E . Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
183
whose vertices all belong to a clique C for some i [ I, and we shall say that D is
i
6 -rejecting if the subsets S J are rejected in D for all a [ A. We will use the same
a
abbreviation, CARS, for a -acyclic and 6-rejecting suborientation of the graph G .
Proposition 21. An EFF 5
kC , S ; I, A; Jl is stable if the graph G has no CARS.
i a
The proof goes along exactly the same lines as for EFFs generated by graphs, in Proposition 18, and we omit it here.
Kernel-solvability also generalizes for an arbitrary effectivity function 5 kC , S ; I,
i a
A; J l as follows.
Proposition 22. A graph G has no CARS if for any linear order of the vertices in the hyperedges C , i [ I, there exists always an a [ A such that the hyperedge S is
i a
dominating, i
.e. for every vertex j [ J\S there is an i [ I such that j [ C , C S ± 5
a i
i a
and there exists a vertex j9 [ C S which is greater than j in the given order of C .
i a
i
The proof goes along the same lines as for Proposition 19 above, and we omit it here. Let us consider an interpretation of the above. Let J denote a set of students organized
into several competing and possibly overlapping teams, denoted by S J, a [ A.
a
Furthermore, let I denote the set of tests, and C J for i [ I, denote the set of students
i
taking the ith test. Then the EFF 5 kC , S ; I, A; Jl is stable iff for every result of
i a
these tests there exists a ‘winning’ team, i.e. a team S such that for every participant
a
j [ ⁄ S there exists a member of the team j9 [ S who is better than j in some tests i [ I.
a a
4. Monotonicity
I A
The family of EFFs :2 3 2 →
h0, 1j admit a natural partial order, defined by 9 iff K, B 9K, B for all K I, and B A, or in other words, iff K,
B 5 1 implies 9K, B 5 1. Thus, for two effectivity functions 5 kK , B ; J; I, Al and
j j
9 9
9 5 kK , B ; J9; I9, A9l, the relation 9 holds iff for every j [ J there exists
j j
9 9
j9 [ J9 such that K K and B B . In particular, an EFF 5 kK , B ; J; I, Al will
j 9 j
j 9 j
j j
remain the same if we remove all the pairs K , B which are not inclusion-minimal.
j j
9 9
Analogously, for EFFs 5 kC , S ; I, A; Jl and 9 5 kC , S ; I9, A9; J9l we have
i a
i a
9
9 iff for every i [ I there exists i9 [ I9 such that C C . and for every a [ A
i 9 i
9
there exists a9 [ A9 such that S S . In particular, the EFF 5
kC , S ; I, A; Jl, will
a 9 a
i a
remain the same if we remove all the players C and outcomes S which are not
i a
inclusion-minimal. It is both obvious and well known that stability is antimonotone.
Lemma 23. If EFF is stable and 9 then EFF 9 is stable
, too. Let us apply this to EFFs generated by graphs. Given a graph G and an induced
subgraph G9 in G, the above definitions imply that . Thus, according to Lemma
G 9 G
184 E
. Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
23, is stable whenever is stable. In other words, if G9 has a CARS then G also
G 9 G
has one, i.e. G9 is kernel-solvable whenever G is kernel-solvable. Analogously, we can define ‘induced subEFFs’ of an arbitrary EFF. Given an EFF
9 9
5 kC , S ; I, A; Jl and a subset J9 J, the subEFF 5 kC , S ; I9, A9; J9l induced
i a
J 9 i
a
by J9 is defined by setting I9 5 hi [ IuC J9 ± 5j, and A9 5 ha [ AuS J9 ± 5j, and
i a
9 9
defining C 5 C J9 for all i [ I9, and S 5 S J9 for all a [ A9. Obviously,
i i
a a
J 9
for any J9 J, thus the following is implied by Lemma 23.
Lemma 24. An induced subEFF is stable whenever the EFF is stable.
J 9
As an application, let us consider an EFF 5 kC , S ; I, A; Jl for which I 5 hi , i j,
i a
1 2
A 5 ha , a j, and J 5 h1, 2j, and where C 5 S 5 h1j, and C 5 S 5 h2j. It is easy to
1 2
1 1
2 2
verify that T 5 hhi j, ha j, ha j; hi j, ha j, ha jj is a rejecting table for , with no
1 1
2 2
2 1
common player cycles, and hence is not stable, by Theorem 16. Together with Lemma 24 this implies Corollary 9 of Theorem 8.
5. Proofs of main theorems and their applications Proof of Theorem 8. Given an EFF 5
