62 M
. Sotomayor Mathematical Social Sciences 38 1999 55 –70
and p is matched to hq ,q j then p gets 28 instead of 21 and the rest of the players in
2 1
4 1
the coalition gets 11 instead of 6. Hence the matching x is not setwise-stable. j
4. Existence of stable matchings for the case with separable preferences: A negative result
This section addresses the problem of the existence of stable matchings. In the negative direction, one-to-one matching markets with one or three sides may have no
stable matching, as in the roommates problem Gale and Shapley, 1962 and the man–woman–child marriage problem Alkan, 1986. The non-existence of stable
matchings in the roommates problem, for example, is due to the existence of odd cycles in the preferences of the players. Hence, since this fact cannot be observed in the
one-to-one matching market with two sides, the Marriage model always has a stable matching. Likewise markets with many-to-one matching and non-substitutable prefer-
ences may have no stable matching Kelso and Crawford, 1982 and Example 2.7 of Roth and Sotomayer, 1990. However, when preferences are substitutable the many-to-one
matching market always has stable matchings Kelso and Crawford, 1982. Example 3 below shows that existence theorems, which hold for two-sided one-to-one and many-to-
one matching markets, may fail to hold in the discrete many-to-many cases. In these cases the substitutability property is not enough to guarantee the existence of stable
matchings. Even the separability condition, which in continuous matching games with transferable utility makes existence of stable outcomes possible for the three kinds of
matchings is not sufficient to preserve the existence of stable outcomes in the many-to- many case with non-transferable utilities. The interested reader can see Sotomayor,
3
1992; Sotomayor, 1998, where a continuous model with separable utilities is discussed.
Example 3. Nonexistence of stable matchings. Consider Example 1 again. We will
show that this example does not have any setwise-stable matching. First, observe that p
3
prefers hq ,q j to any other set of players and p is the second choice for q and q ; q
1 2
3 1
2 3
prefers h p , p j to any other set of players and q is the second choice for p and p .
1 2
3 1
2
Then, in any stable matching x, p must be matched to q and q while q must be
3 1
2 3
matched to p and p . Separate the cases by considering the possibilities for the second
1 2
partner for q , under a supposed stable matching x:
1
Case 1. q is matched to
hp ,p j. Then x 5 0 and we have that p ,q causes an
1 2
3 24
2 4
instability in the matching, since p prefers q to q and p is the second choice for q .
2 4
1 2
4
Case 2. q is matched to
hp ,p j. The following possibilities may occur:
1 3
4
i q
is matched to hp ,p j. Then h p , p ,q ,q j causes an instability in the
2 3
4 1
2 1
2
matching. This matching is pairwise-stable but is not in the core.
3
In that model, for each pair i, j [P 3Q there is a non-negative number a which can be divided between i
ij
and j in any way they want.
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
