Tgk
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/sequentialdecisiOOofek
HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Sequential Decision Making:
How
Prior
Choices Affect Subsequent
Valuations
Elie
Ofek
Muhamet
Yildiz
Ernan Haruvy
Working Paper 02-40
November 2002
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssm.com/abstract_id=353421
Sequential Decision Making:
How
Prior Choices Affect Subsequent
Valuations
Elie Ofek,
Muhamet
Yildiz
and Ernan Haruvy*
November 2002
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
LIBRARIES
*Elie Ofek:
Harvard Business School, Soldiers
Department of Economics, MIT, Cambridge,
at Dallas School of
authors would
MA
02163.
thank Al Roth, Daron Acemoglu, George
and Brian Gibbs
for detailed
Muhamet
Yildiz:
02142. Ernan Haruvy: University of Texas
Management, 2601 North Floyd Road, Richardson,
like to
helpful suggestions,
MA
Field, Boston,
Wu
comments on an
TX
75083-0688.
The
and Vikram Maheshri
earlier version.
for
The paper
has also benefited from comments of participants at the Behavioral Research Council conference
(Great Barrington,
MA,
July 2002).
Abstract
This paper develops and tests a model of sequential decision making where a
of ranking a set of alternatives
of these
same
maker who
is
alternatives.
is
first
stage
followed by a second stage of determining the value
The model assumes a boundedly
rational Bayesian decision
uncertain about his/her underlying preferences over the relevant attributes,
and who has to exert
costly cognitive effort to resolve this uncertainty.
Compared
to
when
only valuation takes place, the analysis reveals that ranking a set of alternatives prior
to determining their value has three primary effects:
a) the
spread (or dispersion) of
valuations between most and least preferred alternatives increases, b) decision makers
will,
on expectation, exert more
effort in the valuation phase,
and
c)
the
more each
attribute contributes to overall utility the greater the relative impact of ranking
valuation spread.
for a product.
with actual
The
analysis also sheds light
The
series of controlled lab
findings have implications for
situations ranging from auctions,
on
on how prior ranking impacts the demand
These results are then corroborated in a
prizes.
is
where there
is
many
real
a tendency to
life
experiments
decision
making
prioritize items before
determining a bid, to the ranking of job candidates prior to determining wages and
benefits to be offered.
More
generally, the results bear
decisions can affect future related decisions.
on our understanding of how past
Introduction
1
Past decisions are often used as input to guide future related decisions. This
as beneficial
light
when the information conveyed
in a previous decision
on dimensions associated with the decision at hand, even
identical. In the context of
would
affect the
subsequent determination of
maximum
who
downtown
is
reminded of the
if
decisions.
willingness to pay for a given
later, this
same
downtown
A
similar question
may
choose
How
would
urban job
arise even within a single
Examples of such procedures are common: an employer might
consumer
among
at
an auction
sellers before
projects before deciding
site
with
determining
many
similar items
how much
how much
R&D
In an ideal world where individuals
first
to bid until an item
them out
effortlessly, Individual's
first
preference structure with
full
rank the set
made (Roth
might have to repeatedly
is
secured (Peters
rank order product development
resources to devote to each one (Keefer 2001).
know
their preferences with certainty or can
previous choices would not have an informational
value for subsequent decisions. In reality, however, individuals seldom
know
their
may need
contingencies. Hence,
most decision making tasks regarding multiple alternatives
costly effort, in the
to anticipate the likelihood of future usage/consumption
form of cognitive thinking or time-consuming research, and
How
past choices potentially useful input.
in determining valuations to
ordering of the alternatives?
when only
How do
these questions both theoretically
Social-science literature has
The
then should we expect the
effort
will
entail
render
expended
be impacted by the knowledge of a prior choice or rank
valuation takes place?
preference elicitation.
own
confidence (for example, the exact trade-off between two
product attributes) or
to
is
the decision maker divides the complex decision problem into multiple sub-
and Severinov 2001), and a management team might
figure
located
or suburbia change
of candidates interviewed before determining the details of each offer to be
1984), a
is
individual
fact that she has chosen the lower-paying
over the higher paying suburban job?
decision
previous choices
or a big house in the suburbs.
the willingness to pay (and thus bid) for a house located in
the individual
how
has chosen a job offer with certain wages
suburban town. Imagine that several months
considering buying a small house
if
expected to shed
an identical job that pays more wages but
in a big metropolitan city over
regarded
these decisions are not
such dynamic decision-making, we ask
good. For example, consider an individual
in a small
if
is
is
previous choices impact final valuations, compared
The
and
goal of this paper
is
to provide an answer to
empirically.
examined the implications of various decision tasks on
focus has been on
how performing
various tasks, such as
choice, rating
Tversky
and matching can
Montgomery
1988,
et al.
yield different
et
al.
outcomes when performed separately
Bazerman
1994,
1992,
et al.
Huber
(e.g.,
et al.
However, the implications of intertemporally combining a set of tasks on
2002).
elicited preferences
have not been studied. In this context, our paper focuses on
previous ranking task, where the output required
is
how a
an ordinal relationship between the
where the output required
alternatives, affects a subsequent valuation task,
final
is
a measure
of willingness to pay for each of the alternatives.
To examine
sequence of evaluating alternatives, we construct a model
this prevalent
of individual decision-making over a set of two alternatives defined over two attributes
The
that need to be traded-off.
for
central features of the
examining the above decision sequence
preferences
(much
in the vein of
March
model that make
are: a) individuals are uncertain
1978,
forgetful of specific details
outcome of this decision phase
(i.e.,
emerging from cognitive
structure,
own previous
and
That
said,
c)
effort
As
though individuals
during ranking, the
such, our boundedly rational
own
choices as a source of information about their
and may be perceived to have a preference for consistency
in Yariv (2002)).
about their
the rank ordering of alternatives) can be incorporated
in the subsequent determination of willingness to pay.
agents use their
relevant
and Keeny and Raiffa 1976), b) through
costly cognitive effort they can resolve part of this uncertainty,
may be
it
utility
(similar to the agents
our agents also exhibit other specific patterns of behavior
(also observed in our experimental setting) that
cannot be explained by such preference
for consistency.
The
model
analysis of the
reveals three interesting findings.
the spread of valuations for the two alternatives
precedes valuation. This
is
is,
is
as a result of ranking
is
overall utility
is
is
on expectation, greater when ranking
may
induce the agent to think more
perceived to be more valuable.
the information
Second, this increased spread
more pronounced when the contribution
higher. Lastly,
we
find that the effort
(canonically) expected to be higher
expended
when ranking information
of each attribute to
in valuing alternatives
is
present, even though
previous effort has obviously already been expended in the ranking stage.
ine
how
prices)
.
We
also
exam-
prior ranking affects the likelihood of purchase (with any given distribution of
In particular,
we show through a canonical example that
the probability of a sale
when
are centered around the
mean expected
A
shown that
not only because of the extra information embodied in the
rankings, but, interestingly, also because rankings
when
First, it is
series of
prior ranking increases
prices are either very low or very high (but not
when they
value).
experiments designed to allow comparison of valuation and ranking deci-
combined
sions (and their
for future
were carried out. The experiments used actual prizes
effect)
consumption of a familiar product category, namely dining at local restau-
and were constructed to induce
rants,
(1964) mechanism.
The
truth-telling
through a Becker-DeGroot-Marschak
empirical results strongly confirm the implications of the theory,
and were designed to rule out possible alternative explanations (such as learning or task
familiarity)
.
In particular,
we confirm
that the effect of ranking on valuation spread be-
comes more pronounced as the stakes involved
of the prizes
higher)
is
This provides an example
.
for
(i.e.,
as the decisions themselves
rest of the
paper
is
the average value
a general possibility that certain
generated by bounded rationality (Rubenstein 1998)
effects that are
The
in the task increase
may
get even larger
become more important.
organized as follows: In the next section
we develop theory
that allows the modeling of a sequence of decisions regarding the same alternatives,
in particular, ranking
model as hypotheses, which we then
findings of the
A
To shed
utility
We
formulate the central
through a
test
series of controlled
The paper ends with concluding remarks.
experiments.
2
and subsequent monetary valuation.
Model of Sequential Decision Making
light
on our questions of
interest, in this section
with uncertain preferences.
an agent confronted with two
We
then analyze the
we develop a simple model
utility
of
maximizing behavior of
rank in
different alternatives that she either needs to 1)
order of preference, 2) provide an exact monetary value for each, or 3)
first
rank and
then value each of the alternatives.
Notation:
dom
We
variable x,
E [x\G]
E [x]
E [x\G] Pr(G).
TZ +
/",
and
2.1
/"' for
We
The
respectively.
,
is
and Var (x\G)
x given event G.
the expected value of x and
Var
for the conditional expected value
also write
sets of real
Pr(G)
first,
(x) the variance of x.
and non-negative
real
We
write
and the conditional variance of
G
for the probability of event
Given any three-times
the
Given any ran-
use the following standard notation throughout.
will
and
let
E [x
numbers are denoted by
differentiable function
/
:
TZ
—
>
TZ,
we
:
G]
1Z
=
and
write
/',
second, and third derivatives, respectively.
Set-Up
Consider a boundedly rational agent who wishes to maximize the expected value of
a
utility function u, the
parameters of which she does not know with certainty.
The
agent
is
make a normatively optimal
rational in the sense that she wishes to
but at the same time
is
constrained by the costliness of effort needed to resolve the
This principle
uncertainty.
decision
prevalent in models of
is
bounded
rationality (e.g.,
Simon
1982 and Gabaix and Laibson 2000) and has been generally accepted as a factor not to
be ignored in
to
real or laboratory settings
become more informed
in
(Smith 1985). 2 Thus,
it is
possible for the agent
making her decision by introspectively accessing information
associated with the alternatives at hand. This introspection requires cognitive effort and
can be thought of as a mental cost that
is
accompanied by
disutility.
3
Ergin (2002)
uses a few plausible axiomatic assumptions regarding decision makers that are consistent
with our characterization above. In addition, while the agent
details arising
from cognitive
become
what
clear in
Assume our agent
is
E
into account
wa
random
:
TZ
—
variable
and u(Y)
V,
The
and
i
= w a (ay) +
a
b
X
ax
bx
Y
ay
by
2
This principle
deciding,
and
is
we
:
in
1Z
—
1Z are increasing functions, 6
>
this notation,
whose expected value
are concerned with cases
also related to
i is:
b (bi)
we
write
knows
8wb{by). Thus, while the agent
6,
and Y, each defined
value the agent attaches to obtaining good
E (X, Y). With
not with certainty know
to be non-trivial,
wb
and
X
b:
wa (ai) + Sw
where
by the agent (the updating process
presented with two indivisible goods,
1Z.
tasks, the
follows).
terms of two attributes a and
Here, ax, ay, bx, by
forgetful about specific
expended in any previous related decision
effort
outcome of such decisions can be taken
will
is
E
[5] is
is
u(X)
wa (•)
taken to be
a non-negative real
= w a (ax) + Svjb{bx)
and
l.
4
w\, (•)
she does
For the decision
where no good dominates the
what Marschak (1968) termed
other.
5
as the cost of thinking, calculating,
acting.
3
It may also be possible in some cases for the agent to expend resources in obtaining information
by seeking out costly external sources. For example, speaking to friends, getting an expert opinion,
conducting a survey of relevant literature,
4
Our
results generalize to the cases
value for obtaining good
we
i is:
S a w a (ai)
etc.
where there are more than two attributes and where the agent's
+
focus on the case where only Sb{= 6)
8bVJi,(bi),
is
random,
where 6 a ,6b are independent. For expositional ease
reflecting the situation whereby the contribution of
one attribute to overall utility is ex ante known with less certainty.
5
It is worth noting that the attribute conflict hypothesis of Fischer
tradeoff in attributes will lead to preference uncertainty.
et. al
(2000) predicts that such a
.
Hence, without loss of generality, we assume
— u>b{bx) >
Wbiby)
good
We
would be
structure, she
prefer
0.
X
to good
indifferent
Y
X
if
<
8
such that for each c G
our agent
if
and only
knew her
8
if
=
p,
=
true utility
and
strictly
p.
associated with
V (c),
and variance
8.
the agent
8,
may expend effort
c (measured
For some c
is
a
<5
of
<5
=
CDF
We
F(-;c).
agent obviously obtains an estimate
her utihty for each of the goods.
(i.e.,
The
1Z+ x
[0, c]
—> [0,1]
(1)
5,
emphasize that 8
e
is
and
6 are stochastically
treated as an estimator
been expended. After thinking, the
a number) that she can then use to establish
net utility from each good with estimator 8 can
as:
lUa(Ot)
=
can easily be shown that Var(e)
variance of the remaining uncertainty
precision of
:
6e,
variable) before actual thinking has
expressed
F
such that
the remaining error associated with the estimator
random
now be
consider a function
c]
independent, and 8 has
(i.e.,
0,
(•;
6
where e
>
F c) is a cumulative distribution function (CDF) with mean
where V (c) = Far (6). By expending c G [0,
units of effort, our
[0, c],
agent can get an estimate
It
if
and A&
terms of utility). In our model, the agent has the following simple, non-adaptive way of
acquiring information about
1
j^ > 0. Note that
between
and Y
and only
if
To reduce the uncertainty
in
=
write p
A a = w a (ax) — w a (ay) >
+ 8w h (bi) (Var(8)
is
—
c.
Var{8))/{\
+
Var(6)).
Hence, the
decreasing with Var(8), which measures the
8.
In subsequent analysis, determining the variance of the estimators plays a central
role.
We
will
assume that
Assumption
1
V
satisfies the following condition:
The function
V
increasing, strictly concave,
is strictly
and three-times
continuously differentiate
The
condition that
V
is
increasing
she gets a more precise idea about 8
the error term e).
is
The
concavity of
sufficient for optimization,
V
(i.e.,
(i.e.,
(i.e.,
V > 0) implies that as the agent thinks more
with lower noise, measured by the variance of
V" <
0) ensures that the first order condition
and more importantly,
is
consistent with the agent having
an incentive to learn only part of the information about
of the noise term positive
when making her
V has an inverse, which will be denoted by
8,
decision. Since
h(-).
leaving the variance
V" <
0,
the
first
Var
(e)
derivative
Subsequent Effort and the Updating of Estimators
2.1.1
In order to be able to trace the impact of a previous decision regarding the two alternatives
on a subsequent
decision,
from the former into the
importance of attribute
this estimate,
£o,
if
we need
to specify
Through
latter.
how
the agent would incorporate information
the agent obtains an estimate
b,
the agent then expends
thinking about the relative
initial effort Co in
of 8 (with 8
<5o
=
SqEo).
Given
c\
units thinking about the residual uncertainty
=
6 8iEi,
she gets an estimate Si such that
8
independent, and
8\
has
constrained so that the variance of the error term
is
not
where (again) the estimators
CDF
(Here
F(-;c\).
c\
is
So,
Note that E{8
rendered negative).
and
<5i,
=
]
e\ are stochastically
unbiased estimators. Note also that Var(8
Using the model described above, we
A
2.1.2
hence E[eq]
)
how information
is
=
E[e{\
how our
will analyze
we
this,
=
yielding
1,
= V (ci).
and Var(8i)
)
subsequent valuation of goods. Before
will affect her
to illustrate
= 1,
= V (c
E[8i]
agent's previous rankings
present a canonical example
processed in accordance with the model.
Canonical Example
Consider the stochastic process
D = e Zt
E
(t
t
[0, 1]),
Z
where
t
is
given by the stochastic
differential equation
1
9
dZ = —z<rdt +
t
and
B
that the expected future change in
of as a
ZQ =
,
D
a standard Brownian motion. Thus defined,
is
t
crdBt
D is nil,
random walk the agent goes through
for the task at
hand. In this context,
E(D =
and
mind
that
E
[8]
=
and
1
log 8
aggregate effect of
\to,t c ]
of length
iV(— \a
,
all associations,
Upon
of each other.
~
(t c
a
t.
D
t
can be thought
to obtain relevant information
should be thought of as an abstract point along the
t
continuum of information she can potentially consider.
2
any
for
1
t)
in her
a martingale, which implies
is
t
2
).
That
is,
8 can
We
can now express 8 as D± so
be completely determined by the
which are assumed to be stochastically independent
thinking c units, she learns incremental information in an interval
—
to)
= a log (1 +
c);
the resulting estimate
is
given by 8
=
j^-,
a
fc
log-normal random variable with
Var{8)
For
{oca
random
2
)
<
1,
V
=
V(c)
satisfies
variable (with
=
mean E{8) =
1
+
c)
exp [a
log (1
Assumption
mean E(e) =
1
1.
The
and variance
a2] -
error
1
=
(1
term e
and variance Var
(e)
=
a°2
+
=
c)
8/8
exp
[(1
is
-
1.
also a log-normal
— a log (1 +
c))a
—
2
]
.
1
=
exp
[a
2
+
(1
]
c)~
aa
—
Note that
1).
thinking c units, the agent would learn
in this example, c
all
=
exp (1/a)
—
1,
so that
the information relevant for determining
by
6.
In the above example, the support of the estimators was unbounded. For simplicity,
and without qualitatively affecting any of our
findings,
we
will
assume
in
what
follows
that estimators are uniformly bounded:
Assumption 2
D>
There exists some
such that
F (D, c) =
1
for each c 6
[0, c].
Ranking
2.2
We now describe how the agent ranks goods X
and
Y in order of preference. We explicitly
problem and present a basic comparative
define the agent's optimization
static
about the
choice of introspection length.
When
asked to rank goods
receive, the agent first decides
units, she obtains
of goods
X
<
p.
6r
i.e.,
X
and
Y
in
on the length
terms of which good she would prefer to
c r of her
an estimate 5 r and based on
and Y.
thought process.
this estimate
,
X
if
Therefore, her expected utility from choosing cr
is
that she prefers
Ur (c r = E[wa (a x + 6w b (bx
)
)
)
:
K
<
p] -f
if
E[w(X)|5r
E[wa (a Y ) + Sw b (b Y )
Note that the expectation operator, E, depends on
cv
thinking cv
makes a preference ordering
and only
It is clear
Upon
:
]
K>
(through F(-;cr )).
>
-E[u(Y)|5 r ],
p]
-
We
cr
(2)
.
compute
that
UT {Cr) = A b E[p -8 r
Using
(3),
we obtain the
Proposition
and
—|^—
<
Assume
at each
d
that
>
1
.
F
(3)
c,..
is
continuously differentiable,
—^^
>
at each d
<
1,
Then,
tr
= argmax c6
2.
tr
increases whenever both
[o,c]
<p} + E[u(Y)} -
following proposition, which states a basic comparative static.
1.
The
Ur (c;p)
decreases with
wa
and
w
b
\p
—
1|,
and
are multiplied by a constant
A
>
1
condition assumed in the proposition states that as the agent thinks longer, she
obtains a
is
1
:8T
more
precise estimator in the sense of second order stochastic dominance. (This
somewhat stronger than the condition that
V is increasing.)
Under
this condition, the
optimal choice of introspection length before the ranking decision decreases with
\p
—
1|,
which measures the degree by which one good ex ante dominates the other. 6 In addition,
else equal, introspection length increases
all
when the
stakes involved
contribution of each attribute to overall utility) are higher.
is
(i.e.,
The proof of this
the unit
proposition
relegated to the Appendix.
we analyze our
In the next two subsections,
valuation decisions.
To
expended, we
will
effort
when confronted with
agent's behavior
simplify our expressions and ensure an interior solution for the
assume two additional regularity conditions. The
first
condition
is:
V'(0)>l/(A(E[6\c. r <p})
where
A
is
a constant (that
be defined
will
2
(4)
),
This condition guarantees that the
later).
agent will always think some positive amount in a subsequent valuation task, even
The second
she knows the outcome from a previous ranking of the two alternative goods.
condition
when
is:
V'(c)<l/(A(E[8\6 r >p}) 2 )
where
c
V (c) =
given by
is
dition guarantees that
Var
(e r )
= Var (6) —
(
some uncertainty
will
Var(8 r )
(5)
j
/
1
(
+ Var(8r
remain unresolved even
) )
.
This con-
after the valuation
decision.
2.3
Valuation
We now analyze how the agent decides on the value of X and Y, when required to provide
the highest amount that she would be willing to pay for each of the goods. In a valuation
decision, the agent
must again first decide on the length
6 V for the relative importance of attribute
u(X) and u(Y),
for
effort to
will
determine estimates
ux and uy
finally
the exact amount of
is costly,
be expended prior to making a decision depends on how the valuation estimates
be used to determine the agent's
goods
X
some known
and Y,
et
respectively.
probability n.
7
al.
One
payoffs.
(1964).
For instance,
let
p
>
1,
in
In this paper,
we
consider the following
The agent provides estimates ux and uy
of the goods (say
good X)
we draw a monetary
Subsequently,
uniform distribution on the interval [0,m]
6
and
an estimate
Given that thinking
respectively.
mechanism, proposed by Becker
for
b,
of thought, c„, obtain
for
which case ex ante
X
some
is
large integer
better than Y.
is
then selected with
prize (say
m.
If
px) from a
the estimate for
The assumption
—g^
<
0,
leads to g c gr < 0. By Milgrom and Roberts (1994), this shows that the optimal choice of introspection
length decreases as p increases.
T
That
is,
each good
is
selected with probability
selected are mutually exclusive.
7r;
and the events that
X
is
selected
and that
Y
is
the good
is
higher than the monetary prize (ux
otherwise, she receives the
monetary
m > max {w a (a x
so that
77i
> max {u x ,Uy}
a dominant strategy.
Wai^y)
+ S v w b (by)
That
where
8V
>Px), the agent
receives the
good (X);
prize (px)- Consistent with
Assumption
2,
+ Dw b (b x ,w a
)
)
with probability
is,
+ Dwb (by)}
(a Y )
Under
1.
this
6v
)
the expectation of 6 conditional on
is
(6)
mechanism, truth-telling
= w a (a x +
the agent submits u x
we take
all
wb (b x
)
and u Y
is
=
the information the
agent has by the end of the valuation phase.
Note that
(i.e.,
in reality
pays p(X))
if
when the agent
and only
our mechanism minus px
reflecting the
same
.
if
.
Thus, her payoffs
for
case, she
good X, she buys the good
would get what she gets under
be the same as here minus a constant,
will
states that the agent
multiplied by a constant that
c„ units
u x > px In that
px
preferences.
Our next proposition
cost of thinking.
faces a price
We
is
will write
maximizes the variance of her estimator,
determined by the specifics of the mechanism, minus the
Uv (c„)
for the agent's
and submits valuations u x and uy
take
2,
utility
(as defined above), to
mechanism described above to determine her
Proposition 2 Under Assumption
expected
when she
thinks
be followed by the
payoff.
m
as in (6).
Then, given any cv 6
[0, c],
we
have
Uv (c) = AVar(8 v + B )
c„
= AV (c) + B -
c,
where
A =
^H(bX )+wl(by)},
B = ~[E[u 2 {X)]+E[u 2 {Y)]]+nm-AVar{6),
and 6 V
is
the estimator she obtains
The proof
is
relegated to the Appendix.
U'v
Since
V
is
upon thinking
increasing,
it
{cv
c v units.
Prom Proposition
2
we
have,
)=AV'{cv )-l.
(7)
follows that the optimal length of thought in valuation (prior
to providing utility estimates), cv
=
argmaxc € [o z£]Uv (c;A),
other parameters affect c„ only through A.
Note that
A
is
is
The
increasing with A.
increasing with
probability that the estimate will be used for a given good), with
w
2
(b x )
n
(i.e.,
and
w
2
the
(by)
the coefficients that translate the variance of 8 V to the variances of
(i.e.,
and with 1/m
respectively)
(i.e.,
u x and uy,
the probability that the price will be in an interval of
unit length, measuring the need for precision).
V
Finally, since
is
concave, using the
Cy
=
arg
first
max Uv
order condition
=
(c)
h (1/A)
we obtain
(7),
(8)
,
ce[o,e]
=
where h
{V')~
.
From
conditions (4) and (5)
evident that
it is
G
c\,
(0, c).
Ranking and Valuation
2.4
We now
describe
ranked.
Our boundedly
member
the details of her introspection. In other words, by the nature of the ranking
decision she
how
the agent determines the value of goods that she has previously
rational agent
knows which good
is
same
(cv),
effort
she must
but would need to go through the same
effort again) to
the relative importance of attribute
prefers but does not re-
and the optimal amount of
preferred
have expended to rank the alternatives
trospection (expending the
knows which good she
in-
recapture the pertinent details regarding
b.
During her ranking decision, the agent obtained an estimator 8 r with
8
=
8r e r
(9)
,
where 8 T and e T are stochastically independent, and 8 r has
argmaxc6
[o ie]
Ur
(as explained in §2.2). Let us first
knows that she has ranked
8r
<
p.
X
at least as high as
<5 r
,
she has no information about
with 8 T ). Given the concavity of
it is
and
6r
is
V and the fact that
is
Furthermore,
.
Hence, based on
subsequent introspection for purposes of valuation yields an estimator of e r that
independent of 8 r and
8r
,
8 £r
,
and
=
SrS^Era,
e rv are stochastically independent
obtained through an introspection of length
P,K T =
K
r
E[8\8 r
<
is
satisfies
8
]
.
sufficiently high so that
the agent will choose to resolve only part of the uncertainty in e r
where
the
the same, in any decision
optimal for the agent to think about e r
our regularity condition (5) guarantees that the variance of e r
2.1.1,
=
cr
Y. This implies she must have found
cost structure for resolving uncertainty regarding e r
task subsequent to ranking,
F(-;cr ) with
examine the case whereby the agent
Thus, while the agent has some information on
e r (the residual error associated
CDF
Cr V
p].
10
.
(10)
and 8 Er
is
Her new estimator
the estimator of e r
for 8 is 8„,
=
.E^l^r
,
<
Similar to the proof of Proposition
2,
we compute the expected
c rv units, given the ranking information, to
Urv (crv \8 T <
where Bt~g <p
i
is
same
the
<
variances are conditional on {8 r
this
p)
+ i%
B defined in Proposition 2,
as
p}.
from thinking
be
= AVar(6 rv \6 r <
p)
utility
r
<p
-
i
c rv
,
except that the expectations and
<
Since Var(8 TV \8 r
p)
=
(E[8\8 r
2
<
p])
V {crv )^
becomes
Urv {crv \8 r <p) =
<
Since (E[8\8 r
c rv [8 T
2
<
p})
<
p}
=
1,
< P }fV {crv +
A{E[8\8 r
and
c vo
UV0 (Cu
previous ranking.
)
The
p|
c rv
(11)
.
we must have
arg
max Urv (crv \6 T <
<
p)
arg
c£[0,c]
where
S| ir <
)
max Uvo (cy
=
)
c vo
(12)
,
c6[0,c]
are introspection effort and expected utility of valuation without
implication of the inequality (12)
as follows:
is
knowing that she
X at least as high as Y, the agent infers that she must have found attribute b
has ranked
not so important. Thus she chooses to think
associated with
its relative
weight. Together with (4)
crv [5 r
In similar fashion,
if
<p}
=
h(l/(A(E[6\6T
the agent were to
than good X, then she would
infer that
remaining uncertainty
less in resolving the
know she
and
<
(5), (11)
implies that
2
(13)
p}) )).
Y
previously ranked good
she must have found 8 r
>
p,
and the
higher
utility
from
introspection of length crv would be
Urv {Cr v \8r >
where
Bn
ances are
>
i
now
is
the
same
as
p)
=
B
A{E[8\8 r
>
2
P))
V (c^) + Br$r>p "
l
(14)
Crv,
in Proposition 2, except that the expectations
conditional on {8 r
>
p}. In this case, the agent chooses to
and
vari-
expend thinking
effort
Cr V [8 r
Analogously to (12),
is
because the agent
8
it
is
>p}
=
h(l/(A(E[8\8 r
>
2
(15)
p}) )).
straightforward to establish that crv [8 r
now knows
>
p]
>
cvo
.
This
she must have found attribute b relatively important,
Note that Var{8 rv \6 r < p) = Var{6 £r E[6\S T <
< p}) 2 V {c rv ), where the penultimate equality
(E[6\S r
independent.
11
p}\6 T
is
<
p)
due to the
=
(E[6\6 T
2
=
p]) Var(6 ST \6 r < p)
and 6 Er are stochastically
<
fact that 6 r
—
x
and has a greater incentive to expend
<
Appendix we show that
effort is positive, yet
<
c„,
c
make a more informed
effort to
and V(0) < V(cTV )
some uncertainty
still
(satisfying the regularity conditions (4)
<
In the
decision.
Var(e). Hence the optimal
remains unresolved after the valuation phase
and
(5)).
Clearly, the presence of ranking information impacts the optimal effort in the valuation
What
stage.
with ranking information
E
[c rv ]
Since Pr
=
Pr (o r
ft <
we have the
<
<
p]
+
Pr
2
<
p) h(l/(A(E[6\8r
p) E[6\6 r
+ Pr
p]) ))
ft >
(o T
>
p) E[6\S r
p]
>
p) h{\ / (A(E[6\6 r
=
E[8]
=
l
and
(1 /
2
)
is
and
(4)
(5),
E [crv >
]
expend more
effort (in expectation) in
convex. Note that h (l/x 2 )
increasing function of x.
it
h (l/x
=
c„
h (1/A),
cvo (resp.,E
is
As we
makes the agent
a convex function ofx
will discuss later,
if
place and provided that h (l/x 2 )
and only
if
x
v'(h(i/x 2 ))
1S
anon_
—V"/V is typically decreasing in x; we
to decline faster than l/x for convexity. In the canonical case described in §2.1.2,
2
)
is
indeed a convex function of
x,
9
hence
E [c™] >
cvo
and the presence of ranking
information increases the optimal effort during the valuation stage (in expectation).
proposition reflects the fact that
when
attribute b
expend
is
when
effort
How
2.5
[c rv ]
subsequently generating valuations for these same
when no ranking has taken
goods, compared to the case
need
p]) )).
a convex (resp., concave) function ofx.
In words, Proposition 3 implies that rank ordering a set of goods
is
2
>
following comparative statics result:
whenever h
)
effort
is:
Proposition 3 Under Assumption 2 and conditions
c\j
The ex ante expected
the direction of this impact in expectation?
is
when
found important
8r
<
/i(l/x
8T
(i.e.,
2
)
>
is
p)
The
convex, the greater returns to effort
overshadow the lowered incentives to
p.
Ranking Affects Subsequent Valuation
Having established how the agent incorporates her knowledge of previous rankings when
selecting the optimal effort to allocate in the valuation decision,
how
we
to
the valuations themselves are impacted by such a sequence of decisions.
will first establish
when
If
V{c)
=
C t for
^,1/(1-7)3.2/(1-7)^
j
s
some 7 €
examine
To do
so,
the findings from introspection in the valuation phase leave
the findings from the ranking phase intact, and alternatively,
9
we now turn
(0,1),
V
(c)
=
indeed a convex function of
7 c^
x.
12
1
and h(z)
=
(z/ 7 )
when they can
1/(7_1)
.
lead to
Hence, h (l/x 2 )
=
<
a preference reversal
when the good found
show that knowledge
will
valuations, as
to have a higher value
was previously
Then, confining ourselves to a very large class of canonical functions,
ranked lower).
we
(i.e.,
of the previous ranking decision leads to
measured by the variance of the
estimator for
final
more accurate
This occurs not
6.
only because of the extra information embodied in the rankings, but also because the
agent
induced to think more when the information
is
Using this
we
result,
will
show that knowledge
perceived to be more valuable.
is
of a previous ranking decision increases
mean squared
the spread of valuations, as measured by the
difference
between the
utility
estimators for the goods, and that this effect becomes more pronounced as the stakes
involved increase, as measured by the contribution of each attribute to overall
Using the relationship in
(10),
we write
/ E[6\6 r
-
E[6\8 r
for the estimator for 8
>
Wa{ a y)
+
for the values of
when the agent
u (X) and u (Y).
no previous ranking information
e
u ^, Uy
derivations to define
estimator 6 Er
§2.4).
,
i.e.,
when she
T
Note that we have u£
[8 r
We
p\), respectively.
5 rtJ Wb(6y)
<
>
p}6 Er [K
p]6 Sr
when the agent needs
she has already ranked, and where 8Er
and F(-;crv [6 r
utility.
<
P
]
p]
to provide
and
p]
6 er
also write
if
K<p
if
6r
>
,..,
p
monetary values
>
[S r
p]
for the
goods that
CDFs F(-; crv [5 r < p))
+ Kv w b(bx) and Uy =
have
u^ = w a( a x)
incorporates ranking information in her estimators
Similarly,
is
[6 r
<
>
we
write
v
u ^, Uy (with estimator
It will also
available.
5 V0 )
when
prove useful in subsequent
as the agent's valuation estimates
she were to use the
if
(hypothetically) ignores the ranking decision outcome (see
- u? = A b (p-STV ), ue£-Uy- = A b (p-6Er ), andu^ -Uy =
A b (p-6 V0 )}°
For purpose of exposition,
ranked good
yields 6 Sr
<
X
p,
let
as least as high as
the
new findings
clearly keeps valuing
us focus on the case where the agent knows she has
good Y.
When
introspection in the valuation phase
are aligned with the ranking information, hence the agent
X higher than Y.
When p <
no longer favor X. But, since the difference
is
8 Er
> p/E[8 r \6 r <
.
£t
p],
the
her rankings by giving
Y
new
When
findings strongly favor Y,
a higher value than
to the ranking information). In this last case,
X
p/E[6 r \8 r
small, the agent
value than Y, rendering the previous ranking intact.
<5
<
(albeit
<
the
new
findings
gives
X
a higher
p],
still
the result of introspection
and the agent
is
in effect reverses
with a decreased difference due
we observe a
preference reversal between
the two decision tasks.
10
6 rv
To
see why, take for
w b (b x )}
[w a (a y )
+
6 rv
example
w b {b y )] =
u™ — £™. One can
(w a (a x - w a (a y )) )
13
easily establish that
6 TV i"Wb(b y )
-
ui b (b x ))
u^ — u™ = w a(p. x ) +
= A b (p - 6 rv ).
\
»
worth pointing out that when the agent knows she has ranked
It is also
than
Y
<
8T
(i.e.,
she incorporates the ranking information into her
p),
new
X
higher
findings by
< p] < 1 (see (16)). By doing this, she lowers her valuations
ur£ < u££ and Uy < up. 11 The reverse is true when 8 T > p.
multiplying 8£r with E[8 T \8 r
X
for both
and Y,
i.e.,
The Impact
2.5.1
of Rankings on the Variance of the Estimators
and the
Likelihood of Purchase
In a valuation decision, the variance of the estimator measures
is
when she
provides her estimates. Thus,
from previous rankings
To
—V"/V
is
non-increasing.
express the variances of estimators for
a function
$
:
—
7£+
understand how information
critical to
affects the variance of the estimator for 8. In this discussion,
on the case when
will focus
it is
how informed the agent
1Z
8,
with
CDF F {-;h (^)),
one can define
by
V(x)=x 2 v(h(-^\Y\/x>0.
When
the agent expends effort
have ^(E[8\8 r
ty(E[6\6 r
of
^
>
<
Var(6 V0 )
p\)
=
p\)
=
Var(8rv \8 T <
Var(8 rv \8 T >
p)
1
—V"/V
is
p) otherwise.
The
Under Assumption
$
1,
is
final
mators as
follows.
One might
When
risk aversion
X. This
c),
(— V"/V')
and
£>,
a utility function,
is
functions with constant
+
p,
has taken place we
ranked higher than good Y, and
following
increasing.
(8)
results in estimator
light
Lemma
describes the shape
on how information from a
8.
Moreover,
^
is
convex whenever
cM
,
which
it
is
affects the variance of valuation esti-
ranking leads the agent to think
think that the information that
the relative value of attribute
is
<
8T
source of uncertainty. In our case,
log (1
is
and by
non-increasing.
increase her estimate for
When V
(before valuation)
estimator for
Based on the above lemma, previous ranking
12
X
when good
under the assumptions of our model, and sheds
Lemma
1
(17)
= h (1/A) in the valuation phase. This
= ^ (1). Analogously, if prior ranking
c\j
ranking decision affects the variance of the
11
=
no prior ranking has taken place, we have E[6]
8 V0 with variance
we
12
not true.
X
has been ranked higher than
will lead her to
value of attribute
non-increasing. In Economics,
decreasing
a,
but
to
would lead her to
is
as well as the
uncertain about
decrease her estimate.
—V"/V measures the absolute risk aversion.
satisfy the
Y
compared
The answer depends on the parameters
when the agent knows the
(CARA) and
less
much work has
(DARA)
above property.
14
Canonically, the absolute
focused on the family of utility
absolute risk aversion, such as
1
- e~ QC
,
when no ranking information
the case
is
Given that the agent obtains
available.
additional information in the valuation phase, her ultimate decision
informed,
agent
is
i.e.,
Var(8 rv \8 r <
prompted
p)
dominates
%(E[6 r \5r <
and
to think more,
also that ranking information
effects
=
in this case
(1)
When
Var(8 Vo ).
>
Var(8 rv \8 r
Since
ty.
^
is
a sense
is in
>
8r
> Var(8 VQ
p)
p,
less
the
Recall
).
Which
incorporated into 8 rv according to (16).
is
determined by the shape of
is
< *
p])
=
less
of these
convex, ex ante, ranking
information will lead to a more informed decision, as stated by our next Proposition.
Proposition 4 Under Assumptions
(6),
and 2 and the
1
regularity conditions (4), (5),
we have
Var(6 TV ) > Var(5 V0 )
—V"/V
whenever
(18)
non-increasing.
is
Proof. Under our hypothesis, the conditional variances are determined by
Lemma
1,
^
is
convex whenever
Var(8 TV )
is
<
=
Pr(<5 r
< p)^(E[8 r \8 r <
inequality holds
(Pr(<5 r
+
Pr(8 r
> p)Var(8 rv \8 r >
p\)
+
Pr{8 r
> p)^{E\8 T \8 T >
<
p)E[6 r \6 r <p}
+
Pr{6 r
>
= Var(8 V0
y(l)
due to the
penultimate equahty
p)
p)E[8 T \K
>
and by
(19)
p)
p\)
p])
).
facts that
due to the
is
<
p)Var(8 rv \8 r
\&,
we have
non-increasing. In that case,
Pv(8 r
=
p); the
—V"/V
=
> *
The
and
^
is
convex and that
fact that
E[8r
]
—
Pr(<5 r
>
p)
= 1 — Pr(<5 r <
1.
Proposition 4 establishes that, ex ante, prior rankings lead to more informed decisions.
As we
discussed earlier, this
not only because of the information contained in the
is
rankings but also because such information leads the agent to think longer
decision
is
turn to examine the implications of Proposition 4 for the ex ante likelihood
Consider good X, with price px-
of the agent purchasing a good.
8
the agent
>
q
=
(p x
is
buy
willing to
— wa (ax)) /w b (bx).
X
if
and only
= w a (ax) +
if iix
and x
good
=
E[8 T
\
8wb(bx)
>
Px,
i-e.,
Then x =
1
when
there
is
= E[8 r 8 r < p] when the agent knows that he ranked X higher than
8 T > p] otherwise. The probability that the agent is willing to buy the
no prior ranking, x
,
Given an estimate
Let us write x for the conditional expectation of 8 prior
to the valuation decision (before expending cognitive effort).
Y
the
more important.
We now
8,
when
\
is
ii{x- q )
=
(
i-F(^h
Ax
x'
!
x
15
\
2
We
on the canonical example described in
will focus
= l-#(
n(,;,)
where
$
The behavior
of
is
=
v
V
is
approximately
^i
example, we have
+ ^/2)
2
r^- log (-yAx )
depends on whether 7
II
large, the function
effort,
CDF,
the standard normal
is
1
§2.1.2. In that
the variance, and 7
is
With non-decreasing marginal
linear.
the agent has an incentive to invest in cognitive effort until virtually
resolved
she invests at
(if
As
all).
it is
we
highly likely
II (x; q)
decreases, causing
changes as x increases
On
to increase.
II (x; q)
now
as x increases, as the agent
may
to
because 6 converges to
initially increase
From the
>
(provided q
<
for
each
We
q.
also check that
II (x; q) is
q
Hence, as plotted in Figure
at
x
=
q
- 2 "'
1,
{
7 > 1/2
1/2
if
7
1
if
7 <l/2
when 7 <
- 7)
2(1
/ (7 j4)
2 < 1 - 2 t)
x
1/2, II
and approaches
=
1/2
easy to verify that
if
.
(•;<?) is
—
x
1 as
!-»
>
a U-shaped function minimized
^Ax 1 >
Since
00.
> (jAy
7~ "
1, II
g) is
(-;
an
.
the impact of prior rankings on the ex ante probability of sale taking place?
is
Recall that x can take the values
ante probability of sale
Pr (Sr
When 7 <
1/2
and only
if
increasing function in the allowable region whenever q
What
it is
II (x; q)
2-7-1
3-r-2
,
q/x
properties of the log function,
if
increasing
(7^4)
we
sale occurring,
x) but will eventually decrease
37-2
l-T
1
>
uncertainty
On the one hand,
1/2).
the latter effect from increasing variance will be small. In fact,
=
returns to
focus on this case.
in probability as its variance goes to infinity.
iim Tl{x;q)
is
the other hand, the variance v also increases
thinks longer.
n (x; q)
this
the region 7
(in
all
.
end up in a corner solution
will
when 7 > 1/2, we believe our theory is more relevant for 7 < 1/2 and
To understand the impact of prior ranking on the probability of
examine how
When 7
greater than or less than 1/2.
is
= aa 2
<
pj
n
is
1,
<
E[6r 8 r
\
higher with ranking
(E[S r \Sr
<
p];
q)
+ Pr
(s r
p],
if
>
and E[8 r
and only
p)
II
\
>
6r
p].
Therefore, the ex
if
(E[6r \K
>
p]\
q)
>n
(1; q)
.
(20)
Since
Pr (ST < p) E[Sr \6r
(20) holds
E[6 T \6 r
>
if
and only
if II (•; q) is
and
When
p),
1).
II
<
p]
+
Pr
(K >
p) E[6 r \K
>
p]
= 1,
convex with respect to these three points (E[6 r \5r
(-;
g)
is
convex
16
(resp.,
<
p],
concave) with respect to these
Y=
^^--^__^
\
0.3,
A=
20.9
q =
'
0.9
1
q = 0.2
q =
0.8
03
-
0.7
0.6
q = 0.9
0.5
"q^i
v^^^—
"*"
0.4
-
qTl.5
__
0.3
^<^^^
/s
0.2
q = 4
0.1
q = 8
4r ~-^
Figure
.
i
II
1:
—
1—
i
as a function of
t
x
for
7 <
0.5. (.4
=
1/(0.167)).
three points, prior rankings increase (resp., decrease) the ex ante probability of sale. In
particular,
assuming that both E[6 r 6 r
\
to check whether
II
g) is
(-;
<
p]
and E[8 r
convex or concave around
consider the case that the price (and hence q)
figure). In that case, II (; q) is
convex at x
=
1,
is
8r
\
1.
>
p]
We
very small
are close to
1,
refer to Figure
(e.g.,
when
q
<
we need
1.
First,
0.3 in the
hence prior rankings increase the ex ante
When the price of the good is around the ex ante value of the good
(e.g., when 0.9 < q < 1.5 in the figure), II (; g) is concave at x = 1, hence prior rankings
decrease the ex ante probability of sale. When the price is very high (e.g., when q > 4 in
probability of sale.
the figure),
II
(•;
q) is again
ante probability of
sale.
convex at x
=
1,
hence prior rankings again increase the ex
In summation, prior ranking tends to increase the probability
of a sale occurring at extreme prices (that are relatively high or low) in our canonical
example.
The Impact
2.5.2
of Rankings on the Dispersion of Valuations
Having established
in Proposition 4 that prior rankings increase the variance of the esti-
we can
derive a testable hypothesis about the spread of valuations submitted
mator
for 6,
for the
two goods.
We
measure the spread by the mean squared difference between the
valuations for the two goods. Hence,
of the valuations with
E [{ii™ — v.™) 2
}
and
E [(u v^ —
Uy) 2 are the spreads
and without information from a previous ranking
17
]
decision, respec-
tively.
= A b (p —
u ^ — Uy
7
Since
[(t#
- uTYv
E
[(t#
- u vJ) 2 =
2
]
— uVy =
v
and u £
= A 2 E[(p -
E
)
6 rv )
6 rv )
A
= A2
2
}
-
<5„
),
we have
[Var(8 rv )
+
(p
b
(p
2
-
l)
and
The
}
following proposition
A 2 E[(p - 6 V0
= A2
2
}
)
[Var(8V0 )
+
(p
an immediate corollary to Proposition
is
2"
-
l)
4.
.
states that
It
under our usual regularity conditions, knowledge of previous rankings increases the ex
ante spread between the
maximum amounts the
Proposition 5 Under Assumptions
and
1
agent
and the
2,
willing to
is
pay
for the
two goods.
and
regularity conditions (4), (5),
we have
(6),
E
whenever
—V"/V
The above
is
on the
We
-
>E
2
fi?)
]
-
[(«£
2
fi"?)
(21)
]
non-increasing.
proposition establishes that by
them
valuation spread between
The next
[(&£
proposition examines
is
first
higher than
how
if
ranking a set of goods, the subsequent
valuation alone were to be performed.
the effect of ranking on valuation dispersion depends
relative contribution to overall utility of each unit of the attributes (a,
first
define
Notation
hi).
some notation.
>
Given any A
obtaining good
and
i
is
\(w a (a
wa
multiply
0,
+ 6w b {bi)).
l )
wb
and
by A so that the agent's
utility
from
Use superscript A to indicate that the payoffs
are multiplied with A.
Proposition 6 Assume:
decreases
is
when we
(i)
F
increase the
-V
non-increasing, and (Hi)
(•; •)
is
such that E[6 r \8
amount of
(h (1/x
2
))
p]
increases and E[8 T \6
effort c to obtain the
/V"
under the same assumptions of Propositions
(h (1/x
1
and
Var{tTV - Var(8
)
is
>
2
)) is
estimator
6, (ii)
<
p]
—V"/V
a convex function of x. Then,
5,
X
J
(22)
increasing in A.
The
first
assumption
estimate in a sense that
(ii),
and the fact that h
(i)
is
is
states that as the agent exerts
more
effort
she gets a better
stronger than the second-order stochastic dominance. Given
decreasing, (Hi) can be replaced by the assumptions that
18
V'/V"
(weakly) concave and h(l/x
is
An
V (c) =
when
clearly satisfied
(i),
is
)
c7
a convex function of
and 7 €
>
E\8 T \8 T
Var(8 rv ), as $
the agent exerts
1
convex by
— Var(8
creases [Var(8 TV )
)].
— see
(ii)
x.
These two assumptions are
as in our canonical example.
more
two ways.
in
)]
effort in
becomes larger while E[8T \8T <
p]
is
(0, 1),
— Var(8 VQ
increase in A increases [Var(8 rv ]
then from Proposition
by
2
Since Var(8 Vo )
(19).
A gets larger
the ranking decision.
becomes
p]
First, as
13
Hence,
smaller. This increases
is
not affected, this in-
Second, an increase in A increases A, and hence leads the
agent to exert more effort in the valuation stage as well. This increases both Var(8 rv )
and Var(8 vo ). Under
difference [Var(8 rv )
(Hi), it
impacts Var(8 rv ) more and thereby further increases the
- Var(8 VQ )}.
Proposition 6 also implies that the greater the contribution of each additional unit
more pronounced the
of the attributes, the
That
tions.
is
E
is,
E [(u £ - u rYvX
r x
[(u
x
r x 2
£ - u^
will test in
)
]
We now
]
/A
2
-
E.[(v%
x
2
)
f
- uvYoX
ranking on the spread of valua
2
)
}
=
A
2
A
2
,
{Var{8^)
-
Var{8.>V
true even without condition (Hi).)
is
oA
- My
[{u
effect of
v
]
/A
2
is
also increasing in A,
J
Finally,
an implication we
our experiments.
Testing
3
)
- E
(This statement
increasing in A.
T
2
Model Implications
present the results of a series of experiments designed to examine the model
findings in actual decision
making
situations.
Given the interest in understanding the
impact of ranking on subsequent valuation (compared to when valuation alone
is
con-
ducted) we focus on the findings of §2.4-2.5. In particular, one can state the primary
testable implications in the form of three hypotheses as follows:
Hypothesis 1 When valuation comes
is
known, the squared
the goods
is
(or absolute) difference
higher than
Hypothesis 2 The
after ranking
if
and the rank ordering of alternatives
between the monetary values stated
for
valuation alone takes place.
relative
impact of ranking on valuation
is
increasing in the con-
tribution of a unit of each product attribute to overall utility.
Hypothesis 3 When
is
known, the
13
effort
The assumptions
valuation comes after ranking and the rank ordering of alternatives
expended in determining valuations
(i-iii)
of the proposition are
all satisfied
19
for all
goods
is
higher than
by the canonical case described
if
in §2.1.2.
valuation alone takes place.
In addition,
we
will
examine the implications of our analysis
in §2.5.1 for the effects
of prior ranking on the likelihood of purchase in a given price range.
Experimental Design and Method
3.1
To
test the
above hypotheses, we conducted three experiments.
All three presented
individuals with decisions regarding pairs of restaurants that were described in terms of
two or three
attributes.
The
MA), Type
set of possible attributes included
Location (different areas
Food Quality
of
Food
and Decor; see Appendix
B
(Table Bl) for a more detailed description. Subjects were
recruited from the general
Cambridge and Boston (MA) areas and included both students
in Cambridge,
(Asian, Indian, Seafood), Service Level,
(undergraduate and graduate) and non-students. Subjects were promised a
$10
for their willingness to participate
explained below.
The main
and a chance to earn more
goal of Experiment
1
was to
test
to replicate the results of Experiment 2
and
of
in cash or prizes as
Hypothesis
valuation spread, and together with Experiment 2 to test Hypothesis
also allowed a test of Hypothesis 3 regarding effort expended.
minimum
2.
1
regarding
Experiment 2
Experiment 3 was designed
rule out alternative explanations (we describe
these issues in greater detail below).
3.2
Experiment
1
Participants were presented with eight separate decisions, each describing two restaurants (one pair at a time). Table
Bl (Appendix B)
lists
the set of alternatives. Stimuli
were presented using a computer interface and responses were recorded through the
same
interface.
Subjects were randomly assigned into one of two conditions- a "valu-
ation only" condition
(VO) and a "ranking and valuation" condition (RV). In the
VO
condition (N=44), subjects were asked to state their dollar value for a dinner- for-two
(which included an appetizer, main course and dessert) at each restaurant.
Subjects
were told up-front that at the end of the experiment prizes would be awarded as
lows: one of the sixteen (2x8) restaurants for
fol-
which they provided a dollar value would
be selected with equal probability. In addition, the computer would randomly draw a
number between
0-50. If the dollar value the individual stated for the dinner-for-two at
the selected restaurant was greater than the randomly drawn number, a dinner-for-two
voucher (non-transferable) at that specific restaurant would be granted to the individual.
20
Otherwise, the individual would get the
see that this
Marschak
mechanism induces
(1964).
random number
and
truth-telling
in actual dollars. It
is
easy to
consistent with Becker, Degroot
is
and
Participants were presented with several examples before beginning
the experiment to help familiarize them with the above mechanism.
RV
In the
in the
VO
same
condition (N=44), subjects faced the
condition in two separate stages. In the
eight pairs of restaurants as
first stage,
participants were asked
to rank the two restaurants in each decision in order of dining preference (again for
T
a dinner-for-two), by designating with a
preferred alternative.
The two
and were asked to state
I
decision
was
first
first stage
randomly
selected, they
RV
selected, a
invoked.
I
On
mechanism
select either
a stage
same
a dinner-for-two
ranking designations.
I
prizes
or a stage II decision.
would receive a dinner-for-two voucher at
identical to that described
'1').
above
If
for the
a stage
VO
their
II
If
a
most
decision
condition was
average, 20 minutes elapsed between a particular ranking decision in stage
and the need to value the same
alternatives in stage
and most
eight observations per individual
details
their least
condition, subjects were informed that
preferred restaurant (the restaurant they designated with a
was
'2'
by an explanation of the mechanism by which
would be awarded (with examples). In the
stage
their dollar value for
Subjects were reminded of their
stages were separated
the computer would
most preferred and a
In the second stage, subjects were sequentially shown the
eight pairs of restaurants
at each alternative.
their
likely
II.
This process provided us with a
reduced the recollection of any specific
from introspection in a particular ranking decision. 14
3.2.1
Results of Experiment 1
Given the valuations supplied by subjects
for
each of the decisions, we could test whether
or not prior ranking affected the squared spread of valuations using the following regression model:
{u A
where q
is
- uB
an intercept term,
specific decision,
RANK
is
a
prior to the valuation phase,
2
)
=a +
dj, j
6
dummy
and
e is
[1,2,
Note that the same attributes appear
decisions. In the decisions used in
...,
+ a r RANK + e,
8]
are
dummy
(23)
variables to control for the
variable denoting whether ranking
had taken place
a standard normal error term. The results of the
regression analysis are presented in Table
14
ctjdj
1.
in only
Experiment
2,
two cases that are separated by several intervening
there were no two exactly overlapping attributes per
decision.
21
Table
1:
Ranking on Valuation Spread
Effect of Prior
Parameter
Estimate
t-stat
p- value
69.71
6.01
<0.01
1
11.34
0.73
0.46
Decision 2
-19.93
-1.29
0.20
Decision 3
-32.72
-2.05
0.04
Decision 4
-25.61
-1.66
0.10
Decision 5
-40.11
-2.59
0.01
Decision 6
-22.53
-1.46
0.15
Decision 7
-37.80
-2.44
0.02
RANK
30.526
3.95
<0.01
Intercept
Decision
As can be seen from the above
table, ranking alternatives prior to determining will-
ingness to pay significantly increases the spread of valuations for any two alternatives.
Experiment
1
gression (23)
difference.
thus strongly supports Hypothesis
we use the absolute
The
This result also holds
1.
decision pairs)
all
was
in the re-
between valuations instead of the squared
difference
average absolute valuation difference between
alternatives (across
if
$5.2 in the
VO
first
and second ranked
condition and $7.1 in the
RV
condition (see Table B2).
Experiment
3.3
In this experiment
and N=39
in the
2
we had
VO
and
subjects consider dinner-for-one dining alternatives
RV
(N=37
conditions, respectively). Given that the contribution of
each attribute to overall willingness to pay (or utility)
is
expected to be lower than when
the same alternatives are considered for a dinner-for-two, 15 using responses from both
experiments enables us to test Hypothesis
2.
were a subset of those used in Experiment
we wanted
15
It
The restaurant
1 (see
pairs used in this experiment
Table Bl). To examine Hypothesis
to compare the time subjects spent thinking in the valuation phase
3,
when
could be the case that a dinner-for-two prize introduces considerations not present with a dinner-
pay given the same restaurant alternatives (and hence same attribute
between Experiments 1 and 2 (comparing same decisions and same conditions, see
Appendix B), we believe that we are capturing a formulation consistent with Proposition
for-one, but since willingness to
levels) is higher
Table B4
6.
in
Vouchers
for dinner-for-two cost exactly twice as
restaurant.
22
much
as dinner-for-one vouchers for the
same
prior ranking
was or was not performed. In order to do
so,
we equated
the
number
of
VO and RV conditions to avoid confounding effects associated
16
in the experiment and/or learning effects.
Thus, in the VO
decisions performed in the
with overall time spent
RV
condition
we had
subjects state dollar values for ten pairs of restaurants, while in the
condition
we had
subjects rank five pairs of restaurants and then state dollar values for
same
these
five pairs (ten decisions altogether).
the
five pairs in
VO
pairs of the
first five
The
construction was such that the last
condition corresponded to the five pairs in the
VO
to that of Experiment
condition were ignored).
1,
RV
condition (the
The reward mechanism was
identical
with subjects able to receive dinner-for-one vouchers or cash
(and the computer randomly drawing a number between 0-35).
3.3.1
Results of Experiment 2
The average
absolute valuation difference between
first
(across all decision pairs) with dinner-for-one options
$4.1 in the
RV condition
(see
Table B3).
and second ranked alternatives
was $3.3
A similar regression to
in the
condition and
(23), again confirms that
prior ranking increases the spread of valuations (significant at the
how
VO
10%
level).
To examine
increasing /decreasing the relative contribution of each attribute to overall utility
spread of valuations between the two conditions, we compute the following
affects the
variable:
- uB f
+ u B y/2
(u A
V
(u A
[M)
This variable "normalizes" the valuation spread by an average measure of the
(or wiUingness to pay)
relative
from the two alternative restaurants. Thus, a comparison of the
impact of ranking on valuation between the dinner-for-one and dinner-for-two
scenarios
is
y
16
utility
possible.
To do
so,
we estimate the
following regression equation:
= p + 0,(1, + /3 RVl RV*D1 + (3 HV2 RV * D2 + /3V02 V0 *D2 +
In Experiment
1
we observed
that on average subjects monotonically decreased the
e,
amount
(25)
of time
they spent with each subsequent decision (perhaps due to boredom with the experiment or learning
effects).
Hence, because subjects in the
phase while those in the
VO
RV
condition had eight ranking decisions prior to the valuation
condition immediately valued the eight pairs, there would be a confound
between the number of decisions previously encountered and whether more or less time was being spent
as a result of having ranking information available. Theoretically, since the number of decisions is
doubled in RV condition, the probability that a particular decision will be chosen for prize is divided
by two, substantially decreasing the optimal effort in RV condition by (7). The set-up in Experiment 2
was designed to overcome these concerns. In addition,
trade-offs between different attributes (see Table Bl).
23
in this
experiment
all
relevant decisions involved
where
an intercept term, dj,j €
is
fi
RV
specific decision,
are dinner-for-one
and
VO
are
[1, 2, ..., 8]
dummies
and dinner-for-two
are
for the
dummy
variables to control for the
experimental condition, arid
indicators, respectively.
The
results
D1,D2
from estimat-
ing the above regression model are given in the table below.
Table
Effect of Prior
2:
Ranking and Attribute Importance on Valuation
Spread
Parameter
Estimate
t-stat
p- value
0.15
4.65
<0.01
-0.08
-2.20
0.03
Decision 2
-0.124
-3.15
<0.01
Decision 3
-0.01
-0.58
0.56
Decision 4
-0.02
-2.47
0.014
RV*D1
0.04
0.94
0.35
RV*D2
0.18
4.35
<0.01
VO*D2
0.001
0.02
0.98
Intercept
Decision
As the
1
results indicate, the higher the "stakes" of the decision at
each attribute
is
expected to increase overall
Thus Hypothesis
valuation spread.
validity
RV
from a
pilot
2
is
utility)
hand
the
(i.e.,
the more a prior ranking impacts
strongly confirmed.
17
This result also derives
study we conducted whereby we assigned respondents to
conditions with no incentive mechanism in place
more
VO
and
subjects were paid for their
(i.e.,
participation in the study, but there were no prizes allocated that depended on their
responses). In this pilot, where the "stakes" for the valuations supplied were in a sense
nil,
there was no significant overall effect for ranking. In fact, in five of the eight decisions
the spread was actually greater in the
Experiments
1
and
2,
condition. It
is
also
when examining the average amounts
in each decision for their
common
VO
most preferred
vs.
noteworthy that in both
subjects were willing to pay
least preferred alternative, there
was no
pattern of divergence for the increased spread across conditions. In some cases
both most and
compared
least preferred alternatives
to the
VO
were given a higher value in the
condition (though the spread
still
increased), in
RV
some
condition
cases both
valuations were lower, and in other cases the most preferred alternative was given a higher
value in the
17
Once
RV
than in the
VO
condition and the least preferred a lower value.
again, these results were corroborated
if
in (24)
24
we used absolute values instead
Though
of squares.
we
realize
our experiments are between subjects, we believe these results are consistent
with our model
on how valuations
(see §2.5,
result of a prior ranking).
As we measured the duration
we
both goods can increase or decrease as a
of time subjects spent
making each
of their decisions,
could also determine the effect of prior ranking on the effort expended in valuating
pairs of dinner-for-one alternatives.
T=
where
7
for
18
is
T is the time the subject
an intercept term,
We
l0
estimated the following regression model:
+ lj d + lr RANK,
(26)
J
spent determining valuations for any given decision pair,
dj are as in (23)
and
RANK designates
if
a ranking decision had
preceded valuation. Estimation results are given in the table below
Table
3:
Effect of Prior
Parameter
Ranking on Time Spent
Estimate
t-stat
p- value
15.09
5.54
<0.01
20.09
5.74
<0.01
Decision 2
6.00
1.71
0.087
Decision 3
1.41
0.40
0.69
Decision 4
5.63
1.61
0.11
10.90
4.92
<0.01
Intercept
Decision
1
RANK
As
clearly evident
from Table
3,
significantly increases the time spent
alternatives (compared to
of time spent (across
all
in Valuation
Phase
having performed a prior ranking of alternatives
on determining willingness to pay
when no such ranking takes place).
for the
In fact, the average
same
amount
decisions) determining valuations for a decision pair in the
RV
50% higher than that in the VO condition (32.16 and 21.72 seconds,
19
Table B5 in Appendix B). This result is consistent with Hypothesis 3.
condition was almost
respectively; see
18
Given that related behavioral decision theory has mainly been concerned with how different decision
tasks, when examined separately, would yield different outcomes, it is not clear they would entirely bear
on the focus of our study (explicitly combining two decision tasks). For example, it is not clear that
'anchoring and adjustment' (Tversky and Kahneman 1974) is relevant in this case, because a ranking
is a different output measure altogether than a valuation (or rating) measure. Even if subjects were
to 'anchor' on the rankings, one might expect the most preferred alternative to be 'adjusted' to have a
higher (average) value in the RV condition than in the VO condition, and the opposite be true for the
least preferred alternative. As indicated, this pattern was not prevalent in our experiments.
We believe that in the last decision subjects were perhaps hurried to conclude the experiment, and
25
One
additional point regarding the connection between the experimental results and
the theory developed
is
worth
In our model, the ex ante
stressing.
same
estimates for a given alternative should remain the
has taken place, since E[6]
=
support for this property in
regardless of whether ranking
]
1
-E
restaurant across subjects in each condition (see Table
statistically significant difference for only
B
value of the
^™] = 1- We find strong, between subject,
Experiment 2. Comparing the mean valuation for each
E[8 vo
=
mean
B4
Appendix B),
in
reveals a
one restaurant out of ten (namely, alternative
in decision pair 3).
3.3.2
The Impact
We
examined how the demand
also
affected
of Prior
Ranking on Likelihood of
for
by prior ranking. Recall from
each restaurant, as a function of price, would be
§2.5.1 that,
under certain conditions, ranking
expected to increase
demand when
To examine whether
this holds in our experimental setting,
m
p =
defined by price points: a)
m
p - = one
Table B4), b)
VO)
m
when p - < v k
restaurant,
mean
i.e.
We
is
either very low or very high.
we looked
at three regions
value for a restaurant in any decision (see
standard deviation below the mean,
deviation above the mean.
each treatment (RV,
prices are extreme,
the
sales
c)
p
m
=
one standard
then separately counted the number of individuals in
that would be willing to pay for a dinner-for-one at a given
l
,
p
m <
v k and pm
l
,
<
v\
,
where
v\
for restaurant k (there are 10 restaurants). In Table 4 below,
is
individual
i's
we present the
valuation
difference
between the two treatments in the percent of individuals willing to buy at each price
(denoted
A%
sales are
more
).
As Table
likely to
4 shows, our experimental results support a pattern whereby
take place at low or high prices as a result of alternatives
being rank ordered and then valued. Around the mean, willingness to pay
is
first
relatively
unaffected by a prior ranking.
hence the times are
levels
is
less divergent across
the two conditions.
again strongly confirmed. Note that this
2
3 that h (l/a; )
is likely
is
See also §3.4 where the result on
effort
also consistent with the discussion after Proposition
to be a convex function.
26
Table
4: Effect
of Prior
Ranking on Demand
at Different Price Points
A%
p
m-
p
m
p
m+
Decision
Restaurant
1
A
5.1%
B
A
B
A
B
A
B
A
B
5.4
2.7
2.6
5.1
2.7
2.7
2
3
4
5
0%
7.7%
12.8
21.6
7.7
5.4
8.1
8.1
7.7
10.3
2.7
8.1
1%
4.5%
Average
7.1%
Experiment 3
3.4
While in Experiment 2 we controlled
by having
five "filler"
decisions in the
RV
for the overall
valuation decisions in the
number
VO
of decisions each subject faced
condition (in lieu of the five ranking
condition), one cannot entirely rule out alternative explanations for
the results presented in Table 3 based on a difference in the nature of tasks in the two
conditions.
about valuations in the
decisions
may be
In particular, there
last five decisions of the
had already taken
place).
used a set-up similar to that in the
with
five relevant
five irrelevant
greater 'task familiarity' or learning specificity
VO
To account
VO
condition (given that five valuation
for such alternative explanations,
condition of Experiment 2
(i.e.,
valuation decisions) except that subjects (N=49)
restaurant pairs.
20
This way, in both the
RV
and
VO
first
we
dinner- for-one
rank ordered
conditions, subjects
faced ten decisions, five ranking and five subsequent valuation decisions.
20
The
first five
ranked pairs used in the
five valuation decisions.
2
and
for suggesting
We thank Al
Experiment 3 to
VO
Roth
condition were different restaurants than those in the last
for pointing out the potential
us.
27
problem arising
in
Experiment
Results of Experiment 3
3.4.1
Table B6
five
(in
Appendix B) presents the amount of time subjects spent
valuation decisions. For convenience,
corresponding
RV
,
was 10.2 and
Hypothesis
RV
dinner-for-one condition (using
re-estimated equation (26) with the
7r
we juxtapose
new
1%
significant at the
data.
level.
in each of the last
these results with those from the
data from Experiment
The parameter estimate
The
We
2).
also
for prior ranking,
results once again strongly corroborate
3.
In addition to replicating the results on effort levels, re-estimating regression equation
new data
(25) with the
yielded a parameter estimate
(significant at the
1%
P RVl and (3 V02 were not significantly different from 0. Thus
As for the likelihood of sales, the results from Experiment 3
while the parameters
level),
supporting Hypothesis
2.
support higher demand in the
the
=0.15
(3 RV2
mean
(in
mean
at the
RV
condition
when
price
is
one standard deviation below
analogy to column 3 of Table 4) and no difference in
analogy to column 4 of Table 4)
(in
.
When
price
is
demand when
price
is
one standard deviation
above the mean, no difference between the conditions was found (as opposed to expected
higher
demand
in the
RV
condition, see
column 5 of Table
21
4).
Conclusion
4
Whether
among
implicit or explicit, the tendency to first obtain a relative sense of ranking
alternatives prior to attaching a specific value to each seems pervasive in
making
decision
contexts.
In this paper,
we
set out to
decisions affects the final valuation of alternatives
examine how
and the amount of
arriving at these valuations.
The model we have proposed
rational optimizing behavior
and bounded
many
this sequence of
effort
expended
in
incorporates aspects of both
rationality.
Several interesting findings, supported through a series of laboratory experiments,
have emerged.
We
find that,
on expectation,
first
ranking a set of alternatives increases
the spread of values generated in a later stage and increases the
in the valuation phase; this
21
That
said,
is
in addition to the effort already
amount of effort expended
expended
our Experiment 3 was conducted almost a year after Experiment
2.
in first ranking
Because we are
VO data from the former with RV data from the latter, there was the potential for inflationary
and/or seasonal effects. Comparing the mean valuation for each restaurant in the VO condition between
comparing
the two experiments reveals that the
new
new
valuations are in fact slightly higher (by 7%). If
valuations for each subject by this factor, the
in fact higher (by
10% on
demand
at
we
average) with prior ranking (in analogy to column 5 of Table 4).
28
deflate the
one standard deviation above the mean
is
Furthermore, we have shown that prior ranking can affect the likelihood
alternatives.
of sales taking place.
we
In particular, using a canonical example,
predict that prior
ranking tends to increase sales when prices are either very low or very high. Given that
the ranking of alternatives
often viewed as a
is
way
to simplify valuation decisions,
believe that these results bear significance for our understanding of
we
many decision-making
contexts.
At a
practical level, our results have implications for real
tendency to rank alternatives prior to determining valuation
attempt to use
this two-stage process to their advantage.
is
life
situations where the
encountered. Sellers
For example, car dealers
may
will of-
ten refuse to discuss the price of any vehicle before they have taken the customer through
the lot and
seem,
is
work to
made him/her rank
The
dealers' reasoning,
it
would
that encouraging a ranking of alternatives (mainly on non- price attributes), will
their
advantage in subsequent price negotiations. In addition, the growing per-
vasiveness of Internet
has
the set of relevant cars.
made
commerce and
this two-stage process far
digital information dissemination in recent years
more pronounced. For
instance, in
consumer
elec-
tronic auctions, consumers are often given a listing of several relevant products in the
desired category. For each alternative, attribute levels are specified in the description of
the product. After reading
first (or if
all
descriptions,
consumers must decide which item to bid on
multiple items are required a complete rank ordering of
all).
They then must
decide on a specific valuation of the alternatives in order to enter a bid.
At a theoretical
level,
our paper provides a framework for capturing the impact of
previous choices on current related decisions.
We
do so
in the context of
assumes boundedly rational agents with uncertain preferences. Though
sible to
model these agents' behavior using some anomalous preference
a preference
for consistency,
we show that
it
a model that
may be
relation,
plau-
such as
there will be other details in their expected
pattern of behavior that could not possibly be the result of such an anomalous prefer-
ence characterization.
Subjects in our controlled lab experiments did, in fact, exhibit
such patterns of behavior.
29
References
Max
Bazerman,
H.,
George F. Lowenstein and Sally Blount White (1992), "Reversals of Pref-
erence in Allocation Decisions: Judging an Alternative versus Choosing
Among
Alternatives,"
Administrative Science Quarterly, 37, 220-240.
Becker,
Gordon M., Morris H. DeGroot, and Jacob Marschak
Single-Response Sequential Method," Behavioral Science,
9,
(1964), "Measuring Utility by a
226-232.
Ergin, H. (2002): "Preference for Flexibility with Contemplation Costs," Princeton University,
mimeo.
Huber,
A
J.,
D. Ariely, and G. Fischer (2002), "Expressing Preferences in a Principal-Agent Task:
Comparison
of Choice,
Rating and Matching," Journal of Behavioral Decision Making, forth-
coming.
Fischer, G., J. Jia,
RandMAU
and M. F. Luce (2000), "Attribute Conflict and Preference Uncertainty: The
Model," Management Science 46, 669-684.
Gabaix, X. and D. Laibson (2000): "Bounded Rationality and Directed Cognition," mimeo.
Keeny, R. L. and H. Raiffa (1976), Decisions with Multiple Objectives: Preferences and Value
Tradeoffs.
New
York: Wiley.
Keefer, Donald L. (2001) "Work-package-ranking system for the Department of Energy's Office
of Science
and Technology,"
Interfaces, 31:4
,
109-111
March, James G.(1978), "Bounded Rationality, Ambiguity, and the Engineering of Choice,"
Bell Journal of Economics
Marschak,
Review
J.
(1968),
587-608.
"Economics of Inquiring, Communicating, Deciding," American Economic
58, 1-18.
Milgrom, P. and Roberts,
84n3,
9,
J.
(1994),
"Comparing
p. 441-459.
30
Equilibria,"
American Economic Review,
V
Montgomery, H. (1983), Decision Rules and
the Search for a
Dominance
structure:
a Process Model of Decision Making, Analyzing and Aiding Decision Processes,
Towards
Amsterdam
North-Holland Publishing.
Montgomery, H.
et al.
(1994),
"The Judgment- Choice Discrepancy: Noncompatibility
structuring?," Journal of Behavioral Decision Making, Vol.
Payne,
J.,
Bettman
J.
7,
or Re-
145-155.
and Johnson, E. (1993), The Adaptive Decision Maker, Cambridge Uni-
versity press.
Peters, Michael
and Sergei Severinov (2001), "Internet Auctions with Many Traders," working
paper, University of Toronto.
Roth, A. E. (1984), "The Evolution of the Labor Market for Medical Interns and Residents:
Case Study in
Game
A
Theory," Journal of Political Economy, 92:991-1016.
Rubenstein Ariel (1998), "Modeling Bounded Rationality,"
Simon, H. A. (1982), Models of Bounded Rationality. 2
MIT Press.
vols.
Cambridge,
Cambridge, Mass.:
MA.
MIT
Press.
Smith, V. L. (1985), "Experimental Economics: Reply," American Economic Review 75: 265272.
Tversky,
Amos and
Kahneman
Daniel
(1974),
"Judgment Under Uncertainty: Heuristics and
Biases," Science, 185, 1124-1131.
Tversky, A., Sattath,
S.
and
Slovic, P. (1988), "Contingent
Weighting
in
Judgment and Choice,"
Psychological Review, Vol. 95, No.3, 371-384.
Yariv, L. (2002),
"I'll
See
It
When
I
Believe
It
-
A
Cowles Foundation Discussion Paper, Number 1352.
31
Simple Model of Cognitive Consistency,"
A
Appendix - Omitted Proofs
Proof. (Proposition l)Under our model
Ur
where /
dUT
(c;
=
(c;
= A b j" (r-
p)
we
set-up,
Sr
write (3) as
+ E{u {¥)} -
f [K] c) dS
)
,
dF/dc. Then,
p)
p d_
= A,
%{p-'^)f{fr,C
dp
Jo
r=P
= A
(p-p)f(p;c)
fc
+
M
f
(<5 r
|E
;cJ d8 T
i»(
y )'-| c
= A b F(p;c).
f(6T c)d6
;
J
[f>-
dp
Thus,
d 2 UT
d
{p; c) _
dcdp
Then, by hypothesis,
is
decreasing with
argmax c6 [o )e fr
]
To
(
and so
p,
c ;p)
prove, part
any p
for
2,
is
>
1,
with
dc[
we have
\p
—
1|
A F(p;c)]=A b
dF{p;c)
b
—g^g'
= p— 1.
Qc
= A& —^^ <
Similarly, for
1,
hence argmax c€
<
any p
increasing with p, and hence decreasing with |p
is
note that when
wa
and
(r
-
wb
is
multiplied with A,
—gig
1,
—
0) c]
>
= —p +
1|
[
1>
UT (c; p)
hence
1.
we have
cp
Ur
(c;
A)
= AA
fc
/
Sr
+ XE [u (Y)} -
f (K; c) d8
)
c.
Hence,
dUr
(c;
A)
dX
which
is
tP
= Ab
l
(r-6 r )f{6 r ;c^d8 + E[u(Y)},
increasing in c under the hypothesis of the proposition.
Proof.
ux
estimate
Uv
Here, the
(Proposition 2) Given u(X), and given that
is
selected, the value of having
is
=
{ux)
first
—
m
/""*
1
Jo
—
m
1
u {X) dp x
+
f
m
\
m
If
=—
^u
m (u(X)ux- 2 x +
1
Pxdpx
Jux
~2
V
/
equality expresses the fact that the agents will receive
otherwise, while
calculation,
X
px
and the
is
distributed uniformly on
final
[0,
m].
We
one by adding and subtracting
32
X
if
px < ux and px
,
obtain the second equality by simple
^u (X)
2
.
Similarly,
Given u (Y), and
given that
Y
selected, the value of having
is
m/2. Therefore, given the true value
~
+ nUv
irUv (ux)
U(8v)
*-
2m
(fa
Wb
TV
i
- » po) 2 -
+w
)
- , cnf + *j
(» y
8V
-
where
A=
2
2m Wb
tt-
{i>x)
E
(K - sf
On
[U{8 V )
+ Wb {by)
(8e v
-
[.
(x)
2
is
+ u yf + irm
(
u{X) 2 +u{Y) 2 +
2m
irm
-Km.
is
AE
Cv
= E \fv (e -
}
~ u ?)) + 2m u ?) +
{K-6) + ^[u(xf + u{ Y)
andB '=2^ E
2
2
8)
2
8
Therefore, the value of introspection of length c
Uv (c) = E
(^ Y
the utility of having an estimate 8 V
(by)
b
Uv (uy) — — 5m
is
-^^y)
(bx?{K~s)
w b (bx
~2m
{uy)
8,
uy
l)
2
]
=
(8-8) 2
+ B'-c,
U (xy
+ 7rm.
,(Yy
2
E[6 V ]E
(27)
[(e v
-
l)
2
]
Now,
= E^Var (e v
)
the other hand,
Var(8)
= E
=
-1
{8 v e v
E[8 v }Var(e v )
=E
+
E[eI]-\ = e\8 v ]{E[eI]-1)+E
Var(8 v ).
Substituting the previous equation in this one,
-
(8 V
)
where
B = B'
Var
{8)
- AVar
When ux >
m
,
- Var{8v
the agent gets
—
Uv (c)
when the
tails of
and argmax[/„
= Var {8) -
Var(8 v ).
)
+ B' - c = AVar(8v + B - Cy =
)
AV{cy)
+B-
c,
{8).
not bounded, we need to correct
Therefore,
obtain
we obtain
Substituting this in (27),
Uv {cv = -A
2
8)
we
[E
Uv
u(X) independent
,
by adding a term
[u x :ii x
of her estimate ux- Hence,
be
>m]+E[u Y
:u Y
sufficiently close to the
2.
33
8V
is
in the order of
> m]]
.
the distributions are sufficiently thin, and
(cy) will
when
m
is
sufficiently large,
ones computed through Proposition
(Boundary Conditions on c™ and V(cr V )) The boundary conditions (which are
Proof.
By
omitted) are not binding:
>
V{ct V [8 t
<
p\)
follows that the
V (0) <
(4),
V'(c TV [8 r
c^^r >
Var(e), hence
p]
<
p})
<
<
Cr
<
V'(0), hence Cr V [8 r
c.
<
Since c rv [8 r
V
<
<
p})
< V(crv [8 r >
V^Cyo)
<
p])
>
p]
<
p]
boundary conditions are not binding. Moreover, since
V(cr V [8 r
<
<
Cy
is
From
0.
(5),
>
Cr V [8 r
by
p], it
we have
increasing,
Var(e). Hence, some uncertainty remains
unresolved in each decision.
Proof. (Lemma
and V", which
V and 1/ (Ax
will
2
1) In this
always be h (l/ (Ax
*' (x)
will
2
= 2xV + x 2 (V)
(h (1/
-
V (h (1/ {Ax
Since
)).
V
omit the arguments of the functions V,
To show that
interchangeably.
),
we
proof
^
is
(Ar 2 )))'
2
increasing,
)))
we
=
first
= 2xV + j(h
(1/
(Ax
1/
2
we
),
V"
',
,
will write
compute that
(Ax 2 )))'
,
where
(Ax 2 )))'
(h (1/
Since
is
V" <
we have
0,
=
h>
(1/
(^x 2 )))'
(h (1/
>
(Ar 2 )) (-2/ (Ax 3 ))
>
Hence, *' (x)
0.
--^.
=
at each
x
>
(28)
0,
showing that
*
increasing.
Towards showing
^
convex,
is
we
further
compute that
= 2V + 2x(V>).(h(l/(Ax 2 )))' + ±(h(l/(Ax 2 )))"
*"(x)
= 2V + ^(/ (l/(^x 2 )))' + \(h(l/
l
By
(28),
Using
3
fc
we
further
compute that
<v (*")))* =
_
6
Substituting (30) and (31) in (29),
'
*
Since
{x)
[x)
zv
-2V
V >
4{V )2
v"
and V" <
0,
V
3x
+ x 3 V"
A 2 x 6 {V") 2
Ax 3 V"
4V"
_ 6(y') 2 _ AV" (V'f
~~
(Ax 2 Y(V"Y
V" ~
(V"f
j(-a^)'
{Ax 2 ) 2 V"
*»
V" > 2V
(V'/V")' =
-^ ut
1
'
wnen
v
(V'\3
)3
4V '"- {V
'
•
is
- 2V'"V'/{V"f <
*" >
non-increasing,
0,
showing that
proof.
will
now prove
Proposition
~
(vy
this implies that
—V"/V
'
we obtain
6(r)2
I
^
'
firff\2
We
(29)
we have
(28),
(
(Ax 2 )))".
6. Firstly,
34
2
V
(V
whenever
V /V"
is
2
1
)
V"
1
2V"
f
\
"
(V")
V
2
- 2V'"V'/(V"f <
also non-increasing, hence
V" > %$- >
0,
i.e.,
we have
!%$-, and completing the
Fact 7
y
<
y'
Given any convex function f, any x,x' ,y,y' £ R, and any 9,6' 6
and with 9x
+
(1
—
Of
Proof. (Proposition
9)
y
(x)
=
+
6'x'
—
(l
and 9 with Ox
(x, y,0;
By
+
)
-
(1
V (h(l/(Ax
9
_
0')
(y')
f
.
2
)))
(32)
A 2 V"{h(l/{Ax 2 )))
dA
x, y,
f
0'f (x
note that
6) First
d*(x;A)
Given any
<x <
we have
9') y',
+ (l-0)f (y) <
with x'
[0,1]
+ (1 —
9) y, define
A) =0y(x;A)
+ (l-0)V(y; A) - *
(1;
A)
(32),
y(h (i/(Ax* )))
'^2
dA
Since
y// f
Ax
;
<M
is
v
_
j
V"{h(l/(Ax 2 )))
concave in
Now, given any A and
A'
z,
>
we have
dcf>/dA
<
we have
with A
A',
%X.~~X
Varied) - VartfL]
(i/
(/i
#[£!£
0,
hence
;.V
X
=
By assumption
-A -A
E[8 r
\8 T
>
(E[o r
X
Var(6
v
<
is
increasing in A.
%v
,
xA'
X
X
\o r
>
p},Pv(o r
\6r
>
/>],
X
X
\5
</>
p},E[o r
in the hypothesis
< E[6 r
\6 T
>
p].
second inequality holds because
<j>
'
;
P);
AX
)
X
)-Var(8V0 ).
and by Proposition
Together with Fact
is
a
<p) a
< p);A x ')
Pr(# <
-A' -A'
(i)
-A' -A'
p]
X
4>
(h (im))
V"[h{l/A))
<p],£[5;i(5; >p],Pr(£
(i5[£|£ < p},E[o T
>
y
V)))
(
V"(h(l/(Ay 2 )))
7,
1,
35
\5 T
<
-A -A
p]
< E[6 r \8 r <
this gives the first inequality.
increasing in A, which
definition.
E[5 T
is
p]
<
The
increasing in A. Equalities are
by
Appendix — Supplement to Experimental Findings
Lj
Table Bl: Restaurant Pairs
7
Decision
Alternative''
Type c
Location"
A
Indian
Kendall Sq.
B
Seafood
Inman
1
2(1)
3(2)
4
5
6(3)
Food Quality 6
Service Quality
'
Decor
Sq.
A
Harvard Sq.
12
13
B
Central Sq.
17
16
A
20
14
B
17
16
A
21
16
B
17
19
A
Kendall Sq.
15
16
B
Harvard Sq.
17
13
A
Indian
17
B
Asian
12
7(4)
A
16
13
20
6
8(5)
B
A
B
"Number
''In
Seafood
15
Indian
18
in paranthesis denotes decision
used
in
Experiment 2 (which only had 5 restaurant
pairs).
each decision, subjects were presented with two options designated "A" or "B" Restaurants were
.
not identified by
name
to reduce chances of personal biases.
c
Denotes the type of cuisine served at each of the alternatives (Asian, Indian or Seafood).
d
AU
restaurants are in Cambridge, Massachussetts (at either one of the following squares: Central,
Harvard, Inman, Kendall).
This allowed prizes to be relevant to the subject pool recruited from the
general Boston area.
e
(see
Denotes the quality of food at the given restaurant based on the restaurant guide/critic "Zagat"
www.zagat.com). The scale for food grade ranges from 0-30, where 30 in the Zagat scale implies
highest possible food quality. Subjects were informed of the scale and
its
meaning
prior to the beginning
of the experiment.
-^Denotes the quality of service at the given restaurant based
(see
www.zagat.com). The scale
for service
on the restaurant guide/ critic "Zagat"
grade ranges from 0-30, where 30 in the Zagat scale implies
highest possible quality of service.
9 Denotes the state of interior furnishings and the emphasis on decorative style at the given restaurant
based on the restaurant guide/critic "Zagat" (see www.zagat.com). The scale for decor ranges from 0-30,
where 30 in the Zagat scale implies an excellent/extraordinary level.
36
9
Table B2:
Mean Absolute
Decision Pair
VO
Difference in Valuation (Experiment 1)
Condition
RV
Condition
(N-44)
(N=44)
1
7.71
8.98
2
4.02
7.75
3
4.59
6.91
4
5.20
4.98
5
4.36
5.96
6
4.84
7.59
7
4.36
6.23
8
6.11
8.48
Average
5.15
7.11
Table B3:
Mean Absolute
Decision Pair
VO
Difference in Valuation (Experiment 2)
Condition
RV
Condition
(N=37)
(N=39)
1
2.97
4.10
2
2.97
4.00
3
3.23
4.31
4
3.40
3.54
5
3.75
4.31
Average
3.26
4.05
37
Tale B4: Average Willingness to Pay For Each Restaurant 22
Experiment
Decision
23
3(2)
6(3)
7(4)
8(5)
22
Only the
Number
2
RV
VO
RV
A
23.25
20.52
11.94
12.90
B
A
B
A
B
A
25.73
24.01
14.64
15.72
31.64
27.71
17.94
18.08
27.14
22.89
15.13
15.05
24.61
23.32
13.45
14.46
21.73
21.68
10.76
12.82
24.23
24.05
14.16
14.15
B
25.23
23.91
14.97
15.80
A
29.64
25.43
15.30
15.74
B
27.75
24.36
14.35
14.05
Table shows the average willingness to pay (across
tion).
23
Experiment
vo
Restaurant
2(1)
1
five
common
all
respondents in a particular study and condi-
decisions across experiments are presented.
in parenthesis is the
corresponding Experiment 2 decision number.
38
Table B5: Time Spent in Valuation Phase (Experiment 2)
Decision Pair
VO
Condition
RV
Condition
p- value
(N=37)
(N=39)
1
27.41
53.46
<0.01
2
17.78
35.13
0.02
3
20.78
23.33
0.42
4
20.89
31.46
0.02
5
21.73
19.70
0.45
Average
21.72
32.16
Table B6: Time Spent in Valuation Phase (Experiment 3)
Decision Pair
VO
Condition
RV
Condition
p- value
(N=49)
(N=39)
1
35.27
53.46
<0.01
2
21.56
35.13
0.04
3
17.78
23.33
0.03
4
19.98
31.46
<0.01
5
17.49
19.69
0.26
Average
22.4
32.16
39
Date Due
3
^
Lib-26-67
MIT LIBRARIES
3 9080 02527 9727