Characterizing properties of stochastic objective functions

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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

1

INTRODUCTION

Thispaperstudies optimizationproblems wherethe objectivefunctioncan bewritteninthe form V(8)

=

\jt{s)dF{s\6), where;risapayofffunction,

F

is a probabilitydistribution, and

6

andsare real vectors. For example,thepayofffunction

n

mightrepresentanagent'sutilityora firm'sprofits, thevectorsmightrepresentfeaturesofthecurrent stateoftheworld,andtheelementsof6 might

represent an agent's investments, effort decisions, other agent's choices, or the nature of the

exogenousuncertainty in the agent'senvironment.

The economic

problem underconsideration often determines

some

propertiesofthe payoff

function; forexample,autilityfunctionmight be

assumed

tobe nondecreasing andconcave, whilea

multivariate profit function might have sign restrictions

on

cross-partial derivatives. These assumptionsthen determine asetIT of admissiblepayofffunctions.

We

mightthenwishtoanswer

questionssuch as: Isasetof investments

6

isworthwhile?

Are

there decreasingreturns tothose investments?

Does

one investment increase thereturns to anotherinvestment?

To

answerthose questions,

we

needto

know

whetherV(Q) satisfiestheappropriate properties,i.e., nondecreasing, concave, or supermodular.1

The

goalofthispaperistodevelopmethods suchthat,given aparticularproperty

P

(suchas

nondecreasing),asetof admissiblepayoff functionsII,andaparameterizedprobability distributionF,

we

can determinewhetherthefollowing statementistrue:

J7i(s)dF(s;6)

satisfiesproperty

P

in

6

forall

n

inIL (1.1)

Thatis,given

P

andII,

we

wishtodescribethesetofprobability distributionswhichsatisfy(1.1).

Thispaperrestrictsattention tothecase

where

the properties

P

andsets IIare"closedconvex

cones."

We

definea property

P

tobea"Closed

Convex Cone" (CCC)

propertyifthesetof functions

whichsatisfy

P

isclosed(under an appropriate topology),positivecombinations of functionsintheset are also intheset,andconstant functionsare in theset. Importantexamplesof

CCC

propertiesinclude nondecreasing,concave,supermodular,anypropertywhichisdefined

by

placingasign restrictionon

apartialderivative,and combinations oftheseproperties. Thus,ifthepayofffunction representsa

consumer'sutility,IIcouldbethe setofunivariate,nondecreasing,concave payofffunctions;ifthe

1_Intuitively,

afunctionissupermodularif.givenanytwoarguments ofthe function, increasingoneincreases the returns toincreasing the other.

When

thefunctionisdifferentiable,thisamountstopositive cross-partial derivativesbetween every pairof arguments. Supermodularityisimportantinthecontext of comparativestatics(seeTopkis(1978),

payoff functionrepresentsafirm's profits asa functionof

two

complementaryqualityinnovations,

n

couldbethesetofbivariate,nondecreasing,supermodularpayofffunctions.

Forsets IIandforproperties

P

whichare

CCC,

thispaperdevelops a characterization ofwhich

probability distributions satisfy(1.1). Checking(1.1) directlyisdifficultbecause,ingeneral,Ifmight bea verylargeset.

Thus

we

ask,

When

isitpossibletofinda smallersetofpayofffunctions,denoted

T,sothatforanyprobability distributionF, statement(1.2)belowwillbetrueifandonlyif(1.1)is

true?

J

7t(s)dF(s;8)satisfiesproperty

P

in6forall

%

in

T

( 1.2)

We

canthinkof

T

asa"testset" forII: ideally,

T

isasetofpayofffunctionswhichissmallerand

easier tocheckthanII,butitcan beusedtotestwhether

J7C(s)dF(s;0)satisfiesproperty

P

in6,given

onlytheinformationthat

K

isinthe largersetII.

Usingthese ideas,

we

canrestatethegoalofthepaper:

we

wanta theorywhichhelps us determine

thebest"testset"fora givenII.

We

proceedin

two

steps. First,

we

examinethecasewherethe

property

P

is"nondecreasing"; second,

we

study other

CCC

properties.

The

firstcasehasbeenthe

focusofthe literatureonstochasticdominance, wheredifferentauthorshavestudieddifferentsets II. 2

Thispaperunifies andextends the existing literatureonstochasticdominance,providing an exact characterizationof themathematical structureunderlyingall stochastic

dominance

theorems.

We

furtherprovideanalgorithmforgenerating

new

stochasticdominance theorems whichrelaxesthead hocdifferentiabilityandcontinuityassumptions whichare

common

inthis literature.3 In thesecond

partofthepaper

we

show

thatthemethodsfromstochasticdominance can beappliedtocharacterize

when

V{6)satisfiesotherpropertiesP,includingconcavityandsupermodularity.

We

now

provideanoverview of ourresults. In thefirstpartofthispaper,

we

show

thatforthe

property nondecreasing,ifIIisa closedconvex cone andcontains constantfunctions,thenthebest

T

isthe setof"extremepoints"ofII. Just asabasisgenerates alinearspacevia linearcombinations, so asetofextremepointsgeneratesa closedconvex conevia positivelinearcombinations andlimits.

Thus,

we

show

thatif

we know

onlythatapayofffunction

k

liesintheclosedconvex coneII,and

we

wantto

know

ifV(6)isnondecreasing,itisequivalenttocheckthatV(0)isnondecreasing

on

a smaller

2

Inparticular,the univariate stochasticdominanceproblem has beenstudiedby Rothschild andStiglitz(1970,1971)and

HadarandRussell (1971); notable contributionstothe multivariateproblemincludeLevyand Paroush(1974),Atkinson and Bourguignon(1982),andMeyer(1990). Shakedand Shanthikumar (1994) provide areferencebookonthe subjectof stochasticordersandtheirapplicationsineconomics,biology,andstatistics.

3_{Brumelle and Vickson (1975)}

takeafirststeptowardsrelaxing theseassumptionsandidentifying themathematical structurebehind stochasticdominance; in contrast to theirwork, whichprovides only sufficientconditions fora stochasticdominancerelationship,thispaperprovidesanexact characterizationofallstochasticdominancetheorems.

setofpayofffunctions, theextremepointsof n. Itwilloftenbe

much

easier to verifymonotonicityof V(6) forthe setofextremepointsof

n

thanfor

n

itself. Inthispaper, theprocedureof usingthe

extreme pointsof

n

asatestsetfor II willbe referredtoas the "closedconvex cone"

method

of provingstochasticdominancetheorems.

Examples

fromthe existing stochasticdominanceliterature,whichare specialcasesofthis result,

includeFirstOrderStochastic

Dominance (FOSD),

where

n

isthe setofunivariate,nondecreasing payofffunctions,andSecond OrderStochastic

Dominance

(SOSD), whereTIisthe setofunivariate,

concave payofffunctions. In the case of

FOSD,

the setofextreme pointsisthe setofone-step functionswhicharezeroupto

some

constant,and onethereafter. ThesearepicturedinFigure 1. In

thecase of

SOSD,

thesetofextremepointsisthe setof "angle," or"min"functions,picturedinFigure

2,where eachfunctiontakes the

minimum

ofitsargumentand

some

constant.

The

closedconvex cone

method

for stochasticdominancehasbeenrecognizedandexploredinthe

contextofparticular setsofpayofffunctions (Topkis,1968;Brumelle andVickson, 1974; Gollierand

Kimball, 1995).4 However,thispapergoes

beyond

the existing literature in

two

respects. First,

by

developing appropriateabstract definitions todescribestochasticdominancetheorems,

we

areableto

make

general statements aboutthe entire class ofstochastic

dominance

theorems, including those

which havenotyetbeenconsideredintheeconomicsliterature. Second and

more

importantisthe fact that

we

provea

new

resultaboutthisclassof theorems:

we

proveformallythatthe"closedconvex cone" approachtostochasticdominanceisexactlythe rightone.

By

this

we mean

that (1.1)and(1.2) returnthe

same

answerforeveryprobability distribution

F

ifandonlyiftheclosedconvex coneof

T

(uniontheconstantfunctions,ifthesearenotinT)isequaltoIT;

no

otherT'swillalways

make

(1.1)

and(1.2) equivalent. Inthiscase,clearlythesmallestsetwhichgeneratesIIisthebestsettocheck.

The

secondpartofthispaper showsthatthe"closedconvex cone"

method

canalsobeappliedto characterizeotherpropertiesofV(6).

We

ask

two

questions:First,forwhatproperties

P

istheclosed

convex cone approachvalid?

And

second,forwhatproperties

P

istheclosedconvex cone approach

exactly therightone, as in thecase ofstochastic

dominance?

We

first

show

thatif

P

isa

CCC

property,thentheclosedconvex cone approach can always be usedtocharacterize

when

V(6)satisfies P.

We

thenfinda subsetof

CCC

properties,which

we

call"Linear DifferenceProperties," forwhich

we

can

show

thattheclosedconvex cone approachisexactlythe rightoneforcheckingwhetherV(0)

satisfies P.

Examples

of LinearDifference Properties include monotonicity, supermodularity, concavity,andpropertieswhichplace signrestrictions

on

partial derivatives. Combinations ofthese

properties,however,arenotingeneralLinear DifferenceProperties,althoughsuch combinationsare

4_{Independently, Gollier}

CCC

properties. Table Isummarizesproperties whichare

CCC

propertiesandLinear Difference

Properties.

TABLE

I

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

SignRestrictiononaPartial Yes Yes

Derivative

Constant Yes

No

Nondecreasingand Concave Yes

No

ArbitraryCombinationsof Yes

No

CCC

Properties

ArbitraryCombinationsof Yes

No

LDP

Properties

For

many

sets ofpayofffunctions,

n,

whichare

commonly

studiedineconomics, the existing literature

on

stochasticdominancehasimplicitly identified theextremepointsof thosesets. Insuch

cases, theproblemof characterizing a Linear Difference Property (suchassupermodularity)

becomes

quitestraightforward:simply looktothe the existingliteraturetofind theappropriatetest set,T, for the setofpayofffunctionsIIunderconsideration. Then,usingthe resultsofthispaper,

we know

that (1.2) characterizes the setof parameterizedprobability distributions forwhichthe stochastic objective

functionwillbe supermodularforallpayoff functionsinII.

We

illustratethis technique

by

showing

how

the closedconvex cone approach can be usedto

characterize theproperty supermodularityforseveralimportantclassesof payofffunctions. These

resultscaninturnprovidesufficient(andsometimesnecessary) conditionsforcomparativestatics

conclusions. Inparticular,

we

examineapplications inprincipal-agent theory,welfareeconomics,and

This paper proceedsas follows. InSection2,

we

introducea motivatingexample,theproblemof

a risk-averse agent's choice ofeffort. Inthisproblem,

we

characterizethepropertiesnondecreasing, concave,and supermodularforthe agent'sexpectedutilityfunction.

Our

generalanalysisofstochastic

dominancetakes placeinSection3.

We

provideanexact characterization ofstochasticdominance

theorems, highlightingtheimportantroleplayed

by

linearityofthe integral.

We

furtherextendour

resulttoincorporate "conditionalstochasticdominance." Section4developscharacterizationsof other

propertiesof _f7t(s)dF(s;d)andprovides applicationsoftheproperty supermodularity. InSection5,

we

analyze conditions under

which

functions ofthe

form

J

K(x,s)dF(s;6) are supermodularor

concave in (x,d), showing

how

to apply stochasticmonotonicity resultsto this problemas well.

Section6concludes.

2

MOTIVATING EXAMPLE:

A

RISK-AVERSE AGENT'S

CHOICE OF

EFFORT

Inthissection,

we

present a motivatingexample, where

we

analyze

how

a risk-averse agent's choice ofeffort affects herexpected payoffs, and

how

that choice ofeffort interacts with exogenous

parameters

which

describethe probability distribution. Formally,

we

characterize the properties nondecreasing, concave,and supermodularfortheagent'sexpectedutilityfunction. Theseresultsare specific examples

—

examples

where

the payofffunctionis nondecreasing and concave

—

ofa

stochasticdominancetheorem, a"stochasticconcavity theorem,"anda"stochasticsupermodularity theorem." These examplesillustratethe parallel structureunderlyingthethree classesof theorems.

Considerarisk-averse agent

whose

utility {it)dependsontheoutputofastochasticproduction technology,wheretheoutputisdenoteds. Supposethattheagent'seffort(c)affectsthe probability distributionofs. Further,consider a parametertwhichrepresentsexogenous changesinthe stochastic

production technology. For example,achangeintmightrepresentaworker

moving

from one jobto another. Then,

we

canwrite theagent'sproblemasfollows:

max

fk(s)

•dF(s;e,t)-c(e)

Firstobservethatitisnottrivialto verify thatV(e,t)

=

\it(s) dF(s;e,t)isnondecreasingineffort:

anincrease in effortwhichisproductiveonaveragemightnotincreaseexpectedutility ifthe effort also

increasesthe riskinessofthe distribution. Second,notethatifV(e,t)-_c(e)isconcaveineffort,then

thefirstorder conditions

(FOCs)

characterizetheoptimum,a propertywhichisusefulintheanalysis

some

economicproblems,suchasprincipalagent problems. Further,ifV(e,i)failstobe concavein e,thenthere exists

some

linearcostfunction c(e)

=

ae

suchthatthe

FOCs

failtocharacterizethe

nondecreasingint. If V(e,t) failstobesupermodular, thenevenif

V

isconcave, there exists

some

linear costfunction c(e)

=

ae

suchthat e*{i) fails to be nondecreasing in t (this follows from

Milgrom

and

Shannon

(1994);see

Theorem

A.lintheAppendix). Thus,therequirementthatV(e,t) satisfiesthepropertysupermodular(orconcave,respectively)issufficient for thedesiredeconomic

conclusion, andfurtheritcannot berelaxedaslongas

we

requirethattheconclusion holdsforall linear cost functions.

Letus firstidentifyconditionsunder whichV(e,i)isnondecreasingine.

The

following

well-known

resultisadaptedfromtheRothschildandStiglitz(1970,1971)

work

onstochasticdominance (where

we

suppresstinthe notation):

Proposition 2.1 Thefollowing twoconditions are equivalentforallprobabilitydistributions F(-,e):

(i)

For

all

K

nondecreasing

and

concave, \lt(s) dF(s;e) isnondecreasingine. (ii) Thefollowing aresatisfied:

(a) \sdF(s;e) isnondecreasingine. a

(b)

For

all a, -\F(s;e) isnondecreasingine.

Intuitively,forarisk-averseagent

who

likesincome,effortwillincreaseexpectedutility ifandonlyif effortincreasesthe

mean

income(condition(ii)(a))andreducesthe "riskfunction" (condition(ii)(b)).

Thisresultisoftenusedinthefinanceliterature;ithasbeencalledSecond Order MonotonicStochastic

Dominance

(SOMSD).

In this paper,

we

will

work

with a restatement of Proposition 2.1,

which

emphasizes that

conditions(i)and(ii)are actuallysymmetricconditions.

The

following propositionisequivalentto

Proposition2.1 (where

A

1

bethespaceofprobability distributionsdefinedonSR, and Si indicates the

extendedrealline,thatis, SR

u

{-00,00}):

Proposition 2.1' Thefollowingtwoconditionsare equivalentforall F(-;e)eN:

(i)

For

all

n

inthe set YlSOM

=

[ti\k:SR-»

%

nondecreasing, concave},

J7t(s) dF(s;e) is nondecreasingine.

(ii)

For

allyinthe set

T

SOM

=

{y\y(s)

=

min(a,s),a

e

9t}, jy(s) dF(s\e) isnondecreasing

Conditions (i) and(ii)ofthisPropositionsimply restateconditions(i)and (ii)ofProposition2.1,

except that

we

have replaced -JF(s;e) with jmin(a,s)dF(s;e) in condition (ii). It is

straightforwardto verify(usingintegrationbyparts) that theformer termisnondecreasingifandonly

ifthelattertermis. This conditionisnot usually associated with

SOMSD

(seeBrumelle and Vickson

(1975) foranexception). This

way

ofwriting the

SOMSD

theoremillustratesthe mathematical

structureunderlyingthe stochasticdominancetheorem.

As

written inProposition2.1',the resultsays

thatinsteadofcheckingthat _{j n(s)} dF(s;e) isnondecreasingin X for allpayofffunctionsin the

relativelylargeset, II50M,itisequivalenttocheckthat \y(s)dF(s;e) isnondecreasingine forall

payoff functionsin thesmallerset,

T

S0M.

The

twosetsofpayoff functionsarepicturedinFigure2.

The

relationshipbetweenthe

two

setsof payoff functionscanbedescribedas follows: T1S0M isequaltotheclosedconvex coneofthesetwhich

istheunion of

T

S0M

andthefunctionsy{s)

=

1andy(s)

=

-1. Thatis,

by

takingpositivecombinations

andlimitsof sequences(or nets)of elements ofthelatter set,

we

cangenerateanyfunctionin TISOM,

whichis itselfaclosedconvex cone. In Section3,

we

will

show

that this"closedconvex cone"

relationshipbetweenTlSOM and

r

SOM _holds

forallstochasticdominancetheorems.

Now,

letusask adifferent,butrelated, question:

When

is

Jx(s) dF(s;e)concavein effort? This

problem

was

addressed

by

Jewitt (1988),

who

analyzes conditions under

which

the FirstOrder Approach

(FOA)

toanalyzing principal-agentproblemsisvalid.

The

FOA

requires thatiftheagent's

FOCs

aresatisfied,thentheagent'schoiceofeffortmust beoptimal. ExtendingJewitt'sanalysis,

we

can

show

thatthe sufficient conditionshe derivesare in factnecessary.5

_The

_{followingresult is}

analogoustoProposition2.1:

Proposition 2.2 Thefollowing twoconditions are equivalentforall

F(;e)e

A

1

:

(i)

For

all

%

inthe set YlS0M, \k(s)

•dF(s;e) isconcaveine. (ii)

For

allyinthe set

T

SOM',

\y(s) dF(s;e) isconcaveine.

Proof: Thiswillbeestablished asa simplecorollaryof

Theorem

4.1below, together with Proposition2.1

.

Q.E.D.

5Jewittfirstderives conditionsunderwhichthe agent'sutilityfunctionisincreasingand concaveinoutput,giventhe optimalcontract;hethenaddresses the questionof concavity oftheexpectedutilityfunctionin effort. It isonlythelatter

Proposition2.2saysthatexpected payoffsareconcaveine forallpayoff functionsin

U

SOM ifand

onlyifexpected payoffs areconcaveineforallpayofffunctionsin

T

SOM. Again,this theoremis

usefulbecausecondition (ii)is easier to checkthat condition(i): the set

r

SOM is

much

smaller.

Condition (ii) can be interpreted asrequiring thatthere are decreasing returns to e in terms of increasingthe

mean

anddecreasingthe "riskfunction." Notethatthe pairofsetsof payofffunctions,

(

n

SOM

,

T

SOM

),isthe

same

inbothpropositions.

Now

we

turn toask afinalquestion:

When

istheoptimal choice ofeffort

monotone

nondecreasing

int,whichparameterizesthe stochasticproduction technology?

More

precisely,

how

can t affectthe

probability distributionover outputin sucha

way

that monotonicity oftheoptimal effort int is

ensured, without any additional information about the cost of effort function or the agent's preferences? This question has not been answeredin the existingliterature; thus,the following proposition provides a

new

insight into thecomparativestaticsproblem. Notethatthispropositionis

true irrespectiveofwhetherexpectedutilityismonotoneineffort.

Proposition 2.3 Thefollowing three conditions are equivalentforall F(-;e,f)e

A

1

:

(MCS) For

allcostfunctions c

and

all

K

in IlSOM, e'(t)

=

argmax\K(s)dF(s;e,t)-c(t) is

e Js

monotone

nondecreasingint.6

(i)

For

all

n

e

TlS0M, \K{s)dF{s;e,t) issupermodularin{e,t\ (ii)

For

all

ye

T

S0M, jy(s) dF(s;e,t) issupermodularin(e,t).

Proof:

The

equivalence ofparts

(MCS)

and(i)followsdirectlyfrom

Milgrom

and

Shannon

(1994), as stated in

Theorem

A.1intheAppendix.

The

equivalence of(i)and(ii)willbe

established asacorollaryof

Theorem

4.1belowtogetherwith Proposition2.1.

Q.E.D.

The

formaldefinitionof supermodularity (andthecomparativestaticstheoremwhichreliesuponit) canbefoundintheAppendix. Intuitively, V(e,t)issupermodulariftincreasesthe returns toeffort.

Proposition2.3providesnecessaryandsufficientconditionsfor

monotone

comparativestaticsinthis

problem. If(ii)isviolated,then

we

canconstructpayofffunctionsandcost functionssuchthatthe

choiceofeffortisnotmonotonicint. Thus,

we

haveidentified theexact conditionswhichensure

monotone

comparativestatics. Condition (ii)requiresthateandtarecomplementaryinterms of increasingthe

mean

ofthe probability distributionandinterms of reducingtherisk.

The

intuitionis

straightforward:sincearisk-averse,income-lovingagentlikeshigh expectedreturnsandlowrisk (as

"In general, theoptimal emaybe aset. Then,thistheoremrequires that thesetbe nondecreasingintheStrongSet Order, as definedintheAppendix.

shown

in

SOMSD),

variableswhicharecomplementaryinincreasingthe

mean

anddecreasingthe risk arecomplementaryinincreasingexpectedutilityofsuch anagent. Notethatifeitheroneofeand/ does notaffectthe

mean

orthe riskinessof the agent'sincome,thecorrespondingcomplementarity

conditionsare satisfiedtrivially.

Notice that Propositions 2.2 and 2.3 have astructure

which

is very similarto the existing

stochasticdominanceresult,as stated inProposition2.1. In Section3,

we

willbuild aframeworkfor

analyzing Proposition2.1 andotherstochasticdominancetheorems. In Section4,

we

willformalize

the relationshipbetweenPropositions2.1through2.3.

3

MONOTONICITY OF STOCHASTIC

OBJECTIVE

FUNCTIONS

The

goal ofthis section is to provide a unified

framework

for analyzing stochastic

dominance

theorems. In Section 3.1,

we

introduce a

framework which

incorporates the existing stochastic

dominanceliterature,arguingthateachstochasticdominance theoremdescribesarelationshipbetween twosetsofpayofffunctions. InSection3.2,

we

provearesult

which

characterizesamathematical

relationshipbetweentwosetsofpayoff functionswhichisequivalenttothe relationshipdetermined

by

the stochasticdominancetheorem. Section3.3extendsthe result tothecaseofconditionalstochastic

dominance.

3.1

A

Unified

Framework

forStochastic

Dominance

In this section,

we

introducethe

framework which

we

willuse to discuss stochastic

dominance

theoremsasanabstract classof theorems,andtodrawpreciseparallelsbetweenstochasticdominance

theoremsandothertypesof theorems.

Letusfirstconsideranotherwell-known exampleof astochasticdominancetheorem,FirstOrder

Stochastic

Dominance

(FOSD). Thistheorem can bestated asfollows(where I

A(s) istheindicator

functionforthesetA):

Proposition 3.1 Thefollowingtwoconditions are equivalentforall

F(;0)e

A

1

:

(i)

For

all

n

e

n

TO

s

{k\k:9i-»

%

nondecreasing},

JK(s)dF(s;6)isnondecreasinginft

(ii)

For

all

_y

e

T

FO

_m

{y\y(s)

=

I

M

(s),a

e

Sfi}, jy(s)dF(s;6) isnondecreasinginft

Thistheoremhasthe

same

structure as the

SOMSD

theorem, Proposition2.

1

'.

The

setsofpayoff functionsare illustrated inFigure 1. Thistheoremsaysthatinsteadof checkingthatexpected payoffs

arenondecreasing in

8

forallnondecreasing payofffunctions,

IF

,itisequivalenttocheckthat

functionsofupperintervals.

The

latter setofpayoff functionsis

much

smaller,andso condition(ii)is

easier tocheckthatcondition(i); condition(ii)can be reducedto a restrictionwhichrequiresthat

1

-

F(a;6),the

complement

ofthecumulativedistributionfunction,isnondecreasingin

8

forae9?.

The

latterrequirementisthe

more

standard

way

ofstating

FOSD.

Thereare

many

otherexamplesofstochasticdominancetheorems,

some

with multiple

random

variables;

we

will discussotherexamples belowin Section4.5. Ingeneral, stochasticdominance theoremsallhaveaparallel structure, as illustrated inPropositions2.1'and3.1. However,different stochasticdominance theoremsarecharacterized

by

different_(11,0pairs.

The

SOMSD

theoremstates thatthe pair

(U

SOM,

T

SOM

)satisfiesaparticular relationship; the

FOSD

theoremstatesthatthe pair

(IT™,

r

F0

)satisfiesthe

same

relationship.

To make

thisrelationship precise,

we

introduce a formal

definitionofthestatement

"A

pairofsetsofpayofffunctions,(n,D,satisfiesastochasticdominance

theorem." Ifthatstatementistrue,then

we

willsaythat(11,0isa stochastic

dominance

pair.

We

willusethe abstract definition to

make

statementsaboutthe classofstochasticdominancetheorems,

and

how

thatclassoftheoremsrelatestootherclassesof theorems.

We

allowformultidimensional

payofffunctions andprobabilitydistributions, usingthe followingnotation:the setofprobability

distributions

on

SR" is denoted A", with typical element

F:SR

n

_—

»[0,1]. Further, for a given parameterspace 0,

we

will use the notation A"_etorepresent the setofparameterized probability

distributions

F

:9?"

x

-»[0,1] suchthatsuchthat F(;0)eA" forallds 0.

Definition 3.1 Consider a pair ofsets of payofffunctions(TI,r), with typical elements

K

:

9T

—

>SR

and

_y

:9?"

—

> SR. Thepair(II,T) isastochastic

dominance

pair ifconditions (i)

and

(ii)are equivalentforallparameter spaces withapartialorder

and

all

F

€

A

n e:

(i)

For

all

ttgU,

jn(s)dF(s;6)is

nondecreasing

in 6.

(ii)

For

all

_{y eT,}

Jy(s)dF(s;6)

is

nondecreasing

in 6.

Further,

we

definetheset I._{SDT to}bethe setofall_(11,0pairswhichare stochasticdominancepairs,as follows:

2

_5Dr

=

{(II,r)|(n,r)isastochasticdominancepair}

Thus,

when

agiven (11,0pairisastochasticdominancepair,

we

write (II,T)

e

Z

SDr.

Definition3.1 clarifiesthestructureofstochastic

dominance

theorems. Stochasticdominance theoremsidentify pairs ofsets of payofffunctions

which

havethe following property: givena

parameterized probability distribution F, checking that all of the functions in the set

(fy(s)dF(.s;0)|y

er|

are nondecreasing. Stochastic

dominance

theorems are useful because, in general, theset

T

issmaller thantheset II.

We

canthinkofthisdefinition asa statement aboutthe equalityoftwosets. First, letus definethe set ofadmissible parameter spaces for the property "nondecreasing" together with probability

distributionsparameterized

on

those spacesas follows:

V

ND

=

{(F,0)|©has apartialorderand

F

e

A

n

ej

Now,

we

canrephrasethe definition as follows:(11,0isastochasticdominancepairif

j(F,0)€

V

n

N

M

7r(s)dF(s;d)isnondecreasingon V/reIli

=

|(F,0) e

V

_{ND \JY(s)dF(s;0)}n isnondecreasingon

Vy

e

Ij

Definition3.1 differsfromthe existing literature(i.e.,Brumelleand Vickson, 1975),inthatthe existingliteraturegenerallycomparestheexpected valueof

two

probability distributions,say

F

andG,

viewing

J7t(s)dF(s) andJ7t(s)dG(s) as

two

different linearfunctionals

mapping

payoff

functionsto Si. In contrast,byparameterizingthe probability distributionand viewing

Jn(s)dF{s\6)asabilinear

functional

mapping

payofffunctionsand(parameterized)probability distributions tothereals,

we

are

ableto createan analogy betweenstochastic

dominance

theorems andstochasticsupermodularity theorems,an analogywhich

would

notbeobvious usingthestandardconstructions.

The

utilityofthis definition will

become more

clear

when

we

formalizethe relationshipbetweenstochasticdominance andstochastic

P

theorems,forotherproperties

P

(suchassupermodularity).

To

provide specific examples ofpairs ofsets ofpayoff functions

which

satisfy stochastic

dominancetheorems,

we

summarizethree univariate stochasticdominance theoremsinTableIt. There

are potentially

many

otherunivariate stochasticdominance theorems(forexample,theorems wherethe setofpayofffunctionsimposesrestrictions

on

the third derivativeof thepayofffunction);however,

we

willsimplyreportthe threemostfamiliar univariate stochasticdominance theoremshere.

TABLED

COMPONENTS

OF UNIVARIATE STOCHASTIC

DOMINANCE

THEOREMS"

SetsofPayoffFunctions,II SetsofPayoffFunctions,

T

(i)

U

F0

₌

{n\n:SR->

%

nondecreasing}

r

F

°={

r

\y(s)

=

I [a„)

(s),aeX}

(ii)

n

5°

s

{n\n:SR->

%

concave}

r

S0

={Y\Hs)

=

-s} i

u{y|y(.y)

=

min(a,s),

aeSRJ

(iii)

YlSOM

=

\n

k

:SR

—

» SR, nondecreasing, 1 concave

r«w

_

|y|

y

(j)

_

min(fl,s), ae9?}

Each(II,r)pairinTableII isa(univariate)stochasticdominancepair.

TableII (iii)correspondstoa

SOMSD

theorem,as

shown

inProposition2.

1

',whileTableII(i)

correspondstoa

FOSD

theorem, Proposition3.1. Let us

now

illustratethe interpretationof Definition

3.1 andTableIIwith athirdexample:Second OrderStochastic

Dominance

(SOSD),

shown

inTableII (ii). Itis

known

that

JK{s)dF{s;8)

isnondecreasingin

6

forallunivariate,concave payofffunctions

ifand onlyif \min(a,s)dF(s;d) is nondecreasingin

8

forall

aeSi,

and

\sdF(s;6)

does not

depend on 6. Observethatboth y(s)

=

s and y(s)

=

-s

areincludedin

T

so;thisforcesthe

mean

of

the distribution tobeboth nonincreasingandnondecreasing,and henceconstantin6.

Thereare

many

otherstochasticdominance theoremsinadditiontothe univariateexamplesgiven above.

Levy

and Paroush(1974) deriveresultsforbivariate functions,while

Meyer

(1990) extends theseresultsand examines

some

multivariatestochasticdominance theoremsas well.

We

will report

some

of theseresults inSection4.5,wherethe

main

objectiveistoapply theseresults toproblemsof

stochasticsupermodularity.

3.2 ExactConditionsfor aStochastic

Dominance Theorem

Inthissection,

we

studynecessaryandsufficientconditionsforthe pair(11,1")tobeastochastic

dominancepair.

We

wanttospecify theexactmathematicalrelationshipwhichisequivalenttothe

statementthat (II,T)

e

Z

_sor.

We

willfirstdiscussourresultanditsimplications;then, inSections

3.2.1and3.2.2,

we

willprovidethemathematicalargumentsunderlyingtheresult.

The main

resultofthissectionisthat (II,T)

e

I,_SDT ifandonlyifthefollowing statementistrue: theclosure(underanappropriate topology) oftheconvex cone ofII

u

{1,-1} isequaltotheclosure

(underthattopology) oftheconvex coneof

Tu

{1,-1}, where {1,-1} denotesthesetcontainingthe

twoconstantfunctions, {n{s)

=

1}

u

{it{s)

=

-1}.

We

formalizethisusingthefollowingnotation:

cc(nu

{1,-1})

=

cc(ru

{1,-1}) (3.1)

In thecontext ofspecific setsofpayofffunctions

n,

the existing literature identifies similar,but stronger sufficientconditions for thecorresponding stochastic

dominance

theorems, usinga

more

restrictivenotion of closure(i.e.,atopology with

more

opensets)thantheone which

we

will identify

below. For example, Brumelle and Vickson(1975) arguethat (3.1)issufficientforthe {Yl,T) pairs

shown

inTableIItobestochasticdominancepairsunderthetopology of

monotone

convergence. In thispaper,usingthe abstract definition

we

have developedfora"stochasticdominancepair,"

we

are able toformallyprovethat the sufficientconditionshypothesized

by

Brumelleand Vickson(1975)are in fact sufficient foranystochastic

dominance

theorem, notjust particularexamples. Further, the resultthat (3.1)isalsonecessaryfor (Tl,T) tobeastochasticdominancepairisa

new

contributionof

thispaper.

We

now

arguethat this resulttellsus

when we

cannot

do

betterthan theclosedconvex cone method. First,observethatunless

T

isa subsetofn,thereis

no

guaranteethatstochasticdominance theoremsprovide conditionswhichare easier tocheckthan

J7C(s)dF(s;0)nondecreasing in

6

forall

jrell. For example,

n

might beasetwhichisnot aclosedconvexcone. Itsclosedconvex cone might be

much

larger,anditmightnotbepossibletofindasubset,T,ofthatclosedconvex conefor

whichiseasier tocheckthat expectedpayoffsare nondecreasingin 6. Thus, (3.1) indicates that stochasticdominancetheorems,asdefinedinDefinition3.1, aremostlikely tobeuseful

when

IIisa closedconvexcone. For example,inthecaseof

FOSD,

we

considerthe setof payofffunctions IIF0.

Itiseasytoverify thatpositive scalarmultiplesand convex combinationsofnondecreasing functions

arenondecreasingfunctions, as are limitsofsequencesornetsofnondecreasingfunctions. Finally,

constant functionsare also in ITF0.

When

ITcontainstheconstant functionsandisa closedconvexcone,(3.1)becomes:

n

=

cc(ru{l,-l}) (3.2)

In principle,the mostuseful

T

is the smallestset

whose

closed

convex

coneisII.

However,

in general, there willnotbeauniquesmallestset.

To

seethis,consider the caseof

FOSD,

wherethe set

r

FO

contains indicator functions ofupperintervals.

By

taking limitsof sequences of convex

combinations of elements of

T

FO

_u

{1,-1},andappropriatelyscalingthesefunctions,

we

cangenerate

anynondecreasingfunction. However,

we

canalsodefine theset

t

FO

=

{y\y(s)

=

I

[a„,($), a

e

q\, where

_Q

representsthe rationals,andnotethat

n

F0

=

cc(fF0

u

{1,-1}) .

While

f

F0

_c

_T

FO

practicethesmallerset

f

F0 isnot anyeasier tocheck. Thus, stochastic

dominance

theorems are

generallystatedso that

T

isthesmallestclosedset

whose

closedconvex coneis II;

we

willcallsucha setthe"extremepoints"ofII.

Finally,because(3.2)isnecessaryandsufficientfor(H,T)tobeastochasticdominancepair

when

n=cc(IIu{l,-l}),

we know

that

we

cannotdo anybetterthanletting

T

bethe setofextremepoints ofII: thereisnosmaller or easier-to-check closedsetof payofffunctions, f, suchthat (II,f) is a stochasticdominancepair. Thisiswhat

we

mean

when

we

saythat

we

have provedthattheclosed

convex cone methodisthe "right"approach.

In the next

two

subsections,

we

prove that (3.1) characterizes stochastic

dominance

pairs.

Considertheproblemof orderingtwoprobability distributions,

F

1

and

F

2

,where

we

saythatasetof

payoff functionsIIorders

F

1

higherGower)than

F

2if \7r(s)dF\s)

>

(<)\x(s)dF2(s) forall

K

inII; ifneither inequalityholdsforall

n

inII,then

we

saythatthe distributionsarenotordered

by

II. In

Section 3.1.1,

we

first

show

that II orders

two

distributions exactly the

same

as the set

cc(IIu{l,-l}).

We

then

show

that

two

sets, II and T, order arbitrary pairs of probability

distributions in the

same

way

if

and

onlyif(3.1) holds, thatis,theirclosed

convex

conesarethe

same. InSection3.2.2,

we

use theseresults toprovethe characterizationofstochasticdominance

pairsaccordingto(3.1).

3.2.1

The Main

Mathematical

Results

We

begin

by

proving

some

generalmathematicalresults aboutlinearfunctionalofthe form

\Kdfi.

Our

results are variationsonstandardtheoremsfromthe theoriesoflinearfunctional analysis,

topologicalvector spaces, andlinearalgebra; the

main

contributionofthis sectionistodefinethe

appropriatefunction spacesandtopologyandrestatetheprobleminsuch a

way

that

we

canadaptthese

theoremstosolvethe stochasticdominanceproblem.

We

will

work

with aclassof objects

known

as finitesignedmeasures

on

9t",denoted

W.

Any

finite signedmeasure\ihas a"Jordandecomposition" (see

Royden

(1968), pp. 235-236), sothat

fi

=

/i+

—

it

,where each

component

isapositive,finitemeasure.

We

willbeespecially interested in finitesignedmeasureswhich havethepropertythat \d/i

=

0,sothat \d/i+

=

\dfi~. Denotethesetof

allnon-zerofinitesignedmeasures which havethispropertyby^»;

we

are interested inthis setbecause elements ofthis setcan alwaysbewritten as \i

=

—

[F

1

-

F

2],whereitisapositive scalar,and

F

1

and

F

2

areprobability distributions; likewise, forany

two

probability distributions

F

1

and

F

2 , the

measure

F

1

Inthissection,

we

willproveresultswhichinvolveinequalitiesof theform

\n

d/i

>

0, where

fieg*. Sincepositive scalarmultipleswillnotaffectthisinequality,and becausethe integraloperator islinear,

we

canwithoutlossofgenerality interpretthisinequality as jJt(s)dF1

(s)

>

_J7t(s)dF2(s) for theappropriatepairofprobabilitydistributions.

The

formernotationwillbeeasier to

work

within

terms of provingour

main

results, andfurtheritwillbeusefulinprovingresultsinSection4about

stochastic

P

theoremsforproperties

P

other than "nondecreasing."

Now

letus begin our formalanalysis. LetP* bethe setofbounded, measurable payofffunctions

on 9?". Definethe bilinearfunctional f5:P'

x%"

-»SR by

/J(7T,/i)

=

jady..

Then

f}{P",7X«)isa

separated duality: thatis,forany

&*

(J?,thereisa

n

eP*

suchthat j3(7T,/i,)

*

Pix,^),

and for

any it

x

*

n

2,thereis a \i

eW*

suchthat fi{K_vii)

*

/J(7T₂,/0-7

Our

choiceof(P",3£") is

somewhat

arbitrary:allof ourresultsholdif

we

letA" bea subset ofmeasurable payofffunctionson9?" and

we

let

B

nbe anysubsetofTff,solongasp\An_JBn

)isa separatedduality. Forconsistency

we

willusethe pair{P*,WP)inour formalanalysis.

We

now

construct ourtopology,

where

would

like to findthe coarsesttopology (thatis,the topology withthefewestopensets)suchthatthesetofallcontinuouslinearfunctionals

on

P*(the

dualof P*)isexactly the set {/J(-,//)|/ie2£*}; this is the

weak

topology a{P',tKH

)

on

P*.

By

Bourbaki

(1987, p. 11.43), this topology uses as a basis

neighborhoods

of the

form

N(7z;£,(Hi,—,Hk))

=

\ftmax|/J(7T

-7t,fl,)\

<

£

k

wherethereisaneighborhoodcorrespondingtoeach

finiteset (/^,...,^_{t )}<z

W

and each e

>

0.

We

will returnto clarify the relationshipbetweenthis

topologyandothertopologiesinthediscussionfollowing

Theorem

3.4,below.

Let cc(A) denote theconvex cone ofasetA,andlet

A

denote theclosure of

A

(where the topologyisunderstoodtobe a^P*,?^)inthediscussionbelow,unlessnoted).

We

now

usethe fact thatthefunctional P(7t,^i)

=

\ndiiislinearandcontinuousinitsfirstargumenttoprovethefollowing simple

lemma.

The

proof ofthis

lemma

is elementary, but

we

state ithere because all ofthe

mathematicalresults inthispaperbuilduponit.

Lemma

3.2 Considera setof payofffunctionsIIcP».

Then

thefollowing twoconditionsare

equivalentforall jl&g>:

(i)

For

all

n

e

II,

\nd\i>

0.

7_{Theboundedness assumption guarantees}

thatthe integralofthepayofffunctionexists. Itispossible toplaceother

restrictionsonthepayofffunctionsandthespaceoffinitesignedmeasures sothatthe pairisaseparated duality,inwhich case theargumentsbelowwouldbe unchanged;forexample,itispossible torestrictthepayofffunctionsandthesigned measuresusinga"boundingfunction."Formorediscussionsofseparateddualities,seeBourbaki(1987,p.n.41).

(ii)

For

all

7tecc(Uu

{1,-1}),

jnd/i>0.

Proof: First,

we

show

(i)implies(ii).Fix ameasure fieg'.

Then

thefollowing implications

hold:

For

all

K

ell,

jndn>0

=>

Forall

;rellu

{1,-1},

_J;rd/i>0

(Since J d[i

=

-j

dji

=

0).

=>

Forall

ff€cc(nu

{l.-l}),

_J^d//>0

(Since [tt, rf/z

>

and J

n

₂

dfi>0

implies J

[a,^,

+

a

₂7T₂]d/i

=

a^TTid\i

+

a

2

\n

2

dfi>0

when

a,,a₂

>0).

=>

For

a//

K

ecc(IIu{l,-l}), J7rd//

>

(Recall that foracontinuousfunction/,

/(A)

c

/(A).

The

implicationthen followsbecause

thehalf-space [x

e

9^|jc

>

0}isclosed,andthe linear functional P(;/l)iscontinuousforall/*.)

That(ii)implies(i)followsbecauseIT

c

cc(II

u

{1,-1}). £?.£.£>.

Lemma

3.2canberestatedanotherway:thesetofmeasures//

e^»

forwhich jndfi

>

forall7T

e

II

isexactlythe

same

as the setofmeasures [i

e£*

forwhich

Jtt^

>

for all it

e

cc(Il

u

{1,-1}).

Formally,

{/i€^-|J^d/x>0

V;renj

=

{//Gf*IJ^rd/t

>0

V;recc(IIu{l,-l})}

As

discussedabove,becausepositive scalarmultiples

do

not reversethe inequalitiesand because

thefunctional_fiislinear initssecondargument,thelatterequalityisequivalenttothefollowing:

FKF^A^JTrdF^JKdF

2

Vn

e

III

=

If^F

2

_eA^TtdF

1

ZJndF

2

V/r

e

cc(II

u

{!,-1})

Thatis,forany

two

probabilitydistributions, IIordersthe

two

distributions exactlythe

same

as

Building

from

this

lemma,

we

turn toprove the

main

mathematical theorem underlyingthe characterizationofstochasticdominancetheorems,aswellasthe characterizationsof otherproperties inSection4. Thistheorem

makes

use ofthe linearityofthefunctional /J(tt,jU) in n.

The

proofthat

(ii) implies (i)relies on

Lemma

3.2,while theproofthat (i) implies (ii)

makes

useofa standard separating hyperplane argument. Notethatthechoiceof topology,whichdeterminesthemeaningof

closure,iscriticalforthe applicationofthe separatinghyperplane theorem.

Theorem

3.3 Consider a pair ofsetsof payofffunctions(II,I*),whereII

and

T

aresubsets_of

P

t

. Thenthefollowingtwoconditions are equivalent:

(i)

lii€$'\J7tdn>0 V;rell} =

j/ie£-|Jy^>0 Vyerj.

(ii)

cc(U

u

{1,-1})

=

cc(r

u

{1,-1}).

Proof: Firstconsider(ii)implies(i). Supposethat

cc(IIu

{1,-1})

=

cc(Tkj{1,-1}).

Then

we

have: j//e2'|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})}

(Byassumption.)

=

{nep\jydfi>0

Vyer}

(By

Lemma

3.2.)

Now

we

provethat(i)implies(ii). Define II

=

cc(U

u

{1,-1})and

F =

cc(T

u

{1,-1}).

Suppose(withoutlossofgenerality) that there existsa

y

e

f

suchthat

_y

e

fl.

We

know

that

the

(XP'&P)

topologyisgeneratedfromafamilyofopen,convexneighborhoods. Recall

from abovethatthesetof continuouslinearfunctionalson P»isexactlythe set

{p(-,H)\fisTX'}. Usingthesefacts,acorollary tothe

Hahn-Banach

theorem8_impliesthat

8_SeeDunford

and Schwartz (1957,p.421),Kothe(1969,p.244)fordiscussionsofthe relevanttheorems aboutthe separationofconvexsets. SeealsoMcAfeeandReny(1992)forarelated applicationoftheHahn-Banachtheorem, where theseparatinghyperplanealsotakes theform of an element ofJ*.

since flisclosedandconvex,there existsa constant candafi.

e

7JF (aseparating

hyperplane)sothat _P(x,n.)

>

cforall iteft,and /J(y,//,)

<

c.

Since {1,-1}

€

flandis flconvex, €fl as well. Thus, /?(0, /x.)

=

>

c.

Now

we

will

argue

we

cantakec

=

withoutlossofgenerality. Supposenot.

Then

there existsa

n

e

fl

c

suchthat c

<

/?(#,//.)

=

c

<

0.

Choose

anypositive scalar

p

suchthat

p

>

—

>

1(which c

impliesthat

pc<c).

Since flisa cone, pit

€

ft. But, f}(pk,iL.)

= pc<c,

contradicting the

hypothesisthat/J(7T,/i.)

>

cforall

K

eft. So,

we

letc

=

0.

Because {1,-1}

e

ft,and fi{n,^u)> forall

n

e

ft,

we

concludethat /3(1,//.)

=

-p\l,/i.)

=

0,

andthus jJ^,

=

0. So, fi.eg', and

we

have

shown

that \7tdfi,

>

forall 7Tgft,but

JyrfyU,

<

0,whichviolatescondition(i). Q.E.D.

The

proofof

Theorem

3.3 is analogous tothe proofofthe "bipolartheorem"

from

the theory of topological vector spaces(seeSchaefer, 1980,p. 126). Thisresultisdifferentbecause

W,

not£",is

thedualofthespaceP";thisaccountsfor theinclusionoftheconstant functionsincondition(ii) of

Theorem

3.3.

As

above,

we

canrestatecondition(i)of

Theorem

3.3as follows,withoutlossofgenerality:

If

,

F

2_e A" I

\k dF

l

>

\iz

dF

2 Vtt

e

n

j

= If

1 ,

F

2

e

A" I

jydF

1

>

jydF

2

_Vy

e

T

j

The

theoremfirststatesthatiftwosets ofpayofffunctionshavethe

same

closedconvexcone,then theywillorderanypairofprobability distributions the

same

way. Second,thetheoremstatesthatif

twosetsofpayoff functions orderallpairsofprobability distributionsthe

same

way,thosetwosetsof payoff functionsmust havethe

same

closedconvexcone.

3.1.2 Exact Characterization of Stochastic

Dominance

Theorems

Building

from

themathematicalresultsofthelastsubsection,

we

now

statethe

main

resultof Section3. Thistheoremapplies

Theorem

3.3 togivenecessaryandsufficientconditionsforapair (Il,r)to satisfyastochasticdominancetheorem.

Theorem

3.4 Considera pair ofsetsofpayofffunctions(II,T),whereII

and

T

are subsetsof

P

1

. Thenthefollowingtwoconditionsare equivalent: (i) Thepair(II,T)isastochasticdominancepair.

Proof:

To

seethat(ii)implies(i),fixapair(11,0and assumethat(ii)holds. Consideran

arbitrary probability distribution F(;0) eA",and choose 8H

>

6L. Define

F

1

=

F(;6

H) and

F

2

₌

F(;0

_L). Let//

=

F

1

-

F

2

,andnotethat/I6£* .

Now,

notethat

Theorem

3.3impliesthat

For

all

Ten,

\itd(F

l

-

F

2

)

>

<=>

For

allyer, \yd{F

x

-

F

2

)

>

Sincethismust betrueforall

6

H

>6

L,then(II,r)isastochasticdominancepair.

To

see that(i)implies(ii),notethatif(ii)fails,then (withoutlossofgenerality) there exists

some y

e

f

suchthat

y

efl. Butthen,by

Theorem

3.3there existsa iu

e

£* sothat

jndfi,

>

forall iteft,but Jyd/x.

<

0. Let fc

=

jd^:

=

jdfc

. Let

=

{6

L,6H},where

6

_H

>6

L,anddefineaparameterizedprobability distributionas follows:

F(;6

_H)

= j/C

and

F(;0

_L)

=

j/z;. Butthen,

Jn(s)dF(s;9)isnondecreasingin forall

^

6

II,while

\y{s) dF(s;d)isstrictlydecreasingin6. Q.E.D.

Theorem

3.4provides an algorithm forgenerating stochastic

dominance

theorems, and for

checking whether (11,0pairs satisfystochasticdominancetheorems. Itgives theweakestpossible

sufficientconditionsonaparticular(n,r)pair toguaranteethatitisastochasticdominancepair:if cc(II

u

{1,-1})

*

cc(T

u

{1,-1}),thenthere willalwaysexistaparameterizedprobability distribution

suchthat

fy(s)dF(s;d)isnondecreasingin

8

forall

y

e

T,but Jn(s)dF(s;6)isstrictlydecreasingin

6

for

some 7ren.

However,

Theorem

3.4

makes

useofthe

weak

topology,which might belessfamiliarthan

some

others;forthisreason,

we

will

now

discussthe relationshipbetweenclosureinthe

weak

topologyand

closureinothertopologies. Inparticular,

some

existing stochastic

dominance

theoremshavebeen provedby showingthattheclosureunder

monotone

convergence oftheconvex cones ofthetwosets are equal (Brumelle and Vickson, 1975; Topkis, 1968). This raises the question,

what

is the

relationshipbetweenthoseresults,andresultsusing the notion of closureunderthe

weak

topology?

The

followingcorollaryanswersthat question.

Corollary 3.4.1 LetT beanytopology

on P>

suchthat thejunctionals {/J(-,//)|/l

€

w\,

are continuous linear Junctionals. Supposethat(11,0isa pair ofsetsofpayofffunctions, eachin

p:

Ifcc(Ilu{l,-l})T

=cc(ru{l,-l})

T

(closuretaken with respectto t),then(11,0

«

astochastic

dominancepair.

Proof: Tisfinerthano(p*pK?) (by Bourbaki,

TVS

n.43). For anyset

A

in"P*,

we know

AcA

r

cA

ff

. Takingclosuresin

(RP'W)

ofeachset,

we

concludethat

A

c

₌

_A

T

. Thus,if

cc(nu{l,-l})

r

_{=cc(ru{l,-l})}

T

,then

cc(nu{l,-l})

a

_{=cc(ru{l,-l})}

ff

. This implies

that (Il,r)isastochasticdominancepairby

Theorem

3.4. Q.E.D.

To

seethe intuitionbehindthis result, firstnotethat,given

two

topologies T, and r2,wherez, is

coarserthan 12 (written T,

c

T_{2 ),}theclosureof aset

A

under %2 (-A*2

)iscontainedinthe closureof

A

undert,(

A

T

>).Thisistruebecausethe finertopology has

more

closedsets;

A

Tl isa closedsetunder T2,butsincex₂ isfiner,theremight bea closedsetwhichisastrictsubsetof

A

T

< butthatstillcontains

A.

Sincethe

weak

topologyiscoarser thananytopologywhich

makes

the linear functionals /?(7T,/x)

continuousinn,theclosureof asetof payoff functionsunder

monotone

convergenceiscontainedin theclosureofthesetunderthe

weak

topology. Thus,iftheclosed(under

monotone

convergence)

convex conesoftwosetsofpayoff functionsarethesame, thentheclosed (underthe

weak

topology)

convex conesof those

two

setsofpayofffunctionswillbethesame. Thistellsusthatcheckingthat theclosed(undermonotoneconvergence)convexcones oftwosetsofpayoff functionsarethesameis

sufficient to establish that the

two

setsofpayofffunctions satisfyastochasticdominancetheorem.

Thus, in practice,

when

checking whether (TIT) satisfy sufficientconditionsforastochastic

dominancetheorem,itispossibletocheck whethertheclosureoftheconvex conesofthetwosetsare equivalent,usinganytopologywhichisconvenient (providedof coursethatthetopology guarantees continuity of the functional /?(•,/*))•

The

topologies of

monotone

convergence, dominated

convergence, and uniform convergence are examples oftopologies

which

might be useful in applications.

Remark

1

We

havecharacterizedthe relationshipbetweenY1S0M and

T

SOM asfollows:

n

sou

₌

cc(

n

soM

_u

!!_!})

=

cc(

T

som

_u

[i-i])

Since

much

ofthe existing literatureinvolvesintegration

by

parts,itisusefultoillustratethe relationshipbetweenintegration

by

partsandourcharacterization.

We

can viewthe integration

by

partsasa

means

of discoveringtheextremepointsof asetofpayoff functionswhich forms

a closedconvexcone.

In thecasewherethedistributionfunctioniscontinuousanddifferentiablewith respectto

support [0,J], both Propositions 2.1' and 3.1 can be proved directly

by

first rewriting

expectedprofits using integration

by

parts, and then taking the partial derivative ofthis

expression withrespect to theparameter(s). It isthe integration

by

partswhichdeterminesthe relationshipbetween

U

S0M and

F

SOM

,andestablishingthis relationshipplays adirect role in

the proofs of both Propositions 2.1'

and

3.1.

We

can rewrite the functional P(7Z,F)

=

J' 7t(s)dF(s) as follows,usingintegration

by

parts:

f

7C(s)dF(s) Js=0

=

\k(s)

-

n\s)

s

+

s _f' n"(s) ds]•

f

dF(s) (3.3) Js=0 _J Js=0

+

7r\s)\

S

sdF(s)-\

S

k"{s)\

S min(s,v)dF(v)ds Jj=0 Jj=0 Jv=0

Notethatthefunctions y(s)

=

s and y(s)

=

min(a,s),

which

determinethe setoffunctional

T

SOM

,appearinthisexpression,asdothefunctions n' and 7t",whichcharacterizethe setof

payofffunctions, TlSOM. Infact,

we

canseethatthefunctional

f K(s)dF(s)isthe limitof a

Js=0

sequenceofpositivecombinationsoftheexpected value ofthefunctionsin

T

SOM,pluslinear

combinationsof

J

dF(s).

The

set TlSOM isdefined sothat %' and

-n"

are positive; thus, the functions y{s)

=

sand y(s)

=

min(a,.s) arepositive in

T

S0M.

The

constant functions {1,-1}areincludedbecause f dF(s)isconstantat1,butthe coefficient

Jj=0

\k(s)

-

7t'(s) s

+

s

J* iz"(s) ds

canbepositiveornegative.

To

see

how

this isused in the study of

mono

tonicity, note that

we

canevaluate the

functional given in (3.1) at a parameterized probability distribution F(;0) (i.e., take

P(n,F(;d))) andthendifferentiate,asfollows:

-§}Jlo n(s)dF(s;d) (3.4)

=

k'(s)

-^jsdFisid)

-H

o n"(s) JqJJ min(s,v)dF(v;6)ds

This expression canbe usedtoprovethe

SOMSD

theoremdirectly. Ifthe

mean

and j^Tmn(v,s)dF(v;0) are nondecreasingin 6, thenfor any nondecreasing,concave payoff

function, this expression is clearly positive.

On

the other hand, if the

mean

or

j^min(v,s)dF(v;9) weredecreasingin

8

somewhere,

we

couldconstructa nondecreasing,

concave payofffunctionwhichputalloftheweightonthefailures.

What

theintegration

by

parts

shows

usis thatinsteadof checkingthat /J(tt,F(;0)) is nondecreasing in

6

forall

it

6

n

S0*f

(the left-handsideof(3.4)),

we

cancheckthat p(y,F(;6)) isnondecreasingin

6

secondsetof conditionsiseasier tocheck.

The

constant functions {1,-1}

from

above

do

not appearin

T

SOMbecause [1-dF(s;&)

=

1isalwaysconstantin0.

We

emphasizethe fact that the

Jj=0 sets

U

S0M and

T

50M

aredeterminedby the integration

by

parts (equation 3.3),not

by

the

subsequentdifferentiationwithrespect to theparametersofthe distribution.

Inananalogous way,

we

canevaluatethe functional

f n(s)dF(s)attheparameterized

Js=0

probability distribution

F(;0

X,62),using equation(3.3),andtake the

mixed

partialderivative:

The

proofofthe stochasticsupermodularity theoremfollows similar argumentsto thatof

SOMSD.

Thisexampleclarifiesthe relationshipbetweenthe

new

stochasticsupermodularity

theoremandthe existing stochasticdominanceresult.

3.3 Conditional Stochastic

Dominance

Inthissection,

we

show

that

Theorem

3.4 can be extendedtothe caseof"conditionalstochastic

dominancetheorems."

We

willusethefollowingnotation:

f dF(t;6)

G(

S;6,K)

=

±£^

J dF(t;6) jx(s)dF(s,6) \dF(s,6) seK

We

now

introduceanobjectwhichisanalogoustoastochasticdominancetheorem:

Definition 3.2 Consider apairofsets of payofffunctions(II,T), with typical elements

it:SR"->SR

and

y

:9?"-»SR. Let

K

beacollectionofsubsetsof9?". Thenthepair (TIX)isa

K-conditionalstochastic

dominance

pair ifconditions(i)

and

(ii)are equivalentforall with

apartialorder

and

all

F

:

9T

x

->[0,1]suchthatsuchthat F{;6)

e

A":

(i)

For

all

neU.

and

all

KeK,

k{G\K)is

nondecr

easingin 6.

(ii)

For

all

yeT

and

all

KeK,

y(6\K)is

nondecreasing

in 6.

Theorem

3.5 Consider apairofsetsof payofffunctions(Yl,T),whereII

and

T

aresubsets of

P

1

. Let

K

heacollection ofsubsetsof9t\ where 9?"€ K. Thenthefollowingtwoconditions

are equivalent:

(i) The pair (TIT)isa ^-conditionalstochastic

dominance

pair.

(ii) cc(II

u

{1,-1})

= ccOTu

{1,-1}).

Proof:

We

canapplytheproofof

Theorem

3.4almostexactly. Let

n^

=

{n

I_K\neII},and

likewisefor Tk.

Then

notethatcc(Ilu{l,-l})

=

cc(T

u

{1,-1})impliesthat

cc(U

_K

u{I

_K,-IK})

=

cc(TK

u

{I_K,-IK})forallK. Then,forevery

K

we

apply

Theorem

3.4,

(ii)implies(i).

To

show

that(i)implies(ii),theargumentsof

Theorem

3.4canbeusedto

show

thatif(ii)fails,then(i)mustfailforthecasewhere

K

=

9?". Q.E.D.

One

exampleof a conditionalstochasticdominance theorem whichhasappearedinvariousformsin theliterature(seeWhitt(1982)) involvesthe setofnondecreasingpayofffunctions, as follows:

Theorem

3.6 The following conditions are equivalent:

(i)

For

all it:9?"

—

»9? nondecreasing

and

allsublattices

#c

91", k(6\K)isnondecreasing

in6.

(ii)

For

allincreasingsets

Ac

9?"

and

allsublattices isfc9?", f dG(s;6,K) is JAnK

nondecreasingin6.

Proof: Apply

Theorem

3.5together withthe fact that

cc({k\k nondecreasing}

u

{1,-1})

=

cc({I

A\IA(s)nondecreasing}

u

{1,-1}). Q.E.D.

Thistheoremisusuallyprovedusing algebraic arguments; but usingthe resultsofthispaper,

we

see thatitfollowsasanimmediatecorollaryof

Theorem

3.5. First, take thecase where

n-1.

Then

condition(ii)isequivalenttorequiringthatF(s;6)satisfiesthe

Monotone

Likelihood Ratio

(MLR)

Order(foraproof,seeWhitt(1980)),definedasfollows:

Definition 3.2 The parameter

6

indexes theprobability distribution F(;6)

e

A

1

accordingto the

Monotone

Likelihood Ratio

Order

(MLR)

if,forall

6

_H

>

_L, thereexist

numbers

-oo<a<b<°°

and

a nondecreasingfunctionh:[a,b]-»9? suchthat F(a;6

H)

=

0, F(b;6

L)

=

1,

and

jKdF(s;dH)

=

jKh(s)dF(s;GL)forall K^[a,b].

When

thesupportof

F

isconstantin6,and

F

has a density/, then an equivalent requirementisthat/

satisfiesthe

Monotone

Likelihood Ratio Property

(MLRP),

as follows:

f(sH;6)

f(sL;6)

isnondecreasingin

6

foralls11

_>

Milgrom

(1981) introduced the

MLRP

to theeconomics literaturein thecontext of signaling problems.

The

density/satisfies the

MLRP

ifandonlyifthelogofthedensityissupermodular.

The

MLR

implies

FOSD

(butnotthe reverse); thatis,ifF(s;6)satisfies

MLR,

thenthe distribution F(s;6)

isdecreasingin flpointwise.

FOSD

does not placeanyrestrictionsonthe

movements

of F(s;6)apart

fromthe restriction thatchangesin6 donotcausethe distribution to cross.

The

MLR,

ontheother hand,requires that foranysublatticeK,the distribution conditionalon

K

satisfies

FOSD.

Now,

supposethatn

>

1. If

we

assumethatthevectorsisa vectorofaffiliated

random

variables

(see

Milgrom

and

Weber

(1982),

who

defineaffiliationand

show

thatifthe distributionhas adensity, thelogofthatdensitymust be supermodular9

),then condition(ii)requires thateachmarginal, Ffaid),

satisfies

MLR.

The

mostgeneralcase,wherethevectorsisanarbitraryvectorof

random

variables,

has nottoourknowledge beenanalyzedintheliterature.

We

havepresented

Theorem

3.6 toillustratethe relationshipbetweenseveral existingresults.

The

framework developedinthispapershowsthattheproof of

Theorem

3.6isanimmediateconsequence; previous analyseshavereliedon

more

complicated arguments. But,

Theorem

3.6isjustoneexample;

inhisanalysisof firm entryandexit inadynamicenvironment,

Hopenhayn

(1992) derives another

conditional stochasticdominanceresult,which heterms

monotone

conditionaldominance. Thisresult

considersthecase of

n

=

IlFOand

K

isthe setofindicatorfunctionsofupperintervals.

Our

theorem shows

how

togenerateother conditional stochasticdominance theoremsasapplicationsarise.

4

OTHER

PROPERTIES

OF

STOCHASTIC OBJECTIVE FUNCTIONS

Thissectionderivesnecessaryandsufficientconditionsfortheobjective_{function, j 7c(s)dF(s;&)}, to

satisfyproperties

P

otherthan"nondecreasing in 0,"forexample,the propertiessupermodularor

concavein8.

We

ask

two

questions:(i)For whatproperties

P

istheclosedconvex cone

method

of

proving stochastic

P

theoremsvalid? (ii)For whatproperties

P

can

we

establish that the closed

convex coneapproachisexactly the right one, as in thecaseofstochasticdominance?

We

proceedas follows. InSection4.1,

we

introduce notationanddefinitionswhichextend our

stochasticdominance frameworkto"stochasticsupermodularity theorems,"andfor arbitrary properties

P,"stochastic

P

theorems." Section 4.2answersquestion(i),showingthattheclosedconvex cone

approachtoprovingstochasticdominance theoremsisvalid forall"stochastic

P

theorems,"where

P

is

a

CCC

property. InSection 4.3

we

introducea

new

classofproperties,Linear Difference Properties

(LDPs),whichisa subset of

CCC

properties.

We

show

thatexamplesof

LDPs

includeconvexity,

9_{In fact,Milgrom and Weber}

(1982) also showthat, for general distributions, affiliationis equivalentto log-supennodularity oftheSR"derivativeon a product measure.

supermodularity,andallpropertieswhichplacea signrestriction

on

aderivative (refer toTableIfora

summary

of

CCC

propertiesand LDPs). Section 4.4 addresses question(ii),provingthattheclosed

convex cone approachisexactlythe rightoneforallstochastic

P

theorems

when

P

isan

LDP.

Section 4.5 applies the resultsofthis sectionto thecaseofsupermodularity, providingexamples of(11,1") pairsfromthe stochasticdominanceliterature(whichare also stochasticsupermodularitypairs,byour

mainresult)anddiscussingapplicationsofstochasticsupermodularity theorems.

4.1 StochasticSupermodularity

Theorems

and

Stochastic

P

Theorems

We

now

introduce a construct

which

we

will callstochastic

P

theorems,

which

are precisely

analogous to stochastic

dominance

theorems. Let us begin witha specific example, stochastic

supermodularity theorems. Using

two

parallel definitions,

we

will be abletoidentifythe close

relationshipbetweenthe stochastic

dominance

theoremsandstochastic supermodularity theorems. Stochastic

dominance

theorems focus

on

monotonicity of expectedprofitswith respecttoasingle

parameter, whilestochastic supermodularitytheoremsexamine whether increasingone parameter

increases the returns(inexpectedprofits) toincreasingtheotherparameter(recallthatsupermodularity

canbecheckedpairwise). Proposition2.3isanexample ofastochasticsupermodularity theorem.

The

followingdefinitionispreciselyanalogoustoDefinition3.1:

Definition 4.1 Consider a pair ofsetsofpayofffunctions(II,F), with typical elements

it:

9T

—

>9?

and

_y

:9?"

—

»91. The pair(II,T)isastochastic supermodularity pairif

conditions(i)

and

(ii)are equivalentforalllattices

©

and

all

F

e

A" e:

(i)

For

all

n

sU,

fJt(s)dF(s;6) is

supermodular

in 6.

Js (ii)

For

all

yeT,

[y(s)dF(s;6) is

supermodular

in6.

Further,

we

definethe set

Z

J5r tobethe setofall(II,r)pairs

which

arestochasticsupermodularity pairs,analogousto "L

_sm

.

To

placeourexample

from

Proposition _{3.3 in this}framework,the_pair

(U

SOM

_,r

SOM

)ismI.

!lsr.

When

comparingDefinition3.1 (stochasticdominance theorem) andDefinition4.1 (stochastic

supermodularity theorem), itishelpfulto recall that stochastic

dominance

theoremsandstochastic

supermodularity theorems are bothcharacterized

by

(11,0pairscorresponding to sets oflinear functionalsoftheformf5(K,-),whicharethen

composed

with parameterizedprobability distributions.

Thus, itis possibleto

compare

the

two

types of theoremsdirectly, even thoughin a stochastic

dominancetheorem,theparameter spaceisanysetwith apartialorder,whileinastochasticdominance

More

generally,

we

can definea classoftheoremscalled Stochastic

P

Theorems, which are

definedforanarbitrarypropertyP.

We

are interested inproperties

P

togetherwithparameterspaces

Q

pwhicharedefined sothat,givenafunction h:

P

—

> 9t,thestatement "h(6)satisfiesproperty

P

on

0/'

iswell-definedandtakesonthefollowingvalues: "true"or"false."Let

Q

p denotethesetof allsuch parameter spaces 0,,. Further, fora givenproperty P,

we

willdefinethe setof admissible

parameter spacestogetherwithprobability distributionsparameterizedonthosespaces:

V

n

P

=

{(F,e

p)\epe

e

pand

F

e

A%

p}

.

Definition 4.2 Consider apairofsetsof payofffunctions (IT,r), withtypical elements

K

:SK"

—

>9?

and

7:9?"

—

>9?. Thepair(II,

O

w

astochastic

P

pairifconditions(i)

and

(ii)are equivalentforall

(F,Q

_P)e

V

p:

(i)

For

all 7teTl,

j7t(s)dF(s;6) satisfies property

P

on

©

p.

(ii)

For

all

yeT,

fy(s)dF(s;6)satisfies property

P

on 0„.

Js

For example,forastochasticconcavitytheorem, 0,, can be any convexset, andcondition (i)of Definition 4.2 isinterpreted, "Forall tte

n,

Jn(s)dF{s;6) is concave in 6."

As

in the case of

stochasticdominance,

we

let I.

SPT bethe setofallstochastic

P

pairs foragiven property P. Also

analogousto stochasticdominance,

we

canrewrite therequirementof Defintion 4.2as follows:

l(F,e) e

V

_{p \J7t(s)dF(s;6)}satisfies

P

on Vtte

nj

=

|(F,0)ev;\jy(s)dF(s;G)satisfies

P

on

V/

€

rl

Inthenextsection,

we

show

thattheclosedconvex cone

method

of provingstochasticdominance

theoremscanbeextendedtoallstochastic

P

theorems,if

P

isa closedconvex coneproperty.

4.2 The Closed

Convex

Cone Approach

isValidforall

CCC

Properties

Inthissection,

we

identifyaclassofproperties,which

we

call"Closed

Convex Cone" (CCC)

properties,suchthatthefollowing statementistrue:forallproperties

P

whichare

CCC,

ifapair(II

_J)

is astochastic

dominance

pair, then(tl.Disastochastic

P

pair;thus, forall

CCC

properties, the

closedconvex cone

method

ofprovingstochastic

P

theoremsisvalid. Thisresultisusefulbecauseit allows ustousethe existingtheoremsfromthe stochasticdominanceliteraturetogenerate

many

new