STABILITY IN MATCHING PROBLEMS WITH WEIGHTED
PREFERENCES
Maria Silvia Pini, Francesca Rossi, Kristen Brent Venable
Department of Pure and Applied Mathematics, University of Padova, 35121 Padova, Italy
Toby Walsh
NICTA and UNSW, Sydney, Australia
Keywords:
Stable marriages, Weighted preferences.
Abstract:
The stable marriage problem is a well-known problem of matching men to women so that no man and woman,
who are not married to each other, both prefer each other. Such a problem has a wide variety of practical
applications, ranging from matching resident doctors to hospitals, to matching students to schools or more
generally to any two-sided market. In the classical stable marriage problem, both men and women express
a strict preference order over the members of the other sex, in a qualitative way. Here we consider stable
marriage problems with weighted preferences: each man (resp., woman) provides a score for each woman
(resp., man). Such problems are more expressive than the classical stable marriage problems. Moreover, in
some real-life situations it is more natural to express scores (to model, for example, profits or costs) rather
than a qualitative preference ordering. In this context, we define new notions of stability and optimality,
and we provide algorithms to find marriages which are stable and/or optimal according to these notions.
While expressivity greatly increases by adopting weighted preferences, we show that in most cases the desired
solutions can be found by adapting existing algorithms for the classical stable marriage problem.
1 INTRODUCTION
The stable marriage problem (SM) (Gusfield and Irv-
ing, 1989) is a well-known problem of matching the
elements of two sets. It is called the stable marriage
problem since the standard formulation is in terms
of men and women, and the matching is interpreted
in terms of a set of marriages. Given n men and n
women, where each person expresses a strict order-
ing over the members of the opposite sex, the prob-
lem is to match the men to the women so that there
are no two people of opposite sex who would both
rather be matched with each other than their current
partners. If there are no such people, all the mar-
riages are said to be stable. In (Gale and Shapley,
1962) Gale and Shapley proved that it is always pos-
sible to find a matching that makes all marriages sta-
ble, and provided a polynomial time algorithm which
can be used to find one of two extreme stable mar-
riages, the so-called male-optimal or female-optimal
solutions. The Gale-Shapley algorithm has been used
in many real-life scenarios (Roth, 2008), such as in
matching hospitals to resident doctors (Roth, 1984;
Irving, 1998), medical students to hospitals, sailors to
ships (Liebowitz and Simien, 2005), primary school
students to secondary schools (Teo et al., 2001), as
well as in market trading (Tesfatsion, 1998).
In the classical stable marriage problem, both
men and women express a strict preference order
over the members of the other sex in a qualitative
way. Here we consider stable marriage problems
with weighted preferences. In such problems each
man (resp., woman) provides a score for each woman
(resp., man). Stable marriage problems with weighted
preferences are interesting since they are more ex-
pressive than the classical stable marriage problems,
since in classical stable marriage problem a man
(resp., a woman) cannot express how much he (resp.,
she) prefers a certain woman (resp., man). Moreover,
they are useful in some real-life situations where it
is more natural to express scores, that can model no-
tions such as profit or cost, rather than a qualitative
preference ordering. In this context, we define new
notions of stability and optimality, we compare such
notions with the classical ones, and we show algo-
rithms to find marriages which are stable and/or op-
45
Pini M., Rossi F., Venable K. and Walsh T..
STABILITY IN MATCHING PROBLEMS WITH WEIGHTED PREFERENCES.
DOI: 10.5220/0003144200450053
In Proceedings of the 3rd International Conference on Agents and Artificial Intelligence (ICAART-2011), pages 45-53
ISBN: 978-989-8425-41-6
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

timal according to these notions. While expressivity
increases by adopting weighted preferences, we show
that in most cases the desired solutions can be found
by adapting existing algorithms for the classical sta-
ble marriage problem.
Stable marriage problems with weighted prefer-
ences have been studied also in (Gusfield, 1987; Irv-
ing et al., 1987). However, they solve these problems
by looking at the stable marriages that maximize the
sum of the weights of the married pairs, where the
weights depend on the specific criteria used to find an
optimal solution, that can be minimum regret crite-
rion (Gusfield, 1987), the egalitarian criterion (Irving
et al., 1987) or the Lex criteria (Irving et al., 1987).
Therefore, they consider as stable the same marriages
that are stable when we don’t consider the weights.
We instead use the weights to define new notions of
stability that may lead to stable marriages that are dif-
ferent from the classical case. They may rely on the
difference of weights that a person gives to two dif-
ferent people of the other sex, or by the strength of
the link of the pairs (man,woman), i.e., how much a
person of the pair wants to be married with the other
person of the pair. The classical definition of stabil-
ity for stable marriage problems with weighted pref-
erences has been considered also in (Bistarelli et al.,
2008) that has used a semiring-based soft constraint
approach (Bistarelli et al., 1997) to model and solve
these problems.
The paper is organized as follows. In Section 2
we give the basic notions of classical stable marriage
problems, stable marriage problems with partially or-
dered preferences and stable marriage problems with
weighted preferences (SMWs). In Section 3 we in-
troduce a new notion of stability, called α-stability
for SMWs, which depends on the difference of scores
that every person gives to two different people of the
other sex, and we compare it with the classical notion
of stability. Moreover, we give a new notion of op-
timality, called lex-optimality, to discriminate among
the new stable marriages, which depends on a vot-
ing rule. We show that there is a unique optimal sta-
ble marriage and we give an algorithm to find it. In
Section 4 we introduce other notions of stability for
SMWs that are based on the strength of the link of the
pairs (man,woman), we compare them with the clas-
sical stability notion, and we show how to find mar-
riages that are stable according to these notions with
the highest global link. In Section 5 we summarize
the results contained in this paper, and we give some
hints for future work.
A preliminary version of this paper has been pre-
sented in (Pini et al., 2010b).
2 BACKGROUND
We now give some basic notions on classical stable
marriage problems, stable marriage problems with
partial orders, and stable marriage problems with
weighted preferences.
2.1 Stable Marriage Problems
A stable marriage problem (SM) (Gusfield and Irv-
ing, 1989) of size n is the problem of finding a stable
marriage between n men and n women. Such men
and women each have a preference ordering over the
members of the other sex. A marriage is a one-to-one
correspondence between men and women. Given a
marriage M, a man m, and a woman w, the pair (m, w)
is a blocking pair for M if m prefers w to his partner in
M and w prefers m to her partner in M. A marriage is
said to be stable if it does not contain blocking pairs.
The sequence of all preference orderings of men
and women is usually called a profile. In the case of
classical stable marriage problem (SM), a profile is a
sequence of strict total orders.
Given a SM P, there may be many stable mar-
riages for P. However, it is interesting to know that
there is always at least one stable marriage.
Given an SM P, a feasible partner for a man m
(resp., a woman w) is a woman w (resp., a man m)
such that there is a stable marriage for P where m and
w are married.
The set of all stable marriages for an SM forms
a lattice, where a stable marriage M
1
dominates an-
other stable marriage M
2
if men are happier (that is,
are married to more or equally preferred women) in
M
1
w.r.t. M
2
. The top of this lattice is the stable mar-
riage where men are most satisfied, and it is usually
called the male-optimal stable marriage. Conversely,
the bottom is the stable marriage where men’s prefer-
ences are least satisfied (and women are happiest, so it
is usually called the female-optimal stable marriage).
Thus, a stable marriage is male-optimal iff every man
is paired with his highest ranked feasible partner.
The Gale-Shapley (GS) algorithm (Gale and
Shapley, 1962) is a well-known algorithm to solve the
SM problem. At the start of the algorithm, each per-
son is free and becomes engaged during the execu-
tion of the algorithm. Once a woman is engaged, she
never becomes free again (although to whom she is
engaged may change), but men can alternate between
being free and being engaged. The following step is
iterated until all men are engaged: choose a free man
m, and let m propose to the most preferred woman
w on his preference list, such that w has not already
rejected m. If w is free, then w and m become en-
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
46

gaged. If w is engaged to man m’, then she rejects
the man (m or m’) that she least prefers, and becomes,
or remains, engaged to the other man. The rejected
man becomes, or remains, free. When all men are en-
gaged, the engaged pairs form the male optimal stable
matching. It is female optimal, of course, if the roles
of male and female participants in the algorithm were
interchanged.
This algorithm needs a number of steps that, in
the worst case, is quadratic in n (that is, the number
of men), and it guarantees that, if the number of men
and women coincide, and all participants express a
strict order over all the members of the other group,
everyone gets married, and the returned matching is
stable.
Example 1. Assume n = 2. Let {w
1
, w
2
} and
{m
1
, m
2
} be respectively the set of women and men.
The following sequence of strict total orders defines a
profile:
• m
1
: w
1
> w
2
(i.e., man m
1
prefers woman w
1
to
woman w
2
),
• m
2
: w
1
> w
2
,
• w
1
: m
2
> m
1
,
• w
2
: m
1
> m
2
.
For this profile, the male-optimal solution is
{(m
1
, w
2
), (m
2
, w
1
)}. For this specific profile the
female-optimal stable marriage coincides with the
male-optimal one. 2
2.2 Stable Marriage Problems with
Partially Ordered Preferences
In SMs, each preference ordering is a strict total or-
der over the members of the other sex. More gen-
eral notions of SMs allow preference orderings to
be partial (Manlove, 2002; Irving, 1994; Halldors-
son et al., 2003; Gelain et al., 2010b; Gelain et al.,
2010a). This allows for the modelling of both indif-
ference (via ties) and incomparability (via absence of
ordering) between members of the other sex.
In this context, a stable marriage problem is de-
fined by a sequence of 2n partial orders, n over the
men and n over the women. We will denote with SMP
a stable marriage problem with such partially ordered
preferences.
Given an SMP, we will sometimes use the notion
of a linearization of such a problem, which is ob-
tained by linearizing the preference orderings of the
profile in a way that is compatible with the given par-
tial orders.
A marriage M for an SMP is said to be weakly-
stable if it does not contain blocking pairs. Given a
man m and a woman w, the pair (m, w) is a blocking
pair if m and w are not married to each other in M and
each one strictly prefers the other to his/her current
partner.
A weakly stable marriage M dominates a weakly
stable marriage M
0
iff for every man m, M(m) ≥
M
0
(m) or M(m)  M
0
(m) ( means incomparable)
and there is a man m
0
s.t. M(m
0
) > M
0
(m
0
). Notice
that there may be more than one undominated weakly
stable marriage for an SMP.
Example 2. Let {w
1
, w
2
} and {m
1
, m
2
} be respec-
tively the set of women and men.
An instance of an SMP is the following:
• m
1
: w
1
> w
2
,
• m
2
: w
1
> w
2
,
• w
1
: m
1
 m
2
,
• w
2
: m
1
> m
2
.
For this instance, both M
1
= {(m
1
, w
2
), (m
2
, w
1
)} and
M
2
= {(m
1
, w
1
), (m
2
, w
2
)} are weakly stable mar-
riages and M
1
dominates M
2
. 2
2.3 Stable Marriage Problems with
weighted Preferences
In classical stable marriage problems, men and
women express only qualitative preferences over the
members of the other sex. For every pair of women
(resp., men), every man (resp., woman) states only
that he (resp., she) prefers a woman (resp., a man)
more than another one. However, he (resp., she) can-
not express how much he (resp., she) prefers such a
woman (resp., a man). This is nonetheless possible in
stable marriage problems with weighted preferences.
A stable marriage problem with weighted pref-
erences (SMW) (Irving et al., 1987) is a classical
SM where every man/woman gives also a numeri-
cal preference value for every member of the other
sex, that represents how much he/she prefers such a
person. Such preference values are natural numbers
and higher preference values denote a more preferred
item. Given a man m and a woman w, the preference
value for man m (resp., woman w) of woman w (resp.,
man m) will be denoted by p(m, w) (resp., p(w, m)).
Example 3. Let {w
1
, w
2
} and {m
1
, m
2
} be respec-
tively the set of women and men.
An instance of an SMW is the following:
• m
1
: w
[9]
1
> w
[1]
2
(i.e., man m
1
prefers woman w
1
to
woman w
2
, and he prefers w
1
with value 9 and w
2
with value 1),
• m
2
: w
[3]
1
> w
[2]
2
,
• w
1
: m
[2]
2
> m
[1]
1
,
• w
2
: m
[3]
1
> m
[1]
2
.
STABILITY IN MATCHING PROBLEMS WITH WEIGHTED PREFERENCES
47

The numbers written into the round brackets identify
the preference values. 2
In (Irving et al., 1987) they consider stable mar-
riage problems with weighted preferences by looking
at the stable marriage that maximizes the sum of the
preference values. Therefore, they use the classical
definition of stability and they use preference values
only when they have to look for the optimal solution.
We want, instead, to use preference values also to de-
fine new notions of stability and optimality.
We will introduce new notions of stability and op-
timality that are based on the weighted preferences
expressed by the agents and we will show how to
find them by adapting the classical Gale-Shapley al-
gorithm (Gale and Shapley, 1962) for SMs described
in Section 2.
3 α-STABILITY
A simple generalization of the classical notion of sta-
bility requires that there are not two people that prefer
with at least degree α (where α is a natural number)
to be married to each other rather than to their current
partners.
Definition 1 (α-stability). Let us consider a natural
number α with α ≥ 1. Given a marriage M, a man m,
and a woman w, the pair (m,w) is an α-blocking pair
for M if the following conditions hold:
• m prefers w to his partner in M, say w
0
, by at least
α (i.e., p(m, w) − p(m, w
0
) ≥ α),
• w prefers m to her partner in M, say m
0
, by at least
α (i.e., p(w, m) − p(w, m
0
) ≥ α).
A marriage is α-stable if it does not contain α-
blocking pairs. A man m (resp., woman w) is α-
feasible for woman w (resp., man m) if m is married
with w in some α-stable marriage.
3.1 Relations with Classical Stability
Notions
Given an SMW P, let us denote with c(P), the classi-
cal SM problem obtained from P by considering only
the preference orderings induced by the preference
values of P.
Example 4. Let us consider the SMW, P, shown in
Example 3. The stable marriage problem c(P) is
shown in Example 1. 2.
If α is equal to 1, then the α-stable marriages of
P coincide with the stable marriages of c(P). How-
ever, in general, α-stability allows us to have more
marriages that are stable according to this definition,
since we have a more relaxed notion of blocking pair.
In fact, a pair (m, w) is an α-blocking if both m and
w prefer each other to their current partner by at least
α and thus pairs (m
0
, w
0
) where m
0
and w
0
prefer each
other to their current partner of less than α are not
considered α-blocking pairs.
The fact that α-stability leads to a larger number
of stable marriages w.r.t. the classical case is impor-
tant to allow new stable marriages where some men,
for example the most popular ones, may be married
with partners better than all the feasible ones accord-
ing to the classical notion of stability.
Given an SMW P, let us denote with I
α
(P) the set
of the α-stable marriages of P and with I(c(P)) the set
of the stable marriages of c(P). We have the following
results.
Proposition 1. Given an SMW P, and a natural num-
ber α with α ≥ 1,
• if α = 1, I
α
(P) = I(c(P));
• if α > 1, I
α
(P) ⊇ I(c(P)).
Given an SMP P, the set of α-stable marriages of
P contains the set of stable marriages of c(P), since
the α-blocking pairs of P are a subset of the blocking
pairs of c(P).
Let us denote with α(P) the stable marriage with
incomparable pairs obtained from an SMW P by set-
ting as incomparable every pair of people that don’t
differ for at least α, and with I
w
(α(P)) the set of the
weakly stable marriages of α(P). It is possible to
show that the set of the weakly stable marriages of
α(P) coincides with the set of the α-stable marriages
of P.
Theorem 1. Given an SMW P, I
α
(P) = I
w
(α(P)).
Proof. We first show that I
α
(P) ⊆ I
w
(α(P)). Assume
that a marriage M 6∈ I
w
(α(P)), we now show that
M 6∈ I
α
(P). If M 6∈ I
w
(α(P)), then there is a pair
(man,woman), say (m, w), in α(P) such that m prefers
w to his partner in M, say w
0
, and w prefers m to
her partner in M, say m
0
. By definition of α(P),
this means that m prefers w to w
0
by at least degree
α and w prefers m to m
0
by at least degree α in
P, and so M 6∈ I
α
(P). Similarly, we can show that
I
α
(P) ⊇ I
w
(α(P)). In fact, if M 6∈ I
α
(P), then there
is a pair (man,woman), say (m, w), in P such that m
prefers w to w
0
by at least degree α and w prefers m
to m
0
by at least degree α. By definition of α(P), this
means that m prefers w to w
0
and w prefers m to m
0
in α(P) and so M 6∈ I
w
(α(P)), i.e., M is not a weakly
stable marriage for α(P). 2
This means that, given an SMW P, every algo-
rithm that is able to find a weakly stable marriage for
α(P) provides an α-stable marriage for P.
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
48

Example 5. Assume that α is 2. Let us consider the
following instance of an SMW, say P.
• m
1
: w
[3]
1
> w
[2]
2
• m
2
: w
[4]
1
> w
[2]
2
,
• w
1
: m
[8]
1
> m
[5]
2
,
• w
2
: m
[3]
1
> m
[1]
2
.
The SMP α(P) is the following:
• m
1
: w
1
 w
2
,
• m
2
: w
1
> w
2
,
• w
1
: m
1
> m
2
,
• w
2
: m
1
> m
2
.
The set of the α-stable marriages of P, that coincides
with the set of the weakly stable marriages of α(P), by
Theorem 1, contains the following marriages: M
1
=
{(m
1
, w
1
), (m
2
, w
2
)} and M
2
= {(m
1
, w
2
), (m
2
, w
1
)}.
2
On the other hand, not all stable marriage prob-
lems with partially ordered preferences can be ex-
pressed as stable marriage problems with weighted
preferences such that the stable marriages in the two
problems coincide. More precisely, given any SMP
problem P, we would like to be able to generate a
corresponding SMW problem P
0
and a value α such
that, in P
0
, the weights of elements ordered in P differ
more than α, while those of elements that are incom-
parable in P differ less than α. Consider for example
the case of a partial order over six elements, defined
as follows: x
1
> x
2
> x
3
> x
4
> x
5
and x
1
> y > x
5
.
Then there is no way to choose a value α and a lin-
earization of the partial order such that the weights of
x
i
and x
j
differ for at least α, for any i,j between 1 and
5, while at the same time the weight of y and each of
the x
i
’s differ for less than α.
3.2 Dominance and
Lex-male-optimality
We recall that in SMPs a weakly-stable marriage
dominates another weakly-stable marriage if men are
happier (or equally happy) and there is at least a man
that is strictly happier. The same holds for α-stable
marriages. As in SMPs there may be more than one
undominated weakly-stable marriage, in SMWs there
may be more than one undominated α-stable mar-
riage.
Definition 2 (dominance). Given two α-stable mar-
riages, say M and M
0
, M dominates M
0
if every man
is married in M to more or equally preferred woman
than in M
0
and there is at least one man in M married
to a more preferred woman than in M
0
.
Example 6. Let us consider the SMW shown in
Example 5. We recall that α is 2 and that
the α-stable marriages of this problem are M
1
=
{(m
1
, w
1
), (m
2
, w
2
)} and M
2
= {(m
1
, w
2
), (m
2
, w
1
)}.
It is possible to see that:
• M
2
does not dominate M
1
since, for m
1
,
M
1
(m
1
) > M
2
(m
1
) and
• M
1
does not dominate M
2
since, for m
2
,
M
2
(m
2
) > M
1
(m
2
). 2
We now discriminate among the α-stable marriages of
an SMW, by considering the preference values given
by women and men to order pairs that differ for less
than α.
We will consider a marriage optimal when the
most popular men are as happy as possible and they
are married with their most popular best α-feasible
women.
To compute a strict ordering on the men where the
most popular men (resp., the most popular women)
are ranked first, we follow a reasoning similar to the
one considered in (Pini et al., 2009; Pini et al., 2010a),
that is, we apply a voting rule (Arrow et al., 2002)
to the preferences given by the women (resp., by the
men). More precisely, such a voting rule takes in in-
put the preference values given by the women over the
men (resp., given by the men over the women) and re-
turns a strict total order over the men (resp., women).
Definition 3 (lex-male-optimal). Consider an SMW
P, a natural number α, and a voting rule r. Let us de-
note with o
m
(resp., o
w
) the strict total order over the
men (resp., over the women) computed by applying r
to the preference values that the women give to the
men (resp., the men give to the women). An α-stable
marriage M is lex-male-optimal w.r.t. o
m
and o
w
, if,
for every other α-stable marriage M
0
, the following
conditions hold:
• there is a man m
i
such that M(m
i
) 
o
w
M
0
(m
i
),
• for every man m
j
≺
o
m
m
i
, M(m
j
) = M
0
(m
j
).
Proposition 2. Given an SMW P, a strict total order-
ing o
m
(resp., o
w
) over the men (resp., women),
• there is a unique lex-male-optimal α-stable mar-
riage w.r.t. o
m
and o
w
, say L.
• L may be different from the male-optimal stable
marriage of c(P);
• if α(P) has a unique undominated weakly stable
marriage, say L
0
, then L coincides with L
0
, oth-
erwise L is one of the undominated weakly stable
marriages of α(P).
STABILITY IN MATCHING PROBLEMS WITH WEIGHTED PREFERENCES
49

Example 7. Let us consider the SMW, P, shown in
Example 5. We have shown previously that this prob-
lem has two α-weakly stable marriages that are un-
dominated. We now want to discriminate among them
by considering the lex-male-optimality notion. Let
us consider as voting rule the rule that takes in in-
put the preference values given by the women over
the men (resp., by the men over the women) and re-
turns a strict preference ordering over the men (resp.,
women). This preference ordering is induced by the
overall score that each man (resp., woman) receives:
men (women) that receive higher overall scores are
more preferred. The overall score of a man m (resp.,
woman w), say s(m) (resp., s(w)), is computed by
summing all the preference values that the women
give to him (the men give to her). If two candidates
receive the same overall score, we use a tie-breaking
rule to order them. If we apply this voting rule to the
preference values given by the women in P, then we
obtain
• s(m
1
) = 8 + 3 = 11,
• s(m
2
) = 5 + 1 = 6,
and thus the ordering o
m
is such that m
1

o
m
m
2
. If
we apply the same voting rule to the preference values
given by the men in P,
• s(w
1
) = 3 + 4 = 7,
• s(w
2
) = 2 + 2 = 4,
and thus the ordering o
w
is such that w
1

o
w
w
2
. The
lex-male-optimal α-stable marriage w.r.t. o
m
and o
w
is the marriage M
1
= {(m
1
, w
1
), (m
2
, w
2
)}. 2
3.3 Finding the lex-male-optimal
α-stable marriage
It is possible to find optimal α-stable marriages by
adapting the GS-algorithm for classical stable mar-
riage problems (Gale and Shapley, 1962).
Given an SMW P and a natural number α, by The-
orem 1, to find an α-stable marriage it is sufficient to
find a weakly stable marriage of α(P). This can be
done by applying the GS algorithm to any lineariza-
tion of α(P).
Given an SMW P, a natural number α, and two
orderings o
m
and o
w
over men and women computed
by applying a voting rule to P as described in Defini-
tion 3, it is possible to find the α-stable marriage that
is lex-male-optimal w.r.t o
m
and o
w
by applying the
GS algorithm to the linearization of α(P) where we
order incomparable pairs, i.e., the pairs that differ for
less than α in P, in accordance with the orderings o
m
and o
w
.
Algorithm 1: Lex-male-α-stable-GS.
Input: P: an SMW, α: a natural number, r: a
voting rule
Output: µ: a marriage
o
m
← the strict total order over the men
obtained by applying r to the preference values
given by the women over the men
o
w
←: the strict total order over the women
obtained by applying r to the preference values
given by the men over the women
P
0
← the linearization of α(P) obtained by
ordering incomparable pairs of α(P) in
accordance with o
m
and o
w
;
µ ← the marriage obtained by applying the GS
algorithm to P
0
;
return µ
Proposition 3. Given an SMW P, a natural number
α, o
m
(resp., o
w
) an ordering over the men (resp.,
women), algorithm Lex-male-α-stable-GS returns the
lex-male-optimal α-stable marriage of P w.r.t. o
m
and
o
w
.
4 STABILITY NOTIONS
RELYING ON LINKS
Until now we have generalized the classical notion of
stability by considering separately the preferences of
the men and the preferences of the women. We now
intend to define new notions of stability that take into
account simultaneously the preferences of the men
and the women. Such a new notion will depend on
the strength of the link of the married people, i.e.,
how much a man and a woman want to be married
with each other. This is useful to obtain a new notion
of stable marriage, that looks at the happiness of the
pairs (man,woman) rather than at the happiness of the
members of a single sex.
A way to define the strength of the link of two
people is the following.
Definition 4 (link additive-strength). Given a man m
and a woman w, the link additive-strength of the pair
(m, w), denoted by la(m, w), is the value obtained by
summing the preference value that m gives to w and
the preference value that w gives to m, i.e., la(m, w) =
p(m, w) + p(w, m). Given a marriage M, the additive-
link of M, denoted by la(M), is the sum of the links of
all its pairs, i.e.,
∑
{(m,w)∈M}
la(m, w).
Notice that we can use other operators beside the
sum to define the link strength, such as, for example,
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50

the maximum or the product.
We now give a notion of stability that exploit the
definition of the link additive-strength given above.
Definition 5 (link-additive-stability). Given a mar-
riage M, a man m, and a woman w, the pair (m, w)
is a link-additive-blocking pair for M if the following
conditions hold:
• la(m, w) > la(m
0
, w),
• la(m, w) > la(m, w
0
),
where m
0
is the partner of w in M and w
0
is the partner
of m in M. A marriage is link-additive-stable if it does
not contain link-additive-blocking pairs.
Example 8. Let {w
1
, w
2
} and {m
1
, m
2
} be, respec-
tively, the set of women and men. Consider the fol-
lowing instance of an SMW, P:
• m
1
: w
[30]
1
> w
[3]
2
,
• m
2
: w
[4]
1
> w
[3]
2
,
• w
1
: m
[6]
2
> m
[5]
1
,
• w
2
: m
[10]
1
> m
[2]
2
.
In this example there is a unique link-additive-stable
marriage, that is M
1
= {(m
1
, w
1
), (m
2
, w
2
)}, which
has additive-link la(M
1
) = 35 + 5 = 40. Notice
that such a marriage has an additive-link higher
than the male-optimal stable marriage of c(P) that
is M
2
= {(m
1
, w
2
), (m
2
, w
1
)} which has additive-link
la(M
2
) = 13 + 10 = 23. 2
The strength of the link of a pair (man,woman),
and thus the notion of link stability, can be also de-
fined by considering the maximum operator instead
of the sum operator.
Definition 6 (link maximal-strength). Given a man
m and a woman w, the link maximal-strength of the
pair (m, w), denoted by lm(m, w), is the value ob-
tained by taking the maximum between the preference
value that m gives to w and the preference value that
w gives to m, i.e., lm(m, w) = max(p(m, w), p(w, m)).
Given a marriage M, the maximal-link of M, denoted
by lm(M), is the maximum of the links of all its pairs,
i.e., max
{(m,w)∈M}
lm(m, w).
Definition 7 (link-maximal-stability). Given a mar-
riage M, a man m, and a woman w, the pair (m, w)
is a link-maximal-blocking pair for M if the following
conditions hold:
• lm(m, w) > lm(m
0
, w),
• lm(m, w) > lm(m, w
0
),
where m
0
is the partner of w in M and w
0
is the partner
of m in M. A marriage is link-maximal-stable if it
does not contain link-maximal-blocking pairs.
4.1 Relations with other Stability
Notions
Given an SMW P, let us denote with Linka(P) (resp.,
Linkm(P)) the stable marriage problem with ties ob-
tained from P by changing every preference value that
a person x gives to a person y with the value la(x, y)
(resp., lm(x, y)), by changing the preference rankings
accordingly, and by considering only these new pref-
erence rankings.
Let us denote with I
la
(P) (resp., I
lm
(P)) the
set of the link-additive-stable marriages (resp.,
link-maximal-stable marriages) of P and with
I
w
(Linka(P)) (resp., I
w
(Linkm(P))) the set of
the weakly stable marriages of Linka(P) (resp.,
Linkm(P)). It is possible to show that these two sets
coincide.
Theorem 2. Given an SMW P, I
la
(P) = I
w
(Linka(P))
and I
lm
(P) = I
w
(Linkm(P)).
Proof. Let us consider a marriage M. We first
show that if M ∈ I
w
(Linka(P)) then M ∈ I
la
(P). If
M 6∈ I
la
(P), there is a pair (m, w) that is a link-
additive-blocking pair, i.e., la(m, w) > la(m, w
0
) and
la(m, w) > la(m
0
, w), where w
0
(resp., m
0
) is the part-
ner of m (resp., w) in M. Since la(m, w) > la(m, w
0
),
m prefers w to w
0
in the problem Linka(P), and, since
la(m, w) > la(m
0
, w), w prefers m to m
0
in the prob-
lem Linka(P). Hence (m, w) is a blocking pair for the
problem Linka(P). Therefore, M 6∈ I
w
(Linka(P)).
We now show that if M ∈ I
la
(P) then M ∈
I
w
(Linka(P)). If M 6∈ I
w
(Linka(P)), there is a pair
(m, w) that is a blocking pair for I
w
(Linka(P)), i.e.,
m prefers w to w
0
in the problem Linka(P), and w
prefers m to m
0
in the problem Linka(P). By defi-
nition of the problem Linka(P), la(m, w) > la(m, w
0
)
and la(m, w) > la(m
0
, w). Therefore, (m, w) is a link-
additive-blocking pair for the problem P. Hence,
M 6∈ I
la
(P).
It is possible to show similarly that
I
lm
(P) = I
w
(Linkm(P)). 2
When no preference ordering changes in Linka(P)
(resp., Linkm(P)) w.r.t. P, then the link-additive-
stable (resp., link-maximal-stable) marriages of P co-
incide with the stable marriages of c(P).
Proposition 4. Given an SMW P,
if Linka(P) = c(P) (Linkm(P) = c(P)) , then
I
la
(P) = I(c(P)) (resp., I
lm
(P) = I(c(P))).
STABILITY IN MATCHING PROBLEMS WITH WEIGHTED PREFERENCES
51

If there are no ties in Linka(P) (resp., Linkm(P)),
then there is a unique link-additive-stable marriage
(resp., link-maximal-stable marriage) with the high-
est link.
Proposition 5. Given an SMW P, if Linka(P) (resp.,
Linkm(P)) has no ties, then there is a unique link-
additive-stable (resp., link-maximal-stable) marriage
with the highest link.
If we consider the definition of link-maximal-
stability, it is possible to define a class of SMWs
where there is a unique link-maximal-stable marriage
with the highest link.
Proposition 6. In an SMW P where the prefer-
ence values are all different, there is a unique link-
maximal-stable marriage with the highest link.
4.2 Finding Link-additive-stable and
Link-maximal-stable Marriages
with the Highest Link
We now show that for some classes of preferences it
is possible to find optimal link-additive-stable mar-
riages and link-maximal-stable marriages of an SMW
by adapting algorithm GS, which is usually used to
find the male-optimal stable marriage in classical sta-
ble marriage problems.
By Proposition 2, we know that the set of the
link-additive-stable (resp., link-maximal-stable) mar-
riages of an SMW P coincides with the set of the
weakly stable marriages of the SMP Linka(P) (resp.,
Linkm(P)). Therefore, to find a link-additive-stable
(resp., link-maximal-stable) marriage, we can sim-
ply apply algorithm GS to a linearization of Linka(P)
(resp., Linkm(P)).
Algorithm 2: link-additive-stable-GS (resp.,
link-maximal-stable-GS)
Input: P: an SMW
Output: µ: a marriage
P
0
← Linka(P) (resp., Linkm(P));
P
00
← a linearization of P
0
;
µ ← the marriage obtained by applying GS
algorithm to P
00
;
return µ
Proposition 7. Given an SMW P, the marriage re-
turned by algorithm link-additive-stable-GS (link-
maximal-stable-GS) over P, say M, is link-additive-
stable (resp., link-maximal-stable). Moreover, if there
are not ties in Linka(P) (resp., Linkm(P)), M is link-
additive-stable (resp., link-maximal-stable) and it has
the highest link.
When there are no ties in Linka(P) (resp.,
Linkm(P)), the marriage returned by algorithm link-
additive-stable-GS (resp., link-maximal-stable-GS) is
male-optimal w.r.t. the profile with links. Such a mar-
riage may be different from the classical male-optimal
stable marriage of c(P), since it considers the happi-
ness of the men reordered according to their links with
the women, rather than according their single prefer-
ences.
This holds, for example, when we assume to have
an SMW with preference values that are all different
and we consider the notion of link-maximal-stability.
Proposition 8. Given an SMW P where the prefer-
ence values are all different, the marriage returned
by algorithm link-maximal-stable-GS algorithm over
P is link-maximal-stable and it has the highest link.
5 CONCLUSIONS AND FUTURE
WORK
In this paper we have considered stable marriage
problems with weighted preferences, where both men
and women can express a score over the members of
the other sex. In particular, we have introduced new
stability and optimality notions for such problems and
we have compared them with the classical ones for
stable marriage problems with totally or partially or-
dered preferences. Also, we have provided algorithms
to find marriages that are optimal and stable accord-
ing to these new notions by adapting the Gale-Shapley
algorithm.
We have also considered an optimality notion (that
is, lex-male-optimality) that exploits a voting rule to
linearize the partial orders. We intend to study if this
use of voting rules within stable marriage problems
may have other benefits. In particular, we want to
investigate if the procedure defined to find such an
optimality notion inherits the properties of the voting
rule with respect to manipulation: we intend to check
whether, if the voting rule is NP-hard to manipulate,
then also the procedure on SMW that exploits such a
rule is NP-hard to manipulate. This would allow us to
transfer several existing results on manipulation com-
plexity, which have been obtained for voting rules,
to the context of procedures to solve stable marriage
problems with weighted preferences.
ACKNOWLEDGEMENTS
This work has been partially supported by the MIUR
PRIN 20089M932N project “Innovative and multi-
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52

disciplinary approaches for constraint and preference
reasoning”.
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