Testing Ambiguity Theories in a New Experimental Design
with Mean-Preserving Prospects
BY
Chun-Lei Yang and Lan Yao
OCTOBER 2012
Abstract
Ambiguity aversion can be interpreted as aversion against second-order risks associated with
ambiguous acts, as in MEU/α-MP and KMM. In our design the decision maker draws twice
with replacement in the typical Ellsberg two-color urns, where a different color wins each
time. Consequently, all conceivable simple lotteries share the same mean, while the variance
increases with the color balance. MEU/α-MP, KMM and Savage’s SEU predict
unequivocally that risk-averse (-seeking) DMs shall avoid (choose) the 50-50 urn that
exhibits the highest risk conceivable. While this is true for many subjects, we also observe a
substantial number of violations. It appears that the ambiguity premium is partially paid to
avoid the ambiguity issue per se (the source), consistent with both experimental findings on
source dependence and the CEU weighting function model. This finding is robust even when
there is only partial ambiguity. We also show in an excursion that Machina’s paradox in the
reflection example disappears once the preference theories are formulated with our notion of
source-specific act.
KEYWORDS: Ambiguity, Ellsberg paradox, expected utility, experiment, Machina paradox,
second-order risk, source premium, source-specific act, weighting function
JEL classification: C91, D81
Acknowledgements: We are grateful to Jordi Brandts, Yan Chen, Soo-Hong Chew, Songfa
Zhong, Dan Houser, Ming Hsu, Jack Stecher, and Dongming Zhu for helpful comments.

Research Center for Humanities and Social Sciences, Academia Sinica, Taipei 115, Taiwan; e-mail:
cly@gate.sinica.edu.tw, fax: 886-2-27854160, http://idv.sinica.edu.tw/cly/

School of Economics, Shanghai University of Finance and Economics, Shanghai, China 200433; email:
yao.lan@mail.shufe.edu.cn
1
1. Introduction
The Ellsberg Paradox refers to the outcome from Ellsberg’s (1961) thought experiments, that
missing information about objective probabilities can affect people’s decision making in a way that is
inconsistent with Savage’s (1954) subjective expected utility theory (SEU). Facing two urns
simultaneously in Ellsberg’s two-color problem, one with 50 red and 50 black balls (the risky urn) and
the other with 100 balls in an unknown combination of red and black balls (the ambiguous urn), most
people prefer to bet on the risky urn, whichever the winning color is. This phenomenon is often called
ambiguity aversion. Many subsequent experimental studies confirm Ellsberg’s finding, as for example
surveyed in Camerer and Weber (1992).
Many extensions to SEU have been proposed to rationalize the Ellsberg paradox and applied to
economic analysis. Among the most prominent ones, Gilboa and Schmeidler (1989) develop the
maxmin expected utility (MEU) theory, generalized to the so-called α-MP (multi-prior) model by
Ghirardato, Maccheroni, and Marinacci (2004). MEU solves the paradox and has been applied to
studies on asset pricing in Dow and Werlang (1992) and Epstein and Wang (1994) among others.
Another theory that has found broad applications because of its convenient functional form is the
smooth model of ambiguity aversion by Klibanoff, Marinacci, and Mukerji (2005, KMM). Chen, Ju,
and Miao (2009), Hansen (2007), Hansen and Sargent (2008), and Ju and Miao (2009) successfully
applied KMM to studies of asset pricing and the equity premium puzzle. The third is the model of
Choquet expected utility (CEU) by Schmeidler (1989), where the DM uses a weighting function
called capacity to evaluate prospects.1 Mukerji and Tallon (2004) survey application of CEU in
various areas of economics such as insurance demand, asset pricing, and inequality measurement.
Given the success in the applied fields, many new experimental studies have been conducted to
test these models and characterize subjects’ behavior accordingly. However, all previous experiments
on ambiguity aversion we are aware of share the feature that the ambiguous prospect can be
associated with a first-order lottery that is of either lower mean or higher variance than the benchmark
risky prospect. As such, one cannot distinguish whether the observed ambiguity aversion reflects
willingness to pay an ambiguity premium for the second-order risk associated with the uncertain act,
1
For further theoretical models of multi priors, second-order sophistication, and rank-dependent utility, see
Segal (1987, 1990), Casadesus-Masanell, Klibanoff, and Ozdenoren (2000), Nau (2006), Chew and Sagi (2008),
Ergin and Gul (2009) and Seo (2009) among others. Wakker (2008) and Eichberger and Kelsey (2009) offer
excellent surveys.
2
which α-MP (MEU) and KMM predict, or for the issue of ambiguity per se, which turns out to be
consistent with CEU and seems to be behind the ideas of source dependence studies initiated by Heath
and Tversky (1991) and Fox and Tversky (1995). We have a lottery design that is a simple
modification of Ellsberg’s two-color problem that enables this separation.
In our design, the DM draws twice with replacement from a two-color urn. With the novel rule of
each color winning exactly one of the draws, ours has the unique feature that all conceivable color
compositions yield the same expected value and differ only in the variance that increases with the
balance of color in the urn. The payoff is risk free if all balls in the urn are of the same color.
Consequently, according to SEU, α-MP (MEU) and KMM, a risk-averse DM is to prefer both the
ambiguous and the objective uniformly compound urn when pitched against the objective 50-50 urn;
while a risk-seeking DM’s preference displays the exactly reversed order. In fact, even without
precise knowledge of risk attitude, these theories predict that the DM is to consistently show the same
order of preference in these two decisions. Note, to avoid Machina’s paradox (Machina, 2009), the
preference models for testing are formulated with our new notion of source-specific act.2 This enables
us to identify partial ambiguity in a straightforward manner, and to design a partial ambiguity
treatment (PA) where the color composition in the urn is only partly unknown, in addition to the full
ambiguity one (FA), as robustness check for our basic finding of persistent violation to second-order
risk models of ambiguity. The predictions do not change when the extent of ambiguity varies from
full to partial.
It turns out that 22-39% subjects violate the above-mentioned theoretical predictions after
eliciting their risk attitude with a simple multi-price-list (MPL) method, depending on decision issues
and treatment conditions. Disregarding the risk attitude, 23-43% violate the consistency prediction.
Interestingly, CEU proves to be sufficiently general to not be tied down to any specific prediction for
testing, within our design. In particular, it is not bound to evaluating the utility function with a virtual
lottery (via weighting function) that is mean preserving, which the other theories require in our design.
To the extent that CEU’s weighting leads to a lower virtual mean, we may explicitly identify its
difference to the original mean as the premium for the source, besides the premium for second-order
risk associated with ambiguity postulated by the other theories mentioned.
2
For detailed discussion of Machina’s paradox with his reflection example and how it goes away with the
notion of source-specific act, see the Excursion in Section 4.
3
In the next section, we discuss the relevant preference models, our experimental design, and the
associated theoretical predictions. Data analysis is in Section 3. We then further interpret our results in
relation to findings in the literature in Section 4, as well as discuss Machina’s reflection example and
how our notion source-specific act helps to solve Machina’s dilemma in an excursion, before
concluding the paper with Section 5.
2. Theoretical Models and Experimental Design
Models of decision under uncertainty
Let Ω be a state space with a sigma algebra ∑, and X be an outcome space. An act is a mapping
𝑓: Ω → X. An individual is assumed to have a preference ordering over the space of all acts, for
making decisions under uncertainty. For our purpose, assume the outcome space consists of finite real
numbers that represent monetary payoffs, 𝑋 = {𝑥1 , 𝑥2 , … , 𝑥𝑛 }, with 𝑥1 > 𝑥2 , … , > 𝑥𝑛 . For any set
Z, let Δ(𝑍) denote the space of probability distributions, i.e. lotteries, on Z. An act f and a probability
distribution on Ω induce a unique probability distribution 𝑝 ∈ Δ(𝑋). However, the specific lottery
device, or the source that governs the circumstances of the underlying uncertainty, might involve
higher-order compound lotteries in Δ𝑘 (𝑋) for arbitrary k that the DM may or may not reduce to their
first-order forms before evaluation. Suppose k is the highest relevant order of stochastic elaboration
by the DM, then different sources of uncertainty can be associated with different sets of admissible
order-k compound lotteries, 𝑆 ⊆ Δ𝑘 (𝑋). In the spirit of the revealed preference approach, we assume
that the pair (f, S) summarizes all relevant aspects of a decision option and the DM is to be indifferent
between (f, S) and (g, S) for any acts f and g. In other words, additional information details as reflected
in the sub sigma algebra on Ω induced by 𝑓 −1 are considered irrelevant.
Note that this notion of (f, S) is an attempt to explicitly identify the source of uncertainty, and
hence is called a source-specific act subsequently. The motivation for this new notion comes from our
insight that Machina’s paradox can be avoided if the preference models below are defined on the
outcome space X with explicit recognition of partial ambiguous set 𝑆 ⊆ Δ𝑘 (𝑋), instead of their
standard definitions on the state space Ω where partial ambiguity is implicit in the act f.3 The
3
Details on our solution of Machina’s paradox can be found in the Discussions. Note that Gajdos, Hayashi,
4
ambiguous Choice C in our FA treatment, for example, has 𝑆𝐶 consisting of the 11 simple lotteries
listed in Table 1, or of its convex hull alternatively. All choices with objective lotteries have singleton
S. In the case of our compound lottery Choice D, for example, 𝑆𝐷 = {𝜇𝐷 } with 𝜇𝐷 ∈ Δ2 (𝑋)
representing the uniform distribution over 𝑆𝐶 . Source dependence as discussed in Fox and Tversky
(1995), Hsu, Bhatt, Adolphs, Tranel and Camerer (2005), and Abdellaoui et al. (2011) among others
can be reinterpreted as different natural sources leading to different subjective specification of S,
presumably in the first-order space Δ(𝑋). Ergin and Gul’s (2009) issue dependence can be interpreted
as referring to differentiations in 𝑆 ⊆ Δ2 (𝑋) associated with different choices. Also, one advantage
of source-specific formulation is to explicitly discuss partial ambiguity, which motivates our PA
treatment.
With the term (f, S) defined, we now turn to well-known models that are relevant for our
experimental tests. Savage’s (1954) subjective expected utility theory assumes that there is a
monotone utility function on the outcome space, 𝑢: 𝑋 → ℝ, such that for each source-specific act (f, S)
with 𝑆 ⊆ Δ(𝑋), there is a subjective belief 𝑝 ∈ 𝑆 so that
𝑛
(1)
SEU(𝑓, 𝑆) = ∑
𝑖=1
𝑝(𝑥𝑖 )𝑢(𝑥𝑖 )
Ghirardato et al. (2004) have the so-called α-MP (multi-prior) model, a generalization to MEU,
as follows. Given that the DM has a compact set K for each (f, S),
(2)
α-MP(𝑓, 𝑆) = α min ∑
𝑝∈𝐾⊆𝑆
𝑛
𝑝(𝑥𝑖 )𝑢(𝑥𝑖 ) +(1-α) max ∑
𝑖=1
𝑝∈𝐾⊆𝑆
𝑛
𝑝(𝑥𝑖 )𝑢(𝑥𝑖 )
𝑖=1
In the extreme case of α = 1, we obtain the original MEU expression proposed by Gilboa and
Schmeidler (1989).
(2a)
MEU(𝑓, 𝑆) = min𝑝∈𝐾⊆𝑆 ∑𝑛𝑖=1 𝑝(𝑥𝑖 )𝑢(𝑥𝑖 ).
The smooth model of ambiguity aversion (KMM) by Klibanoff et al. (2005) considers the space
of second-order compound lotteries as the relevant space for decision under uncertainty and assumes
that there is a monotone function 𝑣: ℝ → ℝ, with which the DM evaluates the certainty equivalents of
Tallon, and Vergnaud (2008) also attach an admissible set to an act in their model, which is defined on the state
space, while ours is on the outcome space. Chew and Sagi (2008) have a model that identifies sources with
small worlds in the form of sub sigma algebra on Ω.
5
first-order lotteries evaluated with u. For each (f, S) with 𝑆 ⊆ Δ2 (𝑋), there is a second-order
subjective belief 𝜇 ∈ 𝑆 so that
𝑛
(3)
KMM(𝑓, 𝑆) = ∫
𝑣 (∑
𝑝(𝑥𝑖 )𝑢(𝑥𝑖 )) d𝜇(𝑝)
𝑖=1
𝑝∊𝛥(𝑋)
Unlike the above models where the DM is to evaluate the act (f, S) with some admissible
probability distribution in S, Choquet expected utility by Schmeidler (1989) evaluates it with a
weighting function, called capacity, instead. Let 𝐸𝑖 = [𝑥 = 𝑥𝑖 ] denote the event that yields the
monetary payoff 𝑥𝑖 , which increases with i = 1, …, n. A weighting function w defined on the sigma
algebra generated by these events is a capacity, if it is non-negative, 𝑤(∅) = 0, 𝑤(⋃𝑛𝑖=1 𝐸𝑖 ) = 1,
and 𝑤(𝐴) ≤ 𝑤(𝐵) whenever 𝐴 ⊆ 𝐵. The payoff under CEU is then the following.
𝑖
𝑛
(4)
CEU(𝑓, 𝑆) = ∑
[𝑤 (⋃
𝑖=1
𝑗=1
𝑖−1
𝐸𝑗 ) − 𝑤 (⋃
𝑗=1
𝐸𝑗 )] 𝑢(𝑥𝑖 )
𝑛
Note with 𝑞𝑖 : = 𝑤(⋃𝑖𝑗=1 𝐸𝑗 ) − 𝑤(⋃𝑖−1
𝑗=1 𝐸𝑗 ), we have ∑𝑖=1 𝑞𝑖 = 1 and 𝑞𝑖 ≧ 0, for all i. Thus, CEU
can be interpreted to evaluate the utility function u with a more flexible distribution q that may not be
in the set of admissible lotteries S. In our experimental study, this added degree of freedom proves to
be crucial to distinguish CEU from the other models. Note that in the special case that the DM assigns
a probability distribution p over the act, or when dealing with an objective lottery, the weighting takes
the form of an increasing function, 𝑤: [0,1] → [0,1], that is at the center of prospect theory by
Tversky and Kahneman (1992). We then can work with the following instead.
𝑛
(4a)
CEU(𝑓, 𝑆) = ∑
𝑛
[𝑤(𝑝(𝑥 ≤ 𝑥𝑖 )) − 𝑤(𝑝(𝑥 ≤ 𝑥𝑖−1 ))]𝑢(𝑥𝑖 ) = ∑
𝑖=1
𝑞𝑖 (𝑥𝑖 )𝑢(𝑥𝑖 )
𝑖=1
We refer to Wakker (2008) for more detailed discussion.
Decision problems of the experiment
There are three urns labeled B, C, and D. Each urn has 2N balls, each of which can be red or
white colored. The novel feature of our design is to have subjects draw from the selected urn twice
with replacement, with a different color winning 50 Yuan each draw. If the first draw is red and the
second is white, he gets 100 Yuan; if the two draws are of the same color, he gets 50 Yuan; but if the
6
two colors are in the order of white first and red second, he gets 0. Urn B is the 50-50 risky one with
exactly N red and N white balls. Urn C is the ambiguous urn where the number of red balls could be
any in a subset 𝑆 ⊆ 𝐻𝑁 : = {0,1, … ,2𝑁}. Urn D is a compound lottery with uniform distribution over S.
The options associated with urns B, C, D are subsequently denoted Choice B, C and D respectively.
Subjects face three simple decision problems one after another. Problem 1 is meant to test their
risk attitude. On a list of 20 cases of sure payoffs that range from 5 to 100 Yuan in steps of 5 Yuan,
subjects have to choose either the sure payoff or the risky one, Choice B, for every case.4 Problem 1
is in fact a simple form of the MPL procedure that can also be viewed as a modified version of the
BDM procedure.5 Problem 2 is our main test for theoretical predictions regarding ambiguity aversion.
In this problem, subjects have to decide between Choice B and Choice C. Problem 3 is a test on
preference over objective compound lotteries, where subjects are to choose between (the first-order
risk) Choice B and (the second-order risk) Choice D.
We have two main treatments that differ both in sizes of the urn and in whether there is full or
partial ambiguity in Choice C. In the full ambiguity treatment (FA), N = 5 and 𝑆FA = 𝐻5. In the
partial ambiguity treatment (PA), N = 8 and 𝑆PA = {0,1,2,3,4,5; 11,12,13,14,15,16} ⊆ 𝐻8 . By
definition, FA and PA also differ in Choice D due solely to the difference between 𝑆FA and 𝑆PA.
Note, however, that the feature of a different color winning each round ensures that the mean of the
lottery is always 50 Yuan, independent of the color composition in the urn. In fact, all compound
lotteries can be ranked regarding their variances, with Choice B being associated with the highest
possible variance. As illustration, Table 1 summarizes the statistical characteristics of all physically
feasible first-order lotteries in our design, for N = 5.
Theoretical predictions
Let 𝜋ℎ𝑁 ∈ Δ(𝑋), with X = {0,50,100} being the outcome space, denote the induced simple
lottery that associates with a hypothetical urn with h red and 2N-h white balls, according to our
double-drawing rule. Let 𝐻𝑁 = {𝜋ℎ𝑁 }2𝑁
ℎ=0 denote, with slight abuse of notations, the physically
4
We aim at revealing individual certainty equivalent values of Choice B. Though we may alternatively replace
Choice B with its reduced form (100, 1/4; 50, 1/2; 0, 1/4) here, it would lose the structural congruence to Choice
C and D, which we consider eminently crucial to our design.
5
Sapienza, Zingales and Maestripieri (2009) use a similar method. See Becker, DeGroot, and Marschak (1964)
for BDM procedure. See Holt and Laury (2002) for multi-price-list (MPL) procedure. See also Harrison and
Rutström (2008) and Trautmann, Vieider and Wakker (2011). Detailed discussion can be found in Appendix C2.
7
feasible set of first-order lotteries under our design. Generically, the outcome probabilities are
𝜋ℎ𝑁 (0) = 𝜋ℎ𝑁 (100) = ℎ(2𝑁 − ℎ)/4𝑁 2 =: 𝑝ℎ0 and 𝜋ℎ𝑁 (50) = 1 − 2𝑝ℎ0 , respectively. Due to our
symmetrical design, {h-red, (2N-h)-white} and {(2N-h)-red, h-white} urns induce equivalent
prospects, in all aspects relevant for decision under uncertainty. The mean for 𝜋ℎ𝑁 is the same 50 for
all h. But the variance, var 𝜋ℎ𝑁 = 2 ∗ 502 ℎ(𝑁 − ℎ)/𝑁 2 , increases from h = 0 to h = N and then
symmetrically decreases from h = N to h = 2N, with max var 𝜋ℎ𝑁 = var 𝜋𝑁𝑁 = 1250. The crucial
h
feature for our design is that a more color-balanced urn constitutes a mean-preserving spread to a less
balanced one. As illustration, Table 1 summarizes the stochastic characteristics of all 11 elements in
𝐻5 .
Table 1: Complete list of feasible first-order lotteries,6 N = 5
𝝅𝟎
𝝅𝟏
𝝅𝟐
𝝅𝟑
𝝅𝟒
𝝅𝟓
𝝅𝟔
𝝅𝟕
𝝅𝟖
𝝅𝟗
𝝅𝟏𝟎
Red
0
1
2
3
4
5
6
7
8
9
10
White
10
9
8
7
6
5
4
3
2
1
0
p(0)
0
.09
.16
.21
.24
.25
.24
.21
.16
.09
0
p(50)
1
.82
.68
.58
.52
.50
.52
.58
.68
.82
1
p(100)
0
.09
.16
.21
.24
.25
.24
.21
.16
.09
0
mean
50
50
50
50
50
50
50
50
50
50
50
variance
0
450
800
1050
1200
1250
1200
1050
800
450
0
Though there are only 2𝑁 + 1 lotteries in 𝐻𝑁 physically feasible, it is nonetheless conceivable
that more complicated compound lottery devices can be used to determine which of them gets
chosen.7 To be on the safe side, let us assume that the relevant set of lotteries for the ambiguous
Choice C under SEU, α-MP, MEU and CEU is in the convex hull of 𝐻𝑁 , i.e., 𝑆𝐶 ⊆ co 𝐻𝑁 ⊆ Δ(𝑋).
At the heart of our design is the feature that, for any 𝜋, 𝜋 ′ ∈ 𝑆𝐶 , mean(𝜋) = mean(𝜋 ′ ) = 50 and
Table 1 summarizes all possible first-order lotteries given this payoff rule, with 𝜋ℎ coding for the lottery
with h red balls and 10-h white balls. There are exactly 11 of them. Each column lists the distribution of
monetary outcome, its mean and its variance. For example, the urn with 4 red and 6 white balls, 𝜋4 , gives us the
probabilities of .24, .52, and .24 to earn the prize of 0, 50, and 100 Yuan, respectively; with a mean of 50 Yuan
and a variance of 1200. Obviously, our modified Ellsberg risky prospect, 𝜋5 , has the highest variance of 1250,
while all color compositions yield the same mean payoff.
7
For example, Stecher, Shields and Dickhaut (2011) have an ingenious method to generate virtual ambiguity
via objective but mathematically involved compound lotteries, which illustrates the need to consider the convex
hull here.
8
6
one of 𝜋 and 𝜋 ′ is a mean-preserving spread to the other due to the nature of 𝐻𝑁 . In fact, let 𝑝𝜋0
denote the probability π assigns to x=0 for any 𝜋 ∈ 𝑆𝐶 , then 𝑝𝜋0 ∊ [0, .25], i.e. SC can be represented
as a one-parameter family by the compact interval [0, .25].
Let 𝑛 = #𝑆𝐶 , 𝜋𝐷 = ∑ℎ∈𝑆𝐶 𝜋ℎ /𝑛 ∊ 𝑆𝐶
is the reduced first-order distribution for Choice D, with var 𝜋𝐷 = ∑ℎ∈𝑆𝐶 var 𝜋ℎ /𝑛 < 𝑣𝑎𝑟 𝜋𝑁𝑁 . 8
More specifically, for treatments FA and PA, 𝑝𝜋0FA = .15, 𝑝𝜋0PA ≈ .12, var 𝜋𝐷FA = 750, var 𝜋𝐷PA ≈
𝐷
𝐷
602, respectively.
Once the DM reveals his risk attitude associated with u in Problem 1 as being risk averse, risk
neutral or risk seeking (corresponding to CE < 50, = 50, or > 50, i.e., 𝑢(0) + 𝑢(100) − 2𝑢(50) < 0,
= 0, or > 0), specific predictions can be derived for his decision in Problems 2 and 3 based on the
above-mentioned models, which we can test experimentally. First, Problem 3 only involves singleton
sources 𝑆𝐵 = { 𝜋𝐵 } and 𝑆𝐷 = {𝜋𝐷 }, i.e., there is no ambiguity. It is obvious that both SEU and
α-MP predict preferences of D over B for a risk-averse DM as well as B over D for risk-seeking DMs,
because 𝜋𝐵 = 𝜋𝑁𝑁 is a mean preserving spread of 𝜋𝐷 . In fact, as 𝜋𝐵 is a strict mean preserving
spread to any lottery in 𝑆𝐶 but itself, a risk-averse (-seeking) DM in Problem 2 is also to prefer C to
B (B to C), as long as he does not put all weight on 𝜋𝑁𝑁 when evaluating Choice C. In the PA
treatment, the potential indifference is ruled out by design as 𝜋𝑁𝑁 ∉ 𝑆𝐶 = 𝑆PA . In the FA, the latter is
the case if 𝜋𝐵 ∉ 𝐾 ⊆ 𝑆𝐶 in the MEU formulae (2a) or α < 1 even when 𝐾 = 𝑆𝐶 with α-MP. This
exactly illustrates the fundamental difference from Ellsberg’s original design, where a subjective
probability can be associated with the ambiguity prospect that may yield a higher mean or a lower
variance than the benchmark risky prospect.
The same prediction is also true for KMM. For Choice C under KMM, we assume 𝑆𝐶KMM =
{𝜑 ∈ Δ2 (𝑋): supp 𝜑 ⊆ 𝑆𝐶 } , whose first-order reduction trivially coincides with 𝑆𝐶 . Let 𝑐𝜋 : =
𝑝𝜋0 [𝑢(0) + 𝑢(100)] + (1 − 2𝑝𝜋0 ) 𝑢(50) denote the expected value for 𝜋 ∈ 𝑆𝐶 . For any 𝜋, 𝜋′ ∈ 𝑆𝐶 ,
(5)
𝑐𝜋 − 𝑐𝜋′ = (𝑝𝜋0 − 𝑝𝜋0′ )[𝑢(0) + 𝑢(100) − 2𝑢(50)].
For any 𝜋′ ∈ 𝑆𝐶 with 𝜋′ ≠ 𝜋𝐵 , since 𝑝𝜋0𝐵 = .25 > 𝑝𝜋0′ , 𝑐𝜋𝐵 − 𝑐𝜋′ ≶ 0 iff 𝑢(0) + 𝑢(100) −
2𝑢(50) ≶ 0 , i.e. iff CE ≶ 50 . Now, for any strictly increasing 𝑣(. ) and any 𝜇 ∊ 𝑆𝐶KMM ⊆
Δ2 (𝑋) without degenerately putting all weight on 𝜋𝐵 , we conclude from the definition of KMM that
Note for any compound lottery 𝑦 = (𝑝ℎ : 𝑥ℎ )ℎ , var y = ∑ 𝑝ℎ var 𝑥ℎ + ∑𝑝ℎ (𝑥̅ℎ − 𝑦̅)2. Due to mean preserving,
the second term vanishes in our design.
9
8
(6)
KMM(B) ≶ KMM(C) ⇔ KMM(B) ≶ KMM(D) ⇔ CE ≶ 50
The proof is straightforward in that, due to monotonicity, 𝑣(𝑐𝜋𝐵 ) is either the maximum or the
minimum on {𝑐𝜋 : 𝜋 ∊ 𝑆𝐶 }, depending on whether the DM is risk seeking or averse. Note that this is
true whether the support for 𝜇 is restricted to 𝑆𝐶 or co 𝑆𝐶 . In summary, we have the following
theoretical predictions to test.
Hypothesis In both FA and PA treatments, SEU, α-MP (MEU in limit case), and KMM predict that
risk-averse individuals (CE<50 in Problem 1) are to choose C over B in Problem 2 and D over B in
Problem 3, while risk-seeking individuals (CE>50 in Problem 1) are to choose B over C or D in both
Problem 2 and 3.
Note that any decision in Problems 2 and 3 by a Problem-1 risk-neutral individual is trivially
consistent with the theory prediction, as is obvious from equation (5) above. And the theories predict
that people with non-neutral risk attitudes should have a strict preference among the two choices in
both Problems 2 and 3, which makes it redundant to provide the option of indifference between the
two choices in Problems 2 and 3 in the design. The above discussion also reveals that risk aversion or
seeking is exactly equivalent to the choice of either D or B in Problem 3. Thus, a weaker consistency
requirement restricted to behavior in Problem 2 and 3 is that the DM shall do either BB or CD there.
Hypothesis* (Weak consistency) To be consistent with models of SEU, α-MP (MEU in limit case)
and KMM, individuals shall choose either BB or CD in Problems 2 and 3, in both FA and PA.
In contrast, such sharp behavior predictions cannot be made with CEU. It turns out that any
decision profile across Problem 1-3 can be rationalized within the CEU model.
Lemma 1 For any combination of decisions in Problems 1, 2 and 3 in FA and PA, there is a
weighting function 𝑤 under the CEU model that rationalizes them.
A detailed proof can be found in Appendix C1. One way to understand this difference between
CEU and α-MP/KMM is to recall the fact that each weighting function 𝑤 induces a virtual lottery
𝑞 ∈ ∆(X) not necessarily in the mean-preserving class of co 𝑆𝐶 , so that CEU is exactly the expected
utility weighted with 𝑞 . In fact, 50 − ∑𝑛𝑖=1 𝑞𝑖 (𝑥𝑖 )𝑥𝑖 can be roughly interpreted as the source
premium, which would be zero under α-MP/KMM in similar terms due to our special prospect design.
10
Note, although our design is not intended to discriminate among different shapes of 𝑤, the partial
ambiguity approach may be useful for this in future studies.
Experimental procedure
We ran two treatments that differ in the number of balls in the urn and whether the color
composition is partially unknown. The full ambiguity one (FA) has 10 balls in the urn with full
uncertainty over the 11 color compositions in Choice C. The partial ambiguity one (PA) has 16 balls
in the urn with uncertainty over a set of 12 of the total 17 color compositions, presented with the
explanation to subjects that the absolute difference of the two colors is at least 6. The design is chosen
so that the size of the ambiguous set is similar (n=11 vs. n=12), there is some difference in maximal
risk between PA and FA ( max var 𝜋ℎ8 = 1074 < 1250 = var 𝜋𝑁𝑁 ), but the former is not too small to
ℎ∈𝑆PA
make the ambiguity issue irrelevant (e.g. under N=1000, n=12). Note that the primary purpose of
designing two treatments this way is to check whether and how any potential violation to the main
hypotheses is robust.
Our instructions were done with a PowerPoint presentation (Appendix A). Subjects were to hand
in their decisions on one problem before they got instructions for the next one. To increase credibility,
we demonstrated drawings with the urn to be used later in Choices B and D during instructions.
Choice C urn was prepared before the session and placed on the counter for all to see.9 After all
decision sheets were collected, subjects were called upon to have their decisions implemented one by
one.10 For both FA and PA, subjects drew randomly from one of the three decision problems and
were paid in cash according to the realization of their decisions in that problem. Note, in an initial
study, we ran sessions for FA with only about 10% of subjects randomly chosen for payment. To add
to data robustness, we also present its results here and call it FAR treatment henceforth.
A total of 269 subjects from Shanghai University of Finance and Economics participated in the
experiment. All participants were first-year college students of various majors ranging from
9
Note, our double-draw alternate-color-win design conceptually removes the subjects’ fear of possible
manipulation of color composition by the experimenter. Nevertheless, students still regularly asked to inspect
the content of the ambiguous urn C after the decision implementation.
10
After subjects handed in their decisions, they were given the option to have the payment procedure
implemented later in the experimenter’s office, if they did not wish to wait. Only two of them made use of this
option.
11
economics and management to science and language. 160 students participated in the treatments PA
and FA and all of them were paid. We ran two sessions in each treatment. 109 subjects participated in
the treatment FAR of three sessions, and only 11 subjects were paid randomly. Average payoff for all
171 subjects with real payment in the three treatments was 62.2 Yuan, and average duration for a
session was 40 min.11 Note we also ran an auxiliary session with 30 subjects on incentivized
comprehension tests. Details on its motivation, design and outcomes can be found in Appendix B1.
3. Experimental Results
Problem 1 elicits individuals’ risk attitude. The certainty equivalent value (thereafter CE) of the
risky lottery (Choice B) in our experiment is defined as the lowest value at which one starts to prefer
sure payoff to the lottery. The majority of subjects (77, 72, and 100 in PA, FA, and FAR respectively)
revealed monotone behavior of switching from B to A with increasing sure payoffs. Subsequent
analyses are restricted to these samples only.12 Note the incentivized comprehension tests show that
subjects from the cohort have no problem understanding the statistical implications of our unique
double-draw lottery design. In addition, most people displayed preferences over different urns of our
design that are consistent with standard theory of risk. Details are in Appendix B1.
The average CE values are 50, 49.65 and 46.1 for the treatments PA, FA and FAR with standard
deviations of 12.46, 11.11 and 15.22, respectively. In PA, we have 28.57%, 42.86% and 28.57% of
the subjects with CE<50, CE=50, and CE>50, respectively, which correspond to risk aversion,
neutrality, and seeking. The numbers in FA are 34.72%, 34.72%, and 30.56%, and those in FAR are
38%, 32% and 30%, respectively.13 Chi square test reveals no significant difference between PA and
11
Note that 1 USD = ca. 6.8 Yuan. Regular student jobs paid about 7 Yuan per hour and average first jobs for
fresh graduates paid below 20 Yuan per hour. The duration of 40 minutes is the average time spent by all
subjects including the long waiting time for payoff implementation.
12
Only 8 out of 85 subjects (9.41%) in the PA treatment, 3 out of 75 subjects (4%) in the FA treatment, and 9
out of 109 subjects (8.26%) in the FAR treatment switched back from A to B, which is deemed anomalous and
excluded from our data analysis. We also run an additional session of the treatment FAR (41 subjects) with the
alternate order of problems 1, 3 and 2, to control for potential order effects. Chi-square test confirms no
existence of order effects, with p=0.831, 0.640, and 0.759 for Problem 2, 3, and both combined, in comparison
with the order used in our design. Also, Arló-Costa and Helzner (2009) has a similar order-independence finding
like ours. We did not include the session in this paper.
13
Note that this kind of distribution of risk attitude is common in the literature. Halevy (2007), using the
standard BDM mechanism, has 31.7%, 30.5%, and 38.5% of the 105 subjects in his sample as risk averse,
12
FA regarding subject risk attitudes among the categories of risk averse, neutral and seeking (p=0.569).
Figure B1 in the appendix shows the distributions of subjects’ CE values.
As summarized in the Hypothesis, for risk-averse (-seeking) individuals for Problem 2 and 3, the
theories of α-MP (MEU) and KMM predict the choice of C and D (B and B), respectively. Figure 1
illustrates the case of violations. Note that [-,-] in the brackets refers to the 95% confidence interval
defined by percentage, throughout this paper. In PA, we observe 22.08% ([12.60, 31.55]) in Problem 2
and 3, respectively, when all samples are considered. In FA, these numbers are 27.78% ([17.18, 38.38])
in both Problem 2 and 3. In FAR, we observe 39% ([29.40, 49.27]) and 36% ([26.64, 46.21]) of
violations in Problem 2 and 3, respectively. Details can be found in Table B2 in Appendix B2.14 The
equality of proportion test on the difference in violation rate between PA and FA yields p-values
0.2105 and 0.2105 for Problem 2 and 3 respectively.
Figure 1: Proportion of violations in Problem 2 and 3
0.6
0.5
0.4
0.3
0.2
0.1
0
PA FA FAR
Averse (CE<50)
PA FA FAR
PA FA FAR
PA FA FAR
Seeking (CE>50)
Neutral (CE=50)
All included
Problem 2
Problem 3
So far, we have discussed decision consistency comparing Problem 1 with 2 and 1 with 3. In fact,
even without Problem 1, the theories also have a clear prediction on joint decisions within Problem 2
and 3, as set forth in the Hypothesis* (weak consistency).
The proportion of the types BB, BD, CB and CD are listed in Table 2, where numbers in brackets
indicate sample sizes. The proportion of inconsistent types (BD and CB) in PA and FA are 23.38%
neutral, and seeking, respectively.
14
For comparison, proportions for decisions in favor of the 50-50 risky, indifferent, or ambiguous urn in a
standard Ellsberg 2-color problem are (31.43%, 31.43%, 37.14%) in Stecher, Shields and Dickhaut (2011),
(46%, 10%, 44%) among Halevy’s (2007) risk-averse subjects, and (86.05%, 9.3%, 4.65%) among Halevy’s
(2007) risk-seeking subjects, respectively.
13
([13.71, 33.05], 18 out of 77 observations,) and 33.33% ([22.18, 44.49], 24 out of 72 observations),
respectively. The former is significantly lower than the latter, with p=0.0885 in the equality of two
proportion test. We also observe inconsistent type (BD and CB) in FAR is 43% ([33.14, 53.29], 43 out
of 100 observations). Such large scales of inconsistency here suggest that people may inherently treat
the ambiguous and the compound-risk issues differently.15 The difference between PA and FA also
implies that people behave more consistently when facing less ambiguous situations. Besides, it is
interesting to observe that the proportion of inconsistency weakly decreases from risk-averse, to
risk-neutral and risk-seeking subjects, (40, 32, 27) for FA and (27.27, 24.24, 18.19) for PA, as well as
from FA to PA. In combination with Figure 1, the latter observation suggests that violation of theories
of SEU, MEU and KMM might decrease with reduction of ambiguity, which is the case from FA to
PA.
Another pattern of behavior inconsistency is reflected in the relative frequency of people
switching action from B in Problem 2 to non-B in Problem 3, and vice versa. In the treatment FA, we
find that the switch rates are 16/38 = 42.11% ([26.31, 59.18]) (BD/(BD+BB)) and 8/34 = 23.53%
([10.75, 41.17]) (CB/(CB+CD)). The former is significantly higher than the latter, with p=0.0475 in
the equality of two proportion test. In the treatment PA, the switch rates are 11/32 =34.38% ([18.57,
53.19]) (BD/(BD+BB)) and 7/45 = 15.55% (CB/(CB+CD)) ([6.49, 29.46]). And the former is
significantly higher than the latter, with p=0.0272 in the equality of two proportion test. Thus, it is
interesting to observe that in both treatments people with preference for the ambiguous option in
Problem 2 turn out to be more consistent than those with Choice B. One way to understand this result
is to assume that decision for C or D carries with itself some sort of biased selection for DMs that are
more predisposed to follow second-order risk models.
Table 2. Proportion (%) of the types BB, BD, CB and CD
Risk averse
15
Risk neutral
Risk seeking
All
PA
[22]
FA
[25]
FAR
[38]
PA
[33]
FA
[25]
FAR
[32]
PA
[22]
FA
[22]
FAR
[30]
PA
[77]
FA
[72]
FAR
[100]
BB
13.64
20
28.95
27.27
32
28.13
40.91
40.91
23.33
27.27
30.56
27
BD
13.64
24
26.32
18.18
24
31.25
9.09
18.18
16.67
14.29
22.22
25
CB
13.64
16
18.42
6.06
8
18.75
9.09
9.09
16.67
9.09
11.11
18
CD
59.08
40
26.32
48.49
36
21.88
40.91
31.82
43.33
49.35
36.11
30
The results from the comprehension tests as reported in Appendix B3 rule out the concern that subjects may
not have properly understood the statistical implications of our double-draw design.
14
Note that we also clearly reject the randomization hypothesis of uniform distribution over the
four decision types BB, BD, CB and CD, with p=0.000 for PA, and p=0.0168 for FA, and p=0.3735
for FAR using the chi-square goodness-of-fit test. Moreover, the equality of proportion test rejects the
hypothesis that the inconsistent types result from 50-50 randomization, with p= 0.0000, 0.0023 and
0.0163 in PA, FA and FAR, respectively.
4. Discussions
Following Savage’s tradition of subjective expected utility, MEU and KMM also posit that the
DM evaluates the uncertain prospect with a feasible distribution on outcomes. Ambiguity aversion is
traditionally interpreted as willingness to pay a premium to avoid the variability of the range of
ambiguity behind the prospect, i.e. premium to avoid the additional, second-order risk beyond that
attached to any single objective distribution. However, when the ambiguous prospect is only
associated with mean-preserving contractions over the risky one, thus without any reason to pay
premium based on a wider range of unwanted risks, as in our design, a substantial share of subjects
still chose to avoid the ambiguous prospect, in violation of predictions by MEU and KMM. In
comparison, CEU has no problem in explaining these violations.
Technically, it is due to the fact that CEU is equivalent to weighting the given
von-Neumann-Morgenstern utility 𝑢(. ) with virtually arbitrary distributions not restricted to the
mean-preserving class of 𝑆𝐶 or 𝑆𝐶𝐾𝑀𝑀 , which is in contrast a binding condition for MEU and KMM.
Thus, a CEU DM may overweigh the x=0 event with some 𝑞 ∉ 𝑆𝐶 as if he is willing to pay a
premium to avoid the issue of ambiguity per se, even when it implies nothing but a mean-preserving
contraction over the simple-risk prospect. In standard Ellsberg-type studies, however, the admissible
set of distributions in the problem design of ambiguity is often the whole space ∆(𝑋), as in Ellsberg’s
two-color problem. Here, the CEU-induced weighting distribution 𝑞 also must fall within ∆(𝑋),
similar to those for MEU or KMM. In Ellsberg’s three-color problem or Machina’s (2009) Reflection
example involving partial ambiguity, there are still sufficient variations in mean and variance in both
directions associated with the ambiguous prospect, so that it is easy to overlook the case of potentially
𝑞 ∉ 𝑆𝐶 . In this sense, our novel prospect design offers a way to cleanly separate premium for avoiding
15
risk variability from that for avoiding the issue per se, associated with the ambiguous prospect. CEU
appears to be able to better deal with the latter. This view of premium for the issue per se is indeed
consistent with the source-dependence interpretation of ambiguity and recent neuroimaging studies.
Source preference and ambiguity premium
Many studies feature natural events in their design of ambiguous prospects, and find that decision
under uncertainty depends not only on the degree of uncertainty but also on its source. In a series of
studies, Heath and Tversky (1991) find support for the so-called competence hypothesis that people
prefer betting on their own judgment over an equiprobable chance event when they consider
themselves knowledgeable, but not otherwise. They even pay a significant premium to bet on their
judgments. These data cannot be wholly explained by aversion to ambiguity, as in the second-order
preference theories, because judgmental probabilities are more ambiguous than chance events.16 Fox
and Tversky (1995) find that ambiguity aversion is produced by the shock from the source of either
less ambiguous events or more knowledgeable individuals, and stated the comparative ignorance
hypothesis.17
As a further illustration, Fox and Weber (2002) observe that preference for the ambiguous bet
takes a huge boost when preceded by a quiz of equally unfamiliar and ambiguous background rather
than by a quiz of rather familiar background. This is as if the high-familiarity quiz reminds him of the
existence of the rational brain condition where he actively weighs different pieces of relevant
information before making the judgment. Being framed this way, facing an unfamiliar judgment may
cause a great sense of unease not present in its absence. In fact, having already made his judgment in a
similarly unfamiliar quiz helps trick his brain, say, the amygdala, to lower the suspicion and fear
against too uncertain issues. In some sense, the willingness to pay an ambiguity premium for the
source is greatly reduced in the latter, comparatively. This view is consistent with both the
competence and the comparative ignorance hypotheses.
Abdellaoui, Baillon, Placido and Wakker (2011) introduce a novel source method to
16
Fox and Tversky (1998) also find consistent evidence on the competence hypothesis.
Tversky and Fox (1995) and Tversky and Wakker (1995) study likelihood sensitivity, another important
component of uncertainty attitudes that depends on source. They provide theoretical and empirical analyses of
this condition for ambiguity. Chow and Sarin (2002) make a distinction between the known, unknown and
unknowable cases of information, which is consistent with the comparative ignorance hypothesis.
16
17
quantitatively analyze uncertainty. They show how uncertainty and ambiguity attitudes for uniform
sources can be captured conveniently by graphs of preference functions. They fit the CEU model with
individual decisions and find support for the source preference hypotheses.18
Recent neuroimaging studies like Hsu, Bhatt, Adolphs, Tranel and Camerer (2005), Huettel,
Stowe, Gordon, Warner and Platt (2006), and Chew, Li, Chark and Zhong (2008) compare brain
activation of people who choose between ambiguous vs. risky options and suggest that these two
types of decision making follow different brain mechanisms and processing paths. For example,
evidence in Hsu et al. (2005) suggests that, when facing ambiguity, the amygdala, which is the most
crucial brain part associated with fear and vigilance, and the OFC activate first and deal with missing
information independent of its risk implication. 19 In a study with treatment variations between
strategic vs. non-strategic and cooperative vs. competitive conditions, Chark and Chew (2012) also
find that activities in the amygdala and OFC are positively correlated with the level of ambiguity
associated with the decisions. In some sense, when the brain switches modes facing ambiguity or a
different source of ambiguity, the DM may become much less probabilistically sophisticated. This
sudden change in sophistication may be captured by the issue or source-specific premium induced by
a CEU weighting function that reflects their pure preference for sources without elaboration on
associated variability of risk. As further support, psychological studies in general identify multiple
processes (some more effortful and analytic, others automatic, associative, and often emotion-based)
in play for decisions under risk or uncertainty (Weber and Johnson, 2008).
In fact, reducing the extent of ambiguity may increase the chance and intensity that the
calculating process of the brain becomes involved as compared to the emotional process around
amygdala activity. As an extreme mind experiment, if ambiguity is restricted to up to 10 red balls in a
N=1000 urn, we expect people to treat this prospect as if it were the one with no red balls. This is
consistent with the observed reduction of inconsistency from FA to PA in our study. Along this line of
thinking, we conjecture that there must be a threshold point between almost zero to full ambiguity in
18
Early studies by Einhorn and Hogarth (1985, 1986) and Hogarth and Einhorn (1990) hold the same idea, but
do not fit within the revealed preference approach. Abdellaoui, Vossmann, and Weber (2005) analyze general
decision weights under uncertainty as functions of decision weights under risk.
19
However, Huettel et al. (2006) show that activation within the lateral prefrontal cortex was predicted by
ambiguity preference, while activation of the posterior parietal cortex was predicted by risk preference, without
implicating the amygdala. Hsu, in private communication, pointed out that the difference in implicated brain
parts between Hsu et al. (2005) and Huettel et al. (2006) might be due to a design difference, as learning might
have occurred while repeatedly facing the same task in the latter. In this sense, our design is closer to that of Hsu
et al. See Dolan (2007) for further study of the behavioral role of the amygdala and OFC.
17
our design where the source premium stops to be relevant. This is however up to future studies to
clarify.
In general, if the perceived mental burden is too high, people are more likely to switch to simple
heuristics such as ambiguity aversion. Force-feeding people with tutorials in favor of following
Savage’s sure thing principle, or other similar priming, also have good chance to reduce ambiguity
aversion per se. One way conceivable is to present a simple version of Table 1 to the subjects after
introduction of the double-draw rule. Priming subjects into thinking risks may indeed reduce the
impact of the said source premium, which may be worthwhile pursuing in further follow-up studies.
In light of the neural study findings, this would be similar to investigating the specific triggering
mechanisms of the brain switching between different decision pathways.
Literature on estimating ambiguity models
Given the success of the prominent models of ambiguity in economic applications, many
experimental studies have attempted to estimate how well they fit lab data. For example, Halevy
(2007) tests the preference models for consistency, where subjects were asked to price four different
prospects including the original Ellsberg ones. He concludes that the actions of the majority of
subjects in his experiment are best explained by KMM (35%) and RDU/CEU20 (35%), followed by
SEU (19%) and MEU (1%) respectively. The remaining 10% cannot be explained by any theories and
are considered noise. In light of our study, we might speculate that CEU was the best fitting for those
35% people in Halevy’s design because they were more sensitive to the source per se rather than to
second-order risks. In this sense, our studies can be considered complementary.
Subsequently, Baillon and Bleichrodt (2011) find that models predicting uniform ambiguity
aversion are clearly rejected, and those allowing different ambiguity attitudes for gains and losses are
able to accommodate models such as prospect theory, α-maxmin (i.e. α-MP), CEU and a
sign-dependent version of the smooth model. In some sense, the domain-induced source shock may
require a higher source premium than the familiarity-induced one. Ahn, Choi, Gale and Kariv (2011)
find that people take measures to reduce the second-order risk induced by ambiguity and that most
20
Note that, according to Wakker (2008), what Halevy (2007) calls RNEU is virtually the same as RDU and
CEU, all of which are related to Quiggin (1982). Schmeidler (1989) proved however that under the convexity
condition CEU is equivalent to MEU.
18
subjects’ behavior is better explained by the α-maxmin model than by two-stage models such as
KMM. Further tests on the α-MP model include Chen, Katuscak and Ozdenoren (2007) and Hayashi
and Wada (2011).
Hey, Lotito, and Maffioletti (2008) generate ambiguity from a Bingo blower in an open and
non-manipulable manner in the lab and find that sophisticated models (such as CEU) did not perform
sufficiently better than simple theories such as SEU. In a follow-up study, Hey and Pace (2011)
evaluate the prediction power of various theoretical models and claim that “sophisticated theory does
not seem to work”, particularly the two-stage models.
Andersen, Fountain, Harrison, and Rutström
(2009) use variations of the model developed by Nau (2006) to estimate subject behavior. They show
that subjects behave in an entirely different qualitative way towards risk as towards uncertainty. In
some sense, these papers raise questions about universal validity of second-order sophisticated class
of models, from different angles than ours.
All in all, these studies find mixed support for different models in various experimental designs
of ambiguous environments. Given their design purpose, they are not directly conducive to the
discussion of separation between premium for the source and premium for second-order risk, which is
the domain of our current study. However, it may be interesting to think of designs that connect our
idea with those in the above discussed literature to identify explicit premium for ambiguity issue per
se in a less restrictive setup than ours.
Modeling the source
Abdellaoui et al. (2011) assume that different sources of uncertainty lead to different weighting
functions. Alternatively, we can identify the source with a more detailed specification of the act under
consideration. Chew and Sagi (2008) model different sources as different small worlds within the
universal state space (Ω, Σ). Gajdos, Hayashi, Tallon, and Vergnaud (2008) attach explicit set
description of admissible distributions on the state space to an act. In contrast, we identify the source
with the set of admissible (first- or second-order) distributions on the outcome space naturally
associated with the prospect. Given our source-specific acts, one may imagine that an MEU or KMM
DM does not change his functional form when making decision under uncertainty. For different
natural sources 𝑓, however, the admissible set 𝑆𝑓 may change subjectively. With CEU, since the
19
choice of weighting distribution is not limited to 𝑆𝑓 , it is indeed more convenient to think of the
source as causing different 𝑤.
Note that the state space is an auxiliary notion that greatly facilitates axiomatic approaches to
modeling decisions under uncertainty. For purpose of empirical studies, however, the more intuitive
and primitive notion of the outcome space is the primary object of concern. The notion of the
source-specific act is an attempt to translate empirically relevant statements from the former to the
latter with minimal loss of generality.
4.1. Excursion: Machina’s Reflection paradox and source-specific act
Machina (2009) points out that standard formulation of CEU satisfies the tail-event separability
property, which may cause decision paradoxes. In his Reflection example (Table 3), theory predicts
𝑓5 ≿ 𝑓6 ⟺ 𝑓7 ≿ 𝑓8 . L’Haridon and Placido (2010) show that 72-88% of subjects behaved in
violation of this prediction, in various experimental setups. Baillon, L’Haridon and Placido (2011)
subsequently demonstrate that this paradox persists also in many other models, including MEU and
KMM, with the most popular assumption of a concave second-order utility index 𝑣.
We believe the reason behind this paradox is hidden in the fact that standard derivations of these
preference notions are based on axioms stated on the general space of abstract acts, without explicitly
taking into account the source-specific informational circumstances associated with the acts. For this
reason, we introduced the term source-specific act that enhances a standard act with a set of lotteries
that are admissible under the circumstance, and based our formulation of MEU, KMM, and CEU on
it.
Table 3. Machina’s (2009) Reflection example
50 balls
f5
f6
f7
f8
50 balls
E1
E2
E3
E4
$400
$400
$0
$0
$800
$400
$800
$400
$400
$800
$400
$800
$0
$0
$400
$400
20
For MEU (and α-MP respectively) for example, Machina’s original acts 𝑓5 , 𝑓6 , 𝑓7 , 𝑓8 will be
properly enhanced to source-specific ones as (𝑓5 , 𝑆), (𝑓6 , 𝑆 ′ ), (𝑓7 , 𝑆′), ( 𝑓8 , 𝑆) , where for 𝑋 =
{0, 400, 800},
𝑆 = {𝑝 ∈ ∆(𝑋): 𝑝0 ∊ [0, .5], 𝑝400 ∊ [0,1], 𝑝0 + 𝑝400 ≤ 1} and
𝑆′ = {𝑝 ∈ ∆(𝑋): 𝑝0 ∊ [0, .5], 𝑝400 = .5}
with 𝑝 = (𝑝0 , 𝑝400 , 1 − 𝑝0 − 𝑝400 ). It is trivial to conclude that MEU(𝑓5 , 𝑆) = MEU( 𝑓8 , 𝑆) and
MEU(𝑓6 , 𝑆 ′ ) = MEU (𝑓7 , 𝑆′) by definition (2a), i.e. (𝑓5 , 𝑆) ∼ (𝑓8 , 𝑆) and(𝑓6 , 𝑆 ′ ) ∼ (𝑓7 , 𝑆′), as the
Ω-based function 𝑓𝑘 is rendered irrelevant because it contains no extra information not already in S.
Consequently, 𝑓5 ≿ 𝑓6 ⟺ 𝑓8 ≿ 𝑓7 , in consistency with observed behavior. In other words, the
Reflection example causes no paradox for our source-enhanced notion of MEU in contrast to Baillon
et al. (2011).
In our study, the empirical violation against α-MP (MEU) and KMM persists despite this
source-specific modification. And in the case of KMM, it is even true for all monotone second-order
utility index 𝑣, while Baillon et al.’s (2011) only challenges the class of concave 𝑣’s. In summary,
while the paradox around the Reflection example suggests we should incorporate more information
details into the preference analysis independent of the specific models, our results suggest that
probabilistic models like α-MP (MEU) and KMM might not be as robust as the weighting function
approach of CEU in general.
5. Conclusion
When people reveal ambiguity aversion as in Ellsberg-type decisions, the conventional belief as
expressed in α-MP/MEU and KMM, among many others, is that their ambiguity premium is paid to
avoid the variability of a wider range of risk, or the second-order risk. The source dependence
approach postulates a source premium at the core of ambiguity aversion, which is consistent with
CEU and has been corroborated in recent neuroimaging studies. By restricting attention to a
mean-preserving class of prospects via a novel design, we manage to cleanly disentangle these two
effects. The new design allows us to separate people who are avoiding ambiguity per se from those
21
avoiding second-order risk. Although many people’s decisions are consistent with the prediction of
α-MP/MEU and KMM, we robustly find that a substantial number of people still show ambiguity
aversion that cannot be attributed to aversion to second-order risks, which indicates that their
premium is paid to avoid the issue of ambiguity per se.
Note that in our design with full ambiguity CEU appears to be a more complex and flexible
model than the competing ones, by possessing the additional dimension of variation for source
premium. But it might not always be the best fitting model for statistical analysis due to issues like
over fitting. With the insight of our study, further designs are needed to explore the conditions for
when a simpler model without a parameter for source premium may be methodologically more
appropriate. Also, the normal brain is endowed with decision pathways of both the rational
(second-order risk) and the emotional (source aversion) kind, either of which could be triggered under
certain conditions. Thus, behavior in real-world decision making situations may be pointedly
manipulated by either priming them into aversion to ambiguity per se or explicitly training them into
thinking second-order risks. Future studies are required to further explore the boundary and extent of
the source premium identified in the current study.
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26
Appendix A: Instructions (Slides translated from Chinese original)
This is an economic decision experiment supported by the National Research Fund. Please listen to
and read the instruction carefully, and make your choices seriously. Depending on your choice and
luck, you will have the chance to earn different amounts of money in the experiment. Payments are
confidential and no other participant will be informed about the amount you make. From now on and
till the end of the experiment any communication with other participants is not permitted. If you have
a question, please raise your hand and one of us will come to your desk to answer it.

[used in the FA/PA treatment] The experiment comprises three decision problems. At the end
of the experiment, you will make a random draw to select one from the three decision problems
in today’s experiment. We will pay you fully based on the realization of your decision in that
problem.

[used in the FAR treatment] The experiment comprises three decision problems. Because of
the time constraint, at the end of the experiment, we will randomly choose 3 students for real
monetary payment. Every selected student will make a random draw to select one from the three
decision problems in today’s experiment. We will pay you fully based on the realization of your
decision in that problem.
We will now start with Problem 1.
[Slide 1]
[used in the FA/FAR treatment]
Problem 1: Making a choice between option A and urn B
 Urn B contains 5 red balls and 5 white balls.
B
 Payoff rule for urn B: Two balls are to be drawn from urn B with replacement. You get 50 Yuan
if the first ball drawn is red and nothing if it is white. Conversely, you get 50 Yuan if the second
ball drawn is white and nothing if it is red. You get paid the sum of money earned in the two
draws.
27
[used in the PA treatment]
Problem 1: Making a choice between option A and urn B
 Urn B contains 8 red balls and 8 white balls.
B
 Payoff rule for urn B: Two balls are to be drawn from urn B with replacement. You get 50 Yuan
if the first ball drawn is red and nothing if it is white. Conversely, you get 50 Yuan if the second
ball drawn is white and nothing if it is red. You get paid the sum of money earned in the two
draws.
[Slide 2] Decision sheet for Problem 1
Make a choice by checking either option A or urn B in each row
Situation
Payoff of Option A
1
5 Yuan
2
10 Yuan
3
15 Yuan
…
…
9
45 Yuan
10
50 Yuan
…
…
19
95 Yuan
20
100 Yuan
Option A
Urn B
[Slide 3] At the end of the experiment, if your payoff is decided by Problem 1, the process of realizing
the payment is as follows.

You are asked to randomly draw one of the twenty situations in option A, and your choice (A or
B) in this situation will decide how you are paid. For example, if you draw situation 1 and your
choice in situation 1 is “option A” (to accept fixed payoff of 5 Yuan and give up drawing balls
from urn B), then you will be paid 5 Yuan immediately. If you draw situation 1 and your choice
in situation 1 is “urn B” (to draw balls from urn B and give up fixed payoff of 5 Yuan), then we
will let you draw balls from urn B to realize your payoffs. In another example, if you draw
28
situation 20 and your choice in situation 20 is “option A” (to accept fixed payoff of 100 Yuan and
give up drawing balls from urn B), then you will be paid 100 Yuan immediately. If you draw
situation 20 and your choice in situation 20 is “urn B” (to draw balls from urn B and give up
fixed payoff of 100 Yuan), then we will let you draw balls from urn B to realize your payoffs. If
you draw other situations, your payoff will be realized in a similar method.
[Slide 4]
[used in the FA/FAR treatment]
Problem 2: Make a choice between urn B and urn C
 Urn C contains a mixture of 10 red and white balls. The number of red and white balls is unknown;
it could be any number between 0 red balls (and 10 white balls) to 10 red balls (and 0 white
balls).
 Payoff rule for urn C: same as Payoff rule for urn B.
B
C
[used in the PA treatment]
Problem 2: Make a choice between urn B and urn C
 Urn C contains a mixture of 16 red and white balls. The number of red and white balls is unknown
and satisfy the constraints that either there are at least 6 more white balls than red balls or there
are at least 6 more red balls than white balls in the urn, in other words | number of the red –
number of the white| ≥6.
 Payoff rule for urn C: same as Payoff rule for urn B.
B
C
29
[Slide 5]
[used in the PA treatment]
Quiz: Is there any possibility that urn C contains 9 red balls and 7 white balls?
The answer: there is NOT, because of |9-7|<6
[Slide 6] Decision sheet for Problem 2
Question: If you are asked to make a choice between urn B and urn C, which urn will you choose?
□Urn B
□Urn C
[Slide 7]
[used in the FA/FAR treatment]
Problem 3: Make a choice between urn B and urn D
 Urn D contains a mixture of 10 red and white balls. The number of red and white balls is
determined as follows: one ticket is drawn from a bag containing 11 tickets with the numbers 0 to
10 written on them. The number written on the drawn ticket will determine the number of red
balls in the urn. For example, if the number 3 is drawn, then there will be 3 red balls and 7 black
balls in the urn.
 Payoff rule for urn D: same as Payoff rule for urn B.
Draw the number of red balls in urn D
2 3 4
0
1
5
6 7
8
9
10
B
0
D
[used in the PA treatment]
Problem 3: Make a choice between urn B and urn D
 Urn D contains a mixture of 16 red and white balls. The number of red and white balls is
determined as follows: one ticket is drawn from a bag containing 12 tickets with the numbers 0 to
5, and 11 to 16 written on them. The number written on the drawn ticket will determine the
number of red balls in the urn. For example, if the number 3 is drawn, then there will be 3 red
balls and 16 white balls in the urn.
30
 Payoff rule for urn D: same as Payoff rule for urn B.
Draw the number of red balls in urn D
2
1
0
12
3
5
13
11
14
15
B
4
16
D
[Slide 8] Decision sheet for Problem 3
Question: If you are asked to make a choice between urn B and urn D, which urn will you choose?
□Urn B
Gender
□Urn D
□Male
□Female
31
Appendix B1: The Comprehension Tests
B1.1 Motivation and design
At the core of our study here is the double-draw, alternate-color-win lottery design associated with
Choice B, C, and D. Since it is novel in the literature, it is legitimate to question whether subjects may
not have fully understood its statistical implications and whether they would behave consistently
when comparing simple objective lotteries such as Choice B.
To test for comprehension, we conducted an additional session with freshmen students of the cohort
as in our PA treatment, with the Instructions given below in section B1.1. They first faced Problem 1
where they were asked to match outcome distributions (those in Table 1) with urns of 6 different color
compositions in the N=5 condition. Correct answers are rewarded with money payments. After
handing in their Problem 1 decisions, they faced the task of revealing their preferences over four
different objective urns with our novel double-draw rule. And three out of a total of 30 participants
were randomly chosen to receive monetary payment according to their decisions, the detail of which
is given in B1.1. The session lasted 15 min excluding time for payment. Average payoff was 12 Yuan.
B1.2 The results
In Problem 1, 28 out of 30 subjects answered all 6 questions correctly. The remaining two
subjects made 2 mistakes each. The observed 6.67% (=2/30, [0.82, 22.07]) failure ratio is similar to
6.88% (=11/160, [3.48, 11.97]), which is the ratio of people with anomalous decision in Problem 1 of
our PA and FA treatments). The equality of proportion test yields p=0.4835. Thus, we conclude that
aside from regular noise there is no reason to believe that subjects in our study had abnormal
comprehension issue that jeopardizes our main conclusions in this paper.
In Problem 2, 24 out of 30 subjects ranked the four urns in a standard manner that make them
clearly identifiable as either risk averse (12 obs.) with A(1) = A(9) > A(3) >A(5), or risk seeking (10
obs.) with A(5) > A(3) > A(1) = A(9), or risk neutral (2 obs.) with A(5) = A(3) = A(1) = A(9). Two had
the ranking (A(5) > A(1) = A(9) >A(3), which can be viewed as consistent with prospect theory with
an unusual reference point. The remaining four displayed the rankings of A(5) > A(9) > A(3) >A(1),
A(5) > A(3) > A(1) >A(9), and A(9) > A(5) > A(1) >A(3). They failed to realize that statistically A(1)
= A(9). All in all, 86.67% (=26/30, [69.28, 96.24]) of subjects could be considered consistent with
standard theories on decision over simple lotteries.
B1.3 Instruction to the Comprehension Tests
This is an economic decision experiment supported by the National Research Fund. Please listen
to and read the instruction carefully, and make your choices seriously. Depending on your choice and
luck, you will have the chance to earn different amounts of money in the experiment. Payments are
confidential and no other participant will be informed about the amount you make. From now on and
till the end of the experiment any communication with other participants is not permitted. If you have
a question, please raise your hand and one of us will come to your desk to answer it.
The experiment comprises two decision problems. After Problem 1 finishes, we will give you the
instruction for Problem 2.
32
Problem 1
Now we start from Problem 1.
Urn A contains a mixture of 10 red and white balls. The number of red balls in urn A is denoted by n,
and accordingly, an urn containing N red balls and 10-N white balls is denoted by A(n).
Suppose we play the following game of drawing balls from urn A(n). The payoff rule is as follows.
Two balls are to be drawn consecutively from urn A(n) with replacement. You get 20 Yuan if the first
ball drawn is red and nothing if it is white. Conversely, you get 20 Yuan if the second ball drawn is
white and nothing if it is red. You get paid the sum of money earned in the two draws. Following the
rule, for any urn like A(n), one of the payoffs 0, 20 and 40 will be realized under a certain
probabilities.
Suppose there are 6 urns A(0), A(1), A(2), A(3), A(4) and A(5), as listed below, which contain n=0, 1,
2, 3, 4 and 5 red balls in the 10-ball urn A(n) respectively. The 6 profiles of probabilities for
respective payoffs are listed below. Please find the profile of probabilities that fits the urn correctly.
A(0)
A(1)
A(3)
A(4)
A(2)
A(5)
In Problem 1, you earn money by correctly matching the profile of probabilities with the fitting urn.
You earn 1 Yuan by making 1 correct match. Besides, you get a bonus of 4 Yuan, in other words a
total of 10 Yuan, if you make all 6 matches correctly.
33
Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
Probability of
0 Yuan
Probability of
40 Yuan
Probability of
20 Yuan
Your choice
0
0.25
0.16
0.24
0.09
0.21
0
0.25
0.16
0.24
0.09
0.21
1
0.5
0.68
0.52
0.82
0.58
A(
A(
A(
A(
A(
A(
)
)
)
)
)
)
Problem 2
Suppose you have 4 urns A(1), A(3), A(5) and A(9), as described in Problem 1. The rule of payoffs for
drawing balls is exactly the same as the one we used in Problem 1. Which urn would you prefer most
to draw balls from? Please rank the four urns from the highest (the most preferred) to the lowest (the
least preferred), and fill the four numbers 1, 3, 5, 9 into ( ) respectively. If any two urns are
indifferent to you, please use “=” to connect the urns.
Because of the time constraint, 3 of you will be randomly selected for payment. Once selected, you
will randomly draw two urns out of the four. Then, we will let you to draw balls from the urn which
you revealed to like better. We will pay you fully based on the realized payoff resulting from your
drawings. If you are indifferent between the randomly selected two urns, which would be connected
with “=” on your decision sheet, the tie will be broken randomly for you. Please rank the four urns:
The most
preferred
Gender
A( )
A( )
□Male
A( )
□Female
34
A( )
The least
preferred
Appendix B2: Further Data
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
PA
FA
FAR
Figure B1. Distribution of Certainty Equivalent Value in Problem 1
Table B2. Violations to hypotheses and 95% confidence intervals
Obs.
Problem 2
Problem 3
Problem 2 and
3
Problem 2 or 3 or
both
Partial
Ambiguity
risk averse
22
27.27% [6]
[7.06, 47.48]
27.27% [6]
[7.06, 47.48]
13.64%[3]
[-1.94, 29.21]
40.91%[9]
[18.60, 63.22]
risk seeking
22
50% [11]
[27.31, 72.69]
50% [11]
[27.31, 72.69]
40.91%[9]
[18.60, 63.22]
59.09%[13]
[36.78, 81.40]
All
77
22.08% [17]
[12.60, 31.55]
22.08% [17]
[12.60, 31.55]
15.58%[12]
[7.30, 23.87]
28.57%[22]
[18.25, 38.89]
Full
Ambiguity
risk averse
25
44% [11]
[24.40, 65.07]
36% [9]
[17.97, 57.48]
20%[5]
[3.15, 36.85]
60%[15]
[39.36, 80.64]
risk seeking
22
40.91% [9]
[20.71, 63.65]
50% [11]
[28.22, 71.78]
31.82%[7]
[10.68, 52.96]
59.09%[13]
[36.35, 79.29].
All
72
27.78% [20]
[17.86, 39.59]
27.78% [20]
[17.86, 39.59]
16.67%[12]
[8.92, 27.30]
38.89%[28]
[27.62, 51.11]
FA with
35
Random-pay
risk averse
38
55.26% [21]
[38.70, 71.83]
47.37% [18]
[30.74, 64.00]
28.95%[11]
[13.84, 44.05]
73.68%[28]
[59.02, 88.35]
risk seeking
30
60% [18]
[41.39, 78.61]
60% [18]
[41.39, 78.61]
43.33%[13]
[24.51, 62.15]
76.67%[23]
[60.60, 92.73]
All
100
39% [39]
[29.27, 48.73]
36% [36]
[26.43, 45.57]
24%[24]
[15.48, 32.52]
51%[51]
[41.03, 60.97]
Note: Pairs of numbers in square brackets [-, -] refer to 95% confidence intervals defined by
percentage. Single numbers in square brackets [-] refer to size of violation observations. Under risk
aversion, violation refers to choices of B in Problem 2 or 3. Under risk seeking, violation refers to
choices of C in Problem 2 or D in Problem 3. Violation in both Problem 2 and 3 refers to choices BB
under risk aversion and CD under risk seeking respectively. Violation in Problem 2 or 3 or both refers
to choices BD, CB and BB under risk aversion and BD, CB and CD under risk seeking.
36
Appendix C: Miscellaneous
C1: Proof of Lemma 1
Lemma 1 For any combination of decisions in Problems 1, 2 and 3, there is a weighting function 𝑤
under the CEU model that rationalizes them.
Proof: Note that any 𝜋 ∈ 𝑆𝐶 is uniquely represented by some 𝜃 = 𝜋(0) = 𝜋(100) ∊ [0, .25]. For
uniform comparisons between ambiguous and objective-risk choices in our design under CEU, let us
make the simplifying assumption that the DM facing ambiguity assigns subjective probability for all
relevant events. Thus, the DM can be modeled as using the same weighting function 𝑤: [0,1] → [0,1]
to evaluate all three prospects of B, C, and D, as in (4a). In the ambiguous Choice C, this implies that
he directly chooses some 𝜋 ∈ 𝑆𝐶 to apply w on it without the detour of defining the weights on sets
first. Given normalization 𝑢(0) = 0, we have
(7)
CEU(𝐶) = 𝑤(𝜃)𝑢(100) + [𝑤(1 − 𝜃) − 𝑤(𝜃)]𝑢(50)
for some 𝜃 ∊ [0, .25]. Consequently, we have
(8) CEU(𝐵) − CEU(𝐶) = [𝑤(. 25) − 𝑤(𝜃)][𝑢(100) − 𝑢(50)] + [𝑤(. 75) − 𝑤(1 − 𝜃)]𝑢(50)
𝑤(.25)−𝑤(𝜃)
𝑢(50)
Let𝑊𝜃 ≔ 𝑤(1−𝜃)−𝑤(.75), 𝜂: = 𝑢(100)−𝑢(50), then CEU(𝐵) − CEU(𝐶) > 0 iff 𝑊𝜃 > 𝜂. Similarly
in Problem 3, CEU(𝐵) − CEU(𝐷) > 0 iff 𝑊.15 > 𝜂 , as 𝑝𝜋0𝐷 = .15 . Note that any decision
combination for Problem 2 and 3 is consistent with a risk-neutral DM. The remaining 8 combinations
(risk averse or seeking, B vs. C, B vs. D) impose different joint conditions on the utility function u
and weighting function w, as displayed in Table C1 below. It is straightforward to check that proper
parameters for u and w can be found for each behavior profile to make it consistent under CEU.
For example, suppose 𝜂 > 1, which is equivalent to CE < 50 in Problem 1, then he prefers B to
D in Problem 3 if 𝑊.15 is sufficiently greater than 1, which means he is sufficiently more sensitive to
changes in small-probability events than those in large-probability ones. If 𝑊.15 ≤ 1, however, he
prefers D to B. For Problem 2, CEU has the additional maneuver room in the form of picking any 𝜃 ∊
[0, .25]. Note that the argument in the proof equally applies to both PA and FA.
η>1
Table C1
B≻D
D≻B
η<1
B≻C
C≻B
B≻C
C≻B
Wθ > 𝜂
Wθ < 𝜂
Wθ > 𝜂
Wθ < 𝜂
W.15 > 𝜂
W.15 > 𝜂
W.15 > 𝜂
W.15 > 𝜂
Wθ > 𝜂
Wθ < 𝜂
Wθ > 𝜂
Wθ < 𝜂
W.15 < 𝜂
W.15 < 𝜂
W.15 < 𝜂
W.15 < 𝜂
37
C2: Elicitation of risk attitudes: BDM vs. MPL
As a methodological note, most experiments on the Ellsberg paradox used to use the standard
BDM mechanism in which the subject is asked to state a minimum certainty-equivalent selling price
to give up the lottery he has been endowed with. This auction procedure provides a formal incentive
for the subject to truthfully reveal their CE of the lottery. However, in its original form it appears hard
for some subjects to comprehend. In a pilot study where subjects were to make binary decisions first
and to reveal a BDM price for their preferred choices second, 26 out of 89 subjects (29.2%) displayed
inconsistent evaluations. More specifically, aside from the Problem 2 and 3 binary decisions as in this
paper, subjects in the pilot faced another choice between urn B and an urn with equal likelihood of
either 3 or 7 red balls. After the binary decision is made, the subjects have to announce their selling
price for their preferred prospect. The inconsistency comes from the fact that they evaluate the same
choice with different values in different problems. Additionally, Stecher, Shields and Dickhaut (2011)
also studied an Ellsberg-type problem by making a choice between risk and ambiguity accompanied
with the standard BDM mechanism for both prospects. Among the 60 subjects, only 40% (24 subjects)
had clear, consistent decisions on choice and price, in other words, to choose the prospect with a
higher BDM price. About 23% had clear conflict between choice and price, and 37% of subjects
priced both prospects the same but preferred one of them.21
Thus, we choose to use a modified version of the BDM mechanism, Multiple Price List (MPL),
to elicit subjects’ risk attitude. The MPL is a relatively simple procedure for eliciting values from a
subject and has been widely used in experimental economics. First, instead of asking subjects to
reveal a single selling price, we ask them to make 20 simple binary decisions, where a randomizing
device determines which of them is realized. Compared with the standard BDM, the attraction is not
only how easy it is to explain to the subjects, but also the fact that if the subject believes that his
responses have no effect on which row is chosen, then the task collapses to a binary choice in which
the subject gets what he wants if he answers truthfully. Andersen, Harrison, Lau and Rutström (2006)
studied the properties of the MPL method by a series of experimental designs. Also, Sapienza,
Zingales and Maestripieri (2009) use a similarly modified BDM method, which they consider an
adaptation from the mechanism used in Holt and Laury (2002). The elicitation process of the certainty
equivalent associated with a bet is also one of the basic steps in Abdellaoui et al. (2011) for elicitation
of risk and ambiguity attitudes.22
21
Stecher, Shields and Dickhaut (2011) for example made their subjects take a quiz on the procedure and
reviewed them with the experimenter before being admitted into the experiment, to minimize the problem
associated with difficulties comprehending the experimental procedure.
22
Also see Trautmann, Vieider and Wakker (2011) for further comparisons between BDM and certainty
equivalent measurements under risk and ambiguity.
38
In addition, the binary decision in our modified BDM is similar in shape to the subsequent parts
of the experiments, which facilitates the comparison to ambiguity attitudes. Weber and Johnson (2008)
argue that, when measuring levels of risk taking with the objective of predicting risk taking in other
situations, it is important to use a decision task that is as similar as possible to the situation for which
behavior is being predicted. To quote Harrison and Rutström (2008), “For the instrument to elicit
truthful responses, the experimenter must ensure that the subject realizes that the choice of a buying
price does not depend on the stated selling price. If there is reason to suspect that subjects do not
understand this independence, the use of physical randomizing devices (e.g., a die or bingo cages)
may mitigate such strategic thinking.” And the 29.2% inconsistency rate encountered in a pilot to the
present study using the original BDM fittingly echoes this reasoning.
When comparing the distribution of risk attitudes in the present study to some related studies in
the literature, we find quite consistent results. In our experiments for PA (FA), there are 28.57%
(34.72%) risk-averse, 42.86% (34.72%) risk-neutral, and 28.57% (30.56%) risk-seeking subjects,
respectively. In comparison, using standard BDM, Halevy (2007) has for the small (big) incentive
treatment 31.73% (44.74%) risk-averse, 0.77 % (44.74%) risk-neutral, and 37.5% (10.52%)
risk-seeking subjects, respectively.
39