M . Sotomayor Mathematical Social Sciences 38 1999 55 –70
63
ii q is matched to
hp ,p j. Then p ,q causes an instability in the matching,
2 3
1 1
4
since p is the first choice for q and p prefers q to q .
1 4
1 4
2
iii q is matched to
hp ,p j or hp j. Then p ,q causes an instability in both
2 3
2 3
4 2
cases, since q is the second choice for p and q prefers p to p and prefers p to
2 4
2 4
2 4
have an unfilled position.
Case 3. q is matched to
hp ,p j or hp j. Then p ,q causes an instability in both
1 1
3 3
4 1
cases, since q is the first choice for p and q prefers p to p and prefers p to have an
1 4
1 4
1 4
unfilled position. Hence there is no possible arrangement among the players which forms a stable
matching. j
In the Blair example players’ preferences do not satisfy the condition of being responsive. Since separability implies responsiveness and consequently substitutability,
Example 3 holds for these three kinds of preferences. In Example 3 the core is non-empty and it is disjoint from the set of pairwise-stable
matchings. In fact, the matching described in Case 2i is the only pairwise stable matching and it is not in the core. The core is non-empty because the matching where
h p , p j is matched to hq ,q j and h p , p j is matched to hq ,q j is in the core.
1 2
1 2
3 4
3 4
5. The case with substitutable and non-necessarily strict preferences
5.1. Description of the model This section describes the many-to-many matching model when players’ preferences
are substitutable but not necessarily strict. This is an extension of the many-to-one matching model with substitutable and strict preferences presented Roth and
4
Sotomayor, 1990 . There are two disjoint and finite sets of players, P and Q. P has m elements and Q has
n elements. Let y [ P Q. If A and B are any sets of potential partners for player y we write
A . B to mean y prefers A to B; A B to mean y likes A at least as well as B and
y y
| A
B to mean y is indifferent between A and B. Faced with a set A of possible 5
y
partners, player y can determine which subsets of A he she likes best. We denote the set of all such subsets of A by Ch A. That is:
y
P1: T [ Ch A if and only if T A and T T 9 for all T 9 A.
y y
4
In the many-to-many firm-worker models of Roth 1984 and Blair 1988 the preferences are strict. Furthermore, wages are negotiated and are modeled as a discrete variable. In the model we are describing here
we treat salaries as an implicit part of the job description and we allow non strict preferences. When preferences are strict, our definition of substitutability Definition 4 is equivalent to the definition presented
in Roth 1984 which is a reformulation of the definition proposed in Kelso and Crawford 1982.
64 M
. Sotomayor Mathematical Social Sciences 38 1999 55 –70
Ch also has the following property:
y
P2: If T [ Ch A and T B A, then T [ Ch B .
y y
In particular, if y chooses T from A and j [ A, so the choices of y from T h jj
contains T. The general notion of substitutable preferences is given in Definition 4 below. When
preferences are strict, to say that y has substitutable preferences has the following meaning:
Let F and G be sets of potential partners of y such that F G 5 f. Then: A If y chooses w [ F when G is available, then y still chooses w when G is not
available. Equivalently:
B If y does not choose w [ F when G is not available then y does not choose w when G is available.
A and B can also be stated in the following way: A Let F 9 F. If F 9 Ch F G then F 9 Ch F .
y y
B Let F 9 F and S 5 Ch F . If F 9 5 F 2 S then Ch F G F 9 5 f.
y y
A natural extension of A and B to the case with non-strict preferences is: A9 Let F 9 F and S9 [ Ch F G. If F 9 S9 then F 9 S for some S [ Ch F .
y y
B9 Let F 9 F and S [ Ch F . If F 9 5 F 2 S then S9 F 9 5 f, for some S9 [
y
Ch F G.
y
A and B are equivalent under strict preferences. However, A9 is not necessarily equivalent to B9 under non-strict preferences. Then A9 and B9 are necessary to
characterize the substitutability of y’s preference. That is:
Definition 4. Player y [ P Q has substitutable preferences if i for all S9 [ Ch F
y
G there is some S [ Ch F such that S9 F S and ii for all S [ Ch F there is
y y
some S9 [ Ch F G such that S9 F S.
y
When preferences are strict, conditions i and ii are equivalent to require that if Ch F G 5 S9 then S9 F Ch F , so Definition 4 is equivalent to Definition 6.2 of
y y
Roth and Sotomayor 1990. This is the sense in which y regards the players in S9 F more as substitutes than complements: y continues to want to work with S9 F even if
some of the other possible partners become unavailable. The proposition below extends Proposition 2.3 of Blair, 1988 to the case where
preferences are not necessarily strict.
Proposition 1. Suppose the preferences are substitutable . Let F and G be sets of
possible partners for player y [ P Q. Let S [ Ch F. If T [ Ch S G then T [
y y
Ch F G.
y
Proof. By substitutability [Definition 4ii] there is some S9 [ Ch F G such that
y
S9 F S. Since S9 G G we may write that S9 S G F G, where the second inclusion follows from the assumption that S [ Ch F . By P2, S9 [ Ch S G. Now
y y
| let T [ Ch S G. Then S9
T and so T [ Ch F G. j
5
y y
y
M . Sotomayor Mathematical Social Sciences 38 1999 55 –70
65
Definition 5. The matching x is feasible if for every player y [ P Q we have that
5
C y,x [ Ch C y,x .
y
Note that if the preferences are separable and player y has quota q, then Ch C y,x
y
contains either the set of the q most preferred acceptable partners in C y,x or all the acceptable players in C y,x, whichever set has the smaller size. Hence, Definitions 1
and 5 are equivalent when preferences are separable.
Definition 6. The matching x is pairwise-stable if it is feasible and there is no pair i, j with x 5 0 such that if T [ Ch Ci,x
h jj and S [ Ch C j,x hij then T . Ci,x
ij i
j i
and S . C j,x.
j
If there is such a pair i, j , P1, P2 and the feasibility of x imply that j [ T and i [ S. In fact, by P1, T is a subset of Ci,x
h jj; if j [
⁄ T then T Ci,x; P2 applied to
| T Ci,x Ci,x
h jj implies that T [ Ch Ci,x; by the feasibility of x, T Ci,x,
5
y y
which contradicts that T . Ci,x. Hence j [ T. Analogously we show that i [ S. We say
i
that i, j causes an instability in x. When preferences are separable, one verifies that Definitions 3 and 6 are equivalent.
5.2. Existence of pairwise-stable matching For every j [ Q and every feasible matching x set:
A j,x ; hi [ P; i, j causes an instabilityin xj.
Remark 1. If A j,x±f, then for all f± S A j,x and for all T[ Ch C j,x S we
j
| must have that T. C
j,x. In fact, if not we would have that T C
j,x, by P . Now let 5
j j
1
i [ S. Then using that C j,x C j,x
hij C j,x S and applying P , we get that
2
C j,x[ Ch C j,x
hij, which contradicts the fact that i,j causes an instability at x.
j
Set: E
; hx feasible; if A j,x ± f for some j [ Q, then for all set S with f ± S A j,x, there is some B S such that C j,x B [ Ch C j,x S
j.
j
Remark 2. It follows from Remark 1 that the set B in the definition of E is non-empty.
Thus, if a feasible matching x is in E and i, j causes an instability for all i [S, then j will be better off by forming new partnerships with some elements of S, without
dissolving any of her old partnerships. Observe that if x is pairwise-stable then x is in E. When preferences are separable then E is the set of feasible matchings x such that if
x is unstable and j causes an instability with some firm i
, then j has at least one unfilled position and p. i for all p
[ C j,x.
j
Lemma 1. Let x be a feasible matching . Consider j[ Q. Suppose that for all i[B P
5
If x is feasible then x is individually rational, but the converse is not true. In fact, a matching x may not be feasible without being individually irrational, since it might still be that C y,x. f.
y
66 M
. Sotomayor Mathematical Social Sciences 38 1999 55 –70
we have that x 5 0 and i,j does not cause an instability in x. Furthermore, for all i[B
ij
and all T [ Ch Ci,x j we have that T . Ci,x. Then C j,x[ Ch C j,xB.
i i
i i
j
Proof. It follows from Definition 6 that C j,x[Ch C j,x
hij, for all i [B. If uBu51
j
we are done. If not take hi ,i jB. By Proposition 1, making F 5C j,xhi j, S5
1 2
1
C j,x, G 5 hi j and T5S then C j,x[Ch C j,xhi jhi j. By repeated application
2 j
1 2
of Proposition 1 we get that C j,x[Ch C j,xB . j
j
Lemma 2. Let x [ E. Consider j[ Q. Suppose that for all i[B P we have that x 50.
ij
Furthermore , for all i[B and all T [ Ch Ci,x j we have that T . Ci,x. Then
i i
i i
there is some T [ Ch C j,xB such that C j,x T.
j
Proof. Set B
;B B , where no i [B causes an instability with j in x and B A j,x.
1 2
1 2
By Lemma 1 C j,x[Ch C j,xB . If B 5f then we are done. Otherwise, by
j 1
2
definition of E there is some T [Ch C j,xB such that C j,xT. We claim that
j 2
T [Ch C j,xB B . But this is immediate from Proposition 1 applied to F 5
j 1
2
C j,xB , G 5B and S 5C j,x. j
1 2
It is clear that the set E is non-empty, since it contains the matching where every one is unmatched. Since E is finite and the preferences are transitive, there is a matching x
there might be more than one which is maximal for the players in P among all matchings in E. That is, if x [E and some i [P prefers x to x, then there is some i9[P
6
such that i9 prefers x to x.
Theorem. The matching x is pairwise-stable.
Proof. Suppose, by contradiction, that x is unstable via some pair i, j [P 3Q. Then A j,x±f. By definition of E and Remark 2, by making S 5 A j,x, there is some
f ±B A j,x such that C j,xB [Ch C j,x A j,x. Now construct a new
j
matching x9 as follows: match j with C j,xB and if i [B choose C [Ch Ci,x
i i
h jj and match i to C . [There might be some dissolved partnerships i,k where i [B and
i
k [Ci,x]. It follows from Definition 6 that j [C . Ci,x for all i [B. Now keep the
i i
partnerships of the remaining players. Therefore, x9 is weakly preferred by all players in P and strictly preferred by all players in B. We are going to show that x9[E which
contradicts the definition of x. First observe that j cannot cause an instability with any player p in P. In fact, it suffices to verify the instabilities caused by pairs p, j , with
p [ A j,x. Then let p [ A j,x. But it is immediate from P applied to C j,xB
2
C j,xB h pjC j,x A j,x that C j,xB [Ch C j,xB h pj. Thus,
j
C j,x9[Ch C j,x9 h pj and p, j does not cause an instability. Now we only need
j
to check that in all dissolved partnerships i,k, with Ak,x9±f, if f ±B9Ak,x9
then there is some S [Ch Ck,x9B9 such that Ck,x9S. Then let k belong to
k
some dissolved partnership, with Ak,x9±f, and let f ±B9 Ak,x9. First observe that
9
if x 50 and x 51 then i [ ⁄ B9. In fact, if i,k was dissolved then
ik ik
6
When preferences are separable we can choose the matching x which maximizes o a x , for all x[E.
i , j i j
i j
M . Sotomayor Mathematical Social Sciences 38 1999 55 –70
67
Ci,x9 [ Ch Ci,x h jj
1
i
by construction of x9. If S [Ch Ci,x9
hkj then S Ci,x9 and S Ci,x9hkj
i i
i i
i
Ci,x h jj, where the last inclusion follows from assertion 1. So S Ci,xh jj,
i
|
and so Ci,x9 S by assertion 1. Hence, S Ci,x9. Therefore, i,k does not cause
5
i i
i i
an instability at x9 and so i [ ⁄ B9.
Hence, x 50 for all i [B9. Therefore, we only need to establish that for all i [B9
ik
and all T [Ch Ci,x hkj we have that T . Ci,x. If this is done, then Lemma 2
i i
i i
applied to B9 and k implies that there exists some T [Ch Ck,xB9 such that
k
Ck,xT. Since Ck,xB95[Ck,x9B9][Ck,x2Ck,x9] we can use sub- stitutability property [Definition 4i], making F 5Ck,x9B9, S95T, G 5Ck,x2
Ck,x9, to get that there is some S [Ch Ck,x9B9 such that T [Ck,x9B9]
k
S. But Ck,x9Ck,xT, so Ck,x9T. Furthermore, Ck,x9Ck,x9B9, so Ck,x9T [Ck,x9B9]S and so Ck,x9S, and we have the desired result. Then
suppose by contradiction that there exists some i [B9 and some T [Ch Ci,x hkj
i i
| such that T
Ci,x. Then 5
i i
Ci,x [ Ch Ci,x hkj,
2
i
by the definition of Ch . Assertion 2 and the fact that i [ Ak,x9 imply that Ci,x±
i
Ci,x9, so Ci,x9[Ch Ci,x h jj by construction of x9. Using assertion 2 and
i
making F 5Ci,x hkj, G5h jj and S5Ci,x, we can apply substitutability property
[Definition 4ii] to get that there is some S9[Ch Ci,x
hkjh jj such that S9
i
[Ci,x hkj]Ci,x. Hence, x 50 implies k[
⁄ S9. Thus, S9Ci,x
h jj
ik
Ci,x
h jjhkj. P can be applied yielding that S9[Ch Ci,xh jj. But Ci,x9[
2 i
| Ch Ci,x
h jj, so S9 Ci,x9 and hence Ci,x9[Ch Ci,x
hkjh jj. Also 5
i i
i
Ci,x9[Ci,x9 hkj][Ci,xh jjhkj]. Then P can be used and it implies that
2
Ci,x9[Ch Ci,x9 hkj, which contradicts the fact that i [ Ak,x9.
i
Therefore, the matching x9 is in E, which contradicts the definition of x. Hence, x is pairwise-stable and the proof is complete.
j
6. Final conclusions
