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
The following simple criterion of stability was obtained by Keiding 1985. Given an EFF 5
kK , B ; J; I, Al and a utility function u:I 3 A∞R, let us denote by R for j [ J
j j
j
a subset of the outcomes not necessarily all of them which are strictly worse than any one from B for all the players of the coalition K . In other words,
j j
R ha [ Auui, a , ui, b for all b [ B and i [ K j
R
j j
j
By this definition, B R 5 5 for every j [ J, and hence the coalition K being effective
j j
j
for B , can reject all the outcomes in R . Consequently, these outcomes cannot belong to
j j
the core C , u. Let us consider now an extended list T 5
hK , B , R u j [ Jj such that K is effective
j j
j j
for B and rejects R for every j [ J. The core C , u is empty if all the outcomes are
j j
rejected, i.e. R 5 A. In this case we shall call the list T 5
hK , B , R u j [ Jj a
j [ J j
j j
j
rejecting table for the pair , u. Let us denote the family of such rejecting tables by RT , u.
Let us consider next the case when we are given only an EFF but no utility u is specified. We shall call a list of the form T 5
hK , B , R u j [ Jj a rejecting table of the
j j
j
EFF , if K , B 5 1 for every j [ J and R 5 A. Let us denote the family of
j j
j [ J j
such lists by RT . Let us call a rejecting table T [ RT utilizable if T [ RT , u for some utility function u. By definition of stability, an EFF is not stable iff it has a
utilizable rejecting table T [ RT .
E . Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
181
To have a combinatorial characterization of utilizability of rejecting tables, we shall need the following definitions. Let us call a subset J 5
h j , . . . , j j J, r 2 of the
C 1
r
rows of a rejecting table T [ RT a cycle, if B R ± 5, B R ± 5, . . . , and
j j
j j
1 2
2 3
B R ± 5. Such a cycle J will be called a common-player cycle if all the
j j
C
r 1
corresponding coalitions have a player in common, i.e. if K ± 5.
j [ J j
C
Theorem 16. Keiding, 1985 An EFF is stable if and only if every rejecting table T [ RT contains a common-player cycle.
The proof results immediately from the following lemma.
Lemma 17. A rejecting table T [ RT is utilizable if and only if it contains no common-player cycle.
Proof. If there exists a common-player cycle J in T [ RT , then T can not be
C
utilizable because the preference of any common player i [ K , according to Eq.
j [ J j
C
R, would be cyclic over the outcomes a [ B R , a [ B R , . . . , a [ B
1 j
j 2
j j
r j
1 2
2 3
r
R .
j
1
Conversely, if there are no common-player cycles in T [ RT , then the inequalities in Eq. R induce an acyclic partial preference over the outcomes for every player i [ I.
Due to the acyclicity, we can choose a utility ui, p for every player i [ I, which
realizes the same preferences. For such a utility function u we shall have T [ RT ,u, completing the proof of the lemma.
h Keiding’s theorem can be reformulated in terms of the hypergraphs and 6
corresponding to the effectivity function . Let us begin with the special case of EFFs generated by graphs. Given a graph G 5
kV, E
l, let us direct some of its edges. We assume that some edges may remain non-directed but no edge can be bidirected. The obtained partially directed graph D is
called a suborientation of G. We say that a vertex v [ V of the suborientation D rejects a subset V 9 V if v [
⁄ V 9 and every edge v, v9 for v9 [V 9 is directed from v9 to v in D.
A suborientation D of G is called rejecting if every maximal stable set of G is rejected in D by a vertex. A directed cycle of D whose vertices form a clique in G is called a
clique-cycle of D. A rejecting suborientation without clique cycles is called a clique- acyclic rejecting suborientation or in short a CARS.
Proposition 18. Boros and Gurvich, 1994 A graph G has no CARS if the corre- sponding EFF
is stable.
G
Proof. In fact this proposition is ‘dual’ to Keiding’s theorem. Let us suppose that the EFF
is not stable and consider a rejecting table T [ RT which has no
G G
common-player cycles. Every outcome a [ A is rejected by some coalition according to T, hence there exists a corresponding row K, B, R in T for which K, B 5 1, and
G
a [ R. This implies, according to the definition of , the existence of a vertex v [ V for
G
182 E
. Boros, V. Gurvich Mathematical Social Sciences 39 2000 175 –194
which K K and B B. Then v [ ⁄ S , where S is the maximal stable set corre-
v v
a a
sponding to the outcome a, since all such maximal stable sets are included in B by the
v
definition of , and B R 5 5 by the definition of a rejecting table. Let us orient all the
G
edges of G between the stable set S and vertex v towards vertex v. By repeating the
a
same for all the outcomes, we obtain an orientation D of some of the edges of G, and we claim that it is a CARS. Indeed, D is rejecting because T [ RT is a rejecting
G
table and D is clique-acyclic and in particular, no edge of G is bidirected because T [ RT has no common-player cycles.
G
For the reverse direction, we can construct a rejecting table of the form T 5 hK , B ,
v v
R uv [V j [ RT from any CARS D of G, by defining R as the set of maximal
v G
v
stable sets rejected in D at vertex v. It is easy to verify that T is indeed a rejecting table for , since D is a rejecting suborientation, and that T has no common player cycles,
G
since such a cycle in T would correspond to a clique-cycle in D. Thus, the above constructions provide a correspondence between CARS of a graph G
and rejecting tables without common-player cycles of the corresponding EFF , in both
G
directions. h
There is a simple relation between CARS and kernel solvability of graphs. Given a graph G 5
kV, El, let us direct all of its edges, allowing some edges to be bidirected.
1
The obtained ‘overdirected’ graph D is called a superorientation of G. Given a
1
superorientation D , a subset S V is called a kernel if S is a stable set of G and every
1
vertex outside of S has a successor in S, according to D . In other words, if no edge of G has both endpoints in S and for every vertex v [
⁄ S there exists a vertex v9 [ S such
1
that the edge v, v9 in D is directed, from v towards v9 or bidirected. Obviously,
only a maximal stable set of G can be a kernel. If the graph G is complete i.e. G is a clique then only a single vertex can be a kernel, and a vertex v is a kernel iff v is a sink
1
in D , i.e. iff all the edges incident to v are directed towards v or are bidirected.
1
Finally, a graph G is called kernel-solvable if a superorientation D of G has a kernel
whenever every clique of G has a kernel. There is a simple one-to-one correspondence between sub- and superorientations of a
graph G, via interchanging unoriented edges with bidirected ones, and vice versa. It is easy to observe that a suborientation is a CARS iff the corresponding superorientation
has no kernel. Thus, we obtain the following claim.
Proposition 19. A graph G has no CARS if G is kernel-solvable. Theorem 20. Boros and Gurvich, 1994 A perfect graph has no CARS.
By Proposition 19, this theorem implies Theorem 4. According to Conjecture 5, the converse, i.e. that graphs with no CARS are perfect, is conjectured to hold too.
Let us now consider a general EFF 5 kC , S ; I, A; Jl, and let us assign to the
i a
hypergraph 5 hC ui [ Ij a graph G 5 kJ, E l on the vertex set J, whose edges
i
are those pairs j, j9 [ J, j ± j9 for which h j, j9j C for some i [ I. Given a
i
suborientation D of G , we shall call it -acyclic if there exists no directed cycle
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
