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o
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http://www.archive.org/details/valueofinformatiOOathe
'(DBAfEYj
working paper
department
of
economics
The Value Of Information In
Monotone Decision Problems*
Susan Athey
Jonathan Levin
No. 98-24
November, 1998
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
The Value Of Information In
Monotone Decision Problems*
Susan Athey
Jonathan Levin
No. 98-24
November, 1998
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
The Value
of Information in
Siisan
Monotone Decision Problems*
Athey
M.I.T. and
Jonathan Levin
NBER
First Draft:
M.I.T.
September 1997
This Draft: November 1998
Abstract
an
A seminal theorem due to Blackwell (1951) shows that every Bayesian decision-maker prefers
informative signal Y to another signal X if and only if Y is statistically sufficient for X.
Sufficiency
is
an imduly strong requirement in most economic problems because
incorporate any structure the model might impose.
In this paper,
we develop a
it
does not
general the-
ory of information that allows us to characterize the information preferences of decision-makers
based on how their margiual returns to acting vary with the imderlyiug (unknown) state of the
world.
Our
whereby
analysis imposes one central restriction:
all
we
consider "monotone decision problems,"
decision-makers in the relevant class choose higher actions
the signal are realized.
We
show how
this restriction
when
higher values of
can be exploited to characterize
infor-
mation preferences using stochastic dominance orders over the distributions of posterior behefs
generated by different signals. Of particular interest for appUed modeling,
we
identify condi-
tions under which one decision-maker has a higher marginal value of information than another
decision-maker, and thus will acquire
more information. The
results are
appUed to oligopoly
models, labor markets with adverse selection, hiring problems, and a coordination game.
JEL
Classification
Numbers: 044, C60, D81.
Keywords: Bayesian decision problem, value of information, stochastic dominance, stochastic
orderings, decision-making imder uncertainty.
*We would
Lrhmann,
like to
thank Scott Ashworth, Kyle Bagwell, Glenn Ellison, John Geanakoplos, Bengt HolmstrOm, Eric
as well as seminar participants at
MIT,
Yale, the 1998
Summer Meetings
of the Econometric Society,
and
the Pacific Institute of Mathematical Sciences for helpful discussions. Athey thanks the Cowles Foundation at Yale
University and
NSF
Grant SBR-9631760
for support.
Introduction
1
who
In a Bayesian decision problem, an agent
is
uncertain about the true state of the world must
choose an action after observing an imperfectly informative signal. BlackweU (1951, 1953) proved
the seminal result that every agent faced with such a decision problem will prefer (ex ante) an
informative signal y to another signal
of "better information"
himself).^ For
is
useful
no
restrictions
In this paper,
intuitive,
but
y
if
it is
—
in sharp contrast to
on the decision-maker's payoff
we show
is
statistically sufficient for x.
also quite
demanding
x by
all
noted by Blackwell
far stronger
than needed to
most economic models, Blackwell's theorem
function.
and derive comparisons among a richer
different classes of decision-makers,
to another signal
is
(as
This notion
that by exploiting properties of the decision-maker's preferences,
possible to relax sufficiency
many
and only
if
economic modeling, we might expect that sufficiency
compare information structures
places
and
x
members
we
it is
set of information structures.
derive conditions under which a signal y
is
For
preferred
of the class. In most cases, the informativeness order can be
represented as a stochastic dominance ordering over the distribution of posteriors that might arise
from each information source.
We also provide conditions under which one decision-maJcer wiU have
a higher marginal value of information than another decision-maker, and thus
information.
The
hiring problems,
The
results are
apphed to oligopoly models, labor markets with adverse
starting point of our analysis
is
to focus attention on
on the state of the world H{uj), a
admissable signals {£}.
problems,
in the state of the
it is
selection,
The
class of payoff functions
is
monotone decision problems, each
class of payoff functions
defined by
how
incremental returns to taking an action (a) vary with the state of the world
many economic
more
and a coordination game.
consisting of a prior belief
set of
will acquire
world
it is
(i.e.,
U, and a
the decision-meiker's
{u>).
For example, in
assumed that the incremental returns to an action are nondecreasing
the payoff function
ti(a;,a) is
supermodular). Given such a
class,
possible to (partially) order posterior beUefs about the state of the world so that "higher"
beliefs
induce higher actions for
all
decision-makers in the class.
to a
monotone decision problem, we require that
from
different realizations of the signal
all
For a signal to be admissable
of the posteriors that might be generated
can be ordered in
this way.
To
illustrate, for
the class of
decision-makers with supermodular payoffs, higher posteriors in the sense of First Order Stochastic
See Blackwell and Girshik (1954).
For y to be sufficient for x,
all
of the posteriors, no matter
generated by x must be in the convex hull of the set of posteriors generated by y. There are
examples of distributions which cannot be ranked according to
X cannot be more informative than a normally distributed
further examples.
sufficiency.
signal
Unless a signal x
is
hew
many
unlikely,
unsatisfying
normally distributed,
y in the Blackwell order. See Lehmann (1988)
for
Dominance (FOSD) induce higher
set of posteriors
eictions,
and the
x
signal
is
admissable only
can be totally ordered by FOSD. In general,
if
the corresponding
different classes of decision-makers
induce different "stochastic dominance" orderings over posteriors (for instance,
will
returns to acting are concave in the state of the world, the relevant order
is
if
the marginal
Second Order Stochastic
Dominance (SOSD)).
We
show that a natural information ordering over
problems.
We
a signal y
find that
"more informative"),
posteriors induced
preferred to
x
a
for
is
available for
monotone decision
class of decision-makers
(and thus y
is
the "high" posteriors induced by y are (on average) higher than the "high"
if
by
is
signals
and the "low" posteriors induced by y are (on average) lower than the
x,
"low" posteriors induced by x.
The terms
"high" and "low" refer to the stochastic dominance order
induced by the restriction on the decision-maker's payoff function.
To
fix ideas,
return again to the example of supermodular payoff functions. Recalling that the
posteriors of the signal
x
are totally ordered,
we can index them with a parameter
FOSD)
higher values of the index correspond to higher ordered (by
be comparable across
will
signals,
normahzed
it is
probability a^, a posterior lower (in the
We
x
similarly index the posteriors
for all
supermodular payoff functions
expression G{oj\ay
>
is
conditions
We
also
tpoSD
is
realized after observing x.
signal
y more informative than
— ct
fraction of the posteriors
[0, 1],
F{uj\ax
more informative than x
we
will
be relevant
derive are sufficient for
signal to another; they are necessary
structure.
a)
this index
>
a).
if
high realizations of w are more likely
posteriors are realized. For other classes of payoff functions, other stochastic
dominance orders (such as SOSD)
The
>
aG
Then a
y.
a) represents the average over the highest 1
generated by y. Thus, a signal y
when highly-ranked
order) than F{ii)\ax)
for all
if,
So that
so that
to have a uniform distribution ex ante: with
G{u\ay) generated by
G{uj\ay
The
FOSD
posteriors.
ctx,
all
for the
comparison of average posteriors.
decision-makers in a given class to prefer one
when we compare
small (differential) changes in the signal
show that the theory can be generalized by considering orders based on
single
crossing properties rather than stochastic dominance.
Our second
objective
is
to derive conditions under which one decision-maker will have a higher
marginal value for information than another, and thus will acquire more information
mation
is costly.
We
find that
if
u and v
are in the
same
class of payoflF functions,
when
infor-
and the two
decision-makers consider purchasing signals ranked according to our criteria, then decision-maker
u buys more information than v if «'s
preferences over the distribution of posteriors
optimal decision rule are "more sensitive" than
u's.
The meaning
of "sensitive"
is
when using an
determined by
the class of payoff functions under consideration. Although our conditions depend on the optimal
policies
Our
of
many
applications.
by Lehmann (1988). He considered one
specific class
chosen by the agents, and thus are not primitive, they can be verified in
work
results are related to
monotone decision problems
property,
and where the
Lehmann
(1988) derived a
in statistics
— those where the decision-maker's payoffs
signals satisfy the
monotone
new information
ordering that relaxes Blackwell's sufficiency criterion.
1997), principal-agent problems (Jewitt, 1997),
on
statistical
and
While our methods are quite
Tirole, 1997).
hypothesis testing -
Lehmann's work
related to
likelihood ratio property. For such problems,
effectiveness ordering has already entered economics in the context of auctions (Persico,
Lehmann's
and
satisfy a single crossing
is
we obtain
implicit incentive
different
models (Dewatripont, Jewitt
- Lehmarm uses an approach based
the eflPectiveness ordering as a special case. Closely
that of Persico (1996),
who
studied the same specific class of decision
problems. His paper develops an approach to ranking decision-makers in terms of their incentives to
Our approach
acquire information."^
to information acquisition builds directly on his,
and provides
a significant generalization to other classes of monotone decision problems.
The paper develops
as follows. In the next section
we
describe the model and briefly review the
standard approach to information and some preliminary results on stochastic orderings. In Section
3,
we introduce the
classes of
MDPs.
characterizes the
Section 4 includes ovu:
economic appUcations
2.1
and
and a
for information. Section 6 gives
some
firms, adverse selection in labor markets,
hiring problem.
The
last section discusses
some
and concludes.
The Bayesian Decision Problem
decision-maker
after observing
an
structures,
The Model
2
A
terms of their marginal value
uncertainty,
some important
for several classes of MDPs. Section 5 presents results
— to information gathering by
game under
possible extensions
in
discuss
main theorems on ordering information
monotone information order
on ordering payoff functions
a coordination
monotone decision problems (MDPs), and
idea of
interval.
action a
€
(DM) who
is
an informative
Let
V
A C R.
uncertain about the true state of the world must take an action
signal.
denote the set of
Her payoff
u{u;, a)
The
all
state of the world
is
probability distributions
denoted
on
Cl.
a;
€
The
fi,
where
DM must
depends on both her action and the true
state;
fi
CR
is
choose an
we assume u
Persico (1997) further established that similar techniques can be used to rank the revenue to the auctioneer under
different auction formats.
is
a bounded measurable function taking
f2
x
R ^ R.
Throughout, we
will
maintain the following
assumption about the set of available actions.
(A) Either
The
A
is finite,
or yl
We
is
a compact interval of
R and u(cj, a)
continuous in
is
o.
DM has prior distribution H{uj). Before axiting, the DM observes some informative random
variable x, with support Af
{u, x)
is
C
R, and forms a posterior distribution F{uj\x). The joint distribution of
then written F{uj, x), while the marginal distributions are denoted F{x) and F{lo)
will refer to
F
Observe that
as
=
H(uj).^
an "information structure."
many
different information structures
can be equivalent from the perspective of
decision-making, since only the posterior generated by a signal realization affects behavior (the
value of the signal does not). In particular,
so long as F{u)\T{x)=T(x))
structure, F, can
= F{co\x = x)
it
does not matter
for all x.
The
if
the
DM observes x or some T{x),
payofi'-relevant features of
an information
be uniquely characterized in terms of the probability measure induced on the space
of posteriors V, which
we write
as fxp.^
of an information structure before
We will begin by working with this abstract characterization
mapping back
to the
formulation in terms of the joint
first
distribution F(uj,x).
The
decision-maker's problem, given a posterior distribution
max f
aeA
to obtain an optimal action a*{P),
of the decision
and a
The
The
classical
7-',
is
to solve
u(aj,a)dP{aj)
(1)
realized payoff u(a;, a* (P)).
We define the
(ex ante) value
problem as
V*(F,u)
2.2
PG
Classical
= [ [ u{cv,a*{P))dPiu)dfxAP)JvJn
(2)
Approach to Information
approach to information, due to Blackwell (1951, 1953), begins by writing V*{F,u)
as
V*{F,u)
= f {max / u{u},a{P))dP{u})\dfip{P) = / u*{P)dnp{P).
Jv <P) Jn
Jv
I
'Note that F{ui)
=
H{u!)
is
needed to ensure that the expectation of the posterior
joint distribution of (w,x) as primitive is
(3)
J
is
the prior.
Taking the
a departure from the analyses of Blackwell (1951) and Lehmann (1988),
which take as primitive the distribution of x|w. This means that their information orders are the same for
distributions
H. Our information rankings
will
be given
relative to
a fixed prior H. Thus, we
may
all
prior
sensibly consider
averages over subsets of the posterior distributions.
Endow V with the weak topology.
= fx-piuMeB'^^^'''^'
*Tliis construction is introduced in Blackwell's original article (Blackwell, 1951).
Then F(w\x)
is
measurable with respect to the Borel o--algebra and
/Xp.(JE)
Using revealed preference,
observe that for A €
it is
1],P^,P'^£ V,
[0,
max \x f udP^ +
°-
V
Now suppose
(1
-
Jo.
G is
< max A
°-
)
+ max(l - A)
udP^
/
"
Jo.
/ udP"^.
and that
ijlq
stochastically dominates fip for convex functions,
vex functions, then V*{G,u)
(p
V
:
-^
> V*{F,u)
M..
for
Cleaily,
fiQ stochastically
^q
over poste-
J^ fdfiQ
i.e.
dominates
any payoff function u and action
G more valuable than F;
decision-maker will find
if
(4)
7n
an alternative information structure which induces a measure
any convex function
ipdfip for
A) / udP'^ \
Jo.
riors distributions,
Jp
straightforward to show that u*(P) will be convex in P. Simply
fip for con-
A,
set
>
i.e.
every
Blackwell (1951) showed the converse via a sepa-
ration argument. Thus, Blackwell's order can be thought of as a multivariate generahzation of the
(perhaps more famiUar) mean- preserving spread of Rothschild and Stiglitz (1970).
The geometric
representation of a mean-preserving spread in multiple dimensions can be formalized using
known
is
y to be
as a "dilation," so that
/j,q is
a dilation of
the set of posteriors generated by
sufficient for x,
can further be shown that
fip.^ It
x must
lie
in the
what
for a signal
convex hull of those
generated by y.
Sufiiciency can be understood in the context of the following three state,
Q,
=
{uii,U2,0J3},
signal
that
y that
is
is
where wi
< 0^2 < 0J3,
and suppose the prior on
cv is
two signal example. Let
imiform
Consider a
(5, 5, 5).
equally hkely to induce posterior beliefe (|, ^, ^) and (0, ^, ^), and another signal
equally likely to induce posterior beliefe (^j 5, ^)
and
(g)
^s ^)- As
x
illustrated in Figure 1,
the posteriors generated by y are a mean-preserving spread of the posteriors generated by x. So y
be
will
But
sufficient for x.
another signal z only
The ranking
will
if
this is clearly
very special, since y can be sufficiency ranked relative to
the posteriors generated by z
lie
on the
line
between
This lack of robustness
is
possible).
inferences. For instance,
And
(V3.
(0, |, ^).
Figure 1 also highhghts the fact
imappealing.^
that no two-point information structiure can be sufficient for y (the prior
is
and
be upset by even the sfightest perturbation of z leading to a potential posterior
belief off of the fine.
spreading
(|, |, ^)
In spite of this, y
is
clearly not uniquely suited to
y provides no information at
all
as to the relative
while y provides significant information as to whether
point information structures might be
is fixed,
more informative on
u
is
and so no further
making
all
types of
HkeUhood of 0^2 versus
low or high,
this question. Since
it
many
other two-
seems reasonable
A dilation D P —* Dp is a mapping from V into the set of probability measures on V such that /_ QdDp{Q) = P.
is, D associates with each P £P & non-degenerate probability measure on V with mean P. For our problem,
:
That
Ha
=
S-pDpdfj,p(P). Strassen (1965) has
shown that other
stochastic dominance relations also have this sort of
representation.
Indeed,
it
has motivated work in statistics on "approximate sufficiency" see Le
;
Cam
(1964).
to suppose that particular classes of decision-makers
that appear in economic models
suggests that
it
should be possible to derive other informativeness
pose on the decision-maker's preferences.
many economic
r(u))
=
u{uj,a^)
u{a},a^),
how
—
some
critical level wq-
for instance r(uj)
Using the above approach
payoff functions (r(a;) increasing in
=
and from that
if
might be increasing, or
and only
it is
if
the state variable
is
hard to see how to incorporate
For example, suppose one restricted attention to the class of supermodular
such a restriction.
u*{cj,P)
perspective,
the returns to taking a higher action,
concave, or polynomial in the state of the world, or positive
greater than
example
criteria.
Most important from our
depend on the state
inferences, the
to exploiting any structure one might im-
is ill-suited
models, are assumptions about
—
classes of decision-makers
— care only about making certain types of
Unfortunately, the classical approach
of
— especially the
cv).
and second, even
u{u!,a*{P));
First,
it
if it
did,
is
not clear that this imphes anything about
we would
still
need to connect this property
with a meaningful characterization of "better" information. In the next few sections, we will show
that by placing a total order over the posteriors (such as a stochastic dominance order),
it is
possible
to overcome these difficulties.
Monotone Decision Problems and Stochastic Orders
3
Preliminaries: Stochastic
3.1
Since
we
facts.^
will
make
Assume
extensive use of stochastic orderings,
that
U is some set
probabiUty distributions on R".
written
Dominance and
P" ysD-u P^
we
U
is
first
We
say that
P^
R" —+
R, and
stochastically dominates
P^
let
known
P^, P^ be two
with respect to U,
if
for
the set of nondecreasing univariate functions, then
Stochastic
review a few definitions and
of measurable functions taking
f u{z)dP"{z) > fu{z)dP^{z)
If
Single Crossing
Dominance (FOSD)
relationship. If
U
is
allueU.
ysD-u
is
(SD)
the standard First Order
the set of concave univariate functions, then
>~SD-u corresponds to Second Order Stochastic Dominance (SOSD).
Any set of functions
dominance orders
U induces a stochastic dominance order.
for our
Some critical features of stochastic
purposes can be understood using the notion of a closed convex cone, as
follows.
^For further treatments of this material, see inter
Border (1991) and Athey (1998a,1998b).
alia,
Karlin and Studden (1966), Karlin (1968), Jewitt (1986),
Definition 1
>
any a,j3
If
0,
A
U is a closed convex cone (ccc) if (a)
(h) U is closed under the weak topology.
and
/ udP^ > J udP^
for all
u € U, then the same inequality
closed convex cone generated by
U
A
1
U generates the same stochastic dominance order
weaker notion than stochastic dominance
P" ^sc-u P^
is
U
for
hold for any function v in the
will
P^
and u(z)
= —1
as ccc{U
U
P^
and
are probability
(denoted {1,-1}). Thus,
{1, —1}).^
that of stochastic single crossing.^
We
write
if
^
fu{z)dP^{z)>0
If {1,
=
au + ^v E
implies that
(denoted ccc{U)). Further, since
distributions, then (SD) holds for the functions «(z)
the set
U
u,v E
set
fu{z)dP" {z)>0
—1} € U, then one can show (Athey, 1998a) that )^sc-u
does not contain constant functions, however, )^sc-u
some
sets of payoff functions
crossing payoff functions, which arise in auction
is
will
we wish
games and
(SC)
equivalent to )^sd-u-^^
The
weaker than >^sD-u-
is
between stochastic dominance and stochastic single crossing
of information orders, since
foralluGC/.
become
If
U
distinction
relevant in our analysis
to consider (most notably, single
portfolio problems)
do not contain the
constant functions.
The
stochastic single crossing order is used in our analysis because
statics prediction
Lemma
1 Let U\ he
that for all
P^
about monotonicity of the optimal
a^ >
some
set of functions taking f)
a^, u{oj,a^)
ysc-Ui P^ implies
—
u{(jj,a^)
policy.
—
loosely referred to as
that there exists a selection a*(P)
much
smaller set
implies a comparative
^^
=
x
R— E
>
and
axgraayiaeA jQu{oj,a)dP{u>).
Then
R. Suppose that u{oj,a)
G Ui. Let a*{P)
In fact, just this insight allows us to characterize stochastic
the order induced by a
>
it
:
Jl
from a*{P) such that a*{P^)
dominance orderings, since we can
Eu (for the case of nondecreasing functions, a set
bsU = ccc{Eu U {1, —1}). See Border (1991) or
extreme points, so long
>
a*{P^).
also consider
a
of indicator functions),
Athey (1998a)
for
more
strict inequalities)
have
discussion.
® Various
definitions of single crossing properties (using different combinations of weaJc
and
been proposed by different authors in economics (Milgrom and Shannon, 1994; Shannon, 1995) and
1968).
The
definition in (SC)
involves only
weak
is
inequalities,
statistics (Karlin,
not the most natural variation for comparative statics theorems.
(SC)
is
especially appropriate for working with closed convex cones
However, as
it
and stochastic
orders.
P" and P^ are probability distributions.
To simpUfy the notation and the statement of the result we consider only sufficiency
'"This result hinges on the assumption that
qualifications, the single crossing condition is in fact necessary for
Shannon (1994) and Shannon
(1995).
comparative statics as
here, but
well.
with minor
See Milgrom and
Proof. Define a function v{T,a)
Then v{T,a)
J^u{uj,a)dP^{u)).
{0, 1}
:
is
there exists a selection from a*(T)
x
R—
weak
bivariate
=
=
R, with v{l,a)
>
single crossing in {T,a)
argmaxae^i;(r, a) that
is
=
J^u{iJ,a)dP^{oj) and v{0,a)
and by Shannon (1995),
D
nondecreasing.
Monotone Decision Problems
3.2
Our approach
is
to restrict attention to particular classes of decision problems.
characterized by a restriction on the
DM's
Each
class will
be
payoff function and a corresponding restriction on the
set of admissable information structures.
Definition 2 The pair {U2,T) constitutes a class of monotone decision problems
prior H{ui) and
(MDP-U)
For
moreover,
Ui, then
some
all
if
uG
(MDP-.F) For
u
set
u G
:
fl
Ui of bounded measurable functions taking
U2, if
xM. —^
a^,a^ G
R
and a"
>
all
F eT,
F{u))
satisfy this definition,
U), implies that every
state in
some
(MDP-T),
into R, such that:
a^, then u{u},a^)
-
u{u},a^)
G Ui. And
U2-
=
if(cj),
And
and further ifx^, x^ esupport(F) and x" > x^, then
moreover,
if
F
we
refer to
them
an information structure with prior H{u})
is
and posteriors completely ordered in x by >~sc-Ui
{U21T)
a
some bounded measurable function with incremental returns in
M. is
F{uj\x^) '^sc-Ui F{uj\x^).
If
Q,
if there exists
>
as an
then
F E T.
MDP
pair.
The
first
condition,
DM under consideration has incremental returns to acting that
pre-specified
way
(for
rely
(MDPon the
example, they might be nondecreasing or concave); the second,
implies that for every admissable information structure, the induced posterior beliefs
are completely ordered in the sense of stochastic single crossing.
What do
each
these restrictions
DM u E
buy us? Let
{U2,f^) be an
U2 has an optimal poUcy a*{P) that
by yui-sc- And (MDP-T) implies that
be ordered in this way and indexed by
x.
if
F
G
monotone
is
.F,
MDP pair.
in
P
By (MDP-U) and Lemma
when
posteriors are ordered
then every posterior associated with
Thus, each
DM
u € U2
1,
takes higher actions
F
can
when she
receives higher realizations of the signal.
In this way, the
MDP
restrictions
think of the posteriors generated by
the goal in this paper
is
to
impose an ordinal structure, so that one can heuristically
x going from low
compare
to high along a single dimension. However,
different information structures.
that the posteriors for a given signal can be totally ordered,
compare the posteriors
arising
from
different signals.
8
it
While (MDP-T) implies
provides no guidance as to
how
to
Thus, we introduce a cardinal index for the
by each
posteriors generated
from
This
different signals.
that Tf{x)
=
signal,
is
a€
accomplished
we may "match"
so that
[0,1],
comparison posteriors
a strictly increasing function
vising
Tp
X
:
-^
so
[0, 1],
a.
For an important set of cases, we will show that
it is
appropriate to match posteriors according
to their percentile in the ex ante signal distribution. Formally, for
with marginal distribution F{x), we
F{LJ,F~^{a)) on
f2
X
[0, 1],^'^
=
Tf{x)
let
so that F{oj\a)
on
of two signals,
As illustrated in Figure
[0, 1].
x and
is
F{(jj,
x)
F{u>,a)
=
The probabiUty
of
we can
the "a-percentile posterior."
is
2,
an information structure
F{x). Using this index,
observing a posterior below F{uj\a) in the >^Ui-sc order
distribution
for
given by a, so that F{a)
is
we can use such an index to compare the
the imiform
realizations
y, according their a-percentile.
Using this construction, we can represent a decision-maker's policy function a
terms of the action
let
it
[0,1]
:
—y
A
in
PoHcy functions a{a) can then be
prescribes for an a-indexed posterior.
analyzed without reference to a particular information structure. Using this representation, for any
probability distribution
F onQx
[0, 1]
,
we
define the ex ante expected value from using the policy
a by
V{F,u,a)^ f
f
u{Lj,a{a))dF{u;,a).
(5)
J [0,1] J a
An
important consequence of
(MDP-U)
«(cj,
a{a"))
follows: for
any nondecreasing a{a),
- u{u}, a{a^)) €
C/i
for
This in turn implies that u[uj,a{a)) viewed as a function of
In words, whenever the
DM
uses a
a^ >
(a;,
a^.
a) can be extended to be in
we assumed
for the incremental returns of the primitive
payoff function, and thus u{uj,a{a)) inherits the properties specified by
discussion of this subsection can be formally
Theorem 2
Suppose (C/2,^)
T). Then for any Tj?
Further, there exists
'^We
:
Af
—
>
o-tc
[0, 1]
an
MDP pair
summarized as
and that (A)
holds.
C/2-
follows.
Consider any (u,F) G {U2,
continuous and strictly increasing, F{uj, a)
some nondecreasing a^
can, without loss of generality, take F{x) to
:
[0, 1]
[xojXi]
is
and x*
DM
=x—
be continuous and
not compact).
would experience no payoff
(ii — xo). And F*(x*)
Then F{x) A" — [0, 1] is a
:
»
will
be
strictly increasing.
loss
(F
is
If
from observing i*
strictly increasing (one
bijection
=
F(a;,
Tp^{a)) G
J^.
^ A s^lch that:
provided a construction which shows that one can always take F[x) to be continuous.
interval (xo,a;i], notice that the
C/2.
monotone poUcy, the incremental returns to having a higher
a-indexed posterior retain the properties
The
(6)
Lehmann
F{x)
=
is
(1988, p. 527)
constant o /er some
x on x
<
xo, x*
=
xo on
needs a more involved construction
continuous and strictly monotone), so
if
X
F is well defined.
(i)
For
all
aG
=
(u) V{F,u,a^{-))
a" >
(iii)
For any
(iv)
There exists
Proof,
V*iF,u).
For any
a^ >
So from
Lemma
a^{Q).
(a)
w^
:
x
A
is
R ^ R, u^
G U2, with u^{u},a)
compax;t, for
T
all
P
G
= u{u},a^{a)).
jQu{u},a)dP{u>) attains a
7-*,
maximum on
A.
and (MDP-T) implies that F{u}\a.^) ysC-Ui F{io\Q^).
from
>
The
a<
= maxa^A jQu{uj,a)dF{u}\Tf^{a)). The result then
the definitions of the distributions, (iii) Let a^ = a^{a^), a^ = a^(a^)
But then u{b),oF{a^)) - u(a;,a^(a^)) = u{u^,a^) - u{uj,a^) G Ui by
J^u{iv,a^{a))dF{uj\a)
(i),
and note that a^
Ui.
there will exist a nondecreasing selection from the set of optimizers, denoted
1,
follows immediately
u{uj,a^{Qi)) for
^2
a^, monotonicity of
By
(iv)
- u{oj,a^(a^)) G
a^, u{u},a^{a"))
Because
(i)
(MDP-U).
a^{a) G argmaxa^A Jq u{uj,a)dF{u>\a).
[0, 1],
a^
.
function u^(w,a) can be defined to equal u(a;,a^(a)) for
0,
and to equal
it(a;,a^(l)) for
a>
1. It
follows
a G
to equal
[0, 1],
from (MDP-U) that u^ G U2.
D
3.3
Examples
Our framework covers many problems
classes of
problems and then show
Example
1.
of economic interest. In this section,
how
briefly
is
Nmnerous economic appUcations take
level of
posteriors
this
form
(see
is
Suppose Ui
is
the set of
nondecreasing in the state of the world.
Milgrom and Roberts
FOSD, denoted
we have
P^ ^fosd
P^
>~SC-Ui
P^
if
(1990)); for example,
must be ranked by FOSD, which requires that F{uj\x)
P^
ii F
a^id only if
P^- So (MDP-T) implies that
corresponds to a higher probability that the state of the world
FOSD
u)).
investment for a firm, where the marginal returns are indexed by
Since Ui contains constant functions,
according to
common
the set of functions u{u},a) that are supermodular in (w,a).
In words, the incremental return to a higher action
might represent the
discuss four
the theory can be applied to other cases.
Supermodular (Incremental returns increasing in
nondecreasing functions. Then U2
we
is
is
is
higher than
a
oj.
P^
£ T, the induced
decreasing in x: a higher signal
high.
Figure 3 illustrates the
ordering over posteriors for a 3-state example.
Exeunple
2.
functions.
Then U2
Concave returns (Incremental returns concave in w). Suppose Ui
is
the set of functions
ti(a;,
is
the set of concave
a) that have concave incremental returns.
10
Such a
problems (a G
arise naturally in binary decision
can
payoflF function
decide whether or not to undertake
some
present a hiring problem of this type).
More
generally,
as she raises her action.^^ For this case
P^
SOSD {P^ ysosD
is "less risky."
X and
in
for
So
any
if
F € T,
P^)'- the
then F{uj\x^)
J^^ F(u}\x)du
oj,
is
DM
where the
must
risky venture with concave payoff r{uj) (in Section 5
more risk averse
according to
{0, 1})
u E
ii
P^
U2, then the
ysc-Ui P^
if
DM
we
becomes strongly
and only
dominates
if P-'^
DM takes a higher action when the posterior about
^SOSD
F{lo\x^). This requires that F[a;|x]
nonincreasing in x. Alternatively, for any x^
constant
is
< x^
uj
F{uj\x^)
,
can be attained from F{uj\x^) by a sequence of mean-preserving spreads (Rothschild and Stightz,
1970).
Example
WSC(luo) (Incremental returns weak
3.
functions r{aj) such that r{u})
zero from below at uq.
arise in the context of
on a risky
We
<
for u;
< a;o
single crossing at uq).
and r{u) >
say such a function
for a;
> a;o,
on a
risk-free asset,
a
is
the set of
the functions that cross
WSC(a;o). Payoff fimctions in this class
satisfies
investment imder uncertainty problems, where
asset, cjq is the return
i.e.
Suppose Ui
is
uj
might represent the return
the portfolio weight on the risky asset,
+ (1 — a)wo)- In this case, if P^,P^ have densities p^ ,p^
with respect to Lebesgue measure, then P^ ^sc-Ui P^ if and only iip^(uj) — ^ iV^H p^iui) satisfies
and investor
WSC{uo)
payoflFs are
given by v{auj
ino) (Athey, 1998b).
Thus
if
FG
.;^,
this reduces to
^Hg^j < (>) ^^Ij
w<
as
(>)a;o.
A high signal means that it is more Ukely that the true state u is greater than uq.
Since the class of
WSC(uq)
functions does not contain constant functions, a stronger condition
is
required to order
posteriors
by stochastic dominance. For convenience, define the
set of functions
WSC{(jJq) as the
set of functions r(Lj) satisfying
P^((jS)
—p^{u))
is
WSC{uq).
WSC{ijJq) and
r{u}Q)
=
0.
We
have
P^
^SD-Ui P^
if
and only
if
In particular, the densities must cross at wq, a restriction not imposed
by >sc-Ui
Example
4.
Weak
Single Crossing (Incremental returns single crossing in
set of functions r((J) that cross zero
of
all
from below at some point
to).
Suppose U\
In other words, Ui
cjq-
is
is
the
the imion
the WSC(a;o) sets. Payoflt functions u{u,a) with incremental returns that are single crossing
—
also arise throughout economics
of single crossing functions,
for instance in bidding
we have
P^
>-sc-wsc P^
if
and pricing problems. ^^ For the
and only
Wo, which impUes that (where the densities exist) pP{u)/p^{lJ)
'^Recall that r"{Lj)
is
strongly
more
risk averse
than r^(w)
if
there
is
is
if
P^
>~sc-wsc(wo)
nondecreasing in
some A
>
u).
such that r^(a;)
P^
class
^^^
^
This order
—
Ar^(a»)
is
concave. See Ross (1981) or Jewitt (1986).
''See Milgrom
and Shannon (1994)
for
an extensive discussion of the single crossing property, and Athey (1998b)
for applications in stochastic problems.
11
has received a great deal of attention in economics and
(MLR)
Likelihood Ratio
and
if
order
(P^ ^mlr P^)- Then
a density exists, f(uj\x^)/f{(jj\x^)
Note that the
class of payoff functions
if
F € ^,
nondecreasing in
is
and
statistics,
it is
known
as the
Monotone
F{u\x) must be ordered by
MLR,
ijj}^
with single crossing incremental returns includes the class
of supermodular payoff functions and the class of payoff functions with incremental returns that
But expanding the
are WSC(cjo)-
of information structures that
be ordered by
This
is
comes
set of payoff functions
we can attempt
at
— we have to
a cost
limit the set
to compare. For instance, requiring the posteriors to
MLR is significantly stronger than requiring the posteriors to be ordered by FOSD.^^
illustrated in Figure 3.
More
generally, our theory applies
if
U\
is
some arbitrary closed convex cone of functions, such
as linear functions or quadratic functions or functions with positive n*'^ derivatives. For
case, there
is
a rich theory of stochastic dominance that allows one to characterize the relation
ysD-Ui and hence the
relevant restriction
)
of the cone
any such
We
U\}^
on posterior
beliefe, in
terms of the "extreme points"
leave this pursuit to the reader.
Ordering Information Structures
4
This section contains our main results on ordering information structures.
a sufficient condition for
all
DMs
to another information structure
(MIO)
We
for (f/2,^); it
show that (MIO)
with payoffs u G U2 to prefer an information structure
F gT. We call this condition the
is
not just
sufficient,
when U\
is
but also necessary,
for small (differential)
nondecreasing in
tS)
payoflF
MDPs
log-supermodnlar functions
functions
based on stochastic
(where u
is
inelastic
is price,
and
positive
is
smaller thsin the set of payoff functions with incremental returns
(MDP- U), and
thus our analy-
below does not apply directly to this class of payoffs. To see an example where log-supermodular payoffs
suppose that the action o
firms maximize
(o — c)D{a,iJ),
arise,
and higher states of the world correspond to more
demand.
'*In particular, assuming
all
changes in
induce the same stochastic single crossing order, >~sc-Ui, despite the
that are single crossing. However, the set of log-supermodular payoffs does not satisfy
sis
J^
a closed convex cone that contains the constant functions.
^'Athey (1998b) also shows that the set of log-supermodular
fact that the set of
G
"monotone information order"
Section 4.2 provides a general analysis of informativeness for classes of
is
G
can be characterized using the stochastic dominance order induced by Ui.
the information structure
u{uj,a^)/u{ij,a^)
We begin by identifying
F is
indexed by the
MLR is equivalent to assuming the
F(a;|i)
is
ordered by
FOSD
for
prior distributions H{ui) (Milgrom, 1981).
^^That
is,
we check (SD)
for
some
1966; Border, 1991; Jewitt, 1986;
set of functions
Eu
such that ccc{Eu
Athey 1998a.
12
U {1, —1}) =
U. See Karlin
and Studden,
ysc-Ui
is
when U\ does not contain the constant
Recall from Section 3.1 that
single crossing.
Thus, the ordering based on stochastic single crossing can be
weaker than >~SD-Ui-
when Ui does not
used to compare a greater variety of information structures
functions.
derive
Lehmann's (1988)
and
effectiveness order as a corollary,
contain constant
relate our
work to
Examples are provided.
his.
Monotone Information Orders Using Stochastic Dominance
4.1
We
We
functions,
begin by stating a sufficient condition for
G to be more informative than F to
a given class of
decision-makers.
Theorem 3
V*iG,u)
>
Let (U2,T) he and
V*{F,u) for
all
MDP pair and suppose
ueU2
if for all
>
G{cv\G{y)
If
the assumptions of
greater than
F
returns in Ui.
by
G
in the
Theorem
aG
(A) holds. Consider any
(MIO)
The condition (MIO)
(where "high" means G{y)
(MIO)
a).
we
obtains,
>
is
easy to interpret.
It
write
G
>^miO-Ui F,
i.e.
F}^ This
G
axe lower: observe that
aF(u3\F{x)
<
a)
condition
+ (1 - a)F{cj\F{x) >
< a) ysD-u^
G{u;\G{x)
=
a)
also equivalent to saying
is
H{aj),
is
that for
<
a).
(7)
all
a€
[0, 1],
(8)
Notice that in this condition, signals are implicitly indexed by their ex ante percentile.
motivates us to use the index ax
=
Thus, throughout this subsection,
we
To
see this, note that
/
(MIO)
This
F(x), as described in Section 3.2 and illustrated in Figure
let
An2ilogous to Blackwell, one can interpret
F posteriors.
is
says that the high posteriors induced
the prior. So an equivalent expression to (MIO)
F{u\Fiy)
'
G
a) are on average higher (according to stochastic dominance)
that the low posteriors induced by
H{ijj) is
>
monotone information order induced by payoff functions with incremental
than the corresponding high posteriors induced by
where
Then
J^.
(0, 1)
a) ^sd-u, F{u;\F{x)
3 hold and
F,G G
is
G(ij\G{y)
F{u},a)
= F{u,F~^{a))
(MIO) as saying that the
= a)da
let
G{uJ,a)
G posteriors are
equivalent to saying that for
all
^sD-Ui f F(cj\F{x)
Jo
and
a€
=
2.
G{cj,G~^{a)).
"more spread out" than the
[0, 1],
= a)da
Jo
which can be interpreted in a manner similar to the second order stochastic dominance condition for comparing two
distributions,
F"
and F^, on
[0, 1],
which requires that /^ F"{z)dz <.J^ F^(z)dz with equality when z
13
= \.
Proof of Theorem
imder (MIO),
that
G is
We
3.
any u G
for
C/2,
show that (MIO)
V{G,u,a) > V{F,u,a)
more informative than
F
V*(G,u)
A=
{ai,...,an}
is
ri{u))
with Oi+i
>
Cj.
< a„_i <
q;„
=
V{F,u,a)
1
= Cj
such that a{a)
=11
Jn
imply
see this, observe that
V*{F,u).
Define
= u{u,ai+i) - u{uj,ai).
Consider some arbitrary monotone increasing policy a
...
To
3.
will
nondecreasing, and thus by revealed preference
> ViG,u,a^) > V{F,u,a^) =
is finite,
namely that
any a{a) nondecreasing. This
for
under the conditions of Theorem
a^{a) € axgmaxa jQu{cv,a)dF{(jj\F{x)=a)
Suppose that
actually implies a stronger result,
on
[ai_i,Q:i].
:
—
>
[0, 1]
We
A.
=
can find ao
<
ai
<
Then we have
u{uj,a{a))dF{oj,a)
J[Q,i]
n
.
=
J^u{uj,ai) \F{ai \u;)
=
E[u{u},a{)\
+
I
- F(ai_i |a;)
IYI [«('^'«i+i) - w('^»ai)]
n—
=
E[u{u},ai)]
E[u{uj,ai)]
equalities follow
from (MIO)
since,
+ V(l
/ ri{uj)dF{oj\ax
The
idea in the proof
of action entails a
/ ri{uj)dG{uj\ay
by algebraic manipulation and Bayes'
argument and
new
is
is
>
a,)
>
ai)
for
each
i,
rule.
The
ri{u)
G
=
dF(a;)
V{G,u,a).
inequality follows directly
Ui.
The
case of
A
compact
D
deferred imtil the next section.
very simple. Starting from action ai, every
(interim)
i
-f^
by (MDP-U), we know that
follows via a limiting
- F(ai|w)]
-
- Oi)
i=i
The
1
•
+ ^(1 - ai)
n—l
<
dF{uj)
gamble on
cj,
described by
Formally, payoff's for a given action can be written as the
sum
ri{uj)
=
new increment
u(w,aj+i)
—
to the choice
u{ijj,ai)
G
Ui.
over these incremental gambles:
t-i
u{uj,
tti)
= u{u}, ai) + Y^ rj{Lj).
(9)
3=1
Jumping up to a higher action
taking on the gamble for
all
(say,
from
Oj to Oi+i) at
some posterior indexed by F{x)
higher-ranked posteriors as well, since the
14
= a^ means
DM uses a monotone policy.
The
on ex ante payoffs
effect
is
are described by 'i^sD-Ui-,
ri(Lo)
G than under F.
As
is
no possible way
F
instead of G.
this
for
then given by E[ri{uj)\F(x)
nondecreasing. Since
ai\.
Preferences over the gambles
and (MIO) requires that every such gamble
argument applies when the
more favorable under
is
DM uses the optimal policy for signal 5, there
a decision-maker in the class to do better, from an ex ante standpoint, using
we showed that (MIO)
In the proof,
>
V{G, u, a) > V{F, u, a)
implies
(MIO) has such powerful consequences,
informativeness order. At a
minimum,
it
it
for
any u € U2 and a{a)
might seem to be "too strong" as an
> V{F, u, a^)
seems reasonable to compare V{G, u, a^)
only for policies a^ which are optimal for information structure F. However, for important classes
of
monotone decision problems, we show that
restricting the policy function in this
way does not
permit a weakening of (MIO).
To
why
see
this
might be true, consider the simplest case of
A=
{0, 1}
and some u EU2- The
= Otoa = lata posterior indexed by
F(x) = a^''^. Now consider utility functions of the form w{(j},a) = u{u},a) + aK, where
is
7^
an arbitrary constant. There is no particular reason for the policy of jumping to a = 1 at a^'" to
be optimal for w imder F; however, a DM with payoff w who does use a^'^ (suboptimally) will
optimal policy for u (denoted a^'") entails jumping from o
ii!"
,
prefer
G to F
if
and only
if
the
DM with payoff u does as well:
V{G, u, a^'"") - V{F, u, a^'") - [V{G,w, a^-") - V{F, w, a^-")]
=
Similarly, a
(1
- a) / KdGiu\G{y) >
F{x)
K, we
>a
G to F
-
(1
-
a)
/ KdFiuj\F{x) >
DM with payoff u who uses a^''^ (suboptimally) prefers G to F
V{F,w,a^'^)}^ To extend
varying
a)
this idea,
suppose that
for every
some wk- And thus
using their optimal response to F,
it
will
=
0.
(11)
and only if V{G, w, a^''^)
wk = u + aK G
a =
for F{x) <
K,
potentially can recover any policy of the form
as the optimal policy of
if
a)
(10)
for all payoff functions
{wk €
be necessary that V{G, u, a)
U2-
a, a
Then by
=
1 for
C/2} to prefer
> V{F, u, a)
for all
monotone poUcies a{a).
This heuristic argument can be
made
exact imder the following condition:
(Ui-C)Ui = ccc{UiU{l,-l}).
'^Note that this argviment turns critically on our decision (implicit in
policy functions that
functions a
:
[0, 1]
depend on a
— A based
>
for information structure
posterior's
{MIO))
to
rank in the ex ante distribution, Oy
compare
=
G{y).
G
to
F
using (fixed)
Had we used
policy
on some other index (not the marginal), the policy of increasing the action at ay
G would no
longer correspond to choosing a higher action
faa.
15
when G{y) >
=a
ai and (11) would
-
That
r
€.
is,
Ui
is
a closed convex cone, and
Ui implies that r
u + aK e
The
+K
E Ui
contains the constant functions. Under this condition,
it
any K. Thus - crucially
ioT
our purposes -
for
if it
G
U2, then
U2.
when {U\-C)
following theorem establishes that,
in the information structure,
(MIO)
is
holds, then for small (differential) changes
both necessary and
sufficient for a ranking of information
structures.
Theorem 4
Let {U2,T) be an
parametrized by
6,
with
F^ E
MDP pair such that (U\-C) and (A) holds.
T for
Then ^V*{F^,u) >
all 9.
for
LetF^{co,x) be smoothly
u €
all
U2, if
and only
if
F'^"" yMio-u, F'.
We
Proof. Sufficiency follows from above.
V(G,u,a^)
> V{F,u,a^),
by assuming that (MIO)
there
is
some
a,
A=
fails
G
Then
)f-MlO-Ux F.
<a)> f f{uj)dF{u\F(x) < a).
f{uj)dGiuj\Giy)
=
E[f{uj)\F{x)
u{lo, a)
By
We proceed
the optimal policy for information structure F.
and constructing a contradiction. Suppose
Then compute Ka
{0,1}.
is
necessary for
is
and f £ Ui such that
/
Let
where a^
prove a stronger result, that (MIO)
(f/i-C), since
f eUi, then u GU2.
and zero otherwise. Then V{G,u,a^)
An
=
a],
and define
= a (r(a;) — Ka)
optimal poficy for this
> V{F,u,a^),
if
f u{u,a^{a))dGiu>,G-\a))> f
/
J[o,i]
(12)
Jn
and only
DM
is
to set a^{a)
=
1 if
a>a
if
u(uj,a^{a))dF(uj,F-\a))
f
(13)
J[o,i] Jci
f [f(w) - Ka]dG{uj\G{y) < a) < f [f{uj) - Ka]dF{u\F{x) > a)
Ja
(14)
Jq
But, the expected value of a constant function
is
simply that constant. So (13) and (14) contradict
(12).
For the case of small changes, consider F^, a smoothly parameterized family, and
be the optimal
policy.
Then, by the envelope theorem,
Then apply the above argument
It is also possible to
to compare F^'^^
^ V{F^,u,a^)\g^a
and F^
Hq
:
r(a;)
>
16
a^{a)
^ V{F^,u,a^)
D
at the policy a^.
view the necessity result through the lens of
Consider testing the null hypothesis
=
let
statistical hypothesis testing.
against the alternative
Hi
:
r{u))
<
0, for
some
G
r(a;)
1
—
f/i.
That
a.
—
is 1
However, we impose a constraint on the decision- maker: the test has an average size of
the ex ante probability (over
is,
A decision-maker
a.
all
would then
facing this problem
max
f
/
a(x)G{0,
solve:
a{x)r(uj)dF{uj,x)
=1—a
a{x)dF{x)
/
s.t.
possible signals) of accepting the null hypothesis
Jx
Assuming the conditions of Theorem 4 apply, an optimal policy
a{x)
=
1 if
and only
which, by (MIO),
(1
—
a)
is
if
larger
E[r{(v)\G{y)
•
F{x)
>
>
a.
The
for this testing
ex ante expected payoff will be (1
— a)
problem
will set
>
E[r(uj)\F{x)
•
a],
than the payoff from solving the inference problem associated with G,
a].
Thus, (MIO) can be interpreted as requiring that better information
allows (on average) better inference about the returns to taking a higher action.^"
Examples
4.1.1
We now revisit
two of our examples from above.
Example
Supermodular (Incremental returns increasing in
1.
decreasing functions, the relation
ysD-Ui corresponds
supermodular monotone information order (MIO-SPM)
>
"^FOSD F{u}\ax
oc).
That
is,
If
G
FOSD
F
yMlO-Ui
as
a
if
FOSD. Thus
and only
eictions
increases.
is
aG
So a higher signal
is
to relate
{MIO-SPM)
example. Recall the
to sufficiency.
G
see
why
G bringing about
To do
this,
we return
y be a signal that put equal likelihood on posteriors
MDPs
>
a)
a better match
to our earlier three-
since (0,^,^)
ypoSD
(|, ^,
< (J2 < £^^3. As
i)
(fj^)^)(0,
this alternative
approach
is
essentially equivalent to our necessity proof, let
f,
a
+ A).
17
(defined as in the proof).
Then
li
and
(0, \, \).
Consider an
|,|).
supermodular MDPs. Further, from the discussion in Section
multipUer on the constrained optimization problem for
a{f{w)
G{u}\ay
than under F. Since
DM has uniform prior over the three states ui
admissable for supermodular
also admissable for
To
(0, 1),
in the
"good news" about the state of the world.
information structure z that puts equal likelihood on posteriors (§,5,0) and
is
F
higher than
and the true state of the world.
illustrated in Figure 4, let
Note that y
is
for all
then high signals are (on average) "better news" imder
It is interesting
state, two-signal
if
G
the set of non-
is
G lead on average to higher posterior beliefe than
if F, G € !F, then F{ij}\a), G{ui\a) are increasing in
high signals lead to high actions, one can think intuitively about
between
Since U\
high signals from
high signals from F. Recall from above that
the sense of
to
uj).
Clearly z
3.3,
y and z
A be the Lagrange
in the proof equals
cannot be compared by Blackwell's sufficiency
prefer z to y,
according to
z
i.e.
)~mio-SPM
FOSD. When
The reason
V-
However,
criteria.
is
supermodular decision-makers
all
DM sees a low signal realization from f
the
more spread out
5's posteriors are
simple:
,
her beliefs about the state
are truly pessimistic, while the converse holds for high signal realizations.
we
Figure 5 illustrates this shift in a different way. For each signal,
posterior L,
and the higher posterior H.
SPM)
more probabihty weight on
place
stochastic dominance,
way.
it is
We
{u}2-,L),
and
F using a
that y, z
and
consider the signal structure
(0, |, 5). It follows easily
yMlO-SPM
perturb x so that
it
probability (where e
G
that
x
is
— e,e -f
>
<
0,
x that puts equal
and only
77
>
0,
and e
-f 277
77,
^
—
then we
^),
77)
and
still
(e,
|
—
e:
is
—
x.
77,
^
Moreover,
+
7/)
if
we
with equal
have y ^^mio-spm x, even though
4).
So there
fact that
when
there are three or
more
is
a sense in which
states of the world, there
significant restriction in Blackwell's requirement that the posteriors of the
"bad" signal must
the convex hull of the "good" signal posteriors. If there are only two states of the world,
is
G
robust.
Our examples highhght the
state of the world
if
on posteriors
likelihood
is sufficient for
the two signals are not Blackwell comparable (illustrated in Figure
is
if
admissable for supermodular decision problems and
generates posteriors (|
our information order
F
)~mio-spm
Yet no two-point information structure
X.
be characterized this
in general
"marginal-preserving spread" of the form illustrated in Figure 3.^^
As a further comparison,
(|,0, 5)
(MIO-
In fact, using ideas from bivariate
{(jj^,H).
Results from Meyer (1991) can be used to show that
obtained from
FOSD)
see that signals that are higher according to
show that (MIO-SPM) can
possible to
label the lower (by
a "dichotomy"), then this restriction
can be summarized in a single dimension.
It
is
no longer
|ri|
=
is
lie
a
in
2 (the
severe, since all posteriors
can be shown that on dichotomies (MIO-SPM)
is
equivalent to sufiiciency.
Example
2.
Concave (Incremental returns concave in w).
the relation ysD-Ui corresponds to
becomes
"less risky"
>-SOSD F{uj\ax
>
(by
a):
SOSD)
as
SOSD.
a
When U\
Recall from above that
So
increases.
G
>-mio-CV
is
the set of concave functions,
{MDP-T) impUes
F
if,
for
any a,
high signals lead on average to less risky posteriors imder
that F{u}\a)
G{u:\b.y
G
>
a)
than under
F. Since high signals lead to high actions and high actions lead to interim preferences that are
more
risk-averse,
low risk
will
Interestingly
one can see immediately that having a better match between high signals and
be valuable to the
(MIO-CV)
DM. We
give an example in Section 6 where such preferences arise.
implies sufiiciency
when
there are only two or three possible states of the
^'See also Levy ajid Paroush (1974) and Shaked and Shantikumar (1997) for more discussion of the stochastic
dominance order associated with supermodular functions.
18
world.
we could
Just as
an
find
MDP
pair (172,^) corresponding to any set of incremental return
functions Ui that forms a closed convex cone,
order condition.
Once
again, >^mjo-Ui says that
dominate the average posteriors under
mative than
F
for the class of
relationship induced
we can
F
if
describe the relevant
monotone information
the average posteriors under
(with respect to functions in Ui), then
And we know
problems in {U2,J^)-
by a closed convex cone can be described
G stochastically
G is more infor-
that any stochastic dominance
terms of the extreme points of
in
the cone.^^
4.2
Monotone Information Orders Using Stochastic
In the previous section
we indexed
Single Crossing
and then compared
posteriors by their ex ante percentile
infor-
mation structures, finding that informativeness rankings could be derived using famiUar notions of
be
stochastic dominance. For this approach to
negative constant functions.
function f and
its
When
we
"tight,"
Ui does not contain the constants, but does contain some other
— r, we can extend our
additive inverse
required Ui to contain the positive and
results using stochastic single crossing in
place of stochastic dominance.
We
introduce an additional pair of restrictions (jointly denoted (R)) in order for {U2,J^) to be
admissable:
(R-U) Ui
is
a closed convex cone, and there some
(R-.F) For an r satisfying (R-U), for
that jB[r(a;)|x]
>
b for all
x
F
E
T with
>
such that
functions, then (R) holds with r
containing
We
F
—r G
C/i.
>
such
in support(x).
=
l{u,!»tjo}
(that
r
is,
is
f(uj)
=
1. If
Ui
is
an indicator function
the class of WSC(a;o)
for
a "small" interval
uiq).
begin with a
Lemma
r,
associated signal x, there exists 6
Ui contains the constant functions, then (R) holds with
If
5 Let U2
Lemma
—
G(cj),
and continuous
to observe in passing that
can concatenate our conditions
DM would be better
(according to
that generalizes a key step in the proof of sufficiency above.
(MDP-U) and assume (A)
satisfy
and G, where F{u))
It is interesting
a
all
f(a;)
oflF fifter
FOSD) and one
— as
is
if
the
holds.
Consider two information structures
strictly increasing
DM's marginal
functions
Tp
:
X
^^
[0,1],
returns are both nondecreasing and concave,
we
often done for straightforward preferences over gambles. For example, such
a sequence of two changes to F{ij\ax
that reduces risk (according to
19
>
SOSD).
a),
one that makes higher states more Ukely
-^
Tc-.y
[0, 1].
Define Tn(a;,a)
=
G{u},T°-\a)) - F{u},Tp^{a)). Then
u{oj,a{a))dm(uj,a)
//
>0
(15)
In J [0,1]
for
u EU2 and
all
all
A
a :[0,i\ -^
nondecreasing, if and only if for
I
Jo,
An
important thing to note
when
checking (16)
=
r{cv)
E Ui only
=
and Tciv)
When
f,
if
Pr(rG(y)
<
=
ct)
a€
(0, 1),
(16)
<
a)
<
choice of index in the last subsection can be immediately understood:
for all r
all
not necessarily a probabiUty distribution on u;
is
checking that Pr(rG(y)
1 entails
eUi and
r{uj)d^m{co,a)<0.
that m{-,a)
is
r
all
Pt{Tf(x)
<
a),
which
Ft{Tf{x)
if
<
a). Thus, our
{1,-1} € Ui,
(16) holds
—
always true when Tf{x)
is
F{x)
G(y).
—f G
C/i,
then (16) requires that
f
=
rd^m{uj,a)
0.
(17)
Jn
This motivates our new choice of indexing functions, Tp, Tq:
^-(^)This choice of
Tf,Tg guarantees
that (17) will hold, meaning that
condition under which (16) will hold for
If
Ui
Tp
:
is
r
G
C/i,
r 7^ r. If r
the class of WSC(a;o) functions, then Tf{x)
X —^
[0, 1] is
Theorem
V*{G,u)
>
6 Let {U2,T) be an
V*{F,u) for
Tf and To
=
=
1,
it
remains only to identify a
= F{x)
then Tf{x)
as above.
F{x\uq). In general, {R-J^) implies that
strictly increasing.^^
all
MDP
ueU2
pair,
if for all
G{oj\TG{y)
where
all
^''^
-^^^^-wr-
/,r(.)dF(.)
>
suppose (A) and (R) hold.
ae
Let
F,G G
Then
J^.
(0, 1),
a) ysc-u, Fiu\TF{x)
>
(MIO')
a)
are defined by (18).
{MIO') generaUzes {MIO) by requiring that the average posteriors be ordered using stochastic single crossing rather
than the (stronger) stochastic dominance order. To interpret
stochastic single crossing requirement can be restated as follows: for each a,
E[r{uj)\TG{y)
And
>
a]
>
=>
E[r{ui)\TF{x)
>
a]
continuity can be ensured (without changing the information content) using
20
>
0.
and
for all r
this,
€
the
C/i,
(19)
Lehmann's (1988) construction.
>
This contrasts with stochastic dominance, which requires E[r{o:)\TG{y)
If
ot]> E[r{u>)\TF{x)
>
a].
Ui contains the constant functions, then stochastic dominance and stochastic single crossing
To apply Theorem
it
6,
is
useful to have
checking only a single inequality: for
E[r{uj)\TG{y)
>
Lemma
Proof.
only
if
It
all
first
can be shown^"* that when Ui
>
G
of (20)
is
>
>
>
Pt{Tf{x)
a]
result
is
is
>
equal to Pr(rG(y)
So, (20)
G.
and {MIO"), and
implied by (MIO").
a closed convex cone, then
each a, (19) holds
for
>
-
a]
X{a)E[r{ij) \Tf{x)
>
a]
>
if
0.
and
(20)
>
a]
>
0,
and the left-hand
— r,
the minimized value of the objective). But, checking (20) for r and
likewise for
{MIO')
a).
establishing the equivalence of {MIO')
subject to the constraint that E[r{uj)\TF{x)
ci]
{R-U), implies that A(a)
(18)).
E[r{u})\TF{x)
(A(a) can be interpreted as the Lagrange multiplier in the problem of minimizing
C/i-
E[r{uj)\TG{y)
and r eUi,
such that
E[r{Lj) \TG{y)
for all r
an
(0, 1)
>
a)
show that the
5 to
there exists a X{a)
a€
>
Pr(TG(y)
a]
The proof of Theorem 6 proceeds by
then applying
G >^mio-Ui F even when appljang (MW).
alternative condition for (MW) that requires
we maintain the notation
coincide. For this reason,
is
=
E[f{u>)\TG{y)
Q:)/Pr(ri?(x)
>
decision policies
a
[0,1]
:
a]/E[f{cj)\TF{x)
>
a].
But, the latter ratio
is
To complete the
proof, let F{uj,a)
equivalent to the following: for
all
mG
> V{F,u,a{-)). So for all u G
> V{G, u, a^) > V*{F, u).
-^ A, V{G,u,a{-))
(monotone) poHcy for u under F, V*{G, u)
We now show that (MW)
is
both in Ui by
is
in turn
a) (simply substitute in from the definitions of Ti? and
equivalent to {MIO").
By Lemma 5, {MIO")
>
tight for a larger class of decision
=
Tg
in
F{u},T^^{a)) and
and
f/2
U2,
side
if
a^
all
is
monotone
an optimal
D
problems than that identified in
Theorem 4 above.
Theorem 7
Let {U2,J^) be an
parametrized by
6,
vnth
F^ £
MDP pair and suppose
T for
all 6.
(A) and (R) hold. Let F^{oj,x) he smoothly
Then ^V{F^,u)
>
for
all
u G
U2, if and only if
F'+'^ yMio-u, F'.
Proof. As in Theorem
V{F,u,a^), where a^
is
4,
we prove a
stronger result, that
(MIO)
is
necessary for V{G,u,a^)
the optimal policy for information structure F. Suppose
*See Jewitt (1986), Collier and Kimball (1995), or Athey (1998a) for alternative proofs.
21
G )/-miO-Ui
>
F.
Using the representation {MIO"),
/
this implies that there
A=
{0,1}.
Then
some
and r E Ui such that
a,
> f f{u:)dF{u:,T^\a)).
Jn
fiu;)dG{uj,Ta\a))
Ju,
Let
is
define
Ka = E[riu)\TFix) = a]/E[f{u)\TF{x) =
The denominator
By (R-U), u G
is
DM
optimal policy for this
V{F,u,a^),
and only
if
/
Then
non-negative by (R-J^).
is
= a{f{cj) — Kaf{uj))
E[f{uj)
—
to set a^{a)
Kar{Lo)\TF{x)
=
f [f{uj)
single crossing in a, so that
is
zero otherwise.
> [
J
[
[0,1]
Tp and Tq
So (23) contradicts
n{^,a''{a))dF{co,T^\a))
(22)
Ja
(21).
As
so that J^f{u)d^G{uj,TQ^{a))
in
Theorem
4,
=
(23)
J^r{u))d^F{u},T^^{a)) for
{U2,T) are an
MDP
pair corresponding to Ui,
information structure are not required by
unless Ui contains the constant functions.
{MDP-T)
If,
all
necessity for the case of a smoothly parameterized
D
distribution follows from an application of the Envelope Theorem.
If
an
Then V{G,u,a^) >
- Kafiu;)]dG{u,TaHa)) <
f [f{uj) - K^f{ijj)]dF{u},Tp\a))
Jn
But, we have defined
a.
a]
if
J [0,1] Jn
JQ
=
a > a and
1 if
u{oj,a''{a))dG{u:,T^Hc^))
/
a].
define
u{LJ,a)
By (MDP-T),
U2-
(21)
and
F,G E
J^,
the posteriors induced by each
to be totally ordered
by stochastic dominance
however, they do happen to be ordered by stochastic
dominance, we can show that (MIO) and (MIO') are equivalent.
(MDP-SD)
For
all
F
E T,
if
x", x^ EsuppoH(F) and
an
MDP
x" >
x^, then F{u\x") ysD-Ui
F{uj\x^).
Proposition 8 Let (f/2,^)
F.GeT.
Then (MIO)
is
be
(MDP-SD)
hold.
Let
equivalent to (MIO).
Proof. Note that J fdF{uj\x) and
that
pair and suppose (A), (R), and
j fdF{u}\x) = JfdF{u))
is
— j rdF{ijj\x)
constant in x.
are both increasing in x, so
And
22
likewise for
it
must be the case
G. Simplifying the expressions
from
(18),
Tf{x)
=
F{x), Tg{x)
equivalent to (MIO'))
The
holds,
is
immediately that (MIO") (which we know
It follows
when the
Tf{x) reduces immediately to F{x), our
in x. It follows that
posteriors are ordered by
ysc-Ui, E[f\x] can vary with
earlier
x.
Examples
4.2.1
We now interpret the monotone information orders corresponding to Examples 3
In the process
Example
we
relate our information orders to
<
r{uj)
(>)0 as
Then (MIO') reduces
<
a;
(>)0.
So r
=
efficiency criteria.
single crossing at u>q).
l{o;«a;o}>
and 4 from above.
=
and Tf{x)
Recall that
r
if
is
F{x\ujo) using Bayes'
to
oj)
<
FT(F{x\uJo)>a
Pr(G(y|a;o)>a;|t<;)
>
Pi{F{x\u}o)>a\oj)
Pr(G(y|a;o)>a
Alternatively, F{F-'^(a\uJo)\uj)
interested in
Lehmann's
WSC(ujo) (Incremental returns weak
3.
WSC(a;o), then
Rule.
is
equivalent to (MIO).
must be constant
index. In contrast,
G{x).
because when posteriors are ordered by stochastic dominance and (R-U)
result obtains
£'[f|a;]
=
knowing whether
-
|
G{G~'^{a\cJo)\uj)
is
\
<
loq
u>
ojq
for co
u>)
for
WSC(ujo). Recall that the
w is above or below uq — and (MDP-T)
DM
essentially
is
ensures that high signals are
u being above uiq. The information condition implies that if the true state of the
then the probability of receiving "good news" under G (defined asy > G~-^(a|c<Jo)) is
"good news" about
world
u)
> uJo,
higher than under F. If
y, f{u}o\x)
= ^(wojy) =
we impose the
additional assumption
h{u}Q) (the prior).
(MDP-SD),
is
on a high
states are
low states are
to F, while neither signal
is
less likely
is
WSC(u}q)
is
ujq) is
more
WSC(u>o) can
also
uq against the
constrained to have a size of 1
—
a:
—
a. Thus,
we
reject
Hq
when
:
F{x\aJo)
equivalent to requiring that F{F~^{a\u)o)\u})
greater for
G
likely
That
under
is,
conditional
G as
opposed
be interpreted in terms of hypothesis
=
have the following interpretation.
<
for all a.
u}
rejecting the null is 1
(w
for
Consider testing the null hypothesis
However, the test
and high
WSC(u!o)
informative about the state uq.
The monotone information order
testing.
follows that for all x,
Then, conditions (MIO) and (MIO') are equivalent, and
we can simply check that g{Lu\G{y)>a) — f{u}\F{x)>a)
posteriors,
it
The
conditional
>
—
a,
and
alternative
:
u <
loq.
on Hq, the probability of
likewise for G.
G{G~^{q.\u}q)\u})
probability of rejecting the null
Hi
when the
if a;
Since
>
MIO-
(<)a;0)
alternative
is
we
true
than F; and the probability of rejecting the null when the alternative
23
is
false
>
(w
powerful for a given
which
To
will
size.
This
is
Gis uniformly more
test using
Lehmann
exactly the interpretation given by
example where
problem discussed in Section
F
(1988) in his analysis,
more informative than
is
4.
can be apphed, return to the portfolio allocation
this order
G if
worthwhile.
is
Single Crossing (Incremental returns single crossing in
Ukelihood has been derived by
Lehmann
problems (G )^l F)
For this
provides a more powerful inference about whether or
it
tion order for the case of single crossing payoff functions
this class of
= v{cujJ + (1 — a)ujQ).
where payoffs axe given by u{u,a)
3.3,
not investment in the risky asset
Example
than F. Thus, the hypothesis
be discussed in more detail below.
see a very simple
problem,
G
smaller for
ujo) is
(1988).
for all
if
G
a;
and signal distributions with monotone
He showed
A",
that
G is more informative than F for
G~^{F{x\uj)\lj)
We now
(1997) has given further equivalent characterizations.
The monotone informa-
uj).
is
nondecreasing in uj?^ Jewitt
show that Lehmann's order can be
obtained as a special case of our result.
we noted above that the
Recall that
some
ijjQ
was the union of
all
the WSC(a;o) sets.
recover the information preferences of
Theorem
(Hi)
By
quantifying over
all
crossing points
ujq,
we can
with singe crossing payoff functions.
G >l
(i)
F;
G
(ii)
>"Af/0-VKSC(i^o)
^ /'^^
^^^
^^ ^ ^'
G ^MIO-SPM F for all prior distributions H{uj).
then F{u}\x)
is
ordered by MLR..
priors
G
yMlO-wsc(u,o) ^-
WSC{a}o) to hold
and taking
Now
And
is
increasing in ysc-wsc(u}o) ^^^
similarly,
if
F{uj\x)
and only
if
FOSD, we have
F{F~^{a)\C:
F{o}\F{x)
<
ui)
<
<
-
(i)
and
is
and
a)
-
by
(ii)
in
cj.
FOSD
as
as
x
is
Letting
is
increases,
x increases
are equivalent.
G-'^{F{x\ujo)\uo)
Suppose
WSC(uo).
x
=
for
For
F~^{a\ujo),
WSC{ujo). This
exactly Lehmann's
increasing in
cj,
which
Suppose
G
)^mio-SPM F. Then using
(iii).
<
G{G~^{a)\il)
G{oj\G{y)
<
io)).
<
Letting
is
Applying Bayes' Rule,
a).
x
=
F~^{a) and taking
(1988) also considered necessity, although his theorem formally answers a slightly diflFerent question.
finds that the efficiency order
is
necessary and sufficient for one signal to return higher payoffs than another in
every possible state (instead of on average, given the prior). The discussion following
outline of
^^^ liJo
G{G-'^ {a\uo)\uj)
must be increasing
G~^(F(x\u!)\lij)
(ii)
increasing
show that
each term, we have G-'^{F{x\oj)\cjj)
ojq if
this is equivalent to
is
This means that F(F-^(a|a;o)|a;)
consider the equivalence of
the definition of
Lehmann
F{uj\x)
for every ljq, the expression
G~'^{-\uj) of
hold for every
condition.
if
F is ordered by MLR.. We first
H, then
all
He
DMs
9 The following are equivalent:
Proof. First note from above that
will
once from below at
set of all functions that cross zero
Lehmann's approach.
24
Example 3 above provides an
G
<
^{\Cb
This
see
we
of both sides,
implied by
is
To
Lo)
G >-l F,
how Lehmann's
see this
and quantifying over
Of
course,
The
order, z
y,
>-£,
is
none of the three are ordered by
two-point priors,
MLR,
so
positive only in
Or
<
<
uj)\ui
it
implies
oj)
<
G~^{F{x)).
D
>-£,.
is
^mio-spm
^.
Using
Lehmann's order does not apply to
it.
sufficiency.
analysis can be extended to other payoflF classes.
again yL.
^{F{x\Cj
example, recall that z )^mio-spm V
but x does not satisfy
Athey (1998b) that when U2
order
all
G
order can depart from the supermodular monotone information order
in practice, return to Figure 4. In that
Lehmarm's
equivalent to
is
For instance,
it
follows from results in
the set of log-supermodular functions the appropriate information
in another example, consider the set of payoffs with incremental returns that
some intermediate range between
loq
and
lj'q,
where
cvq
<uJq. This case works out
similarly to single crossing at a point.
5
Ordering Payoff Functions
We now provide conditions under which one decision-maker
has a higher incremental return to im-
proving her information than another decision-maker. Persico (1997) has investigated this question
for the case of single crossing payoff functions
is
single-crossing in
to,
the
mation than the second
functions.
We
if
§^u{uj,
generalize this result.
family of distributions which
If
is
To do
this,
we
first
Umit our attention to marginal changes
two signal distributions are linked by a smoothly parameterized
information-ranked
all
along the
way from
compare the incremental return to increasing the signal strength from
We need a small amount of notation:
let
9
from J^u{(x},a)dF^{aj\a), and
Suppose {U21T)
parametrized on Q, with
F^ ^
is
an
let
u^(uj,a)
By the
T for all 6.
envelope theorem,
to G,
we can then
F to G.
Let
aP''^{a)
6 e
:
be a nondecreasing
= «(w,a^'"(a)).
MDP pair and that the family {F^{u>,x) lOeGjis
smoothly
Let u,v be bounded measurable payoff functions.
somee, u^{u,a)-v^{u},a) E U2, then §gV{F^,u)
Proof.
F
be a convex subset of R, and suppose {F^{u), x)
6} is a family of information structures smoothly parametrized by 9.
Theorem 10
§^v{u}^ a^{a))
according to Lehmann's information order for single crossing payoflF
in the information structure.
selection
a^ip))—
decision-maker (u) benefits more from a small increase in infor-
first
{v),
and has shown that
> ^V{F^,v)
If,
for
at 6.
we have
d
-V{F',u)= f f
u(a;,a^'"(a))d
«/Sz •/[0,1J
25
i,F^i^,a)
(24)
So letting w{uj,a)
=
u{uj,a^'^{a))
— v{u),a^''"{a))
—
and m(ui,a)
-V{u-F')--^V{v;F')^ [ [
Oif
J CI J
de
^F^(a;,a), we have
w{u,a)d^m{cv,a).
(25)
'
The assumption
[0,1]
F^
that
increasing in
is
^ increases implies that for any r{uj)
^miO-Ui^^
G Ui,
a'e(0,i)
r{uj)d^m{uj,a)>0.
I
(26)
Jn
Lemma
5 then implies that (25) evaluated at 6
Theorem 10
says that
if
—
u^
v^
in U2,
is
is
i.e.
if
u^
is
maker with payoff function u has a higher marginal value
The theorem
function v.
is
does not require that u,v
nonnegative for each agent, although
this is not the case (since policies
If
u
more
is
it is
"more U2", than
for information
v^,
then the decision-
than the
DM with payoff
E U2, or that the marginal value of information
somewhat hard
to imagine applying the
Theorem when
might not be monotone)
sensitive to information
compare changes
D
nonnegative.
than v in response to every signal in the family, we can
Prom
in information that are not marginal.
this,
we can derive comparative statics
on the amount of amount of information acquired.
Theorem
11 Suppose the conditions of Theorem 10 are
U2 for every 6 € @, where
let
6*{u)
Proof.
=
Q
is
argmaxfige V{6;u)
Let ^(7^;^)
=
a closed interval. Let
— C{0). Then
V{u;F^), and ^(7^;^)
supermodular in (^,7), so V(6,'y)
Theorem,
A
0*{'y) is
9*{u)
— C{6)
is
C
satisfied,
— R
>
:
6*{v) (in the strong set order).
=
V{v;F^). Applying Theorem
supermodular in (^,7).
is
By Topkis'
€
and
10, V{e;-f) is
(1978) Monotonicity
nondecreasing in the strong set order.
that the conditions on
u and v
the objective function evaluated at
require
{^,0()
6e the cost of information;
>
few words are in order about the results of this section.
and 11
and thatu^ {uj,a)—v^
its
A
main drawback to Theorems 10
are not primitive, since they depend on the properties of
optimum. In
general, verifying the necessary condition
may
knowing something about the shape of the optimizer, or about the curvature of the payoff
functions.
To take an apparently simple example, suppose
supermodular.
One can
where information
that a^{a;u)
>
is
verify that sufficient conditions for
u{u},a)
u
=
to acquire
+ g{a),
where v
is
more information than
v,
v{u},a)
ranked by the supermodular information order, are that a^{a;u)
a^{a;v), and v^aa
>
0, (i.e.
marginal returns are supermodular).
26
>
a^(a;v),
The
first
condition will hold
if
g{a)
last is (reasonably) transparent,
the second condition requires significant assumptions on the curvature of u.
linear,
should we draw from this? While
may be
necessary to place a fair
On the other hand,
this area.
we think
but even
What
if
g
conclusion
that these results hold promise for applied modehng,
amount of structure on the model
to
make them
is
it
operational.
the existing hterature (prior to Persico (1997)) provides minimal guidance in
For example, an increase in the Blackwell information order increases the ex ante expected
value of any function which
is
ordered information than v
is
and the
increasing,^^
is
convex in the posterior
if
and only
if
- v*{P) =
w*(P)
Thus, agent u
beliefs.
will
J^ [u(a;,a*'"(P))
buy more Blackwell-
- v{a!,a*'''{P))] dP{u)
convex in the posterior P{uj). While convexity of u*(P) and v*{P) follows as a simple consequence
of optimality, convexity of the difference
not at
is
all
transparent. Checking the condition would
almost certainly involve non-primitive assumptions similar to the ones described in this section.
Applications
6
We now present
several applications of the results developed above.
The
first
two applications axe
standard decision problems; the third examines adverse selection in a labor market equilibrium,
The examples demonstrate both
while the fourth studies a coordination game.
weaknesses of the approach.
strengths and
In particular, while comparisons of information structures
may be
obtained immediately in a broad range of cases, comparative statics on information acquisition
often require additional assumptions.
Application: Cost Uncertainty for Producers
6.1
A
growing literature in Industrial Organization considers the value of information to firms in
oligopoly models (see for instance
The methods outUned above
Mirman, Samuelson, and Schlee (1994) and
allow for
some new
Consider the problem faced by a producer
jective.
results in this general class of problems.
who must
choose quantity,
We will compare the importance of information to firms
However, we
will restrict attention to covert
of other agents fixed
when we analyze the
we
will consider /3
social planner's total surplus.
To
see this, let u{ui,a,T)
for every w, so
The
= i;(ai,a)
to maximize
diflterent
some ob-
market structures.
we hold the
strategies
of gathering information.
Write the firm's gross surplus as a general function, R{q;
structure. In particular,
under
q,
information gathering, whereby
eflFects
references therein).
€ {M,D,S},
/3),
where
(3
parameterizes the market
representing monopoly, duopoly, and the
firm faces imcertainty about its cost: the total cost of producing
when t
= 0,
and
v{uj,a)
a*(a,T) will be increasing in (a,T). That
is,
27
+g{a) when t
o*(a;«)
>
=
1.
Then u
a'{a;v) for every a.
is
supermodulax in {a,T)
q
=
given by C{q;uj). Thus, the firm's profits are given by Tr{q,uj;P)
is
R{q\P)
consider a smoothly parameterized family of information structures, {F^{u},a)
F^{a)
= a
denoted
tt (g,
for
each 6 and
a; P)
=
R{q;
a e
all
Expected
[0, 1].
— E[C{q, oj)\q]- Assume
/?)
profits given
:
—
^
C{q;uj).
We
€ G}, where
an a-percentile posterior are
that each of these functions
is
differentiable in
Under appropriate assumptions, the optimal choice of quantity is then determined by MR{q; P)
q.
j-^R{q;P)
=
=
f^E[C{q,u)\a]=E[MC{q,u:)\a\.
Suppose that 6 indexes F^(a;, a) according to {MIO-SPM), and that a indexes F^{ui\q) according to
FOSD
for this problem. If
{q,(^).
Since
We first
for all 6.
apply our analysis from above to characterize increasing information
we assume that MC{q; a;)
{MDP-T)
information (according to
decreasing in
cu,
then
7r(g', uj;
(MIO-SPM))
cording to
SPM.
FOSD. Assume
Finally,
higher for firm
nondecreasing in w.
nondecreasing in
sider
is
9,
a^ indexes F^(a;|a^) ac-
>
is
decreasing in w, increasing in
q^{a) and q^'{a)
>
q^'{a).
Then
q,
is
ordered by
MIO-
and submodular in
the marginal value of 6 is
than firm P^.
Proof. By Theorem
MR{ql^{a); P)
MC{q; u)
assume that q^{a)
p^
supermodular in
under different market
that the smoothly parameterized family {F^(a;,a)}
In addition, assume that
(uj,q).
is
model described above, where for each
the
is
leads to higher ex ante expected payoff's.
We now turn to consider how the marginal value of information changes
conditions. We begin with the following result.
Proposition 12 Consider
P)
the optimal choice, q^{a;6), will be nondecreasing in a. Better
satisfied,
is
is
10,
it
suffices to
If g^' (a)
u
>
show that Kg{q^{a),uj;P^)q"'{a)-TTq{q^(a),u);P^)q^'{a)
^^'(q:),
(since each terra
— MC{q^{a),u)
whether MC{q^{a),Lo)
for
then our result obtains
is
if
7rq(g^ (a), w;/3''^)—7rg(9^ (a), a;;
^^)
separately nondecreasing). Recall that 'nq[q^(a),w,P)
each firm. Since the
— MC{q^{a),u))
is
first
is
=
term does not vary with w, we can con-
nondecreasing in u, which follows since MC{q,cj)
submodular.
To
To
interpret the conditions
interpret the conditions
it is
useful to consider
Now
it
satisfied if C{q,u))
= q^/u).
on the quantities chosen, which of course are not primitive conditions,
some examples.
of the social planner. Let R{q;
As usual,
on the cost function, note that they axe
P^)
First consider
= P(g)
follows directly that q^{a)
q and R{q; p^)
> q^(a), that is,
consider the terms g^'(a) and q^'{a).
«/,,_
=
/'(a)
•
comparing a monopolist's problem with that
The
=
J^ P{t)dt.
the monopolist imderprovides quantity.
implicit function
theorem yields
£E[MC{qP{a),u;)\a]
^^MR{qP{a);P) - ^^E[MC{qncc),u)\a\
28
>
Thus, q^'{a)
P'(q^{a))
q^'{a)
if
MC{q,uj)
submodular
is
> 2P'{q^{a))+q^{oL)P"{q^{a)).
latter inequality
is
the
If
P is
in (g,a;),
if
MC{q,oj)
convex
is
in q,
and
if
linear or convex, a sufficient condition for the
commonly assumed requirement that the marginal revenue curve
is
steeper
than the demand curve.
We can also compare the value of information for a monopoUst and a duopolist. We suppose that
both duopolists have the same
initial signal
structure as the monopolist, but only one duopoUst.
has the opportunity to (covertly) purchase additional information
information acquisition game, but
assumptions, the result q^{oi)
Further, q^'{a)
>
q^'{a)
if
2P'{q^{a))
This condition
is
>
we
will
(it is
not consider that case here).
With
q^{oi) obtains so long as marginal revenue
marginal cost
is
submodular, and
+ q^{a)P"iq^{a)) >
study the
also possible to
is
these
symmetry
downward
sloping.
if
2P'(2g^(a)) +g^(a)P"(2g^(a)).
satisfied for the constant elasticity
demand
curve.
Summarizing, we see that imder several reasonable conditions, the social planner has a larger
centive to acquire information than the monopolist, but the monopolist
may
in-
have a larger incentive
to acquire information than the duopoUst.
Application: Screening for a "TEu-get" Hire
6.2
An
entrepreneur or manager needs to hire a worker or subcontractor to perform a very specific
task.
She interviews an interested party, who appears to have roughly the right
The manager now has
to
gleaned from the interview.
a large
oflFer?
make a
What
And what would
To model such a
it
situation,
single take-it-or-leave
sort of signal
mean
it's
co
moves away
in
monetary
from the interview
we assume that there
both directions. So r(w)
is
be more
which achieves a
which
is
increasing.
effective?
maximum
The manager has a
A
a way that the manager
shape of p and
r, it
will offer
will
—a]. In order
more money following a "good"
K the
Hiring
at uj*,
and
offer
x about
a
cu
will
be
(arising
— a\x]. We can write
for signals to
be ordered in such
signal, regardless of the exact
must be the case that x orders the posteriors by
Section 3.3 for interpretations).
a^{x)
= p{a)[r(oj)
wage
signal
from an information structure F{u},x)), and chooses a to maximize p{a)E[r(u})
the payoff function u{u!,a) as u{u},a)
co*.
concave (but not necessarily symmetric
not just the distance from perfection that matters, but the direction).
ax:cepted with probability p{a),
manager to make
some optimal task outcome
r{iv),
is
based on information
offer
will cause the
for the screening process to
the agent would result in outcome w, and a payoff of
decreases as
it
characteristics.
SOSD
(see
Example 2
in
potential signals from the interview satisfy this condition,
be monotone regardless of the exact shape of p and
29
r.
Now
for the question of
what constitutes better information. Applying (MIO-CV),
F if Va €
more informative interview that
G{oj\G{x)
The
intuition
but there
is
is
>
[0, 1]
a)
residual uncertainty.
the less risk she believes there
informative screening process
Fiuj\Fix)
is,
is
if
the agent
is
hired
like this residual risk
one where high signals (which are more
6.3
Application: Adverse Selection
suggests
a).
is
always
how our
in particular,
likely to lead to hires)
information orders can be applied to adverse selection mar-
is
unknown with
prior H(uj),
—a
the top 1
if
(u),
x)
is
observed by the school, but not by the general labor market. Instead, only
is
fraction of the class "graduates," while the rest
— the labor market observes
do not
a given worker graduated. Each firm has production function J{uj) increasing in u, and the
,
labor market
is
available to the
competitive so that workers receive a wage £J[J(cj)|J], where 3
Now
the information
graduates will be E[J{u))\F{x)
for
>
a],
and
for
non-graduates E[J{uj)\F{x)
<
suppose that the schooling process becomes more informative about the worker's abiUty,
in the sense that
F increases to G in the supermodular monotone information order
E[J{uj)\G{y)
while the wage decreases for those
interprets a failure to graduate as
is
economy
(MIO-SPM).
immediately that the wage for graduates will increase since
It follows
process
is
market about productivity.
The market wage
.
first
and schooling
generates a noisy signal x about underlying ability. Suppose that the joint distribution of
a]
more
and Labor Markets
period training (in school). Each worker's productivity
only
A
Consider the following stylized situation. Workers hve for two periods and spend the
F{uj,x). This signal
u)*,
—
the more aggressively she will pursue the candidate.
less residual risk.
Our next example
>
outcome
The manager does not
imply
kets.
^sosD
simple: after the interview the expected
some
G will be a
more
is
revealing.
constant at
To extend
this,
are "skill-sensitive"
>a]> E[J{u)\F{x) > a],
who do
not graduate.
bad news about
=
is
that the labor market
and as worse news when the schooling
is,
1
— /? of workers,
have production function
E\J{ijj)\
for the
whole
but inequality increases with information.
suppose now that a fraction
suppose that E[S{u!)\
point
Note that the average wage (and average production)
£'[J(a;)],
— that
ability,
The
5(a;),
< a can be hired into jobs that
with S"(w) > J'{u>) for all u. And
/?
so that "on average" productivity
30
is
the same at the two jobs.
Assuming that each worker gets her expected marginal product as a wage, the equiUbrium
two-job labor market has a fraction
>
jobs at a wage E[S{oj)\F{x)
a],
j3
of workers,
all
whom
of
in this
graduated, going to "skill-sensitive"
the remaining graduates taking less skill-sensitive jobs at a lower
wage (they are rationed), and non-graduates receiving E[J{ij)\F{x)
<
a] as before.
Consider an increase in the information generated by school screening in the sense of (MIO-
SPM). This
for
graduates and decrease the wage for non-graduates. But
it
production of the economy by leading to better matching between high-
will also increase the total
skill
wage
will increase the
workers and skill-sensitive jobs. Moreover, as the fraction of skill-sensitive jobs,
/?,
increases
toward a, the social returns to better screening by the schools increase (assuming the planner cares
only about gross production and not inequality). Similarly,
supermodular
S{(jj,t) is
if
in r,
then
the social returns to better screening will be increasing in r.
Application: Coordination
6.4
Under Uncertainty
The game
This section considers a game where both players can choose to acquire information.
involves
signal
two players with symmetric payoffs «(a;, Oi, aj) where
ttj,
=
with joint distribution F^'{u},ai), where F^<(q:j)
Assimie the
ctj's
and
r
restrict
quality (^1,^2)
are independent conditional on u.
> 7 >
0.
We
let u(a;,
their strategies (01,02).
We
aj
i's
and
01,02)
action. Player
i
receives
6i reflects signal quality.
=
ai[a;
+ 702 + K] —
make unobserved
In this supermodular game, players
and then choose
player
Gj is
^ral,
choices of signal
look for symmetric Nash equilibria.
Conditional on the information structure, Player 1 maximizes
+ 'yE[a2\cL>; e]+K]-
ai[cj
Jn
to find
first
question to ask
=
is
be ordered according to
-f-7£?[o2|ck:i;^]
FOSD is sufiicient;
The condition
for
In the unique (essentially)
be monotone in
all
that
is
needed
aj. Clearly requiring
is
that the posteriors
moments. Since any two distributions can be ranked according
{MDP-!F)
is
simply notational.
are of the form Ua{u})
satisfying these restrictions forms a
will
actually
The
linearity of payoffs arises since
= 73^(0; K], and so Ui{ai,u}) = a-:p^[u} K].
= c + duj, where d > 0. The set of marginal returns
j plays her equilibriimi strategy, £^[aj|w]
Thus marginal retioms
+ K].
+ K].
when the optimal poUcy
their first
to their means, assimiing
^ [S[a;|ai;0]
^r^r [£^[a;|ai; Oi]
that the posteriors are ordered by
when
=
an optimal action ai(ai)
symmetric equilibrium, ai{x)
The
-Taj dF^'{uj\ai)
L
-\-
-I-
convex cone, with corresponding
monotonicity of the policy
is
that
31
£?[a;|at]
(MDP) and (MIO)
be increasing in
Oj.
conditions.
The
condition
G is more informative than F for
under which
be derived immediately from Theorem
Va G
3:
is
decision-makers with linear marginal returns can
[0, 1],
<a]> E[oj\G{y) < a].
E[uj\F{x)
This
all
(MIO-LIN)
the linear monotone information order.
We now
consider the information acquisition choice.
{MIO-LIN). Expected payoffs to player
i
are Il.{6i,6j)
metric equilibrium must also satisfy n,(^i,0i) —C'{6i)
acquisition.
Assume that
6i
indexes Fi according to
= E[u{uj,ai{ai),aj{aj))\9i,62], so the sym= 0, the first order condition for information
We are also interested in how changes in the parameters alter the amount of information
gathered in equilibrium. Straightforward algebra yields
{u;
Comparative
statics follow
known marginal
+ K)-^{E[uj\ai;ei]+K)
immediately from an application of Theorem
benefit to acting, has
no
A
10.
change in K, the
on the amount of information gathered.
effect
in the quadratic cost of action, r, decreases the
amount of information gathered. And
An increase
as
is
intuitive,
an increase in 7, the returns to coordination, increases information gathering. Similarly, agent's
actions are increasing
game
reinforces
K and decreasing in r. Endogenizing the information structure in this
known complementarities.
Conclusion
7
In this paper,
We
7 and
we have obtained a new
set of results for the
have provided a general analysis of when one signal
is
standard Bayesian decision problem.
more valuable than another
signal to a
given class of decision-makers, and also provided conditions under which decision-makers within a
given class will differ systematically in their marginal value for information.
We
general framework to several payoff classes of particular economic relevance,
corresponding monotone information orders.
section, the results
what
can be applied to a variety of economic
sort of information
about acquiring
As we have attempted to
is
have applied
and described
this
their
point out in the previous
settings, allowing characterizations of
valuable in a given envirorunent, and which sorts of agents caxe most
it.
32
An
extremely interesting, but potentially
extension of this research,
difficult
value of information in strategic settings. In such a setting, information
to analyze the
is
may have both a
direct
strategic value. In standard incentive problems, information has only strategic value, while
and a
in Persico's (1997)
study of auctions, information
acquired secretly so there
is
is
only a direct value.
In the industrial organization literature discussed in Section 6.1, results combining both direct and
strategic benefits are obtained, but they typically rely
The examples from
may
this literatiure suggest that
many
not be available in
results
settings.
restrictive functional
unambiguous
results
about the value of information
remains to be seen whether general characterization
It
can be obtained in game-theoretic models.
Proof of
Lemma
5.
Assume
first
[q;j_i,q;j].
exist
is finite,
ao
=
<
A=
ai
>
{ai, ...,an} with Oj+i
<
...
< an-\ <
ctn
=
1
=
^^^u{uj,ai) [m{ai\u))
/
Jn
J[Q,l\
^
,
'"
such that a{a)
-1
I
I
-Z)r=i
— u(lj
,
ai)m(0\u))
= -^2
ri(a;),...,r„_i(a))
with
ri{uj)
are zero a.e. with respect to Tn{u)).
Moreover, suppose (16)
= ai
on
=
ai
^^
[0,6:)
and a{a)
The
fails for
=
G
On on
C/j.
J
- u{uj,ai)] m{ai\uj)
>
dm{uj)
ri{uj)dTn^{u},ai )
The
third equality uses the fact that 7n(l|a;), m(0|a;)
last step follows
some a G
[a, 1].
f
^
y~^ [u{uj,ai+i)
a{a)
any
|
[u{oJ,ai+i) -u(u},ai)]m{ai\uj)
'n-l
(15).
for
— m{ai-i\uj)] dm{uj)
u(u! an)'m(l\ui)
I
some
Then
Oj.
Then
u(LO,a{a))dm{u),a)
/
/
Ju
A
that
monotone nondecreasing a{a), there
for
form assumptions.
Appendix
8
on
on
(0, 1),
r
€
from (MDP-U). Then clearly (16) implies
Ui.
Then define u{u,a)
By (MDP-U), u E
U2, and a
is
= af{u)). And let
clearly nondecreasing.
Substituting into the derivation above,
/
/
u(a;,a(Q:))dm(a;,a)
= — (on — oi)
Jn J[o,i]
So
(15)
Now
must imply
suppose
f{uj)dajm(u,&)
<0.
(16).
A is compact. We know from above that
a{a) step functions.
/
Ju
We show that
(16) is equivalent to (15) holding for
(15) holds for all nondecreasing functions a{a)
33
if
and only
aU
if it
holds for
all
step functions. Let a{a) be
some nondecreasing
sequence of step functions converging to a{a). Then since u
And since
be converging to u{u},a{a)).
Theorem
to
u{u),a)
is
and
let a'-(a),a^{a),...
be a
continuous in a, then u(ui,a''{a)) will
is
bounded, we can apply the Lebesgue Convergence
show
u(u},a^(a))dm{uj,a)
/
—
clearly, (15) will
u{ij,a{a))dTn{uj,a).
/
>
Jnx[o,i]
Then
function,
JnxlQ,!]
hold for
all
a nondecreasing
if
and only
if it
holds for
all
step functions.
D
References
[1]
Athey, Susan
(1998a): "Characterizing Properties of Stochastic Objective Functions,"
MIT
Working Paper 96-lR.
[2]
Athey, Susan
(1998b): "Comparative Statics under Uncertainty: Single Crossing Properties
and Log-Supermodularity,"
[3]
Blackwell, David
ley
[4]
Blackwell, David
[6]
(1953): "Ekjuivalent
&
Border, Kim
Proceedings of the Second Berke-
93-102.
Statistics,
Comparisons of Experiments," Annals of Mathemat-
265-272.
Blackwell, David and M. Girshik
John Wiley
Paper 97-llR.
(1951): "Comparisons of Experiments,"
Symposium on Mathematical
ical Statistics, 24,
[5]
MIT Working
(1954):
Theory of Games and
Statistical Decisions.
Sons.
(1991): "Functional Analytic Tools for
CD.
Operators, Riesz Spaces, and Economics, ed.
Expected Utility Theory,"
Aliprantis, K.
Border and
in Positive
W. Luxemburg.
Springer Verlag.
[7]
Dewatripont, M.,
I.
Jewitt, and
J.
Theory and an Application to Mission
[8]
Setting,"
(1997):
"The Economics of Career Concerns:
Mimeo, University of
Bristol.
Collier, C. and M. Kimball (1995): "Toward a Systematic Approach to the Economic
Effects of Uncertainty
[9]
TiROLE
I:
Comparing
Risks,"
Mimeo, University of Toulouse.
Jewitt, Ian (1986): "A Note on Comparative
of Mathematical Economics, 15, 249-254.
34
Statics
and Stochastic Dominance," Journal
[10]
Jewitt, Ian (1997):
"Information and Principal Agent Problems," mimeo, University of
Bristol.
[11]
Karlin, Samuel
[12]
Karlin, Samuel and William Studden
Analysis and Statistics.
[13]
Le Cam, Luc
Volume
(1968): Total Positivity:
New York:
(1964):
I.
Stanford University Press.
(1966): Tchebycheff Systems with Applications in
Interscience.
"Sufficiency
and Approximate
Sufficiency,"
Annals of Mathematical
Statistics, 35,
[14]
Lehmann, E.L.
(1988):
"Comparing Location Experiments,"
Annals of
Statistics,
16
(2):
521-533.
[15]
Levy, Haim and Jacob Paroush, "Toward Multivariate
Economic Theory 7
[16]
Efficiency Criteria,"
Journal of
(2) Feb., pp. 129-42.
Meyer, Margaret
(1990): "Interdependence of Multivariate Distributions: Stochastic
inance Theorems and an Application to the Measurement of
Ex
Dom-
Post Inequality under Uncer-
tainty," Nuffield College Discussion Paper, no. 49.
[17]
MiLGROM, Paul
tions," Bell
[18]
"Good News and Bad News: Representation Theorems and Applica-
Journal of Economics, 12, 380-391.
MiLGROM, Paul and John Roberts
in
[19]
(1981):
Games with
(1990): "Rationalizability, Learning,
and EquiUbrivmi
Strategic Complementarities," Econometrica, 58, 1255-1277.
MiLGROM, Paul and Christina Shannon
(1994):
"Monotone Comparative
Statics,"
Econo-
metnca, 62, 157-180.
[20]
MiRMAN,
L., L.
Samuelson, and E. Schlee
in Duopolies." Journal of
[21]
Economic Theory,
(1994): "Strategic Manipulation of Information
36, 364-384.
Persico, Nicola (1996): "Information Acquisition
in Affiliated Decision Problems,"
mimeo,
UCLA.
mimeo, UCLA.
[22]
Persico, Nicola (1997): "Information Acquisition
[23]
Ross, Stephen (1981): "Some Stronger Measures of Risk Aversion in the Small and the Large
with Applications," Econometrica 49
in Auctions,"
(3), pp. 621-38.
35
[24]
Rothschild, Michael, and Joseph Stiglitz
Journal of Economic Theory 2
[25]
[27]
I.
(1997):
(2),
(1995),
A
Definition,"
"Supermodular Stochastic Order and
Dependence of Random Variables," Journal of Multivariate Analysis,
Shannon, Chris
ory 5
"Increasing Risk:
pp. 225-43.
Shaked, Moshe and George Shantikumar
Positive
[26]
(3),
(1970):
"Weak and Strong Monotone Comparative
Statics,"
61, 86-101.
Economic The-
March, pp. 209-27.
Strassen, V. (1965): "The Existence of Probability Measures with Given Marginals," Annab
of Mathematical Statistics, 36, 423-439.
[28]
TOPKIS,
Donald
(1978):
"Minimizing a Submodular Function on a Lattice,"
Research, 26, 305-321.
36
Operations
S ufficiency with 3
Figure
1:
States:
Q = {6),, Oj.'ys}-
Two signals, x
states
Assume «,
and y where y
,
Each signal has two (equally
is
and 2-point signal
distributions.
Prior: (y,j,j).
<6t}2 <<i)3.
sufficient for x.
likely) possible realizations:
^ = {x'^,x"],1f = {y';y"),
?r{x
= x'')=?v{y = y^)=\
<o,i^^
GO3
00.
Figure 2: Indexing Posteriors by
Ex Ante Percentile
a=F(xJ,a = G(>'J.
/-
^,
y
3^
Figure 3: Illustration of Monotone Decision Problem
States: Q.
= {(Dy,C02,C0^}.
yi:
(MDP)
Conditions.
= {x';x"}. The posteriors generated by x
are illustrated in
the diagram.
Case
1: f/2 is set
of supennodular functions.
F(co\ x) satisfies
Given
Case 2: U^
X
ailx"> x*-.
restriction.
is
is set
admissable.
of fiinctions with single crossing incremental
Fi(0\x) satisfies
(MDP) only if F((o\x") J>-^„
Given F{o)\x^) as
satisfy
Thus,
F((Ol x^) for
F(0)I x^) as illustrated in diagram, only posteriors within solid lines satisfy
(MDP)
Thus,
(MDP) only if F(0)\x") >>05o
3c
illustrated in diagram,
returns.
Fi,(0\x^) for all
x" > x^.
only posteriors within dotted lines
(MDP) restriction.
is
not admissable.
=^.0J
CD.
jr
Figure 4; S upermodulai Monotone Information Order with 3 states and 2-point
signal distributions.
States:
Q = {(O^,(O2,C0^}.
Three signals,
Assume
6),
x,y,andz, where
No two signals
are ordered
Each
two (equally
signal has
Robustness: If F((0\x'')
is
<«2
<6>3- Prior: (j.j.j)-
z yuio-spu
5'
>-mio-spm
^
-
by sufficiency.
likely) possible realizations
anywhere within the dotted
which
region,
t4%i0}
5'
satisfy
MDP-SPM.
>-mio-spu
^•
=(o,-i,l)
Figure 5: Illustration of the Supermodular Monotone Information Order.
States: Q.
= {0)^,0)2,(02}. Assume
Two signals, y and
z
,
where z
o,
< Oj < 6)3.
>-mio-spm V-
Moving from ^ to z moves probability weight onto
(Low signal,a)^ and {High signal.O)^), and away from
(High signal,0)^ and {Low signaUo)^).
Change in Probability of event from j
High Signal
No Change
—
+
No Change
+
—
ffl.
©,
Realization
Low Signal
Realization
©,
31
to z
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