2. Theoretical Models and Experimental Design

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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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15273.

Savage,
L
eonard J.
1954
.

The Foundations of S
tatistics
. John Wiley & Sons, New York.

Schmeidler, D
avid
.

1989
.

“Subjective Probability and Expected Utility without

Additivity
.

Econometrica

57
(3)
: 571
-
587.

Segal
,

U
zi.
1987
.

“The Ellsberg Paradox and Risk Aversion:
An Anticipated Utility Approach
.

International Economic Review
,

28
(1):

175

202.


Segal,

U
zi.


1990
.

“Two
-
Stage Lotter
ies without the Reduction Axiom
.

Econometrica
,

58
(2):

349

377.


Seo
,

K
youngwon.

2009.

“Am
biguity and Second Order Belief
.

Econometrica
, 77(5)
:

1575
-
1605.

Stecher, J
ack Douglas,

T
imothy W.

Shields
,

and J
ohn Wilson

Dickhaut
.
201
1
.


Generat
ing Ambiguity
in the Laboratory.


Management

Science
,

f
orthcoming
.

Trautman
n, Stefen

T
.
,

Ferdinand

M. Vie
ider
, and Peter

P. Wakker. 2011.


Preference Reversals
f
or
Ambiguity Aversion.


Management

Science
,

57(7): 1320
-
33
.

Tversky, Amos
and

Craig R. Fox
.
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
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,
63
(6):

1255

1280.

Wakker
, Peter P. 2008. “Uncertainty.” In Lawrence Blume & Steven N. Durlauf (eds.),
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Palgrave: A Dictionary of Economics
, 6780

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Weber
, E
lke

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ric

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.
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.

“Decisions under Uncertainty: Psychological, Economic,

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.


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P. Glimcher, C. Camerer, E. Fehr,
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. 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.