College Admissions and the Stability of Marriage
Author(s): D. Gale and L. S. Shapley
Reviewed work(s):
Source: The American Mathematical Monthly, Vol. 69, No. 1 (Jan., 1962), pp. 9-15
Published by: Mathematical Association of America
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COLLEGE
ADMISSIONS
AND THE STABILITY
OF MARRIAGE
andtheRAND Corporation
D. GALE*AND L. S. SHAPLEY,BrownUniversity
1. Introduction.The problem with which we shall be concerned relates to
the followingtypical situation: A college is consideringa set of n applicants of
which it can admit a quota of only q. Having evaluated theirqualifications,the
admissions officemust decide which ones to admit. The procedure of offering
admission only to the q best-qualifiedapplicants will not generallybe satisfactory, for it cannot be assumed that all who are offeredadmission will accept.
Accordingly,in orderfora college to receiveq acceptances, it willgenerallyhave
to offerto admit morethan q applicants. The problemof determininghow many
and which ones to admit requires some ratherinvolved guesswork.It may not
be known (a) whethera given applicant has also applied elsewhere; if this is
known it may not be known (b) how he ranks the colleges to which he has
applied; even ifthis is knownit will not be known (c) whichof the othercolleges
will offerto admit him. A result of all this uncertaintyis that colleges can expect only that the enteringclass will come reasonably close in numbersto the
desired quota, and be reasonably close to the attainable optimumin quality.
The usual admissions procedurepresentsproblemsforthe applicants as well
as the colleges. An applicant who is asked to list in his application all other
colleges applied forin orderof preferencemay feel,perhaps not withoutreason,
that by tellinga college it is, say, his thirdchoice he will be hurtinghis chances
of being admitted.
One elaboration is the introductionof the "waiting list," wherebyan applicant can be informedthat he is not admitted but may be admitted later if a
vacancy occurs.This introducesnew problems.Suppose an applicant is accepted
by one college and placed on the waiting list of another that he prefers.Should
he play safe by accepting the firstor take a chance that the second will admit
him later? Is it ethical to accept the firstwithout informingthe second and
then withdrawhis acceptance if the second later admits him?
We contend that the difficulties
here describedcan be avoided. We shall describe a procedureforassigningapplicants to collegeswhich should be satisfactory to both groups,whichremovesall uncertaintiesand which,assuming there
are enough applicants, assigns to each college preciselyits quota.
2. The assignmentcriteria.A set of n applicants is to be assigned among m
colleges,whereqi is the quota of the ith college. Each applicant ranksthe colleges
in the orderof his preference,omittingonly those collegeswhich he would never
accept under any circumstances.For conveniencewe assume there are no ties;
between two or more colleges he is neverthethus, if an applicant is indifferent
less requiredto list themin some order.Each college similarlyranksthe students
who have applied to it in orderof preference,having firsteliminatedthose appli* The workofthefirst
ofNaval ResearchunderTask
authorwas supportedin partbytheOffice
NRO47-018.
9
10
COLLEGE ADMISSIONS AND STABILITY OF MARRIAGE
[January
cants whom it would not admit under any circumstanceseven if it meant not
fillingits quota. From these data, consistingof the quotas of the colleges and
the two sets of orderings,we wish to determinean assignmentof applicants to
colleges in accordance with some agreed-uponcriterionof fairness.
Stated in this way and looked at superficially,the solution may at first
appear obvious. One merelymakes the assignments "in accordance with" the
given preferences.A little reflectionshows that complications may arise. An
example is the simplecase of two colleges,A and B, and two applicants,a and $,
in which a prefersA and , prefersB, but A prefers, and B prefersa. Here, no
assignmentcan satisfyall preferences.One must decide what to do about this
sort of situation. On the philosophy that the colleges exist for the students
rather than the other way around, it would be fittingto assign a to A and ,3
to B. This suggeststhe followingadmittedlyvague principle:otherthingsbeing
equal, students should receive consideration over colleges. This remark is of
little help in itself,but we will returnto it later after taking up another more
explicit matter.
The key idea in what followsis the assertion that-whatever assignmentis
finallydecided on-it is clearly desirable that the situation described in the
followingdefinitionshould notoccur:
DEFINITION. An assignment
ofapplicantsto colleges
willbecalledunstableif
A andB, respectively,
thereare twoapplicantsa and ,Bwhoare assignedtocolleges
A toB and A prefers: toa.
although
,3prefers
Suppose the situation described above did occur. Applicant ,3 could indicate to college A that he would like to transferto it, and A could respond
by admitting,B,lettinga go to remain within its quota. Both A and f3would
consider the change an improvement.The original assignment is therefore
"unstable" in the sense that it can be upset by a college and applicant acting
togetherin a mannerwhich benefitsboth.
Our firstrequirementon an assignment is that it not exhibit instability.
This immediatelyraises the mathematical question: will it always be possible
answer to this question will be given
to findsuch an assignment?An affirmative
the result seems not enin the next section,and while the proofis not difficult,
tirelyobvious, as some examples will indicate.
Assuming for the moment that stable assignmentsdo exist, we must still
decide which among possibly many stable solutionsis to be preferred.We now
return to the philosophical principle mentioned earlier and give it a precise
formulation.
is calledoptimalif every
DEFINITION. A stableassignment
applicantis at least
as welloffunderit as underanyotherstableassignment.
Even grantingthe existence of stable assignmentsit is far fromclear that
there are optimal assignments. However, one thing that is clear is that the
1962]
COLLEGE ADMISSIONS AND STABILITY OF MARRIAGE
11
optimal assignment,ifit exists,is unique. Indeed, if therewere two such assignments, then, at least one applicant (by our "no tie" rule) would be better off
under one than under the other; hence one of the assignmentswould not be
optimal afterall. Thus the principlesof stabilityand optimalitywill, when the
existencequestionsare settled,lead us to a unique "best" methodof assignment.
3. Stable assignments and a marriage problem. In trying to settle the
question of the existence of stable assignmentswe were led to look firstat a
special case, in whichthereare the same numberof applicants as colleges and all
quotas are unity. This situation is, of course, highlyunnatural in the context
of college admissions, but there is another "story" into which it fits quite
readily.
A certain communityconsists of n men and n women. Each person ranks
those of the opposite sex in accordance with his or her preferencesfora marriage
partner.We seek a satisfactoryway of marryingoffall membersof the community.Imitatingour earlierdefinition,we call a set of marriagesunstable(and
here the suitabilityof the termis quite clear) if under it thereare a man and a
woman who are not marriedto each other but prefereach other to theiractual
mates.
QUESTION:
marriages?
For any patternof preferences
is it possible tofind a stableset of
Before givingthe answer let us look at some examples.
Example 1. The followingis the "rankingmatrix"of threemen,a, ,3,and 7,
and threewomen,A, B, and C.
A
B
C
a
1,3
2,2
3,1
p
3,1
1,3
2,2
y
2,2
3,1
1,3
The firstnumberof each pair in the matrixgives the rankingof women by the
men, the second number is the ranking of the men by the women. Thus, a
ranks A first,B second, C third,while A ranks, first,y second, and a third,etc.
There are six possible sets of marriages; of these, three are stable. One of
these is realized by givingeach man his firstchoice, thus a marriesA, A marries
B, and y marries C. Note that although each woman gets her last choice, the
arrangementis neverthelessstable. Alternativelyone may let the women have
their firstchoices and marrya to C, ,Bto A, and y to B. The third stable arrangementis to give everyonehis or her second choice and have a marryB,
f marryC, and y marryA. The reader will easily verifythat all otherarrangements are unstable.
12
COLLEGE ADMISSIONS AND STABILITY OF MARRIAGE
[January
Example 2. The rankingmatrixis the following.
A
B
C
D
113
2)3
@)
4,3
83 1,.4
r 2
4, 1
1,4
3,3
3,4
)~
4,1
a
a
4,1
ci2) 3),1
1,4
There is only the one stable set of marriagesindicated by the circled entries
in the matrix.Note that in this situation no one can get his or her firstchoice if
stabilityis to be achieved.
Example 3. A problem similar to the marriage problem is the "problem of
the roommates."An even numberof boys wish to divide up into pairs of roommates. A set of pairingsis called stableif underit thereare no two boys who are
not roommatesand who prefereach other to theiractual roommates.An easy
example shows that therecan be situations in which thereexists no stable pairing. Namely, considerboys a, ,B,y and 3, where a ranks:3 first,,Branks 'y first,
,yranks a first,and a, 3 and y all rank a last. Then regardlessof S's preferences
there can be no stable pairing, for whoever has to room with a will want to
move out, and one of the othertwo will be willingto take him in.
The above examples would indicate that the solutionto the stabilityproblem
is not immediatelyevident. Nevertheless,
THEOREM 1.
Therealwaysexistsa stablesetofmarriages.
Proof.We shall prove existenceby givingan iterativeprocedureforactually
findinga stable set of marriages.
To start, let each boy propose to his favoritegirl. Each girl who receives
more than one proposal rejects all but her favoritefromamong those who have
proposedto her. However,she does not accept himyet,but keeps himon a string
to allow forthe possibilitythat someone better may come along later.
We are now ready forthe second stage. Those boys who were rejected now
propose to theirsecond choices. Each girlreceivingproposalschooses herfavorite
fromthe groupconsistingof the new proposersand the boy on her string,ifany.
She rejects all the rest and again keeps the favoritein suspense.
We proceed in the same manner.Those who are rejected at the second stage
propose to theirnext choices,and the girlsagain reject all but the best proposal
they have had so far.
Eventually (in fact,in at mostn2- 2n+2 stages) everygirlwill have received
a proposal, foras long as any girl has not been proposed to there will be rejectionsand new proposals,but since no boy can proposeto the same girlmorethan
once, every girl is sure to get a proposal in due time. As soon as the last girl
1962]
COLLEGE ADMISSIONS AND STABILITY OF MARRIAGE
13
gets her proposal the "courtship"is declared over, and each girlis now required
to accept the boy on her string.
We assert that this set of marriagesis stable. Namely, suppose John and
Mary are not marriedto each other but John prefersMary to his own wife.
Then John must have proposed to Mary at some stage and subsequentlybeen
rejected in favorof someone that Mary liked better. It is now clear that Mary
must preferher husband to Johnand thereis no instability.
The reader may amuse himselfby applying the procedure of the proof to
solve the problemsof Examples 1 and 2, or the followingexample whichrequires
ten iterations:
A
B
C
D
a
1, 3
2, 2
3, 1
4, 3
3
1,4
2,3
3,2
4,4
y
3,1
1,4
2,3
4,2
1,4
4,1
5
2,2
3,1
The conditionthat therebe the same numberof boys and girlsis not essential. If thereare b boys and g girlswith b <g, then the procedureterminatesas
soon as b girls have been proposed to. If b>g the procedureends when every
boy is either on some girl's stringor has been rejected by all of the girls. In
eithercase the set of marriagesthat resultsis stable.
It is clear that thereis an entirelysymmetricalprocedure,with girlsproposing to boys, whichmust also lead to a stable set of marriages.The two solutions
are not generallythe same as shown by Example 1; indeed, we shall see in a
moment that when the boys propose, the result is optimal for the boys, and
when the girlspropose it is optimal forthe girls.The solutionsby the two procedures will be the same only when thereis a unique stable set of marriageo.
4. Stable assignments and the admissions problem. The extension of our
"deferred-acceptance"procedureto the problemofcollegeadmissionsis straightforward.For conveniencewe will assume that ifa college is not willingto accept
a student underany circumstances,as describedin Section 2, then that student
will not even be permittedto apply to the college. With this understandingthe
procedure follows: First, all students apply to the college of their firstchoice.
A college with a quota of q then places on its waiting list the q applicants who
rank highest,or all applicants if there are fewerthan q, and rejects the rest.
Rejected applicants then apply to their second choice and again each college
selects the top q fromamong the new applicants and those on its waiting list,
puts these on its new waitinglist,and rejects the rest.The procedureterminates
when every applicant is either on a waiting list or has been rejected by every
college to which he is willingand permittedto apply. At this point each college
14
COLLEGE ADMISSIONS AND STABILITY OF MARRIAGE
[January
admits everyoneon its waitinglist and the stable assignmenthas been achieved.
The proofthat the assignmentis stable is entirelyanalogous to the proofgiven
forthe marriageproblemand is leftto the reader.
5. Optimality.We now show that the "deferredacceptance" procedurejust
described yields not only a stable but an optimal assignment of applicants.
That is,
THEOREM 2. Everyapplicant is at least as well offundertheassignmentgiven
by thedeferredacceptanceprocedureas he would be underany otherstableassignment.
Proof. Let us call a college "possible" fora particularapplicant if there is a
stable assignmentthat sends him there.The proofis by induction.Assume that
up to a given point in the procedure no applicant has yet been turned away
froma college that is possible for him. At this point suppose that college A,
having received applications from a full quota of better-qualifiedapplicants
pi, * * *, 3, rejects applicant a. We must show that A is impossible fora. We
know that each j3ipreferscollege A to all the others,except forthose that have
previously rejected him, and hence (by assumption) are impossible for him.
Consider a hypotheticalassignment that sends a to A and everyone else to
colleges that are possible forthem.At least one of the fPiwill have to go to a less
desirable place than A. But this arrangementis unstable, since j3 and A could
upset it to the benefitof both. Hence the hypotheticalassignmentis unstable
and A is impossible for a. The conclusion is that our procedure only rejects
applicants fromcolleges which they could not possibly be admitted to in any
stable assignment.The resultingassignmentis thereforeoptimal.
Parentheticallywe may remark that even though we no longer have the
symmetryof the marriageproblem,we can still invertour admissionsprocedure
to obtain the unique "college optimal" assignment.The invertedmethod bears
some resemblanceto a fraternity"rushweek"; it startswitheach college making
bids to those applicants it considers most desirable, up to its quota limit,and
then the bid-forstudents reject all but the most attractive offer,and so on.
6. Concludingremarks. The reader who has followedus this far has doubtless noticed a certain trendin our discussion.In makingthe special assumptions
needed in order to analyze our problemmathematically,we necessarilymoved
furtheraway fromthe originalcollege admissionquestion,and eventuallyin discussing the marriageproblem,we abandoned realityaltogetherand enteredthe
world of mathematical make-believe. The practical-mindedreader may rightfullyask whetherany contributionhas been made toward an actual solution of
the originalproblem.Even a roughanswer to this question would requiregoing
into matterswhich are nonmathematical,and such discussion would be out of
place in a journal of mathematics.It is our opinion,however,that some of the
ideas introducedhere mightusefullybe applied to certain phases of the admissions problem.
1962]
GRADUATED INTEREST RATES IN SMALL LOANS
15
Finally, we call attentionto one additional aspect of the precedinganalysis
which may be of interestto teachers of mathematics.This is the fact that our
result provides a handy counterexampleto some of the stereotypeswhich nonmathematiciansbelieve mathematicsto be concernedwith.
Most mathematiciansat one time or another have probably found themselves in the position of tryingto refutethe notion that they are people with
"a head forfigures,"or that they "know a lot of formulas."At such timesit may
be convenientto have an illustrationat hand to show that mathematicsneed
not be concernedwith figures,eithernumericalor geometrical.For this purpose
we recommendthe statement and proof of our Theorem 1. The argument is
carried out not in mathematicalsymbolsbut in ordinaryEnglish; there are no
obscure or technical terms. Knowledge of calculus is not presupposed. In fact,
one hardlyneeds to knowhow to count. Yet any mathematicianwillimmediately
recognize the argument as mathematical, while people without mathematical
in followingthe argument,though not betrainingwill probably finddifficulty
cause of unfamiliaritywith the subject matter.
What, then, to raise the old question once more, is mathematics? The
answer, it appears, is that any argumentwhich is carried out with sufficient
precision is mathematical,and the reason that your friendsand ours cannot
understandmathematicsis not because they have no head forfigures,but because they are unable to achieve the degree of concentrationrequiredto follow
a moderatelyinvolved sequence of inferences.This observation will hardly be
news to those engaged in the teaching of mathematics,but it may not be so
readilyaccepted by people outside of the profession.For themthe foregoingmay
serve as a usefulillustration.
GRADUATED INTEREST RATES IN SMALL LOANS
HUGH E. STELSON, MichiganState University
Many small loan companies charge a graduated interestrate in accordance
withvarious state laws. For example, 3% per monthis chargedon the first$150
of a loan, and 2% on the portionof the loan in excess of $150. Rates may be
graduated in two, threeor more brackets.A three-bracketloan mightbe at the
rate of 2'% on that part of the loan or loan balance whichis $100 or less, at the
rate of 2% on that part of a loan which is in excess of $100 but less than $200,
and at the rate of 1% on that part of a loan which is in excess of $200. Such a
graduated rate is written:21%/2%/1%/$100/$200.
The main problem considered in this paper is that of findingthe level
monthlyrentpaymentwhichwill amortizea loan in a given time at a graduated
rate.