Random Matching and Aggregate Uncertainty

Munich Personal RePEc Archive

Random Matching and Aggregate

Uncertainty

Molzon, Robert and Puzzello, Daniela

University of Illinois

2008

Online at

https://mpra.ub.uni-muenchen.de/8603/

ROBERTMOLZON

†

ANDDANIELAPUZZELLO

‡

Abstrat. Randommathingisoftenusedineonomimodels asa meansof introduingunertainty insequential

deisionproblems. Weshow that randommathingshemes thatsatisfy standard onditions onproportionality are

notunique. Heneeonomimodelsthat userandommathingasashokmaynotbewelldenedunless additional

onditionsareimposedonthemathingsheme.Twoexamplesshowthatinasimplegrowthmodel,radiallydierent

optimal behavior anresult fromdistint mathingshemessatisfying idential proportionality onditions. Wegive

onditionsontherewardandtransitionstruturesofsequentialdeisionmodelssothemodelsarewelldenedinspite

ofnon-uniquenessofthemathingsheme.Finally,informationentropyisintroduedasamethodforseletingunique

mathingstruturesforthesemodels.

KeywordsandPhrases: randommathing,aggregateunertainty,informationentropy

JEL Classiation Numbers: C00,C73,C78,D83

1. Introdution

Randommathing ofagentsplaysafundamental rolein modelsofeonomi, soialand biologialsystems. These

models often attempt to explain how interation among individuals will impat the system as a whole. Sine the

mannerinwhihtheindividualsinteratdependsuponhowtheyaremathedintherstplae, alearunderstanding

ofthemathingproessisessential. Importantexamplesoftherolerandommathingplaysinmodelsanbefoundin

[18℄,[17℄, and[15℄. 1

Onefrequentobjetivein randommathing modelsis adeterministimodel that apturestheaggregatebehavior

oftheagents. Theearly paperbyHardyandWeinberg[15℄ approximatedastohastisystemwithrandommathing

in populationgenetisbyadeterministisystemsothattheproblemof omputinglimitingpopulationtypesbeame

quite straightforward. Indeed, one of the main objetives of the extensiveliterature on random mathing has been

to showthat it isreasonableto approximate aompliateddisrete dynamialsystemin whih agentsarerandomly

mathedbyadeterministisystemthatisusefulforomputation. Theattempttogetatratabledeterministisystem

hasledmanyauthors to onsideraontinuumofinteratingagents. Thisis theimpliitassumptionin thepaperby

HardyandWeinberg.

Inordertoobtainadeterministimodel,anindependeneonditionwithrespettotypesisoftenimpliitlyassumed.

Iftheagentsareofdistint types,thentheeventthat agent

g

is mathedto anagentof apartiulartypeshould,in

somesense, be independentof theeventthat agent

e

g

is mathed to anagentof apartiular type. This assumption impliesthattherandombilateralmathinggeneratesaproessofi.i.d. randomvariables(indexedbythesetofagents

takingvaluesintheset oftypes). Thenon-existeneof measurableontinuoustimestohastiproessesofnontrivial

i.i.d. randomvariables is ontainedin the work of Doob[9℄. Judd[16℄elaborated onthese issues foreonomistsby

showingthat arandommathingproess foraontinuum ofagentsthatsatises anindependene oftypesondition

isnotmeasurablewithrespettothenaturalprodut

σ

−

algebra

plaedontheprodutofmathesandagents. R. Boylan [7℄ used another approah to obtaindeterministi systemsthrough mathing models. He onstruted

speialmathing shemes forbothnite andountablyinnitesets ofagents. His randommathing shemes satisfy

propertiesthatmakeitpossibletoapproximatestohastidynamialsystemsbydeterministisystems. Deterministi

systemsarethereforeobtainedasthelimitofthestohastisystemsasthenumberofagentstendstoinnity.

C.Al

´

os-Ferrer[4℄ showedthat itispossibleto onstrutrandommathingmodelsforaontinuum ofagentssuh

that theprobabilitythat agivenagentismathed to anagentof agiventypeis proportionalto thefrequenywith

whih thetypeis presentin the population. Thestohasti proess he onstrutsis measurable withrespetto the

standardprodutmeasure ontheset of agentsand theset ofmathes. This proessis observablewith thestandard

Theauthorswouldliketothankpartiipantsatthe2007SAETMeetinginKos,Greeeandthe2008WorkshoponGeneralEquilibrium

inRiodeJaneiro,Brazilforhelpfulommentsandsuggestions.

†

RobertMolzon,DepartmentofMathematis,UniversityofKentuky,Lexington,KY40506,USA,molzonms.uky.edu.

‡

DanielaPuzzello,DepartmentofEonomis,UniversityofIllinois,Champaign,IL61820,USA,dpuzzelluiu.edu. 1

Intheeonomisliterature,randommathingappearsinmanysubelds,inludinggametheory,monetarytheory,laboreonomisand

experimentaleonomis. Sinemanyofthepapersweiteinthisworkgiveexellentreviewsoftheuseofrandommathingineonomis,

mathed with a partiular type. D. Due and Y. Sun [10℄ were ableto onstruta mathing proess that satises

anindependene intypesondition. Theirproessismeasurable withrespetto aprodut

σ

−

algebra

that islarger than thestandard

σ

−

algebra

onthe produt of agents and mathes, and theirexistene resultuses methods from nonstandardanalysis.

Sine a signiant amount of work has been done that attempts to settle the foundational problems of random

mathing, itis fairto ask ifthe issue,at thispoint,is simplyatehnialone. Eonomists haveagood sense ofhow

randommathing should work in theirmodels, and areunderstandably relutanttodrop amodel ortheorybeause

the foundations might haveminor tehnial diulties. Weshall showhow themehanisof the random mathing

proess an have signiant impliations for eonomi models. In this sense, our ontribution omplements that of

Aliprantis,Camera,Puzzello[2,3℄where theanonymitypropertiesofrandommathingshemesareemphasized.

In this paper we disuss the role nonuniqueness plays in random mathing models. In setion 5 wegive simple

examples of growth models that use random mathing as shok. The mathing shemes used in the examples are

identialfromthepointofviewofthestandardpropertiesimposedonmathingintheeonomisliterature. However,

in our examples,the behavior ofthe systemsthese growthmodels desribe areradially dierent from one another.

Theseexamplesdesribereasonablesituationswherethemehanisofmathinghavesubstantiveeonomiimpliations,

andtheyanbeextendedinseveraldiretions.

Thefundamentalproblemthatourexamplesillustratestemsfromthenon-uniquenessofrandommathingshemes

thatsatisfystandardassumptionsaboutthemathing. Sinetheexisteneproblemforrandommathingisnontrivial,

[7, 4, 13, 10, 11℄, it is perhapsnot surprising that theuniqueness question hasnot been examinedup to this point.

Wethinkthat itisimportanttostudynonuniquenesssineitimprovesourunderstandingoftheworkingsofrandom

mathingsystems.

Before oneattempts to address the problems posed by non-uniquenessof randommathing shemes that satisfy

proportionality onditions, it is neessary to have a lear understanding of the aggregate unertainty onept for

random mathingmodels. An extensivedisussion ofaggregateunertaintyfor thease ofshoksthat do notome

frommathinganbefoundinAl

´

os-Ferrer[6℄. Wemaketheoneptofnoaggregateunertainty fordynamirandom

mathingmodels preiseinDenition 3.5.

Thispaperpresentstwomethodsfordealingwiththeuniquenessissue. Therstmethodexaminestherelationship

betweentherandommathingstrutureandtherewardandtransitionstruturesofthemodels. Thisapproahprovides

suientonditionsfortheformulationofrandommathingmodelswithnoaggregateunertaintywherenonuniqueness

isnolongerrelevant. Insomeasesthemodelofinterestmayexhibitaggregateunertainty,and nonuniqueness may

be relevant. Our seond method introdues the onept of information entropy to obtain uniqueness of a random

mathingsheme.

2. Outline of Paper

Wepresentabrief outlineof the papersothat the reader anquikly see themain ideas withouthaving to read

throughthetehnialdetailsofeahsetion. Ifoneiswillingtoaeptsomeoftheintermediateresults,oneanstart

with Setion3,and thengoonto read theSetions5,6,and 8. Oneanthenread thesetionsthat disussspei

models ofdierentardinalitiesforthepopulations,4.1, 4.2,and4.3. Finally, thesetionontheinformationentropy

ofrandommathing,Setion7,anberead.

InSetion3,SequentialDeisionProblemswithRandomMathing, wedesribetypiallassesofeonomimodels

thatuserandommathingasshok. Thesetofagentsinthesemodelsanbenite,ountablyinniteoraontinuum.

Inthissetion,wedonotassumeanyspeialardinalityforthesetofagents,butdesribethethreeessentialstrutures

ofthesemodels: reward,transition,andmathing. Wedesribehowthemathingstrutureinteratswiththereward

struture and the transition struture, and we larify why onemust be onerned with the joint measurablity of

theagentand mathingmeasures. Wealsodesribethreestandardassumptionsthat relatethetype of anagentto

therandom mathing, and weshallrefer to them asProportionality assumptions in this overview. We alsogive a

denitionofaggregate unertainty forthesemodels.

In Setion 4, Mathing Strutures and Proportionality, we desribe in more detail the methods of onstruting

random mathing shemes that satisfy the proportionality onditions. We deal with eah of the three possibilities

for theardinalityof theset of agentsin separate subsetions. Foreahof thethree ardinalities, weshowthat the

set ofmathingshemesthat satisfyproportionalityis anonemptyonvexsetin somelargedimensional spae(large

anmeaninnite). Inpartiular, theset ofrandom mathingshemesthat satisfyapartiularset ofproportionality

onditionsisfarfrom unique. Infat,theproblem ofnon-uniquenessbeomesmorediultastheardinalityofthe

withRandom Mathing. Herewepresentaverysimplegrowthmodel, derivedfrom astandardwellknownexample,

thatdemonstratestwopoints.

(1) Twomathing shemesthat satisfy idential proportionalityonditions angive radially dierent eonomi

results.

(2) Forourmodel,thestrutureoftheprobabilitydistributiononthesetofmathesisimportanttothe

determi-nationof optimalpoliies.

Although weuse amodel with a nite set of agentsto demonstratethe problem, asimple modiation shows that

the probem persists if the set of agents is innite (ountable or a ontinuum). The problem is the result of the

non-uniquenessofthemathingstrutureanditsrelationshipwiththerewardandtransitionstrutures.

We next desribe, in Setion 6, No Aggregate Unertainty, how one an mitigate the problem aused by

non-uniqueness. Weshowthatiftherewardandtransitionstrutures ofamodeldependonlyonthetypesoftheagents,

then tworandom mathing shemes that satisfy thesameproportionality onditionswill give thesame well dened

deterministi model. Thus unertainty will disappear in the aggregate. The results are again independent of the

ardinality of the set of agents and level of sophistiation of the mathing sheme. Although one must makefairly

strongassumptions onthe rewardand transitionstrutures, it will alwaysbepossibleto usethis tehniqueifone is

willingtoassumethatthesystemanbedesribedbyanitesetofstates.

Setion7,TheEntropyofaMathingStruture,introduesanewideatoeonomimodelingwithrandommathing.

Insomeasesitwillnotbedesirabletolimitthestatespaeofindividualagentstonitesets. Forexample,onemight

wanttoallowanindividualtopossessanyrealnumberamountofapitalupto somebound. Thiswillmakethestate

spae for the individual agentsunountable, and the method of dealingwith non-uniqueness desribed in Setion 6

will notwork. Tomake suh amodel well dened oneneedsawayto singleout auniquerandommathing sheme

that satises the proportionalityonditions. In someases onean do this by using theinformation entropy ofthe

mathingsheme.

Wenally summarize ourresultsand indiate somefuture diretions forresearh in Setion 8. Althoughrandom

mathingshemesaretypiallyassumedtobeexogenous,onemightonsiderrandommathingasaontrolvariable.

Indeedthisisalreadybeingdoneforompliatedbiologialandeonomisystems. Forexample,ontrolofepidemis

isoftenlinkedtoontroloverarandommathingstruture.

3. Sequential DeisionProblems withRandom Mathing

Aneonomimodelwithrandommathingofagentsfrequentlyhasthestrutureofasequentialdeisionproblem.

Weonsider disretetimemodelsdenedoveraninnitetimehorizon. Thestateofthesystemisdetermined asthe

aggregateof the statesof the individuals that make upthe population. In this setion, we takea generalapproah

to theproblemofmathing andsothesetof agentsanbenite, ountablyinnite,oraontinuum. Wedenotethe

stateof anindividual agentat time

t

by

s

(

t, g

)

. We write

s

(

g

)

to indiate the stateofthe agent

g

whenthe timeis

understoodbytheontext. Generally

s

(

g

)

willbeavetoronsistingofvariousattributesoftheagentsuhastypeof

good produed, apitalholdings,buyeror seller,andsoon. Thestateofthepopulationasawholeis thendesribed

bythefuntion

s

(

∗

)

that givesthestateofeahindividual. Thesequentialdeisionproblemanberoughlydesribed asfollows.

(1) Agents are randomly mathed and interat. The interation resultsin areward (or ost) forthe individual

agents. Theindividualrewardsaredeterminedbythestateoftheagent,thestateoftheagent'spartner,and

aontrolvariable. Thestateoftheeonomyistheaggregateofthestatesofindividuals.

(2) Aftertheinterationor mathingofagents,theindividualstatesofagentshange. Thenewstatesare

deter-minedasaresultoftheinterationandhenedepend uponthemathingsheme.

Tomakethesemodelspreise,itisneessarytodesribetherewardproess,themathingproess,andthetransition

proess. Therearegenerallytwowaysof seletingvaluesfortheontrolparameter. Perhapsthesimplest approahis

tostipulatethattheontrolparameterisseletedbyaentral planner. Thustheinterationofindividualagentsand

thesubsequenttransitiontoanewstateisindependentofdeisionsmadebyagents. Theentralplannermightselet

values ofthe ontrol to optimize totalexpeted rewardfor theentire populationsubjetto some set of onstraints.

Alternatively,theontrolparametermightbeseletedbyarulelinkedtotheinformationavailabletoindividualagents.

Inthisaseindividualagentsseletvaluesoftheontroltooptimizeexpetedindividualreward.

Sine bothmethods of seleting ontrol valuesresultin veryompliated stohasti dynamial systems, one tries

to simplify the model through assumptions on all three parts of the sequential model. One goalof the simplifying

gregateeonomy that depend only onthe proportionalityof types. The resulting model is oftensaid to ontainno

aggregate unertainty.

Weintroduesomenotationto maketheideasmorepreise. Let

A

denote theset ofagents,with atypialagent

in

A

denoted by

g

. Assume we havea way of measuringthe size of subsets of

A

so that wehave ameasure spae,

(

A,

L

, µ

)

. Inase

µ

isnotanitemeasureon

A

(e.g.

A

is theset ofnaturalnumbersand

µ

isountingmeasure)one mightenounteronvergeneproblems. Tohandlethisproblemweassumewehaveanexhaustionof

A

bysets,

A

n

of nite

µ

measure,suhthat

A

=

∪

A

n

with

A

n

⊂

A

n

+1

. Let

Φ

denotetheset ofbilateralmathesofagents,thatisthe set of involutionsof

A

with noxed point. Let

ϕ

denote atypialelementof

Φ

. In subsequent setions,wedisuss

the set

Φ

in detail, but for now theonly additionalstruture weplae on

Φ

is aprobabilitydistribution,

Q

, with a

σ

−

algebra

ofsubsetsthatwedenoteby

F

,sothat

(Φ

,

F

, Q

)

isaprobabilityspae.

Therewardstrutureforthemodelisdeterminedbyrewardfuntions fortheagentswhihwedenote by

R

g

(

s

(

g

)

, s

(

ϕ

(

g

))

, u

)

.

Thevariable

u

istheontrol. Thisnotationisonsistentwithourassumptionthatrewardsforindividualagentsdepend

onlyon thestate ofthe agent, thestate ofthe agent's partner,andthe ontrol. Theexpetedaggregaterewardfor

thepopulationforasingletimeperiodis

E

Q

[

R

(

s, u

)]

.

Aomputationofthisexpetedrewardwillinvolveintegration(orsummation)overtheprodutmeasurespae

A

×

Φ

. Weuse integration withthe understanding that theintegralis traditionally written asasum in thease ofdisrete

distributions on disreteprobabilityspaes. Inase themeasure

µ

isnotnite the aggregaterewardoverallagents

would notgenerallybenite, sowedene expetedrewardasalimitwithrespet totheexhaustion of

A

. Inase

µ

isnitethelimitisnotneeded. Theaggregateexpetedrewardisomputedas

(3.1)

E

Q

[

R

(

s, u

)] = lim

n

→∞

1

µ

(

A

n

)

Z Z

A

n

×

Φ

R

g

(

s

(

g

)

, s

(

ϕ

(

g

))

, u

) (

dµ

(

g

)

×

dQ

(

ϕ

))

.

Remark 3.1. Oneould allowmoreompliated rewardstruturesthat inlude stohasti shoks independent ofthe

shokintroduedbyrandommathing. Wewanttofousonthestohastishokintroduedbythemathingproess,

andsowedonotinludethisompliatingfeature inourdisussion.

Wenext desribethe transition strutureof thegeneralmodel. Weassumethat thestateof atypial agent

g

at

time

t

+ 1

isadeterministifuntion ofthestateof

g

attime

t

,thestatetheagentwithwhom

g

ismathed attime

t

, andtheontrol variable

u

. Inotherwords

s

(

t

+ 1

, g

) =

f

(

s

(

t, g

)

, s

(

t, ϕ

(

g

))

, u

)

.

Therefore,thetransitionprobabilitiesforagentsaregivenby

Prob

Q

(

s

(

t

+ 1

, g

)

|

s

(

t, g

)

, u

) =

Q

{

ϕ

|

s

(

t

+ 1

, g

) =

f

(

s

(

t, g

)

, s

(

t, ϕ

(

g

))

, u

)

}

.

Thetransitionprobabilitiesdependexpliitlyontheprobabilitydistribution

Q

denedon

Φ

. Thetransition

probabil-itiesfortheentiresystemaretheaggregatesofthetransitionprobabilitiesforindividualagents. Intermsofthestate

funtion thisanbewritten

Prob

Q

(

s

(

t

+ 1

,

∗

)

|

s

(

t,

∗

)

, u

) =

Q

{

ϕ

|

s

(

t

+ 1

,

∗

) =

f

(

s

(

t,

∗

)

, s

(

t, ϕ

(

∗

))

, u

)

}

wheretheequalityintheexpression

Q

{

ϕ

|

s

(

t

+ 1

,

∗

) =

f

(

s

(

t,

∗

)

, s

(

t, ϕ

(

∗

))

, u

)

}

mustberegardedasequalityoffuntions denedontheset ofagents,

A

.

Atthis point, wehavedesribed thegeneral sequentialdeision model that uses random mathing asshok. We

havementionedtwoapproahesto solvingamodel ofthistype. Eitheraentral plannerseletsvaluesoftheontrol

to optimizetotalexpeted(disounted) rewards,ortheontrolisseleted by somerule basedonstatesofindividual

agentsand information available to individual agents. (Perhapsthe history of the entire system upto the present

time.) We now turn ourattention to the mathing struture and theassumptions that are made in anattempt to

simplifythemodel.

Perhapsthemostommon assumptionmade abouttherandom mathingstruture involvesthetype ofan agent.

Suppose thereexistsasetoftypes

S

=

{

τ

1

, τ

2

, τ

3

,

· · ·

, τ

L

}

, andameasurablefuntion

τ

:

A

⇒

S

that assignsatypeto eahagent. Ingeneral,theset oftypesneednotbenite;however,this additionalassumption

randommanneraftermathing,itmightbemoreappropriatetodene

τ

asarandomvariable. Anexamplesofmodels

ofthistypeanbefoundinLagosandWright[19℄. Here weshallrestritthedisussiontotheaseofadeterministi

typeassignmentbeausewewishtoonentrateontherandommathingasshok.

Thetype ofanagentbeomesaomponentofthestatevariablefortheagent. Hene,oneanwritethestateofthe

agentat time

t

as

s

(

t, g

) = (

τ

(

t, g

)

, k

(

t, g

))

where we inorporate state information other than typein the state variable

k

(

t, g

)

. Forexample, in anumber of

mathingmodelsinmonetarytheory,agentsare assignedatypebasedonagoodproduedbytheagent. Theagents

are alsoassigned an amount ofmoney viaanendowmentorthroughatrade, prodution, and onsumption proess.

Seefor examplethepaperof Kiyotakiand Wright [18℄. As before, weshalldrop the

t

from thenotationin asethe

timeperiod isunderstoodbyontext.

We need a way to measure the relative size of theset of agents of various types. The relative size of the set of

agentsofagiventypeanbeomputedbyexhaustingthesetofagents

A

byaninreasingsequeneofsubsets,

A

n

,of nitemeasure,andthenomputingtheproportionofagentsinthesets

A

n

ofagiventype. Wesay

A

n

exhausts

A

if

A

=

∪

A

n

with

A

n

⊂

A

n

+1

. Denotetheproportionofagentsinthepopulationoftype

τ

k

by

P

k

. Hene,wedene

P

k

by

(3.2)

P

k

= lim

n

→∞

µ

{

g

∈

A

n

|

τ

(

g

) =

τ

k

}

µ

{

g

∈

A

n

}

.

wheneverthelimitexists.

Remark 3.3. Ifthemeasure

µ

ontheset ofagentsisbounded,thereisnoneedforthelimit. Thisistheasefornite setsof agentsoriftheset ofagentsis

[0

,

1)

and

µ

isLebesguemeasure. Inthease

A

=

N

,there isanaturalhoie forthesequeneof exhaustingsets;let

A

n

=

{

1

,

2

,

3

,

· · ·

, n

}

,with

µ

theountingmeasure.

Thedesirefor adeterministi model with mathing leadsto aset of assumptionson themathing proess. Eah

mathshouldpreservethemeasureofsetsofagents. Next,theprobabilitythatagivenagentismathedtoanagentof

sometypeshouldequaltheproportionofagentsofthattypeinthepopulation. Finally,foreahmath,theproportion

of agents ofa giventype that are mathedto a seond typeshould equalthe produtof the proportions ofthe two

typesinthepopulation. Weformalizethiswiththefollowingsetofthreerandommathingpropertiesrelatedtotypes.

WefollowthelabelsusedbyAl

´

os-Ferrerin[5℄here.

Denition3.4. Supposetheproportions

P

k

aredenedbyequation3.2 foranexhaustion of

A

by

A

n

. Therandom mathing shemedened by

(

A,

L

, µ

)

and

(Φ

,

F

, Q

)

, satisestheProportionality Laws ifthefollowingonditionsare satised.

M1. MeasurePreserving

µ

(

E

) =

µ

(

ϕ

(

E

))

forallmeasurable

E

for

ϕ

M2. ProportionalLaw

Q

{

ϕ

|

τ

(

ϕ

(

g

)) =

τ

l

}

=

P

l

fora.e.

g

∈

A

M3. Quasi-independene

lim

n

→∞

µ

{

g

∈

A

n

|

τ

(

g

) =

τ

i

and

τ

(

ϕ

(

g

)) =

τ

j

}

/µ

(

A

n

)

=

P

i

P

j

fora.e.

ϕ

∈

Φ

Notethatin theasethat

µ

(

A

)

isnite, thereisnoneedforthelimitinpropertyM3. 2

We an now make the onept of aggregate unertainty for the dynami random mathing model preise. This

onepthasalonghistoryin theeonomisliterature,and several methodshavebeenproposed to onstrutmodels

thatexhibit thenoaggregateunertaintyproperty. Seeforexample[16℄.

Denition3.5. Therewardstrutureofthemodelontainsno aggregate unertainty if

E

Q

[

R

(

s, u

)] =

E

Q

e

[

R

(

s, u

)]

whenever

Q

and

Q

e

satisfythesameProportionalityLaws. Similarly,thetransitionstrutureofthemodelontainsno aggregate unertainty if

Prob

Q

(

s

(

t

+ 1

,

∗

)

|

s

(

t,

∗

)

, u

) =

Prob

e

Q

(

s

(

t

+ 1

,

∗

)

|

s

(

t,

∗

)

, u

)

whenever

Q

and

Q

e

satisfythesameProportionalityLaws. Ifbothofthese onditionsare satised,wesaythemodel ontainsnoaggregateunertainty.

2

Al

´

os-FerrerreferstopropertyM3astheMixingProperty.Wepreferadierenttermsinemixingouldbeonfusedwithauniform

expetedrewardanbeomputedintermsoftheproportionalityonstants

P

i

. Heneourdenitionisonsistentwith theonept ofno aggregateunertainty in aserandomshoksare applied toindividual agents(without mathing).

Wenextdisussthemathingstruture ofthese modelsinmoredetail. Weonsidereahofthethreepossibilitiesfor

theardinalityoftheset ofagents,andshowwhynon-uniquenessofthemathingshemeoursin eahase.

4. Mathing Strutures andProportionality

Reallthatthesetofmathes,

Φ

,isthesetofinvolutionsdenedonthesetofagentssuhthatnoagentismathed

with herself. So if

ϕ

∈

Φ

then

ϕ

is a bijetion,

ϕ

(

ϕ

(

g

)) =

g

, and

ϕ

(

g

)

=

6

g

. A random mathing struture is a probabilityspaewith

Φ

astheset ofelementsofthespae,andwedenotethisspaeby

(Φ

,

F

, Q

)

.

Inthissetion,wedisusstheexisteneandnon-uniquenessofprobabilitydistributionson

Φ

that satisfythethree

onditions M1, M2, and M3 of Denition 3.4. Most of the work that has been done in trying to establish rigorous

foundations formathing models havefousedon theexistenequestion. It doesnotseemto bewidely known that

oneanonstrutrandommathing strutures fornitesets of agentsthat will satisfytheProportionalityLaws. In

partiular,thismakesitpossibletoonstrutmathingmodelswithnitelymanyagentsthatexhibitthenoaggregate

unertaintyondition. Weexpetthat theonstrutionbelowwillalsobeusefulinexperimentaleonomissineone

mustwork withnitesetsofagents. Someexamplesmaybefoundin[8,12℄.

4.1. Mathing for Finite Sets of Agents. Takeasthe setof agents,

A

=

{

1

,

2

,

3

,

· · ·

, N

}

, with

N

= 2

n

. Wean representanybilateralmath,

ϕ

,asfollows. Write

ϕ

∼

(

a

11

, a

12

)(

a

21

, a

22

)

· · ·

(

a

n

1

, a

n

2

)

tomeanthat

ϕ

(

a

i

1

) =

a

i

2

and

ϕ

(

a

i

2

) =

a

i

1

for

1

≤

i

≤

n

. Inotherwords,

a

i

1

and

a

i

2

aremathed. Wefurtherorderthepairssothatthefollowinginequalitieshold.

a

11

< a

21

<

· · ·

< a

n

1

,

a

j

1

< a

j

2

.

Weshall refertothis orderingonthe pairsasthe lexiographiordering, andit makestherepresentationunique. It

also givesus an easywayto ountthe numberof elements in theset,

Φ

, of involutions. Wedenote this numberby

m

(2

n

)

andwehave

m

(2

n

) =

ard

Φ = (2

n

−

1)(2

n

−

3)(2

n

−

5)

· · ·

(3)(1)

.

This follows bynotingthat wemust have

a

11

= 1

and that there are

2

n

−

1

hoies for

a

12

. Reursivelyapply this observation untiltheset

A

isexhaustedtogettheresult.

It will be onvenient to have an alternate expression for the number of involutions when we disuss probability

distributions ontheset

Φ

. Ifweselettheelementsformathing twoatatimeand notethat theorderin whihwe

seletpairsisunimportant,thenweimmediatelyhave

m

(2

n

) =

ard

Φ =

2

n

2

2

n

−

2

2

· · ·

2

2

n

!

.

Hereweusethenotation,

p

q

=

p

!

/

(

q

!(

p

−

q

)!)

for

p

hoose

q

.

Additionalpropertiesofinvolutions,bilateralmathing,andequivalentountingargumentsaredisussedin[1,2,17℄.

Wenowdesribeprobabilitydistributionson

Φ

thatsatisfytheonditionsM1,M2 andM3.

Theorem 4.1. Let

S

=

{

τ

1

, τ

2

, ..., τ

L

}

be a nite set of types andlet

P

1

, P

2

, ..., P

L

be positive rational numbers with

P

P

i

= 1

. Then thereexistsa niteset of agents,

A

, atypeassignment,

τ

:

A

→

S

,and aprobability distribution

Q

onthe setof bilateral mathes,

Φ

,of

A

suhthat the following onditionshold with

µ

the ountingmeasureon

A

.

(i)The proportion of agentsof type

τ

l

equals

P

l

,so

µ

{

g

|

τ

(

g

) =

τ

l

}

/µ

(

A

) =

P

l

; (ii)

Q

{

ϕ

∈

Φ

|

τ

(

ϕ

(

g

)) =

τ

l

}

=

P

l

for all

g

∈

A

;

(iii)

µ

{

g

|

τ

(

g

) =

τ

i

and

τ

(

ϕ

(

g

)) =

τ

j

}

/µ

(

A

) =

P

i

P

j

for a.e.

ϕ

∈

Φ

.

Proof. Write

P

l

=

p

l

/q

l

where

p

l

and

q

l

areintegers,and thequotients

P

l

arein reduedform. Theset

A

willonsist ofatotalof

N

= 2

q

1

q

2

· · ·

q

L

agents. Weshalldeneasubset

Ψ

⊂

Φ

ofbilateralmatheshavingaspeialform. The numberofagentsoftypes

τ

l

willbe

n

l

= 2

p

l

q

1

q

2

· · ·

q

ˇ

l

· · ·

q

L

wherethe

q

ˇ

l

indiatesthat

q

l

isnotpresentintheprodut. Sine

P

P

l

= 1

wehave

indiatingtype. Denoteanagentoftype

τ

l

by

g

l

∗

where

∗

runsfrom

1

through

n

l

. Noweahmath,

ϕ

∈

Ψ

,willmath theagentsasfollows. Eah

ϕ

willmath

u

l

agentsoftype

τ

l

to agentsoftype

τ

l

andwillmath

v

ij

agentsoftype

τ

i

to agentsoftype

τ

j

where

i

6

=

j

. So forexample thenumberof agentsof type

τ

1

mathed to agentsof type

τ

1

will equal

u

1

, and thenumberoftype

τ

1

agentsmathed totype

τ

2

agentsis

v

12

. Forinstane, ifthere are twotypes, a typialmathanbedenotedas

g

1

∗

g

∗

1

· · ·

g

1

∗

g

∗

2

· · ·

g

∗

2

g

∗

1

g

1

∗

· · ·

g

∗

1

g

1

∗

g

∗

1

· · ·

g

1

∗

g

∗

2

· · ·

g

∗

2

g

∗

2

g

2

∗

· · ·

g

∗

2

wheretheagentsin therstrowaremathedtotheorrespondingagentin theseond row.

Weassignthefollowingvaluestothenumbers

u

l

and

v

ij

for

i

6

=

j

. Let

u

l

=

p

l

q

l

(

n

l

) =

p

l

q

l

(2

p

l

q

1

q

2

· · ·

q

ˇ

l

· · ·

q

L

)

v

ij

=

p

j

q

j

(2

p

i

q

1

q

2

· · ·

q

ˇ

i

· · ·

q

L

) =

= 2

p

i

p

j

q

1

q

2

· · ·

q

ˇ

i

· · ·

q

ˇ

j

· · ·

q

L

=

p

i

q

i

p

j

q

j

2

q

1

q

2

q

3

· · ·

q

L

=

p

i

q

i

p

j

q

j

N

Thenumbers

v

ij

are learlyintegersandfrom equation4.1 itfollowsthat

u

l

is aneven integer. It isalso learfrom theexpressionfor

v

ij

that

v

ij

issymmetriin

i

and

j

asrequired. Fromthelastexpressionfor

v

ij

wealsoseethatthe proportionofagentsoftype

τ

i

thataremathedwithagentsoftype

τ

j

equals

P

i

P

j

. Heneweobtainproperty(iii)of thetheoremwiththesehoies.

It is easy to omputethe totalnumber ofmathes in the set

Ψ

desribed above. First let

m

(

k

)

denote thetotal number of bilateral mathes of

k

agents as before. Then the total number of distint mathes of the above form,

denoted

M

(

P

1

, P

2

, ..., P

L

)

,isgivenby

M

(

P

1

, P

2

, ..., P

L

) =

ard

Ψ =

n

1

u

1

n

1

−

u

1

v

12

n

1

−

u

1

−

v

12

v

13

· · ·

n

1

−

u

1

−

...

−

v

1

,L

−

1

v

1

L

· · ·

· · ·

n

L

u

L

n

L

−

u

L

v

L

1

. . .

n

L

−

...

−

v

L,L

−

2

v

L,L

−

1

Y

l

m

(

u

l

)

Y

i<j

v

ij

!

Weassign equalprobabilityto eah mathin

Ψ

. Inother words,weputtheuniform disretedistribution,

Q

,onthe

set

Ψ

of bilateral matheswehave desribed above. Thedistribution

Q

has the property given in statement(ii) of thetheorem. Tosee thislet

g

∗

∗

denote anyagent. Forexample,onsider agent

g

1

1

. Theprobabilitythat this agentis mathedtoanagentoftype

τ

1

equalsthenumberofmathesinwhih

g

1

1

ismathedtoatype

τ

1

agentdividedbythe totalnumberofmathes. Butthis isexatlyequaltotheproportionoftheagentsseletedtobemathed totype

τ

1

agents,orin otherwords,

P

1

asdesired. Similarly, theprobabilitythat agent

g

1

1

ismathedtoatype

τ

i

agentequals

theproportionoftype

τ

1

agentsseletedtobemathed withtype

τ

i

agentsor

P

i

.

Example 4.2. An example will help explain the aboveonstrution and ounting argument. Suppose we want to

onstrut amathing model with twotypesand thepropertythat the probabilitythat anygiven agentbemathed

withanagentofeithertypeequal

1

/

2

. Then

P

1

=

P

2

= 1

/

2

. Thenumberof agentsweneedis ard

A

= 2

q

1

q

2

= 2(2)(2) = 8

with

4

agentsof eah type. Wedenote thetypesas

a

and

b

to makethe notationabit leanerin thisexample. We

maththeagentsasfollows.

a∗

a∗

a∗

b∗

a∗

b∗

b∗

b∗

where the

∗

subsripts rangefrom

1

through

4

. Agentsin the toproware mathedwith theorrespondingagentin theseondrow. Thereare

4

2

(

4

hoose

2

)waystoseletthe

a

agentsto bemathedto

a

agents. Onethesetwo

a

agentsareseleted,theremaining

2

type

a

agentsmustbemathedto

b

agents. Similarly,thereare

4

2

waysto

seletthe

b

agentstobemathedwith

b

agents. Onethepairsofagentsofeahtypeareseleted,thereare

2!

waysto

ofountingthewaystomaththe

a

agentswiththe

a

agentssinethereareonlytwoofthemandheneonlyoneway

tomaththem.) Thusthereare

4

2

2

2

4

2

2

2

2! = 72

mathesof this form. We assignprobability

1

/

72

to eahof themathes. Theprobabilitythat anygivenagent,for

exampleagent

a

1

, willbemathed to atype

a

agentis

1

/

2

andthe probability that

a

1

will be mathed toa type

b

agentis

1

/

2

. Foreahmath ofthe72mathes, exatlytwoofthetype

a

agentsaremathedto type

a

agentssine

twoof thetype

a

agentsaremathed to oneanother. Also, exatlytwoof thetype

a

agentsare mathed totype

b

agents. Hene,onditionsM2andM3aresatised. InthisaseM1istrivial,sotheProportionalityLawshold.

Theaboveprobabilitydistributionis notuniquein thesense that thereare otherprobabilitydistributions onthe

setofallmathesofthe

8

agentssuhthat theProportionalityLawsaresatised. Infat,weneedonlytwomathes,

eah equallylikely,to aomplishthis. Let

ϕ

1

= (

a

1

, a

2

)(

a

3

, b

3

)(

a

4

, b

4

)(

b

1

, b

2

)

ϕ

2

= (

a

1

, b

1

)(

a

2

, b

2

)(

a

3

, a

4

)(

b

3

, b

4

)

andassignprobability

1

/

2

toeahofthesetwomathesandprobability

0

toallotherbilateralmathesofthe

8

agents.

It is easy to see that the probability that anygivenagentbe mathed to anagentof either type equals

1

/

2

just as

before. Thequasi-independene propertyisalsoobvious.

If

Q

and

Q

e

satisfyonditionsM2 and M3, then any onvexombination of

Q

and

Q

e

will satisfy M2 andM3. In theaseofnitely manyagentswiththeountingmeasure onthesetof agents,M1 holdstrivially. Hene,thesetof

probabilitydistributions,

Q

, suh thattheProportionalityLawshold(withountingmeasureontheset ofagents)is

aonvexpolyhedrawithnonemptyrelativeinterior. SothedistributionsthatsatisfytheProportionalityLawsarefar

fromunique.

4.2. CountablyInniteSetsofAgents. Let

Φ

denotethesetofallinvolutionswithoutxedpointof

N

,thenatural numbers. As in thenite ase, everyelement

ϕ

of

Φ

has aunique representationasan innitesequene of pairsof

naturalnumbers.

(

a

11

, a

12

)

,

(

a

21

, a

22

)

,

(

a

31

, a

32

)

,

· · ·

,

(

a

j

1

, a

j

2

)

,

(

a

j

+1

,

1

, a

j

+1

,

2

)

,

· · ·

wherethepairsofnaturalnumberssatisfytheonditions

(4.2)

a

11

< a

21

< a

31

<

· · ·

(4.3)

a

j

1

< a

j

2

forall

j

andthesetofall

a

ji

exhausts

N

. Therelationshipbetweenthesequeneandtheinvolution

ϕ

is

ϕ

(

a

k

1

) =

a

k

2

forall

k

asin thenite ase. Itwillbeonvenienttohaveaformalnameforthis representation.

Denition 4.3. Let

ϕ

be a bilateral math of either a nite set

{

1

,

2

, ...,

2

n

}

or of the natural numbers

N

. The representationof

ϕ

bythesequene(niteorinnite) ofpairs

(

a

11

, a

12

)(

a

21

, a

22

)(

a

31

, a

32

)

· · ·

(

a

j

1

, a

j

2

)(

a

j

+1

,

1

, a

j

+1

,

2

)

· · ·

satisfyingonditions(4.2)and(4.3)is thelexiographirepresentationof

ϕ

.

It follows immediatelyfrom the denition that the lexiographirepresentation isunique. Weuse theabove

rep-resentation to show that

Φ

is unountable by using aCantor diagonalization argument. We rstgive apreliminary

resultonthelexiographirepresentationofinvolutions.

Lemma4.4. Let

ϕ

= (

a

11

, a

12

)(

a

21

, a

22

)(

a

31

, a

32

)

· · ·

(

a

j

1

, a

j

2

)(

a

j

+1

,

1

, a

j

+1

,

2

)

· · ·

beany involutionof

N

writtenasthe uniquepairwiserepresentation. Then

a

11

= 1

andfor

k >

1

,

a

k

1

= inf

{

a

11

, a

12

,

· · ·

, a

k

−

1

,

1

, a

k

−

1

,

2

}

c

.

where

{}

c

denotesthe omplementin

N

.

Proof. Theresultfollowsdiretlyfromthedenitionofthelexiographiorder.

Proof. Let

F

:

N

→

Φ

beanyinjetionfrom

N

into thesetofinvolutionsof

A

. (Weuse

A

to denotethesetofagents sineweuse

N

toindex mathesintheountingargument.) Wewanttoshowthat

F

annotbeonto, andtodothis we use the standard diagonalizationargument together with the aboveunique representation of an involution. Let

ϕ

n

=

F

(

n

)

andwrite

ϕ

n

in theuniquepairwiserepresentationgivenabove.

ϕ

n

= (

a

n

11

, a

n

12

)(

a

n

21

, a

n

22

)

· · ·

Weshalldeneaninvolutioninuniquepairwiserepresentationformthat diersfrom eahofthe

n

involutionsinthe

n

th

plae. Thisdiagonalizationthenimpliesthatthesetofinvolutionsisunountable.

Let

ψ

be the math

ψ

= (

b

11

, b

12

)(

b

21

, b

22

)

· · ·

(

b

k

1

, b

k

2

)

· · ·

, with the

b

kl

to be dened. We put

b

11

= 1

. Put

b

12

= inf

{

b

11

, a

12

1

}

c

. Soforexampleif

a

1

12

6

= 2

thenwemusthave

b

12

= 2

. Nowdene

b

k

1

reursivelyby (4.4)

b

k

1

= inf

{

b

11

, b

12

, b

21

, b

22

,

· · ·

, b

k

−

1

,

1

, b

k

−

1

,

2

}

c

(4.5)

b

k

2

= inf

{

n

:

n > b

k

1

and

n

∈ {

b

11

, b

12

, b

21

, b

22

,

· · ·

, b

k

−

1

,

1

, b

k

−

1

,

2

, a

k

k

2

}

c

}

.

Note that (4.4) mustbe satisedifweuse theunique pairwise representationfor

ψ

, and the denition of

b

k

2

fores the

k

th

pair in therepresentation of

ψ

to dier from the

k

th

pair in the representation of

ϕ

k

. Sine the

k

th

pairof

the representationof

ψ

diersfrom the

k

th

pair ofthe representation of

ϕ

k

, theinvolution

ψ

diers from every

ϕ

k

.

Thereforethemap

F

annotbeontoandtheresultfollows.

Remark 4.6. The un-ountability of the set of bilateral mathes of

N

is impliitin the work of Boylan [7℄ sinehe onstrutsuniquebilateralmathesfrom sequenesofi.i.d. random variableswithvaluesin anite setof morethan

oneelement. Theardinalityofthesetofsuh sequenesisunountable.

Weareinterestedindesribingdistributionsonthesetofinvolutions. TheaboveresulttogetherwiththeContinuum

Hypothesis,tellsusthatthesetof involutionsof

N

anbeidentiedwiththeunit intervalofrealnumbers. However, ifwewish touse propertiesof standardontinuousdistributions ontheinterval, weneedapreisedesriptionofthe

identiation. Fortunately, the unique pairwise representation gives an easy way to do this. We rst write

Φ

as a

sequeneofdisjointunions. Let

Φ

n

=

{

(1

, n

+ 1)(

∗

,

∗

)

· · · }

betheinvolutionswith

(1

, n

+ 1)

astherstpairintherepresentation. Thesets

Φ

n

aredisjointandtheunionofthe

Φ

n

givesus

Φ

. Next,deomposeeah

Φ

n

intoadisjointunion,

Φ

i

1

=

∪

Φ

i

1

n

with

Φ

i

1

n

=

{

(1

, i

1

)(

a

21

, a

22

)

· · · }

where

a

21

is uniquelydened bythe value of

i

1

and Lemma 4.4. Thesmallest possible value of

a

22

orresponds to

n

= 1

and soon. Continuein thismanner, andreursivelydenedisjointsets ofinvolutions

Φ

i

1

i

2

i

3···

i

k

. Forinstane,

Φ

1

=

{

(1

,

2)

...

}

Φ

11

=

{

(1

,

2)(3

,

4)

...

}

,

Φ

12

=

{

(1

,

2)(3

,

5)

...

}

,andsoon. Weolletthepropertiesofthese setsinthefollowinglemma.

Lemma4.7. The sets

Φ

i

1

i

2

i

3···

i

k

satisfythe followingonditions. (1) If

Φ

i

1

i

2

i

3···

i

k

=

{

(

a

11

,

a

12

)(

a

21

, a

22

)

· · ·

(

a

k

1

, a

k

2

)

· · · }

where the

a

ij

appearing in the rst

k

pairsare xed by thereursivedenition,then

Φ

i

1

i

2

i

3···

i

k

n

=

(

a

11

,

a

12

)(

a

21

, a

22

)

· · ·

(

a

k

1

, a

k

2

)(

a

(

k

+1)1

, b

n

)

· · ·

where

a

(

k

+1)1

isuniquelydened byLemma4.4and

b

n

denotesthe

n

th

smallestpossiblevaluefortheseond

elementofthe

(

k

+ 1)

st

pair.

(2)

Φ

i

₁

⊃

Φ

i

₁

i

₂

⊃

Φ

i

₁

i

₂

i

₃

⊃ · · ·

(3) Forxed

k

,thesets

Φ

i

₁

i

₂

i

3···

i

_k

aredisjoint. (4) Forxed

k

,

∪

Φ

i

1

i

2

i

3···

i

k

= Φ

.

(5) Foreah

ϕ

∈

Φ

there existsauniquesequeneofsets

Φ

i

1

⊃

Φ

i

1

i

2

⊃

Φ

i

1

i

2

i

3

⊃ · · ·

with

{

ϕ

}

= Φ

i

1

∩

Φ

i

1

i

2

∩

Φ

i

1

i

2

i

3

∩ · · ·

Proof. Therst propertyis just the formal restatementof the reursivedenition of the sets. The seond property

followsdiretlyfrom the rst. The third propertyfollowssine twoinvolutionsthat diersomewhere in the rst

k

pairsaredistint. Thefourthpropertyholdssinewelistallpossiblesequenesofpairsoflength

k

inthedenitionof

Φ

i

1

i

2

i

3···

i

k

. Finally, suppose

ϕ

= (

a

11

,

a

12

)(

a

21

, a

22

)

· · ·

(

a

k

1

, a

k

2

)

· · ·

. Bytheonstrutionof thesets

Φ

i

1

i

2

i

3···

i

k

, there existsexatlyoneset

Φ

i

1

ontaining

ϕ

. Next,thereexistsexatly(disjointproperty)onesubset,

Φ

i

1

i

2

⊂

Φ

i

1

ontaining

ϕ

. Continueinthismannertogeneratethesequeneofnestedsets

Φ

i

existsaninvolution

ψ

6

=

ϕ

in theintersetion. Thentherepresentationof

ψ

mustdierfromtherepresentationof

ϕ

. Iftheydieratthe

k

th

pair,then

ϕ

and

ψ

annotbeontainedinthesameset

Φ

i

1

i

2

i

3···

i

k

.

With theabovedeomposition of

Φ

,weareready todene aoneto one andontomap from

Φ

tothehalf open unit

interval

I

= [0

,

1)

. We dene adeomposition of

[0

,

1)

into disjoint nested subintervalsthat orrespond to the sets

Φ

i

1

i

2

i

3···

i

k

. Firstfor

n

= 1

,

2

,

3

,

· · ·

put

I

n

=

n

−

1

n

,

n

n

+ 1

.

This gives

[0

,

1)

as a disjoint union of half open subintervals. Now suppose we have reursively dened half open

subintervals

I

i

1

i

2

i

3···

i

k

satisfyingproperties(2),(3), and(4)ofLemma4.7with

I

replaing

Φ

. Let

I

i

1

i

2

i

3···

i

k

= [

a, b

)

denoteoneofthesesubintervals. Wedeompose

I

i

1

i

2

i

3···

i

k

= [

a, b

)

intoadisjointunionofsubintervalsasfollows. Let

I

i

1

i

2

i

3···

i

k

n

=

a

+

n

−

1

n

(

b

−

a

)

, a

+

n

n

+ 1

(

b

−

a

)

Thesehalf openintervalsaredisjointand

∪

n

I

i

1

i

2

i

3···

i

k

n

=

I

i

1

i

2

i

3···

i

k

.

Lemma4.8. Foreah

x

∈

[0

,

1)

,thereisauniquesequeneof half open intervals

I

i

1

⊃

I

i

1

i

2

⊃ · · · ⊃

I

i

1

i

2

i

3···

i

k

⊃ · · ·

suhthat

{

x

}

=

I

i

1

∩

I

i

1

i

2

∩ · · · ∩

I

i

1

i

2

i

3···

i

k

· · ·

Proof. Bythe onstrution of the disjoint sets

I

i

1

i

2

i

3···

i

k

there is aunique

I

i

1

ontaining

x

. There is now a unique

I

i

1

i

2

⊂

I

i

1

ontaining

x

. Continue in this manner to generate the uniquesequene of sets. The intersetion of the intervals ontains

x

. If theintersetion ontainsaseond point,

y

, thenthere exists an

ε >

0

with

|

x

−

y

|

> ε

. But sinethelengthofthehalfopensubintervals

I

i

1

i

2

i

3···

i

k

tendsto

0

thisisimpossible.

Finally,withtheaboveresultsweare abletoidentify theset

Φ

ofbilateralmathingsofthenaturalnumberswith

thehalf openinterval

[0

,

1)

. Thisidentiationwillallowustobepreiseinourdisussionofprobabilitydistributions

onthesetofinvolutions.

Theorem4.9. Let

ϕ

∈

Φ

with representation

(

a

11

, a

12

)(

a

21

, a

22

)(

a

31

, a

32

)

· · ·

(

a

j

1

, a

j

2

)(

a

j

+1

,

1

, a

j

+1

,

2

)

· · ·

Letthe sequene

(

i

1

, i

2

, i

3

,

· · ·

)

bethe uniquesequeneofnatural numbers assoiatedwith

ϕ

by Lemma4.7sothat

{

ϕ

}

= Φ

i

1

∩

Φ

i

1

i

2

∩

Φ

i

1

i

2

i

3

∩ · · ·

Let

x

∈

[0

,

1)

with

{

x

}

=

I

i

1

∩

I

i

1

i

2

∩ · · · ∩

I

i

1

i

2

i

3···

i

k

· · ·

.

Then the map

F

(

ϕ

) =

x

isabijetion.

Proof. Theresultfollowsimmediatelyfromtheonstrutionoftheuniquerepresentationsof

ϕ

and

x

giveninLemma

4.7andLemma 4.8.

When we disuss the problem of uniqueness for probability distributions on the set of bilateral mathes in the

ountably innite ase, it will be more onvenient to havea slightly dierent representation of the set of bilateral

mathes of the naturalnumbers. The following orollary is obtained by mapping the half open interval

[0

,

1)

onto

[0

,

∞

)

bythemap

x

→

x/

(1

−

x

)

.

Corollary4.10. Thereexistsabijetionfromthe set

Φ

ofbilateral mathesofthe naturalnumbers

N

tothe halfopen interval

[0

,

∞

)

.

Foraountableinnitesetofagentsmodeledas

N

wenowhaveagooddesriptionoftheset

Φ

ofbilateralmathes. Thenextstepistheonstrutionofprobabilitydistributions onthissetthat satisfytheonditionsM1,M2, andM3.

Boylan[7℄provedtheexisteneofaprobabilitydistribution on

Φ

that satisestheseproperties. Inthis aseproperty

M1istrivialifwetake

µ

tobetheountingmeasureon

N

. Ofoursethisisnotanitemeasureandthereforeweneed thelimitin propertyM3. Inhisexisteneresult,BoylanusestheCesaroaveragetodenetheproportionofagentsof

Itiseasyto seethattheprobabilitydistributiononstrutedbyBoylanon

Φ

that satisesonditionsM1,M2,and

M3isnotunique. Werealljustenoughofhisonstrutiontoexplainthenon-uniqueness.

Let

S

=

{

τ

1

, τ

2

, ..., τ

L

}

be a nite set of types. Let

µ

be the ounting measure on

N

. Exhaust

N

by sets

A

n

=

{

1

,

2

,

3

,

· · ·

, n

}

. Assumetheproportionofeahtype,

τ

l

,inthepopulationis

P

l

asdenedbyequation3.2. Let

τ

bethe typeassignmentmap. Let

X

i

bei.i.d. randomvariables(indexed bythe agents) takingvaluesin

S

with distribution givenbythe

P

i

. Inotherwords,

Prob

(

X∗

=

τ

l

) =

P

l

.

Thevalueof

X

l

will tellusthetypeoftheagenttobemathed withagent

l

. So,mathagent

1

withagent

k

where

k

= inf

{

j

|

X

1

=

τ

(

j

)

and

X

j

=

τ

(1)

}

.

Continuetomaththeremainingunmathedagentsinthesamemanner.

Thisdenesaprobabilitydistributiononthesetofallbilateralmathesof

N

thatsatisestheProportionalityLaws. Furthermore everyset ofbilateralmathesoftheform,

{

ϕ

|

ϕ

= (

a

11

,

a

12

)(

a

21

, a

22

)

· · ·

(

a

k

1

, a

k

2

)

· · · }

wheretherst

k

pairsarespeiedandtheremaining(innitelymany)pairsareunspeied,haspositiveprobability.

Thisprobabilitytendstozeroasthenumberofspeiedpairs,

k

,tendsto innity. Oneanusethese propertiesand thebijetiondenedin Corollary4.10,to identifythisdistribution withaontinuousdistributionon

[0

,

∞

)

.

It iseasy to seethat this distributionis notunique. Toonstrutaseond distribution that satisesthe

Propor-tionality Laws, simply interhange two(or more) agentsof the sametypeand reapply theproedure. Forexample,

assumetherearetwotypesofequalproportion. Supposeagents

1

,

2

andagent

1000

havethesametype. Assumethere

aremanyagentsofeahtypefrom

3

through

999

. Swithagent

2

andagent

1000

. Thiswillnothangetheproportion

ofagentsofeahtypeinthepopulation. Withtheoriginalorder,itismuhmorelikelythat agent

1

will bemathed

withagent

2

thanwithagent

1000

. Butwiththeseondorder,thisisreversed. Henewehaveadierentdistribution.

Thesetofprobabilitydistributionson

Φ

,thesetofbilateralmathesof

N

,thatsatisfymathingpropertiesM1,M2, andM3 formaonvexset intheset ofallprobabilitydistributionson

Φ

. There doesnotappearto beanynatural

waytoseletoneoveranother. Weshalldisusstheonsequenesofthisfatbelowandgiveonepossibleapproahto

seletingadistinguishedprobabilitydistribution.

Finally,wementionthattheresultsoftheprevioussetiononrandommathingdistributionsfornitesetsofagents

anbeusedtoonstrutprobabilitydistributionsforountablyinnitesetsofagents. Supposeour(ountableinnite)

populations onsists of nitely many types and the proportion of eah type is a rational number. Find a random

mathingshemeonaniteset ofagentsthatsatises thepropertiesM1, M2, andM3. Suh ashemeexistsbythe

resultsoftheprevioussetion. Nowregroup theagentsof theinniteountable populationinto aountable number

ofsubgroups, eah oneof whih isaopyofthisnite set. Randomlymath theagentsin theinnitepopulationby

randomlymathingtheagentsin eahsubgroup. Therandommathingshemeontheinnitepopulationinheritsthe

propertiesfrom themathing shemeontheniteset ofagents.

The aboveonstrution should makelear the problem onefaes when onstrutingmathing shemes satisfying

theonditionsM1,M2,and M3. Thereadermight,at thispoint,antiipatethatthesameonstrutionis possiblein

theaseofaontinuumofagents.

4.3. RandomMathingfor a Continuumof Agents. Wenowonsider theaseof aontinuum ofagents. The

set of agentsis perhaps mostfrequently modeled astheunit interval

[0

,

1]

orthe half open interval

[0

,

1)

. It is also

frequentlyassumed thatsubsets ofagentsaremeasuredusing Lebesguemeasure ontheinterval. Thismakesiteasy

to makesense of the oneptof the proportion of agentsof agiven type. Theunit interval model ould ofourse

bereplaed withanyother modelwiththeardinalityofthereals,

R

. Forexample,ifoneused theinnitehalf open interval,

[0

,

∞

)

, withanon-nite measure,thenoneouldmakesenseofproportionalityasalimit. Whihevermodel oneuses,itis importanttokeepin mind thatthereal numberidentifyinganagentissimplyanameorlabelforthe

agent. Ifoneusesadierentnameorlabel,thebehaviorofthemodelshouldnothange. Wewill usethe

[0

,

1)

model

inourdisussionhere.

A random mathing sheme for the set of agents,

[0

,

1)

, requires a distribution on theset of involutions of

[0

,

1)

.

Asbefore, weassumetheinvolutionshavenoxedpointso thateveryagentis mathed. Let

S

=

{

τ

1

, τ

2

, τ

3

,

· · ·

, τ

L

}

beanite set oftypes and assumethat eahagentis assigned atype. Suppose theproportionof agentsof type

τ

k

equals

P

k

asbefore. Forthismodel,Alos-Ferrer

´

[4℄onstrutsaprobabilitydistributiononthebilateralmathesthat satisesthepropertiesM1,M2, andM3. Justasin theaseofaountablyinnitenumberofagents,thisprobability

distributionis notunique. Although thesame basiideathat weused in the ountable innitease anbeused to

ontinuumofagentswitheahagentassignedatype,andtheproportionofagentsofeahtypeisidentialinthetwo

opies. Construtrandommathingshemesthat satisfyM1,M2, andM3 oneahopyseparately. Nowshrinkeah

opydownbysomefator. Gluethetwoopiestogethertomakeanintervaloflengthone. Theproportionsofagents

of eah typeonthe twoparts equalstheoriginal proportions. Applytherandom mathing shemeoneah part. In

this waywe geta newprobabilitydistribution sinewe havehangedthe set of involutions. Weprovidethe details

below.

Let

Q

bea probability distributionon somesubset

Ψ

of theset

Φ

of all bilateralmathes of theagentset

[0

,

1)

.

Let

ϕ

∈

Ψ

denoteatypialbilateralmath. Assumethedistribution

Q

satises theonditionsM1, M2,andM3. Fix some

α

∈

(0

,

1)

. Foreah

ϕ

∈

Ψ

dene anewbilateralmathasfollows.

e

ϕ

(

x

) =

αϕ

(

x

α

)

if

0

≤

x < α

α

+ (1

−

α

)

ϕ

(

x

₁

−

₋

_α

α

)

if

α

≤

x <

1

We have dened anew set of bilateral mathes

Ψ

e

and a bijetion with the original set of mathes

Ψ

. Dene a

σ

-algebraon

Ψ

e

usingthebijetionofmathes. Deneaprobabilitydistributiononthesetsinthe

σ

-algebraofsubsetsof

e

Ψ

by

Q

e

(

E

e

) =

Q

(

E

)

if

E

and

E

e

orrespondtooneanotherunderthebijetionofmathes. Sinetheshrinkingfuntion

x

→

αx

doesnothangetheproportionofagentsofagiventype,

Q

e

willalsosatisfypropertiesM1, M2,andM3. But we now havetwo distintprobabilitydistributions (infat aoneparameterfamily) of distributions onthe set ofall

bilateralmathesof

[0

,

1)

. Notethatthemathingshemeweonstrutedbytheglueingproesshasthepropertythat

agentsin therstinterval

[0

, α

)

arenevermathedwithagentsintheseondinterval

[

α,

1)

.

The above onstrution an be applied to any random mathing sheme for a ontinuum of agents. Hene, the

standardonditionsM1,M2,andM3arenotenoughtoisolateasinglemathingshemeinthesetofbilateralmathes

ofaontinuumofagents.

Remark 4.11. The ardinalityofthe setof mappings

R

⇒

R

is knownas

i

2

(Beth 2). It is fairlystraightforwardto showthatthisistheardinalityofthesetofbilateralmathesofaontinuumofagentsaswell. Thegeneralontinuum

hypothesis says that

ℵ

2

=

i

2

. So the set of bilateral mathes of aontinuum of agents is indeed a very large set. Thereforethenonuniquenessproblem forbilateral mathesofaontinuumofagentsseemstobemuhlesstratable

thannon-uniqunessforbilateral mathesofaountableset ofagents.

We nally mentionthat the random mathing sheme we onstruted on the nite set of agents an beused to

onstrut a mathing sheme on the ontinuum of agents

[0

,

1)

. Let

g

denote an agent in a nite set of agents

A

e

for whih we have onstruted a mathing sheme satisfying M1, M2, and M3. Suppose ard

(

A

e

) =

N

. Let

G

be a half open interval of agents of length

1

/N

with all of the same type as agent

g

. We now have a ontinuum of

agentswiththesameproportionof agentsofeah type(usingLebesguemeasure tomeasureproportion)asthenite

set. Nowrandomly math the agentsin theontinuum by mathing the intervals (whih all havethe same length)

aordingto theprobabilitylawusedtomaththeagentsin thenite set. Anintervaloflength

1

/N

ismappedonto

aseond intervalof length

1

/N

in the obvious way. This mathing sheme onthe ontinuum of agents satisesthe

ProportionalityLawsM1, M2,andM3.

4.4. Summary. Foreahof thethreepossibilitiesfortheardinalityof theset of agentsin thepopulation,wehave

shown thattheprobabilitydistributions,

Q

,that satisfytheProportionalityLaws(foraxedmeasure,

µ

, ontheset

of agents) arenot unique. Infat theset of suh distributions is aonvexset in a largedimensional spae (innite

dimensional if

A

isinnite). Inthease

A

is innite,thenon-uniquenessis morediultto quantify sinethesetof

bilateralmathesis

ℵ

1

or

ℵ

2

(assumingthegeneralontinuumhypothesis).

If random mathing is used in an eonomi model, and one only knows that the mathing sheme satises the

Proportionality Laws, the non-uniqueness may make it impossible to ompute even fundamental quantities suh as

transitionprobabilitiesandexpetedrewards. Thenextsetionshowsthis problemanbeserious.

5. A Growth Modelwith Random Mathing

Inthelast setionweshowedthatrandommathingshemesthat satisfyproportionalitylawsarefarfrom unique.

Inpartiular,weshowedhowtoonstrutamultipliityofrandommathingstruturesthatsatisfyassumptionsM1,

M2,andM3foragentssetsofnite,ountable,oraontinuumardinality. Inthissetionwepresentanexamplethat

shows thenon-uniquenessofthemathingsheme anpresentaserious problem foreonomi models. The example

anbeeasily modiedtodemonstratethatthesameproblempersistsintheaseofinnitelymanyagents(ountable

orunountable). We show that in amodiedversionof abasi stohasti growth model (e.g., [21℄), the mehanis

of random mathing have substantiveimpliations for the evolution of the growth model as well as optimality and