LIBRARIES
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/allocationofatteOOgaba
HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
THE ALLOCATION OF ATTENTION:
THEORY AND EVIDENCE
Xavier Gabaix
David Laibson
Guillermo Moloche
Working Paper 03-31
August 29, 2003
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Network
http://papers.ssrn.com/abstract
Paper Collection
id=XXXXXX
at
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
GUILLERMO MOLOCHE
David Laibson
Xavier Gabaix*
MIT and NBER
Harvard University and
NBER
NBER
Stephen Weinberg
Harvard University
Current Draft: August 29, 2003
Abstract.
A
host of recent studies show that attention allocation has important
economic consequences. This paper reports the
first
empirical test of a cost-benefit model of the
endogenous allocation of attention. The model assumes that economic agents have
processing speeds and cannot analyze
all
of the elements in
makes tractable predictions about attention
in
our environment.
allocation
measured
The model
complex problems.
is
mental
The model
allocation, despite the high level of complexity
successfully predicts the key empirical regularities of attention
in a choice experiment. In the experiment, decision time
and attention allocation
finite
is
a scarce resource
continuously measured using Mouselab. Subject choices correspond
well to the quantitative predictions of the model,
which are derived from cost-benefit and option-
value principles.
JEL
classification:
C7, C9, D8.
Keywords: attention,
*Email
satisficing,
Mouselab.
xgabaix@mit.edu, dlaibson@arrow.fas.harvard.edu, gmoloche@mit.edu, sweinWe thank Colin Camerer, David Cooper, Miguel Costa-Gomes, Vince Crawford, Andrei Shleifer, Marty Weitzman, three anonymous referees, and seminar participants at Caltech, Harvard, MIT, Stanford, University of California Berkeley, UCLA, University of Montreal, and the Econometric
Numerous research assistants helped run the experiment. We owe a particular debt to Rebecca
Society.
Thornton and Natalia Tsvetkova. We acknowledge financial support from the NSF (Gabaix and Laibson
SES-0099025; Weinberg NSF Graduate Research Fellowship) and the Russell Sage Foundation (Gabaix).
addresses:
ber@kuznets.fas.harvard.edu.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
Introduction
l.
1.1.
Attention as a scarce economic resource.
assume that cognition
is
2
instantaneous and costless.
Standard economic models conveniently
The
that are based on the fact that cognition takes time.
current paper develops and tests models
Like players in a chess tournament or
students taking a test, real decision-makers need time to solve problems and often encounter trade-
Time almost always has a
because they work under time pressure.
offs
whether or not formal time limits
Because time
tively thinking
is
about others.
is
some elements
in decision
problems while
Such selective thinking exemplifies the voluntary
and involuntary processes that lead to attention
Economics
allocation.
(i.e.
value,
of attention allocation, with its consequences for decision-making,
Indeed, in recent years
many
selec-
conscious)
2
often defined as the study of the allocation of scarce resources.
economic research.
shadow
exist.
scarce, decision-makers ignore
1
positive
seems
like
The
process
a natural topic for
authors have argued that the scarcity of attention
has large effects on economic choices. For example, Lynch (1996), Gabaix and Laibson (2001), and
Rogers (2001) study the
premium
Mankiw and
puzzle.
study the
effects
effects of limited attention
Gumbau
Reis (2002),
on consumption dynamics and the equity
(2003), Sims (2003),
and Woodford (2002)
on monetary transmission mechanisms. D'Avolio and Nierenberg (2002), Delia
Vigna and Pollet (2003), Peng and Xiong (2003), and Pollet (2003) study the
and Teoh (2002) and
pricing. Finally, Daniel, Hirshleifer,
Hirshleifer,
the effects on corporate finance, especially corporate disclosure.
limited attention from a theoretical perspective.
studying
The
1
its
associated market consequences.
current paper
Simon (1955)
is
part of a
new
effects
on
asset
Lim, and Teoh (2003) study
Some
of these papers analyze
Others analyze limited attention indirectly, by
None
of these papers measures attention directly.
literature in experimental economics that empirically studies
—
—
analyzed decision algorithms
e.g. satisficing
that are based on the idea that calculation
See Kahneman and Tversky (1974) and Gigerenzer et al. (1999) for heuristics that
simplify problem-solving. See Conlisk (1996) for a framework in which to understand models of thinking costs.
Attention "allocation" can be divided into four categories: involuntary perception (e.g., hearing your name spoken
is
first
time-consuming or
across a noisy
operations
(e.g.
room
(e.g.
costly.
at a cocktail party, see
1995); involuntary central cognitive
Wegner
1989); voluntary perception
searching for a cereal brand on a display shelf, see Yantis 1998); and voluntary central cognitive operations
planning and control of action, see
excellent overview of research
Kahneman
on attention.
categories of attention allocation, but
last
Moray 1959 and Wood and Cowan
trying to not think about white bears, but doing so anyway, see
two categories).
we
(e.g.
1973, and Pashler and Johnston 1998).
See Pashler (1998) for an
The model that we analyze in this paper can be applied to all of these
discuss an application that focuses on conscious attention allocation (the
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
the decision-making process by directly measuring attention allocation (Camerer
Gomes, Crawford, and Broseta 2001, Johnson
In the current paper,
we
report the
first
et al 2002,
3
et al.
1993, Costa-
and Costa-Gomes and Crawford 2003).
an economic cost-benefit model of
direct empirical test of
endogenous attention allocation.
This economic approach contrasts with more descriptive models of attention allocation in the
psychology literature
(e.g.
Payne, Bettman, and Johnson 1993).
The economic approach provides
a general framework for deriving quantitative attention allocation predictions without introducing
extraneous parameters.
1.2.
Directed cognition:
An economic
approach to attention allocation.
the economic approach in a computationally tractable way,
by Gabaix and Laibson (2002), which they
we apply a
cost-benefit
the directed cognition model.
call
To implement
model developed
In this model, time-
pressured agents allocate their thinking time according to simple option-value calculations.
agents in this model analyze information that
likely to
be redundant.
For example,
it is
surely dominated by an alternative good.
likely to
is
be useful and ignore information that
not optimal to continue to analyze a good that
It is also
The
is
is
almost
not optimal to continue to investigate a good
about which you already have nearly perfect information.
The
directed cognition
model calculates
the economic option-value of marginal thinking and directs cognition to the mental activity with
the highest expected shadow value.
The
directed cognition model comes with the advantage of tractability.
model can be computationally solved even
in highly
complex
settings.
The
directed cognition
To gain
this tractability,
however, the directed cognition model introduces some partial myopia into the option-theoretic
calculations,
and can thus only approximate the shadow value of marginal thinking.
In this paper
we show
myopic calculations
that this approximation comes at relatively
in the directed cognition
little cost,
since the partially
model generate attention allocation choices that ap-
proximate the payoffs from a perfectly rational
—
i.e.
infinitely
forward-looking
— calculation of
option values. Because infinite-horizon option value calculations are not generally computationally
tractable
we
and because directed cognition
is
close to those calculations
focus our analysis on the directed cognition model.
anyway
(cf.
Appendix C),
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
4
Perfect rationality: an intractable model in generic high-dimensional problems.
1.3.
Perfect rationality represents a formal (theoretical) solution to any well-posed problem.
But per-
fectly rational solutions
cannot be calculated in complex environments and have limited value as
modeling benchmarks
such environments. Researchers in
this limitation
Among
ple.
in
and taken the extreme step of abandoning perfect rationality
artificial intelligence researchers,
predictive accuracy, and generality
In contrast to
modeling framework
for
is
as
have acknowledged
an organizing
based on
economic analysis.
when
rationality.
economists use perfect rationality as their pri-
most
But alternatives to perfect
perfect rationality
The
not computationally tractable.
princi-
its tractability,
— not by proximity to the predictions of perfect
Perfect rationality continues to be the
topics of economic study either
is
the merit of an algorithm
artificial intelligence researchers,
mary modeling approach.
rationality
artificial intelligence
is
and
useful
disciplined
rationality should
a bad predictive model or
be active
when
perfect
latter case applies to our attention allocation
analysis.
The complex environment
that
we study
in this
paper only partially
reflects
the even greater
complexity of most attention allocation problems. But even in our relatively simple setting perfectly
rational option value calculations are not computationally tractable. 3
on a solvable
range of
cost-benefit
model
— directed cognition — which can
realistic attention allocation
cognition model
is its
tractability, not
of attention allocation.
We
problems.
Hence we focus our
easily
analysis
be applied to a wide
For our application, the merit of the directed
any predictive deviations from the perfectly rational model
do not study such deviations due to the computational intractability
of the perfectly rational model.
1.4.
An
qualitative
empirical analysis of the directed cognition model.
and quantitative predictions
of the directed cognition
This paper compares the
model to the
results of
iment in which subjects solve a generalized choice problem that represents a very
economic decisions: choose one good from a
set of
N goods.
Many economic
an exper-
common
set of
decisions are special
cases of this general problem.
3
Of course, even when
perfect rationality cannot be solved, its solution might still be partially characterized and
do not take this approach in the current paper because the predictions of the infinite-horizon model that
we have been able to formally derive have been quite weak (e.g., ceteris paribus, analyze information with the highest
tested.
We
variance).
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
In our experiment, decision time
continuously.
We make
subjects an exogenous
is
a scarce resource and attention allocation
time scarce in two different ways.
amount
5
of time to choose one
is
measured
In one part of the experiment
good from a
set of
goods
we
give
— a choice problem
with an exogenous time budget. Here we measure how subjects allocate time as they think about
each good's attributes before making a
final selection of
In a second part of the experiment
problems
like
until a total
is
the one above.
we
give the subjects
an open-ended sequence of choice
In this treatment, the subjects keep facing different choice problems
budget of time runs out. The amount of time a subject chooses
now an endogenous
variable.
But moving too quickly reduces the quality
and
quantity.
for
each choice problem
Because payoffs are cumulative and each choice problem has a
positive expected value, subjects have an incentive to
off quality
which good to consume.
move through the
As a
of their decisions.
choice problems quickly.
result, the subjects trade
Spending more time on any given choice problem
raises the quality of
the final selection in that problem but reduces the time available to participate in future choice
problems with accumulating payoffs.
Throughout the experiment we measure the process
attention allocation within each choice problem
(i.e.
of attention allocation.
how
is
In addition, in the open-ended sequence design,
choice problems
(i.e.
how much time does
a final selection in that problem and
we measure
is
made
attention allocation across
moving on to the next choice problem?).
et at.
1993 and Costa-Gomes, Crawford,
and Broseta 2001), we use the "Mouselab" programming language to measure
is
4
subjects' attention
Mouselab tracks subjects' information search during our experiment.
hidden "behind" boxes on a computer screen.
boxes.
one screen box to be open at any point
4
is
in time, the
Information
Subjects use the computer mouse to open the
Mouselab records the order and duration of information
information the subject
in that choice
the subject spend on one choice problem before making
Following the lead of other economists (Camerer
allocation.
measure
time allocated in thinking about
the different goods within a single choice problem before a final selection
problem?).
We
acquisition.
Since
we
allow only
Mouselab software enables us to pinpoint what
contemplating on a second-by-second basis throughout the experiment. 5
Payne, Bettman, and Johnson (1993) developed the Mouselab language in the 1970's. Mouselab is one of many
For example, Payne, Braunstein, and Carrol (1978) elicit mental processes by asking
subjects to "think aloud." Russo (1978) records eye movements.
Mouselab has the drawback that it uses an artificial decision environment, but several studies have shown that
"process tracing" methods.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
6
In our experiment, subject behavior corresponds well to the predictions of the directed cognition
model.
Subjects allocate thinking time
when the marginal value
because competing goods are close in value or because there
information to be revealed.
We demonstrate this
in
of information acquisition within choice problems.
is
of such thinking
is
high, either
a "large" amount of remaining
numerous ways.
First,
we
evaluate the pattern
Second, in the endogenous time treatment,
evaluate the relationship between economic incentives and subjects' stopping decisions:
i.e.
we
when
do subjects stop working on one choice problem so they can move on to the next choice problem?
We
find that the
acquisition models.
economic model of attention allocation outperforms mechanistic information
The marginal value of information turns out
to be the
most important predictor
of attention allocation.
However, the experiment also reveals one robust deviation from the predictions of the directed
cognition cost-benefit model.
ically,
subjects
Subject choices are partially predicted by a "boxes heuristic." Specif-
become more and more
likely to
end analysis of a problem the more boxes they open,
holding fixed the economic benefits of additional analysis.
insensitivity to the particular
We
also test
In this sense, subjects display a partial
economic incentives in each problem that they
models of heuristic decision-making, but we find
ied heuristics, including satisficing
and elimination by
aspects.
little
face.
evidence for
commonly
stud-
Instead, our subjects generally
allocate their attention consistently with cost-benefit considerations, matching the precise quantitative attention-allocation patterns generated
by the directed cognition model.
Hence, this paper
demonstrates that an economic cost-benefit model predicts both the qualitative and quantitative
patterns of attention allocation.
Section 2 describes our experimental set-up.
Section 3 summarizes an implementable one-
parameter attention allocation model (Gabaix and Laibson 2002).
results of our experiment
and compares those
Section 4 summarizes the
results to the predictions of our model.
Section 5
concludes.
the Mouselab environment only minimally distorts final choices over goods/actions
and Broseta 2001 and Costa-Gomes and Crawford 2002).
"left-to-right" search biases, which we discuss in subsection
(e.g., Costa-Gomes, Crawford,
Mouselab's interface does generate "upper-left" and
(4.4) below.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
Experimental Set-up
2.
Our experimental design
facilitates
decision problem, a choice
An
2.1.
among
An
TV-good game.
second-by-second measurement of attention allocation in a basic
TV goods.
TV-good game
Each box contains a random payoff
1).
is
an TV-row by M-column matrix of boxes (Figure
generated with normal density and zero
(in units of cents)
After analyzing an TV-good game, the subject makes a final selection and "consumes" a
mean.
The
row from that game.
single
7
subject
is
paid the
Consuming a row represents an abstraction
sum
of a very
of the boxes in the
wide
consumed row.
The
problem an TV-good game, since the TV rows conceptually represent TV goods.
this
chooses to consume one of these TV goods.
For example, consider a shopper
The consumer
to choose).
warranty, etc.
number
faces a fixed
The
who
analogy, the TV
metric) appear in the
M
TV's
represent
M
subject
different attributes.
Walmart
different attributes
—
are the rows of Figure 1
TV's from which
(TV different
e.g.
size, price,
and the
remote control,
M attributes
(in
a
utility
M columns of each row.
In our experiment, the importance or variability of the attributes declines as the columns
from
left
For example,
then the variance for column 2
So columns on the
100.
left
screen size or price in our
far
if
the variance used to generate column one
is
900,
and so on, ending with a variance
TV
Columns on the
example.
minor clauses
in the warranty.
we make the task harder by masking the contents
columns 2 through
that
all
initializes
in
column
l.
7
of boxes in
(i.e.,
row)."
columns
However, a subject can
M to unmask the value of that box (Figure
rows had the
game by breaking
same value (zero).
the
for
column 10
of
6
the eight-way
tie
2
To measure
through
left-click
M.
attention,
Subjects are
on a masked box
in
2).
In our experiment, all of the attributes have been demeaned.
We reveal the value of column one because it helps subjects remember which row
column one
1000 (squared
right represent the attributes with the
our game sounds simple: "Consume the best good
shown only the box values
is
represent the attributes with the most (utility-metric) variance, like
least (utility-metric) variance, like
So
move
In particular, the variance decrements across columns equal one tenth of the
to right.
variance in column one.
cents),
call
has decided to go to Walmart to select and buy a television.
of television sets at
television sets have
By
The columns
We
class of choice problems.
that would exist
if
is
which.
In addition, revealing
subjects began with the expectation
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
Only one box from columns
M can be unmasked at a time.
2 through
us to record exactly what information the subject
This procedure enables
analyzing at every point in time. 8
is
the contents of a box does not imply that the subject consumes that box. Note too that
picked for consumption, then
all
8
Revealing
if
a row
is
boxes in that row are consumed, whether or not they have been
previously unmasked.
We
to
introduce time pressure, so that subjects will not be able to
unmask
—
all
chooses/consumes
We study
Of
to unmask.
course,
we
also record
a setting that reflects realistic
—
high
i.e.
frequently face attention allocation decisions that are
—
not choose
N(M — 1)
which row the subject
levels of decision complexity.
The Walmart shopper
is
picking.
selectively attends to information
The shopper may
This com-
Real consumers in real markets
much more complex.
Masked boxes and time pressure capture important aspects
among which she
will
for his or her final selection.
plexity forces subjects to confront attention allocation tradeoffs.
ence.
— or
Mouselab enables us to record which of the
of the boxes in the game.
masked boxes the subject chooses
unmask
also face
of our
Walmart shopper's
experi-
about the attributes of the TV's
some time
pressure, either because she
has a fixed amount of time to buy a TV, or because she has other tasks which she can do in the
store
if
she selects her
TV
quickly.
We
explore both of these types of cases in our experiment.
Games with exogenous and endogenous time
2.2.
budgets.
In our experiment, subjects
play two different types of JV-good games: games with exogenous time budgets and games with
endogenous time budgets.
We
will refer to these as
"exogenous" and "endogenous" games.
For each exogenous game a game-specific time budget
over the interval [10 seconds, 49 seconds].
for
A
is
generated from the uniform distribution
clock shows the subject the
each exogenous-time game (see clock in Figure 2)
.
This
is
amount
of time remaining
the case of a Walmart shopper with
a fixed amount of time to buy a good.
In endogenous games, subjects have a fixed budget of time
many
different
A^-good games as they choose.
— 25 minutes —
in
which to play as
In this design, adjacent A^-good games are separated
When we designed the experiment, we considered but did not adopt a design that permanently keeps boxes open
once they had been selected by the subject. This alternative approach has the advantage that subjects face a reduced
memory burden. On the other hand, if boxes stay open permanently then subjects have the option to quickly
and mechanically
that
we
lost
— open many boxes and only afterwards analyze their content.
the ability credibly to infer what the subject
is
—
Hence, leaving boxes open implied
attending to at each point in time.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
by 20 second buffer screens, which count toward the total budget of 25 minutes.
to spend as
little
or as
much time
an endogenous choice variable.
purchases
We
if
she selects her
as they
This
TV
Subjects are free
want on each game, so time spent on each game becomes
the case of a Walmart shopper
who can move on
to other
quickly.
study both exogenous time games and endogenous time games because these two classes
of problems
commonly
arise in the real
to handle both situations robustly.
attention allocation decisions.
for
is
9
world and any attention allocation model should be able
Both types
problems enable us to study within-problem
of
In addition, the endogenous time games provide a natural framework
studying stopping rules, the between-problem attention allocation decision.
Experimental
2.3.
Subjects receive printed instructions explaining the structure of
logistics.
an iV-good game and the setup
for the
exogenous and endogenous games.
Subjects are then given
a laptop on which they read instructions that explain the Mouselab interface. 9
Subjects play three
games, which do not count toward their payoffs.
test
Then
set of
subjects play 12 games with separate exogenous time budgets.
endogenous games with a joint 25-minute time budget.
the order of the exogenous and endogenous games.
At the end
Finally, subjects play a
For half of the subjects
we
reverse
of the experiment, subjects answer
demographic and debriefing questions.
Subjects are paid the cumulative
sum
of all rows that they consume.
reports the running cumulative value of the
3.
The
Directed cognition model
must decide which boxes to unmask before
jointly decide
We
We
consumed rows.
previous section describes an attention allocation problem.
must
After every game, feedback
their time runs out.
In exogenous games, subjects
In endogenous games, subjects
which boxes to unmask and when to move on to the next game.
want to determine whether economic
principles guide subjects' attention allocation choices.
compare our experimental data to the predictions
by Gabaix and Laibson (2002).
of an attention allocation
model proposed
This 'directed cognition' model approximates the economic value
of attention allocation using simple option value analysis.
Interested readers can view
all
of our instructions on the web: http://post.economics.harvard.edu/faculty/laibson/papers.html
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
The option
value calculations capture two types of effects.
compared and a particular choice gains a
are being
information
when marginal
the standard deviation of marginal information
(i.e.
continued analysis declines.
The
when many
different choices
large edge over the available alternatives, the
Second,
option value of continued analysis declines.
First,
10
is
analysis yields little
low), the economic value of
directed cognition model captures these two effects with a formal
makes sharp quantitative predictions about attention
search-theoretic framework that
new
allocation
choices.
Summary
3.1.
we
first
Step
1:
summarize and then describe
first
2:
For example, what
in detail.
is
the expected economic benefit of unmasking three boxes in the
Execute the cognitive operation with the highest ratio of expected benefit to
if
unmask those two
For
boxes.
Return to step one unless time has run out
of expected benefit to cost falls
We now
cost.
exploration of the 'next' two boxes in the sixth row has the highest expected ratio
of benefit to cost, then
3:
which
row?
example,
Step
into three iterative steps,
Calculate the expected economic benefits and costs of different potential cognitive opera-
tions.
Step
The model can be broken down
of the model.
(in
exogenous time games) or until the ratio
below some threshold value (endogenous time games).
describe the details of this search algorithm.
We
first
introduce notation and then
formally derive an equation to calculate the expected economic benefits referred to in step one.
Notation.
3.2.
We
Since our games
will use lower case letters
—
all
have eight rows (goods), we label the rows A, B,...,H.
a,b, ...,h
— to track a subject's expectations
of the values of the
respective rows.
Recall that the subject knows the values of
at the beginning of the
cell of
each row.
subject
all
boxes in column
game, the row expectations
For example,
if
row
unmasks the second and third
would be updated to 29
=
23
+
17
+
C
will
has a 23 in
cells in
-11.
1
when the game
begins.
Thus,
equal the value of the payoff in the left-most
its first cell,
row C, revealing
then at time zero
cell
c
=
23. If the
values of 17 and -11, then c
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
3.3.
Step
In step
1:.
1
11
the agent calculates the expected economic benefits and costs of
ferent cognitive operations.
For our application, a cognitive operation
unmasking/analysis of boxes in a particular row of the matrix.
is
V
additional boxes in row A."
a partial or complete
Such an operation enables the
decision-maker to improve her forecast of the expected value of that row.
represents the operation, "open
In our notation,
O rA
Because tractability concerns lead
we assume that
us to limit agents' planning horizons to only a single cognitive operation at a time,
individual cognitive operations themselves can include one or
dif-
more box openings.
Multiple box
openings increase the amount of information revealed by a single cognitive operator, increase the
option value of information revealed by that cognitive operator, and
make the
(partially
myopic)
model more forward-looking. 10
The operator 0\
words, the
variances
O vA
(i.e.
selects the
boxes to be opened using a maximal-information
operator would select the
with the most information).
openings (skipping any boxes that
We assume that
single box.
T unopened boxes
(in
row
.A)
cost to include
many components,
is
memory
to right
box
the cost of unmasking a
including the time involved in opening
a box with a mouse, reading the contents of the box, and updating expectations.
includes an addition operation as well as two
left
already).
an operator that opens T boxes has cost T-k, where k
We take this
In other
that have the highest
In our game, this corresponds with
may have been opened
rule.
Such updating
operations: recalling the prior expectation
and rememorizing the updated expectation.
The expected
benefit
(i.e.
option value) of a cognitive operation
is
given by
= <7*g)-|*|*(-M),
w (x,v)
,) S
*4>{±)-\x\*\-!f\,
(1)
:i
where
4>
represents the standard normal density function,
distribution function, x
its
is
<&
represents the associated cumulative
the estimated value gap between the row that
next best alternative, and a
is
is
under consideration and
the standard deviation of the payoff information that would be
10
To gain intuition for this effect, consider two goods A and B, with E(a) = 3/2 and E(b) = 0. Suppose that
opening one box reveals one unit of information so that after this information is revealed E(a') = 5/2 or E(a') = 1/2.
Hence, a partially myopic agent won't see the benefit of opening one box, since no matter what happens E(a ) > E(b).
However, if the agent considers opening two boxes, then there is a chance that E(a') will fall below 0, implying that
gathering the information from two boxes would be useful to the agent. Analogous arguments generalize to the case
of continuous information densities.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
We
revealed by the cognitive operator.
motivate equation
below, but
(1)
first
12
present an example
calculation.
In the
game shown
row H. The
initial
in Figure 2, consider a cognitive operator
H
expected value of
has a current payoff of c
= 23.
is
h
A box in column n will reveal a payoff
r\
=
that explores three boxes in
—28. The best current alternative
So the estimated value gap between
xo
of
=
z
OH
\h
—
c\
row C, which
H and the best alternative
is
= 51.
Hn with variance
(40.8)
(1
— n/10), and the updated
value
H after the three boxes have been opened will be
=
h'
-28
+
r]
H2
+ Vm + VH4-
Hence the variation revealed by the cognitive operator
is
o
(Q +
To
oo
i.e.
To k =
•
=
maker
is
that row
63.2.
So the benefit of the cognitive operator
is
To
+
7 \
To)'
w(xo,&o) —
7.5,
and
its
cost
is
3k.
We now
to
is
motivate equation
analyzing row
A
and
To
(1).
will
fix ideas,
consider a
new game. Suppose
that the decision-
then immediately use that information to choose a row. Assume
B would be the leading row if row A were eliminated,
so
row
B
is
the next best alternative
row A.
The agent
is
considering learning more about row
Executing the cognitive operator
a to
a'
If
=
a
+ e,
where e
is
the
will
sum
A
by executing a cognitive operator O^.
enable the agent to update the expected payoff of row
of the values in the
T newly unmasked boxes
in
the agent doesn't execute the cognitive operator, her expected payoff will be
max (a, b)
row A.
A
from
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
If
13
the agent plans to execute the cognitive operator, her expected payoff will be
E [max (a', b)]
.
This expectation captures the option value generated by being able to pick either row
The
B, contingent on the information revealed by cognitive operator 0^.
cognitive operator
A
or row
value of executing the
the difference between the previous two expressions:
is
E
[max
(a', 6)]
— max (a, b)
This value can be represented with a simple expression.
(2)
Let a represent the standard deviation
of the change in the estimate resulting from applying the cognitive operator:
a2
Appendix
A
=
E{a!
-
a)
2
.
shows that the value of the cognitive operator
is
11
E [max (a', &)] — max (a, b) — w (a — b,a)
To help gain
intuition about the
deduced from the
fact that
w
function, Figure 3 plots
(3)
.
w (x, 1). The
w (x, a) = aw (x/a, 1)
The option value framework captures two fundamental comparative
of a
row exploration decreases the
w (x,a)
is
decreasing in
\x\.
larger the
is
likely to
statics.
First,
the value
gap between the active row and the next best row:
Second, the value of a row exploration increases with the variability
of the information that will be obtained:
information that
general case can be
w (x, a)
is
increasing in a.
In other words, the
more
be revealed by a row exploration, the more valuable such a path
exploration becomes.
3.4.
Step
to cost.
2.
Step 2 executes the cognitive operation with the highest ratio of expected benefit
Recall that the expected benefit of an operator
the implementation cost of an operator
is
is
given by the w(x,a) function and that
proportional to the
This result assumes Gaussian innovations, which
is
number
of boxes that
it
unmasks.
the density used to generate the games in our experiment.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
The
12
subject executes the cognitive operator with the greatest benefit/cost ratio
G = m ax
o
where k
is
w{x
14
,
,(To)
...
^f^'
the cost of unmasking a single box.
Since k
(4)
constant, the subject executes the
is
cognitive operator
O*
Appendix
3.5.
Step
B
=
argmaxiu(xo,o"o)/ro-
contains an example of such a calculation.
Step 3
3.
is
a stopping rule.
In games with an exogenous time budget, the subject
keeps returning to step one until time runs out. In games with an endogenous time budget, the
subject keeps returning until
3.6.
G
falls
below the marginal value of time, which must be calibrated.
We use
Calibration of the model.
two
different
methods
to calibrate the marginal value
of time during the endogenous time games.
First,
we estimate the marginal
value of time as perceived by our subjects.
Advertisements for
the experiment implied that subjects would be paid about $20 for their participation, which would
take about an hour.
In addition, subjects were told that the experiment would be divided into
two halves, and that they were guaranteed a $5 show- up
fee.
Using this information, we calculate the subjects' anticipated marginal payoff per unit time
during games with endogenous time budgets.
This marginal payoff per unit time
opportunity cost of time during the endogenous time games.
is
the relevant
Since subjects were promised $5
of guaranteed payoffs, their expected marginal payoff for their choices during the experiment
about $15.
was
Dividing this in half implies an expectation of about $7.50 of marginal payoffs for the
endogenous time games. Since the endogenous time games were budgeted to take 25 minutes, which
was known to the subjects, the perceived marginal payoff per second of time
(750 cents)/ (25 minutes
12
Our model thus
gives a crude but
processing. See Aragones et
al.
•
60 seconds/minute)
=
in the experiment
0.50 cents/second.
compact way to address the "accuracy vs simplicity" trade-off
much more sophisticated treatment.
(2003) for a
was
in cognitive
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
Since subjects took on average 0.98 seconds to open each box,
shadow
cost per
box opening
it
— k — was chosen to make the model partially
matches the average number of boxes explored
cents/box. Here k
to
is
match the order
3.7.
=
0.49 cents/box.
also explored a 1-parameter version of the directed cognition model, in
cognition
so
we end up with an implied marginal
of
(0.50 cents/second) (0.98 seconds /box)
We
Conceptual
plying that
assumptions.
issues.
First, the
shadow value
Calibrating the model
endogenous games implies k
This model
is
easy to analyze and
The
simplicity of the
is
=
0.18
computationally tractable, im-
model follows from three
model assumes that the agent calculates only a
who assumes
directed cognition
the data.
of search or the distribution of search across games.
gain from executing each cognitive operator.
hiel (1995),
in the
fit
which the cost of
chosen only to match the average amount of search per open-ended game, not
can be empirically tested.
it
15
partially
special
myopic expected
This assumption adopts the approach taken by Je-
a constrained planning horizon in a game-theory context.
model assumes that the agent uses a
fixed positive
shadow value
Second, the
This
of time.
of time enables the agent to trade off current opportunities with future opportunities.
Third, the directed cognition model avoids the infinite regress problem
(i.e.
about thinking about thinking, etc.), by implicitly assuming that solving
the costs of thinking
for
O*
is
costless to the
agent.
Without some version
of these three simplifying assumptions the
Without some
in practice.
calculations) the
,
partial
myopia
(i.e.
model would not be
useful
a limited evaluation horizon for option value
problem could not be solved either analytically or even computationally. Without
the positive shadow value of time, the agent would not be able to trade off her current activity
with unspecified future activities and would never finish an endogenous time game without
(counterfactually) opening
up
all
of the boxes.
Finally, without eliminating cognition costs at
primitive stage of reasoning, maximization models are not well-defined.
We
return
now
to the
first
first
some
13
of the three points listed in the previous paragraph: the perfectly
See Conlisk (1996) for a description of the infinite regress problem and an explanation of why it plagues
We follow Conlisk in advocating exogenous truncation of the infinite regress of thinking.
decision cost models.
all
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
rational attention allocation
rationality
model
is
An
not solvable in our context.
model requires the calculation
16
exact solution of the perfect
of a value function with 17 state variables: one expected
value for each of the 8 rows, one standard deviation of unexplored information in each of the 8
suffers
R 17
This dynamic programming problem in
rows, and finally the time remaining in the game.
from the curse of dimensionality and would overwhelm modern supercomputers. 14
model
contrast, the directed cognition
which has only two state
is
equivalent to 8 completely separable problems, each of
variables: x, the difference
and the current expected value
By
between the current expected value of the row
of the next best alternative row;
and
a, the standard deviation of
So the "dimensionality" of the directed cognition model
unexplored information in the row.
is
only 2 (compared to 17 for the model of perfect rationality).
Although
this
cation (or to test
paper does not attempt to solve the model of perfectly rational attention
it)
,
we can compare the performance
model and the performance
of the perfectly rational model.
the perfectly rational model assumes that examining a
optimal search strategy
Appendix C
gives lower
is
of the partially
costless (analogous to our
bounds on the payoffs
exogenous time games and at
3.8.
least
Other decision algorithms.
71%
new box
rules,
which can be parameterized
two models are
just benchmarks,
—
is
costly
of the directed cognition
and that calculating the
91%
model
The
first
we
O*
is
costless).
relative to the payoffs
endogenous time games.
discuss three other models that
we
two models are simple, mechanical search
model (Simon 1955).
which are presented as objects
The
for
as well as perfect rationality
as well as perfect rationality for
as variants of the satisficing
candidate theories of attention allocation.
1972)
is
assumption that solving
In this subsection,
compare to the directed cognition model.
myopic directed cognition
Like the directed cognition model,
of the perfectly rational model. Directed cognition does at least
for
allo-
third model
for
These
first
comparison, not as serious
— Elimination by Aspects
(Tversky
the leading psychology model of choice from a set of multi-attribute goods.
The Column model unmasks boxes column by column, stopping
either
when time runs out
(in
14
Approximating algorithms could be developed, but after consulting with experts in operations research, we
concluded that existing approximation algorithms cannot be used without a prohibitive computational burden. The
Gittins index (Gittins 1979, Weitzman 1979, Weitzman and Roberts 1980) does not apply here either - for much the
same reason it does not apply to most dynamic problems. In Gittins's framework, it is crucial that one can only do
one thing to a row (an "arm"): access it. In contrast, in our game, a subject can do more than one thing with a row.
She can explore it further, or take it and end the game. Hence our game does not fit in Gittins' framework. We
explored several modifications of the Gittins index, but they proved unfruitful at breaking the curse of dimensionality.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
games with exogenous time budgets) or stopping according to a
endogenous time budgets).
(top to bottom), then in
column
3,...,
etc.
endogenous games, the unmasking continues
A CoiuTan model
the simulations generate an average
number
of empirical
box openings
number
Row model
unmasks
all
;
number
of boxes as a 'yoked' subject. 15
In
a row has been revealed with an estimated
until
an aspiration or
of simulated
satisficing level
estimated so that
box openings that matches the average
row, starting with the "best" row and moving to the
when time runs out
stopping according to a satisficing rule
the
the boxes in column 2
(26 boxes per game).
The Row model unmasks boxes row by
"worst" row, stopping either
all
In exogenous games, this column-by-column unmasking
continues until the simulation has explored the same
value greater than or equal to
games with
satisficing rule (in
Column model unmasks
Specifically the
17
(in
(in
games with exogenous time budgets) or
games with endogenous time budgets).
ranks the rows according to their values in column
the boxes in the best row, then the second best row,
Specifically
Then the Row model
1.
In exogenous games, this
etc.
row-by-row unmasking continues until the simulation has explored the same number of boxes as
a yoked subject (see previous footnote).
In endogenous games, the unmasking continues until a
row has been revealed with an estimated value greater than or equal to
A Row model
or satisficing level estimated so that the simulations generate an average
number
,
an aspiration
box
of simulated
openings that matches the average number of empirical box openings.
A
choice algorithm called Elimination by Aspects
(EBA) has been widely studied
in the psy-
We
apply
framework to analyze games with endogenous time budgets.
We
chology literature (see work by Tversky 1972 and Payne, Bettman, and Johnson 1993).
this algorithm to our decision
use the interpretation that each row
by the ten
a good with 10 different attributes or "aspects" represented
is
different boxes of the row.
Our
aspect by aspect
(i.e.
aspect that
below some threshold value
falls
EBA
column by column) from
left
A EBA
point where the next elimination would eliminate
application assumes that the agent proceeds
to right, eliminating goods
.
all
(i.e.
rows) with an
This elimination continues, stopping at the
remaining rows.
At
this stopping point,
As above, we estimate
pick the remaining row with the highest estimated value.
^4
we
EBA so that
the simulations generate an average number of simulated box openings that matches the average
5
In such a yoking, the simulation
is
the yoked simulation opens TV boxes in
tied to a particular subject.
game
g.
If
that subject opens
N boxes in
game
g,
then
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
number
of empirical
box openings.
Results
4.
Our subject pool
Subject Pool.
4.1.
of the subjects are male (66%).
11% math
21%
or statistics;
social sciences.
The
is
comprised of 388 Harvard undergraduates. Two-thirds
subjects are distributed relatively evenly over concentrations:
natural sciences;
In a debriefing survey
Only 15% report taking an advanced
45%
18
20%
we asked our
29%
humanities;
20%
economics; and
other
subjects to report their statistical background.
statistics class;
40%
report only an introductory level class;
report never having taken a statistics course.
Subjects received a
mean
range from $13.07 to $46.69.
total payoff of $29.23, with a standard deviation of $5.49.
16
All subjects played 12
games with exogenous times.
On
Payoffs
average
subjects chose to play 28.7 games under the endogenous time limit, with a standard deviation of
The number
7.9.
of
games played ranges from 21
to 65.
To
Payoffs do not systematically vary with experimental experience.
we
calculate the average payoff
games.
We
X A (k)
for the
regression
/3
is
calculate
X B (k)
X (k)
for the
of
games played
in
= 1,
round k
exogenous games played After the endogenous games.
X (k) = a + fik
for the
not significantly different from
We
Before and After datasets.
0,
12 of the exogenous time
exogenous games played Before the endogenous games and
which indicates that there
exogenous time games. Learning would imply
/?
>
0.
is
We
no
fail
both regressions,
significant learning within
Also, the constant
any significant learning in exogenous time games. Hence we
estimate a separate
find that in
tinguishable across the two samples. Playing endogenous time games
in
...,
get a sense of learning,
first
a
is
statistically indis-
does not contribute to
to detect any significant learning
our data. 17
4.2.
for
Statistical
our empirical
methodology.
statistics.
samples of 388 subjects.
16
Payoffs show
little
of the current paper, but
See
e.g.
Camerer
The standard
errors are calculated
Hence the standard
all
subjects have identical strategies.
In light of this,
from the particular
we have chosen to adopt
is beyond the scope
Relaxing this assumption
we regard such a relaxation as an important future research goal.
Camerer and Ho (1999) and Erev and Roth (1998) for some examples
(2003),
where learning plays an important
role.
errors
by drawing, with replacement, 500
errors reflect uncertainty arising
systematic variation across demographic categories.
the useful approximation that
17
Throughout our analysis we report bootstrap standard
of
problems
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
sample of 388 subjects that attended our experiments.
Our
19
point estimates are the bootstrap
means. In the figures discussed below, the confidence intervals are plotted as dotted lines around
those point estimates.
We
also calculated
Monte Carlo means and standard
errors for our
associated standard errors tend to be extremely small, since the strategy
determined by the
is
In the relevant figures (4-9), the confidence intervals for the model predictions are visually
model.
indistinguishable from the means.
To
calculate measures of
and calculate a
fit
18
of our models,
fit
statistic that
bootstrap
mean
fit
statistic
we generate a bootstrap sample
compares the model predictions
the associated data for that bootstrap sample.
We
We
budgets.
patterns in the games with exogenous time budgets.
openings across columns and rows.
We
of 388 subjects
for that bootstrap
sample to
repeat this exercise 500 times to generate a
and a bootstrap standard error
Games with exogenous time
4.3.
The
model predictions.
for the
fit
statistic.
begin by analyzing attention allocation
Our
analysis focuses on the pattern of
compare the empirical patterns
of
box
box openings to the
patterns of box openings predicted by the directed cognition model.
We
begin with a trivial prediction of our theory: subjects should always open boxes from
right, following a declining variance rule.
variance rule 92.6% of the time
unopened box
in a given row,
unopened boxes
may
arise
in that row.
(s.e.
left
to
In our experimental data, subjects follow the declining
0.7% 19 ).
Specifically,
when
subjects open a previously
92.6% of the time that box has the highest variance of the
as-yet-
For reasons that we explain below, such left-to-right box openings
because of spatial biases instead of the information processing reasons implied by our
theory.
Now we
number
consider the pattern of search across columns and rows. Figure 4 reports the average
of boxes opened in columns 2-10.
column by column,
The
for
report the average
number
empirical profile
is
calculated by averaging together subject responses on
Specifically,
In Figures 4-9, the width of the associated confidence intervals for the models
The
unmasked,
all
of the exoge-
each of our 388 subjects played 12 exogenous games,
is
on average only one-half of one
percent of the value of the means.
19
of boxes
both the subject data and the model predictions.
nous games that were played.
1
We
units here refer to percentage points, not to a percentage of the point estimate.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
yielding a total of 388 x 12
= 4656
was assigned a subset
games from a
of 12
was played an average of 4656/160
To
the
calculate the empirical
~
opening.
if
value,
(and
profile
all
of the profiles that
we analyze) we count only
a subject unmasks the same box twice, this counts as only one
relatively rare in our
Memory
We
do not study repeat
data and because we are interested in building
Hence, we only implicitly model
a simple and parsimonious model.
each subject
Hence, each of the 160 games
Approximately 90% of box openings are first-time unmaskings.
unmaskings because they are
form.
unique games.
set of 160
this total,
30 times in the exogenous time portion of the experiment.
column
unmasking of each box. So
first
To generate
exogenous games played.
20
costs are part of the (time) cost of opening a
memory
costs as a reduced
box and encoding/remembering
its
which includes the cost of mechanically reopening boxes whose values were forgotten. 20
Figure 4 also plots the theoretical predictions generated by yoked simulations of our model. 21
by simulating the directed cognition model on the
Specifically, these predictions are calculated
exact set of 4656 games played by the subjects.
of 4656
We simulate the model on each game
games and instruct the computer to unmask the same number
of boxes that
from
this set
was unmasked
by the subject who played each respective game.
The analysis compares the particular boxes opened by the
subject to the particular boxes opened
by the yoked simulation of the model. Figure 4 reports an R' 2 measure, which captures the extent
to
which the empirical data matches the theoretical predictions.
statistic
22
This measure
is
simply the
R2
from the following constrained regression: 23
Boxes(col)
= constant +
Boxes{col) +e(col).
Here Boxes(col) represents the empirical average number of boxes unmasked in column
An extension of this framework
might consider the case in which the memory technology include
memory
col
and
capacity
constraints or depreciation of memories over time.
See footnote 15 for a description of our yoking procedure.
22
In other words,
R =
Y^jcoI
\Boxes{col)
^
- (Boxes) —
:
/
Ylicol
-
[Boxes(col)
+
Boxes(col)
/_—
(Boxes)
]
\\2
— (Boxes)
I
where (} represents empirical means.
3
In this section of the paper, the constant
independent variable.
However,
is
redundant, since the dependent variable has the same mean as the
we will consider cases in which this equivalence does not hold,
in the next subsection
necessitating the presence of the constant.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
Boxes(col) represents the simulated average number of boxes opened in column
varies
from 2 to
— oo
below by
above by
10, since the
boxes in column
Intuitively, the R'
.
around the mean explained by the model.
1.7%), implying a very close
(s.e.
2
is
bounded
squared deviations
For the column predictions, the R' 2
match between the data and the predictions
statistic is
86.6%
of the model.
Figure 5 reports the number of boxes opened
on average by row, with the rows ranked by their value
never masked.)
is
col
constrained equal to unity) and bounded
statistic represents the fraction of
Figure 5 reports analogous calculations by row.
Note that
col.
always unmasked. This R' 2 statistic
on Boxes(col)
(since the coefficient
1 (a perfect fit)
1 are
21
in
column
one.
column one
(Recall that
is
We report the number of boxes opened on average by row for both the subject data
and the model predictions. As above, the model predictions are calculated using yoked simulations.
Figure 5 also reports an R' 2 measure analogous to the one described above.
now the
that
in
row row. For our row predictions our R' 2 measure
is
is
unmaskings on the top ranked rows and
coefficient
much
less selective
far too
unity.
This
9.2%), implying a poor match
(s.e.
is
The R' 2
is
negative because
we
number
each game.
4.4.
of boxes
For our row predictions the R' 2 statistic
cussed in section
3,
endogenous games.
We
we
is
86.7%
calibrate one variant
With
we
at the
end of
1.4%).
As
when analyzing
dis-
the
by exogenously setting k to match the subjects' ex-
=
0.49 cents/box
this calibration, subjects are predicted to
In the data, however, subjects
frequency of box opening,
k,
(s.e.
consider two variants of the directed cognition model
We
discussion in section 3).
0.01).
Figure 6 reports
repeat the analysis above for the endogenous games.
pected earnings per unit time in the endogenous games: k
(s.e.
constrain the
opened on average by row, with the rows ranked by their values
Endogenous games.
The
the only bad prediction that the model makes.
Figure 6 reports similar calculations using an alternative way of ordering rows.
the
many
simulations predict too
few unmaskings on the bottom ranked rows.
than the model.
on simulated boxes to be
-16.1%
The model
between the data and the predictions of the model.
subjects are
only difference
Boxes(row), the empirical average number of boxes opened
is
variable of interest
The
open 26.53 boxes per game
consider a second calibration with k
=
opened
(see calibration
open 15.67 boxes per game
(s.e.
0.18.
0.55).
With
To match
this
this lower level of
the model opens the empirically "right" number of boxes.
We
begin with an evaluation of the declining variance rule.
In our endogenous games, subjects
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
91.0% of the time
follow the declining variance rule
(s.e.
0.8%).
when
Specifically,
22
subjects open
a previously unopened box in a given row, 91.0% of the time that box has the highest variance of
the as-yet-unopened boxes in that row.
Figure 7 reports the average number of boxes unmasked in columns 2-10 in the endogenous
games.
We
k
the model predictions with
The empirical data is
Specifically,
=
The
we use the
games that the subjects played.
number
the
of
0.49
of boxes
and n
=
unmasked by column
all
one.
4.2%)
of boxes
We report
The model generates
its
own stopping
and 64.6%
(s.e.
we no
longer yoke
R
statistic for these
column comparisons
in the
endogenous
for
«
statistic is
96.3%
(s.e.
the
1.4%) for n
mean number
of boxes
opened by row
for
Figure 8 reports
both the subject data and the model
=
(s.e.
1.3%) for k
These
subjects
0.49
figures
across rows
effect that
=
of boxes opened
and 84.5%
on average by row, with the rows ranked by
(s.e.
0.8%) for k
=
0.49
Figure 9
their values at the
statistics are
91.8%
(s.e.
0.18.
show that the model explains a very
and columns.
=
0.18.
end of each game. For these endogenous games, the alternative row R' 2
0.5%) for k
0.49
opened on average by row, with the rows ranked by their values in column
2.4%) for k
number
=
= 0.18.
Figure 9 reports similar calculations using the alternative way of ordering rows.
reports the
of our
rule (so
For these endogenous games, the row R' 2 statistics are 85.3%
predictions.
endogenous
box openings).
(s.e.
number
of the
by yoked simulations
Figure 8 reports analogous calculations by row for the endogenous games.
the
for
directed cognition model to simulate play of the 10,931 endogenous
For these endogenous games, the column R' 2
and 73.1%
data and
0.18.
theoretical predictions are generated
Figure 7 also reports the associated
games.
for the subject
calculated by averaging together subject responses on
games that were played.
model.
number
report the average
large fraction of the variation in attention
However, a subset of the results in this section are confounded by an
Costa-Gomes, Crawford, and Broseta (2001) have found in their analysis. In particular,
who
use the Mouselab interface tend to have a bias toward selecting
corner of the screen and transitioning from
The up-down component
left
cells in
the upper
to right as they explore the screen.
of this bias does not affect our results, since our rows are
ordered with respect to their respective payoffs.
left
The
left-right bias affects
randomly
only our column results
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
(figures 4
and
7)
since information with greater economic relevance
,
is
23
located toward the left-hand
side of the screen.
One way
to evaluate this
column bias would be to
flip
the presentation of the problem dur-
ing the experiment so that the rows are labeled on the right and variance declines from right
to
Alternatively, one could rotate the presentation of the problem,
left.
Unfortunately, the internal constraints of Mouselab
rows.
sible.
24
Finally,
of our analyses of either
we note that
neither the
up-down nor the
in attention allocation.
The
subjects allocate their attention between games.
First,
we compare the
all
a predicted
mean
we
focus on several measures
game
to the predicted variation
By
Our experiment
g.
24
utilized 160
first
(s.e.
moments.
0.01).
unique
g.
Let Boxes(g) represent the average number of
In this subsection
The
we
we analyze the
first
and second
.
Note
are analyzing game-specific averages.
empirical (equally weighted)
contrast the 0-parameter version of our
of 15.67
correctly predict
sample {Boxes(g)}^1 and the simulated sample {Boxes(g)} 1 ^1
begin by comparing
0.55).
how
160 games. Let Boxes(g) represent the average number
by subjects who played game
of the empirical
(s.e.
most
also enables us to evaluate
In this subsection
that these respective vectors each have 160 elements, since
26.53
of the analysis above reports
Most importantly, we ask whether the model can
boxes opened by the model when playing game
We
all
empirical variation in boxes opened per
games, though no single subject played
moments
any
and rows within a game, matching the patterns
which games received the most attention from our subjects.
of boxes opened
left-right biases influence
variation.
opened per game.
in boxes
Almost
Our experimental design
predicted by the directed cognition model.
between-game
such spatial
analysis above shows that subjects allocate
of their attention to economically relevant columns
of such
facilitates
row openings (above) or endogenous stopping decisions (below).
Stopping Decisions in Endogenous Games.
within-game variation
is
either of these relabelings impos-
Future work should use a more flexible programming language that
rearrangements.
4.5.
make
swapping columns and
model (with k
mean
=
of
Boxes(g)
0.49) generates
Hence, unless we pick k to match the empirical
mean
(i.e.
However, one could leave the labels on the left and have the unobserved variances decline from right to left. In
would have the minimal variance instead of the maximal variance. Such a design
would eliminate the old left-right bias, but it would create a new bias by making the row "label" appear on the
opposite side of the screen from where search should (theoretically) begin.
this design, the displayed cells
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
re
=
0.18), our
We
turn
model only crudely approximates the average number
now
The
to second moments.
deviation of 12.05
Moreover, when we set
0.03).
(s.e.
the standard deviation rises to 15.85
0.03).
(s.e.
opened per game.
empirical standard deviation of Boxes(g)
0-parameter version of our model (with
0.14), while the
of boxes
re
The
=
re
=
24
is
6.32
(s.e.
0.49) generates a predicted standard
0.18 to
match the average boxes per game,
relatively high standard deviations
simulations reflect the model's sophisticated search strategy
The model continuously
from the
adjusts
its
search strategies in response to instantaneous variation in the economic incentives that the subjects
By
face.
contrast, the subjects are less sensitive to high frequency variation in economic incentives.
Despite these shortcomings, the model successfully predicts the pattern of empirical variation
in the
number
0.66
(s.e.
=
0.18.
re
of boxes
opened
when we
0.02)
set
re
game.
in each
=
The
correlation between Boxes(g)
Similarly, the correlation
0.49.
is
0.61
(s.e.
See Figure 10 for a plot of the individual (160) datapoints for the
re
and Boxes(g)
0.02)
= 0.18
when we
case.
is
set
These
high correlations imply that the model does a good job predicting which games the subjects will
analyze most thoroughly.
The model
good job predicting the relationship between economic incentives and
also does a
Figure 11 plots a non-parametric kernel estimate of the expected number of
depth of analysis.
additional box openings in a game, conditional on the current value of the ratio of the benefit of
marginal analysis to the cost of marginal analysis (see
G=
Recall that
xo
is
The
w{xo,ao)
maxO
kTo
the gap between the expected value of the current row and the expected value
of the next best row,
operator O, and
eq. 4)
To
ao
is
is
the
the standard deviation of the information which
number
of boxes
confidence intervals.
re
=
0.18).
For most levels of
to the pattern in the subject data.
G
The
The
light
figure also
dashed
line represents the relationship
shows bootstrap estimates of the 99%
(the benefit-cost ratio), the model's predictions are close
Subjects do more analysis
economic incentives to do so are high.
revealed by mental
opened by mental operator O.
solid line represents the subject data.
predicted by the model (with
is
(i.e.
open up more boxes) when the
Moreover, the functional form of this relationship roughly
matches the form predicted by the theory.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
We
also evaluate the directed cognition
game and move on
decide to stop working on the current
logit (1
=
=
stop,
G
model by asking whether
predicts
to the next game.
We
25
when
subjects
run a stopping
continue) with explanatory variables that include the measure of the economic
value of continued search (or G), the number of different boxes opened to date in the current
game
(boxes), the expected value in cents of the leading
and subject
Note that
fixed effects.
row
in the current
models predict that
satisficing
G
/V-good game (leader),
and boxes should not have
predictive power in this logit regression, but that a higher value of the leader variable will increase
the stopping probability.
Each observation
for this logit is a decision-node
dogenous games, generating 330,873 observations. 25
economic significance are
cient
G
on
is
G
G
and boxes, with
We
is
0.0064 (0.0002).
clicks) in
The
playing the most important role.
is
G
coeffi-
0.0404 (0.0008);
At the mean values of the explanatory
one-standard deviation reduction in the value of
our en-
find that the variables with the greatest
-0.3660 (with standard error 0.0081); the coefficient on boxes
the coefficient on leader
A
a choice over mouse
(i.e.
more than doubles the probability
variables, a
of stopping.
one-standard-deviation increase in the value of boxes has an effect roughly 1/2 as large and a
one-standard-deviation increase in the value of leader has an effect less than 1/4 as large.
The
logistic regression
shown most importantly by the strong predictive power
subjects place partial weight on the
analyzing the current A^-good
game
u
as
boxes heuristic"
calculate G, the boxes heuristic
reliance on boxes as an
The
is
far
the most
of the boxes variable.
If
increasing the propensity to stop
a useful additional decision input.
example of sensible
by
— they
who can
will
be
less likely
only imperfectly
We view our subjects'
partial
— perhaps constrained optimal — decision-making.
predictive power of the boxes variables supports experimental research on "system neglect"
In other terms,
Wu
(2002).
find that subjects use
5 x)
"
= — expf
1
is
These authors
we estimate:
Probability of continuation
where x
i.e.
For an unsophisticated player
by Camerer and Lovallo (1999) and Massey and
~5
—
more and more boxes are opened
to spend a long time on any one game.
is
However, our subjects are also using other
important predictor of the decision to stop searching.
information, as
—G—
shows that the economic value of information
a vector of decision-node attributes, and
/3 is
+ exp(P'x)
the vector of estimated coefficients.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
sensible rules, but
heuristic
is
fail
to adjust those rules adequately to the particular problem at hand.
a good general rule, but
incentives generated
26
by each
specific
not the first-best rule since
it is
The boxes
iV-good game.
it
The boxes
neglects the idiosyncratic
heuristic
a type of imperfectly
is
sophisticated attention allocation.
Finally, our
uncertain about
too
experiment reveals another bias that we have not emphasized because we are
its
generalizability to other classes of games.
much time per game
had generally collected
in the
less
endogenous games.
is
the directed cognition model.
Table
Subject payoffs would have been higher
if
they
a goal for future research.
Comparisons with Other Models.
their performance.
our subjects allocated
information in each game, thereby enabling them to play more games.
Exploring the robustness of this finding
4.6.
Specifically,
1
The
In this subsection
analysis to date focuses on the predictions of
we
reports R' 2 measures for
consider alternative models and evaluate
all
of the alternative models
summarized
in
subsection 3.8.
None
benchmarks does nearly
of these
as well as the zero-parameter directed cognition model.
For the exogenous games, the directed cognition model has an average R' 2 value of 52.4% (with
a standard deviation of 3.4 percentage points).
averages of -18.8%
(s.e.
6.4%) and -137.4%
(s.e.
The Column and Row models have
respective
13.9%).
For the endogenous games, the zero-parameter directed cognition model has an average R' 2 of
91.2%
(s.e.
(s.e.
0.7%).
0.7%).
0.4%).
We
The Column model with a
The Row model with
satisficing stopping rule
a satisficing stopping rule has an average R' 2 of 55.4%
The Elimination by Aspects model has an average R' 2
also evaluate the different models' ability to forecast
time allocation. 26
Table
1
has an average Rl 2 of 42.9%
shows that
all
of the satisficing
of -6.3%
(s.e.
6.2%).
game-by-game variation
models yield
(s.e.
effectively
in average
no correlation
between the game-by-game average number of box openings predicted by these models and the
game-by-game average number of box openings
actually has a large negative correlation.
By
in the data.
The Elimination by Aspects model
contrast, the zero-parameter directed cognition
model
generates predictions that are highly correlated (0.66) with the empirical average boxes opened by
26
number of boxes opened per game for the 160 unique games
compiled from the endogenous time segment of the experiment.
Recall that this analysis uses the average
dataset.
This data
is
in
the
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
27
game.
5.
A
Conclusion
rapidly growing literature shows that limited attention can play an important role in economic
Most
analysis.
of the papers in that literature have
In this paper
according to cost-benefit principals.
and we
cost-benefit analysis,
direct empirical test of
directly
assumed that scarce attention
we develop a
is
allocated
tractable framework for such
We report
measure the attention allocation process.
the
first
an economic cost-benefit model of endogenous attention allocation.
Our research was conducted with a
stylized choice
attention allocation with a well-understood laboratory
problem that makes
method
it
possible to measure
Mouselab).
(i.e.
In this sense, our
choice problem was a natural starting point for this research program.
The
directed cognition model successfully predicts the key empirical regularities of attention
allocation
measured
in our experiment.
relying on extraneous parameters.
Moreover, the model makes these predictions without
Instead, the
model
is
based on cost-benefit principles.
Understanding how decision-makers think about complex problems
in
We
economic science.
is
The study
level of cognitive processes poses a
fundamental
the study of attention allocation.
way
Stripping back decision-
is still
in its infancy, particularly
But we are optimistic that such process-based research
for a science of
decision-making with
much
greater predictive
the classical "as if modeling that treated cognition as a closed black box.
growing body of work that pries open that box
Crawford, and Broseta 2001).
box more completely.
frontier
set of long-run research goals.
of cognitive processes in economic decision-making
ultimately pave the
new
believe that understanding the cognitive processes underlying economic
choices will further economists' understanding of the choices themselves.
making to the
a relatively
We
(e.g.
Camerer
et al.
will
power than
This paper joins a
1993 and Costa-Gomes,
look forward to future process-based research that will open the
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
Appendix A: Derivation of equation
6.
To gain
intuition for equation (3), begin
.
max
[a ,b)
by assuming that
— max (a, b) =
is
drawn from a Normal(0, a 2 )
the right-hand-side of equation
j
a
+£—
KcrJ
°"
=
b
<
6>a + E
6
if
6
<
(e
_| o _
il>
6
a
+e
(i)*
)0
|
V
J\a-b\
can re-express the option value formula by noting that
when
if
distribution, integrating over the relevant states yields
/"(.+«_»),(£)*_'/-
Likewise,
In this case,
a.
(3):
Jb-a
We
>
(3)
<
|
Because e
b
(|a
—
6|
,a)
x(j)
=
(x)
CT
—(f)'
y
cr
(x), so that:
.
a,
max
,,. —
max (a, 6) =
6J
(a
!;6
—
aif6>a + e
<
,
le
b
if
<
a
+e
Integrating over the relevant states again yields the right-hand-side of equation
=
7.
variance of payoffs in column
box located
in
row
u; (\b
— a\
,
n
is
then a\
m and column n.
=
a
2
are
1
(II
Using
G
drawn from a N(0,a 2 )
— n)
/10, for
(3):
a)
Appendix B: Example calculation of
Consider a game where the payoffs in column
of the
28
n E
{1...10}.
this notation, the current
distribution.
Call
rj
mn the
The
payoff
expected value of row
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
29
A is
a
=
Suppose that
at the current
that boxes {3,
5, 9,
time
in
A
row
27
10} remain unopened.
a
VAn
J2
already opened boxes
n
row
in
a subject has already opened boxes {1,2,4,7,8} and
The
current value of
leading row, except
if
A
the leading row
is itself
row
A
is
= VAl + VA2 + VA4 + VA7 + VA8
In most instances a
the value of the current best alternative to row A.
Call a
A
which case a
itself, in
is
is
the value of the
the value of the second
leading row.
The
'open boxes
So,
if
x
A
relevant cognitive operators for exploring row
and
3, 5,
=a—a
,
9'
and 'open boxes
3, 5, 9,
and
10.'
then the value of the benefit/cost ratio
Ga =
max{ii> (2,0-3)
,w( x,Jcrl+al
box
are 'open
3,'
'open boxes 3 and
Their costs are respectively
Ga
of
row
A
3 and
4.
is,
/2,w( x,\Io\-\-o\+g\
)
1, 2,
5,'
\
/3,
w\
>(x,yj al+al+4+a\A /4}
'
x
and the value of
G
=
a
—
a
at the current
time
is
simply the highest possible benefit/cost ratio across
all
rows:
G = maxG n
.
n
Finally, note that the highest benefit/cost ratio
that opens
in a
game
more than one box.
in
if
=
0.0013)
is
—
—
differ
by
3<7i.
Hence,
if
only one box
required to generate a rank reversal in the rows.
two boxes are opened, the chance of reversal
2T
associated with a cognitive operator
For example, consider a case in which no boxes have been opened
which the leading row and the next best row
opened, a three-sigma event (p
contrast,
may be
is
11 time greater.
Specifically,
is
By
the standard
This pattern of box openings
would not arise under the model, since the model would never
{1,2, 4, 7, 8}
However, such skips can arise in experimental play, and hence the directed cognition model must be
able to handle such cases if it is used to predict continuation play (e.g., stopping decisions).
skip a column.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
deviation of aggregated noise in the two box case
3/\/l^9
=
2.2,
we only
is
a
=
require a 2.2-sigma event (p
\Jo\
=
(0.9) a\. Since
3ui/'y'a\
+
(0.9)
a\
=
0.0146) in the two-box case to generate a
may be much more
Hence, the two-box cognitive operator
rank reversal.
+
30
(when
cost effective
evaluated by the directed cognition algorithm) than the one-box cognitive operator, since the two-
box cognitive operator increases the chance
of reversal
by a factor of 11 but only has twice the
cognitive cost.
Appendix C: Lower bounds for the performance of the directed cognition
8.
ALGORITHM
We
wish to evaluate how well directed cognition does compared to the Perfectly Rational solution
of a
model with calculation
costs.
Perfect Rationality's complexity
Here we provide bounds on
to evaluate.
performs at least
91%
its
performance.
We
will
makes
it
prohibitively hard
show that directed cognition
as well as Perfect Rationality for exogenous time
games and
at least
71%
as
well as Perfect Rationality for endogenous time games.
A
useful
benchmark
could inspect
a
is
all
that of Zero Cognition Costs,
the boxes at zero cost.
game with standard
a normal with
is
mean
deviation of the
We
first
the performance of an algorithm that
evaluate Zero Cognition Costs' performance for
column
a.
Each
and standard deviation a times
Costs will select the best of those 8 values.
variables
first
i.e.
of the 8 rows will have a value
w
(
Vj/10
V
[
The expected value
of the
which
/2
=
5.5
1 /2
maximum
.
Zero Cognition
of 8
i.i.d.
N
(0, 1)
is:
v
~
1.423.
So the average performance of Zero Cognition Costs
will
(5)
be
X 2
5.5 / ZA7
~
3.34a.
We
gather this as
the following Fact.
Fact:
The
average payoff V* of the optimal algorithm will be
V* <
We
less
than
3.34<t.
use this result to analyze exogenous and endogenous time games.
3.34cr.
(6)
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
8.1.
We
Exogenous time games.
rithm on a large number of
and unit standard deviation of the
49sec], as in the experiment.
simulate the performance of the directed cognition algo-
games with the same stochastic structure
i.i.d.
first
column. The time allocated
The performance
at least 3.064/3.34
is
=
is
of directed cognition
inequality (6), the performance of Perfect Rationality
ative performance
31
than
is less
as in the experiments,
randomly drawn from
is
on average 3.064. Applying
3.34, so directed cognition's rel-
Hence, directed cognition does
0.917.
[lOsec,
91%
as well as
Perfect Rationality in exogenous time games.
Endogenous time games.
8.2.
in
games of standard deviation
Monte-Carlo simulations determine the performance of V DC
a, defined as the expected value of the chosen
cost of time, times the time taken to explore the games.
Rationality.
The
relative
performance of directed cognition
V*
Call
(a, k)
rows minus
k,
(<x,
the
the performance of Full
is:
V DC (a,K)
By
generic
definition
game
G
7r (<r,
k)
with a
<
first
1.
We now
column of unit variance. The cost
the expected value under the optimal strategy. Call
1,
and £
=
,
expected value of G"
a
rj
max{r] A1 ,Tj Hi } the highest valued box
lower value than the
game G" that has
game G'
is
A i,
...,rj
Hi
the expected payoff of the
zeroes in the
first
column.
column,
V G = V G + v.
t
boxes
consider a
nt
is
.
V*
is
the values of the 8 rows in column
column. The game
in the first
first
We
(a, k).
of exploring
that would have equal values of £ in the
V* <
Game G"
bound on V*
look for an upper
E
[£]
G
has thus a
VG
first
column: V*
<
=
~
1.42, plus
the value of
It
has 8 rows.
v
.
The
So
V G" + v
has the structure of our games, with zeros in column
1.
Thus,
it
has lower value than a game G'" that has an infinite number of rows, and the same stochastic
structure otherwise:
variance
column
(N+l-n)/N
with
1
has boxes of value
0,
N = 10 columns. V G "
and column n 6
< V G '" The
.
property similar to a Gittins index. Indeed, suppose that one
row. Without loss of generality call that row A.
One
is
value
{2,
...,
10} has boxes with
M ~ VG
'"
of G'" has the
thinking about exploring a
will stop exploring this
row
new
after stopping in
k)
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
column
r. Call at
the value of that row after exploring column 2 through
i-l
So we have:
t.
\ 1/2
—n
Ely
(rir)
/
32
AT
\
where the
Ar^s are
r\
standard normals.
i.i.d.
VA,n+l
game G'" we have
In
=
a\
At the end of the
0.
exploration of this row, one can either pick the box, which has a value a T or drop the current row,
,
and
start the
row
is
game from
max (aT M).
,
which has a value
scratch,
So the value
M=
M
of
game G'"
M = VG
So the payoff after exploring the
.
satisfies:
E [max-(.aTj M) — k (r - 1)
max
stopping time t
|
ai
=
0]
i.e.
=
E [max (a T - M, 0) - « (r - I)
max
.
a\
.
|
= 0]
(8)
stopping time r
game 28 that has explored up
Call u (a,t, k) the value of the
and that at the end of the search r
yields a payoff
max (a r
,
to
column
t,
has current payoff
minus the cognition costs k(t
0)
a,
—
t).
k)
=
Formally:
v
(a, t,
k)
=
max
stopping time r>t
a 6 M,
for
0,
t
G
{1,
...,
10},
«;
>
0.
Because of
E [max (a r
(8),
Using
we conclude
(5),
(k)
=
a
game
a general
a
— k(t —
s.t.
game
t)
\
at
M (k)
=
of a unit standard deviation.
satisfies v
(— M,
1,
the smallest real with that property:
v (-x,
1,
=
k)
0}
1.42
As V*
+ M(«).
{a, k) is
homogenous
of degree
is:
In terms of Gabaix
a]
that
V*
3
it is
min {x >
V* <
for
0)
the value of the
and by standard properties of value functions
M
,
and Laibson
(a, k)
(2002), this
is
<
1.42(7
+ crM (k/<t)
.
the value function of an "a vs 0" game.
1,
the inequality for
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
33
Full Rationality does less well that Zero Cognition Costs, so (6) gives:
V*{a,n) <
We
combine those two inequalities using the
3.34o-
fact that
V
is
convex
in k,
which comes from the
usual replication arguments. This gives:
V*
where
V
(a, k) is
{a, k)
< V*
(a, k)
the convex envelope 29 of 3.34ct and 1.42<r
+ aM (k/ct),
seen as functions of k.
Eq. (7) finally gives
Tr{a,K)>7L:=miV DC (a,K)/V*(a,K)
(9)
<r,re
We
evaluate
71%
29
n
numerically, and find
as well as Full Rationality in
(a, k)
=
=
0.71.
We
conclude that directed cognition does at least
endogenous games.
As we take the lower cord between and
V
tt_
3.34cr
min (l -
and
1.42cr
+ aM (ko/c),
—
3.34c7
+
)
—
(1.42ct
this gives, formally:
+ aM (so/ff))
.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
REFERENCES
9.
Aragones,
E., Gilboa,
I.,
Complex Trade-Off
34
Postlewaite, A., and Schmeidler, D. (2003), "Accuracy vs. Simplicity:
"
A
(Mimeo).
Camerer, C. (2003), Behavioral
Game
Theory: Experiments in Strategic Interaction (Princeton:
Princeton University Press).
Camerer, C. and Ho, T-H. (1999), "Experience Weighted Attraction Learning in Normal-Form
Games", Econometrica,
67, 837-874.
Camerer, C, Johnson, E.J., Rymon, T., and Sen,
S. (1993),
"Cognition and Framing in Sequential
Bargaining for Gains and Losses", in K. Binmore, A. Kirman, and P. Tani
of
Game
MA: MIT
Theory (Cambridge,
(eds.)
Frontiers
Press), 1994, 27-47.
Camerer, C. and Lovallo, D. (1999), "Over confidence and Excess Entry: an Experimental Approach", American Economic Review, March, 89, 306-318.
Conlisk,
J.
(1996),
"Why Bounded
Rationality?" Journal of Economic Literature, June, 34 (2),
669-700.
Costa-Gomes, M., Crawford, V., and Broseta, B. (2001), "Cognition and Behavior in Normal-Form
Games:
An
Experimental Study", Econometrica,
69, 1193-1235.
Costa-Gomes, M. and Crawford, V. (2003), "Cognition and Behavior
Games: an Experimental Study" (Mimeo, University of
Daniel, K., Hirshleifer, D.,
California,
in
Two
Person Guessing
San Diego).
and Teoh, S.H. (2002), "Investor Psychology
in Capital Markets:
Evidence and Policy Implications", Journal of Monetary Economics 49(1) ,139-209.
D'Avolio, G. and Nierenberg, E. (2003), "Investor Attention: Evidence from the Message Boards"
(Mimeo, Harvard).
Delia Vigna,
S.
and
Pollet, J. (2003), "Attention,
(Mimeo, University of California
Erev,
I.
at Berkeley).
and Roth, A. (1998), "Predicting
September, 88(2), 848-81.
Demographic Changes, and the Stock Market"
How
People Play Games", American Economic Review,
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
Gabaix, X. and Laibson, D. (2001), "The
Bernanke and K. Rogoff
NBER
(eds.)
6D
Bias and the Equity
Macroeconomics Annual
,
35
Premium Puzzle",
in B.
257-312.
Gabaix, X. and Laibson, D. (2002), "Bounded Rationality and Directed Cognition," (Mimeo,
MIT
and Harvard).
Gabaix, X. and Laibson, D. (2003), "Some Industrial Organization with Boundedly Rational
Consumers", (Mimeo,
MIT and
Harvard).
ABC
Gigerenzer, G., Todd, P. and the
Smart (Oxford: Oxford University
Research Group (1999), Simple Heuristics that
Make Us
Press).
Gittins, J.C. (1979), "Bandit Processes
and Dynamic Allocation Indices", Journal of the Royal
Statistical Society, 5-41(2), 148-77.
Gumbau,
vard)
F. (2003), "Inflation Persistence
and Learning
in
General Equilibrium" (Mimeo, Har-
.
Hirshleifer, D.,
Lim,
and Teoh, S.H.
S.S.,
of Limited Attention"
Jehiel, P. (1995),
(2003), "Disclosure to a Credulous Audience:
The Role
(Mimeo)
"Limited Horizon Forecast in Repeated Alternate Games", Journal of Economic
Theory, 67, 497-519.
Johnson, E.J., Camerer,
C,
Sen,
S.,
and Rymon, T. (2002), "Detecting
Induction: Monitoring Information Search in Sequential Bargaining"
Theory,
May
Kahneman, D.
Failures of
,
Backward
Journal of Economic
2002, 104(1), 16-47.
(1973), Attention
Kahneman, D. and Tversky, A.
and Effort (New
(1974),
Jersey: Prentice Hall Inc.).
"Judgment Under Uncertainty: Heuristics and Biases",
Science, 185, 1124-1131.
Lynch, A. (1996), "Decision Frequency and Synchronization Across Agents: Implications
gregate Consumption and Equity Returns", Journal of Finance 51(4): 1479-97.
for
Ag-
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
Mankiw, N.G. and
Replace the
Reis, R. (2002),
New Keynesian
36
"Sticky Information Versus Sticky Prices:
Phillips Curve", Quarterly Journal of
a Proposal to
Economics 117(4), 1295-
1328.
Massey, B. C. and Wu, G. (2002), "Detecting Regime Shifts: The Psychology of Over- and Underreaction" (Mimeo, University of Chicago Graduate School of Business).
Moray, N. (1959), "Attention in Dichotic Listening: Affective Cues and the Influence of Instructions", Quarterly Journal of Experimental Psychology, 11, 56-60.
Pashler, H. (ed) (1998), Attention. (UK: Psychology Press).
Pashler, H.
and Johnston, J.C. (1998), "Attentional Limitations
in
Dual Task Performance,"
in
H. Pashler (ed) Attention (UK: Psychology Press), 155-190.
Payne, J.W., Bettman, J.R, and Johnson, E.J. (1993), The Adaptive Decision Maker (Cambridge:
Cambridge University Press).
Payne, J.W), Braunstein, M.L., and Carroll,
J.S. (1978),
"Exploring Predecisional Behavior: an
Alternative Approach to Decision Research", Organizational Behavior and
Human
Perfor-
mance, 22, 17-44.
Peng, L. and Xiong,
W.
(2003) "Capacity Constrained Learning and Asset Price
Comovement",
(Mimeo, Princeton).
Pollet, J. (2003), "Predicting
Asset Returns with Expected Oil Price Changes" (Mimeo, Harvard).
Rogers L.C.G. (2001) "The relaxed investor and parameter uncertainty." Finance and Stochastics
5(2): 131-54.
Russo,
J.
E. (1978),
"Eye Fixations Can Save the World: a Critical Evaluation and a Compari-
son between Eye Fixations and Other Information Processing Methodologies"
,
Memory and
Cognition, 3(3), 267-276.
Simon, H. (1955), "A Behavioral Model of Rational Choice", Quarterly Journal of Economics,
Feb, 69(1), 99-118.
THE ALLOCATION OF ATTENTION: THEORY AND EVIDENCE
37
Sims, C. (2003), "Implications of Rational Inattention", Journal of Monetary Economics, 50(3),
665-690.
Tversky, A. (1972), "Elimination by Aspects:
A
Theory
of Choice", Psychological Review, 79,
281-299.
Wegner, D.M. (1989), White Bears and Other Unwanted Thoughts: Suppression, Obsession, and
the Psychology of
Mental Control (New York: Viking/Penguin).
Weitzman, M. (1979), "Optimal Search
for the Best Alternative",
Econometrica, 47(3), 641-54.
Weitzman, M. and Roberts, K. (1980), "On a General Approach to Search and Information
Gathering",
MIT Working
Wood, N. and Cowan, N.
Paper 263.
Attention Shifts to One's
Psychology: Learning
Name
J. Stiglitz,
in
Cognition, 21(1), 255-60.
Common Knowledge
in
Frequent Are
,
and M. Woodford
Modern Macroeconomics:
How
an Irrelevant Auditory Channel" Journal of Experimental
Memory and
Woodford, M. (2002), "Imperfect
Aghion,
"The Cocktail Party Phenomenon Revisited:
(1995),
Honor
of
(eds.)
and the
Effects of
Monetary Policy",
in P.
Knowledge, Information, and Expectations in
Edmund
S.
Phelps (Princeton: Princeton University
Press).
Yantis, S. (1998), "Control of Visual Attention," in Harold Pashler (ed.) Attention
ogy Press), 223-256.
(UK: Psychol-
.
Table
Ga mes
with Exogisnous
2
2
R'
R'
2
for
for
for
column
row
alt.
Average
R'
Evaluation of Alternative Models
Games
Time Budgets
with
Endogenous Time Budgets
Column
Row
Directed
Model
Model
Cognition
Column
Row
Directed
Cognition
Model
Model
Cognition
No
No
No
No
k = 0.18
S = 42
S = 43
S = 45
A
86.6%
-107.7%
39.3%
96.3%
73.1%
79.7%
34.5%
75.3%
-70.4%
(1.7%)
(18.7%)
(0.9%)
(1.4%)
(4.2%)
(1.8%)
(0.9%)
(1.5%)
(18.4%)
-16.1%
26.2%
-540.6%
85.3%
64.6%
25.1%
88.6%
65.6%
-31.1%
(9.2%)
(0.5%)
(0.4%)
(1.3%)
(2.4%)
(0.4%)
(0.7%)
(1.0%)
(0.7%)
86.7%
25.0%
89.1%
91.8%
84.5%
23.8%
43.0%
53.3%
82.5%
(1.4%)
(0.4%)
(1.5%)
(0.5%)
(0.8%)
(0.3%)
(0.6%)
(0.8%)
(0.8%)
52.4%
-18.8%
-137.4%
91.2%
74.1%
42.9%
55.4%
64.7%
-6.3%
(3.4%)
(6.4%)
(13.9%)
(0.7%)
(1.7%)
(0.7%)
(0.4%)
(0.6%)
(6.2%)
0.66
0.61
0.00
0.02
0.03
-0.39
(0.02)
(0.02)
(0.02)
(0.02)
(0.02)
(0.02)
profile
profile
row
:
Directed
Free parameters?
R'
1
profile
2
Directed
Elimination
Cognition Satisficing Satisficing Satisficing by Aspects
= -6
Correlation with Empirical
Number of Boxes
per
Game
...
...
Bootstrapped standard errors
2
R'
refers to the
plus the
R2
statistic
in
...
...
parentheses.
from a regression of the empirical number of boxes opened by subjects
number of boxes predicted by the models, where the
Correlation refers to the correlation between the
openings predicted by the models.
number
of
co-efficient
on the predictions
boxes opened by subjects
in
is
each
in
each row or column on a constant
fixed to equal
of the
1
160 games and the number of box
Figure
1
:
Sample Game with All Values Unmasked
box at
column are drawn independently
with the same variance, with variances
In the actual experiment, the subjects see only the value of one
a time (see Figure 2).
Values
in each
from a normal distribution
declining linearly from left to right. In this sample, the left-most
column is generated with a standard deviation of 30.6 cents, which is
explained to subjects as a 95% confidence interval of -60 to 60 cents.
Figure 2: Sample
This
is
how
a sample
Game
with Values Concealed
game would appear
to subjects,
with values
concealed by boxes. Subjects can use the mouse to open one box
time. In this
game
at a
the subject faces a set time limit; the clock in the
upper right corner reveals the fraction of time remaining.
Figure
3:
w(x,o) for
a=
1
0.35
0.25
0.15-
0.05-
Figure 3 plots the expected benefit from continued
search on an alternative if the difference between
the value of the searched alternative and its best
alternative is x, and the standard deviation of the
information gained is c = 1 as defined in Equation 3.
This w function is homogeneous of degree one.
Figure 4:
6r
Column
Profiles for
Games
with
Exogenous Time Budgets
DC
5
6
7
Model
(R'
2
= 86.6%)
10
Column
Figure 4 plots the mean number of boxes opened in each column by subjects
and by the directed cognition model, for games in which time budgets are
imposed exogenously. Dotted lines show the bootstrapped 95% confidence
intervals for the data.
Figure
5:
Row
Profiles for
Games
with
Exogenous Time Budgets
DC
Model (FT = -16.1%)
Data
Row
mean number
boxes opened in each row by subjects and
by the directed cognition model, for games in which time budgets are imposed
Figure 5 plots the
of
exogenously. Dotted lines show the bootstrapped 95% confidence intervals for
The rows are ordered according to the values in the first column.
the data.
Figure
6:
Row
Profiles (alternative ordering) for
Games
with
Exogenous Time Budgets
8r
DC
Model (FT = 86.7%)
Row
Figure 6 plots the
mean number
the directed cognition model,
of
for
boxes opened in each row by subjects and by
games in which time budgets are imposed
exogenously. Dotted lines show the bootstrapped 95% confidence intervals for the
data. The rows are ordered according to all the information available to subjects
after
they have concluded their search, just prior to making a choice.
Figure
6r
7:
Column
Profiles for
Games
with
DC
Endogenous Time Budgets
Model, k = 0.18 (R'
2
= 73.1%)
Data
t,5
C
DC
Model, k = 0.49 (R'
2
= 96.3%)
w
X
O 3
CD
r>
£**.
.a
£
-
E
6
8
10
Column
mean number of boxes opened
each column by subjects
and by both calibrations of the directed cognition model, for games in which
agents choose how much time to allocate from a fixed time budget. Dotted
Figure 7 plots the
lines
Figure
show
8:
the bootstrapped
Row
Profiles for
95%
in
confidence intervals for the data.
Games
with
Endogenous Time Budgets
Data
DC
DC
Model, K
Model, K
:=
:=
0.18 (R'
0.49 (R'
2
2
= 64.6%)
=
85.3%)
Row
Figure 8 plots the mean number of boxes opened in each row by subjects and
by both calibrations of the directed cognition model, for games in which agents
choose how much time to allocate from a fixed time budget. Dotted lines show
the bootstrapped 95% confidence intervals for the data. The rows are ordered
according to the values in the initial column.
Figure 9: Row Profiles (Alternative Ordering) for
8r
Games
DC
with
Model, K
Endogenous Time Budgets
=
2
0.18(R' = 84.5%)
=
0.49 (R'
:
Data
DC
Model, K
:
2
=
91.8%)
Row
Figure 9 plots the
mean number
of
boxes opened
in
each row by subjects
games in which
and by both calibrations of the directed cognition model, for
agents choose how much time to allocate from a fixed time budget. Dotted
lines show the bootstrapped 95% confidence intervals for the data. The rows
are ordered according to all the information available to subjects after they
have concluded their search, just prior to making a choice.
Figure 10: Boxes
Opened by Game
45
40
10
20
30
Box openings predicted by the model (k= 0.18)
Figure 10 takes each of the 160 game-types played by
number of boxes opened by
number of box openings predicted by the
model, for the games in which agents choose their crossgame time allocations. The correlation between the
subjects and compares the
subjects to the
model and
the data
is .61.
—
'
Figure 11: Expected Measure of Economic Gain "G" vs. Expected Remaining
Time
40r
....^^MimniniHi— ««iu(iiwn«»'""—
,tt
™"
35
CD
^^^^^
30-
I 25
CO
E
CD
o
Bo
CD
/
1
20-/
sy —
/y
:.,*-----
~
"
///..•••"
1
15 In/
h/
8-10
HI
/
./'
—
—
li'il
Lii
Empirical
-
DCVTk = 0.18
Empirical
\l
99%
C.I.
DCVTK = 0.18 99%C.I.
L*
::
(
J
?
i
«
1—
1
1
4
6
8
10
12
Expected measure of economic gain "G"
1_
_
14
Figure 11 plots non-parametric (kernel) estimates of the expected
of additional
is
box openings
in
in
equation
(4).
16
number
a game, conditional on the current "G" value. "G"
in the directed cognition model,
the benefit-to-cost ratio of marginal analysis
as defined
i
3337
013
Date Due
MIT LIBRARIES
3 9080 026
y
:
:
liill