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. Sotomayor Mathematical Social Sciences 39 2000 119 –132
3 proves the existence of stable outcomes. Section 4 shows that the core is a complete lattice and that it has a unique optimal stable payoff for each side of the market. Two
illustrative examples are presented at the end of this section.
2. The mathematical model
There are two finite and disjoint sets of players P 5 h1, . . . , i, . . . , mj and Q 5 h1, . . . ,
j, . . . , n j. The players in P and Q will be called firms and workers, respectively. There
are two classes of players: F and R with F R 5 P Q. Set F ; hi, j [ P 3 Q; i [ F
and j [ F j and R ; hi, j [ P 3 Q; i [ R or j [ Rj. Therefore, P 3 Q 5 R F. For
each pair i, j [ P 3 Q there is a pair of numbers a , b , where a 1 b can be
ij ij
ij ij
interpreted as being the productivity of the pair i, j . If the partnership i, j is formed then the productivity a 1 b can be divided between i and j so that i receives the payoff
ij ij
u , j receives the payoff v and u 1v 5a 1b . If the partnership i, j [R then u 5a
i j
i j
ij ij
i ij
and v 5b .
j ij
Definition 1. A matching is a matrix x 5 x of zeros and ones that satisfies
o
ij i [P
x 1 and o
x 1. If x 51 resp. x 50 we say that i and j are matched resp.
ij j [Q
ij ij
ij
unmatched at x. If o
x 5 0 resp. o
x 5 0 we say that j resp. i is unmatched
i [P ij
j [Q ij
at x. If x 51 we can write xi 5j or x j 5i. Given S P we write xS 5 h j [Q; x 51
ij ij
for some i [S j. Analogously we define xS for S Q.
Definition 2. An outcome is a matching x and a pair of vectors u, v called payoff, with
m n
u [R and v [R . An outcome will be denoted by u, v; x.
Definition 3. An outcome u, v; x is feasible if
i u 0, v 0, for all i, j [ P 3 Q and u 5 0 resp. v 5 0 if i resp. j is
i j
i j
unmatched. ii u 1 v 5 a 1 b if x 51, and
i j
ij ij
ij
iii u 5 a and v 5 b if x 5 1 and i, j [ R.
i ij
j ij
ij m
n
A feasible payoff is a pair of vectors u, v with u [ R and v [ R such that for
some matching x, u, v; x is a feasible outcome; in this case we say that x is compatible with u, v.
Given some matching x, we say that i is acceptable to j if x 51 and v 0. Similarly
ij j
j is acceptable to i if x 51 and u 0.
ij i
Definition 4. An outcome u, v; x is stable if it is feasible and for all i, j [ P 3 Q we have that:
i u 1 v a 1 b if i, j [ F , and
i j
ij ij
ii u a or v b if i, j [ R.
i ij
j ij
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123
A payoff vector u, v is stable if u, v; x is stable for some matching x. A blocking pair of a feasible outcome u, v; x resp. a feasible payoff u, v is a pair
i, j [ P 3 Q such that either i, j [ F and u 1 v , a 1 b , or i, j [ R and
i j
ij ij
u , a and v , b ; in any case we also say that i and j are blocking partners. Therefore,
i ij
j ij
a feasible outcome resp. payoff is stable if and only if it does not have any blocking
pair. Observe that the existence of a blocking pair is equivalent to the existence of a pair i,
9 9
9
j and a feasible outcome u9, v9; x9, with x 51, such that u .u and v .v . This is to
ij i
i j
j
say that u, v is stable if it is not dominated by any feasible payoff via some pair i, j and matching x9. Since the pairs i, j are the only essential coalitions, a payoff is stable
1
if and only if it is in the core , which will be denoted by C.
Definition 5. Let A be the set of all i’s blocking partners for the feasible outcome u, v;
i j
j j
x. For each j [ A define a feasible outcome u , v ; x such that:
i j
i x 5 1,
ij j
ii u 5 a if i, j [ R, and
i ij
j
iii u 5 a 1 b 2 v if i, j [ F .
i ij
ij j
j
u means the highest payoff that j would pay to i if they break their current partnership
i
and work together.
j k
We say that j is an i ’s favorite blocking partner if u u for all k [ A .
i i
i
Definition 5 implies that if j is an i’s favorite blocking partner for u, v; x then i
j j
j
cannot form a blocking pair for u , v ; x .
Definition 6. An outcome u, v; x is strongly stable if it is stable and there is no pair i,
9
j [ P 3 Q with x 5 0 and a feasible outcome u9, v9; x9 such that x 51, and
ij ij
9 9
i u . u and v 5 v , or
i i
j j
9 9
ii u 5 u and v . v .
i i
j j
Similarly we define a strongly stable payoff. Definition 6 says that u, v is strongly stable if it is stable and it is not weakly
dominated by any feasible payoff via some coalition i, j . The pair i, j is called a weak blocking pair
. Thus, if i, j [ R and i, j weakly blocks u, v; x then x 50
ij
and: i a .u and b 5v ; or ii a 5 u and b . v .
ij i
ij j
ij i
ij j
1
An outcome u, v; x is in the core if it is not dominated by any other outcome via some coalition.
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. Sotomayor Mathematical Social Sciences 39 2000 119 –132
It follows from Definitions 4 and 6 that if i, j [ F then i, j weakly blocks u, v; x if and only if it blocks u, v; x. Thus, if all players are in F then u, v; x is strongly
stable if and only if it is stable. Since the pairs i, j are the only essential coalitions, the set of strongly stable payoffs
coincides with the core defined by weak domination, which is a subset of the core and will be denoted by C.
The following models are well known special cases of the model we are treating here. 2.1. Marriage model
Here F 5 f. Player i is acceptable to player j if b 0. Analogously, player j is
ij
acceptable to player i if a 0. If a 0 and b 0 we say that i and j are mutually
ij ij
ij
acceptable. The players can list their potential partners, in order, in a finite list of preferences. Let Li denote the list of preferences of player i. Thus, Li 5 h, [ j, k], . . . ,
i, p means that a . a 5 a . 0 . a . That is, i prefers h to j and is indifferent
ih ij
ik ip
between j and k. Agent p is not acceptable to i, so i prefers to be unmatched to be matched to p.
In this model, any outcome u, v; x is completely specified by the matching x. Such a matching is called feasible resp. stable if the corresponding outcome is feasible resp.
stable. Therefore, a matching is feasible if every matched pair is mutually acceptable. Here feasibility means individual rationality. Hence, x is a stable matching if it is
individually rational and there is no pair i, j [ P 3 Q such that u , a and v , b . If
i ij
j ij
we consider that an unmatched agent is self-matched then: The matching x is stable if it is individually rational and there is no pair i,
j [ P 3 Q such that i and j prefer each other to their respective mates. The matching x is strongly stable if there is no pair i, j [ P 3 Q such that
a i prefers j to xi and j is indifferent between i and x j , or b i is indifferent between j and xi and j prefers i to x j .
When preferences are strict, it follows from the definitions above that the set of stable matchings coincides with the set of strongly stable matchings. In this case the resulting
model is the well known Marriage model, introduced in Gale and Shapley 1962.
2.2. The assignment game In this case R 5 f. If we define a 5 max
ha 1 b , 0j then:
ij ij
ij
A feasible outcome u, v; x is stable if and only if u 1 v a for all i, j [ P 3 Q.
i j
ij
In this model the set of stable outcomes coincides with the set of strongly stable outcomes. In fact, if i, j weakly blocks u, v; x then i can transfer some payoff to j or
j can transfer some payoff to i, so that i, j blocks u, v; x. This model is the assignment game introduced in Shapley and Shubik 1972.
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3. Existence of stable outcomes
