M . Sotomayor Mathematical Social Sciences 38 1999 55 –70
59
This paper is organized as follows. Section 2 describes the many-to-many matching model with cardinal separable preferences and defines setwise-stability. Section 3 shows,
through examples, that setwise-stability is strictly stronger than pairwise-stability plus core. Section 4 presents an example which shows that setwise-stable matchings may not
exist. Section 5 redefines substitutability and describes a model in which preferences are substitutable and not necessarily strict. We then present a general proof of the existence
of pairwise stable matchings. Section 6 concludes the paper.
2. Description of the model with separable preferences
There are two disjoint and finite sets of players, P and Q. P has m elements and Q has n elements. Each player from one set is allowed to form partnerships with players from
the opposite set. Each player i [ P may form at most r partnerships and each player
i
j [ Q may form at most s partnerships. Denote by uAu the number of elements of the set
j
A. We say that a set of players A is an allowable set of partners for i [ P resp. j [ Q if uAu r resp. s and A Q resp. P. For each pair i, j there are two numbers a and
i j
ij
b . The preferences of the players i and j over allowable sets of partners are determined
ij
by these numbers. Therefore, say, i prefers j to k if and only if a . a and i is
ij ik
indifferent between j and k if and only if a 5 a . We can interpret a resp. b as
ij ik
ij ij
being the payment i resp. j can get in case i and j form a partnership. Under this interpretation it is natural to require that player i prefers the set S Q to the set S9 Q
if and only if o
a . o
a , where S and S9 are allowable sets of partners for i.
j [S ij
j [S 9 ij
Thus, for example, if player i has quota r 5 2 and a . a 1 a this means that player
i i 1
i 2 i 3
i prefers to form a partnership with player 1 and to have one unfilled position than to have both positions filled with players 2 and 3. For simplicity we will consider that the
reservation utilities of i [ P and j [ Q are 0. For our purposes a player will only compare allowable sets of partners. We say that a set of players A is acceptable to player
i resp. j if it is allowable to i resp. j and
o a 0 resp.
o b 0. If i and j are
j [ A ij
i [ A ij
acceptable to each other we say that they are mutually acceptable.
Definition 1. A feasible matching x is an m 3 n matrix x of 0’s and 1’s, defined for all
ij
pairs i, j [ P 3 Q, such that o
x r , for all i [ P and o
x s for all j [ Q.
q [Q iq i
p [P pj j
2
Furthermore, for all i [ P and j [ Q, if x 5 1 then i and j are mutually acceptable. If
ij
x 5 1 resp. x 5 0 we say that i and j are resp. are not matched at x.
ij ij
Thus, a feasible matching is individually rational. We will denote by Ci,x the set of j [ Q such that x 5 1. Thus, if player i has quota five, the expression Ci,x 5
hq ,q ,q j
ij 1
2 3
denotes that player i forms partnerships with q , q and q
and has two unfilled
1 2
3
positions. Similarly we will denote by C j,x the set of i [ P such that x 5 1. It is clear
ij
that Ci,x r and C j,x s for all i, j [ P 3 Q and all feasible matching x.
i j
Setwise-stability was defined in Section 1. The formal definition is the following:
Definition 2. A matching x is setwise-stable if it is feasible and there are no feasible
2
For stability purposes, Definition 1 could require individual rationality instead of mutual acceptance. The reason is that under stability, if two players form a partnership then they are mutually acceptable.
60 M
. Sotomayor Mathematical Social Sciences 38 1999 55 –70
matching x9 and coalitions R P and S Q, with R ± f and S ± f, such that for all i [ R, and for all j [ S:
9 9
i if x 5 1 then q [ S Ci,x and if x 5 1 then p [ R C j,x;
iq pj
9 9
ii o
a x . o
a x and
o b x .
o b x .
q [Q iq
iq q [Q
iq iq p [P
pj pj
p [P pj pj
That is, every new partner of any player in the coalition R S belongs to this coalition. Every player y in R S may continue to be matched with some old partners
from C y,x. Furthermore, everyone in the coalition strictly prefers the new arrangement given by x9.
If there are such coalitions R and S we say that R S causes an instability in the
matching x via x9. When the essential coalitions are pairs
i,j [ P 3 Q, the coalitions which cause instabilities are restricted to these pairs of players
. In this case the concept of stability is called pairwise-stability. Thus pairwise-stability is the concept of stability for the
Marriage and the College Admission games. Formally we have:
Definition 3. The matching x is pairwise-stable if it is feasible and there is no pair i, j , with x 5 0, such that i and j prefer each other to some of their partners or to an unfilled
ij
position, if any.
3. Stability and counterexamples for the case with separable preferences
