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CHARACTERIZING PROPERTIES OF STOCHASTIC OBJECTIVE FUNCTIONS Susan Athey 96-1
Oct. 1995
massachusetts institute of
technology 50 memorial drive Cambridge, mass. 02139
CHARACTERIZING PROPERTIES OF STOCHASTIC OBJECTIVE FUNCTIONS Susan Athey 96-1
Oct. 1995
INTRODUCTION
1
This paper studies optimization problems where the objective function can be written in the form
V(8)
=
\jt{s)dF{s\6),
payoff function,
;ris a
For example, the payoff function
real vectors.
the vector s
where
F
is
a probability distribution, and 6 and s are
n might represent
might represent features of the current
state
an agent's
utility
or a firm's profits,
of the world, and the elements of 6 might
represent an agent's investments, effort decisions, other agent's choices, or the nature of the
exogenous uncertainty in the agent's environment.
The economic problem under consideration often determines some properties of function; for example, a utility
the payoff
function might be assumed to be nondecreasing and concave, while a
multivariate profit function might have sign restrictions on cross-partial derivatives.
assumptions then determine a set questions such as:
investments? questions,
Is
IT
a set of investments 6
Does one investment increase
we need
know whether
to
is
These
We might then wish to answer
of admissible payoff functions.
worthwhile? Are there decreasing returns to those
the returns to another investment?
V(Q) satisfies the appropriate properties,
To answer
i.e.,
those
nondecreasing,
concave, or supermodular. 1
The goal of
paper
this
is to
develop methods such
nondecreasing), a set of admissible payoff functions
we can J
determine whether the following statement
7i(s)dF(s; 6) satisfies property
That
is,
given
P
and
II,
II,
that,
given a particular property
(1.1)
describe the set of probability distributions which satisfy (1.1).
This paper restricts attention to the case where the properties cones."
which
P
and
sets II are "closed
convex
We define a property P to be a "Closed Convex Cone" (CCC) property if the set of functions
satisfy
P is closed (under an appropriate topology), positive combinations of functions in the set
are also in the set,
and constant functions are
in the set. Important
nondecreasing, concave, supermodular, any property which
is
examples of CCC properties include
defined by placing a sign restriction on
a partial derivative, and combinations of these properties. Thus,
consumer's
1
(such as
is true:
P in 6 for all n in IL
we wish to
P
and a parameterized probability distribution F,
utility, II
Intuitively, a function is
to increasing the other.
if
the payoff function represents a
could be the set of univariate, nondecreasing, concave payoff functions;
supermodular
When
if.
if the
given any two arguments of the function, increasing one increases the returns
the function
is differentiable, this
amounts to positive cross-partial derivatives between
every pair of arguments. Supermodularity is important in the context of comparative statics (see Topkis (1978), Milgrom and Roberts (1990, 1994), Milgrom and Shannon (1994)).
payoff function represents a firm's profits as a function of two complementary quality innovations,
n
could be the set of bivariate, nondecreasing, supermodular payoff functions.
For
sets II
P
and for properties
which
probability distributions satisfy (1.1).
be a very large
set.
Thus we
ask,
are
CCC,
Checking
When is
it
this
paper develops a characterization of which
(1.1) directly is difficult because, in general, If
might
possible to find a smaller set of payoff functions, denoted
T, so that for any probability distribution F, statement (1.2)
below
will
be true
if
and only
if (1.1) is
true?
7t(s)dF(s;8) satisfies property J
P in 6 for all % in T
We can think of T as a "test set" easier to check than II, but
only the information that
Using these
ideas,
it
P is
T
is
a set of payoff functions which
whether
test
J
is
7C(s)dF(s;0) satisfies property
.2)
smaller and
P in
6,
given
K is in the larger set II.
we can restate the goal
the best "test set" for a given II.
property
for II: ideally,
can be used to
( 1
We
of the paper:
proceed in two
"nondecreasing"; second,
we
study other
we want
steps.
a theory which helps us determine
First,
we examine
CCC properties.
The
the case
first
where the
case has been the
focus of the literature on stochastic dominance, where different authors have studied different sets
II. 2
This paper unifies and extends the existing literature on stochastic dominance, providing an exact characterization of the mathematical structure underlying all stochastic further provide an algorithm for generating
hoc
differentiability
part of the paper
when V{6)
we show
that the
We now provide
stochastic
an overview of our is
results.
We
dominance theorems which relaxes the ad
common
in this literature. 3 In the
methods from stochastic dominance can be applied
satisfies other properties P, including
property nondecreasing, if II is
new
and continuity assumptions which are
dominance theorems.
second
to characterize
concavity and supermodularity. In the first part of this paper,
we show
that for the
a closed convex cone and contains constant functions, then the best
T
the set of "extreme points" of II. Just as a basis generates a linear space via linear combinations, so
a set of extreme points generates a closed convex cone via positive linear combinations and limits.
Thus,
want
2
we show that if we know only that a payoff function k lies in the closed convex cone II, and we know if V(6) is nondecreasing, it is equivalent to check that V(0) is nondecreasing on a smaller
to
In particular, the univariate stochastic dominance problem has been studied by Rothschild and Stiglitz (1970, 1971) and Hadar and Russell (1971); notable contributions to the multivariate problem include Levy and Paroush (1974), Atkinson and Bourguignon (1982), and Meyer (1990). Shaked and Shanthikumar (1994) provide a reference book on the subject of stochastic orders and their applications in economics, biology, and statistics. 3 Brumelle and Vickson (1975) take a first step towards relaxing these assumptions and identifying the mathematical structure behind stochastic dominance; in contrast to their work, which provides only sufficient conditions for a stochastic dominance relationship, this paper provides an exact characterization of all stochastic dominance theorems.
set
of payoff functions, the extreme points of n.
V(6) for the set of extreme points of
extreme points of
n
n
It
than for
will often
n
be much easier to verify monotonicity of
In this paper, the procedure of using the
itself.
as a test set for II will be referred to as the "closed
convex cone" method of
proving stochastic dominance theorems.
Examples from the existing include First Order Stochastic
dominance
stochastic
which are special cases of this
literature,
Dominance (FOSD), where
n is the
set
of univariate, nondecreasing
payoff functions, and Second Order Stochastic Dominance (SOSD), where TI
concave payoff functions.
In the case of
functions which are zero up to the case of 2,
SOSD,
some
the set of extreme
FOSD,
result,
the set of univariate,
is
the set of extreme points is the set of one-step
constant, and one thereafter.
points is the set
These are pictured
in Figure
1
.
In
of "angle," or "min" functions, pictured in Figure
where each function takes the minimum of its argument and some constant.
The closed convex cone method
for stochastic
dominance has been recognized and explored
in the
context of particular sets of payoff functions (Topkis, 1968; Brumelle and Vickson, 1974; Gollier and
Kimball, 1995). 4 However, this paper goes beyond the existing literature in two respects.
developing appropriate abstract definitions to describe stochastic dominance theorems,
make
we are
by
able to
general statements about the entire class of stochastic dominance theorems, including those
which have not yet been considered that
First,
we prove
a
new
in the
cone" approach to stochastic dominance return the
same answer
and (1.2) equivalent. In
is
if
we prove
exactly the right one.
these are not in T)
is
shows
characterize other properties of V(6).
that the "closed
the fact
By this we mean that (1.1) and
IT;
no other T's
which generates
(1.2)
will always
make
(1.1)
II is the best set to check.
convex cone" method can also be applied
to
We ask two questions: First, for what properties P is the closed
convex cone approach valid? And second, for what properties
P is
exactly the right one, as in the case of stochastic dominance? property, then the closed
is
formally that the "closed convex
F if and only if the closed convex cone of T
equal to
this case, clearly the smallest set
part of this paper
Second and more important
literature.
for every probability distribution
(union the constant functions,
The second
economics
result about this class of theorems:
the closed
We
first
convex cone approach
show
that if
P
is
convex cone approach can always be used to characterize when V(6)
a
CCC
satisfies
We then find a subset of CCC properties, which we call "Linear Difference Properties," for which we can show that the closed convex cone approach is exactly the right one for checking whether V(0)
P.
satisfies P.
Examples of Linear Difference Properties include monotonicity, supermodularity,
concavity, and properties which place sign restrictions
on
partial derivatives.
Combinations of these
properties, however, are not in general Linear Difference Properties, although such combinations are
4Independently, Gollier and Kimball (1995) argue for what they
4
call the "basis
approach" to stochastic dominance.
I
CCC
properties.
Table
I
summarizes properties which are
CCC
properties and Linear Difference
Properties.
TABLE LINEAR DIFFERENCE PROPERTIES AND CLOSED CONVEX CONE PROPERTIES Property
Closed Convex Cone
Linear Difference Property
Nondecreasing
Yes
Yes
Supermodular
Yes
Yes
Concave/Convex (Multivariate)
Yes
Yes
Sign Restriction on a Partial
Yes
Yes
Constant
Yes
No
Nondecreasing and Concave
Yes
No
Arbitrary Combinations of
Yes
No
Yes
No
Derivative
CCC Properties Arbitrary Combinations of
LDP
Properties
For many literature
cases, the
on
sets
of payoff functions, n, which are
stochastic
dominance has
commonly
studied in economics, the existing
implicitly identified the extreme points of those sets. In such
problem of characterizing a Linear Difference Property (such as supermodularity) becomes
quite straightforward: simply look to the the existing literature to find the appropriate test set, T, for the set
of payoff functions II under consideration. Then, using the results of this paper,
(1.2) characterizes the set
function will be supermodular for
We
illustrate this
we know
that
of parameterized probability distributions for which the stochastic objective all
payoff functions in
technique by showing
how
II.
the closed convex cone approach can be used to
characterize the property supermodularity for several important classes of payoff functions. These results
can in turn provide sufficient (and sometimes necessary) conditions for comparative
conclusions. In particular,
we examine
statics
applications in principal-agent theory, welfare economics, and
the study of coordination problems in firms.
This paper proceeds as follows.
In Section 2,
we
introduce a motivating example, the problem of
a risk- averse agent's choice of effort. In this problem,
concave, and supermodular for the agent's expected
dominance takes place
in Section 3.
We provide
we characterize
utility function.
the properties nondecreasing,
Our general
an exact characterization of stochastic dominance
We further extend our
theorems, highlighting the important role played by linearity of the integral. result to incorporate "conditional stochastic
properties of
we
dominance." Section 4 develops characterizations of other
7t(s)dF(s;d) and provides applications of the property supermodularity. In Section 5, f
K(x,s)dF(s;6) are supermodular or
analyze conditions under which functions of the form
concave
analysis of stochastic
in (x,d),
showing how
J
to apply stochastic monotonicity results to this
problem
as well.
Section 6 concludes.
2
MOTIVATING EXAMPLE:
A RISK-AVERSE AGENT'S CHOICE OF EFFORT In this section,
we present
a motivating example, where
how
of effort affects her expected payoffs, and
we
analyze
Formally,
nondecreasing, concave, and supermodular for the agent's expected
examples
stochastic
—
examples where the payoff function
is
illustrate the parallel structure
Consider a risk-averse agent whose technology, where the output
is
denoted
s.
Suppose t
utility function.
another. Then,
t
we can write the agent's problem as max
observe that
it is
not
an increase in effort which
results are
nondecreasing and concave
—
of a
is
underlying the three classes of theorems.
that the agent's effort (c) affects the probability
k(s) dF(s; e,t)-c(e) •
f
=
\it(s) dF(s;e,t) is
some
-
c(e) is
= ae such
that the
FOCs
effort:
utility if the effort also
concave is
in effort, then
useful in the analysis
as principal agent problems. Further, if V(e,i) fails to
linear cost function c(e)
optimum. Finally, observe
nondecreasing in
productive on average might not increase expected
order conditions (FOCs) characterize the optimum, a property which
then there exists
to
follows:
trivial to verify that V(e,t)
some economic problems, such
in the stochastic
might represent a worker moving from one job
increases the riskiness of the distribution. Second, note that if V(e,t)
e,
These
which represents exogenous changes
production technology. For example, a change in
first
exogenous
characterize the properties
depends on the output of a stochastic production
utility {it)
distribution of s. Further, consider a parameter
the
we
dominance theorem, a "stochastic concavity theorem," and a "stochastic supermodularity
theorem." These examples
First
a risk-averse agent's choice
that choice of effort interacts with
parameters which describe the probability distribution.
specific
how
fail to
be concave
in
characterize the
that if V{e,i) is supermodular, then the optimal choice of effort, e*(t), is
nondecreasing in
If V(e,t) fails to
t.
linear cost function c(e)
=
ae
Milgrom and Shannon (1994); satisfies the
be supermodular, then even
such that e*{i)
fails to
Theorem A.l
see
in the
cannot be relaxed as long as
it
V is
concave, there exists some t
follows from
(this
Appendix). Thus, the requirement that
property supermodular (or concave, respectively)
conclusion, and further
if
be nondecreasing in
we
is
sufficient for the desired
V(e,t)
economic
require that the conclusion holds for
all
linear cost functions.
Let us
first
identify conditions under
known
result is adapted
(where
we suppress
Proposition 2.1
t
which
from the Rothschild and
nondecreasing in
V(e,i) is
Stiglitz (1970,
e.
The following
well-
1971) work on stochastic dominance
in the notation):
The following two conditions are equivalent for
all probability distributions
F(-,e): (i)
For all K nondecreasing and concave, The following are
(ii)
dF(s;e)
is
nondecreasing in
e.
satisfied:
\sdF(s;e)
(a)
\lt(s)
is
nondecreasing
in
e.
a
For all a, -\F(s;e)
(b)
Intuitively, for a risk-averse agent
mean income
effort increases the
This result
is
is
nondecreasing
who likes income,
(condition
(ii)(a))
often used in the finance literature;
it
in e.
effort will increase
expected
utility if
and only
and reduces the "risk function" (condition
if
(ii)(b)).
has been called Second Order Monotonic Stochastic
Dominance (SOMSD).
we
In this paper,
conditions
(i)
and
(ii)
will
A
1
Proposition 2.1 (where
extended real
Proposition (i)
For
all
nondecreasing (ii)
in e.
u {-00
,
the set Yl
which emphasizes
The following proposition SR,
is
that
equivalent to
and Si indicates the
00 }):
The following two conditions are equivalent for
n in
2.1,
be the space of probability distributions defined on
line, that is, SR
2.1'
work with a restatement of Proposition
are actually symmetric conditions.
SOM
= [ti\k
:
SR
-»
all
F(-;e)eN:
% nondecreasing, concave},
J
7t(s)
dF(s;e)
is
in e.
For all yin
the set
T SOM = {y\y(s) = min(a,s), a e 9t},
jy(s) dF(s\e)
is
nondecreasing
.
Conditions
(i)
and
(ii)
we have
except that
of this Proposition simply restate conditions
straightforward to verify (using integration by parts) that the former term if
the latter term
This condition
is.
(1975) for an exception).
This
way of
structure underlying the stochastic that instead of
relatively large set, II
50M ,
sets of
the union of
is
set,
T
is itself
relationship
y{s)
Now,
a closed convex cone.
between
let
Tl
and
r
nondecreasing
=
and
1
is
illustrates the 1',
mathematical the result says
for all payoff functions in the
X
is
nondecreasing in e for
all
SOM
all
=
between the two
of
relationship
That
-1.
latter set,
In Section 3,
holds for
The
2.
we
we can
will
stochastic
is,
show
by taking
to analyzing principal-agent
generate any function in TI SOM , that this "closed
convex cone"
dominance theorems.
When is
problems
are satisfied, then the agent's choice of effort
sets
positive combinations
J
x(s) dF(s;e) concave in effort? This
problem was addressed by Jewitt (1988), who analyzes conditions under which the
FOCs
and only
equal to the closed convex cone of the set which
y(s)
us ask a different, but related, question:
Approach (FOA)
if
is
It
.
and the functions
SOM
is
(ii).
2.1,
S0M
and limits of sequences (or nets) of elements of the
which
of Proposition
in Proposition 2.
nondecreasing in
payoff functions are pictured in Figure
T S0M
theorem
dominance theorem. As written
S0M payoff functions can be described as follows: T1 is
SOMSD
equivalent to check that \y(s)dF(s;e)
it is
payoff functions in the smaller
The two
writing the
checking that j n(s) dF(s;e)
(ii)
SOMSD (see Brumelle and Vickson
not usually associated with
is
and
jmin(a,s)dF(s;e) in condition
with
replaced -JF(s;e)
(i)
is valid.
The
First
Order
FOA requires that if the agent's
must be optimal. Extending
Jewitt' s analysis,
we
can show that the sufficient conditions he derives are in fact necessary. 5 The following result analogous to Proposition
Proposition 2.2 (i)
(ii)
For
The following two conditions are equivalent for
all
For
all
% in
is
2.1:
the set Yl
yin the set
S0M
\k(s) dF(s;e)
is
concave
in e.
y(s) dF(s;e)
is
concave
in
•
,
T SOM
all
',
\
F(;e)e A
1
:
e.
Proof: This will be established as a simple corollary of Theorem 4. 1 below, together with Proposition 2. 1
Q.E.D.
5 Jewitt first derives conditions under which the agent's utility function is increasing and concave in output, given the optimal contract; he then addresses the question of concavity of the expected utility function in effort. It is only the latter question which we study here.
Proposition 2.2 says that expected payoffs are concave in e for
only
expected payoffs are concave in e for
if
useful because condition
Condition
(
n
SOM ,
T
SOM
T SOM
payoff functions in
easier to check that condition
(ii) is
(i):
U SOM
payoff functions in .
the set
Again,
r
SOM
this
much
is
if
and
theorem
is
smaller.
can be interpreted as requiring that there are decreasing returns to e in terms of
(ii)
increasing the
all
all
mean and
), is
the
decreasing the "risk function." Note that the pair of sets of payoff functions,
same
in
both propositions.
Now we turn to ask a final question: When is the optimal choice of effort monotone nondecreasing in
t,
which parameterizes the stochastic production technology? More
way
probability distribution over output in such a
that
precisely,
how can
t
affect the
monotonicity of the optimal effort in
is
t
ensured, without any additional information about the cost of effort function or the agent's preferences?
This question has not been answered in the existing literature; thus, the following
proposition provides a
new
Proposition 2.3
monotone
cost functions c
and all K in
Il
SOM ,
e'(t)
all F(-;e,f)e
S0M
in
(i)
For all n e
(ii)
For all ye T S0M jy(s)
Tl
,
t.
is
supermodular
:
is
Js
in {e,t\
dF(s;e,t) is supermodular in
Proof: The equivalence of parts (1994), as stated in
1
6
\K{s)dF{s;e,t)
,
A
= argmax\K(s)dF(s;e,t)-c(t) e
monotone nondecreasing
that this proposition is
in effort.
The following three conditions are equivalent for
(MCS) For all
Note
insight into the comparative statics problem.
true irrespective of whether expected utility is
Theorem A. 1
(MCS) and in the
(i)
(e,t).
follows directly from
Milgrom and Shannon
Appendix. The equivalence of (i) and
established as a corollary of Theorem 4.1
below together with Proposition
(ii)
will
be
2.1.
Q.E.D.
The formal
definition of supermodularity (and the comparative statics
can be found in the Appendix. Intuitively,
V(e,t) is
supermodular
Proposition 2.3 provides necessary and sufficient conditions for
problem. If
(ii) is
choice of effort
is
violated, then
not monotonic in
monotone comparative increasing the
we
statics.
mean of the
if t
theorem which
relies
statics in this
can construct payoff functions and cost functions such that the t.
Thus,
Condition
we have
(ii)
identified the exact conditions
requires that e and
t
are
which ensure
complementary
probability distribution and in terms of reducing the risk.
may be a
Order, as defined in the Appendix.
set.
it)
increases the returns to effort.
monotone comparative
Then,
this
theorem requires
that the set
be nondecreasing
in terms of
The
straightforward: since a risk-averse, income-loving agent likes high expected returns and
"In general, the optimal e
upon
intuition is
low
risk (as
in the Strong Set
shown
SOMSD),
in
complementary
are
variables
which
in increasing
complementary
are
expected
utility
in increasing the
mean and
of such an agent. Note that
decreasing the risk
one of e and
if either
/
does not affect the mean or the riskiness of the agent's income, the corresponding complementarity conditions are satisfied trivially.
Notice that Propositions 2.2 and 2.3 have a structure which
dominance
stochastic
result, as stated in
analyzing Proposition 2.1
very similar to the existing
is
Proposition 2.1. In Section 3,
we
will build a
and other stochastic dominance theorems. In Section
we
4,
framework
for
will formalize
the relationship between Propositions 2.1 through 2.3.
MONOTONICITY OF STOCHASTIC OBJECTIVE FUNCTIONS
3
The goal of theorems.
dominance
two
sets
this section is to
provide a unified framework for analyzing stochastic dominance
we
introduce a framework which incorporates the existing stochastic
In Section 3.1, literature,
arguing that each stochastic dominance theorem describes a relationship between
we prove
of payoff functions. In Section 3.2,
relationship
between two
the stochastic
sets
of payoff functions which
a result which characterizes a mathematical equivalent to the relationship determined by
is
dominance theorem. Section 3.3 extends the
result to the case
of conditional stochastic
dominance.
3.1
A
Unified
In this section,
Framework for
we
Stochastic
Dominance
introduce the framework which
we
will use to discuss stochastic
dominance
theorems as an abstract class of theorems, and to draw precise parallels between stochastic dominance theorems and other types of theorems.
Let us Stochastic
first
consider another well-known example of a stochastic dominance theorem, First Order
Dominance (FOSD). This theorem can be
(where IA (s)
stated as follows
is
the indicator
function for the set A):
Proposition 3.1 (i)
(ii)
The following two conditions are equivalent for
For all n e
n
For all y e
TO
s {k\k
:
9i
-»
% nondecreasing},
M
T FO m {y\y(s) = I
(s),
This theorem has the same structure as the functions are illustrated in Figure are nondecreasing in
8
1.
ae
Sfi},
J
all
F(;0)e A
1
K(s)dF(s;6)
jy(s)dF(s;6)
is
is
:
nondecreasing
nondecreasing
SOMSD theorem, Proposition 2. 1
'.
The
in ft
in ft
sets
of payoff
This theorem says that instead of checking that expected payoffs
for all nondecreasing payoff functions,
expected payoffs are nondecreasing in 6 for
all
10
IF
,
it is
payoff functions in the set
equivalent to check that
T FO
,
the set of indicator
The
functions of upper intervals.
easier to check that condition 1
- F(a;6),
The
of payoff functions
condition
is
is
much
smaller, and so condition
(ii) is
can be reduced to a restriction which requires that
(ii)
complement of the cumulative
the
requirement
latter
latter set
(i);
distribution function,
nondecreasing in 8 for a e
is
9?.
more standard way of stating FOSD.
the
There are many other examples of stochastic dominance theorems, some with multiple random
we
variables;
theorems
all
will discuss other
examples below
stochastic dominance theorems are characterized that the pair
(IT™,
r
F0
(U SOM T SOM )
satisfies the
)
same
theorem." If that statement
is true,
will use the abstract definition to
and how
that class of
then
make
and
different (11,0 pairs.
To make
relationship.
"A pair of sets
definition of the statement
by
satisfies a particular relationship; the
,
In general, stochastic dominance
in Section 4.5.
illustrated in Propositions 2.1'
have a parallel structure, as
FOSD
3.1.
The
However,
different
SOMSD theorem states
theorem
states that the pair
we
introduce a formal
this relationship precise,
of payoff functions, (n,D, satisfies a stochastic dominance
we
dominance
will say that (11,0 is a stochastic
pair.
We
statements about the class of stochastic dominance theorems,
We allow for multidimensional
theorems relates to other classes of theorems.
payoff functions and probability distributions, using the following notation: the set of probability distributions
on
SR" is
we
parameter space 0, distributions
F
:
x
9?"
denoted A", with typical element F:SR n
—»[0,1].
will use the notation A"
e to represent the set
-»
such that such that F(;0) e A" for
[0,1]
all
Further, for a given
of parameterized probability
ds 0.
Definition 3.1 Consider a pair of sets of payoff functions (TI,r), with typical elements
K 9T —> SR and y :
and
:
9?"
—> SR.
The pair
(II, T) is
are equivalent for all parameter spaces
(ii)
(i)
For all ttgU,
(ii)
For all y eT,
Further,
we define the
j
n(s)dF(s;6) y(s)dF(s;6)
is
is
J
set I. SDT to
be the
set
a stochastic dominance pair with a partial order
nondecreasing nondecreasing
and all
conditions
if
(i)
n
F € Ae
:
in 6.
in 6.
of all (11,0 pairs which are stochastic dominance
pairs, as
follows:
2 5Dr = {(II,r)|(n,r) Thus,
when
a given (11,0 pair
is
is
a stochastic dominance pair}
a stochastic dominance
pair,
we write
(II, T)
Definition 3.1 clarifies the structure of stochastic dominance theorems.
e Z SDr
.
Stochastic dominance
theorems identify pairs of sets of payoff functions which have the following property: given a
parameterized
probability
lAn(s)dF(s\6)\iz
e
II
>
distribution
F, checking that
all
of the functions in the set
are nondecreasing is equivalent to checking that all of the functions in the set
11
(fy(s)dF(.s;0)|y
er|
T is
general, the set
Stochastic dominance theorems are useful because, in
are nondecreasing.
smaller than the set
II.
We can think of this definition as a statement about the equality of two sets. set
First, let
us define the
of admissible parameter spaces for the property "nondecreasing" together with probability
distributions parameterized
VND = {(F,0)|© has a Now, we can
on those spaces
partial order
and
as follows:
F e Ane j
rephrase the definition as follows: (11,0
j(F,0) €
VN M 7r(s)dF(s;d) is nondecreasing on
|(F,0) e
VND \JY(s)dF(s;0) is nondecreasing on
V/r e Ili
n
Vy e Ij
existing literature generally
J
7t(s)dF(s) and
Si. In contrast,
functional
a stochastic dominance pair if
n
Definition 3.1 differs from the existing literature
viewing
is
J
(i.e.,
=
Brumelle and Vickson, 1975),
compares the expected value of two 7t(s)dG(s) as
two
different linear functionals
by parameterizing the probability
distribution
J
when we
distributions to the reals,
become more
and stochastic
P theorems, for other properties P (such as supermodularity). specific
set of
many
are
The
utility
of this
formalize the relationship between stochastic dominance
examples of pairs of
sets
of payoff functions which satisfy stochastic
dominance theorems, we summarize three univariate stochastic dominance theorems are potentially
we
dominance theorems and stochastic supermodularity
definition will
To provide
to
n(s)dF{s\6) as a bilinear
theorems, an analogy which would not be obvious using the standard constructions. clear
F and G,
mapping payoff functions
and viewing
mapping payoff functions and (parameterized) probability
able to create an analogy between stochastic
in that the
probability distributions, say
in
Table
It.
There
other univariate stochastic dominance theorems (for example, theorems where the
payoff functions imposes restrictions on the third derivative of the payoff function); however,
we will
simply report the three most familiar univariate stochastic dominance theorems here.
12
TABLED COMPONENTS OF UNIVARIATE STOCHASTIC DOMINANCE THEOREMS"
Sets of Payoff Functions, II
U F0 = {n\n
(i)
n 5° s {n\n
(ii)
:
:
SR
SR
% nondecreasing} -> % concave}
r F°={ r \y(s) = I
->
[a
Yl
SOM
= \n
k
SR
:
—»
SR,
nondecreasing,
„
(s),aeX} )
r S0 ={Y\Hs) = -s} u{y|y(.y)
(iii)
T
Sets of Payoff Functions,
i
= min(a,s), aeSRJ
r «w _ |y| y (j) _ min(fl,s),
1
a e 9?}
concave
Each
(II,r) pair in
Table
II (iii)
corresponds to a 3.1
and Table It is
(ii).
Table
II is
a (univariate) stochastic dominance pair.
corresponds to a
SOMSD theorem,
FOSD theorem, Proposition 3.1.
and only
K{s)dF{s;8)
that
depend on
\min(a,s)dF(s;d)
if
6.
shown
now
Let us
1
in Proposition 2.
',
while Table
illustrate the interpretation
II (i)
of Definition
with a third example: Second Order Stochastic Dominance (SOSD), shown in Table
II
known
is
J
if
as
Observe
is
nondecreasing in 6 for
all
nondecreasing in 8 for
that both y(s)
= s and
the distribution to be both nonincreasing
y(s)
= -s
univariate,
all
II
concave payoff functions
aeSi, and \sdF(s;6) does not
are included in
Tso
this forces the
;
and nondecreasing, and hence constant
mean of
in 6.
There are many other stochastic dominance theorems in addition to the univariate examples given above. Levy and Paroush (1974) derive results for bivariate functions, while these results and examines
some of these
some
multivariate stochastic
results in Section 4.5,
Meyer (1990) extends
dominance theorems as
where the main objective
is to
well.
We will report
apply these results to problems of
stochastic supermodularity.
Exact Conditions for a Stochastic Dominance Theorem
3.2
In this section,
dominance
statement that 3.2.1
and
we
study necessary and sufficient conditions for the pair
(11,1") to
be a stochastic
We want to specify the exact mathematical relationship which is equivalent to the (II, T) e Z sor We will first discuss our result and its implications; then, in Sections
pair.
.
3.2.2,
The main
we will provide the mathematical
result
of this section
is that (II,
arguments underlying the
T) e
the closure (under an appropriate topology) of the
13
I, SDT if
and only
if
result.
the following statement
convex cone of II u {1,-1}
is
is true:
equal to the closure
(under that topology) of the convex cone of
two constant
functions,
{n{s)
=
1}
Tu {1,-1},
where {1,-1} denotes the
set containing the
u {it{s) = -1}. We formalize this using the following notation:
cc(nu {1,-1}) = cc(ru {1,-1})
(3.1)
In the context of specific sets of payoff functions
n, the existing
literature identifies similar, but
stronger sufficient conditions for the corresponding stochastic dominance theorems, using a restrictive notion
of closure
(i.e.,
a topology with
more open
sets)
than the one which
below. For example, Brumelle and Vickson (1975) argue that (3.1)
shown
Table
in
this paper,
II to
we
will identify
sufficient for the {Yl,T) pairs
be stochastic dominance pairs under the topology of monotone convergence. In
we have developed for a
using the abstract definition
able to formally prove that the sufficient conditions hypothesized in fact sufficient for
is
more
we
are
by Brumelle and Vickson (1975)
are
dominance
"stochastic
pair,"
any stochastic dominance theorem, not just particular examples. Further, the
result that (3.1) is also necessary for (Tl,T) to
be a stochastic dominance pair
is
a
new
contribution of
this paper.
We now method.
argue that this result
First,
observe that unless
tells
T is
us
when we cannot do
a subset of n, there
is
theorems provide conditions which are easier to check than jrell. For example,
n
might be much
and
which
is
larger,
might be a it
set
which
is
J
better than the closed
no guarantee
to find a subset, T,
dominance theorems, as defined
in Definition 3.1, are
closed convex cone. For example, in the case of FOSD,
6
most
Its
for all
closed convex cone
of that closed convex cone for
easier to check that expected payoffs are nondecreasing in 6.
stochastic
dominance
7C(s)dF(s;0) nondecreasing in
not a closed convex cone.
might not be possible
convex cone
that stochastic
Thus, (3.1) indicates that
likely to
we consider the
be useful when
of payoff functions
set
II is a II
F0 .
easy to verify that positive scalar multiples and convex combinations of nondecreasing functions
It is
are nondecreasing functions, as are limits of sequences or nets of nondecreasing functions. Finally,
constant functions are also in IT
When
F0 .
IT contains the constant functions
and
is
a closed convex cone, (3.1) becomes:
n = cc(ru{l,-l}) In principle, the
most useful
general, there will not
r FO
(3.2)
T
is
the smallest set
be a unique smallest
set.
To
whose closed convex cone
contains indicator functions of upper intervals.
combinations of elements of
T
FO
Q
However,
in
By
taking limits of sequences of convex
u {1,-1}, and appropriately scaling these functions, we can generate
any nondecreasing function. However,
where
is II.
see this, consider the case of FOSD, where the set
we can
represents the rationals, and note that
also define the set
14
n
F0
t FO = {y\y(s) = I[a „,($),
= cc(f F0 u {1,-1})
.
While f F0
a e q\,
c T FO
,
in
f F0 is not any easier to check. Thus, stochastic dominance theorems are that T is the smallest closed set whose closed convex cone is II; we will call such a
practice the smaller set
generally stated so set the
"extreme points" of II.
Finally, because (3.2) is necessary
n=cc(IIu{l,-l}), we know of II: there stochastic
is
no
we
that
and sufficient for (H,T)
smaller or easier-to-check closed set
dominance
convex cone method
pair.
is
This
is
F
1
higher Gower) than
n in
neither inequality holds for all
we
first
We
cc(IIu{l,-l}).
distributions in the
the set of extreme points
that
we have proved
(II,
f)
is
a
that the closed
show
then
we
that II
show
same way
same. In Section 3.2.2,
II,
if
that
dominance
that (3.1) characterizes stochastic
Consider the problem of ordering two probability distributions,
Section 3.1.1,
T be
of payoff functions, f, such that
the "right" approach.
payoff functions II orders if
be a stochastic dominance pair when
what we mean when we say
two subsections, we prove
In the next
to
cannot do any better than letting
then
F2
we
if
F
1
and
\7r(s)dF\s)
F
2 ,
where we say
> (
< £ k where there
We
0.
is
a neighborhood corresponding to each
will return to clarify the relationship
between
this
topology and other topologies in the discussion following Theorem 3.4, below.
Let cc(A) denote the convex cone of a set A, and topology
is
=
\ndii
The proof of
is
linear
and continuous in
lemma
is
mathematical results in this paper build upon
it.
simple lemma.
3.2
A
denote the closure of
understood to be a^P*,?^) in the discussion below, unless noted).
that the functional P(7t,^i)
Lemma
let
this
its first
elementary, but
Consider a set of payofffunctions jl &g>:
II
we
A
(where the
We now use the fact
argument to prove the following state
it
here because
all
of the
cP». Then the following two conditions are
equivalent for all (i)
For all n e
II,
\nd\i> 0.
7 The boundedness assumption guarantees that the integral of the payoff function exists. It is possible to place other restrictions on the payoff functions and the space of finite signed measures so that the pair is a separated duality, in which case the arguments below would be unchanged; for example, it is possible to restrict the payoff functions and the signed measures using a "bounding function." For more discussions of separated dualities, see Bourbaki (1987, p. n.41).
16
(ii)
For all 7tecc(Uu
Proof:
First,
we show
{1,-1}),
(i)
jnd/i>0.
implies
Fix a measure fieg'. Then the following implications
(ii).
hold:
For all K ell,
=>
(Since
J
d[i
Forall ;rellu
= -j
dji
Forall
[tt, rf/z
>
{1,-1},
J;rd/i>0
= 0).
=>
(Since
jndn>0
and J
ff€cc(nu {l.-l}), J^d//>0
n2 dfi>0
implies
[a,^, J
+ a2 7T2 ]
d/i
= a^TTi
d\i
+ a 2 \n 2 dfi>0
when a,,a 2 >0). =>
For a// K ecc(IIu{l,-l}), J7rd// >
(Recall that for a continuous function/, the half-space [x
That
(ii)
implies
e
(i)
9^|jc
> 0}
is
follows because IT
Lemma 3.2 can be restated another way: is
exactly the
same
/(A)
c /(A).
The implication then follows because
closed, and the linear functional P(;/l) is continuous for all
c cc(II u {1,-1}).
the set of measures
as the set of measures
[i
e£*
for
//
which
/*.)
£?.£.£>.
e^» for which jndfi >
Jtt^ >
for all
it
e
for all
cc(Il
7T
e
II
u {1,-1}).
Formally,
{/i€^-|J^d/x>0 V;renj = {//Gf* IJ^rd/t
As
V;recc(IIu{l,-l})}
discussed above, because positive scalar multiples do not reverse the inequalities and because
the functional
fi is
linear in its second argument, the latter equality is equivalent to the following:
FKF^A^JTrdF^JKdF2 Vn e III That
>0
is,
for any
two probability
= If^F2 eA^TtdF
distributions, II orders the
cc(IIu{l,-l}).
17
1
two
ZJndF
2
V/r e cc(II
u {!,- 1})
distributions exactly the
same
as
we
Building from this lemma, characterization of stochastic in Section 4. (ii)
implies
turn to prove the
dominance theorems,
main mathematical theorem underlying
This theorem makes use of the linearity of the functional
(i)
relies
on
Lemma
3.2,
the
as well as the characterizations of other properties
while the proof that
(i)
implies
/J(tt,jU) in n. (ii)
The proof
that
makes use of a standard
which determines the meaning of
separating hyperplane argument. Note that the choice of topology,
closure, is critical for the application of the separating hyperplane theorem.
Theorem
P
t .
Consider a pair of sets of payofffunctions Then the following two conditions are equivalent: 3.3
(II, I*),
where
(i)
lii€$'\J7tdn>0 V;rell} = j/ie£-|Jy^>0 Vyerj.
(ii)
cc(U u {1,-1}) = cc(r u {1,-1}).
Proof: First consider
we
(ii)
implies
(i).
Suppose
that
II
and T are
subsets of
cc(IIu {1,-1}) = cc(Tkj {1,-1}). Then
have:
j//
e
2'|Jffd!/i
VTrellj
£
= |/X6f*|J^d//>0 V;recc(IIu{l,-l})j (By
Lemma 3.2.) = In e -f ^jydfi >
V76cc(ru{l,-1})}
(By assumption.)
= {nep\jydfi>0 (By
Vyer}
Lemma 3.2.)
Now we prove that (i) implies (ii).
Define II
Suppose (without loss of generality) the
(XP'&P) topology
from above {p(-,H)\fi
that the set
is
= cc(U u {1,-1}) and F = cc(T u {1,-1}).
that there exists a
y e f such that y e
fl
.
We know that
generated from a family of open, convex neighborhoods. Recall
of continuous linear functionals on P»
sTX'}. Using these
facts,
is
exactly the set
a corollary to the Hahn-Banach theorem8 implies that
8 See Dunford and Schwartz (1957, p. 421), Kothe (1969, p. 244) for discussions of the relevant theorems about the separation of convex sets. See also McAfee and Reny (1992) for a related application of the Hahn-Banach theorem, where the separating hyperplane also takes the form of an element of J*.
18
since fl
is
closed and convex, there exists a constant c and a
hyperplane)so that P(x,n.) > c for Since {1,-1} € fl and
argue
we can
take c
hypothesis that
/J(7T,/i.)
Because {1,-1} e and thus j J^,
c
ft,
and
Thus,
for
Ke
all
ft.
€
So,
e 7JF (a separating
c.
> c. Now we
=
/?(0, /x.)
But, f}(pk,iL.)
ft.
we
let
c
=
will
= pc
).
T2 but since x2 ,
T,
cT
2 ),
first
is
the closure of a set
A
ff .
under %2
(
.
Thus,
if
This implies Q.E.D.
3.4.
note that, given two topologies
true because the finer topology has
we know
A = AT
that
-A*
2
) is
more closed
T,
and r2 where ,
z, is
contained in the closure of A
A T is finer, there might be a closed set which is a strict subset of A This
"P*,
c
cc(nu{l,-l}) r =cc(ru{l,-l}) T ,then cc(nu{l,-l}) a =cc(ru{l,-l})
To
«
then (11,0
pair.
Tl
sets;
is
a closed set under
but that
still
contains
the linear functionals
/?(7T,/x)
SR and y
:
9?"
-»
SR.
Let
K be a collection of subsets of
conditional stochastic dominance pair
a partial order and all
F 9T x :
->
[0, 1]
if
conditions
k{G\K)
yeT and all KeK,
y(6\K)
For all neU. and all
(ii)
For all
(i)
and
with typical elements
9?".
(ii)
Then the pair (TIX)
are equivalent for all
such that such that F{;6) e A" :
KeK,
(i)
(II, T),
We now extend Theorem 3.4.
23
is
is
nondecr easing
in 6.
nondecreasing
in 6.
is
a
K-
with
Theorem 3.5 Consider a pair of sets of payofffunctions
P
1 .
Let
K
he a collection of subsets of 9t\ where
(Yl,T),
where
II
and T are
subsets of
€ K. Then the following two conditions
9?"
are equivalent:
The pair (TIT)
(i)
cc(II
(ii)
Proof:
is
a ^-conditional stochastic dominance pair.
u {1,-1}) = ccOTu {1,-1}).
We can apply the proof of Theorem 3.4 almost exactly.
likewise for Tk.
Then note
that
cc(U K u{IK ,-IK }) = cc(TK implies
(ii)
show
(i).
that if
One example of a
To show
(ii) fails,
u {IK ,-IK })
that
then
cc(Ilu{l,-l})
(i)
(i)
implies
must
conditional stochastic
= cc(T u {1,-1})
for all K. Then, for every
IK \n
e
II},
and
K we apply Theorem 3.4,
the arguments of Theorem 3.4 can be used to
(ii),
for the case
fail
n^ = {n
Let
implies that
where
K=
9?".
Q.E.D.
dominance theorem which has appeared
in various
forms in
the literature (see Whitt (1982)) involves the set of nondecreasing payoff functions, as follows:
Theorem 3.6 The following conditions are For all
(i)
it
:
9?"
—» 9?
equivalent:
nondecreasing and
all sublattices
#c 91",
k(6\K)
is
nondecreasing
in 6. (ii)
For
all increasing sets
nondecreasing
Ac 9?"
and all
sublattices isfc 9?",
f JAnK
dG(s;6,K)
is
in 6.
Proof: Apply Theorem 3.5 together with the fact that cc({k\k nondecreasing}
This theorem that
condition (for
(ii) is
A (s)
\I
nondecreasing}
u {1,-1}).
Q.E.D.
usually proved using algebraic arguments; but using the results of this paper,
follows as an immediate corollary of
it
Order
is
u {1,-1}) = cc({IA
Theorem
3.5.
First, take the
equivalent to requiring that F(s;6) satisfies the
case where
we
see
n-1. Then
Monotone Likelihood Ratio (MLR)
a proof, see Whitt (1980)), defined as follows:
The parameter 6 indexes the probability distribution F(;6) e A according the Monotone Likelihood Ratio Order (MLR) if, for all 6 > H L there exist numbers
Definition 3.2
1
to
,
-oo 9? and y
conditions (i)
(i)
and
:
—» 91.
9?"
The pair
(II, T) is
For all n sU,
Jt(s)dF(s;6)
f
is
with typical elements
a stochastic supermodularity pair
are equivalent for all lattices
(ii)
(II, F),
© and all F e A"e
supermodular
if
:
in 6.
Js (ii)
Further, pairs,
For all yeT,
we
define the set
analogous to
"L
sm
(U SOM ,r SOM )ismI. !lsr
When comparing
.
[
y(s)dF(s;6)
Z J5r To
to
is
be the
supermodular
set
of
all (II,r)
in 6.
pairs
which
are stochastic supermodularity
place our example from Proposition 3.3 in this framework, the pair
.
Definition 3.1 (stochastic dominance theorem) and Definition 4.1 (stochastic
supermodularity theorem),
it is
helpful to recall that stochastic
dominance theorems and
stochastic
supermodularity theorems are both characterized by (11,0 pairs corresponding to sets of linear functionals of the
Thus,
it is
form
f5(K,-),
which are then composed with parameterized probability
distributions.
possible to compare the two types of theorems directly, even though in a stochastic
dominance theorem, the parameter space theorem, the parameter space
is
any
is
any
lattice.
26
set
with a partial order, while in a stochastic dominance
.
More
we
generally,
can define a class of theorems called Stochastic
defined for an arbitrary property P.
Qp
which are defined so
on 0/' is all
that,
P
Theorems, which are
We are interested in properties P together with parameter spaces
given a function h
:
P
—> 9t,
the statement "h(6) satisfies property
Qp
well-defined and takes on the following values: "true" or "false." Let
such parameter spaces 0,,
Further, for a given property P,
.
we
P
denote the set of
will define the set of admissible
parameter spaces together with probability distributions parameterized on those spaces: n
VP = {(F,e p )\e p
e
e p and F e A%
p
}
Definition 4.2 Consider a pair of sets of payoff functions (IT,r), with typical elements
K
:
SK"
—> 9? and
7:9?"
—> 9?.
(F,Q P ) e
equivalent for all (i)
For all 7teTl,
(ii)
For
j
yeT,
all
f Js
The pair
Vp
(II,
O w a stochastic P pair
is
y(s)dF(s;6) satisfies property
we
let I. SPT
be the
analogous to stochastic dominance,
l(F,e) e
Vp \J7t(s)dF(s;6)
satisfies
= |(F,0) e v;\jy(s)dF(s;G)
In the next section,
e n,
of
set
P on
satisfies
all
(i)
and
(ii)
are
on
P
on 0„.
can be any convex
n(s)dF{s;6)
all
.
stochastic
is
set,
and condition
concave in 6."
P pairs
As
(i)
of
in the case of
for a given property P.
Also
we can rewrite the requirement of Defintion 4.2 as follows: Vtt e
nj
V/ € rl
P on
stochastic
cone method of proving stochastic dominance
P theorems, if P is a closed convex cone property.
The Closed Convex Cone Approach
we
J
we show that the closed convex
theorems can be extended to
4.2
tt
0,,
©p
P
7t(s)dF(s;6) satisfies property
interpreted, "For all
stochastic dominance,
conditions
:
For example, for a stochastic concavity theorem, Definition 4.2
if
is
Valid for all
CCC Properties
which we
call
"Closed Convex Cone" (CCC)
properties, such that the following statement is true: for all properties
P which are CCC, if a pair (IIJ)
In this section,
is
identify a class of properties,
a stochastic dominance pair, then (tl.D
is
a stochastic
closed convex cone method of proving stochastic
P
pair; thus, for all
P theorems is valid.
CCC
This result
is
properties, the
useful because
allows us to use the existing theorems from the stochastic dominance literature to generate classes of theorems.
CCC properties are properties are defined as follows:
27
it
many new
A property P is a CCC property (written P eCCC) if the set offunctions —> SR which satisfy P forms a closed convex cone, where closure is taken with respect to the topology of pointwise convergence, and if constant functions satisfy P.
Definition 4.3
g
Qp
Note while
we
functions,
The
we
that
are using closure under the topology of pointwise convergence for properties P,
weak topology
are using closure under the
(as defined in Section 3.1.1) for sets of payoff
n and T.
properties nondecreasing, concave, and supermodular are all
CCC
properties, as is the
property "constant." Further, any property which places a sign restriction on a mixed partial derivative is
CCC.
Finally, since the intersection of
two closed convex cones
of these properties can be combined to yield another
"nondecreasing
and
convex"
CCC
CCC
a
is
is itself
For example, the property
property.
This
property.
useful in the context of comparative statics for the following reason.
/i:SR—»SR andg:9?"
—>SR.
If h(x) is
n in II,
and 6
nondecreasing and convex in 6 for
all
supermodular and monotone), then
7t(s)dF(s;g(y))
is
is
is
property
is
Consider two functions,
convex and nondecreasing, and g(y)
monotone (nondecreasing or nonincreasing), then h(g(y))
J
a closed convex cone, any
is
supermodular. Thus,
supermodular and if f K(s)dF{s\6) is
by 6 - g(y) (where g
is
dominance pair, then (H,T)
is
in fact determined
supermodular in
y.
Now we prove a result which builds from Theorem 3.4: Theorem
Suppose property
4.1
a stochastic
P pair.
PeCCC.
If (11,0 is a stochastic
u {1,-1}) = cc{T u {1,-1}),
Equivalently, if cc(Jl
then (FIX)
is
a stochastic
pair.
Proof:
For all
=» (Since
J
dF(s;6)
=>
\[ajz
x
(s)
ell, ^K{s)dF(s;0) satisfies P.
For all n e
II
u {1,-1},
= 1, and constant functions For all n e
jn^s) dF(s;6)
(Since
it
satisfies
cc(Jl
x
=>
{1,
satisfies P.
satisfy P, for all
P eCCC).
-1} ),
J
x(s)dF(s;6) satisfies P.
P and jn2 (s) dF(s;6)
+ a2 K2 {s)] dF(s;6) = a \n
a,,a2 > 0, since
u
f K{s)dF(s;0) Js
{s) x
satisfies
P implies
dF(s;6) + a2 JK2 (s) dF(s;6) satisfies
P is a CCC property). For all K e cc(IIu {1,-1}),
28
J
n(s)dF(s;0) satisfies P.
P for
P
.
(Recall that a subset of a topological space
convergent nets of elements in that all 11,
for
JKa (s) [
dF(s;6
7t(s)dF(s;6)
satisfies
Ka
any net
P
)
is
in cc(Il
u {1,-1})
-» $7i(s) dF(s;6
)
is
closed
if
and only
if
it
contains the limits of all the
Since the linear functional fi(;fi)
na —> n
such that
(Kothe, 1969, p.l
,
6
then given
Since
1).
the pointwise limit of a net of functions
which
P
a
is
is
continuous for
,
CCC property,
For all yecc(ru{l,-l}),
(By Theorem
3.6, since
=>
K(s)dF(s;0)
satisfy P, then j
(n.O
is
J
y(s)dF(s;6) satisfies P.
a stochastic dominance
For all y e T, j y(s)dF(s;6)
pair.)
satisfies P.
T Q cc(T u {1,-1}))
Precisely analogous arguments establish the symmetric implication.
This theorem generates
many new
been identified in the large case, this
and
as well.)
=>
(Since
set. 10
literature
theorem generalizes a
to nondecreasing functions
classes of stochastic
Q.E.D.
P theorems, where the
(11,0 pairs which have
on stochastic dominance are also stochastic
result
by Topkis (1968), who proves
and indicator functions of nondecreasing
P pairs. As a special
that the (11,0 pair corresponding sets, respectively, is
a stochastic
P pair for P in CCC. What we have shown stochastic
P pairs,
Y SDT = E J/T
.
In
then
in this section is that if
£ JDr c Z Jpr However,
Remark 2
.
at
Z JDr c ^ SFr
-
property, the property "constant in 6."
nondecreasing payoff functions, and the set constant theorem," since j 7t(s)dF(s;d)
However, (I1 F0 T) ,
properties
which
note that not
then end of Section 4.4,
"nondecreasing and convex," then
CCC
P is a CCC property,
is
For now,
T = -IT
properties
.
dominance
10 See Kothe (1969, pp. 10-11) for a proof of this statement.
spT
P
denotes the set of in
CCC
all
are such that
P
argue that
let
us consider a simpler example of a
The
pair.
IT
F0
pair (IT
6 if and only
will satisfy equivalence relationships.
29
ro
l,
we
Take the case of
constant in
clearly is not a stochastic
all
and
if
when
the property
the set of
,
FO ,
T)
univariate,
satisfies a "stochastic
-\n(s)dF(s;d)
The next
all
is
is
constant in
6.
section identifies a class of
Linear Difference Properties
4.3
This section characterizes a subset of
CCC properties, which we call Linear Difference Properties we
(LDPs). These properties are interesting because, as
will
show
in Section 4.4, for all
P in LDP,
=£
T.
(4.1)
Combining the "closed
(4. 1)
with Theorem 3.4,
convex cone" method
P is an LDP.
That
is, if
Pe LDP, LDPs
Important examples of
Supermodularity predictions.
We
is
is
we can
then
(II,
OeZ „ s
are supermodular
important because of
also
conclude
that, just as in the
case of stochastic dominance,
P theorems when = cc(ru{l,-l}).
exactly the right one for the study of stochastic
emphasize the
if
and only
if
cc(IIu{l,-l})
and concave; others are summarized
role in the analysis of
its
in
Table
monotone comparative
I.
statics
about concavity, since concavity and convexity are also
result
frequently encountered in economic contexts; for example, if an objective function is concave, then the First
Order Conditions characterize the optimum. Further, concavity can be used to establish the
existence of supporting prices in a resource allocation problem.
The
result described in equation (4.1) is useful
(II, T) is
because
dominance; by
(4.1),
checking to see
if (II
,D
is
this set II
easier to verify whether or not
same
are determined by an economic problem, and
has been analyzed in the stochastic dominance literature, then the corresponding stochastic
supermodularity theorem
r have
may be
a stochastic dominance pair answers the
question. For example, if the characteristics of a set II
not
it
a stochastic supermodularity pair by drawing from the existing literature on stochastic
the
is
immediate. This allows us to bypass the step of checking whether II and
same closed convex cone direcdy
(further,
most of the stochastic dominance
literature
does
(LDPs). The
first
make such a statement explicitly).
We now begin to build our formal definition of Linear Difference Properties important feature of LDPs
is that
they can be represented in terms of sign restrictions on inequalities
involving linear combinations of the function evaluated at different parameter values. For example, H L L if and only if g(6 ) - g(d ) > for all 6" > 6 in 0. This statement g(0) is nondecreasing on specifies a set of inequalities,
where each inequality
high parameter value and a low parameter value. if
and only
x
if
g(6
a difference between the function evaluated another example, g(d)
v 6 2 )- g(6') + gtf a0 2 )- g(d 2 )>0 1
specifies a set of inequalities, this time involving the
In both cases,
is
To take
for all
sum of two
we can represent every inequality by
6\6
2
in
0.
is
at
a
supermodular on
Again,
this
statement
differences.
the parameter values
and the coefficients which
on the corresponding function values. For example, for the property nondecreasing, each inequality involves the coefficient vector (1,-1) and a parameter vector of the form (d H ,8 L ), where 1 is are placed
the coefficient
on g(d") and -1
is
the coefficient
30
on g(6
L ).
Thus, g(6)
is
nondecreasing on
if
and
only
for
if
^aig
{
3 9080 03130 2398 | HB31 .M415
working paper department of economics
CHARACTERIZING PROPERTIES OF STOCHASTIC OBJECTIVE FUNCTIONS Susan Athey 96-1
Oct. 1995
massachusetts institute of
technology 50 memorial drive Cambridge, mass. 02139
CHARACTERIZING PROPERTIES OF STOCHASTIC OBJECTIVE FUNCTIONS Susan Athey 96-1
Oct. 1995
INTRODUCTION
1
This paper studies optimization problems where the objective function can be written in the form
V(8)
=
\jt{s)dF{s\6),
payoff function,
;ris a
For example, the payoff function
real vectors.
the vector s
where
F
is
a probability distribution, and 6 and s are
n might represent
might represent features of the current
state
an agent's
utility
or a firm's profits,
of the world, and the elements of 6 might
represent an agent's investments, effort decisions, other agent's choices, or the nature of the
exogenous uncertainty in the agent's environment.
The economic problem under consideration often determines some properties of function; for example, a utility
the payoff
function might be assumed to be nondecreasing and concave, while a
multivariate profit function might have sign restrictions on cross-partial derivatives.
assumptions then determine a set questions such as:
investments? questions,
Is
IT
a set of investments 6
Does one investment increase
we need
know whether
to
is
These
We might then wish to answer
of admissible payoff functions.
worthwhile? Are there decreasing returns to those
the returns to another investment?
V(Q) satisfies the appropriate properties,
To answer
i.e.,
those
nondecreasing,
concave, or supermodular. 1
The goal of
paper
this
is to
develop methods such
nondecreasing), a set of admissible payoff functions
we can J
determine whether the following statement
7i(s)dF(s; 6) satisfies property
That
is,
given
P
and
II,
II,
that,
given a particular property
(1.1)
describe the set of probability distributions which satisfy (1.1).
This paper restricts attention to the case where the properties cones."
which
P
and
sets II are "closed
convex
We define a property P to be a "Closed Convex Cone" (CCC) property if the set of functions
satisfy
P is closed (under an appropriate topology), positive combinations of functions in the set
are also in the set,
and constant functions are
in the set. Important
nondecreasing, concave, supermodular, any property which
is
examples of CCC properties include
defined by placing a sign restriction on
a partial derivative, and combinations of these properties. Thus,
consumer's
1
(such as
is true:
P in 6 for all n in IL
we wish to
P
and a parameterized probability distribution F,
utility, II
Intuitively, a function is
to increasing the other.
if
the payoff function represents a
could be the set of univariate, nondecreasing, concave payoff functions;
supermodular
When
if.
if the
given any two arguments of the function, increasing one increases the returns
the function
is differentiable, this
amounts to positive cross-partial derivatives between
every pair of arguments. Supermodularity is important in the context of comparative statics (see Topkis (1978), Milgrom and Roberts (1990, 1994), Milgrom and Shannon (1994)).
payoff function represents a firm's profits as a function of two complementary quality innovations,
n
could be the set of bivariate, nondecreasing, supermodular payoff functions.
For
sets II
P
and for properties
which
probability distributions satisfy (1.1).
be a very large
set.
Thus we
ask,
are
CCC,
Checking
When is
it
this
paper develops a characterization of which
(1.1) directly is difficult because, in general, If
might
possible to find a smaller set of payoff functions, denoted
T, so that for any probability distribution F, statement (1.2)
below
will
be true
if
and only
if (1.1) is
true?
7t(s)dF(s;8) satisfies property J
P in 6 for all % in T
We can think of T as a "test set" easier to check than II, but
only the information that
Using these
ideas,
it
P is
T
is
a set of payoff functions which
whether
test
J
is
7C(s)dF(s;0) satisfies property
.2)
smaller and
P in
6,
given
K is in the larger set II.
we can restate the goal
the best "test set" for a given II.
property
for II: ideally,
can be used to
( 1
We
of the paper:
proceed in two
"nondecreasing"; second,
we
study other
we want
steps.
a theory which helps us determine
First,
we examine
CCC properties.
The
the case
first
where the
case has been the
focus of the literature on stochastic dominance, where different authors have studied different sets
II. 2
This paper unifies and extends the existing literature on stochastic dominance, providing an exact characterization of the mathematical structure underlying all stochastic further provide an algorithm for generating
hoc
differentiability
part of the paper
when V{6)
we show
that the
We now provide
stochastic
an overview of our is
results.
We
dominance theorems which relaxes the ad
common
in this literature. 3 In the
methods from stochastic dominance can be applied
satisfies other properties P, including
property nondecreasing, if II is
new
and continuity assumptions which are
dominance theorems.
second
to characterize
concavity and supermodularity. In the first part of this paper,
we show
that for the
a closed convex cone and contains constant functions, then the best
T
the set of "extreme points" of II. Just as a basis generates a linear space via linear combinations, so
a set of extreme points generates a closed convex cone via positive linear combinations and limits.
Thus,
want
2
we show that if we know only that a payoff function k lies in the closed convex cone II, and we know if V(6) is nondecreasing, it is equivalent to check that V(0) is nondecreasing on a smaller
to
In particular, the univariate stochastic dominance problem has been studied by Rothschild and Stiglitz (1970, 1971) and Hadar and Russell (1971); notable contributions to the multivariate problem include Levy and Paroush (1974), Atkinson and Bourguignon (1982), and Meyer (1990). Shaked and Shanthikumar (1994) provide a reference book on the subject of stochastic orders and their applications in economics, biology, and statistics. 3 Brumelle and Vickson (1975) take a first step towards relaxing these assumptions and identifying the mathematical structure behind stochastic dominance; in contrast to their work, which provides only sufficient conditions for a stochastic dominance relationship, this paper provides an exact characterization of all stochastic dominance theorems.
set
of payoff functions, the extreme points of n.
V(6) for the set of extreme points of
extreme points of
n
n
It
than for
will often
n
be much easier to verify monotonicity of
In this paper, the procedure of using the
itself.
as a test set for II will be referred to as the "closed
convex cone" method of
proving stochastic dominance theorems.
Examples from the existing include First Order Stochastic
dominance
stochastic
which are special cases of this
literature,
Dominance (FOSD), where
n is the
set
of univariate, nondecreasing
payoff functions, and Second Order Stochastic Dominance (SOSD), where TI
concave payoff functions.
In the case of
functions which are zero up to the case of 2,
SOSD,
some
the set of extreme
FOSD,
result,
the set of univariate,
is
the set of extreme points is the set of one-step
constant, and one thereafter.
points is the set
These are pictured
in Figure
1
.
In
of "angle," or "min" functions, pictured in Figure
where each function takes the minimum of its argument and some constant.
The closed convex cone method
for stochastic
dominance has been recognized and explored
in the
context of particular sets of payoff functions (Topkis, 1968; Brumelle and Vickson, 1974; Gollier and
Kimball, 1995). 4 However, this paper goes beyond the existing literature in two respects.
developing appropriate abstract definitions to describe stochastic dominance theorems,
make
we are
by
able to
general statements about the entire class of stochastic dominance theorems, including those
which have not yet been considered that
First,
we prove
a
new
in the
cone" approach to stochastic dominance return the
same answer
and (1.2) equivalent. In
is
if
we prove
exactly the right one.
these are not in T)
is
shows
characterize other properties of V(6).
that the "closed
the fact
By this we mean that (1.1) and
IT;
no other T's
which generates
(1.2)
will always
make
(1.1)
II is the best set to check.
convex cone" method can also be applied
to
We ask two questions: First, for what properties P is the closed
convex cone approach valid? And second, for what properties
P is
exactly the right one, as in the case of stochastic dominance? property, then the closed
is
formally that the "closed convex
F if and only if the closed convex cone of T
equal to
this case, clearly the smallest set
part of this paper
Second and more important
literature.
for every probability distribution
(union the constant functions,
The second
economics
result about this class of theorems:
the closed
We
first
convex cone approach
show
that if
P
is
convex cone approach can always be used to characterize when V(6)
a
CCC
satisfies
We then find a subset of CCC properties, which we call "Linear Difference Properties," for which we can show that the closed convex cone approach is exactly the right one for checking whether V(0)
P.
satisfies P.
Examples of Linear Difference Properties include monotonicity, supermodularity,
concavity, and properties which place sign restrictions
on
partial derivatives.
Combinations of these
properties, however, are not in general Linear Difference Properties, although such combinations are
4Independently, Gollier and Kimball (1995) argue for what they
4
call the "basis
approach" to stochastic dominance.
I
CCC
properties.
Table
I
summarizes properties which are
CCC
properties and Linear Difference
Properties.
TABLE LINEAR DIFFERENCE PROPERTIES AND CLOSED CONVEX CONE PROPERTIES Property
Closed Convex Cone
Linear Difference Property
Nondecreasing
Yes
Yes
Supermodular
Yes
Yes
Concave/Convex (Multivariate)
Yes
Yes
Sign Restriction on a Partial
Yes
Yes
Constant
Yes
No
Nondecreasing and Concave
Yes
No
Arbitrary Combinations of
Yes
No
Yes
No
Derivative
CCC Properties Arbitrary Combinations of
LDP
Properties
For many literature
cases, the
on
sets
of payoff functions, n, which are
stochastic
dominance has
commonly
studied in economics, the existing
implicitly identified the extreme points of those sets. In such
problem of characterizing a Linear Difference Property (such as supermodularity) becomes
quite straightforward: simply look to the the existing literature to find the appropriate test set, T, for the set
of payoff functions II under consideration. Then, using the results of this paper,
(1.2) characterizes the set
function will be supermodular for
We
illustrate this
we know
that
of parameterized probability distributions for which the stochastic objective all
payoff functions in
technique by showing
how
II.
the closed convex cone approach can be used to
characterize the property supermodularity for several important classes of payoff functions. These results
can in turn provide sufficient (and sometimes necessary) conditions for comparative
conclusions. In particular,
we examine
statics
applications in principal-agent theory, welfare economics, and
the study of coordination problems in firms.
This paper proceeds as follows.
In Section 2,
we
introduce a motivating example, the problem of
a risk- averse agent's choice of effort. In this problem,
concave, and supermodular for the agent's expected
dominance takes place
in Section 3.
We provide
we characterize
utility function.
the properties nondecreasing,
Our general
an exact characterization of stochastic dominance
We further extend our
theorems, highlighting the important role played by linearity of the integral. result to incorporate "conditional stochastic
properties of
we
dominance." Section 4 develops characterizations of other
7t(s)dF(s;d) and provides applications of the property supermodularity. In Section 5, f
K(x,s)dF(s;6) are supermodular or
analyze conditions under which functions of the form
concave
analysis of stochastic
in (x,d),
showing how
J
to apply stochastic monotonicity results to this
problem
as well.
Section 6 concludes.
2
MOTIVATING EXAMPLE:
A RISK-AVERSE AGENT'S CHOICE OF EFFORT In this section,
we present
a motivating example, where
how
of effort affects her expected payoffs, and
we
analyze
Formally,
nondecreasing, concave, and supermodular for the agent's expected
examples
stochastic
—
examples where the payoff function
is
illustrate the parallel structure
Consider a risk-averse agent whose technology, where the output
is
denoted
s.
Suppose t
utility function.
another. Then,
t
we can write the agent's problem as max
observe that
it is
not
an increase in effort which
results are
nondecreasing and concave
—
of a
is
underlying the three classes of theorems.
that the agent's effort (c) affects the probability
k(s) dF(s; e,t)-c(e) •
f
=
\it(s) dF(s;e,t) is
some
-
c(e) is
= ae such
that the
FOCs
effort:
utility if the effort also
concave is
in effort, then
useful in the analysis
as principal agent problems. Further, if V(e,i) fails to
linear cost function c(e)
optimum. Finally, observe
nondecreasing in
productive on average might not increase expected
order conditions (FOCs) characterize the optimum, a property which
then there exists
to
follows:
trivial to verify that V(e,t)
some economic problems, such
in the stochastic
might represent a worker moving from one job
increases the riskiness of the distribution. Second, note that if V(e,t)
e,
These
which represents exogenous changes
production technology. For example, a change in
first
exogenous
characterize the properties
depends on the output of a stochastic production
utility {it)
distribution of s. Further, consider a parameter
the
we
dominance theorem, a "stochastic concavity theorem," and a "stochastic supermodularity
theorem." These examples
First
a risk-averse agent's choice
that choice of effort interacts with
parameters which describe the probability distribution.
specific
how
fail to
be concave
in
characterize the
that if V{e,i) is supermodular, then the optimal choice of effort, e*(t), is
nondecreasing in
If V(e,t) fails to
t.
linear cost function c(e)
=
ae
Milgrom and Shannon (1994); satisfies the
be supermodular, then even
such that e*{i)
fails to
Theorem A.l
see
in the
cannot be relaxed as long as
it
V is
concave, there exists some t
follows from
(this
Appendix). Thus, the requirement that
property supermodular (or concave, respectively)
conclusion, and further
if
be nondecreasing in
we
is
sufficient for the desired
V(e,t)
economic
require that the conclusion holds for
all
linear cost functions.
Let us
first
identify conditions under
known
result is adapted
(where
we suppress
Proposition 2.1
t
which
from the Rothschild and
nondecreasing in
V(e,i) is
Stiglitz (1970,
e.
The following
well-
1971) work on stochastic dominance
in the notation):
The following two conditions are equivalent for
all probability distributions
F(-,e): (i)
For all K nondecreasing and concave, The following are
(ii)
dF(s;e)
is
nondecreasing in
e.
satisfied:
\sdF(s;e)
(a)
\lt(s)
is
nondecreasing
in
e.
a
For all a, -\F(s;e)
(b)
Intuitively, for a risk-averse agent
mean income
effort increases the
This result
is
is
nondecreasing
who likes income,
(condition
(ii)(a))
often used in the finance literature;
it
in e.
effort will increase
expected
utility if
and only
and reduces the "risk function" (condition
if
(ii)(b)).
has been called Second Order Monotonic Stochastic
Dominance (SOMSD).
we
In this paper,
conditions
(i)
and
(ii)
will
A
1
Proposition 2.1 (where
extended real
Proposition (i)
For
all
nondecreasing (ii)
in e.
u {-00
,
the set Yl
which emphasizes
The following proposition SR,
is
that
equivalent to
and Si indicates the
00 }):
The following two conditions are equivalent for
n in
2.1,
be the space of probability distributions defined on
line, that is, SR
2.1'
work with a restatement of Proposition
are actually symmetric conditions.
SOM
= [ti\k
:
SR
-»
all
F(-;e)eN:
% nondecreasing, concave},
J
7t(s)
dF(s;e)
is
in e.
For all yin
the set
T SOM = {y\y(s) = min(a,s), a e 9t},
jy(s) dF(s\e)
is
nondecreasing
.
Conditions
(i)
and
(ii)
we have
except that
of this Proposition simply restate conditions
straightforward to verify (using integration by parts) that the former term if
the latter term
This condition
is.
(1975) for an exception).
This
way of
structure underlying the stochastic that instead of
relatively large set, II
50M ,
sets of
the union of
is
set,
T
is itself
relationship
y{s)
Now,
a closed convex cone.
between
let
Tl
and
r
nondecreasing
=
and
1
is
illustrates the 1',
mathematical the result says
for all payoff functions in the
X
is
nondecreasing in e for
all
SOM
all
=
between the two
of
relationship
That
-1.
latter set,
In Section 3,
holds for
The
2.
we
we can
will
stochastic
is,
show
by taking
to analyzing principal-agent
generate any function in TI SOM , that this "closed
convex cone"
dominance theorems.
When is
problems
are satisfied, then the agent's choice of effort
sets
positive combinations
J
x(s) dF(s;e) concave in effort? This
problem was addressed by Jewitt (1988), who analyzes conditions under which the
FOCs
and only
equal to the closed convex cone of the set which
y(s)
us ask a different, but related, question:
Approach (FOA)
if
is
It
.
and the functions
SOM
is
(ii).
2.1,
S0M
and limits of sequences (or nets) of elements of the
which
of Proposition
in Proposition 2.
nondecreasing in
payoff functions are pictured in Figure
T S0M
theorem
dominance theorem. As written
S0M payoff functions can be described as follows: T1 is
SOMSD
equivalent to check that \y(s)dF(s;e)
it is
payoff functions in the smaller
The two
writing the
checking that j n(s) dF(s;e)
(ii)
SOMSD (see Brumelle and Vickson
not usually associated with
is
and
jmin(a,s)dF(s;e) in condition
with
replaced -JF(s;e)
(i)
is valid.
The
First
Order
FOA requires that if the agent's
must be optimal. Extending
Jewitt' s analysis,
we
can show that the sufficient conditions he derives are in fact necessary. 5 The following result analogous to Proposition
Proposition 2.2 (i)
(ii)
For
The following two conditions are equivalent for
all
For
all
% in
is
2.1:
the set Yl
yin the set
S0M
\k(s) dF(s;e)
is
concave
in e.
y(s) dF(s;e)
is
concave
in
•
,
T SOM
all
',
\
F(;e)e A
1
:
e.
Proof: This will be established as a simple corollary of Theorem 4. 1 below, together with Proposition 2. 1
Q.E.D.
5 Jewitt first derives conditions under which the agent's utility function is increasing and concave in output, given the optimal contract; he then addresses the question of concavity of the expected utility function in effort. It is only the latter question which we study here.
Proposition 2.2 says that expected payoffs are concave in e for
only
expected payoffs are concave in e for
if
useful because condition
Condition
(
n
SOM ,
T
SOM
T SOM
payoff functions in
easier to check that condition
(ii) is
(i):
U SOM
payoff functions in .
the set
Again,
r
SOM
this
much
is
if
and
theorem
is
smaller.
can be interpreted as requiring that there are decreasing returns to e in terms of
(ii)
increasing the
all
all
mean and
), is
the
decreasing the "risk function." Note that the pair of sets of payoff functions,
same
in
both propositions.
Now we turn to ask a final question: When is the optimal choice of effort monotone nondecreasing in
t,
which parameterizes the stochastic production technology? More
way
probability distribution over output in such a
that
precisely,
how can
t
affect the
monotonicity of the optimal effort in
is
t
ensured, without any additional information about the cost of effort function or the agent's preferences?
This question has not been answered in the existing literature; thus, the following
proposition provides a
new
Proposition 2.3
monotone
cost functions c
and all K in
Il
SOM ,
e'(t)
all F(-;e,f)e
S0M
in
(i)
For all n e
(ii)
For all ye T S0M jy(s)
Tl
,
t.
is
supermodular
:
is
Js
in {e,t\
dF(s;e,t) is supermodular in
Proof: The equivalence of parts (1994), as stated in
1
6
\K{s)dF{s;e,t)
,
A
= argmax\K(s)dF(s;e,t)-c(t) e
monotone nondecreasing
that this proposition is
in effort.
The following three conditions are equivalent for
(MCS) For all
Note
insight into the comparative statics problem.
true irrespective of whether expected utility is
Theorem A. 1
(MCS) and in the
(i)
(e,t).
follows directly from
Milgrom and Shannon
Appendix. The equivalence of (i) and
established as a corollary of Theorem 4.1
below together with Proposition
(ii)
will
be
2.1.
Q.E.D.
The formal
definition of supermodularity (and the comparative statics
can be found in the Appendix. Intuitively,
V(e,t) is
supermodular
Proposition 2.3 provides necessary and sufficient conditions for
problem. If
(ii) is
choice of effort
is
violated, then
not monotonic in
monotone comparative increasing the
we
statics.
mean of the
if t
theorem which
relies
statics in this
can construct payoff functions and cost functions such that the t.
Thus,
Condition
we have
(ii)
identified the exact conditions
requires that e and
t
are
which ensure
complementary
probability distribution and in terms of reducing the risk.
may be a
Order, as defined in the Appendix.
set.
it)
increases the returns to effort.
monotone comparative
Then,
this
theorem requires
that the set
be nondecreasing
in terms of
The
straightforward: since a risk-averse, income-loving agent likes high expected returns and
"In general, the optimal e
upon
intuition is
low
risk (as
in the Strong Set
shown
SOMSD),
in
complementary
are
variables
which
in increasing
complementary
are
expected
utility
in increasing the
mean and
of such an agent. Note that
decreasing the risk
one of e and
if either
/
does not affect the mean or the riskiness of the agent's income, the corresponding complementarity conditions are satisfied trivially.
Notice that Propositions 2.2 and 2.3 have a structure which
dominance
stochastic
result, as stated in
analyzing Proposition 2.1
very similar to the existing
is
Proposition 2.1. In Section 3,
we
will build a
and other stochastic dominance theorems. In Section
we
4,
framework
for
will formalize
the relationship between Propositions 2.1 through 2.3.
MONOTONICITY OF STOCHASTIC OBJECTIVE FUNCTIONS
3
The goal of theorems.
dominance
two
sets
this section is to
provide a unified framework for analyzing stochastic dominance
we
introduce a framework which incorporates the existing stochastic
In Section 3.1, literature,
arguing that each stochastic dominance theorem describes a relationship between
we prove
of payoff functions. In Section 3.2,
relationship
between two
the stochastic
sets
of payoff functions which
a result which characterizes a mathematical equivalent to the relationship determined by
is
dominance theorem. Section 3.3 extends the
result to the case
of conditional stochastic
dominance.
3.1
A
Unified
In this section,
Framework for
we
Stochastic
Dominance
introduce the framework which
we
will use to discuss stochastic
dominance
theorems as an abstract class of theorems, and to draw precise parallels between stochastic dominance theorems and other types of theorems.
Let us Stochastic
first
consider another well-known example of a stochastic dominance theorem, First Order
Dominance (FOSD). This theorem can be
(where IA (s)
stated as follows
is
the indicator
function for the set A):
Proposition 3.1 (i)
(ii)
The following two conditions are equivalent for
For all n e
n
For all y e
TO
s {k\k
:
9i
-»
% nondecreasing},
M
T FO m {y\y(s) = I
(s),
This theorem has the same structure as the functions are illustrated in Figure are nondecreasing in
8
1.
ae
Sfi},
J
all
F(;0)e A
1
K(s)dF(s;6)
jy(s)dF(s;6)
is
is
:
nondecreasing
nondecreasing
SOMSD theorem, Proposition 2. 1
'.
The
in ft
in ft
sets
of payoff
This theorem says that instead of checking that expected payoffs
for all nondecreasing payoff functions,
expected payoffs are nondecreasing in 6 for
all
10
IF
,
it is
payoff functions in the set
equivalent to check that
T FO
,
the set of indicator
The
functions of upper intervals.
easier to check that condition 1
- F(a;6),
The
of payoff functions
condition
is
is
much
smaller, and so condition
(ii) is
can be reduced to a restriction which requires that
(ii)
complement of the cumulative
the
requirement
latter
latter set
(i);
distribution function,
nondecreasing in 8 for a e
is
9?.
more standard way of stating FOSD.
the
There are many other examples of stochastic dominance theorems, some with multiple random
we
variables;
theorems
all
will discuss other
examples below
stochastic dominance theorems are characterized that the pair
(IT™,
r
F0
(U SOM T SOM )
satisfies the
)
same
theorem." If that statement
is true,
will use the abstract definition to
and how
that class of
then
make
and
different (11,0 pairs.
To make
relationship.
"A pair of sets
definition of the statement
by
satisfies a particular relationship; the
,
In general, stochastic dominance
in Section 4.5.
illustrated in Propositions 2.1'
have a parallel structure, as
FOSD
3.1.
The
However,
different
SOMSD theorem states
theorem
states that the pair
we
introduce a formal
this relationship precise,
of payoff functions, (n,D, satisfies a stochastic dominance
we
dominance
will say that (11,0 is a stochastic
pair.
We
statements about the class of stochastic dominance theorems,
We allow for multidimensional
theorems relates to other classes of theorems.
payoff functions and probability distributions, using the following notation: the set of probability distributions
on
SR" is
we
parameter space 0, distributions
F
:
x
9?"
denoted A", with typical element F:SR n
—»[0,1].
will use the notation A"
e to represent the set
-»
such that such that F(;0) e A" for
[0,1]
all
Further, for a given
of parameterized probability
ds 0.
Definition 3.1 Consider a pair of sets of payoff functions (TI,r), with typical elements
K 9T —> SR and y :
and
:
9?"
—> SR.
The pair
(II, T) is
are equivalent for all parameter spaces
(ii)
(i)
For all ttgU,
(ii)
For all y eT,
Further,
we define the
j
n(s)dF(s;6) y(s)dF(s;6)
is
is
J
set I. SDT to
be the
set
a stochastic dominance pair with a partial order
nondecreasing nondecreasing
and all
conditions
if
(i)
n
F € Ae
:
in 6.
in 6.
of all (11,0 pairs which are stochastic dominance
pairs, as
follows:
2 5Dr = {(II,r)|(n,r) Thus,
when
a given (11,0 pair
is
is
a stochastic dominance pair}
a stochastic dominance
pair,
we write
(II, T)
Definition 3.1 clarifies the structure of stochastic dominance theorems.
e Z SDr
.
Stochastic dominance
theorems identify pairs of sets of payoff functions which have the following property: given a
parameterized
probability
lAn(s)dF(s\6)\iz
e
II
>
distribution
F, checking that
all
of the functions in the set
are nondecreasing is equivalent to checking that all of the functions in the set
11
(fy(s)dF(.s;0)|y
er|
T is
general, the set
Stochastic dominance theorems are useful because, in
are nondecreasing.
smaller than the set
II.
We can think of this definition as a statement about the equality of two sets. set
First, let
us define the
of admissible parameter spaces for the property "nondecreasing" together with probability
distributions parameterized
VND = {(F,0)|© has a Now, we can
on those spaces
partial order
and
as follows:
F e Ane j
rephrase the definition as follows: (11,0
j(F,0) €
VN M 7r(s)dF(s;d) is nondecreasing on
|(F,0) e
VND \JY(s)dF(s;0) is nondecreasing on
V/r e Ili
n
Vy e Ij
existing literature generally
J
7t(s)dF(s) and
Si. In contrast,
functional
a stochastic dominance pair if
n
Definition 3.1 differs from the existing literature
viewing
is
J
(i.e.,
=
Brumelle and Vickson, 1975),
compares the expected value of two 7t(s)dG(s) as
two
different linear functionals
by parameterizing the probability
distribution
J
when we
distributions to the reals,
become more
and stochastic
P theorems, for other properties P (such as supermodularity). specific
set of
many
are
The
utility
of this
formalize the relationship between stochastic dominance
examples of pairs of
sets
of payoff functions which satisfy stochastic
dominance theorems, we summarize three univariate stochastic dominance theorems are potentially
we
dominance theorems and stochastic supermodularity
definition will
To provide
to
n(s)dF{s\6) as a bilinear
theorems, an analogy which would not be obvious using the standard constructions. clear
F and G,
mapping payoff functions
and viewing
mapping payoff functions and (parameterized) probability
able to create an analogy between stochastic
in that the
probability distributions, say
in
Table
It.
There
other univariate stochastic dominance theorems (for example, theorems where the
payoff functions imposes restrictions on the third derivative of the payoff function); however,
we will
simply report the three most familiar univariate stochastic dominance theorems here.
12
TABLED COMPONENTS OF UNIVARIATE STOCHASTIC DOMINANCE THEOREMS"
Sets of Payoff Functions, II
U F0 = {n\n
(i)
n 5° s {n\n
(ii)
:
:
SR
SR
% nondecreasing} -> % concave}
r F°={ r \y(s) = I
->
[a
Yl
SOM
= \n
k
SR
:
—»
SR,
nondecreasing,
„
(s),aeX} )
r S0 ={Y\Hs) = -s} u{y|y(.y)
(iii)
T
Sets of Payoff Functions,
i
= min(a,s), aeSRJ
r «w _ |y| y (j) _ min(fl,s),
1
a e 9?}
concave
Each
(II,r) pair in
Table
II (iii)
corresponds to a 3.1
and Table It is
(ii).
Table
II is
a (univariate) stochastic dominance pair.
corresponds to a
SOMSD theorem,
FOSD theorem, Proposition 3.1.
and only
K{s)dF{s;8)
that
depend on
\min(a,s)dF(s;d)
if
6.
shown
now
Let us
1
in Proposition 2.
',
while Table
illustrate the interpretation
II (i)
of Definition
with a third example: Second Order Stochastic Dominance (SOSD), shown in Table
II
known
is
J
if
as
Observe
is
nondecreasing in 6 for
all
nondecreasing in 8 for
that both y(s)
= s and
the distribution to be both nonincreasing
y(s)
= -s
univariate,
all
II
concave payoff functions
aeSi, and \sdF(s;6) does not
are included in
Tso
this forces the
;
and nondecreasing, and hence constant
mean of
in 6.
There are many other stochastic dominance theorems in addition to the univariate examples given above. Levy and Paroush (1974) derive results for bivariate functions, while these results and examines
some of these
some
multivariate stochastic
results in Section 4.5,
Meyer (1990) extends
dominance theorems as
where the main objective
is to
well.
We will report
apply these results to problems of
stochastic supermodularity.
Exact Conditions for a Stochastic Dominance Theorem
3.2
In this section,
dominance
statement that 3.2.1
and
we
study necessary and sufficient conditions for the pair
(11,1") to
be a stochastic
We want to specify the exact mathematical relationship which is equivalent to the (II, T) e Z sor We will first discuss our result and its implications; then, in Sections
pair.
.
3.2.2,
The main
we will provide the mathematical
result
of this section
is that (II,
arguments underlying the
T) e
the closure (under an appropriate topology) of the
13
I, SDT if
and only
if
result.
the following statement
convex cone of II u {1,-1}
is
is true:
equal to the closure
(under that topology) of the convex cone of
two constant
functions,
{n{s)
=
1}
Tu {1,-1},
where {1,-1} denotes the
set containing the
u {it{s) = -1}. We formalize this using the following notation:
cc(nu {1,-1}) = cc(ru {1,-1})
(3.1)
In the context of specific sets of payoff functions
n, the existing
literature identifies similar, but
stronger sufficient conditions for the corresponding stochastic dominance theorems, using a restrictive notion
of closure
(i.e.,
a topology with
more open
sets)
than the one which
below. For example, Brumelle and Vickson (1975) argue that (3.1)
shown
Table
in
this paper,
II to
we
will identify
sufficient for the {Yl,T) pairs
be stochastic dominance pairs under the topology of monotone convergence. In
we have developed for a
using the abstract definition
able to formally prove that the sufficient conditions hypothesized in fact sufficient for
is
more
we
are
by Brumelle and Vickson (1975)
are
dominance
"stochastic
pair,"
any stochastic dominance theorem, not just particular examples. Further, the
result that (3.1) is also necessary for (Tl,T) to
be a stochastic dominance pair
is
a
new
contribution of
this paper.
We now method.
argue that this result
First,
observe that unless
tells
T is
us
when we cannot do
a subset of n, there
is
theorems provide conditions which are easier to check than jrell. For example,
n
might be much
and
which
is
larger,
might be a it
set
which
is
J
better than the closed
no guarantee
to find a subset, T,
dominance theorems, as defined
in Definition 3.1, are
closed convex cone. For example, in the case of FOSD,
6
most
Its
for all
closed convex cone
of that closed convex cone for
easier to check that expected payoffs are nondecreasing in 6.
stochastic
dominance
7C(s)dF(s;0) nondecreasing in
not a closed convex cone.
might not be possible
convex cone
that stochastic
Thus, (3.1) indicates that
likely to
we consider the
be useful when
of payoff functions
set
II is a II
F0 .
easy to verify that positive scalar multiples and convex combinations of nondecreasing functions
It is
are nondecreasing functions, as are limits of sequences or nets of nondecreasing functions. Finally,
constant functions are also in IT
When
F0 .
IT contains the constant functions
and
is
a closed convex cone, (3.1) becomes:
n = cc(ru{l,-l}) In principle, the
most useful
general, there will not
r FO
(3.2)
T
is
the smallest set
be a unique smallest
set.
To
whose closed convex cone
contains indicator functions of upper intervals.
combinations of elements of
T
FO
Q
However,
in
By
taking limits of sequences of convex
u {1,-1}, and appropriately scaling these functions, we can generate
any nondecreasing function. However,
where
is II.
see this, consider the case of FOSD, where the set
we can
represents the rationals, and note that
also define the set
14
n
F0
t FO = {y\y(s) = I[a „,($),
= cc(f F0 u {1,-1})
.
While f F0
a e q\,
c T FO
,
in
f F0 is not any easier to check. Thus, stochastic dominance theorems are that T is the smallest closed set whose closed convex cone is II; we will call such a
practice the smaller set
generally stated so set the
"extreme points" of II.
Finally, because (3.2) is necessary
n=cc(IIu{l,-l}), we know of II: there stochastic
is
no
we
that
and sufficient for (H,T)
smaller or easier-to-check closed set
dominance
convex cone method
pair.
is
This
is
F
1
higher Gower) than
n in
neither inequality holds for all
we
first
We
cc(IIu{l,-l}).
distributions in the
the set of extreme points
that
we have proved
(II,
f)
is
a
that the closed
show
then
we
that II
show
same way
same. In Section 3.2.2,
II,
if
that
dominance
that (3.1) characterizes stochastic
Consider the problem of ordering two probability distributions,
Section 3.1.1,
T be
of payoff functions, f, such that
the "right" approach.
payoff functions II orders if
be a stochastic dominance pair when
what we mean when we say
two subsections, we prove
In the next
to
cannot do any better than letting
then
F2
we
if
F
1
and
\7r(s)dF\s)
F
2 ,
where we say
> (
< £ k where there
We
0.
is
a neighborhood corresponding to each
will return to clarify the relationship
between
this
topology and other topologies in the discussion following Theorem 3.4, below.
Let cc(A) denote the convex cone of a set A, and topology
is
=
\ndii
The proof of
is
linear
and continuous in
lemma
is
mathematical results in this paper build upon
it.
simple lemma.
3.2
A
denote the closure of
understood to be a^P*,?^) in the discussion below, unless noted).
that the functional P(7t,^i)
Lemma
let
this
its first
elementary, but
Consider a set of payofffunctions jl &g>:
II
we
A
(where the
We now use the fact
argument to prove the following state
it
here because
all
of the
cP». Then the following two conditions are
equivalent for all (i)
For all n e
II,
\nd\i> 0.
7 The boundedness assumption guarantees that the integral of the payoff function exists. It is possible to place other restrictions on the payoff functions and the space of finite signed measures so that the pair is a separated duality, in which case the arguments below would be unchanged; for example, it is possible to restrict the payoff functions and the signed measures using a "bounding function." For more discussions of separated dualities, see Bourbaki (1987, p. n.41).
16
(ii)
For all 7tecc(Uu
Proof:
First,
we show
{1,-1}),
(i)
jnd/i>0.
implies
Fix a measure fieg'. Then the following implications
(ii).
hold:
For all K ell,
=>
(Since
J
d[i
Forall ;rellu
= -j
dji
Forall
[tt, rf/z
>
{1,-1},
J;rd/i>0
= 0).
=>
(Since
jndn>0
and J
ff€cc(nu {l.-l}), J^d//>0
n2 dfi>0
implies
[a,^, J
+ a2 7T2 ]
d/i
= a^TTi
d\i
+ a 2 \n 2 dfi>0
when a,,a 2 >0). =>
For a// K ecc(IIu{l,-l}), J7rd// >
(Recall that for a continuous function/, the half-space [x
That
(ii)
implies
e
(i)
9^|jc
> 0}
is
follows because IT
Lemma 3.2 can be restated another way: is
exactly the
same
/(A)
c /(A).
The implication then follows because
closed, and the linear functional P(;/l) is continuous for all
c cc(II u {1,-1}).
the set of measures
as the set of measures
[i
e£*
for
//
which
/*.)
£?.£.£>.
e^» for which jndfi >
Jtt^ >
for all
it
e
for all
cc(Il
7T
e
II
u {1,-1}).
Formally,
{/i€^-|J^d/x>0 V;renj = {//Gf* IJ^rd/t
As
V;recc(IIu{l,-l})}
discussed above, because positive scalar multiples do not reverse the inequalities and because
the functional
fi is
linear in its second argument, the latter equality is equivalent to the following:
FKF^A^JTrdF^JKdF2 Vn e III That
>0
is,
for any
two probability
= If^F2 eA^TtdF
distributions, II orders the
cc(IIu{l,-l}).
17
1
two
ZJndF
2
V/r e cc(II
u {!,- 1})
distributions exactly the
same
as
we
Building from this lemma, characterization of stochastic in Section 4. (ii)
implies
turn to prove the
dominance theorems,
main mathematical theorem underlying
This theorem makes use of the linearity of the functional
(i)
relies
on
Lemma
3.2,
the
as well as the characterizations of other properties
while the proof that
(i)
implies
/J(tt,jU) in n. (ii)
The proof
that
makes use of a standard
which determines the meaning of
separating hyperplane argument. Note that the choice of topology,
closure, is critical for the application of the separating hyperplane theorem.
Theorem
P
t .
Consider a pair of sets of payofffunctions Then the following two conditions are equivalent: 3.3
(II, I*),
where
(i)
lii€$'\J7tdn>0 V;rell} = j/ie£-|Jy^>0 Vyerj.
(ii)
cc(U u {1,-1}) = cc(r u {1,-1}).
Proof: First consider
we
(ii)
implies
(i).
Suppose
that
II
and T are
subsets of
cc(IIu {1,-1}) = cc(Tkj {1,-1}). Then
have:
j//
e
2'|Jffd!/i
VTrellj
£
= |/X6f*|J^d//>0 V;recc(IIu{l,-l})j (By
Lemma 3.2.) = In e -f ^jydfi >
V76cc(ru{l,-1})}
(By assumption.)
= {nep\jydfi>0 (By
Vyer}
Lemma 3.2.)
Now we prove that (i) implies (ii).
Define II
Suppose (without loss of generality) the
(XP'&P) topology
from above {p(-,H)\fi
that the set
is
= cc(U u {1,-1}) and F = cc(T u {1,-1}).
that there exists a
y e f such that y e
fl
.
We know that
generated from a family of open, convex neighborhoods. Recall
of continuous linear functionals on P»
sTX'}. Using these
facts,
is
exactly the set
a corollary to the Hahn-Banach theorem8 implies that
8 See Dunford and Schwartz (1957, p. 421), Kothe (1969, p. 244) for discussions of the relevant theorems about the separation of convex sets. See also McAfee and Reny (1992) for a related application of the Hahn-Banach theorem, where the separating hyperplane also takes the form of an element of J*.
18
since fl
is
closed and convex, there exists a constant c and a
hyperplane)so that P(x,n.) > c for Since {1,-1} € fl and
argue
we can
take c
hypothesis that
/J(7T,/i.)
Because {1,-1} e and thus j J^,
c
ft,
and
Thus,
for
Ke
all
ft.
€
So,
e 7JF (a separating
c.
> c. Now we
=
/?(0, /x.)
But, f}(pk,iL.)
ft.
we
let
c
=
will
= pc
).
T2 but since x2 ,
T,
cT
2 ),
first
is
the closure of a set
A
ff .
under %2
(
.
Thus,
if
This implies Q.E.D.
3.4.
note that, given two topologies
true because the finer topology has
we know
A = AT
that
-A*
2
) is
more closed
T,
and r2 where ,
z, is
contained in the closure of A
A T is finer, there might be a closed set which is a strict subset of A This
"P*,
c
cc(nu{l,-l}) r =cc(ru{l,-l}) T ,then cc(nu{l,-l}) a =cc(ru{l,-l})
To
«
then (11,0
pair.
Tl
sets;
is
a closed set under
but that
still
contains
the linear functionals
/?(7T,/x)
SR and y
:
9?"
-»
SR.
Let
K be a collection of subsets of
conditional stochastic dominance pair
a partial order and all
F 9T x :
->
[0, 1]
if
conditions
k{G\K)
yeT and all KeK,
y(6\K)
For all neU. and all
(ii)
For all
(i)
and
with typical elements
9?".
(ii)
Then the pair (TIX)
are equivalent for all
such that such that F{;6) e A" :
KeK,
(i)
(II, T),
We now extend Theorem 3.4.
23
is
is
nondecr easing
in 6.
nondecreasing
in 6.
is
a
K-
with
Theorem 3.5 Consider a pair of sets of payofffunctions
P
1 .
Let
K
he a collection of subsets of 9t\ where
(Yl,T),
where
II
and T are
subsets of
€ K. Then the following two conditions
9?"
are equivalent:
The pair (TIT)
(i)
cc(II
(ii)
Proof:
is
a ^-conditional stochastic dominance pair.
u {1,-1}) = ccOTu {1,-1}).
We can apply the proof of Theorem 3.4 almost exactly.
likewise for Tk.
Then note
that
cc(U K u{IK ,-IK }) = cc(TK implies
(ii)
show
(i).
that if
One example of a
To show
(ii) fails,
u {IK ,-IK })
that
then
cc(Ilu{l,-l})
(i)
(i)
implies
must
conditional stochastic
= cc(T u {1,-1})
for all K. Then, for every
IK \n
e
II},
and
K we apply Theorem 3.4,
the arguments of Theorem 3.4 can be used to
(ii),
for the case
fail
n^ = {n
Let
implies that
where
K=
9?".
Q.E.D.
dominance theorem which has appeared
in various
forms in
the literature (see Whitt (1982)) involves the set of nondecreasing payoff functions, as follows:
Theorem 3.6 The following conditions are For all
(i)
it
:
9?"
—» 9?
equivalent:
nondecreasing and
all sublattices
#c 91",
k(6\K)
is
nondecreasing
in 6. (ii)
For
all increasing sets
nondecreasing
Ac 9?"
and all
sublattices isfc 9?",
f JAnK
dG(s;6,K)
is
in 6.
Proof: Apply Theorem 3.5 together with the fact that cc({k\k nondecreasing}
This theorem that
condition (for
(ii) is
A (s)
\I
nondecreasing}
u {1,-1}).
Q.E.D.
usually proved using algebraic arguments; but using the results of this paper,
follows as an immediate corollary of
it
Order
is
u {1,-1}) = cc({IA
Theorem
3.5.
First, take the
equivalent to requiring that F(s;6) satisfies the
case where
we
see
n-1. Then
Monotone Likelihood Ratio (MLR)
a proof, see Whitt (1980)), defined as follows:
The parameter 6 indexes the probability distribution F(;6) e A according the Monotone Likelihood Ratio Order (MLR) if, for all 6 > H L there exist numbers
Definition 3.2
1
to
,
-oo 9? and y
conditions (i)
(i)
and
:
—» 91.
9?"
The pair
(II, T) is
For all n sU,
Jt(s)dF(s;6)
f
is
with typical elements
a stochastic supermodularity pair
are equivalent for all lattices
(ii)
(II, F),
© and all F e A"e
supermodular
if
:
in 6.
Js (ii)
Further, pairs,
For all yeT,
we
define the set
analogous to
"L
sm
(U SOM ,r SOM )ismI. !lsr
When comparing
.
[
y(s)dF(s;6)
Z J5r To
to
is
be the
supermodular
set
of
all (II,r)
in 6.
pairs
which
are stochastic supermodularity
place our example from Proposition 3.3 in this framework, the pair
.
Definition 3.1 (stochastic dominance theorem) and Definition 4.1 (stochastic
supermodularity theorem),
it is
helpful to recall that stochastic
dominance theorems and
stochastic
supermodularity theorems are both characterized by (11,0 pairs corresponding to sets of linear functionals of the
Thus,
it is
form
f5(K,-),
which are then composed with parameterized probability
distributions.
possible to compare the two types of theorems directly, even though in a stochastic
dominance theorem, the parameter space theorem, the parameter space
is
any
is
any
lattice.
26
set
with a partial order, while in a stochastic dominance
.
More
we
generally,
can define a class of theorems called Stochastic
defined for an arbitrary property P.
Qp
which are defined so
on 0/' is all
that,
P
Theorems, which are
We are interested in properties P together with parameter spaces
given a function h
:
P
—> 9t,
the statement "h(6) satisfies property
Qp
well-defined and takes on the following values: "true" or "false." Let
such parameter spaces 0,,
Further, for a given property P,
.
we
P
denote the set of
will define the set of admissible
parameter spaces together with probability distributions parameterized on those spaces: n
VP = {(F,e p )\e p
e
e p and F e A%
p
}
Definition 4.2 Consider a pair of sets of payoff functions (IT,r), with typical elements
K
:
SK"
—> 9? and
7:9?"
—> 9?.
(F,Q P ) e
equivalent for all (i)
For all 7teTl,
(ii)
For
j
yeT,
all
f Js
The pair
Vp
(II,
O w a stochastic P pair
is
y(s)dF(s;6) satisfies property
we
let I. SPT
be the
analogous to stochastic dominance,
l(F,e) e
Vp \J7t(s)dF(s;6)
satisfies
= |(F,0) e v;\jy(s)dF(s;G)
In the next section,
e n,
of
set
P on
satisfies
all
(i)
and
(ii)
are
on
P
on 0„.
can be any convex
n(s)dF{s;6)
all
.
stochastic
is
set,
and condition
concave in 6."
P pairs
As
(i)
of
in the case of
for a given property P.
Also
we can rewrite the requirement of Defintion 4.2 as follows: Vtt e
nj
V/ € rl
P on
stochastic
cone method of proving stochastic dominance
P theorems, if P is a closed convex cone property.
The Closed Convex Cone Approach
we
J
we show that the closed convex
theorems can be extended to
4.2
tt
0,,
©p
P
7t(s)dF(s;6) satisfies property
interpreted, "For all
stochastic dominance,
conditions
:
For example, for a stochastic concavity theorem, Definition 4.2
if
is
Valid for all
CCC Properties
which we
call
"Closed Convex Cone" (CCC)
properties, such that the following statement is true: for all properties
P which are CCC, if a pair (IIJ)
In this section,
is
identify a class of properties,
a stochastic dominance pair, then (tl.D
is
a stochastic
closed convex cone method of proving stochastic
P
pair; thus, for all
P theorems is valid.
CCC
This result
is
properties, the
useful because
allows us to use the existing theorems from the stochastic dominance literature to generate classes of theorems.
CCC properties are properties are defined as follows:
27
it
many new
A property P is a CCC property (written P eCCC) if the set offunctions —> SR which satisfy P forms a closed convex cone, where closure is taken with respect to the topology of pointwise convergence, and if constant functions satisfy P.
Definition 4.3
g
Qp
Note while
we
functions,
The
we
that
are using closure under the topology of pointwise convergence for properties P,
weak topology
are using closure under the
(as defined in Section 3.1.1) for sets of payoff
n and T.
properties nondecreasing, concave, and supermodular are all
CCC
properties, as is the
property "constant." Further, any property which places a sign restriction on a mixed partial derivative is
CCC.
Finally, since the intersection of
two closed convex cones
of these properties can be combined to yield another
"nondecreasing
and
convex"
CCC
CCC
a
is
is itself
For example, the property
property.
This
property.
useful in the context of comparative statics for the following reason.
/i:SR—»SR andg:9?"
—>SR.
If h(x) is
n in II,
and 6
nondecreasing and convex in 6 for
all
supermodular and monotone), then
7t(s)dF(s;g(y))
is
is
is
property
is
Consider two functions,
convex and nondecreasing, and g(y)
monotone (nondecreasing or nonincreasing), then h(g(y))
J
a closed convex cone, any
is
supermodular. Thus,
supermodular and if f K(s)dF{s\6) is
by 6 - g(y) (where g
is
dominance pair, then (H,T)
is
in fact determined
supermodular in
y.
Now we prove a result which builds from Theorem 3.4: Theorem
Suppose property
4.1
a stochastic
P pair.
PeCCC.
If (11,0 is a stochastic
u {1,-1}) = cc{T u {1,-1}),
Equivalently, if cc(Jl
then (FIX)
is
a stochastic
pair.
Proof:
For all
=» (Since
J
dF(s;6)
=>
\[ajz
x
(s)
ell, ^K{s)dF(s;0) satisfies P.
For all n e
II
u {1,-1},
= 1, and constant functions For all n e
jn^s) dF(s;6)
(Since
it
satisfies
cc(Jl
x
=>
{1,
satisfies P.
satisfy P, for all
P eCCC).
-1} ),
J
x(s)dF(s;6) satisfies P.
P and jn2 (s) dF(s;6)
+ a2 K2 {s)] dF(s;6) = a \n
a,,a2 > 0, since
u
f K{s)dF(s;0) Js
{s) x
satisfies
P implies
dF(s;6) + a2 JK2 (s) dF(s;6) satisfies
P is a CCC property). For all K e cc(IIu {1,-1}),
28
J
n(s)dF(s;0) satisfies P.
P for
P
.
(Recall that a subset of a topological space
convergent nets of elements in that all 11,
for
JKa (s) [
dF(s;6
7t(s)dF(s;6)
satisfies
Ka
any net
P
)
is
in cc(Il
u {1,-1})
-» $7i(s) dF(s;6
)
is
closed
if
and only
if
it
contains the limits of all the
Since the linear functional fi(;fi)
na —> n
such that
(Kothe, 1969, p.l
,
6
then given
Since
1).
the pointwise limit of a net of functions
which
P
a
is
is
continuous for
,
CCC property,
For all yecc(ru{l,-l}),
(By Theorem
3.6, since
=>
K(s)dF(s;0)
satisfy P, then j
(n.O
is
J
y(s)dF(s;6) satisfies P.
a stochastic dominance
For all y e T, j y(s)dF(s;6)
pair.)
satisfies P.
T Q cc(T u {1,-1}))
Precisely analogous arguments establish the symmetric implication.
This theorem generates
many new
been identified in the large case, this
and
as well.)
=>
(Since
set. 10
literature
theorem generalizes a
to nondecreasing functions
classes of stochastic
Q.E.D.
P theorems, where the
(11,0 pairs which have
on stochastic dominance are also stochastic
result
by Topkis (1968), who proves
and indicator functions of nondecreasing
P pairs. As a special
that the (11,0 pair corresponding sets, respectively, is
a stochastic
P pair for P in CCC. What we have shown stochastic
P pairs,
Y SDT = E J/T
.
In
then
in this section is that if
£ JDr c Z Jpr However,
Remark 2
.
at
Z JDr c ^ SFr
-
property, the property "constant in 6."
nondecreasing payoff functions, and the set constant theorem," since j 7t(s)dF(s;d)
However, (I1 F0 T) ,
properties
which
note that not
then end of Section 4.4,
"nondecreasing and convex," then
CCC
P is a CCC property,
is
For now,
T = -IT
properties
.
dominance
10 See Kothe (1969, pp. 10-11) for a proof of this statement.
spT
P
denotes the set of in
CCC
all
are such that
P
argue that
let
us consider a simpler example of a
The
pair.
IT
F0
pair (IT
6 if and only
will satisfy equivalence relationships.
29
ro
l,
we
Take the case of
constant in
clearly is not a stochastic
all
and
if
when
the property
the set of
,
FO ,
T)
univariate,
satisfies a "stochastic
-\n(s)dF(s;d)
The next
all
is
is
constant in
6.
section identifies a class of
Linear Difference Properties
4.3
This section characterizes a subset of
CCC properties, which we call Linear Difference Properties we
(LDPs). These properties are interesting because, as
will
show
in Section 4.4, for all
P in LDP,
=£
T.
(4.1)
Combining the "closed
(4. 1)
with Theorem 3.4,
convex cone" method
P is an LDP.
That
is, if
Pe LDP, LDPs
Important examples of
Supermodularity predictions.
We
is
is
we can
then
(II,
OeZ „ s
are supermodular
important because of
also
conclude
that, just as in the
case of stochastic dominance,
P theorems when = cc(ru{l,-l}).
exactly the right one for the study of stochastic
emphasize the
if
and only
if
cc(IIu{l,-l})
and concave; others are summarized
role in the analysis of
its
in
Table
monotone comparative
I.
statics
about concavity, since concavity and convexity are also
result
frequently encountered in economic contexts; for example, if an objective function is concave, then the First
Order Conditions characterize the optimum. Further, concavity can be used to establish the
existence of supporting prices in a resource allocation problem.
The
result described in equation (4.1) is useful
(II, T) is
because
dominance; by
(4.1),
checking to see
if (II
,D
is
this set II
easier to verify whether or not
same
are determined by an economic problem, and
has been analyzed in the stochastic dominance literature, then the corresponding stochastic
supermodularity theorem
r have
may be
a stochastic dominance pair answers the
question. For example, if the characteristics of a set II
not
it
a stochastic supermodularity pair by drawing from the existing literature on stochastic
the
is
immediate. This allows us to bypass the step of checking whether II and
same closed convex cone direcdy
(further,
most of the stochastic dominance
literature
does
(LDPs). The
first
make such a statement explicitly).
We now begin to build our formal definition of Linear Difference Properties important feature of LDPs
is that
they can be represented in terms of sign restrictions on inequalities
involving linear combinations of the function evaluated at different parameter values. For example, H L L if and only if g(6 ) - g(d ) > for all 6" > 6 in 0. This statement g(0) is nondecreasing on specifies a set of inequalities,
where each inequality
high parameter value and a low parameter value. if
and only
x
if
g(6
a difference between the function evaluated another example, g(d)
v 6 2 )- g(6') + gtf a0 2 )- g(d 2 )>0 1
specifies a set of inequalities, this time involving the
In both cases,
is
To take
for all
sum of two
we can represent every inequality by
6\6
2
in
0.
is
at
a
supermodular on
Again,
this
statement
differences.
the parameter values
and the coefficients which
on the corresponding function values. For example, for the property nondecreasing, each inequality involves the coefficient vector (1,-1) and a parameter vector of the form (d H ,8 L ), where 1 is are placed
the coefficient
on g(d") and -1
is
the coefficient
30
on g(6
L ).
Thus, g(6)
is
nondecreasing on
if
and
only
for
if
^aig
{
