E . Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
177
2. Results
The prime tools in our analysis are combinatorial. We show that an effectivity function can equivalently be represented by a pair of hypergraphs, forming clique covers of
two graphs on the same set of vertices. We shall start with the interesting special case when the two graphs are complementary to each other. We follow standard graph
theoretical notations, some of them, for completeness, included in Appendix A.
Let us consider a graph G 5 kV, El, and let us assign a player to every maximal clique
and an outcome to every maximal stable set of G. Let us denote by I the set of all the players maximal cliques and by A the set of all the outcomes maximal stable sets. For
every vertex v [ V let us denote by K the family of maximal cliques containing vertex
v
v , and similarly, let B denote the family of maximal stable sets containing vertex v.
v
Finally, let us associate to the graph G an effectivity function by defining K,
G G
B 5 1 for a coalition K I and block B A if and only if K K and B B for some
v v
vertex v [ V and defining K, B 5 0 otherwise.
G
This correspondence was introduced and several results were shown by Boros and Gurvich 1994, 1995, see also Boros et al. 1995. In order to state the new results, we
shall recall first some of the properties shown earlier. For the precise definitions see Appendix A and Sections 3–4, and for more information on perfect and on kernel-
solvable graphs see Appendix A and also Berge 1970, 1975, Berge and Duchet 1983, 1987, Blidia et al. 1988 and Maffray 1988.
Proposition 1. A graph G is perfect if and only if the corresponding EFF is
G
balanced. This follows from a characterization of perfect graphs given by Lovasz 1972a,
Theorem 2.
Proposition 2. A graph G is kernel-solvable if and only if the corresponding EFF is
G
stable. This property is in fact a direct consequence of the criterion of stability stated by
Keiding 1985, see Section 3. Scarf 1967 proved that balanced NTU-games have non-empty cores, and later
Danilov and Sotskov 1987, 1991, 1992 reformulated this result in terms of EFFs, see also Gurvich 1997.
Proposition 3. Balanced EFFs are stable.
As a consequence of all above one can arrive to the following theorem.
Theorem 4. Perfect graphs are kernel-solvable .
178 E
. Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
This statement was conjectured by Berge and Duchet 1983. The converse direction, namely that
Conjecture 5. Kernel-solvable graphs are perfect
was also conjectured in the same paper, and it is still an open problem. Berge and Duchet 1983 observed that Conjecture 5 would result from the Strong Perfect Graph
Conjecture. Boros and Gurvich 1994 proved that Conjecture 5 is equivalent to either of the following conjectures.
c
Conjecture 6. If a graph G is kernel-solvable then its complementary graph G is
kernel-solvable , too.
Conjecture 7. If a graph G is kernel-solvable and a vertex of G is substituted by an edge or by a clique then the resulting graph G9 is kernel-solvable
, too. Thus, assuming that the above conjectures hold true, for EFFs corresponding to
graphs a reasonably simple criterion of stability exists, stated in Proposition 2. However, only some very special EFFs are generated by graphs in the above way.
Let us consider in the sequel the general case, and let us represent an arbitrary EFF by specifying explicitly the list of coalition-block pairs for which takes value 1. More
precisely, let us consider 5 kK , B ; J; I, Al, where I denotes the set of players, and A
j j
stands for the set of outcomes, as above, and J denotes the set of indices of the pairs K I, B A for which K , B 5 1, i.e. for which coalition K is effective for block
j j
j j
j
B . We can interpret the EFF as a pair of hypergraphs _ and on the sets of vertices
j
I and A, respectively, the edges of which have a common set of indices J. Let us consider the dual transposed hypergraphs , and 6 on the common vertex set J
defined by
5 hC Jui [ Ij, 6 5 hS Jua [ Aj
i a
where C 5 h j [ JuK ] ij, and S 5 h j [ JuB ] aj.
i j
a j
An EFF can equivalently be specified by the hypergraphs , and 6, as well. In the sequel, we shall use both notations 5
kC , S ; I, A; Jl and 5 kK , B ; J; I, Al.
i a
j j
d
Given an EFF , Peleg 1984 introduced the dual EFF by exchanging the roles of
d
players and outcomes, i.e. 5 kS , C , A, I, Jl 5 kB , K , J, A, Il using the notations
a i
j j
above. Let us observe that in the special case when an EFF
is generated by a graph
G
G 5 kV, El, we have J 5V and the hyperedges of and 6 are, respectively, the
maximal cliques and maximal stable sets of G. Hence we have the relation
d
5
c
G G
c
where G denotes the complementary graph to G. We can now state some necessary and some sufficient conditions for the stability of an
arbitrary EFF in terms of the corresponding hypergraphs and 6.
E . Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
179
d
Theorem 8. If an EFF 5
kC , S ; I, A; Jl or its dual is stable then,
i a
i for every subset J9 J such that
uC J9u 1 for all i [ I there exists an a [ A
i
such that J9 S , and
a
ii for every J0 J such that uS J0u 1 for all a [ A there exists an i [ I such that
a
J0 C .
i
A simple consequence of this statement is the following corollary.
Corollary 9. If an EFF 5
kC , S ; I, A; Jl 5 kK , B ; J; I, Al is stable then every pair j,
i a
j j
j9 [ J, j ± j9 must belong either to C for some i [ I or to S for some a [ A or both.
i a
In other words , if K , B 5 K , B 5 1 and K K 5 B B 5 5 then the EFF
1 1
2 2
1 2
1 2
cannot be stable. Let us remark that the necessary conditions of stability i and ii of Theorem 8 can
d
be reformulated in dual terms, namely if or are stable, and for some J9 J we
have
d
i K , B 5 1 for all j [ J9 and K K 5 5 for all j ± j9 [ J9, then B ± 5
j j
j j 9
j [ J 9 j
must hold; and similarly
d
ii K ,B 5 1 for all j [ J9 and B B 5 5 for all j ± j9 [ J9 must imply that
j j
j j 9
K ± 5.
j [ J 9 j
d
Remark 10. Gurvich and Vasin 1978 proved that condition i holds if the EFF is playing-minor i
.e. iff holds for some EFF corresponding to a normal game
g g
form g. Later, Boros and Gurvich 1994 gave a shorter proof based on a theorem by Moulin and Peleg 1982 characterizing EFFs generated by game forms as monotone
d
and superadditive. Respectively , we can see that condition ii holds if the dual EFF
d d
is playing-minor. Thus , playing-minority of both EFFs and are necessary for
d
the stability of either one of or . The main contribution of this paper is the following ‘perfect split criterion’.
d
Theorem 11. Both EFFs 5
kC , S ; I, A; Jl and its dual are stable if there exists a
i a
graph G 5 kJ, El such that
i G is perfect; ii every maximal clique of G is a subset of some C , i [ I;
i
ii9 every maximal stable set of G is a subset of some S , a [ A.
a
From Theorems 8 and 11 we will be able to derive that
Theorem 12. To check the stability of an EFF, given as 5
kC , S ; I, A; Jl is an
i a
NP-complete problem.
180 E
. Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
Using Theorems 8 and 11, we can extend the list of equivalent Conjectures 5–7 on kernel-solvability and translate them into game theoretic language.
d
Conjecture 13. If an EFF 5
kK , B ; J; I, Al is stable while its dual not, then there
j j
exist coalitions K , K and blocks B and B such that K , B 5 K , B 5 1,
1 2
1 2
1 1
2 2
K K ± 5, and B B ± 5 hold.
1 2
1 2
In dual terms we can state this conjecture equivalently as
d
Conjecture 14. If an EFF 5
kC , S ; I, A; Jl is stable, and its dual not, then there
i a
must exist a player i [ I and an outcome a [ A such that uC S u 2.
i a
In particular, we conjecture that such an EFF cannot be generated by a graph. The Strong Perfect Graph Conjecture SPGC can also be reformulated in terms of EFFs.
Conjecture 15. If an EFF corresponding to a graph G is not stable then the graph G
G
has an odd hole or an odd antihole. In Section 5 we will prove that Conjectures 13 and 14 are equivalent to Conjecture 5 and
that Conjecture 15 is equivalent to SPGC.
3. Keiding’s theorem and its dual
Kategori