $16.00 29) $30 $24 $18 2 or $30 $29 $15.75 210) $30 $24 $18 2 or $30 $29 $15.50 211) $30 $24 $18 2 or $30 $29 $15.25 212) $30 $24 $18 2 or $30 $29 $15.00 213) $30 $24 $18 2 or $30 $29 $14.50 214) $30 $24 $18 2 or $30 $29 $14.00 215) $30 $24 $18 2 or $30 $29 $13.50 216) $30 $24 $18 2 or $30 $29 $13.00 217) $30 $24 $18 2 or $30 $29 $12.50 218) $30 $24 $18 2 or $30 $29 $12.00 219) $30 $24 $18 2 or $30 $29 $11.50 220) $30 $24 $18 2 or $30 $29 $11.00 221) $30 $24 $18 2 or $30 $29 $10.50 222) $30 $24 $18 2 or $30 $29 $10.00 223) $30 $24 $18 2 or $30 $29 $9.50 224) $30 $24 $18 2 or $30 $29 $9.00 225) $30 $24 $18 2 or $30 $29 $8.50 226) $30 $24 $18 2 or $30 $29 $8.00 227) $30 $24 $18 2 or $30 $29 $7.50 228) $30 $24 $18 2 or $30 $29 $7.00 229) $30 $24 $18 2 or $30 $29 $6.50 230) $30 $24 $18 2 or $30 $29 $6.00 231) $30 $24 $18 2 or $30 $29 $5.50 232) $30 $24 $18 2 or $30 $29 $5.00 233) $30 $24 $18 2 or $30 $29 $4.50 234) $30 $24 $18 2 or $30 $29 $4.00 235) $30 $24 $18 2 or $30 $29 $3.50 236) $30 $24 $18 2 or $30 $29 $3.00 237) $30 $24 $18 2 or $30 $29 $2.50 238) $30 $24 $18 2 or $30 $29 $2.00 2





13

B of Figure 1 provide two tasks eliciting k and k. Appendix F provides the full instructions

given to subjects along with all tasks.

As in the simulations of section 2, our design fixes Y = $24, Z = $18, and m = $5. We

use three values of X > Y +m , {$34, $32, $30}, and three values of X 0 < Y , {$23, $21, $19}.

We use one additional value, X = $25, to check robustness; see the discussion in Appendix

E.21 This set of values allows us to investigate both rank dependence and the predictions that

@k/@X = 0 and @k/@X 0 = 0.

As in the simulations of section 2, we examine three 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}. The cumulative probability of receiving at least

Y ranges from 0.1 + 0.3 = 0.4 to 0.6 + 0.3 = 0.9, which provides broad scope for detecting

the predicted discontinuities. Note that the design varies the relative probabilities of Y and Z

from 0.3/0.1 = 3 to 0.3/0.6 = 1/2, generating a wide range of equalizing reductions.22

With seven values of X/X 0 and three probability vectors, we have 21 equalizing reduction

tasks in total. We organize these tasks into seven blocks, each of which presents the three prob-

ability vectors for a single value of X/X 0. Hence, the tasks within each block are di↵erentiated

by the probability vector, {p, q, 1�p�q}. We distributed task blocks to subjects one at a time,

and collected responses before moving on to the next block. This feature of our design was

intended to limit any tendency to respond mechanically with the same answer as X varies, a

possibility that could artificially generate patterns consistent with @k/@X = 0 and @k/@X 0 = 0

while obscuring discontinuities.

3.2 Prospect Theory Elicitation Tasks

We generate predictions of equalizing reductions for our subjects by eliciting their risk prefer-

ence parameters using the same experimental techniques employed by Tversky and Kahneman

(1992). The approach employs seven tasks, each of which elicits the certainty equivalents for

21Note that for X = $25, adding $5 to Y induces a change of ranks. These tasks allow us to investigate thepossibility that explicit rank changes influence choice. See Appendix E for further detail.

22For example, an expected value decision maker would exhibit values of k, k between $2.50 and $15 acrossthese tasks.

14

a two-outcome lottery, (p, $25; 1 � p, 0), p 2 {0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95}. We grouped

these tasks in a single block. Figure 2 illustrates one of these tasks, and Appendix F describes

the full set. Although these tasks provide no information concerning rank-dependence in prob-

ability weighting, they allow us to determine whether our sample exhibits representative risk

preferences, and they permit us to generate precise sample-specific predictions for equalizing

reductions.

Figure 2: Prospect Theory Elicitation Task

TASK 22On this page you will make a series of decisions between two options. Option A will be a 50 in 100

chance of receiving $25 and a 50 in 100 chance of receiving $0. Initially Option B will be a 100 in 100chance of receiving $25. As you proceed, Option B will change. The amount you receive with 100 in 100chance will decrease. For each row, decide whether you prefer Option A or Option B.

Option A or Option B50 in 100 Chance 50 in 100 Chance 100 in 100 Chance

1) $25 $0 2 or $25.00 22) $25 $0 2 or $24.00 23) $25 $0 2 or $23.00 24) $25 $0 2 or $22.00 25) $25 $0 2 or $21.00 26) $25 $0 2 or $20.00 27) $25 $0 2 or $19.00 28) $25 $0 2 or $18.00 29) $25 $0 2 or $17.00 210) $25 $0 2 or $16.00 211) $25 $0 2 or $15.00 212) $25 $0 2 or $14.00 213) $25 $0 2 or $13.00 214) $25 $0 2 or $12.00 215) $25 $0 2 or $11.00 216) $25 $0 2 or $10.00 217) $25 $0 2 or $9.00 218) $25 $0 2 or $8.00 219) $25 $0 2 or $7.00 220) $25 $0 2 or $6.00 221) $25 $0 2 or $5.00 222) $25 $0 2 or $4.00 223) $25 $0 2 or $3.00 224) $25 $0 2 or $2.00 225) $25 $0 2 or $1.00 226) $25 $0 2 or $0.00 2

15

3.3 Design Details

One hundred fifty three subjects were recruited from the Stanford Economics Research Labo-

ratory subject pool in September, October, and November of 2014. A total of 20 sessions were

conducted with the number of subjects varying between two and sixteen. We varied the order

of the six main equalizing reduction blocks systematically across session. Subjects completed

three equalizing reduction blocks, then the CPT elicitation block, then three more equalizing

reduction blocks. The X = $25 equalizing reduction tasks were always presented last. Table 2

lists all sessions, dates, numbers of subjects and block orders.

Table 2: Experimental SessionsNumber Date Order # Obs

1 09/24/14 {34, 32, 30, CE, 23, 21, 19, 25} 162 09/24/14 {34, 21, 30, CE, 23, 32, 19, 25} 113 09/30/14 {23, 32, 19, CE, 34, 21, 30, 25} 94 09/30/14 {19, 32, 23, CE, 30, 21, 34, 25} 125 10/01/14 {30, 21, 34, CE, 19, 32, 23, 25} 126 10/02/14 {21, 30, 34, CE, 32, 19, 23, 25} 147 10/07/14 {32, 19, 23, CE, 21, 30, 34, 25} 108 10/07/14 {23, 19, 32, CE, 34, 30, 21, 25} 59 10/08/14 {34, 30, 21, CE, 23, 19, 32, 25} 1310 10/09/14 {30, 34, 21, CE, 19, 23, 32, 25} 511 10/14/14 {19, 23, 32, CE, 30, 34, 21, 25} 412 10/16/14 {32, 23, 19, CE, 21, 34, 30, 25} 613 10/26/14 {21, 34, 30, CE, 32, 23, 19, 25} 714 10/28/14 {21, 34, 30, CE, 32, 23, 19, 25} 315 10/29/14 {32, 23, 19, CE, 21, 34, 30, 25} 216 11/05/14 {23, 19, 32, CE, 34, 30, 21, 25} 217 11/07/14 {19, 23, 32, CE, 30, 34, 21, 25} 618 11/10/14 {23, 19, 32, CE, 34, 30, 21, 25} 619 11/14/14 {30, 34, 21, CE, 19, 23, 32, 25} 620 11/18/14 {32, 23, 19, CE, 21, 34, 30, 25} 4

Total 153

Notes: Session number, date, order and number of observations. Order of tasks refers to the value of X/X

0 in

each task block. CE corresponds to the block of tasks with certainty equivalent questions.

To induce truthful revelation of equalizing reductions and certainty equivalents, we incen-

tivized subjects by paying them based on one randomly selected question in one randomly

selected task. On average, subjects earned $26.87. This random-lottery incentive mechanism

16

is widely used in experimental economics, but note that it transforms the experiment into

a single compound lottery. The literature on choice under risk, dating to Holt (1986) and

Karni and Safra (1987), suggests that random mechanisms need not be incentive compatible

in such contexts if either the Independence or Reduction of Compound Lotteries axioms are

violated. As CPT violates independence, this limitation is a potential concern. Importantly,

however, Starmer and Sugden (1991) and Cubitt, Starmer and Sugden (1998) demonstrate

that this mechanism can be used even when individuals deviate from expected utility because,

in practice, subjects appear to narrowly frame each lottery, making each decision e↵ectively

in isolation. Whether their findings apply to our setting is of course an empirical question.

Our CPT-preference elicitation tasks are especially important because they allow us to assess

the validity of the method we use. If isolation fails in this context, then our subjects would

not exhibit standard CPT probability weighting in binary tasks. Conversely, if our subjects

do exhibit standard CPT probability weighting in binary tasks, then one cannot reasonably

attribute the absence of implied discontinuities in the compensating reduction tasks to a failure

of isolation.

Ten of 153 (6.5%) subjects exhibited at least one instance of multiple switching within a

single experimental task. This figure compares favorably to other experiments employing price

lists.23 Because multiple switch points are di�cult to rationalize and may indicate subject

confusion, researchers often exclude such observations or mechanically enforce single switch

points.24 We begin by excluding all subjects exhibiting multiple switch points in any task,

leaving a sample of 143 subjects. In Appendix E we include the ten subjects with multiple

switch points, taking each subject’s first switch point as their relevant choice, and demonstrate

that the results are qualitatively unchanged.

23Around 10 percent of subjects feature multiple switch points in similar price-list experiments (Holt andLaury, 2002; Meier and Sprenger, 2010), and as many as 50 percent in some cases (Jacobson and Petrie, 2009).

24See Harrison, Lau, Rutstrom and Williams (2005) for discussion.

17

4 Results

We present our results in four subsections. We begin with the prospect theory elicitation tasks,

demonstrating the classic probability weighting found in previous studies. The correspond-

ing parameter estimates provide a segue to the second subsection, which explores equalizing

reductions. Although the estimated probability weighting function implies substantial discon-

tinuities, we show that there is no relationship between payo↵ ranks and equalizing reductions.

The third subsection focuses on results for individual subjects, and corroborates the absence

of support for rank-dependent probability weighting. The fourth subsection presents tests of

robustness focusing primarily on between-subject variation. Additional robustness exercises

beyond the results appear in Appendix E.

4.1 Certainty Equivalents: Eliciting CPT Parameters

As in the original experiments of Tversky and Kahneman (1992), we administered seven cer-

tainty equivalent tasks involving lotteries over $25 and $0, with the governing probability

p 2 {0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95}. Figure 3, Panel A summarizes these data. To cap-

ture average behavior, we first estimated an interval regression (Stewart, 1983) describing the

certainty equivalent, C, as a function of indicators for the experimental probabilities, p.25 Fig-

ure 3, Panel A thus presents the estimated mean certainty equivalent for each value of p along

with its 95% confidence interval.26 Following Tversky and Kahneman (1992), the data are

presented relative to a benchmark of risk neutrality so that Figure 3, Panel A would directly

reveal aggregate probability weighting, ⇡(·), if the utility function were linear.

Tversky and Kahneman (1992) and Tversky and Fox (1995) obtain probability weighting pa-

rameters from certainty equivalents by parameterizing both the utility and probability weighting

functions and assuming each observation satisfies the indi↵erence condition u(C) = ⇡(p) ·u(25).25Virtually identical results are obtained when using OLS and the midpoint of the interval.26Standard errors are estimated clustered at the individual level. Appendix Table A2, column (1) provides

corresponding estimates. Column (2) provides estimates of risk premia, demonstrating significant risk toleranceat low probabilities and significant risk aversion at high probabilities.

18

Figure 3: Certainty Equivalents and Equalizing Reductions0

510

1520

25C

erta

inty

Equ

ival

ent

0 20 40 60 80 100Chance of $25

C CPT ModelRisk Neutral 95% CI

Panel A: Certainty Equivalents

05

10Eq

ualiz

ing

Red

uctio

n: k

for X

, Y=2

4+5,

Z=1

8-k

20 25 30 35X

with Y = 24, Z = 18

p = 0.6 p = 0.4 p = 0.195% CI CPT Pred CPT 95% CI

Panel B: Equalizing Reductions

Notes: Panel A: Mean behavior for C estimated from interval regression (Stewart, 1983) of experimental re-sponse on indicators for probability vectors. Standard errors clustered at individual level to provide 95%confidence interval. Appendix Table A2, column (1) provides corresponding estimates. Dashed line correspondsto predicted CPT behavior with ↵̂ = 0.941 (s.e. = 0.019) and �̂ = 0.715 (0.015); standard errors clustered atindividual level. Panel B: Mean behavior for k estimated from interval regression of experimental response onindicators for probability vectors interacted with indicators for value of X. Standard errors clustered at indi-vidual level to provide 95% confidence interval. Appendix Table A4 provides corresponding estimates. Dashedline corresponds to predicted values of (2) and (3) for CPT decision maker with risk preference parameters↵̂ = 0.941 (s.e. = 0.019) and �̂ = 0.715 (0.015). Standard errors clustered at individual level and calculatedusing the delta method to provide 95% confidence interval.

We follow Tversky and Kahneman (1992) by assuming power utility, u(x) = x↵, and a weighting

function ⇡(p) = p�/(p� +(1�p)�)1/�. We then estimate the parameters �̂ and ↵̂ by minimizing

the sum of squared residuals for the non-linear regression equation

C = [p�/(p� + (1� p)�)1/� ⇥ 25↵]1↵ + ✏. (4)

where C is the midpoint of the certainty equivalent interval defined by experimental choice.

When conducting this analysis for our aggregate data with standard errors clustered on the

subject level, we obtain ↵̂ = 0.941 (s.e. = 0.019) and �̂ = 0.715 (0.015). The benchmark model

19

of expected utility, � = 1, is rejected at all conventional confidence levels, (F1,142 = 341.5, p <

0.01). The value of the probability weighting parameter is reasonably close to the estimate of

Tversky and Kahneman (1992) (�̂ = 0.61), and coincides with the findings of Wu and Gonzalez

(1996), who estimate �̂ = 0.71. The dashed line in Figure 3, Panel A, shows the in-sample

model fit, which closely matches actual behavior.27

The empirically parameterized probability weighting function does an impressive job of

matching the patterns exhibited by average certainty equivalents. Given this parameterization,

if the CPT framework is correct, we would expect to observe substantial discontinuities in

equalizing reductions as ranks change. We turn next to the equalizing reductions, and test the

prediction that behavior di↵ers depending on whether the ranks are arrayed as X > Y > Z or

Y > X 0 > Z.

4.2 Average Equalizing Reductions

Figure 3, Panel B presents equalizing reductions, k and k, for each value of X and X 0. We

exhibit separate results for each of the three probability vectors. To determine average behavior,

we performed an interval regression describing the equalizing reduction, k or k, as a function of

indicators for the probability vectors interacted with indicators for the value of X or X 0.28 The

figure exhibits the estimated mean equalizing reductions along with 95% confidence intervals.29

Vertical lines at Y = $24 and Y +m = $29 partition the figure into three regions, one showing

k, another showing k, and a transitional region.

Notable from Figure 3 is the consistency of the equalizing reductions, k and k, for all three

probability vectors. Corresponding statistics are provided in Table 3, Panel A. For {p, q, 1 �

p � q} = {0.6, 0.3, 0.1}, the mean value of k is 9.02 (clustered s.e. = 0.39), while the mean

27The correlation coe�cient for predicted and actual certainty equivalents is 0.93, and a regression of thetrue certainty equivalent on the model’s prediction yields a slope coe�cient of 0.998 (clustered s.e. = 0.020), aconstant of 0.102 (0.214), and an R-squared value of 0.86. The null hypothesis that the constant is 0 and thepredicted value’s true coe�cient is 1 is not rejected (F2,142 = 0.17, p = 0.84).

28Virtually identical results are obtained when using OLS and the midpoint of the interval.29Standard errors are estimated clustered at the individual level. Appendix Table A4 provides corresponding

estimates.

20

Table 3: Equalizing Reductions

Panel A: Mean Behavior Panel B: CPT Predicted Values

{p, q, 1� p� q} k k k - k % Change k k k- k % Change[95% Conf.] [95% Conf.] [95% Conf.] [95% Conf.]

{0.6, 0.3, 0.1} 9.02 (0.39) 9.28 (0.38) 0.26 (0.17) 2.87 (1.92) 7.58 (0.36) 6.06 (0.35) -1.52 (0.03) -20.04 (0.83)[-0.08,0.59] [-0.90,6.65] [-1.57,-1.47] [-21.68,-18.40]

{0.4, 0.3, 0.3} 4.31 (0.12) 4.34 (0.12) 0.04 (0.09) 0.86 (2.00) 4.01 (0.10) 2.52 (0.13) -1.49 (0.04) -37.17 (1.74)[-0.13,0.21] [-3.07,4.78] [-1.57,-1.41] [-40.61,-33.72]

{0.1, 0.3, 0.6} 2.63 (0.08) 2.56 (0.07) -0.07 (0.06) -2.74 (2.06) 2.65 (0.03) 1.87 (0.06) -0.78 (0.04) -29.34 (1.79)[-0.18,0.04] [-6.78,1.29] [-0.86,-0.69] [-32.88,-25.80]

Notes: Panel A: Mean behavior for k and k estimated from interval regression (Stewart, 1983) of experi-mental response on indicators for probability vector interacted with indicator for whether X > Y . Standarderrors clustered at individual level, in parentheses. See Appendix Table A3, column (1) and Appendix TableA4 for detail. Panel B: Predicted behavior calculated from (2) and (3) for CPT decision maker with ag-gregate parameters ↵̂ = 0.941 (s.e. = 0.019) and �̂ = 0.715 (0.015). Standard errors clustered at individuallevel and calculated using the delta method, in parentheses.

value of k is 9.28 (0.38). The di↵erence, k � k = 0.26 (0.17), is not statistically di↵erent

from zero �2(1) = 2.31, p = 0.13. Expressed as a percentage, this corresponds to 2.87%

(1.92%) change. The 95% confidence interval for k � k is [�0.08, 0.59], indicating that we can

reject discontinuities in equalizing reductions more extreme than $-0.08 (-0.90%) at the 5%

significance level. Similar results are obtained for the other probability series.30

Panel B of Figure 3 also displays the predicted values of k and k, based on our estimates

of the CPT parameters, ↵̂ = 0.941 (s.e. = 0.019) and �̂ = 0.715 (0.015). With this parameter-

ization, we obtain closed-form solutions for k and k based on (2) and (3), and derive standard

errors using the delta method. Figure 3 shows the predicted values along with 95% confidence

intervals, shown as dashed lines. Substantial discontinuities are readily apparent.

30 For {p, q, 1 � p � q} = {0.4, 0.3, 0.3}, the mean value of k is 4.31 (0.12), while the mean value of k is4.34 (0.12). The di↵erence, k � k = 0.04 (0.09), is not statistically di↵erent from zero �

2(1) = 0.18, p = 0.67.The 95% confidence interval for k� k is [�0.13, 0.21], indicating that we can reject discontinuities in equalizingreductions more extreme than $-0.13 (-3.07%). For {p, q, 1 � p� q} = {0.1, 0.3, 0.6}, the mean value of kis2.63(0.08), while the mean value of kis2.56 (0.07). The di↵erence, k� k = �0.07(0.06), is not statistically di↵erentfrom zero �

2(1) = 1.70, p = 0.19. The 95% confidence interval for k� k is [�0.18, 0.04], indicating that we canreject discontinuities in equalizing reductions more extreme than $-0.18 (-6.78%). In Appendix Table A5 andAppendix Table A3, column (2) we reproduce the analysis of Table 3, Panel A and Appendix Table A3, column(1) with individual fixed e↵ects and robust standard errors and reach identical conclusions.

21

Corresponding statistics for predicted behavior appear in Table 3, Panel B. For {p, q, 1 �

p � q} = {0.6, 0.3, 0.1}, the predicted values are k = 7.58 (clustered s.e. = 0.36) and k =

6.06 (0.35). The predicted gap, k � k = �1.52 (0.03), is significantly di↵erent from zero,

F1,142 = 3634.43, (p < 0.01). Expressed as a percentage, this discrepancy corresponds to a

20.04% (0.83%) decrease in equalizing reduction as ranks change. Predictions for the other

probability series are even more extreme.31

The contrast between our findings and the theoretical implications of CPT are striking. CPT

predicts that equalizing reductions should change significantly as ranks change, but we find no

evidence of rank dependence, and we can confidently rule out even relatively small changes.

For every probability vector we consider, the 95% confidence intervals for the predicted and

actual di↵erence in equalizing reductions, k � k, are non-overlapping.32 Given that (1) we

have estimated the CPT parameters for the same set of subjects used to study equalizing

reductions, (2) we estimate those parameters using standard methods, and (3) the resulting

parameter estimates coincide with standard findings, we view these results as striking evidence

against the empirical validity of CPT’s assumption of rank-dependent probability weighting.

In principle, the predictive failures documented above could reflect the influence of unrep-

resentative subjects. In the next section we reexamine the data at the subject level, estimating

CPT parameters and evaluating predictive accuracy separately for each individual.

31For {p, q, 1 � p � q} = {0.4, 0.3, 0.3}, the predicted values are k = 4.01 (0.10) and k = 2.52 (0.13). Thepredicted gap, k � k = �1.49 (0.04), is significantly di↵erent from zero, F1,142 = 1363.43, (p < 0.01). Thisdiscrepancy corresponds to a 37.17% (1.74%) decrease in equalizing reduction as ranks change. For {p, q, 1 �p � q} = {0.1, 0.3, 0.6}, the predicted values are k = 2.65 (0.03) and k = 1.87 (0.06). The predicted gap,k � k = �0.78 (0.04), is significantly di↵erent from zero, F1,142 = 310.44, (p < 0.01). This discrepancycorresponds to a 29.34% (1.79%) decrease in equalizing reduction as ranks change.

32With mean estimates and standard errors for both predicted and actual values and assumptions of normalityfor both, hypothesis tests for equality between predicted and actual di↵erences are easily implemented viacalculation of the following test statistic:

z =(k � k)predicted � (k � k)actualq

s.e.(k � k)2predicted + s.e.(k � k)2actual

,

Under the null hypothesis of equality, the distribution of z is standard normal. For {p, q, 1 � p � q} ={0.6, 0.3, 0.1}, z = 10.3, (p < 0.01). For {p, q, 1 � p � q} = {0.4, 0.3, 0.3}, z = 16.1, (p < 0.01). For{p, q, 1 � p� q} = {0.1, 0.3, 0.6}, z = 9.95, (p < 0.01).

22

4.3 Subject-level Analysis

Each subject in our experiment provides us with data on equalizing reductions and certainty

equivalents. Accordingly, we can replicate our analysis at the subject level. For each subject and

each probability vector, we calculate the average k and k for values of X 0 < Y and X > Y +m,

respectively.33 Additionally, we use each subject’s certainty equivalent data to estimate their

CPT risk preference parameters based on (4), and then predict equalizing reductions for each

probability vector based on (2) and (3). The estimates of (4) imply well-defined values of k or

k for 136 of our 143 subjects.34

Figure 4 presents the distributions of predicted and actual values for k � k for each prob-

ability vector, along with their relationship for the subjects considered. For {p, q, 1 � p �

q} = {0.6, 0.3, 0.1}, the median actual value for k � k is $0 and the interquartile range is

[�$0.33, $0.66]. Results for the other probability vectors are similar.35 Thus, the changes in

equalizing reductions for our subjects are concentrated within a small band around zero.

Though the equalizing reductions exhibit almost no rank dependence, substantial di↵erences

between k and k are predicted; see Figure 4, Panel A.36 Panel B shows the relationship between

predicted and actual values of k � k. For every probability vector, the correlation between

predicted and actual behavior is indistinguishable from zero.37

Our subject-level results provide striking evidence against CPT’s assumption of rank-

dependent probability weighting. The majority of subjects exhibit only small changes in equal-

33This calculation is based on the midpoints of the intervals for k or k implied by each subject’s switch point.34Though we obtain estimates of ↵ and � for the remaining seven subjects, the values are extreme. (In five

cases the values of � are extreme, and in two cases the values of both ↵ and � are extreme.) The non-lineartechnique we use to estimate k or k can fail to converge at extreme values.

35For {p, q, 1 � p � q} = {0.4, 0.3, 0.3}, the median actual value for k � k is $0 and the interquartile rangeis [�$0.33, $0.33] while for {p, q, 1 � p � q} = {0.1, 0.3, 0.6}, the median actual value for k � k is $0 and theinterquartile range is [�$0.21, $0.25].

36For {p, q, 1 � p� q} = {0.6, 0.3, 0.1}, the median predicted value for k � k is $-1.12, for {p, q, 1 � p� q} ={0.4, 0.3, 0.3}, it is $-1.24, and for {p, q, 1 � p � q} = {0.1, 0.3, 0.6}, it is $-0.63. Wilcoxon signed rank testsfor equivalent distributions across predicted and actual values of k � k yield the following test statistics: z =7.48, (p < 0.01), z = 8.90, (p < 0.01), and z = 7.45, (p < 0.01) for {p, q, 1�p�q} = {0.6, 0.3, 0.1}, {0.4, 0.3, 0.3},and {0.1, 0.3, 0.6}, respectively.

37 Correlations between predicted and actual values of k� k are ⇢ = �0.05, (p = 0.58), ⇢ = 0.08, (p = 0.37),and ⇢ = 0.05, (p = 0.53) for {p, q, 1 � p� q} = {0.6, 0.3, 0.1}, {0.4, 0.3, 0.3}, and {0.1, 0.3, 0.6}, respectively.

23

Figure 4: Individual Results0

1020

3040

500

1020

3040

500

1020

3040

50

-5 -4 -3 -2 -1 0 1 2 3 4 5

p = 0.60

p = 0.40

p = 0.10

Actual Predicted

Perc

ent

Difference in Equivalent Reductions

Panel A: Predicted and Actual Distributions

-50

5-5

05

-50

5

-3 -2 -1 0 1

p = 0.60

p = 0.40

p = 0.10Ac

tual

Diff

eren

ce

Predicted Difference

Panel B: Predicted and Actual Relationship

Notes : Mean actual value of k � k calculated for each individual in each probability set.Predicted value of k� k calculated for each individual from (2) and (3) for CPT decisionmakerwith parameters estimated from each individual’s certainty equivalent responses. Not includedin graph, but included in statistics, are four observations with actual values for |k � k| > 5.

izing reductions as ranks change from X > Y to X 0 < Y . Further, the individualized CPT

models predict sizable changes for many subjects, and those predictions bear no relation to

the actual magnitudes. Plainly, our main findings are not driven by a few unrepresentative

subjects.

4.4 Robustness: Between-Subject Variation

Our main findings exploit within-subject variation in payo↵ rank. If a subject’s early responses

in the tasks used to elicit equalizing reductions somehow anchor their later responses, that

approach could obscure rank dependence. We note, however, that responses often change

24

considerably at the individual level from one block of tasks to the next. For example, between

the first and second block of tasks, 59% of individual responses di↵er and 37% of responses

di↵er by more than 25 percent.38

To address any residual concerns about anchoring, we replicate our analysis using only

the first task block for each subject. Recall that payo↵ rank is fixed within each block. It

follows that, with this alternative approach, we identify possible rank-dependence entirely from

between-subject variation using responses that are untainted by anchoring. For such between-

subjects analysis, we rely only on the variation in the ordering of tasks across sessions. To

account for selection on observable characteristics, we additionally include measures of gender,

age, and cognitive ability from a post-study questionnaire and each subject’s average certainty

equivalent in their binary lottery tasks to control for the level of risk aversion.39

Table 4 presents between-subjects results based on the first task blocks with and without the

controls noted above.40 We see a hint of rank dependence, particularly for {p, q, 1 � p � q} =

{0.6, 0.3, 0.1}, without controls in Panel A. With controls in Panel B we find essentially no

di↵erences between k and k. In all cases, the degree of observed rank dependence falls far short

of the CPT predictions from Table 3, Panel B.

Can one construe the small di↵erences between k and k without controls as limited evidence

of rank-dependent probability weighting and CPT? In our view, any such inference would be

unwarranted. With 143 subjects in total, the subsamples that first faced X 0 < Y and X > Y

38The 143 subjects make three decisions in each task block yielding 429 potential di↵erences across the firstand second task blocks. Of these, 175 responses (41%) exhibit no change, 100 responses (23%) increase, and 154(36%) decrease. The order of the first two task blocks has no measurable relationship with changes. Forty-foursubjects began with X

0< Y first and then proceeded to X > Y , giving 132 potential di↵erences. Of these,

57 responses (43%) exhibit no change, 34 responses (26%) increase, and 41 (31%) decrease. Forty-one subjectsbegan with X > Y first and then proceeded to X

0< Y , giving 123 potential di↵erences. Of these, 50 responses

(41%) exhibit no change, 33 responses (27%) increase, and 40 (33%) decrease. Under the estimated CPT model,one would expect more frequent decreases for subjects with X

0< Y first and more frequent increases for subjects

with X > Y first. The di↵erences in response across the first two blocks does not seem localized to a limitednumber of subjects with only 28 of 143 subjects (20%) exhibiting no change across any of the three decisions,and 45 (31%) exhibiting a change in all three.

39Cognitive ability is measured with the three question Cognitive Reflection Test introduced and validated inFrederick (2005).

40 See Appendix Table A3, columns (3) and (5) for further detail on these regressions. Of the 143 subjects inthe primary sample, 21 had the X

0 = 19 block first, 23 had the X

0 = 21 block first, 21 had the X

0 = 23 blockfirst, 22 had the X = 30 block first, 19 had the X = 32 block first, and 37 had the X = 34 block first.

25

are of modest size, and consequently, not perfectly matched. Indeed, we find some hints of

selection across these subsamples in observable characteristics such as gender, cognitive ability,

and average risk aversion, all of which correlate highly with equalizing reductions (see Appendix

Table A3, Columns (5) and (6) for detail).41 Absent controls for this potential selection, the

di↵erences in characteristics inflate k relative to k, spuriously producing the appearance of rank

dependence.42

Table 4: Equalizing Reductions Between Subjects

Panel A: First Task Block (without Controls) Panel B: First Task Block (with Controls)


{0.6, 0.3, 0.1} 9.81 (0.65) 8.71 (0.56) -1.10 (0.85) -11.23 (8.15) 9.77 (0.60) 9.13 (0.56) -0.64 (0.83) -6.56 (8.16)[-2.77,0.57] [-27.21,4.76] [-2.26,0.98] [-22.54,9.43]

{0.4, 0.3, 0.3} 4.78 (0.19) 4.41 (0.19) -0.37 (0.27) -7.82 (5.50) 4.65 (0.22) 4.61 (0.21) -0.04 (0.31) -0.87 (6.72)[-0.91,0.16] [-18.60,2.95] [-0.65,0.57] [-14.03,12.29]

{0.1, 0.3, 0.6} 3.16 (0.16) 2.88 (0.12) -0.28 (0.20) -8.91 (6.00) 3.01 (0.20) 3.07 (0.16) 0.06 (0.26) 2.16 (8.84)[-0.68,0.11] [-20.66,2.84] [-0.45,0.58] [-15.17,19.49]

Notes: Mean values of k and k estimated from interval regression (Stewart, 1983) of experimental responseon indicators for probability vectors interacted with indicator for whether X > Y . Standard errors clusteredat individual level in parentheses. See Appendix Tables A3, columns (3) and (5) and Table A8 for detail.Panel A: No controls; 143 total subjects. Panel B: controls include age, gender, Cognitive Reflection Taskscore, and mean certainty equivalent from seven certainty equivalents tasks; 135 total subjects.

41Subjects who first faced X

0< Y (X > Y ) are 51% (45%) male, with Cognitive Reflection Test scores of 2.28

(2.02), and average certainty equivalents of 11.96 (11.35). Of these comparisons, the di↵erence in risk aversionhas a two-sided t-test p-value of 0.05 and the di↵erence in cognitive ability has a two-sided p-value of 0.15. Anomnibus test of selection from the regression of assignment to X

0< Y first on the controls of Table 4, Panel

B yields F (5, 134) = 1.60, p = 0.17), suggestive of the potential for selection on observables. These di↵erencesare of no consequence for the main portion of our analysis, which relies on within-subject variation, but couldbe influencing the results of Table 4, Panel A.

42 In Appendix Table A6, we also present a specification that includes controls for gender, age and cognitiveability, but omits our risk aversion proxy. The latter specification addresses any concerns that the averagecertainty equivalent in binary lottery tasks might be sensitive to the types of three-outcome lottery tasks(X 0

< Y versus X > Y ) the subject encounters first, a possibility we regard as remote. Results for thatspecification are similar to those reported in Table 4, Panel B.

26

4.4.1 Additional Robustness Exercises

We conduct several additional exercises to ensure the robustness of the current findings. In

Appendix E, we provide additional analyses examining alternative formulations for CPT using

the functional forms of Prelec (1998), exploring behavior in tasks where X = 25 (which implies

that adding m = $5 to Y = $24 changes the ranking), and including potentially confused sub-

jects who switch more than once in a given task. These exercises all yield the same conclusion:

rank dependence in risky choice and the predictions of CPT are soundly rejected.

5 Event Splitting and Violations of Dominance

Our analysis casts doubt on the empirical validity of the rank-dependent probability weighting

assumption that lies at the core of CPT. What type of model should behavioral economists

consider in its place? One possibility is that PT is correct, in which case people should actually

exhibit the implied violations of first-order stochastic dominance that motivated the formulation

of CPT in the first place. After we obtained our main results, we fielded an additional treatment

designed to investigate that hypothesis.

We conducted the follow-up experiment at Stanford University and UC San Diego during the

Spring and Fall of 2015. A total of 214 subjects completed the experiment, and 182 exhibited

no instances of multiple switching.43 We elicited certainty equivalents first for binary lotteries,

such as a 40% chance of receiving $30 and 60% chance or receiving $20. In one task the lower

payo↵ was more likely, and in another it was less likely. Then we elicited certainty equivalents

for related “split-event” lotteries, which we created by splitting the more likely event in each of

the binary lotteries. For the preceding example, a “split-event” lottery would take the following

form: a 40% chance of receiving $30, a 30% chance of receiving $20 + ✏, and a 30% chance of

receiving $20� ✏. Across tasks, ✏ took on the following values: $0.50, $1, $2, and $3. Subjects

43We recruited 126 at UC San Diego and 88 at Stanford. Those who exhibited no instances of multipleswitching include 99 subjects from UC San Diego and 83 from Stanford. Subjects from the two locations madequalitatively similar choices.

27

also completed a series of seven prospect theory elicitation tasks involving binary lotteries, as

before.

Most empirical parameterizations of PT imply that a 60% probability receives significantly

less than twice the weight of a 30% probability. Splitting an event occurring with 60% prob-

ability into two similar events, each occurring with 30% probability, should therefore increase

certainty equivalents dramatically. It follows that PT predicts a sharp downward discontinuity

at ✏ = 0. As noted in Section 1, such discontinuities imply violations of dominance. In con-

trast, CPT and EUT predict responses that vary smoothly with ✏ and thereby avoid dominance

violations.

Figure 5 presents our findings. As shown in Panel A, the prospect theory elicitation tasks

exhibit the hallmark pattern of probability weighting. In Panel B, we use the fitted probability

weighting to predict the e↵ect of event-splitting on certainty equivalents.44 The predicted

certainty equivalents vary smoothly with ✏ under CPT, but feature a sharp increase at zero

under PT.45 In contrast, the means of the actual certainty equivalents decrease sharply when

we split the low-outcome event, and then level o↵. Specifically, moving from ✏ = 0 to ✏ = 0.5

reduces the average certainty equivalent by $0.47 (clusted s.e. = 0.11), (z = 4.16, p < 0.01).46

We observe a qualitatively similar though somewhat muted pattern when we split the high-

outcome event.47

The findings in Panel B of Figure 5 are inconsistent with PT, CPT, and EUT, and therefore

call for an alternative explanation. A viable theory must account simultaneously for all three

patterns discussed in this study: (1) the inverse S-shaped certainty equivalent profile, (2)

44 Using the same estimation strategy as before, we obtain the following parameter values: ↵̂ = 0.975(clustered s.e. = 0.019) and �̂ = 0.671 (0.013).

45Notice that under CPT, splitting the low-outcome event leads to decreasing certainty equivalents whilesplitting the high outcome leads to increasing certainty equivalents.

46Test statistics are derived from interval regressions (Stewart, 1983) of certainty equivalents on indicatorsfor ✏. Standard errors are clustered at the subject level.

47 The apparent distaste for splitting an event, which turns a binary lottery into a trinary one, is not anartifact of di↵erences in presentation. As shown in Figure 5, certainty equivalents are essentially the sameregardless of whether we present a binary lottery as a 60%-40% gamble, or as a 30%-30%-40% gamble withidentical payo↵s for the first two events. Recall that we employed a single presentation of binary lotteries foreach subject, so this finding reflects between-subject comparisons.

28

Figure 5: Certainty Equivalents and Split Probabilities0

510

1520

25C

erta

inty

Equ

ival

ent

0 .2 .4 .6 .8 1Chance of $25

CERisk Neutral95% CICPT Model

Panel A: Certainty Equivalents

2325

27C

erta

inty

Equ

ival

ent

0 1 2 3Splitting Value, epsilon

Split Low, $20 Split High, $3095% CI CPT PredictionPT Prediction Version 60-40Version 30-30-40

Panel B: Split Probabilities

Notes: Panel A: Average certainty equivalent, C, estimated from an interval regression (Stewart, 1983) ofelicited certainty equivalents on the probability of winning $25. Confidence intervals based on standard errorsclustered at the subject level. Dashed line corresponds to CPT predictions with ↵̂ = 0.975 (s.e. = 0.019) and�̂ = 0.671 (0.013); standard errors clustered at individual level. Panel B: Average certainty equivalent, C,estimated from interval regressions of elicited certainty equivalents on the value of ✏. Confidence intervals basedon standard errors clustered at subject level. For ✏ = 0, separate averages reported based on presentation style(either 60%-40% or 30%-30%-40% with identical payo↵s for the first two events). Dashed line corresponds toCPT predictions assuming PT at aggregate parameters. Solid line corresponds to prediction assuming CPT ataggregate parameters.

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). Alternatives to CPT that

likewise incorporate rank-dependent probability weighting are also rejected.

One possibility is to reformulate PT probability weighting in terms of normalized weights

– that is, ⇡(pk)⇡(p1)+...+⇡(pK) rather than simply ⇡(pk). That model accounts for the downward

discontinuity observed when splitting the low-payo↵ event, but implies an upward discontinuity

when splitting the high-payo↵ event, which we do not observe. It also precludes the theory from

accounting for the well-known certainty e↵ect, which Allais famously described as a “preference

29

for security in the neighborhood of certainty” (Allais, 2008), for which there is considerable

evidence (see, e.g., Camerer, 1992; Harless and Camerer, 1994).

A second and more promising possibility is that the observed behavior reflects 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 certainty e↵ect as a

special case of this more general phenomenon. Under this hypothesis, su�ciently small values

of epsilon lead subjects to see a lottery as binary rather than trinary, which discretely increases

their certainty equivalents. Because the PT probability-weighting e↵ect works in the opposite

direction, the composite e↵ect of reducing ✏ to zero can be positive or negative.

The aforementioned theory can in principle account for the somewhat di↵erent patterns

observed when we split the high-payo↵ and low-payo↵ events. Higher stakes may reduce the

magnitude of the complexity e↵ect by making subjects more willing to ponder their prospects.

Additionally, if subjects think of discrepancies between payo↵s in relative terms, a higher payo↵

will tend to enlarge the “neighborhood” within which they implicitly “merge” events yielding

similar outcomes. Both implications are consistent with the pattern observed in Panel B of

Figure 5, but we acknowledge that this is an ex post rationalization for a somewhat limited

collection of results rather than a bona fide and systematic test of the theory.

It is important to acknowledge that complexity e↵ects, like PT, potentially give rise to

violations of dominance. However, the implied violations are explicable, because they involve

the selection of options that are easier to understand. To illustrate, in our experiment, splitting

a 60% chance of receiving $20 into a 30% chance of receiving $20.50 and a 30% chance of

receiving $19.50 decreases the certainty equivalent by $0.47. Assuming the addition of $0.50

to the $19.50 payo↵ increases the certainty equivalent by less than $0.47, we plainly have a

violation of dominance: the individual attaches more value to the original lottery than to the

revised split-event lottery even though the latter dominates the former.48

48Viewing complexity as a rationale for indirect violations of dominance between binary and trinary lotteriesmay also yield insights concerning other indirect dominance violations between binary and degenerate lotteries,such as those documented by Gneezy, List and Wu (2006); Andreoni and Sprenger (2011).

30

6 Conclusion

The main lessons of this study concern the empirical validity of rank-dependent probability

weighting, an assumption that lies at the core of Cumulative Prospect Theory (CPT). We

propose and implement an experimental test involving the elicitation of ‘equalizing reductions.’

CPT implies that these equalizing reductions should change discontinuously when a shift in

the payo↵ associated with an unrelated realization alters the ranking of payo↵s, but should

otherwise remain constant. Based on standard parameterizations of CPT as well as our own

estimates for our subject pool, these discontinuities should be substantial. And yet we find

no evidence of the predicted pattern at either the aggregate or individual level, based on both

within and between-subjects analysis. Thus we conclude that an empirical foundation for

rank-dependent probability weighting is absent.

Our findings pose serious challenges for future research on choice under uncertainty. If

CPT has taken the PT agenda in the wrong direction by promoting the assumption of rank-

dependent probability weighting (which several other recent theories embrace), how can we

reconcile PT with the presumed absence of implied dominance violations? As we demonstrate

in a supplemental experiment involving the e↵ects of “event splitting,” those violations indeed

do not arise. Instead, we observe a di↵erent type of violation that is at odds not only with

PT, but also with CPT and EUT. An important direction for future research is to explore

tractable explanations for this finding that do not involve rank-dependence. One potential

explanation involves a general form of aversion to complexity. That notion rationalizes the

behavioral patterns discussed in this study, but whether it survives rigorous and systematic

testing remains to be seen.

A potential defense of rank dependence is that it is an assumption of convenience: it renders

PT more tractable for applications, but is rarely required to account for observed behavior,

including anomalies. This argument strikes us as odd. First, rank dependence undeniably plays

a critical role in a number of applications, and is responsible for generating particular results.

We noted several examples in footnote 5. If there is no empirical support for rank dependence,

31

then plainly those applications require reexamination. Second, in other applications, either (1)

it is known that rank dependence is inessential for generating the results of interest, or (2) it

is not known. Case (1) cannot arise unless the researcher has conducted the analysis without

assuming rank dependence and found it tractable, in which case there is no reason to employ

rank dependence as an assumption of convenience. In case (2), the possibility remains that the

result of interest may be an artifact of an assumption that lacks empirical validity.

Additional research is plainly required to test the robustness of our findings. In twenty-

five years since the publication of Tversky and Kahneman (1992), there have been numerous

studies of probability weighting for binary lotteries, but decidedly few systematic studies of

rank dependence. Further work is needed to resolve the relevance of rank dependence in a

variety of experimental and naturally occurring contexts.

32

References

Abdellaoui, Mohammed, “Parameter-Free Elicitation of Utility and Probability Weighting Func-tions,” Management Science, 2000, 46 (11), 1497–1512.

Allais, Maurice, “Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulatset Axiomes de l’Ecole Americaine,” Econometrica, 1953, 21 (4), 503–546.

, “Allais Paradox,” in Steven N. Durlauf and Lawrence E. Blume, eds., The New Palgrave Dictionaryof Economics, 2nd ed., Palgrave Macmillan, 2008.

Andreoni, James and Charles Sprenger, “Uncertainty Equivalents: Testing the Limits of theIndependence Axiom,” Working Paper, 2011.

and , “Risk Preferences Are Not Time Preferences,” American Economic Review, 2012, 102 (7),3357–3376.

Barberis, Nicholas, Abhiropp Mukherjee, and Baolian Wang, “Prospect Theory and StockReturns: An Empirical Test,” Review of Financial Studies, Forthcoming.

Barseghyan, Levon, Francesca Molinari, Ted O’Donoghue, and Joshua C. Teitelbaum,“Bracketing and Risk Preferences: Identification and Estimation in Field Data,” Working Paper,2015.

Bell, David E., “Disappointment in Decision Making under Uncertainty,” Operations Research, 1985,33 (1), 1–27.

Bleichrodt, Han and Jose Luis Pinto, “A Parameter-Free Elicitation of the Probability WeightingFunction in Medical Decision Analysis,” Management Science, 2000, 46 (11), 1485–1496.

Booij, Adam S. and Gijs van de Kuilen, “A Parameter-Free Analysis of the Utility of Moneyfor the General Population Under Prospect Theory,” Journal of Economic Psychology, 2009, 30,651–666.

, Bernard M. S. van Praag, and Gijs van de Kuilen, “A Parametric Analysis of ProspectTheory’s Functionals for the General Population,” Theory and Decision, 2010, 68, 115–148.

Camerer, Colin F., “Recent Tests of Generalizations of Expected Utility Theory,” in Ward Edwards,ed., Utility: Theories, Measurement, and Applications, Kluwer: Norwell, MA, 1992, pp. 207–251.

Cubitt, Robin P., Chris Starmer, and Robert Sugden, “On the Validity of the Random LotteryIncentive System,” Experimental Economics, 1998, 1, 115–131.

Diecidue, Enrico and Peter P. Wakker, “On the Intuition of Rank-Dependent Utility,” TheJournal of Risk and Uncertainty, 2001, 23 (3), 281–298.

Epper, Thomas and Helga Fehr-Duda, “Balancing on a Budget Line: Comment on Andreoniand Sprenger (2012)’s ’Risk Preferences Are Not Time Preferences’,” American Economic Review,Forthcoming.

Fehr-Duda, Helga and Thomas Epper, “Probability and Risk: Foundations and Economic Im-plications of Probability-Dependent Risk Preferences,” Annual Review of Economics, 2012, 4 (567-593).

33

Fennema, Hein and Peter Wakker, “A Test of Rank-Dependent Utility in the Context of Ambi-guity,” Journal of Risk and Uncertainty, 1996, 13, 19–35.

Frederick, Shane, “Cognitive Reflection and Decision Making,” Journal of Economic Perspectives,2005, 19 (4), 25–42.

Friedman, Milton and Leonard J. Savage, “The Utility Analysis of Choices involving Risk,”Journal of Political Economy, 1948, 56 (4), 279–304.

Gneezy, Uri, John A. List, and George Wu, “The Uncertainty E↵ect: When a Risky ProspectIs Valued Less Than Its Worst Possible Outcome,” The Quarterly Journal of Economics, 2006, 121(4), 1283–1309.

Gonzalez, Richard and George Wu, “On the Shape of the Probability Weighting Function,”Cognitive Psychology, 1999, 38, 129–166.

Harless, David W. and Colin F. Camerer, “The Predictive Utility of Generalized ExpectedUtility Theories,” Econometrica, 1994, 62 (6), 1251–1289.

Harrison, Glenn W., Morten I. Lau, Elisabet E. Rutstrom, and Melonie B. Williams,“Eliciting risk and time preferences using field experiments: Some methodological issues,” in Je↵reyCarpenter, Glenn W. Harrison, and John A. List, eds., Field experiments in economics, Vol. Vol.10 (Research in Experimental Economics), Greenwich and London: JAI Press, 2005.

Holt, CA, “Preference reversals and the independence axiom,” The American Economic Review,1986, 76 (1), 508–515.

Holt, Charles A. and Susan K. Laury, “Risk Aversion and Incentive E↵ects,” The AmericanEconomic Review, 2002, 92 (5), 1644–1655.

Jacobson, Sarah and Ragan Petrie, “Learning from Mistakes: What Do Inconsistent ChoicesOver Risk Tell Us?,” Journal of Risk and Uncertainty, 2009, 38 (2).

Kahneman, Daniel and Amos Tversky, “Prospect Theory: An Analysis of Decision under Risk,”Econometrica, 1979, 47 (2), 263–291.

Karni, Edi and Zvi Safra, “”Preference Reversal” and the Observability of Preferences by Experi-mental Methods,” Econometrica, 1987, 55 (3), 675–685.

Koszegi, Botond and Matthew Rabin, “A Model of Reference-Dependent Preferences,” QuarterlyJournal of Economics, 2006, 121 (4), 1133–1165.

and , “Reference-Dependent Risk Attitudes,” The American Economic Review, 2007, 97 (4),1047–1073.

L’Haridon, Olivie and Laetitia Placido, “Betting on Machina’s Relection Example: An Experi-ment on Ambiguity,” Theory and Decision, 2010, 69, 375–393.

Loomes, Graham and Robert Sugden, “Disappointment and Dynamic Consistency in Choiceunder Uncertainty,” Review of Economic Studies, 1986, 53 (2), 271–82.

Lopes, Lola L., “Risk and Distributional Inequality,” Journal of Experimental Psychology: HumanPerception and Performance, 1984, 10 (4), 465–485.

34

and Gregg C. Oden, “The Role of Aspiration Level in Risky Choice: A Comparison of CumulativeProspect Theory and SP/A Theory,” Journal of Mathematical Psychology, 1999, 43, 286–313.

Meier, Stephan and Charles Sprenger, “Present-Biased Preferences and Credit Card Borrowing,”American Economic Journal - Applied Economics, 2010, 2 (1), 193–210.

Prelec, Drazen, “The Probability Weighting Function,” Econometrica, 1998, 66 (3), 497–527.

Quiggin, John, “A Theory of Anticipated Utility,” Journal of Economic Behavior and Organization,1982, 3, 323–343.

Samuelson, Paul A., “Probability, Utility, and the Independence Axiom,” Econometrica, 1952, 20(4), 670–678.

Savage, Leonard J., The Foundations of Statistics, New York: J. Wiley, 1954.

Sprenger, Charles, “An Endowment E↵ect for Risk: Experimental Tests of Stochastic ReferencePoints,” Journal of Political Economy, Forthcoming.

Starmer, Chris and Robert Sugden, “Does the Random-Lottery Incentive System Elicit TruePreferences? An Experimental Investigation,” The American Economic Review, 1991, 81 (4), 971–978.

Stewart, Mark B., “On Least Squares Estimation when the Dependent Variable is Grouped,” TheReview of Economic Studies, 1983, 50 (4), 737–753.

Tanaka, Tomomi, Colin Camerer, and Quang Nguyen, “Risk and time preferences: Experi-mental and household data from Vietnam,” American Economic Review, 2010, 100 (1), 557–571.

Tversky, Amos and Craig R. Fox, “Weighing Risk and Uncertainty,” Psychological Review, 1995,102 (2), 269–283.

and Daniel Kahneman, “Advances in Prospect Theory: Cumulative Representation of Uncer-tainty,” Journal of Risk and Uncertainty, 1992, 5 (4), 297–323.

von Neumann, John and Oskar Morgenstern, Theory of Games and Economic Behavior, Prince-ton University Press, 1944.

Wakker, Peter, Ido Erev, and Elke U. Weber, “Comonotonic Independence: The Critical TestBetween Classical and Rank Dependent Utility Theories,” Journal of Risk and Uncertainty, 1994,9, 195–230.

Weber, Elke U., “From Subjective Probabilities to Decision Weights: The E↵ect of Asymmetric LossFunctions on the Evaluation of Uncertain Outcomes and Events,” Psychological Bulletin, 1994, 115(2), 228–242.

Wu, George, “An Empirical Test of Ordinal Independence,” Journal of Risk and Uncertainty, 1994,9, 39–60.

and Richard Gonzalez, “Curvature of the Probability Weighting Function,” Management Sci-ence, 1996, 42 (12), 1676–1690.

35

A Re-examination of Prior Prospect Theory ElicitationData

Experiments designed to elicit prospect theory parameters such as Tversky and Kahneman(1992), Tversky and Fox (1995), and Gonzalez and Wu (1999) generally have subjects providecertainty equivalents for binary lotteries. Such data may potentially shed light on rank depen-dence provided an appropriate range of outcomes are considered. For example, Tversky andKahneman (1992) elicit certainty equivalents for a 10%, 50% and 90% chance of receiving $50with the alternative being zero, and also elicit certainty equivalents for a 10%, 50% and 90%chance of receiving $50 with the alternative being $100. Such data can potentially shed lighton whether a given probability of receiving $50 is weighted di↵erently depending on its rank.Under CPT decisionmaking, when the alternative payment is $0, the relevant weighting of ap-probability of receiving $50 is ⇡(p), while when the alternative is $100 it is 1� ⇡(1� p).

When the alternative is $0, Tversky and Kahneman (1992) report median certainty equiva-lents for p 2 {0.1, 0.5, 0.9} of {$9, $21, $37}. When the alternative is $100, the median certaintyequivalents for the same probabilities are {$59, $71, $83}. These data are ideal for examiningwhether the wighting of a given chance of receiving $50 is rank dependent. Given the twoparameter model and non-linear estimation techniques noted in section 4.1, with three obser-vations we can estimate both probability weighting, � and utility function curvature, ↵, withone degree of freedom in each series. Conducting such an exercise on the reported median datawhen the alternative payment is $0, we find � = 0.64 and ↵ = 0.98. The same exercise whenthe alternative payment is $50 yields � = 0.57 and ↵ = 1.01. These findings suggest substan-tial rank dependence in choice. Consider a 10% chance of receiving $50 when the alternativeis $0. With � = 0.64, ⇡(0.1) = 0.18. Now, consider a 10% chance of receiving $50 when thealternative is $100. With � = 0.57, ⇡(0.9) = 0.67 and 1 � ⇡(0.9) = 0.33. The 10% chance of$50 is either treated like 18% or 33% depending on what the alternative is. This conclusion isbarely altered if one imposes a common ↵ = 0.98 across the two data sets (the estimated � forthe alternative of $100 remains at 0.57).

Figure A1, Panel A provides graphical intuition presenting the weighted expected utilitiesfor lotteries over $50 and $0 and over $50 and $100. Panel A assumes a global shape forutility, ↵ = 0.98, and � = 0.64 when the alternative is $0 and � = 0.57 when the alternativeif $50. The blue vertical line corresponds to the expected value of a 10% chance of receiving$50 when the alternative is $0 and the red vertical line corresponds to the expected value of a10% chance of receiving $50 when the alternative is $100. The median certainty equivalents arepresented as the blue and red dots respectively, with their calculated utilities provided as thetwo horizontal bars. That the horizontal and vertical bars intersect at the weighted expectedutility in each case is evidence of the model fitting the data (i.e., predicting well the mediancertainty equivalent). The slight risk tolerance when considering a 10% chance of $50 withan alternative of zero is consistent with overweighting of the 10% chance. The more apparentrisk aversion when considering a 10% chance of $50 with an alternative of $100 is consistentwith sizable underweighting of the 90% chance of $100. But this substantial underweightingimplies even more dramatic overweighting of the 10% chance of $50 in this setting. The dataand corresponding estimates are suggestive of a rank dependence in probability weighting.

The exercise above assumes a global shape to utility, ↵ = 0.98. Importantly, this assumption

1

Figure A1: Rank Dependence in Binary Lotteries

0 20 40 60 80 100

020

4060

80

Panel A: Rank Dependent Interpretation

Outcome

Util

ity

●

●

●

●

Implied Weighting of Chance for $50

10% $50, 90% $0: π(0.1) = 0.18

10% $50, 90% $100: 1−π(0.9) = 0.33

0 20 40 60 80 100

020

4060

80

Panel B: Non−Rank Dependent Interpretation

OutcomeU

tility

●

●

●

●

Implied Weighting of Chance for $50

10% $50, 90% $0: π(0.1) = 0.18

10% $50, 90% $100: 1−π(0.9) = 0.18

is not innocuous. For example, if instead one assumed linear utility up to a certain pointand then applied substantial curvature thereafter, a qualitatively di↵erent conclusion wouldbe reached. Figure A1, Panel B presents results assuming that until x = 50, u(x) = x.Thereafter, u(x) = 50 + (x � 50)0.3, mimicking sharply diminishing marginal utility after acertain point. Under such a simple formulation for utility, one estimates � = 0.63 when thealternative is $0 and � = 0.76 when the alternative is $100. The corresponding weightedexpected utilities are again presented for each of these cases along with expected values andcertainty equivalents for the 10% chance of receiving $50. Once again, the estimated modelsfit the data quite well. In contrast to the prior results, however, there is virtually no evidenceof rank dependence. Consider a 10% chance of receiving $50 when the alternative is $0. With� = 0.63, ⇡(0.1) = 0.18. Now, consider a 10% chance of receiving $50 when the alternative is$100. With � = 0.76, ⇡(0.9) = 0.82 and 1 � ⇡(0.9) = 0.18. The 10% chance of $50 is treatedexactly the same regardless of the alternative.

This exercise demonstrates that interpreting data from binary lotteries as evidence for (oragainst) rank dependence can be quite sensitive. Depending on what assumes about the shapeof utility can lead to qualitative di↵erences in the extent of apparent rank dependence. Oneclear benefit of our proposed test of rank dependence is that at its core it is free from functionalform assumptions both for the shape of utility and probability weighting.

B Equalizing Reductions Under Di↵erent ReferencePoint Formulations

This section investigates the predictions of CPT decisionmaking under alternative formulationsof the reference point. Under CPT, the decisionmaker is assumed to separate gains and losses

2

and weight the corresponding probabilities separately. Gains are weighted according to thecumulative distribution beginning with the best possible outcome, while losses are weightedaccording to the decumulative distribution beginning with the worst possible outcome. CPTalso allows for di↵erences in the extent of probability weighting for gains and losses, ⇡+(·) and⇡�(·), and the shape of utility for gains and losses, u+(·) and u�(·).

A first observation is that when r lies above all possible outcomes, such that any potentialoutcome is a loss, the above insights are qualitatively unchanged. CPT again predicts potentialdiscontinuities in equalizing reductions as ranks change. This is also true for cases where thereference point lies between two outcomes, for cases when the reduction of k from Z induces achange from a gain to a loss, and for cases where adding m to Y induces a change from a lossto a gain. This last case is critical as adding m to Y changes the treatment of gains and losseswhile changing X to X 0 changes the ranks. Simulations identical to those presented in themain text show that even with both events occurring at the same time, substantial di↵erencesin equalizing reductions are observed. This development indicates that the predictions of CPTwith respect to equalizing reductions do not overly depend on the location of a fixed referencepoint. In the next subsections we analyze additional models of reference dependence that allowthe reference point to change as lotteries changes.

B.1 When All Outcomes Are Losses

When r > X > Y > Z and r > X > Y + m > Z � k, the equalizing reduction is identifiedfrom the indi↵erence condition

⇡�(1� p� q)u�(Z|r)+[⇡�(q + 1� p� q)� ⇡�(1� p� q)]u�(Y |r)

+[1� ⇡�(q + 1� p� q)]u�(X|r) =⇡�(1� p� q)u�(Z � k|r)

+[⇡(q + 1� p� q)� � ⇡�(1� p� q)]u�(Y +m|r)+[1� ⇡(q + 1� p� q)�]u�(X|r),

yielding

k = Z � u�1�

✓u�(Z|r)�

⇡�(q + 1� p� q)� ⇡�(1� p� q)

⇡�(1� p� q)

�[u�(Y +m|r)� u�(Y |r)]

◆.

When r > Y > X 0 > Z and r > Y +m > X 0 > Z � k, the equalizing reduction is identifiedfrom the indi↵erence condition

⇡�(1� p� q)u�(Z|r)+[⇡�(p+ 1� p� q)� ⇡�(1� p� q)]u�(X

0|r)+[1� ⇡�(p+ 1� p� q)]u�(Y |r) =

⇡�(1� p� q)u�(Z � k|r)+[⇡�(p+ 1� p� q)� ⇡�(1� p� q)]u�(X

0|r)+[1� ⇡(q + 1� p� q)�]u�(Y |r),

3

yielding

k = Z � u�1�

✓u�(Z|r)�

1� ⇡�(p+ 1� p� q)

⇡�(1� p� q)

�[u�(Y +m|r)� u�(Y |r)]

◆.

Again, k and k will not generally be equal unless ⇡(·) is linear, yielding similar potential fordiscontinuities as observed in our prior development.

B.2 When the Reference Point Lies Between Two Outcomes

When X > Y > r > Z and X > Y + m > r > Z � k, the equalizing reduction is identifiedfrom the indi↵erence condition

⇡+(p)u+(X|r) + [⇡+(p+ q)� ⇡+(p)]u+(Y |r) + ⇡�(1� p� q)u�(Z|r) =⇡+(p)u+(X|r) + [⇡+(p+ q)� ⇡+(p)]u+(Y +m|r) + ⇡�(1� p� q)u�(Z � k|r)

yielding

k = Z � u�1�

✓u�(Z|r)�

⇡+(p+ q)� ⇡+(p)

⇡�(1� p� q)

�[u+(Y +m|r)� u+(Y |r)]

◆.

When Y > X 0 > r > Z and Y +m > X 0 > r > Z � k, the equalizing reduction is identifiedfrom the indi↵erence condition

⇡+(q)u+(Y |r) + [⇡+(p+ q)� ⇡+(q)]u+(X0|r) + ⇡�(1� p� q)u�(Z|r) =

⇡+(q)u+(Y +m|r) + [⇡+(p+ q)� ⇡+(q)]u+(X0|r) + ⇡�(1� p� q)u�(Z � k|r)

yielding

k = Z � u�1�

✓u�(Z|r)�

⇡+(q)

⇡�(1� p� q)

�[u+(Y +m|r)� u+(Y |r)]

◆.

B.3 When Subtracting k from Z Induces a Loss

When X > Y > Z > r and X > Y + m > r > Z � k, the equalizing reduction is identifiedfrom the indi↵erence condition

⇡+(p)u+(X|r) + [⇡+(p+ q)� ⇡+(p)]u+(Y |r) + [1� ⇡+(p+ q)]u+(Z|r) =⇡+(p)u+(X|r) + [⇡+(p+ q)� ⇡+(p)]u+(Y +m|r) + ⇡�(1� p� q)u�(Z � k|r),

yielding

k = Z � u�1�

✓1� ⇡+(p+ q)

⇡�(1� p� q)

�u+(Z|r)�

⇡+(p+ q)� ⇡+(p)

⇡�(1� p� q)

�[u+(Y +m|r)� u+(Y |r)]

◆.

4

When Y > X 0 > Z > r and Y +m > X 0 > r > Z � k, the equalizing reduction is identifiedfrom the indi↵erence condition

⇡+(q)u+(Y |r) + [⇡+(p+ q)� ⇡+(q)]u+(X0|r) + [1� ⇡+(p+ q)]u+(Z|r) =

⇡+(q)u+(Y +m|r) + [⇡+(p+ q)� ⇡+(q)]u+(X0|r) + ⇡�(1� p� q)u�(Z � k|r),

yielding

k = Z � u�1�

✓1� ⇡+(p+ q)

⇡�(1� p� q)

�u+(Z|r)�

⇡+(q)

⇡�(1� p� q)

�[u+(Y +m|r)� u+(Y |r)]

◆.

B.4 When Adding m to Y Induces a Gain

When X > r > Y > Z and X > Y + m > r > Z � k, the equalizing reduction is identifiedfrom the indi↵erence condition

⇡+(p)u+(X|r) + ⇡�(1� p� q)u�(Z|r) + [⇡�(q + 1� p� q)� ⇡�(1� p� q)]u�(Y |r) =⇡+(p)u+(X|r) + ⇡�(1� p� q)u�(Z � k|r) + [⇡+(p+ q)� ⇡+(p)]u+(Y +m|r),

yielding

k = Z � u�1� ((u�(Z|r)�

⇡+(p+ q)� ⇡+(p)

⇡�(1� p� q)

�u+(Y +m|r)

�⇡�(q + 1� p� q)� ⇡�(1� p� q)

⇡�(1� p� q)

�u�(Y |r)).

When r > Y > X 0 > Z and Y + m > r > x01 > Z � k, the equalizing reduction is identified

from the indi↵erence condition

⇡�(1� p� q)u�(Z|r) + [⇡�(p+ 1� p� q)� ⇡�(1� p� q)]u�(X0|r)

+[1� ⇡�(p+ 1� p� q)]u�(Y |r) =⇡�(1� p� q)u�(Z � k|r) + [⇡�(p+ 1� p� q)� ⇡�(1� p� q)]u�(X

0|r) + ⇡+(q)u+(Y +m|r),

yielding

k = Z � u�1�

✓u�(Z|r)�

⇡+(q)

⇡�(1� p� q)

�u+(Y +m|r)�

1� ⇡�(p+ 1� p� q)

⇡�(1� p� q)

�u�(Y |r)

◆.

Note that when adding m to Y induces a gain, two things occur as X is reduced below Y :X switches from gain to loss, while Y switches from loss to gain. In principle, it is possible forthese two forces to o↵set yielding limited discontinuities in practice. In Table A1, we conductanalysis similar to Table 1 assuming a reference point of r = $25. Following the empiricalfindings of Tversky and Kahneman (1992), we assume that gain and loss probability weightingare equivalent, ⇡�(p) = ⇡+(p) = p�/(p� + (1 � p)�)1/�. We also assume a piecewise linearformulation for loss averse utility such that u�(�x) = ��u+(x) with u+(x) = x. Acrosscolumns, we vary the value of � to demonstrate the force of loss aversion for the obtained

5

values. Across values of loss aversion, substantial discontinuities in equalizing reductions areobserved.

Table A1: Simulated Equalizing Reductions When Adding m to Y Induces a Gain

� = 3 � = 2 � = 1.5 � = 1.05

{p, q, 1� p� q} k k k - k k k k- k k k k - k k k k - k(% Change) (% Change) (% Change) (% Change)

{0.6, 0.3, 0.1} 4.78 2.69 -2.09 5.92 3.54 -2.38 7.06 4.39 -2.67 9.01 5.85 -3.16(-44%) (-40%) (-38%) (-35%)

{0.4, 0.3, 0.3} 2.80 1.17 -1.62 3.46 1.52 -1.95 4.13 1.86 -2.27 5.27 2.45 -2.83(-58%) (-56%) (-55%) (-54%)

{0.1, 0.3, 0.6} 1.88 1.02 -0.86 2..33 1.28 -1.05 2.78 1.54 -1.24 3.54 1.98 -1.56(-46%) (-45%) (-45%) (44%)

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 0

, Y, Z} = {$23, $24, $18}, m = $5, r = $25 throughout. CPTcalculations with ⇡(p) = p

�/(p� + (1 � p)�)1/� with � = 0.61; u�(�x) = ��u+(x) with u+(x) = x with �

varying by column

6

C Additional Models

C.1 KR Preferences

Koszegi and Rabin (2006, 2007) (KR) build upon standard reference-dependent preferences inone important way: the referent is expectations-based, taken to be the entire distribution ofexpected outcomes.49

Let r represent a possible reference point drawn according to measure G. Let x be anoutcome drawn according to measure F . Then the KR utility formulation is the double integral

U(F |G) =

ZZu(x|r)dG(r)dF (x)

withu(x|r) = m(x) + µ(m(x)�m(r)).

Every possible outcome of F is compared to every possible reference point of G and gains andlosses are evaluated. The function m(·) represents standard utility and µ(·) represents gain-lossutility relative to the reference point, r. Several simplifying assumptions are made. First, weassume m(x) = x to demonstrate the forces of changing reference points alone for equalizingreductions. Second, following Koszegi and Rabin (2006, 2007), we assume a piecewise-lineargain-loss utility function,

µ(y) =

⇢⌘ · y if y � 0⌘ · � · y if y < 0

�,

where the parameter ⌘ captures sensitivity to gains and losses and � represents the degree ofloss aversion.

Under these preferences, one can construct the utility of lottery L, which yields X > Y > Zwith probabilities p, q, (1� p� q),

UKR(L|L) = p (p[X] + q[Y + ⌘�(Y �X)] + (1� p� q)[Z + ⌘�(Z �X)]) +

q (p[X + ⌘(X � Y )] + q[Y ] + (1� p� q)[Z + ⌘�(Z � Y )]) +

(1� p� q) (p[X + ⌘(X � Z)] + q[Y + ⌘(Y � Z)] + (1� p� q)[Z]) ,

49An additional innovation of Koszegi and Rabin (2006, 2007) is a rational expectations equilibrium concept,the Unacclimating Personal Equilibrium (UPE). The objective of the UPE concept is to represent the notionthat rational individuals will only expect as the referent outcomes that they will definitely consume given theexpectation of said outcomes. Hence, consumption outcomes and referents coincide in UPE. To select among thepotential multiplicity of such equilibria, the KR model features a refinement, the Preferred Personal Equilibrium(PPE). The PPE concept maintains that the UPE with the highest ex-ante expected utility is selected. Anotherequilibrium concept in Koszegi and Rabin (2006, 2007) is the Choice-acclimating Personal Equilibrium (CPE)which applies to decisions made far in advance of the resolution of uncertainty. The operational distinctionbetween the two concepts is that a CPE need not be UPE, but a PPE must be UPE. In the present contextCPE and PPE have similar implications, as both are based on the coincidence of referent and consumptionoutcomes. We implicitly apply CPE/PPE in constructing the predictions of KR. That is, we assume theequalizing reduction corresponds to the point where the the decisionmaker switches from choosing L to Le inCPE/PPE. Sprenger (Forthcoming) demonstrates that such equilibrium behavior may have limited support inrisky choice experiments.

7

or

UKR(L|L) = (p+ pq⌘(1� �) + p(1� p� q)⌘(1� �))X +

(q + pq⌘(�� 1) + q(1� p� q)⌘(1� �))Y +

((1� p� q) + p(1� p� q)⌘(�� 1) + q(1� p� q)⌘(�� 1))Z.

Increasing Y to Y +m < X does not alter any of these gain loss comparisons.50 Hence, theequalizing reduction for increasing Y to Y +m is

k =(q + pq⌘(�� 1) + q(1� p� q)⌘(1� �))

((1� p� q) + p(1� p� q)⌘(�� 1) + q(1� p� q)⌘(�� 1))m.

Similarly, one can evaluate the lottery L0 with which yields Y > X 0 > Z with identical proba-bilities to construct

U(L0|L0) = (p+ pq⌘(�� 1) + p(1� p� q)⌘(1� �))X 0 +

(q + pq⌘(1� �) + q(1� p� q)⌘(1� �))Y +

((1� p� q) + p(1� p� q)⌘(�� 1) + q(1� p� q)⌘(�� 1))Z.

Critically, because Y now lies above X 0, the gain loss comparisons are changed relative to theprior formulation. Increasing Y further induces no further changes to these comparisons.51

Hence, the equalizing reduction, k is identified as

k =(q + pq⌘(1� �) + q(1� p� q)⌘(1� �))

((1� p� q) + p(1� p� q)⌘(�� 1) + q(1� p� q)⌘(�� 1))m.

Note that as X passes below Y , a discontinuity is generated such that k and k will di↵er. Thus,even though KR does not assume rank-dependent probability weighting, its implications arequalitatively similar to those of CPT, and we can test it in the same way.

50 If we consider X > Y + m > Z � k, the ranks remain unchanged and the indi↵erence condition,

(p + pq⌘(1 � �) + p(1 � p� q)⌘(1 � �))X + (q + pq⌘(�� 1) + q(1 � p� q)⌘(1 � �))Y +

((1 � p� q) + p(1 � p� q)⌘(�� 1) + q(1 � p� q)⌘(�� 1))Z =

(p + pq⌘(1 � �) + p(1 � p� q)⌘(1 � �))X + (q + pq⌘(�� 1) + q(1 � p� q)⌘(1 � �))(Y + m) +

((1 � p� q) + p(1 � p� q)⌘(�� 1) + q(1 � p� q)⌘(�� 1))(Z � k),

identifies the equalizing reduction, k.51 If we consider Y + m > X

0> Z � k, the ranks remain unchanged and the indi↵erence condition,

(p + pq⌘(�� 1) + p(1 � p� q)⌘(1 � �))X 0 + (q + pq⌘(1 � �) + q(1 � p� q)⌘(1 � �))Y +

((1 � p� q) + p(1 � p� q)⌘(�� 1) + q(1 � p� q)⌘(�� 1))Z =

(p + pq⌘(�� 1) + p(1 � p� q)⌘(1 � �))X 0 + (q + pq⌘(1 � �) + q(1 � p� q)⌘(1 � �))(Y + m) +

((1 � p� q) + p(1 � p� q)⌘(�� 1) + q(1 � p� q)⌘(�� 1))(Z � k)

identifies the equalizing reduction, k.

8

C.2 DA Preferences

Disappointment Aversion (Bell, 1985; Loomes and Sugden, 1986) (DA) provides an alternativeformulation of expectations-based reference dependence. Instead of the entire distribution ofexpected outcomes, a lottery’s reference point is its expected utility certainty equivalent, c.When constructing the DA utility for the lottery, L, X, Y and Z are all compared to c. AsX > Y > Z, X will be considered a gain and Z will be considered a loss.52 When reducing Xto X 0 < Y , c is reduced to c0. If X 0 remains a gain relative to the induced certainty equivalent,c0, no gain-loss comparisons are altered. This implies that no discontinuity should be presentas ranks change, k = k. Importantly, however, if X is further reduced to X 00, such that X 00 is aloss relative to the induced certainty equivalent, c00, the equalizing reduction will change. Thatis, disappointment aversion has the potential to deliver a discontinuity, but the discontinuityoccurs at a di↵erent location than with CPT and KR. Discontinuities in equalizing reductionswill occur when X passes below the induced lottery’s certainty equivalent, not when X passesbelow Y .

We demonstrate this possibility for a standard formulation of DA decisionmaking. Considerthe lottery, L, which yields X > Y > Z with corresponding probabilities p, q, 1 � p � q. Thedisappointment averse representation of (L) is

UDA(L) = pu(X|c) + qu(Y |c) + (1� p� q)u(Z|c),

wherec = v�1 (pv(X) + qv(Y ) + (1� p� q)v(Z)) ,

with v(·) representing a standard expected utility function. As before, we will assume thatv(x) = x, to demonstrate the forces of changing reference points alone for equalizing reductions,and consider the piece-wise linear formulation for u(x|r). Note that X > c is a gain relative toc, and Z < c is a loss. For simplicity, we first consider the case where Y is also a gain, so thatthe disappointment averse utility is

UDA(L) = p[X + ⌘(X � pX � qY � (1� p� q)Z)] +

q[Y + ⌘(Y � pX � qY � (1� p� q)Z)] +

(1� p� q)[Z + ⌘�(Z � pX � qY � (1� p� q)Z)]

or

UDA(L) = (p+ ⌘p� ⌘p2 � ⌘pq � ⌘�p(1� p� q))X +

(q + ⌘q � ⌘pq � ⌘q2 � ⌘�q(1� p� q))Y +

((1� p� q) + ⌘�(1� p� q)� ⌘p(1� p� q)� ⌘q(1� p� q)� ⌘�(1� p� q)2)Z.

As Y is assumed to be treated as a gain relative to c, increasing Y by m and decreasing Z

52Whether Y is a loss or a gain depends on the exact values, probabilities, and shape of the utility function.In Appendix C, we analyze the case when Y is a gain.

9

by k does not alter the treatment of gains and losses.53 Hence, the equalizing reduction forincreasing Y to Y +m < X is

k =

q + ⌘q � ⌘pq � ⌘q2 � ⌘�q(1� p� q)

(1� p� q) + ⌘�(1� p� q)� ⌘p(1� p� q)� ⌘q(1� p� q)� ⌘�(1� p� q)2

�m.

Now consider the lottery L0, which yields Y > X 0 > Z. If X 0 is a gain, X 0 > c0 = pX 0 + qY +(1� p� q)Z, then the disappointment averse utility is

UDA(L0) = p[X 0 + ⌘(X � pX 0 � qY � (1� p� q)Z)] +

q[Y + ⌘(Y � pX 0 � qY � (1� p� q)Z)] +

(1� p� q)[Z + ⌘�(Z � pX 0 � qY � (1� p� q)Z)].

For additions of m to Y that do not alter the treatment of X 0 as a gain, the equalizing reductionis again

k =

q + ⌘q � ⌘pq � ⌘q2 � ⌘�q(1� p� q)

(1� p� q) + ⌘�(1� p� q)� ⌘p(1� p� q)� ⌘q(1� p� q)� ⌘�(1� p� q)2

�m,

with the implication that no discontinuity should be present as the ranks change, k = k.Importantly, however, if X 0 is reduced further to X 00 < c00 = pX 00 + qY +(1� p� q)Z, then X 00

will be treated as a loss. As such the disappointment averse utility will be

UDA(L00) = p[X 00 + ⌘�(X � pX 00 � qY � (1� p� q)Z)] +

q[Y + ⌘(Y � pX 00 � qY � (1� p� q)Z)] +

(1� p� q)[Z + ⌘�(Z � pX 00 � qY � (1� p� q)Z)].

Provided the addition of m and reduction of k does not further alter the treatment of X 00, theequalizing reduction will be

k =

q + ⌘q � ⌘�pq � ⌘q2 � ⌘�q(1� p� q)

(1� p� q) + ⌘�(1� p� q)� ⌘�p(1� p� q)� ⌘q(1� p� q)� ⌘�(1� p� q)2

�m,

di↵erent from k and k.

53The equalizing reduction, k is identified from the indi↵erence condition

(p + ⌘p� ⌘p

2 � ⌘pq � ⌘�p(1 � p� q))X +

(q + ⌘q � ⌘pq � ⌘q

2 � ⌘�q(1 � p� q))Y +

((1 � p� q) + ⌘�(1 � p� q) � ⌘p(1 � p� q) � ⌘q(1 � p� q) � ⌘�(1 � p� q)2)Z =

(p + ⌘p� ⌘p

2 � ⌘pq � ⌘�p(1 � p� q))X +

(q + ⌘q � ⌘pq � ⌘q

2 � ⌘�q(1 � p� q))(Y + m)

((1 � p� q) + ⌘�(1 � p� q) � ⌘p(1 � p� q) � ⌘q(1 � p� q) � ⌘�(1 � p� q)2)(Z � k)

.

10

D Additional Tables and Figures

The following tables are referenced in the main text and Appendix E

Table A2: Certainty Equivalents

Certainty Equivalents Risk Premia(1) (2)

p = 0.05 2.88 1.63***(0.19) (0.19)

p = 0.10 3.83 1.33***(0.19) (0.19)

p = 0.25 6.45 0.20(0.17) (0.17)

p = 0.50 10.72 -1.78***(0.23) (0.23)

p = 0.75 15.44 -3.31***(0.31) (0.31)

p = 0.90 19.83 -2.67***(0.29) (0.29)

p = 0.95 21.63 -2.12***(0.24) (0.24)

Notes: Coe�cients for certainty equivalents and risk pre-

mia calculated from interval regression of certainty equiva-

lent on indicators for probability. Standard errors clustered

on individual level in parentheses. Result of �2(1) test for

the null hypothesis of risk neutrality (risk premium equal

to zero) presented in column (2). Levels of significance: *

0.10, ** 0.05, *** 0.01.

11

Table A3: Equalizing Reductions Within and Between Subjects

(1) (2) (3) (4) (5) (6)

{p, q, 1� p� q} = {0.4, 3, 0.3} -4.72*** -4.72*** -5.03*** -5.13*** -5.13*** -5.13***(0.31) (0.17) (0.60) (0.60) (0.60) (0.60)

{p, q, 1� p� q} = {0.1, 3, 0.6} -6.40*** -6.40*** -6.65*** -6.77*** -6.77*** -6.77***(0.37) (0.18) (0.68) (0.68) (0.68) (0.68)

(X > Y ) 0.26 0.26 -1.10 -0.93 -0.64 -0.74(0.17) (0.22) (0.85) (0.87) (0.83) (0.83)

(X > Y ) ⇥ {0.4, 3, 0.3} -0.22 -0.22 0.73 0.60 0.60 0.60(0.16) (0.24) (0.75) (0.76) (0.76) (0.76)

(X > Y ) ⇥ {0.1, 3, 0.6} -0.33 -0.33 0.82 0.71 0.71 0.71(0.18) (0.26) (0.88) (0.89) (0.89) (0.89)

19 < Age < 22 -0.10 -0.24(0.41) (0.43)

Age � 22 -0.33 -0.46(0.45) (0.46)

Male 0.89** 0.98**(0.39) (0.39)

Cognitve Reflect Test 0.41** 0.40**(0.17) (0.17)

Avg. Certainty Equivalent 0.19**(0.08)

Constant 9.02 7.44 9.81 9.92 6.37 8.73(0.39) (0.59) (0.65) (0.65) (1.17) (0.77)

Predicted {0.6, 3, 0.1} 9.02 9.02 9.81 9.92 9.77 9.82(0.39) (0.16) (0.65) (0.65) (0.60) (0.60)

H0: No Rank Dependence �2(3) = 4.50 �2(3) = 1.82 �2(3) = 3.76 �2(3) = 2.50 �2(3) = 0.64 �2(3) = 0.86(p = 0.21) (p = 0.61) (p = 0.29) (p = 0.47) (p = 0.89) (p = 0.84)

Fixed E↵ects No Yes No No No NoFirst Block of Tasks Only No No Yes Yes Yes YesDemographic Controls No No No No Yes Yes# Observations 2574 2574 429 405 405 405# Subjects 143 143 143 135 135 135Log-Likelihood -8891.80 -8191.34 -1481.49 -1393.60 -1379.56 -1382.05

Notes: Coe�cients from interval regression of equalizing reduction on indicators for probability series {p, q, 1 � p � q} and order of outcome

X > Y . Standard errors clustered on individual level in columns (1), (3), (4), (5), (6). Robust standard errors in parentheses in column (2).

Column (4) restricts Column (3) sample to 135 Individuals with full control information. Constant, omitted category, is {p, q, 1 � p � q} =

{0.6, 3, 0.1} with X

0< Y . Predicted average for {p, q, 1 � p� q} = {0.6, 3, 0.1} in (2), (5), (6) calculated as average of of fixed e↵ects or at the

average level of controls. Tested null hypothesis of no rank dependence corresponds to test that coe�cients (X > Y ), (X > Y ) ⇥ {0.4, 3, 0.3},(X > Y ) ⇥ {0.1, 3, 0.6} all equal zero. Levels of significance: * 0.10, ** 0.05, *** 0.01.

12

Table A4: Equalizing Reductions for All Conditions

k k(1) (2) (3) (4) (5) (6) (7) (8)

{p, q, 1� p� q} X 0 = 19 X 0 = 21 X 0 = 23 X 0 < 24 X = 30 X = 32 X = 34 X > 24

{0.6, 0.3, 0.1} 9.03 9.03 9.02 9.02 9.24 9.44 9.17 9.28(0.41) (0.40) (0.42) (0.39) (0.41) (0.42) (0.40) (0.38)

{0.4, 0.3, 0.3} 4.33 4.22 4.37 4.31 4.30 4.34 4.38 4.34(0.14) (0.13) (0.14) (0.12) (0.14) (0.15) (0.13) (0.12)

{0.1, 0.3, 0.6} 2.65 2.60 2.64 2.63 2.58 2.52 2.57 2.56(0.09) (0.11) (0.11) (0.08) (0.08) (0.08) (0.09) (0.07)

Notes: Coe�cients calculated from interval regression of equalizing reduction on indicators for probability set, value

of X/X

0 and all interactions. Standard errors clustered on individual level in parentheses. Columns (4) and (8)

provide estimated averages for k and k for columns (1)-(3) and (5)-(7), respectively.

Table A5: Equalizing Reductions with Fixed E↵ects



{0.6, 0.3, 0.1} 9.02 (0.16) 9.28 (0.16) 0.26 (0.22) 2.87 (2.52) 7.58 (0.22) 6.06 (0.21) -1.52 (0.01) -20.04 (0.53)[-0.18,0.70] [-2.07,7.82] [-1.55,-1.49] [-21.07,-19.01]

{0.4, 0.3, 0.3} 4.31 (0.07) 4.34 (0.07) 0.04 (0.10) 0.86 (2.25) 4.01 (0.06) 2.52 (0.08) -1.49 (0.03) -37.17 (1.09)[-0.15,0.23] [-3.55,5.27] [-1.54,-1.44] [-39.31,-35.03]

{0.1, 0.3, 0.6} 2.63 (0.09) 2.56 (0.09) -0.07 (0.13) -2.74 (4.71) 2.65 (0.01) 1.87 (0.03) -0.78 (0.03) -29.34 (1.12)[-0.32,0.17] [-11.99,6.50] [-0.83,-0.72] [-31.53,-27.14]

Notes: Panel A: Mean behavior for k and k estimated from interval regression (Stewart, 1983) of experimentalresponse on indicators for probability vector interacted with indicator for whether X > Y with individual fixede↵ects. Constant taken as mean of fixed e↵ects. Robust standard errors parentheses. Panel B: Predicted behaviorcalculated from (2) and (3) for CPT decision maker with aggregate parameters ↵̂ = 0.941 (s.e. = 0.012) and�̂ = 0.715 (0.010). Robust standard errors calculated using the delta method, in parentheses.

13

Table A6: Equalizing Reductions Between Subjects Alternate Controls

Panel A: First Task Block (without Controls) Panel B: First Task Block (with Alternate Controls)


{0.6, 0.3, 0.1} 9.81 (0.65) 8.71 (0.56) -1.10 (0.85 ) -11.23 (8.15) 9.82 (0.60) 9.09 (0.56) -0.74 (0.83) -7.52 (8.10)[-2.77,0.57] [-27.21,4.76] [-2.36 ,0.88] [-23.40,8.35]

{0.4, 0.3, 0.3} 4.78 (0.19) 4.41 (0.19) -0.37 (0.27) -7.82 (5.50) 4.70 (0.22) 4.56 (0.20) -0.14 (0.30) -2.96 (6.33)[-0.91,0.16] [-18.60,2.95] [-0.73,0.45] [-15.37,9.46]

{0.1, 0.3, 0.6} 3.16 (0.16) 2.88 (0.12) -0.28 (0.20) -8.91 (6.00) 3.06 (0.19) 3.03 (0.15) -0.03 (0.25) -1.09 (8.08)[-0.68,0.11] [-20.66,2.84] [-0.52 ,0.45] [-16.94,14.75]

Notes: Mean values of k and k estimated from interval regression (Stewart, 1983) of experimental response onindicators for probability vectors interacted with indicator for whether X > Y . Standard errors clustered atindividual level in parentheses. See Appendix Table A3, columns (3) and (5) for detail. Panel A: No controls;143 total subjects. Panel B: controls include age, gender, Cognitive Reflection Task score; 135 total subjects.

Table A7: Equalizing Reductions with Multiple Switchers

k k(1) (2) (3) (4) (5) (6) (7) (8)

{p, q, 1� p� q} X 0 = 19 X 0 = 21 X 0 = 23 X 0 < 24 X = 30 X = 32 X = 34 X > 24

{0.6, 0.3, 0.1} 8.72 8.78 8.69 8.73 8.92 9.09 8.76 8.93(0.41) (0.38) (0.41) (0.38) (0.40) (0.42) (0.40) (0.38)

{0.4, 0.3, 0.3} 4.31 4.17 4.29 4.26 4.24 4.32 4.28 4.28(0.14) (0.12) (0.14) (0.12) (0.14) (0.15) (0.14) (0.12)

{0.1, 0.3, 0.6} 2.62 2.56 2.58 2.59 2.59 2.55 2.59 2.58(0.09) (0.11) (0.11) (0.08) (0.09) (0.08) (0.09) (0.07)


of X/X



14

Table A8: Equalizing Reductions First/Last Task Block

k k(1) (2) (3) (4) (5) (6) (7) (8)

{p, q, 1� p� q} X 0 = 19 X 0 = 21 X 0 = 23 X 0 < 24 X = 30 X = 32 X = 34 X > 24

Panel A: First Task Block

{0.6, 0.3, 0.1} 11.10 8.01 10.49 9.81 7.87 9.39 8.85 8.71(1.14) (0.99) (1.13) (0.65) (1.12) (1.21) (0.72) (0.56)

{0.4, 0.3, 0.3} 4.89 4.24 5.27 4.78 4.02 4.61 4.54 4.41(0.32) (0.34) (0.30) (0.19) (0.48) (0.29) (0.24) (0.19)

{0.1, 0.3, 0.6} 3.17 3.08 3.24 3.16 2.62 2.89 3.03 2.88(0.25) (0.25) (0.35) (0.16) (0.20) (0.11) (0.20) (0.12)

Panel B: Last Task Block

{0.6, 0.3, 0.1} 9.46 11.37 6.85 9.12 8.09 8.53 9.72 8.75(0.77) (0.93) (0.95) (0.54) (0.93) (1.11) (1.24) (0.64)

{0.4, 0.3, 0.3} 4.27 4.59 3.84 4.22 4.07 4.15 4.16 4.13(0.25) (0.31) (0.31) (0.17) (0.24) (0.51) (0.42) (0.23)

{0.1, 0.3, 0.6} 2.63 2.37 2.60 2.55 2.51 2.56 2.37 2.48(0.19) (0.23) (0.28) (0.13) (0.14) (0.34) (0.17) (0.14)


of X/X



15

E Additional Robustness Exercises

E.1 Alternative CPT Formulations

Up to this point, we have focused exclusively on the Tversky and Kahneman (1992) parame-terization of CPT. Others have proposed alternative functional forms. One leading alternativeis due to Prelec (1998), who posits a probability weighting function of the form

⇡(p) = exp(�(�ln(p))�).

To explore whether our conclusions are sensitive to functional form, we repeat our analysis forPrelec’s specification. Using our data on certainty equivalents for binary lotteries, we arriveat the following estimates: weighting parameter �̂ = 0.665 (clustered s.e. = 0.021) and utilityparameter ↵̂ = 0.928 (0.019). We then use the parameterized model to predict k and k asbefore. Results appear in Table A9, Panel A. For convenience, we reproduce our results forTversky and Kahneman’s specification in Panel B. Note that the predicted discontinuities areeven larger, and hence less consistent with actual behavior, with the Prelec specification.

Table A9: Equalizing Reduction Predictions for Alternative Functional Forms

Panel A: Prelec Weighting Panel B: Tversky Kahneman Weighting


{0.6, 0.3, 0.1} 8.77 (0.27) 6.50 (0.38) -2.26 (0.13) -25.80 (2.14) 7.58 (0.36) 6.06 (0.35) -1.52 (0.03) -20.04 (0.83)[-2.51,-2.01] [-30.02,-21.57] [-1.57,-1.47] [-21.68,-18.40]

{0.4, 0.3, 0.3} 4.81 (0.05) 2.63 (0.12) -2.18 (0.11) -45.34 (2.37) 4.01 (0.10) 2.52 (0.13) -1.49 (0.04) -37.17 (1.74)[-2.40,-1.96] [-50.02,-40.65] [-1.57,-1.41] [-40.61,-33.72]

{0.1, 0.3, 0.6} 2.95 (0.04) 1.70 (0.05) -1.26 (0.09) -42.49 (2.42) 2.65 (0.03) 1.87 (0.06) -0.78 (0.04) -29.34 (1.79)[-1.43,-1.09] [-47.27,-37.71] [-0.86,-0.69] [-32.88,-25.80]

Notes: Panel A: Predicted behavior calculated from (2) and (3) for Prelec CPT decision maker with parameters↵̂ = 0.928 (s.e. = 0.019) and �̂ = 0.665 (0.021). Standard errors clustered at individual level and calculatedusing the delta method, in parentheses. Panel B: Predicted behavior calculated from (2) and (3) for Kahnemanand Tversky CPT decision maker with parameters ↵̂ = 0.965 (s.e. = 0.021) and �̂ = 0.703 (0.015). Standarderrors clustered at individual level and calculated using the delta method, in parentheses.

E.2 Using Explicit Rank Changes

The last task block in each session featured X = $25 and Y = $24, so that adding m = $5to Y changes its rank. Using the estimated aggregate CPT parameter values, one predictsequalizing reductions of 7.28, 3.71, and 2.49 for {p, q, 1� p� q} = {0.6, 0.3, 0.1}, {0.4, 0.3, 0.3}and {0.1, 0.3, 0.6}, respectively. Note that these values are close to the CPT predictions of kreported in Table 3, Panel B and are substantially higher than those of k.

16

For {p, q, 1 � p � q} = {0.6, 0.3, 0.1}, the mean equalizing reduction is 8.94 (clustered s.e.= 0.41). This value is statistically indistinguishable from the actual value of k for X 0 < Yreported in Table 3, Panel A, �2(1) = 0.27, (p = 0.61), and is significantly lower than the valueof k for X > Y , �2(1) = 3.44, (p = 0.06). For {p, q, 1 � p � q} = {0.4, 0.3, 0.3}, the meanequalizing reduction is 4.12 (0.13), significantly lower than the values of both k and k reportedin Table 3, Panel A, �2(1) = 4.19, (p = 0.04) and �2(1) = 5.36, (p = 0.02), respectively. For{p, q, 1�p�q} = {0.1, 0.3, 0.6}, the mean equalizing reduction is 2.34 (0.08), significantly lowerthan the values of both k and k reported in Table 3, Panel A, �2(1) = 18.82, (p < 0.01) and�2(1) = 11.55, (p < 0.01), respectively.

The pattern described in the previous paragraph is, on its face, somewhat puzzling. Ifthe equalizing reduction does not depend on the ranking of the payo↵ Y , it is di�cult to seewhy it should be systematically lower in the transitional region. Certainly, that implication isinconsistent not only with CPT, but also with PT and EUT. A possible explanation is thatthe X = 25 task block always comes last, and equalizing reductions decline as the experimentprogresses from the first task block to the last (see Table A8, Panel B). Consistent with thishypothesis, the equalizing reductions in the X = $25 tasks are quite close to the values reportedfor the those for last task block (see Table A8, Panel B).

E.3 Multiple Switching

Our main results are derived from the choices of 143 subjects who did not exhibit multipleswitching in any task. For Table A10, we include the remaining subjects, each of whom exhib-ited multiple switching at least once. The results are qualitatively unchanged. As in Table 3,we predict substantial di↵erences between k and k but observe none.54 Thus our conclusionsare robust with respect to the inclusion or exclusion of potentially confused subjects.

54Appendix Table A7 provides estimates of equalizing reductions for each value of X and X

0, and demonstratesthe stability of equalizing reductions across these values.

17

Table A10: Equalizing Reductions with Multiple Switchers



{0.6, 0.3, 0.1} 8.73 (0.38) 8.93 (0.38) 0.20 (0.16) 2.25 (1.88) 7.43 (0.36) 5.88 (0.34) -1.54 (0.03) -20.77 (0.88)[-0.12, 0.51] [-1.42, 5.94] [-1.60, -1.48] [-22.32, -19.22]

{0.4, 0.3, 0.3} 4.26 (0.12) 4.28 (0.12) 0.02 (0.08) 0.52 (1.95) 3.99 (0.10) 2.45 (0.12) -1.54 (0.04) -38.55 (1.67)[-0.14, 0.18] [-3.30, 4.33] [-1.61, -1.46] [-41.85, -35.25]

{0.1, 0.3, 0.6} 2.59 (0.08) 2.58 (0.07) - 0.01 (0.06) - 0.32 (2.34) 2.66 (0.03) 1.85 (0.06) -0.82 (0.04) -30.75 (1.76)[-0.13, 0.11] [-4.91, 4.27] [-0.90, -0.73] [-34.22, -27.27]

Notes: Panel A: Mean behavior for k and k estimated from interval regression (Stewart, 1983) of experimentalresponse on indicators for probability vector interacted with indicator for whether X > Y . Standard errorsclustered at individual level in parentheses. See Appendix Table A4 for detail. Panel B: Predicted behaviorcalculated from (2) and (3) for CPT decision maker with aggregate parameters ↵̂ = 0.941 (s.e. = 0.019) and�̂ = 0.715 (0.015). Standard errors clustered at individual level and calculated using the delta method, inparentheses.

18

F Experimental Instructions

19

Hello and Welcome.

ELIGIBILITY FOR THIS STUDY: To be in this study, you must be a

Stanford student. There are no other requirements. The study will be completely

anonymous. We will not collect your name, student ID or any other identifying

information. You have been assigned a participant number and it is on the note

card in front of you. This number will be used throughout the study. Please inform

us if you do not know or cannot read your participant number.

Participant Number:

EARNING MONEY: Whatever you earn from the study today will be paid in cash at the end of the

study today. In addition to your earnings from the study, you will receive a $5 participation payment. This $5

participation payment will also be paid to you at the end of the study today.

In this study you will complete 28 tasks, each of which asks you to make a series of decisions between two options.

The first option will always be called OPTION A. The second option will always be called OPTION B. Each decision

you make is a choice. For each decision, all you have to do is decide whether you prefer OPTION A or OPTION B.

Once all of the decision tasks have been completed, we will randomly select one decision as the decision-that-

counts. This will done in two steps. First, we will randomly select one of the 28 tasks, and, second, we will randomly

select a decision from that task to be the decision-that-counts. Each decision has an equal chance of being the

decision-that-counts. So, it is in your interest to treat each decision as if it could be the one that determines your

payments.

If you prefer OPTION A in the decision-that-counts, then OPTION A will be implemented. If you prefer

OPTION B, then OPTION B will be implemented.

Throughout the tasks, either OPTION A, OPTION B or both will involve chance. You will be fully informed

of the chance involved for every decision. Once we know which is the decision-that-counts, and whether you prefer

OPTION A or OPTION B, we will then determine the value of your payments.

For example, OPTION A could be a 10 in 100 chance of receiving $20, a 30 in 100 chance of receiving $14 and

60 in 100 chance of receiving $8. This might be compared to OPTION B of a 10 in 100 chance of receiving $20, a

30 in 100 chance of receiving $19 and 60 in 100 chance of receiving $8. Imagine for a moment which one you would

prefer. You have been provided with a calculator should you like to use it in making your decisions.

If this was chosen as the decision-that-counts, and you preferred OPTION A, we would then randomly choose

a number from 1 to 100. This would be done by throwing two ten-sided die: one for the tens digit and one for the

ones digit (0-0 will be 100). If the chosen number was between 1 and 10 (inclusive) you would receive $20. If the

number was between 11 and 40 (inclusive) you would receive $14 . If the number was between 41 and 100 (inclusive)

you would receive $8.

If, instead, you preferred OPTION B, we would randomly choose another number from 1 to 100. This random

number would be completely independent of the random number previously described. If the chosen number was

between 1 and 10 (inclusive) you would receive $20. If the number was between 11 and 40 (inclusive) you would

receive $19. If the number was between 41 and 100 (inclusive) you would receive $8 .

In this example, if you preferred OPTION B and the die read 6-8, how much would you receive (don’t forget

your participation payment!):

In this example, if you preferred OPTION A and the die read 0-9, how much would you receive (don’t forget

your participation payment!):

The tasks are presented in eight separate blocks. In a moment we will begin the first block of tasks.

TASK BLOCK 1

Participant Number:

TASKS 1-3

On the following pages you will complete 3 tasks. In each task you are asked to make a series of

decisions between two uncertain options: Option A and Option B. You may complete the tasks in

any order you wish.

In each task, Option A will be fixed, while Option B will vary. For example, in Task 1 Option

A will be a 10 in 100 chance of receiving $34, a 30 in 100 chance of receiving $24 and 60 in 100

chance of receiving $18. This will remain the same for all decisions in the task. Option B will vary

across decisions. Initially Option B will be a 10 in 100 chance of receiving $34, a 30 in 100 chance

of receiving $29 and 60 in 100 chance of receiving $18. As you proceed, Option B will change. The

amount you receive with 60 in 100 chance will decrease.

For each row, all you have to do is decide whether you prefer Option A or Option B. Indicate

your preference by checking the corresponding box. The first question from Task 1 is reproduced

as an example.

EXAMPLE

Option A or Option B

10 in 100 30 in 100 60 in 100 10 in 100 30 in 100 60 in 100Chance Chance Chance Chance Chance Chance

1) $34 $24 $18 2 or $34 $29 $18.00 2If your prefer Option A, check the green box...

1) $34 $24 $18 2� or $34 $29 $18.00 2If your prefer Option B, check the blue box...

1) $34 $29 $18 2 or $34 $29 $18.00 2�

The other tasks in this block will involve the same payment amounts, but the

chance of receiving the payments will change. Please take a look at all the tasks

and raise your hand if you have any questions.

Remember, each decision could be the decision-that-counts. So, it is in your

interest to treat each decision as if it could be the one that determines your pay-

ments.













TASK BLOCK 2

Participant Number:

TASKS 4-6



any order you wish.









as an example.

EXAMPLEOption A or Option B




1) $32 $29 $18 2 or $32 $29 $18.00 2�






ments.













TASK BLOCK 3

Participant Number:

TASKS 7-9



any order you wish.









as an example.





1) $30 $29 $18 2 or $30 $29 $18.00 2�






ments.













TASK BLOCK 4

Participant Number:

TASKS 10-12



any order you wish.









as an example.





1) $23 $29 $18 2 or $23 $29 $18.00 2�






ments.













TASK BLOCK 5

Participant Number:

TASKS 13-15



any order you wish.









as an example.





1) $21 $29 $18 2 or $21 $29 $18.00 2�






ments.













TASK BLOCK 6

Participant Number:

TASKS 16-18



any order you wish.









as an example.





1) $19 $29 $18 2 or $19 $29 $18.00 2�






ments.













TASK BLOCK 7

Participant Number:

TASKS 19-25


decisions between two options: Option A and Option B. You may complete the tasks in any order

you wish.


A will be a 5 in 100 chance of receiving $25 and a 95 in 100 chance of receiving $0. This will remain

the same for all decisions in the task. Option B will vary across decisions. Initially Option B will

be a 100 in 100 chance of receiving $25. As you proceed, Option B will change. The amount you

receive with 100 in 100 chance will decrease.


your preference by checking the corresponding box.

The first question from Task 19 is reproduced as an example.


5 in 100 Chance 95 in 100 Chance 100 in 100 Chance

1) $25 $0 2 or $25.00 2If your prefer Option A, check the green box...

1) $25 $0 2� or $25.00 2If your prefer Option B, check the blue box...

1) $25 $0 2 or $25.00 2�

The other tasks in this block will involve the same payment amounts for Option

A, but the chance of receiving the payments will change. Please take a look at all

the tasks and raise your hand if you have any questions.



ments.


























chance of receiving $25 and a 5 in 100 chance of receiving $0. Initially Option B will be a 100 in 100 chanceof receiving $25. As you proceed, Option B will change. The amount you receive with 100 in 100 chancewill decrease. For each row, decide whether you prefer Option A or Option B.



TASK BLOCK 8

Participant Number:

TASKS 26-28



any order you wish.









as an example.





1) $25 $29 $18 2 or $25 $29 $18.00 2�






ments.













Documents

Direct Tests of Cumulative Prospect Theory - Harvard … · Direct Tests of Cumulative Prospect Theory ... In contrast to the EU formulations of von Neumann and Mor ... consider a