In this note, we start to claim that established marriages can be heavily destabilized when the population of existing couples is enriched by the arrival of new candidates to marriage. Afterwards, we discuss briefly how stability concepts can be extended to account for entry and exit phenomena affecting the composition of the marriage market.Matching; stability; marriage model; divorce cascades

Filomena, GARCIA

Jean J. GABSZEWICZ

Joana, PAIS

Joana, RESENDE

English

Research Papers in Economics

Département des Sciences Économiques
de l'Université catholique de Louvain
On Gale and Shapley ‘College admissions
and stability of marriage’
J. J. Gabszewicz, F. Garcia, J. Pais and J. Resende
Discussion  Paper   2008-46CORE DISCUSSION PAPER   
2008/67 
 
On Gale and Shapley 'College admissions  
and stability of marriage' 
 
Jean J. GABSZEWICZ1, Filomena GARCIA
2,  
Joana PAIS
3 and Joana RESENDE4  
 
 
December 2008 
 
 
Abstract 
 
In  this  note,  we  start  to  claim  that  established  marriages  can  be  heavily  destabilized  when  the 
population of existing couples is enriched by the arrival of new candidates to marriage. Afterwards, 
we discuss briefly how stability concepts can be extended to account for entry and exit phenomena 
affecting the composition of the marriage market. 
 
Keywords: matching, stability, marriage model, divorce cascades. 
JEL Classification: C78, D00 
                                                           
1  Université  catholique  de  Louvain,  CORE,  B-1348  Louvain-la-Neuve,  Belgium.  E-mail: 
jean.gabszewicz@uclouvain.be. This author is also member of CORE, the newly created association between 
CORE and ECARES. 
2 ISEG/Technical University of Lisbon and UECE, Portugal. E-mail: fgarcia@iseg.utl.pt 
3 ISEG/Technical University of Lisbon and UECE, Portugal. E-mail: jpais@iseg.utl.pt 
4  Université  catholique  de  Louvain,  CORE,  B-1348  Louvain-la-Neuve,  Belgium;  CETE,  University  of  Porto, 
Portugal. E-mail: joana.resende@student.uclouvain.be. 
 
J. Resende greatfully acknowledge financial support by CETE (via FCT – POCI 2010) and by FCT. 
This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the 
Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by 
the authors.  Gale and Shapley (1962) investigate marriage stability within a hetero-
sexual and monogamic community formed by one set of men and one set
of women. Stability (henceforth G￿S stability) in the marriage market is
satis￿ed when women and men are matched so that there are no woman
and no man who are not married to each other though they would prefer
each other to their actual mates. Gale and Shapley (1962) show that, in
such a market, there always exists a stable matching.5
Yet, divorces are a factual evidence and get more and more frequent!
In this note, we start to claim that established marriages can be heavily
destabilized when the population of existing couples is enriched by the
arrival of new candidates to marriage. Using a striking example with n men
and n women, we show that the entrance of a new couple in the marriage
market completely disrupts, however large n may be, the unique stable
matching prevailing before. Afterwards, we discuss brie￿ y how stability
concepts can be extended to account for entry and exit phenomena a⁄ecting
the composition of the marriage market.
. DIVORCE CASCADES: AN EXAMPLE
Consider a marriage market with the same number (n) of women and
men. In this market, both women (as well as men) have homogeneous
preferences over the set of potential mates. More precisely, both men and
women can be ranked in such a way that if a man is higher in the ranking of
men, then each woman strictly prefers him to all others ranked below him,
and similarly for the preferences of men over the set of women. Clearly,
the unique G￿S stable matching for this market consists in matching the
ith-ranked man with the ith-ranked woman, i = 1;::;n.
Consider now that a new single woman and a new single man arrive in
this market. Moreover, assume that this new single woman - the Beauty -
is now ranked by all existing men at the top of the set of women, while the
new single man - the Beast - is ranked by every women at the bottom of the
set of men. In other words, the entry of this couple is as if "The Beauty and
the Beast" had entered the market! Of course now the "top man" starts
to be strongly interested in the Beauty and wishes to divorce from the
5This result illustrates a well-known French proverb, which says "Chaque casserole a
son couvercle"!
2past "top woman" to marry the Beauty. Thus, the past "top woman" now
becomes available to the 2nd-ranked man, who prefers her to his current
mate. Consequently, a new divorce occurs as he wishes to marry the past
"top woman" who is now willing to accept him. And so on and so forth,
until the past "bottom man" decides to divorce from the "bottom woman,"
who now, poor woman, is left with the Beast! Accordingly, and whatever
large the number n, the entry of a single pair of persons of opposite sex
entails a cascade of divorces and fully disrupts all the n previous G￿S
stable matches: this entry generates a domino e⁄ect, where the disruption
of one couple automatically provokes the disruption of the next one in the
ranking, until all previous matches are destroyed.
. EXTERNAL STABILITY
The above example evokes the possibility of de￿ning a concept of exter-
nal stability related to a matching which is G￿S stable. In this example,
we start with a situation that is G￿S stable and we slightly modify the
market by allowing one further man and one further woman to enter the
market. For the particular set-up of preferences introduced above, this
marginal change entails a completely di⁄erent new G￿S stable matching
in the resulting market. For this reason, we argue that the initial matching
is externally unstable. For di⁄erent speci￿cations of preferences, the degree
of external instability could be of course di⁄erent, to the extent that the
number of disrupted couples due to the entry can vary between 0 and n:
This justi￿es the following de￿nition.
Definition 1. Given a marriage market and a G￿S stable matching
in this market, this matching is said to be k￿externally stable whenever,
at any G￿S stable matching in the marriage market consisting of one fur-
ther man and one further woman, at least k existing past matches are not
disrupted.
In the example above, the initial matching is clearly 0￿externally stable
since all existing initial matches are disrupted at the stable matching in
the new marriage market. Notice that it can be the case that no couple is
disrupted by the entry of a new pair: assume, for instance, that the new
man puts the new woman at the top of his ranking and vice versa, while
each existing mate ranks the entrant of the opposite sex in the very bottom
3of his/her previous ordering. In this case, no couple will be disrupted by
the entrants; however, according to the de￿nition, the initial matching is 0-
externally stable, as there exists an entering pair that disrupts all couples.
6
A further observation is that the maximal level of k￿ external stability
that can be achieved is n ￿ 2. In fact, for any preference pro￿le and for
any matching that is stable for that pro￿le, there always exists at least one
pair that disrupts two couples, whilst all others may remain matched with
the same partner.
The degree of external stability of G￿S stable matchings clearly de-
pends on the individual rankings of existing and potential marriage mates.
Given the richness of possible preference rankings, one can hardly make
very general statements about the degree of external stability. Neverthe-
less, when restricting to more particular types of preference rankings, one
might get some insights into this question. To illustrate this point, we con-
sider in the following section, two speci￿c categories of preference set-ups,
namely, vertical and horizontal set-ups.
. VERTICAL AND HORIZONTAL SET-UPS
In real life, individuals￿preferences regarding potential marriage mates
are highly subjective and heterogeneous. In this section, we illustrate that
some preference set-ups engender marriage markets with higher degree of
k￿external stability than others. We analyse two particular preference set-
ups: a horizontal set-up, where each individual has its own preferred match
(peak load preference set-up), and a vertical set-up, where all individuals
have the same preference ordering over the potential marriage partners.
Definition 2. A vertical set-up corresponds to a pro￿le of preference
relations over the potential marriage partners, such that, for every agent
k, k = 1;:::;n, we have i ￿k j , i > j for every i;j = 1;:::;n, i 6= j:
Hence, the vertical set-up case corresponds to the set of preference
relations adopted in our example, with both men and women being ranked
in such a way that if a woman (man) is higher in the ranking of women
6Also notice that the concept of k - external stability applies as well to the case in
which the new marriage market would consist of the previous market with one woman
and one man less, which corresponds to a situation of exit.
4(men), then each man (woman) strictly prefers her (him) to all others
ranked below her (him).
Definition 3. A peak load preference set-up corresponds to a pro￿le
of preferences over the potential marriage partners such that, for every
agent k, k = 1;:::;n, we have i ￿k j , jk ￿ ij < jk ￿ jj and, whenever
jk ￿ ij = jk ￿ jj; i ￿k j , i > j; for every i;j = 1;:::;n, i 6= j:
This de￿nition simply transposes the usual notion of distance used in
continuous models ￿ la Hotelling to de￿ne preference relations character-
ized by the fact that the further distant an option is from the ideal option
of an individual, the least preferred it is. In this case, the ranking of po-
tential marriage mates is no longer unanimous and the top men and top
women will di⁄er from individual to individual. For example, this is the
case when women (as well as men) are di⁄erentiated with respect to some
characteristic and, consequently, they rank potential marriage partners ac-
cording to the similarity with their own characteristic. This implies that
each woman (resp. man) has her (his) speci￿c individual ranking over
marriage partners, with marriage mates whose characteristic is closer to
her (his) own being better positioned in her (his) individual ranking. The
following representation of preferences illustrates the characteristics of peak
load preferences set-up, considering a marriage market with three men and
three women:
￿w1 ￿w2 ￿w3
m1 m2 m3
m2 m3 m2
m3 m1 m1
and
￿m1 ￿m2 ￿m3
w1 w2 w3
w2 w3 w2
w3 w1 w1
Transposing this preference set-up to a marriage market with n women
and n men and ranking men and women according to the characteristic
with respect to which they di⁄er, it can be easily seen that the unique
G￿S stable matching for this market consists in matching the ith-ranked
man with the ith-ranked woman, i = 1;::;n.
53.1. Vertical and horizontal rankings: external stability
In the previous section, we argued that the degree of external stability
in marriage markets is determined by the nature of individuals￿preference
set-ups over potential marriage partners. In this section, we illustrate this
point by contrasting the cases of vertical and peak load preference set-ups,
which exhibit substantially di⁄erent properties in terms of external stabil-
ity. In fact, preference structures corresponding to peak load preference
set-ups are more externally-stable than the ones corresponding to vertical
preference set-ups in the sense that the former produce stable matchings
that have a higher degree of of external-stability than the latter.
Let us consider a marriage market composed of a same number of men
and women. Assume that preference relations over potential marriage
mates are described by the vertical preference set-up. Finally, assume that
an additional man and an additional woman enter this market. If all exist-
ing women rank this additional man between the existing men j and j +1;
while all existing men rank this additional woman between the existing
women j +h and j +h+1; then it is easy to see that all the jhj marriages
corresponding to the existing matches of individuals ranked in-between the
positions of the new members of the market are now disrupted. In the limit,
when we consider the largest possible gap (n), as in the example of "The
Beauty and the Beast," all the existing matches at the initial G-S stable
matching are disrupted at the new G-S stable matching. Hence, vertical
preference set-ups produce stable matchings that are 0-externally stable.
In contrast, peak-load preferences are substantially more stable. Inde-
pendently of the relative positioning of the new members of the marriage
market, the entrance of a new woman and a new man in the market dis-
rupts, at most, two matches of the initial G-S stable matching (all the
remaining couples stay together at the new G-S stable matching). It fol-
lows that, in the case of peak-load preferences, the degree of external
stability is equal to n ￿ 2.
. CONCLUSION
In this note, we have suggested that the notion of G-S stability in a
marriage market could be advantageously complemented with a concept of
stability related to the possibility of entry in this market. Starting from an
6illustrative example, we show how entry can heavily destabilize established
marriages. We propose a tentative concept which could capture the extent
of external stability of a marriage market.
REFERENCES
[1] D. Gale and L.S. Shapley (1962). College admissions and stability of
marriage. American Mathematical Monthly, Vol. 69 (1), 9-15.
7Département des Sciences Économiques
de l'Université catholique de Louvain
Institut de Recherches Économiques et Sociales
Place Montesquieu, 3
1348 Louvain-la-Neuve, Belgique
 ISSN 1379-244X         D/2008/3082/046

7Département des Sciences Économiques de l'Université catholique de Louvain Institut de Recherches Économiques et Sociales Place Montesquieu,

College admissions and stability of marriage.

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