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. 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
This section shows that the concept of pairwise-stability and strong core are not equivalent to the concept of setwise-stability in the many-to-many matching game,
although they are equivalent in the one-to-one and many-to-one cases. Instead, setwise- stability is strictly stronger than pairwise-stability plus core. Example 1 shows that there
may be pairwise-stable matchings which are not in the core, so they are not setwise- stable. Example 2 presents a situation in which the set of setwise-stable matchings is a
proper subset of the intersection of the core with the set of pairwise-stable matchings.
Example 1. A pairwise stable matching which is not in the core. Let P 5
h p , . . . , p j,
1 4
Q 5 hq , . . . ,q j, r 5 s 5 2 for all i, j 5 1, . . . ,4. The pairs of numbers a ,b are given
1 4
i j
ij ij
in Table 1.
Table 1 Number pairs a , b for Example 1 the payoffs of each matched pair are shown in bold
ij ij
Q P
10,1 1,10
4,10 2,10
1,10 10,1
4,4 2,4
10,4 4,4
2,2 1,2
10,2 4,2
2,1 1,1
M . Sotomayor Mathematical Social Sciences 38 1999 55 –70
61
Consider the matching x where p and p are matched to hq ,q j and p and p are
1 2
3 4
3 4
matched to hq ,q j. The payoffs of each matched pair are bold faced in Table 1 above.
1 2
This matching is pairwise stable. In fact, p and p do not belong to any pair which
3 4
causes an instability, because they are matched to their two best choices: q and q ;
1 2
p ,q and p ,q do not cause instabilities since p is the worst choice for q and q is
1 1
1 2
1 1
2
the worst choice for p ; p ,q and p ,q do not cause instabilities since q is the
1 2
1 2
2 1
worst choice of p and p is the worst choice of q . Nevertheless p and p prefer hq ,q j
2 2
2 1
2 1
2
to hq ,q j and q and q prefer h p , p j to h p , p j. Hence this matching is not in the core,
3 4
1 2
1 2
3 4
since it is blocked by h p , p ,q ,q j. j
1 2
1 2
Example 2. A strong corewise-stable matching which is pairwise-stable and is not
setwise-stable . Consider P 5
h p , . . . , p j, Q 5 hq , . . . ,q j, r 5 3, r 5 r 5 2, r 5
1 6
1 7
1 2
5 3
r 5 r 5 1, s 5 s 5 s 5 2 and s 5 s 5 s 5 s 5 1. The pairs of numbers a ,b are
4 6
1 2
4 3
5 6
7 ij
ij
given in Table 2 below. Consider the matching x at which p is matched to
hq ,q ,q j; p is matched to
1 2
3 7
2
hq ,q j; p and p are matched to q ; p is matched to hq ,q j and p is matched to q .
5 6
3 4
1 5
2 4
6 4
This is the matching x of Section 1. The payoffs of each matched pair are bold faced
3
in Table 2 above. This matching is strong corewise-stable. In fact, if there is a matching y which weakly dominates x via some coalition A, then, under y, no player in
A is worse off and at least one player in A is better off. Furthermore, matching y must match all players in A among themselves. By inspection we can see that the only players
that can be better off are p , p , q and q , for all remaining players are matched to their
1 2
1 4
best choices. However, if A contains one player of the set h p , p ,q ,q j then A must
1 2
1 4
contain all four players. In fact, if p [ A then p must form a new partnership with q ,
1 1
1
so q must be in A. If q [ A then q must form a new partnership with p , so p must be
1 1
1 2
2
in A. If p [ A then p must form a new partnership with q , so q must be in A. Finally
2 2
4 4
if q [ A then q must form a new partnership with p , so p must be in A. Thus, if y
4 4
1 1
weakly dominates x via A, then p , p , q and q are in A and p and p form new
1 2
1 4
1 2
partnerships with q and q . Nevertheless, p must keep his partnership with q , his best
1 4
1 2
choice. Then q must be in A, so she cannot be worse off and so p must also be in A.
2 5
But p requires the partnership with q , who has quota of 2 and has already filled her
5 4
quota with p and p . Hence p is worse at y than at x and then y cannot weakly
1 2
5
dominate x via A. The matching x is clearly pairwise-stable. Nevertheless, the coalition
h p , p ,q ,q j causes an instability in x. In fact, if p is matched to hq ,q ,q j
1 2
1 4
1 1
2 4
Table 2 Number pairs a , b for Example 2 the payoffs of each matched pair are shown in bold
ij ij
Q P
13,1 14,10
4,10 1,10
0,0 0,0
3,10 1,10
0,0 0,0
10,1 4,10
2,10 0,0
10,4 0,0
0,0 0,0
0,0 0,0
0,0 10,2
0,0 0,0
0,0 0,0
0,0 0,0
0,0 9,9
0,0 10,4
0,0 0,0
0,0 0,0
0,0 0,0
10,2 0,0
0,0 0,0
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. 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
