Stable Allocation Mechanism

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Stable Allocation Mechanism

Mourad Baïou, Michel Balinski

To cite this version:

Mourad Baïou, Michel Balinski. Stable Allocation Mechanism. 2002. �hal-00243002�

Stable Allocation Mechanism

Mourad Baïou Michel Balinski

Avril 2002

Cahier n° 2002-009

ECOLE POLYTECHNIQUE

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

LABORATOIRE D'ECONOMETRIE

1rue Descartes F-75005 Paris (33) 1 55558215 http://ceco.polytechnique.fr/

Stable Allocation Mechanism

Mourad Baïou

Michel Balinski

Avril 2002 Cahier n° 2002-009

Résumé: Le problème d’allocations stables généralise les problèmes d’affectations stables (" one-to-one ", " one-to-many " ou " many-to-many ") à l’attribution de quantités réelles ou d’heures. Il existe deux ensembles d’agents distincts, un ensemble I " employés " et un ensemble J " employeurs " où chaque agent a un ordre de préférences sur les agents de l’ensemble opposé et chacun a un certain nombre d’heures. Comme dans les cas spécifiques, le problème d’allocations stables peut contenir un nombre exponentiel de stables (quoique dans le cas " générique " il admet exactement une allocation stable).

Un mécanisme est une fonction qui sélectionne exactement une allocation stable pour n’importe quel problème. Le mécanisme " optimal-employés " qui sélectionne toujours l’allocation stable optimale pour les employés est caractérisé comme étant l ‘unique mécanisme " efficace " ou " monotone " ou

" strategy-proof. "

Abstract: The stable allocation problem is the generalization of the well-known and much studied stable (0,1)-matching problems to the allocation of real numbers (hours or quantities). There are two distinct sets of agents, a set I of

"employees" or "buyers" and a set J of "employers" or "sellers", each agent with preferences over the opposite set and each with a given available time or quantity. In common with its specializations, and allocation problem may have exponentially many stable solutions (though in the "generic" case it has exactly one stable allocation).

A mechanism is a function that selects exactly one stable allocation for any problem. The "employee-optimal" mechanism XI that always selects xI, the

"employee-optimal" stable allocation, is characterized as the unique one that is, for employees, either "efficient", or "monotone", or "strategy-proof."

Mots clés : affectation stable, mariage stable, couplage stable, transport ordinal, problème d'admission, many-to-many matching, two sided market

Key Words : stable assignment, stable marriage, stable matching, ordinal trasportation, university admissions, two-sided market, many-to-many matching.

Classification AMS: 91B68, 91B26, 90B06, 91A35

1 Université Blaise Pascal, CUST, B.P. 206 - 63174 Aubière Cedex, and Ecole Polytechnique, Laborattoire d'Econométrie. e-mail: baiou@custsv.univ-bpclermont.fr.

2 CNRS and Ecole Polytechnique, Laboratoire d'Economiétrie, 1 rue Descartes, 75005 Paris. e-mail:

balinski@poly.polytechnique.fr

Thestablemarriage(orstableone-to-one) problemisthesimplestexampleofa

two-sidedmarket. There aretwodistint setsofagents,e.g.,men andwomen,

andeahagentononesideofthemarkethaspreferenesovertheoppositeset.

Mathings between men and women are sought that are stable in the sense

that no man and woman not mathed (to eah other) an both be better-o

bybeingmathed [7℄. Thestable admissions (or stable one-to-many) problem

isamoregeneralexampleofatwo-sidedmarket,againwithtwosetsofagents

eahhavingpreferenesovertheoppositeset. Ononesideofthemarketthere

are individuals, e.g., prospetive students, interns or employees, and on the

other there are institutions, e.g., universities, hospitals or rms, eah seeking

to enroll some given number of individuals [7℄. A still more general ase is

thestablepolygamouspolyandry(orstablemany-to-many) problemwhereevery

agentseekstoenrollgivennumbersofagentsoftheoppositeset[2℄. Allofthese

areproblemsofassignment: agentsaremathedwithagents[7,9,8,10℄.

The stable alloation problem [3℄ is also a two-sided market with distint

setsofagentswhereeahagenthasstritpreferenesovertheoppositeset. But

hereeah agent is endowed with real numbers quantities or hours of work

andinsteadofmathing(oralloating0'sand1's)theproblemistoalloate

realnumbers. For example,oneset of agentsonsists ofworkmeneahwith a

numberofavailablehoursofwork,theotherofemployerseahseekinganumber

ofhours ofwork. Stability asks that nopairof opposite agentsan inrease

theirhourstogether eitherduetounusedapaityorbygivinguphourswith

lesspreferredpartners.

Here,asisoftentrue,thestudyofthemoregeneralproblemlariesandin

someaspetssimpliestheissuesand viewsthatonernthepartiularases.

Setion 1 the problem presents the model and Setion 2 stable

alloations summarizes the salient fats onerning them, their existene,

strutureandproperties(see[3℄).

Analloationmehanism isafuntionthat seletsauniquestable alloa-

tionforanyalloationproblem. Setion3mehanismsuniquelyharater-

izestheemployee-andemployer-optimal(orrow-andolumn-optimal)alloation

mehanisms in termsof three separateproperties: eieny, monotoniity,

andstrategy-proofness. This generalizesto stable alloationssimilarhara-

terizationsrstestablishedforadmissionsorone-to-manymathing[6,1℄,then

formany-to-manymathing[2℄.

1 The problem

Astablealloationproblem( ;s;d;)isspeiedbyadiretedgraph dened

overagrid, andarraysofreal numberss; d>0and 0,as follows. There

are two distint nite sets of agents, the row-agents I (employees) and the

olumn-agents J (employers), and eah agent has a strit preferene order

overtheagentsoftheoppositeset. Eah employeei2I hass(i)units ofwork

themaximumnumberofunits that i2I mayontratwithj 2J. Thisdata

ismodeledasagraph.

The nodesof thepreferenegraph are the pairsof opposite agents(i;j),

i2I andj2J. TheyaretakentobeloatedontheIJ gridwhereeahrow

orrespondsto anemployee orsupplieri 2I and eaholumn toan employer

or aquirer j 2 J. The (direted) ars of , or orderedpairs of nodes, are of

twotypes: ahorizontalar (i;j); (i;j 0

)

expressessupplieri'sprefereneforj 0

overj (sometimeswritten j 0

>

i

j), symmetriallyavertialar (i;j); (i 0

;j)

expresses aquirer j's preferene for i 0

over i (sometimes written i 0

>

j i). If

(i;j) = 0 for some (i;j) then the node may be omitted. Ars implied by

transitivityare omitted. Figure 1givesan examplewhere the values s(i)are

assoiated withrows,the valuesd(j)with olumns, and the values(i;j)are

arbitrarilylarge.

The stable marriage problem is the stable alloation problem with s(i) =

d(j) = 1 and (i;j) = 0 or 1, for all i 2 I; j 2 J; the stable university

admissionsproblem isthestable alloationproblem withs(i)positiveintegers,

d(j)=1and(i;j)=0or 1,foralli2I; j 2J; andthestable many-to-many

problem is the stable alloation problem with s(i) and d(j) positive integers,

and(i;j)=0or 1,foralli2I; j2J (see[4, 5,2℄).

Itisonvenient,andunambiguous,torefertothesuessorsofanodeor

to sayanode follows another in its rowor olumn, meaningthey or itare

preferredorrankedhigher. And,similarly,torefertothepredeessorsofanode

or to saya nodepreedesanother in itsrow or olumn, meaning theyor

itarelesspreferredor rankedlower. Alsoarst,leastpreferred(orlast, most

preferred)node inaroworolumnhasnopredeessors(nosuessors)and

arst(orlast)nodewithertainpropertieshasnopredeessors(nosuessors)

withthoseproperties.

In general, if S is a set and y(s); s 2 S, a real number, then y(S) def

=

P

s2S

y(s); also(r;S) def

= f(r;s):s2Sg. For(i;j)2 , (i;j

) def

= f(i;l):l

i

jgand(i;j

>

) def

= f(i;l):l>

i

jg;thesets(i

;j)and(i

>

;j)aredenedsimilarly.

An alloation x = x(i;j)

of a problem ( ;s;d;) is a set of real-valued

numberssatisfying

x(i;J)s(i); alli2I;

x(I;j)d(j); allj2J;

0x(i;j)(i;j); all(i;j)2 ;

alled,respetively, the row, the olumn and the entry onstraints. In Figure

1both y and z are alloations of the example. It may be and will be

assumedthat(i;j)min

s(i);d(j) .

Analloationx isstableifforevery(i;j)2 ,

x(i;j)<(i;j) implies x(i;j

)s(i) or x(i

;j)=d(j):

l 2 J may together, ignoring others, improve the alloation for themselves.

Speially,thevalueofx(k;l)maybeinreasedbyÆ>0,withx(k;j)>0for

somej <

k

l dereasedbyÆ (or x(k;J)<s(k))andx(i;l)>0forsomei<

l k

dereasedbyÆ(orx(I;l)<d(l)). Otherwise,(k;l)isstableforx. Inpartiular,

ifeitherx(k;l)=(k;l)orx(k;l

)=s(k)then(k;l)isrow-stable;andifeither

x(k;l)=(k;l)orx(k

;l)=d(l)then(k;l)isolumn-stablesoanodemay

bebothrow-andolumn-stable.

Inthe speial aseof marriage,(k;l) blokswhen man k andwomanl are

notmathed x(k;l)= 0

, k is not mathed oris mathed to aless desirable

woman thanl x(k;l

)=0

, l is not mathed or mathed to aless desirable

man thank x(k

;l)=0

, and (k;l)=1: thus together kand l an realize

abettersolutionfor themselves. In Figure1,y is notstable (4;3) bloksy

(theothernodesarestablefory)whereasz isstable.

(1)=10

d(1)=

10 1 11

14 1

2 17

y=

z= 11

14 6 12 7 3

2 1 s

(2)=12 s s

(3)=14 (4)=20 d(2)= d(3)= d(4)=

11 13 15 17

s s

Figure1: An alloationproblem(noupperbounds).

2 Stable alloations

Thissetionsummarizes thepertinentfats onerningstable alloations. For

proofsandamoreompletedesription,see[3℄.

Theemployee-orrow-greedysolutionofaproblem( ;s;d;)isdenedby

assigningtoeahrow-agenti2I his/her/itspreferredsolutionatingasifthere

werenootherrowagents. Itisdened reursively,beginningwithi'spreferred

hoie(the lastnodeinrowi):

(i;j)=min

s(i) (i;j

>

); d(j); (i;j) :

Ifnoolumnonstraintisviolated,isastablealloation. Intermsofmarriage,

assignstoeahmanhisfavoriteavailablewoman,andifnowomanisassigned

morethan oneman itis astableassignment. Theemployer-orolumn-greedy

solution( ;s;d;)isdened similarly.

whoreeivesaproposalmaydisardeverymanlowerinherprefereneswithout

hangingthe problem. A similar fat holds for alloationproblems. If x is a

stablealloationofaproblem( ;s;d;),then

x(i;j)

(i;j) def

= max n

0; min

(i;j); d(j) (i

>

;j) o

;

and the problems ( ;s;d;) and ( ;s;d;

) are equivalent in the sense that

theyadmitexatlythesamesetofstable alloations.

This suggests the generalization of the Gale-Shapley algorithm, the row-

greedyalgorithm: trytherow-greedysolution;ifitisanalloationthenitmust

beastablealloation;otherwise,newstrongerboundsmaybededuedandthe

proessrepeated. For disrete problems when s;d and are integer-valued

theproedure mustterminatewith aproblemwhose row-greedysolutionis

stable,soproves:

Theorem 1 There exist stable alloations for every stable alloation problem

( ;s;d;).

The theorem is proven for arbitrary real-valued data via an indutive al-

gorithm [3℄thatisstronglypolynomial: itrequiresat most3jIjjJj+jJj steps

tondastable alloation, wherejKjis theardinalityofK. Infat,the row-

greedyalgorithm is arbitrarily badfor disrete problems, and it is notknown

whetheritonvergesatallinthegeneralase.

Confronted with any two stable alloations an agent has no hesitation in

deidingwhihhe,sheoritprefers. Formally,anytwostablealloationsxand

ymaybeomparedwiththedenitionthat follows

x def

i

y;i2I; ifx(i;k)<y(i;k)impliesx(i;j)=0forj<

i

k; (1)

read row-agenti prefersx to y or is indierent betweenthem. x def

=

i

y when

x(i;)=y(i;),meaningiisindierentbetweenx andy (impliitly howothers

fareis ofnoimportaneto i),andx def

i

y when x

i

y andx6=

i

y. Symmetri

denitionsholdforolumn-agentsj 2J.

x

i

y implies x(i;j) < y(i;j) is true for at most one x(i;j) > 0. In

partiular, if x

i

y then x(i;k) < y(i;k) and x(i;j) > y(i;j) imply k <

i j.

Sineeah agentis assigned exatlythe sametotalnumber ofhours by every

stablealloation, row-agenti prefersx toy, or x

i

y, implies thaty maybe

transformedinto xbydereasingsomevaluesthat orrespondtoless-preferred

olumn-agentsandinreasingothersthatorrespondtomore-preferredolumn-

agents.

Ineet,thesimplestompletedesriptionofanagent'spreferenesbetween

stablealloationsisthemin-minriterion: thevalueoftheleast-preferredtype

of hourshould be as small as possible. Letting i(x) =j if x(i;j )> 0and

x(i;j)=0forj<j ,thismeans

x

i y if

either i(x)>

i i(y)

or i(x)=i(y)=j andx(i;j )<y(i;j ):

(2)

Theopposition ofinterestsbetweenrowsandolumnsholds heretoo:

Theorem 2 Ifx;y arestable andx(k;l)6=y(k;l),thenx

k

yfor k2I ifand

onlyif x

l

y for l2J.

Morevoer, itis easy to verifythat the set of allstable alloations is adis-

tributivelattie with respet to thepartial order

I

on thepreferenes of all

row-agentsI dened by:

x def

I

y if x

i

y foralli2I:

Surprisingly, onsiderable more is true. When the data s > 0, d > 0 and

(i;j) 0 are arbitrary real numbers it is to be expeted that no sum of a

subsetofthes(i)equalsthesumofasubsetofthed(j),northatsuhsumsare

equalwhenthes(i)andd(j)areeahreduedbyasumofsomeorresponding

(i;j): this isthe generi, strongly nondegenerate problem. In this ase the

problemhasauniquestable alloation.

Aordingly,itisonlyduetothedegeneraiesofthestableone-to-one,one-

to-manyand many-to-manymathing problems that therih lattiestruture

potentiallyinvolvingexponentiallymany stable alloations ours. But

the data of a stable alloation problem is often integer valued and may well

admit degeneraiesand so multiple stable alloations. Thus it is neessaryto

havearationaleforhoosingone stablealloationin thepreseneofmany.

Theexamplegivenaboveissuhaninstane. Ithasexatly7extremestable

alloationsmeaningstable alloationsthatare notaonvexombinationof

others. Theyare givenin Figure 2. The stablealloation z ofFigure 1is not

extreme: z= 3

10 x

3 +

7

10 x

4

. Whenthedata isinteger-valuedthere alwaysexist

stablealloationsin integers.

3 Mehanisms

Analloationmehanismisafuntionthatseletsexatlyonestablealloation

for any problem ( ;s;d;). Three haraterization are given of eah of two

partiularly onspiuous mehanisms. This generalizes known results for the

one-to-many[6,1℄andmany-to-manymathingproblems[2℄.

Theemployee-orrow-optimalstablealloationx

I

ofaproblem( ;s;d;)is

denedby:

x

I

i

x; alli2I; foreverystable alloationx:

x

I

attributes to every row-agentthe best possible alloationamong all stable

alloations. Therow-optimalalgorithm [3℄establishes

xJ= x = 1

10 5 10

12 4 119 5

x = 3

2 5 10

13 14 1

11 x =

2

5 10

13 2 9

14 3 5 10

12 13 2

9 5

14 11

15 10 1

2 3

2 14 1 11

15 3 10

x₄= x₅=

xI=

14 15

2 3

11 1

Figure2: All extremestable alloation: arrowsshowolletivepreferenes

I

(inthisaseaompleteorder).

Theorem 3 Every problem ( ;s;d;) has a uniquerow-optimal stable alloa-

tionx

I .

Bysymmetry, everyproblem hasauniqueemployer-or olumn-optimalstable

alloation x

J

. So two obvious examples of mehanisms are the employee- or

row-optimalmehanism

I

thatalwaysseletsx

I

,andtheemployer-orolumn-

optimalmehanism

J

thatalwaysseletsx

J .

Eieny

Itwouldbeagreeableifitouldbeassertedthattheemployee-optimalmeh-

anism

I

is eient in that no alloation, stable or not, is ever better for

theemployees thanthe employee-optimalstable alloation x

I

. This depends,

ofourse, on what is meant bybetter: in an intuitivesense thealloationy

giveninFigure1isolletivelypreferredtox

I

bytheemployeesI (y isbloked

by(4;3)). But bythe min-min riterion (2) for omparing stable alloations,

row-agent4wouldbeindierentbetween x

I

and y. Thedenition ofbetter

willextendthemin-minriteriontoarbitraryalloations.

Consider now aproblem where the s(i);i 2 I are generous in omparison

with the d(j);j 2 J, asin Figure 3. The employee-optimal stable alloation

x

I

is viiouslyemployee-ineient: everyalloation,stableor not,that gives

a total of 7 hours to employee 1 and a total of 11 to employee 2 is better

forthe employees. Thus if x

I

isin somesense eient this possibilitymust

be exluded: aordingly, when x

I

(i;J) < s(i) any other alloation y with

y(i;J)=x(i;J)willbeonsideredequallypreferredbyi.

Guidedbytheseexamples,extendthedenitionofanemployee'spreferenes

betweenstablealloations(2)topreferenesbetweenarbitraryalloationsxand

yas follows:

3 4 0 0 0 0 5 6 x_I =

6 11

11

3 4 5

Figure3: x

I

ineient.

x def

i y if

i(x)<

i i(y)or

i(x)=i(y)=l ;x(i;j )<y(i;j )

when x(i;J)=y(i;J)=s(i)

x(i;J)>y(i;J) when y(i;J)<s(i):

Also,rowagentiisindierentbetweentwoalloations,x

i

y,ifi(x)=i(y)=

j andx(i;j )=y(i;j )whenx(i;J)=y(i;J)=s(i),orifx(i;J)=y(i;J)<

s(i). Takex

i

y tomeanx

i

y orx

i

y. Asbefore,

x def

I

yifx

i

yforalli2I;

andx def

I

y ifx

I

y andx

i

y isnottrueforalli2I.

Apreliminarylemma onerningstable alloationsisneeded.

Lemma1 Supposethatxisastablealloation andyisanalloation,stableor

not,forwhihy

i

xforalli2I. Thenthereexistsastablealloationy

I x.

Moreover,

(i)x(i;J)=y(i;J)=y

(i;J)=s(i) for everyi2I,

(ii) x(I;j)=y(I;j)=y

(I;j)for every j2J, and

(iii)y

(i;j)>x(i;j)impliesthereexistsh

j

ifor whih y(h;j)>x(h;j).

Proof. Tobegin,supposethatforsomej2J,x(I;j)<y(I;j). Thenx(i;j)<

y(i;j) for some i 2 I. If x(i;J) < s(i) then (i;j) bloks x, a ontradition;

and if x(i;J) = s(i) then y

i

x implies x(i;j 0

) > y(i;j 0

) for some j 0

<

i j,

so (i;j) bloks x, again a ontradition. Therefore, x(I;j) y(I;j) for all

j 2J, andx(I;J)y(I;J). Buty

i

x means x(i;J)y(i;J)for i2I, so

x(I;J)y(I;J)and the inequalitiesare allequations. Finally, x(i;J) <s(i)

impliesx(i;J)<y(i;J)sox(i;J)=y(i;J)=s(i).

Let =

i;i(x)

:i 2 I . Bydenition, x i;i(x)

>y i;i(x)

for every

i;i(x)

2 . From above it follows that for eah i;i(x)

2 there is at

leastone h2 I with x h;i(x)

<y h;i(x)

h;i(x)

. Buty

h

x implies

h;h(x)

preedes h;i(x)

in rowh. Therefore,itmustbethat h<

i(x) i sine

otherwisex wouldnotbestable.