1
Testing Ambiguity Theories
in a New Experimental Design
with Mean

Preserving Prospects
B
Y
Chun

Lei Yang
and Lan
Yao
O
CTOBER
201
2
Abstract
Ambiguity aversion can be interpreted as aversion against
second

order risk
s associated with
ambiguous
acts, as in
MEU
/
α

MP
and KMM
.
I
n 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 so
urce)
,
consistent with
both
experimental
findings
on
source dependenc
e
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.
K
EYWORDS
:
Ambiguity,
Ellsberg paradox,
expected utility,
experiment
,
Machina paradox,
s
econd

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

278
5
4160,
http://idv.sinica.edu.tw/cly/
School of Economics, Shanghai University of Finance and Economics, Shanghai, China 20043
3
;
e
mail:
yao.lan@mail.shufe.edu.cn
2
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 m
aking 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 pr
oposed t
o 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)
. ME
U
solves the
paradox
and has been applied to
studies on asset pricing in Dow and Werlang (1992) and Epstein and Wang (1994) among other
s
.
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 (198
9
)
,
where the DM uses a weighting functio
n
called capacity to evaluate
prospect
s
.
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 stud
ies 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 a
mbiguity aversion
reflects
w
illingness 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
Eichberg
er
and Kelsey (200
9
)
offer
excellent surveys.
3
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)
.
W
e
have a lottery design
that is a simple
modification of Ellsberg’s two

color problem
that
enables this separation.
In our
design
, t
he
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 compoun
d 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
orde
r 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 prediction
s 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
,
w
ithin 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 assoc
iated 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.
4
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
Ω
. An individual is assumed to have a preference ordering over the space of
all acts
, for
making
decision
s
under uncertainty
. For our purpose, assume the outcome space consists of finite real
numbers that represent monetary payoffs,
, with
.
For any set
Z
, let
denote the space of proba
bility distributions, i.e. lotteries, on Z.
An act
f
and a probability
distribution on Ω induce a unique probability distribution
. However, the specific lottery
device, or the source that governs the circumstances of the underlying uncertainty, mi
ght involve
higher

order compound lotteries in
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,
. 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
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
wi
th explicit recognition of partial ambiguous set
, instead of their
standard definitions on the state space
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,
5
ambiguous Choice C
in our FA treatment, for example,
has
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,
with
representing
the uniform distribution over
.
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 s
ubjective specification of
S
,
presumably
in
the first

order
space
. Ergin and Gul’s (2009) issue dependence can be
interpreted
as
referring to
differentiations in
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) s
ubjective expected utility theory assumes that there is a
monotone
utility function
on the outcome space,
, such that for each source

specific act (
f, S
)
with
, there is a
subjective belief
so tha
t
∑
Ghirardato
et al.
(2004)
have
the so

called
α

MP (multi

prior) model
, a
generaliz
ation to
MEU
,
as follows.
Given
that the DM has a compact set
K
for each (
f, S
)
,
α
α
∑
(1
α
)
∑
In the extreme case of
α
, we obtain the
original
MEU expression
proposed
by Gilboa and
Schmeidler (1989).
min
∑
.
The
smooth model of ambiguity aversion
(KMM) by
Klibanoff
et al
.
(2005)
consider
s
the space
of second

order compound lotteries as the relevant space for decision under uncertainty and assumes
that there is a monotone function
, with which the DM evaluates the certainty equivalents of
Tallon, and Vergnaud (2008)
also attach an admissible set to a
n 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
.
6
first

order lotteries
evaluated with
u
. For ea
ch (
f, S
) with
, there is a
second

order
subjective belief
so that
∫
(
∑
)
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
=
denote the event that yields the
monetary
payoff
,
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,
,
⋃
,
and
whenever
.
The payoff under CEU is then the following.
∑
[
(
⋃
)
(
⋃
)
]
Note with
(
⋃
)
(
⋃
)
, we have
∑
and
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 exper
imental stud
y
, 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,
,
that is at the center of prospect theory by
Tversky and Kahneman (1992).
We then can work with the following instead.
∑
∑
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 2
N
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
7
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
. Urn D is a compound
lottery
with uniform distribution over
S
.
The
options associated with
u
rn
s
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 bet
ween
Choi
ce 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
. In the
partial ambiguity treatment (PA),
N
= 8 and
.
By
definition
, FA and PA also differ in Choice D
due solely to the difference between
and
.
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 p
redictions
Let
,
with
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
0
denote
, with slight abuse of notations,
the
physically
4
We aim at revealing individual certainty equ
ivalent values of Choice B.
Though
we may
alternatively
replace
Choice B with its reduce
d
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 B
ecker, 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
Trau
tmann, Vieider and Wakker (2011
)
.
Detailed
discussion can be found in
A
ppendix
C
2
.
8
feasible
set of first

order lotteries under our design
.
Generically,
the
outcome
probabilit
ies
are
0
and
0
, respectively
. Due to our
symmetrical design, {
h

red, (
2
N

h
)

white} and {(
2
N

h
)

red,
h

white} urns induce equivalent
prospects, in all aspects relevant for decision under
uncertainty
.
The m
ean
for
is the same 50 for
all
h
. But the variance
,
∗
,
increases from
h
= 0 to
h
=
N
and then
symmetrically decreases from
h
=
N
to
h
=
2
N
, with
h
.
The crucial
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
.
Though
there are
only
lotteries in
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
, i.e.,
.
At the heart of our design is the feature
that,
for any
,
and
one of
and
is a
mean

preserving
spread to the other
due to the nature of
.
In fact,
let
0
6
Table 1
summarizes all possible first

order lotteries given this payoff rule, with
coding for the lottery
with
h
red balls and 10

h
white balls. There are
exact ly
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,
, gives us the
probabilit ies of .24, .52, and .24 to earn the prize of 0, 50, and 100 Yuan, respectively; with a mean of 50 Yuan
and a varian
ce
of
1200
.
Obviously, our modified Ellsberg risky prospect,
, has the highest variance
of
1250
,
while all color compositions yield the same mean payoff
.
7
For example,
Ste
cher, Shields and Dickhaut (201
1
)
have an ingenious method to generate virtual am
biguity
via objective but mathematically involved compound lotteries, which illustrates the need to
consider the convex
hull here.
Table
1
: Complete list of feasible first

order lotteries
,
6
N
= 5
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
v
ariance
0
450
800
1050
1200
1250
1200
1050
800
450
0
9
denot
e
the probability
π
assigns to
x
=0 for any
, then
0
, i.e.
can be represented
as a one

parameter family
by the compact interval [0,
.25].
Let
∑
is
the
reduced
first

order
distribution
f
or Choice D
, with
∑
.
8
More specifically, for
treatment
s FA and PA,
0
,
0
,
,
, respectively.
Once the DM reveals his risk attitude associated with
u
in Problem 1 as
being risk averse,
risk
neutral
or risk seeking
(
correspond
ing
to CE <
50
,
=
50
, or >
50,
i.e.,
)
,
specific predictions can be derived for his decision in Problem
s
2 and 3 based on the
above

mentioned models, which we can test experimentally. First,
Problem 3
only involves singleton
sources
and
, i.e., there is no ambiguity. I
t 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
DM
s
,
because
is a mean preserving spread of
. In fact, as
is a strict mean preserving
spread to any lottery in
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
when evaluating Choice C
.
In the PA
treatment, t
h
e
potential indifference
is
ruled out by design as
. In the FA, the latter is
the case if
in the MEU formulae
(2a) or
α
<
1 even when
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.
T
he same
prediction
is also true for KMM.
For Choice C under KMM, we assume
, whose first

order reduction
trivially
coincides with
.
Let
0
0
denote the
expected
value for
. For any
,
0
0
.
For any
with
, s
ince
0
0
,
iff
, i.e. iff
.
Now, for any strictly increasing
and any
without degenerately putting all weight on
,
we conclude from the definition of KMM that
8
Note for any compound
lottery
𝑦
,
y
∑
∑
̅
𝑦
̅
. Due to mean preserving,
the second term vanishes
in our design.
10
The proof is straightforward in that
,
due to monotonicity
,
is either the
maximum or the
minimum o
n
, depending on whether the DM is risk seeking or averse.
Note that this is
true whether the support for
is restricted to
or
.
In summary, we have
the
following
theoretical prediction
s
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 theor
ies
predict
that
peo
ple with non

neutral risk attitudes should
have a strict preference among
the two choices in
both Problem
s
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 discussi
on 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 Problem
s
1, 2 and 3
in FA
and PA
, there is a
weighting
function
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
induces
a
virtual lottery
not necessarily in
the mean

preserving class of
, so that CEU is exactly the expected
utility weighted with
.
In fact,
∑
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
.
Note,
although our
design
is not intended
to discriminate among
different shapes of
, the partial
ambiguity approach may be useful for this in future studies
.
11
Experimental p
rocedure
We
ran two treatments that differ
in the number of balls
in the urn
and wh
ether the color
composition
is partially unknown.
The full ambiguity one (FA) has
10
ball
s 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 F
A (
), but the former is not too
small to make the ambiguity issue irrelevant (e.g. under
N
=100
0
,
n
=12).
Note that the
primary
purpose of
design
ing
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
PowerP
oint 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
decisio
n sheets were collect
ed
,
subjects w
ere
call
ed
upon to have their decisions implemented one by
one
.
10
For both
FA
and
PA
, subjects
dr
e
w
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 payme
nt. To add
to data robustness, we also present
its results here and
call
it
FAR treatment henceforth.
A
total of
269
subjects
f
rom Shanghai University of Finance and Economics
participated
in
the
experiment.
All participants were
first

year college students
of various
majors
ranging from
economics
and
management to
science and language
.
160 students participated in the treatment
s PA
and FA
and
all of them were paid. We ran two sessions in each treatment.
1
09
subjects participated in
the treatment FAR
of
three
sessions,
and only
1
1
subjects were paid randomly.
Average payoff for all
9
Note, o
ur
double

draw alternate

color

win
design
conceptually
removes the subjects
’
fear of possible
manipulation of color
composition
by the experimenter.
Nevertheless
, s
tudents
still
regularly
asked
to inspect
the content of
the ambiguous
urn C after the
decision implementation
.
10
After
s ubjects h
anded in their decis ions, they were
given the option
to
have the
payment
procedure
imple ment
ed
later in the experimenter’s office, if they did not wish to wait
. Only two of them
made
use of this
option
.
12
171
subjects with real payment in the t
hree
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
incent iv
ized
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
payoff
s
.
Subsequent
analys
e
s
are restricted to these
sample
s
only
.
12
Note the incentivized
comprehens ion
tests show that
subjects from the cohort have no pr oblem understanding t he statistical implicat ions of our unique
double

draw lottery design. In addit ion, most people displayed preferences over different urns of our
design that are consistent with standar
d theory of risk. Details are in Appendix
B1
.
The average
CE
value
s are
50
, 4
9.65
and
46.1
for the
treatment
s
PA
,
FA
and FAR
with standard
deviation
s
of
12.46
,
11.11
and 1
5
.
22
, respecti
vely
.
In
PA
, w
e 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
,
neutral
ity
,
and
seeking.
The numbers
in
FA
are
34.72%
,
34.72%
, and
30.56%
, and those
in
FAR are
38%,
3
2
%
and 30%
,
respectively.
13
Chi square
test reveals no signif i
cant difference between
PA
and
F
A
regarding subject risk attit udes
among the categories of risk averse, neutral and seeking
(p=0.569
)
.
Figure
B
1
in the appendix
shows the distribution
s
of subjects’ CE values.
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
On ly 8 out of 85 s ubjects (9.41%) in the PA treat ment, 3 out of 75 s ubjects (4%) in the FA treat ment, and 9
out of 109 s ubjects (8.26%) in the FAR treat ment s witched back fro m A to B, which is deemed ano malo
us and
e xcluded fro m our data analys is. We also run a
n additional
s ess ion
of the treatment FAR (41 s ubjects ) with the
alternate
order of proble ms 1, 3 and 2
, to control for potential
order effects
.
C
h i

square
test
confirms no
existence of order effects, with
p=0.831
,
0.640
, and
0.759
for
P
roblem 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 d is tribution of ris k attitude is co mmon in the literature.
Ha levy
(2007)
,
us
ing
the
standard BDM mechanism
, has
31.7%
,
30.5%
, and
38.5% of
the
105 subjects
in his sample as
risk averse
,
neutral
,
and
seeking
, respectively.
13
As summarized in the Hypothesis, for
risk

averse (

seeking)
individuals
for
P
roblem
2 and 3,
the
theor
ies
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
P
A
, 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.
7
8% ([
17
.
18
, 3
8.38
])
in both Problem
2 and 3.
In
F
AR
, we observe 3
9
% ([
29
.
40
, 4
9
.
27
]
) and 3
6
% (
[2
6.64
, 4
6
.
21
]) of
violations in Problem 2 and 3, respectively
.
D
etail
s can be found in Table B2 in
Appendix B
2
.
14
The
equality of proport ion test
on the difference in
violat ion rate
between
PA and
FA
yields p

values
0.2105
and
0.
2105
for Problem
2
and 3 respectively
.
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
F
A are
23.38
%
([
13.71
,
33.05
]
, 18 out of 77 observations,
) and
33
.
33
%
([
22
.
18
,
44.49
]
, 24 out of 72 observations
),
respectively.
T
he former
is significantly
low
er than the latter
, with
p=0.0
885
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
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
H
alevy’s
(2007)
risk

averse subjects
,
and (
86.05%, 9.3%
,
4.65%
) among
H
alevy’s
(2007)
risk

seekin
g
subjects, respectively
.
0
0.1
0.2
0.3
0.4
0.5
0.6
PA
FA
FAR
PA
FA
FAR
PA
FA
FAR
PA
FA
FAR
Averse (CE<50)
Seeki ng (CE>50)
Neutral (CE=50)
All included
Figure 1: Proportion of violations in Problem 2 and 3
Problem 2
Problem 3
14
of 100 observations).
Such
large scales
of inconsistency
here suggest
that
people
may
inherently
treat
the ambiguous
and
the compou
nd

risk issues
differently
.
15
The difference between PA and FA
also
implies that people behave more consistently when facing less ambiguous s ituat ions.
Besides,
it is
interesting to observe that the
proport ion
of incons istency weakly decreases
from risk

averse,
to
risk

neutral
and
r isk

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
P
roblem 2
to non

B
in
P
roblem
3
,
and vice versa
. In the treatment
FA
, we
find that the switch rates
are
16
/3
8
=
42.1
1
%
([26.31, 59.18
]
)
(BD/(BD+BB))
and
8
/3
4
=
23.53%
([10.75, 41.17])
(CB/(CB+CD)).
The former is
significantly
higher
than
the latter
, wit
h
p=0.0
475 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=0.0
272 in the e
quality of two proportion test.
Thus, it is
interesting to observe that
in both treatments
people with preference for the ambiguous option
in
P
roblem 2
turn out to be
more consistent
than those with
C
hoice 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
Risk neutral
Risk seeking
All
PA
[22]
FA
[25]
FAR
[
38
]
PA
[33]
FA
[25]
FAR
[
3
2]
PA
[
2
2]
FA
[22]
FAR
[
30
]
PA
[77]
FA
[72]
FAR
[1
00
]
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
3
0
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
15
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.
15
hypothesis that
the inconsistent
types result from 50

50 randomization, with
p
=
0.0000
,
0.0023
and
0.0163
in
PA
,
F
A
and
FAR
, respectively
.
4
. Discussion
s
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 ambigu
ous 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 pros
pect, in violation
of
predictions by
MEU and KMM
.
In
comparison,
CEU has no problem
in
explain
ing
these violations.
Technically, it is due to the fact that CEU is equivalent to weighting the given
von

Neumann

Morgenstern utility
with virtually arbitrary distributions not restricted to the
mean

preserving class of
or
,
which
is
in contrast
a
binding condition for MEU and KMM.
Thus, a CEU DM may overweigh the
x
=0 event with
some
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 t
he problem design of ambiguity is often the whole space
,
a
s in Ellsberg
’s
two

color problem. Here, the CEU

induced weighting distribution
also must fall within
,
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
.
In this sense, our novel prospect design offers a way to cleanly separate premium for avoiding
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 vie
w
of premium for the issue per se
is indeed
consistent with the source

dependence interpretation of ambiguity a
nd recent neuro
imaging studies
.
16
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 th
eir 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 seco
nd

order
preference theories,
because judgmental probabilities are more ambiguous than chance events.
16
Fox
and Tversky (1995) f
i
nd that ambiguit y aversion is produced by the shock from the source of either
less ambiguous events or more knowledgeable indivi
duals
,
and stated the
comparative ignorance
hypothesis
.
17
As a
further illustrat ion
,
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 uneas
e
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 unce
rtain 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
quantitatively analyz
e
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
16
Fox
and Tversky (1998) also find consistent evidence on the competence hypothesis.
17
Tvers ky and Fox (1995) and
Tvers ky and
Wakker
(1995)
s tudy
l
ikelihood s ens itivity, another important
component of uncertainty attitudes that depends on
s ource
. They provide theoretical and e mpirica l ana lys es of
this condition for a mb iguity.
Chow and Sa rin (2002) ma ke a dis tinction between the known, unknown and
unk
nowable cases of information, which is consistent with the comparative ignorance hypothesis.
18
Early s tudies by Einhorn
and
Hogarth (1985, 1986)
and
Hogarth and Einhorn (1990)
hold th
e s ame idea, but
do not fit within the revealed pre ference approach. Abde
llaoui, Vos s mann, and
Weber
(2005) ana lyze general
17
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 fa
cing 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. competit ive condit ions, Chark and Chew (2012) also
find
that activit ies in
the
amygdala and OFC
are positively correlated
with
t he
level of
ambiguity
associated with the decis ions
.
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 indu
ced 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, other
s 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
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.
F
orce

feeding people with tutorials
in favor of
follow
ing
decision weights under uncertainty as functions of
decision
weights under risk.
19
Ho wever,
Huettel et a l. (2006) s how that activation within the latera l p refrontal corte x was predicted by
amb iguity prefer
ence, while activation of the pos terior parietal corte x was predicted by ris k 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 am
ygdala and OFC.
18
S
avage
’
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 in
to 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 th
e 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 as
ked to price four different
prospect
s
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/CEU
20
(35%)
, followed by
SEU
(19%)
and MEU
(1%)
respectively.
The
remaining 10% cannot be explained by any theor ies 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 t
o
second

order risks.
In this sense, our studies can be considered complementary.
Subsequently,
Baillon and
Bleichrodt
(2011)
find that m
odels 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
re
quire
a
higher source premium than the
familiarity

induced one
.
Ahn, Choi,
Gale and
Kariv
(20
1
1
)
find
that people take measures to reduce the second

order risk induced by ambiguity
and
that
most
subjects’ behavior is better explained by the α

maxmin
model
than by
two

stage
model
s such as
KMM
.
Further tests on the
α

MP
model include
Chen, Katuscak and Ozdenoren (2007)
and
Hayashi
and
Wada
(201
1
)
.
Hey, Lotito, and
Maffioletti
(2008)
generate ambiguity from
a Bingo blower
in an open and
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 (198
9
) proved however that under the convexity
condition CEU is equivalent t
o MEU.
19
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
”
, particular
ly
th
e
two

stage
models.
Andersen, Fountain, Harrison, and
Rutström
(2009)
use
variations of the model
developed by Nau (200
6
)
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 our
s
.
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 seco
nd

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 set
up than ours.
Modeling the source
Abdellao
u
i
et al.
(2011) assume that differen
t
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
.
Gajdos, Hayashi, Tallon, and Vergnaud (2008) attach explicit set
descri
ption 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
, however, the admissible set
may
change subjectively
.
With CEU, since the
choice of weighting distribution is not limited to
, it is indeed more convenient to think of
the
source as causing different
.
Note that the state space is an
auxiliary
notion that greatly facilitates axioma
tic approaches to
modeling
decision
s
under uncertainty
.
For purpose of empirical studies, however, the more intuitive
20
and primitive notion of the outcome space is
the primary object of concern
.
T
he notion of the
source

specific act is an attempt to transla
te 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
paradox
es.
In
his Reflection example (Table
3
)
,
theory predicts
.
L
’
Haridon and Placido
(201
0
)
show that 72

88%
of
subjects behaved in
violation of this prediction, in various experimental setups.
Baillon
,
L
’
Haridon and Placido
(2011)
subseque
ntly demonstrate that th
is
paradox persists also in many other models
,
including MEU and
KMM
,
with the most popular assumption o
f
a
concave second

order utility index
.
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

spec
ific 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
50 balls
E1
E2
E3
E4
f5
$400
$800
$400
$0
f
6
$400
$400
$800
$0
f7
$0
$800
$400
$400
f
8
$0
$400
$800
$400
For MEU
(and
α

MP respectively)
for example, Machina’s original acts
will be
properly enhanced to source

specific ones as
, where
for
,
0
00
0
00
and
0
00
21
with
0
00
0
00
.
It is
trivial to conclude
that
and
by
definition (
2
a
)
,
i.e.
and
,
as the

based
function
is
rendered
irrelevant
because it contains no extra information not already in
S
.
Consequently,
, 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
, while Baillon
et al
.
’s (2011) only challenge
s
the class of
concave
’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 other
s,
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 neuro
imag
ing
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
avoiding second

order risk.
Although many people
’
s decision
s
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 variati
on
for source
22
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, b
ehavior
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 f
urther explore the boundary and extent of
the source
premium
identified in the current study.
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.
1995
.
“Weighing Risk and Uncertainty
.
”
Psychological Review
,
102
(2)
, 269
–
283.
Tversky, Amos, and Daniel
Kahneman. 1992.
“
Advances in Prospect Theory: Cumulative
Representation of Uncertainty.
”
Journal of Risk and Uncertainty
, 5(4): 297

323.
Tversky, Amos, and P
eter
P. Wakker
.
1995. “Risk Attitudes and Decision Weights
.
”
Econometric
a
,
63
(6):
1255
–
1280.
Wakker
, Peter P. 2008. “Uncertainty.” In Lawrence Blume & Steven N. Durlauf (eds.),
The New
Palgrave: A Dictionary of Economics
, 6780
–
6791, Macmillan Press: London.
Weber
, E
lke
U., and
E
ric
J. Johnson
.
2008
.
“Decisions under Uncertainty: Psychological, Economic,
and Neuroeconomic Explanations of Risk Preference
.
”
In:
P. Glimcher, C. Camerer, E. Fehr,
and
R.
Poldrack
(Eds.),
Neuroeconomics: Decision Making and the Brain
. New York: Elsevier.
26
Appendix A:
Instructions
(Slides
translated from Chinese original
)
This is an economic decision
experiment
supported by
the N
ational
R
esearch
F
und
.
Please listen to
and read the instruction carefully, and make your choices seriously. Depend
ing
on your choice and
luck, you will have the chance to earn different amount
s
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, p
lease 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
problem
s
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 problem
s in today’s experiment. We will pay you fully based on the realization of your
decision in that
problem
.
We will now start with
P
roblem 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.
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.
B
27
[
used in the
PA
treatment
]
Problem 1
: Making a choice between option A and urn B
Urn B cont
ains
8
red balls and
8
white balls.
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
[Slide 3]
At
the end of the experiment
, if your payoff is decided by P
roblem 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
Y
uan and give up draw
ing
balls
from urn B
)
,
then you will be paid 5 Yuan
immediately.
I
f 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
Make a choice by checking either option A or urn B in each row
Situation
Payoff of Option A
Option A
Urn B
1
5 Yuan
2
10 Yuan
3
15 Yuan
…
…
9
45 Yuan
10
50 Yuan
…
…
19
95 Yuan
20
100 Yuan
B
28
situation
20
and your
choice in situation
20
is “
option A
”
(to accept fixed payoff of 100 Yuan and
give up draw
ing
balls from urn B)
,
then you will be paid 100 Yuan immediately.
I
f 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
.
[
used in the
PA
treatment
]
Problem 2
: Make a choice between urn B and urn C
Urn C contains a mixture of 1
6
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 t
he red
–
number of the white
≥
6
.
Payoff rule for urn C: same as Payoff rule for urn B
.
B
C
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
.
[
used in the
PA
treatment
]
Problem 3
: Make a choice between urn B and urn D
Urn
D
contains a mixture of 1
6
red and white balls. The number of red and white balls is
determined as follows: one ticket i
s drawn from a bag containing 1
2
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 o
f 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.
2
1
3
4
5
6
7
8
9
10
0
0
Draw t
he number of red balls in urn D
B
D
30
Payoff rule for urn D: same as Payoff rule for urn B
.
[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
□
Urn D
Gender
□
Male
□
Female
0
1
2
3
4
5
11
12
13
14
15
16
D
B
Draw t
he number of red balls in urn D
31
Appendix B
1
:
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
P
roblem 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.
A
fter
handing in their
P
roblem 1 decisions, they faced the task of revealing their preferences over four
different objective urns w
ith 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 p
ayoff was 12 Yuan.
B1.2
The results
I
n P
roblem 1, 28
out of 30 subjects
answer
ed 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
P
roblem 1 of
our
PA and FA
treatments
)
.
T
he 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
conclusion
s in this paper.
In P
roblem 2, 24 out of 30 subjects rank
ed
the four urns
in a
standard
manner that make them
clearly iden
ti
fiable as either
risk avers
e
(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.
T
he
remain
ing
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
theor
ies
on decision
over simple lotteries
.
B1.3
Instruction
to the Comprehension Tests
This is an economic decision
experiment
supported by
the N
ational Research Fund. Please listen
to and read the instruction carefully, and make your choices seriously. Depend
ing
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 commu
nication 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
P
roblem
1 finishes, we will give you the
instruction
for
P
roblem
2.
32
Problem 1
Now we start from P
roblem 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, a
n
urn containing N red balls and 10

N white balls is denoted by A(
n
).
Suppose w
e 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
2
0 Yuan if the first
ball drawn is red and nothing if it is white. Conversely, you get
2
0 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.
Followin
g the
rule, for any urn like A(
n
),
one of the
p
ayoffs 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 c
ontain
n
=0, 1,
2, 3
, 4 and 5 red balls in the 10

ball urn A(
n
) respectively. The 6
profile
s of probabilities
for
respective
payoffs are listed below. Please find the
profile
of
probabilities
that fits
the urn correctly.
In
P
roblem
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.
A(0)
A(1)
A(3)
A(5)
A(4)
A(2)
33
Probability of
0 Yuan
Probability of
40 Yuan
Probability o
f
20 Yuan
Your choice
Question 1
0
0
1
A( )
Question 2
0.25
0.25
0.5
A( )
Question 3
0.16
0.16
0.68
A( )
Question 4
0.24
0.24
0.52
A( )
Question 5
0.09
0.09
0.82
A( )
Question 6
0.21
0.21
0.58
A( )
Problem 2
Suppose you have 4 urns A(1),
A(3), A(5) and A(9)
, as described in
P
roblem 1
. The rule of payoffs for
drawing balls is exactly
the same as the one we used in P
roblem 1. Which urn would you prefer most
to draw balls from? Please rank the four urns from the high
est
(the most preferred) to the low
est
(the
least preferred), and fill the four numbers 1, 3, 5, 9 in
to
( ) respectively. If any two urns are
indifferent
to you, please use
“
=
”
to connect the urn
s.
Because
of the time constraint, 3
of you
will be randomly s
elected for payment.
Once selected, you
will randomly d
raw 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
A( )
A( )
A( )
A( )
The least
preferred
Gender
□
Male
□
Female
34
Appendix B2: Further Data
Figure
B
1
.
D
istribution of Certainty E
quivalent
V
alue in Problem 1
Table
B
2
.
Violations to h
ypothese
s and
95%
c
onfidence
i
nterval
s
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
2
2
5
0
%
[
11
]
[
27.31
,
72.69
]
50%
[
11
]
[27.31, 72.69]
40.
91%[9
]
[
18.60,
63.22
]
59.09%[13
]
[
36.78
, 8
1.40
]
All
7
7
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]
[3
9.36, 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
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
PA
FA
FAR
35
Random

pay
risk averse
38
5
5
.
26
% [2
1
]
[38
.
70
,
71.83
]
4
7
.
37
% [
18
]
[3
0
.
74
,
64.00
]
2
8
.
95
%[11
]
[
13.84
,
44.05
]
7
3
.
68%[28
]
[5
9.02
, 8
8
.
35
]
risk seeking
30
60
%
[
18
]
[
41.39
, 7
8.61
]
60
%
[
18
]
[
41.39
, 7
8.61
]
43.33%[13
]
[24
.
51, 62.15
]
76.67%[23
]
[60.60, 92.73
]
All
1
00
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 n
umbers
in
square brackets
[

,

]
refer to 9
5
% confidence intervals
defined by
percentage
.
Single n
umbers
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
P
roblem 2 and 3 refers to choices BB
under risk aversion and CD under risk seek
ing respectively. Violation in P
roblem 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 Problem
s
1, 2 and 3, there is a weighting function
under the
CEU
model that rationalizes them
.
Proof
:
Note
that
any
is uniquely represented by some
.
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
assign
s
subjective probability for
all
relevant
event
s
.
Thus, the DM can be modeled as using t
he same weighting function
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
to apply
w
on it without the detour of defining the weights on sets
first
.
Given normalization
, we have
for some
. Consequently, we have
Let
,
0
00
0
,
then
iff
. Similarly
in Problem 3,
iff
, as
0
.
Note that any decision
combination for Problem 2 and 3 is consistent with a risk

neutral DM. The remaining 8 combinations
(risk averse or see
k
ing, B vs. C, B vs. D) impos
e 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
,
which is equivalent to CE < 50
in
Problem 1
, then he prefers B to
D in Problem 3 if
is sufficiently greater than
1
, which means he is sufficiently mor
e sensitive to
changes in small

probability events than those in large

probability ones. If
, however, he
prefers D to B. For Problem 2, CEU has the additional
maneuver room
in
the
form of picking any
.
Note that the argument in the proof
equally applies to both PA and
FA.
Table C1
η
η
≻
≻
≻
≻
≻
W
θ
W
θ
W
θ
W
θ
W
W
W
W
≻
W
θ
W
θ
W
θ
W
θ
W
W
W
W
37
C2:
Elicitation of risk attitudes: BDM vs. MPL
As a methodological note,
most
experiments on the Ellsberg paradox use
d 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 incenti
ve
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
preferre
d
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 a
nother choice
between urn B and an urn with equal likelihood of
either 3 or 7 red balls. After the binary decision is made, the subject
s
ha
ve
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. Amon
g the 60 subjects, only
40% (24 subjects)
ha
d
clear
,
consistent decisions on choice and price, in other words, to choose the prospect
with a
higher BDM price. About 23%
ha
d
clear conflict between choice and price, and 37% of subjects
priced both prospects
the same but prefer
red
one of them.
21
Thus, we
choose
to use a modif ied version of the BDM mechanism
, Mult iple 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 ran
domizing
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 cho
sen, then the task
collapse
s
to a binary choice in which
the subject gets what he wants if he answers truthfully.
Anders
e
n
,
H
arrison
,
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
similar
ly
modified BDM method, which they consider an
adaptation from the mechanism used in Holt and Laury (2002)
.
The e
licitation
process of
the
certainty
equivalent associated with a bet is also one of the basic
step
s
in
Abdellaoui
et al
.
(2011) for
elicitation
of
risk and ambiguity attitudes.
22
21
Stecher, Shields and Dickhaut (201
1
)
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 experiment
al
procedure.
22
Als o s
ee
Tra
u
tmann, Vieider and Wakker (2011)
for further
comparison
s
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 attitude
s.
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 bein
g 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
pre
sent study
using the original BDM fittingly echoes this reasoning.
W
hen
compar
ing
the distribution of risk attitudes in the present
study to
some related studies in
the literature
,
we
find quite consistent results.
In our experiments
f
or
PA
(
F
A)
,
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
3
1
.
73
%
(44.74%)
risk

averse
,
0.77
%
(44.74%)
r
isk

neutral
,
and
3
7
.
5
%
(10.52%)
risk

seeking subjects
,
respectively.
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