Stable Families of Coalitions and Normal Hypergraphs

Transcription

1 Stable Families of Coalitions and Normal Hypergraphs E. Boros V. Gurvich A. Vasin March 1997 Abstract The core of a game is defined as the set of outcomes acceptable for all coalitions. This is probably the simplest and most natural concept of cooperative game theory. However, the core can be empty because there are too many coalitions. Yet, some players may not like or know each other, so they cannot form a coalition. The following generalization seems natural. Let K be a fixed family of coalitions. The K-core is defined as the set of outcomes acceptable for all the coalitions from K. Let us call a family K g-stable if the K-core is not empty for any finite normal form game, and similarly, let K be called V -stable if the K-core is not empty for for any compact superadditive NTU-game. We prove that both V - and g-stability of a family K are equivalent with the normality of K. Normal hypergraphs can be characterized by several equivalent properties, e.g. they are dual to clique hypergraphs of perfect graphs. Key words: cooperative game theory, TU-games, NTU-games, core, K-core, coalition, stable family of coalitions, characteristic function, generalized characteristic function, monotonicity, superadditivity, Bondareva-Shapley s theorem, Moulin-Peleg s theorem, Scarf s theorem; effectivity function, game form; perfect graph, normal hypergraph 1 Introduction We say that a coalition rejects an outcome (allocation) if this coalition can guarantee a strictly better outcome (allocation) for all its participants, otherwise we say that the coalition accepts the outcome (allocation). The core of a game is defined for different coopearative models but always in the same way: as the set of outcomes (allocations) acceptable for all coalitions. The simplest model of a cooperative game with transferable utilities (TU-games) is the characteristic function (CF) form. A criterion of non-emptiness of the core for such games was given by Bondareva (1962, 1963; in Russian) and then independently by Shapley (1965, 1967). It is a rather The first and second authors gratefully acknowledge the partial support of AFOSR (Grant F ) and ONR (Grants N J-1375 and N J-4083). The third author acknowledges the financial support by the Russian Fund for Basic Research. RUTCOR, Rutgers University, P.O.Box 5062, New Brunswick, NJ 08903, USA, ({boros,gurvich}@rutcor.rutgers.edu). On leave from the International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Moscow, Russia Moscow State University, Department of Computational Mathematics and Cybernetics, Moscow, Russia Revised version of RUTCOR Research Report, RRR 22-95, Rutgers University, June

2 simple criterion based on Farkas (1902) lemma on duality in linear programming. According to this criterion, a CF has to satisfy a system of linear inequalities, one inequality corresponding to every balanced subfamily of coalitions. A family of coalitions K is called v-stable if the K-core is not empty for any superadditive CF. The Bondareva-Shapley criterion easily generalizes for K- cores. This generalization results immediately in the following criterion given by Gurvich and Vasin (1977, 1978b; in Russian), and then independently by Kaneko and Wooders (1982). A family K is v-stable iff K is partitionable, i.e. if any balanced weighting of K is a positive linear combination of partitions. Examples and interpretations of such families were considered by Gurvich and Vasin (1977, 1978b), Kaneko and Wooders (1982), Greenberg and Weber (1986), Demange(1990), Quint (1991, 1994) Aharoni, Hajnal and Milner (1994), Kuipers (1994). A more general model of cooperative games, with non-transferable utilities (NTU-games), is the generalized characteristic function (GCF) form. A sufficient condition of non-emptiness of the core for these games were given by Scarf (1967). This result is based on an important lemma related to a fix point theorem, see Danilov and Sotskov (1992). However, Scarf s conditions are also stated in terms of balanced families, and formally they are very close to Bondareva-Shapley s inequalities. A family K is called V-stable if the K-core is not empty for any compact superadditive GCF. Scarf s theorem easily generalizes for K-cores. This generalization immediately results in the following criterion given by Gurvich and Vasin (1978a; in Russian). A family K is V-stable iff the extended family K [I] is partitionable, where [I] denotes the family of all the singletons, i.e. coalitions of cardinality 1. For the case K [I] the same result was obtained independtly by Kaneko and Wooders (1982). Here we prove a somewhat stronger statement (see Theorem 7), by showing that to any family K for which K [I] is non-partitionable, one can always construct a finitely generated (and consequently, compact) superadditive GCF which has its K-core empty. Thus the classes of v- and V-stable families of coalitions in fact coincide, provided all the singletons are included. Such an assumption is natural for game theory. Indeed, the main reasons to consider not all the coalitions but only a fixed family was that not all the coalitions can be organized. However, nobody can prevent a single player to organize the coalition which consists only of himself. On the other hand, from a combinatorial point of view, these two conditions: K is partitionable and K [I] is partitionable differ a lot. The second one is much stronger and more natural, in some sense. Namely, Le Breton, Owen and Weber (1992) proved that K [I] is partitionable iff the hypergraph K is normal. The same result was obtained independently by Kuipers (1994), and by Boros and Gurvich (1994). In fact, this theorem is a translation in game theoretic language of the result by Lovász (1972a, Theorem 2) characterizing normal hypergraphs, which were introduced by Berge (1970). In consequence of all the above, for a family K normality and V -stability are equivalent properties. A family K is called g-stable if its K-core is not empty for any finite game in normal form. It was also shown by Gurvich and Vasin (1978a) that V-stability is equivalent to g-stability. Here we provide a shorter proof, based on a result by Moulin and Peleg (1982) characterizing effectivity functions generated by game forms (as monotone and superadditive). We also prove the stronger statement that every finitely generated superadditive GCF is realizable by a finite normal form game (see Theorem 11). For completeness, we collect and reproduce the proofs (some of them in a simplified form) for the mentioned results given by Gurvich and Vasin (1977, 1978a; in Russian). Let us also remark that the criterion of stability for effectivity functions, given by Keiding (1985), connects the notion of g-stability to kernel-solvability of graphs, and in particular to a 2

3 conjecture by Berge and Duchet (1983) which was proved, using some of the above results, by Boros and Gurvich (1994). Recently, a shorter proof was given by Aharoni and Holzman (1995), which, although not game theoretic, is based also on Scarf s (1967) lemma. 2 Balanced weightings and partitionable families of coalitions Let us consider a family K of coalitions and a non-negative integer valued weight function on it, w : K Z +. The function w is called a balanced weighting of K with multiplicity m if the equation K i K K w(k) = m holds for every player i I, or in other words, if each player participates in the same number of coalitions (taking weights into account). We shall call a family K balanced if it has a positive balanced weighting w : K Z +. For an arbitrary non-negative weighting w, the subfamily K w = {K K w(k) > 0} is called the support of w. A balanced weighting is called minimal if its support does not contain the support of some other balanced weighting. A balanced weighting w of multiplicity m = 1 is called a partition weighting (or simply a partition), because in this case the support K w of w is a partition of I. A balanced weighting w is called a sum of partitions if there are partitions w i : K {0, 1} and non-negative integers α i Z +, i = 1,..., l, such that w = l i=1 α i w i. Proposition 1 (Shapley (1965)) The following properties of a family K are equivalent. (a) Any balanced weighting of K is a sum of partitions. (b) Any balanced weighting of K contains a partition. (c) Any minimal balanced weighting of K is a partition. We shall call a family K partitionable if K satisfies the equivalent conditions (a),(b),(c) above. We shall see that there are further equivalent properties to (a),(b), and (c) in terms of perfect graphs and normal hypergraphs. We recall here 3 examples for partitionable and non-partitionable families, from Gurvich and Vasin (1977, 1978b). Example 1 I = {1, 2,..., n}; K = C n = {(1, 2), (2, 3),..., (n 1, n), (n, 1)}, n 3. An even cycle is partitionable (clearly, it is the sum of 2 partitions), but an odd cycle is not. Example 2 I = {1, 2,..., n}, K = {K I i, i K, i < j < i j K}. In other words, I is a finite completely ordered set, and K is the family of all the intervals of I. Clearly, the family K is partitionable. We can interpret the above example such that players are owners of the shops situated along a railway line in a remote region. Then, it is natural to suppose that only a few close neighbors can organize a coalition (and effectively coordinate the prices, for example). In this case the customers living within the corresponding interval cannot resist. But if an intermediate player fails to participate then the whole coalition fails. This example was considered also by Greenberg and Weber (1986). It was generalized by Aharoni, Hajnal and Milner (1994) for the case of an infinite ordered set. The next example is a further generalization. It was considered also by Demange (1990) and by Kuipers(1994). 3

4 Example 3 Let T =< I, E > be a finite tree where I is the set of vertices (identified with the players) and E is the set of edges. The family K, of all the connected subgraphs (i.e. subtrees) of T, is partitionable. This example generalizes the previous one in which the corresponding tree T is a simple path. A similar interpretation can be given. Another interpretation was suggested by Gurvich and Menshikov (1989). Let us fix an arbitrary vertex r T. It will be called the root and the corresponding player will be called the big boss. The obtained rooted tree will be treated as a hierarchical scheme (of a company). It is supposed that if two players i, i I participate in a coalition then the players along the path connecting the corresponding vertices also have to be included in this coalition. The boss of the minimal rank common for the players i and i and also all the intermediate bosses belong to this path. (The path is unique because T is a tree.) The following class of families was suggested by T. Quint (1991, 1994). Example 4 Let I = I 1 I m be a partition of I and let K = {(i 1,..., i m ) i j I j, j = 1,..., m} {(1),..., (n)}. Here K can be interpreted as m-sided assignment families; see T. Quint (1991, 1994). In case m = 2 we get the standard Shapley-Shubik (1972) assignment families, which are partitionable. However, an m-sided assignment family, for m > 2 is not partitionable (c.f. Kaneko and Wooders (1982)). 3 TU-games and v-stability A characteristic function (CF) is defined as a mapping v : 2 I IR. The real values v(k), K I, can be interpreted as anticipated profits of the coalitions. A CF v is called superadditive if v(k K ) v(k ) + v(k ) K, K I, K K =. For any x = (x 1,..., x n ) IR n and K I let x K = i K x i. The set of outcomes (or possible allocations) is defined as A = {x x I v(i)} IR I, i.e. A consists of all the vectors which the coalition of all the players is able to obtain. Let us remark that vectors x A may contain negative components, too. An outcome x is called acceptable for a coalition K I if x K v(k), i.e. if this coalition cannot expect more than given by this outcome. Let us remark that x i v({i}), i I are not required for x A. The core C(v) is the set of outcomes acceptable for all the coalitions K I. In other words, the core is defined by the following system of linear inequalities C(v) = {x IR I x I v(i) and x K v(k) K I}. Given an arbitrary set of coalitions K 2 I, the K-core C(v, K) is defined as the set of outcomes acceptable for all the coalitions K K, i.e. C(v, K) = {x IR I x I v(i) and x K v(k) K K}. According to the above definitions, C(v, K) = C(K) for K = 2 I. A set K is called v-stable if C(K, v) for any superadditive CF v. 4

5 Theorem 1 (Bondareva (1962,1963), Shapley(1965, 1967)) The core C(v) is not empty iff any (minimal) balanced weighting w : 2 I Z+ of all the coalitions of multiplicity m satisfies the inequality K I w(k)v(k) mv(i). Proof follows directly from the characterization of consistent systems of linear inequalities, applied to the system defining the core, see e.g. Farkas (1902). Theorem 2 (Gurvich and Vasin (1977)) The core C(v, K) is not empty iff any (minimal) balanced weighting w : K Z + of K of multiplicity m satisfies the inequality K K w(k)v(k) mv(i). This theorem is an easy generalization of the previous one. The proof follows exactly the same lines. Let us consider 3- and 4-cycles for instance Example 5 I = {1, 2, 3}, K = C 3 = {(1, 2), (2, 3), (3, 1)}. The K-core is given by x 1 +x 2 v(1, 2) x 1 +x 3 v(1, 3) +x 2 +x 3 v(2, 3) x 1 +x 2 +x 3 v(1, 2, 3) There is only one minimal balanced weighting, yielding the inequality v(1, 2) + v(1, 3) + v(2, 3) 2v(1, 2, 3). Hence C(v, K) exactly when v satisfies the above inequality. This inequality is not a consequence of superadditivity, e.g. it is violated by the superadditive (and simple) CF v given by and hence K is not v-stable. v(1) = v(2) = v(3) = 0, v(1, 2) = v(1, 3) = v(2, 3) = v(1, 2, 3) = 1, Example 6 I = {1, 2, 3, 4}, K = C 4 = {(1, 2), (2, 3), (3, 4), (4, 1)}. The K-core is given by x 1 +x 2 v(1, 2) +x 2 +x 3 v(2, 3) +x 3 +x 4 v(3, 4) x 1 +x 4 v(1, 4) x 1 +x 2 +x 3 +x 4 v(1, 2, 3, 4) In this case there are two minimal balanced weightings, yielding the inequalities v(1, 2) + v(3, 4) v(1, 2, 3, 4), and v(2, 3) + v(1, 4) v(1, 2, 3, 4) which hold for any superadditive CF v, and hence K is v-stable. Let us assign a CF v K to a family of coalitions K. For this, let us first add to K all the unions of pairwise disjoint coalitions from K and let us denote the obtained expanded family by K. Then let us define v K by v K (K) = max K : K K K K. 5

6 Lemma 1 The CF v K is superadditive, takes nonnegative integer values and satisfies the relations: v K (K) = K K K, and v K (K) < K K K. Furthermore, if the family K is not partitionable, then C(v K, K) =. Proof. The first four claims are straightforward. Let us consider the last one, and let us assume that K is a non-partitionable family. Then there exists a minimal balanced weighting w : K Z + of multiplicity m, which is not a partition. Let us consider the support K w and the corresponding CF v Kw. Obviously, C(v Kw, K w ) C(v K, K). Let us prove that C(v Kw, K w ) =. Indeed, on the one hand v Kw (I) < I because I K w, and on the other hand K K w w(k)v Kw (K) = K K w(k) K = m I because the weighting w is balanced. Thus the inequality mv w Kw (I) < K K w w(k)v Kw (K) is implied, and hence C(v Kw, K w ) = follows by Theorem 2. Theorem 3 (Gurvich and Vasin (1977), Kaneko and Wooders (1982)) A family K is v-stable iff K is partitionable. Proof. The statement follows directly from Theorem 2 and the previous lemma. ( ): If w : K Z + is a partition, then the inequality of Theorem 2 follows from the superadditivity of the CF v. ( ): Let w : K Z + be a minimal balanced weighting of K which is not a partition. Let us consider the support K w = {K K w(k) 0)} and the CF v Kw. According to the previous lemma, this CF is superadditive and the core C(v Kw, K w ) is empty. Thus K w is not v-stable, and consequently K is not v-stable, either. 4 NTU-games and V -stability A generalized characteristic function (GCF) is defined as a mapping V : 2 I 2 IRI. The subsets V (K) IR I are interpreted as the expected sets of possible allocations for the coalitions K I, provided side payments are not allowed. The following 3 properties of GCFs will be always assumed. P1 V (K) for any K I. P2 V (K) = V 0 (K) IR I\K for any K I, where V 0 (K) IR K, i.e. a coalition K I is not able to influence the allocations of the complementary coalition I \ K. P3 V 0 (K) is a comprehensive subset of IR K for any K I, i.e. x V 0 (K) and x i x i for all i K implies x V 0 (K). A GCF is called compact if it has the following 3 properties. P4 V (K) is a closed subset of IR I for any K I. P5 For all players i I there exists v i IR such that V ({i}) = {x IR I x i v i }, i.e. each player s profit is bounded from above. P6 The set V 0 (K) {x IR K x i v i i K} in IR K is compact. 6

7 Let us denote by V P O (K) the set of Pareto optimal vectors of V 0 (K), i.e. V P O (K) = {x V 0 (K) y V 0 (K) : y x and y i x i i K}. A GCF V is called finitely generated if V P O (K) is a finite not empty set for any K I. Obviously, any finitely generated GCF is compact. A GCF is called superadditive if P7 V (K K ) V (K ) V (K ) for all coalitions K, K I with K K =. It is called monotone if P8 V 0 (K ) P R K V 0 (K ) for all coalitions K K I, where P R K V 0 (K ) denotes the projection of V 0 (K ) on IR K. Proposition 2 Any superadditive GCF is monotone, i.e. P7 implies P8. Proof. Let us fix a superadditive GCF V, a pair of coalitions K and K such that K K, and an arbitrary vector x V 0 (K ). We have to prove P8, that is x P R K V 0 (K ). Let us fix an arbitrary vector x V 0 (K \ K ). Coalitions K and K \ K are disjoint and their union is K. Thus, according to P7, there exists a vector x V 0 (K ) such that x i x i for any i K. The set of outcomes is defined as A = V (I) IR I, i.e. A contains all those allocations which the coalition of all the players is able to guarantee. An outcome x is called acceptable for a coalition K I if this coalition cannot guarantee more for all the participants, i.e. if there is no x V (K) such that x i > x i for all the players i K. The core C(V ) is defined as the set of outcomes acceptable for all the coalitions K I. Given an arbitrary family of coalitions K 2 I, the K-core C(V, K) is defined as the set of outcomes acceptable for all the coalitions K K, i.e. C(V, K) = V (I) \ K K intv (K) (see also Kaneko and Wooders (1982)). Thus in particular, C(V, K) = C(V ) for K = 2 I. A family K will be called V-stable if C(V, K) for any compact, superadditive GCF V. Theorem 4 (Scarf (1967)) The core C(V ) is not empty if V is compact and for any (minimal) balanced weighting w : 2 I Z + of all the coalitions the following inclusion holds: V (K) V (I). (4.1) K I,w(K) 0 Theorem 5 (Gurvich and Vasin (1978a)) The K-core C(V, K) is not empty if the GCF V is compact and for any (minimal) balanced weighting w : K [I] Z + of the extended family K [I] the inclusion (4.1) holds. Proof. The proof is based on Theorem 4. Let us introduce a new GCF V obtained from V by discriminating coalitions which do not belong to the family K, i.e. by defining V (K) = V (K) if K K and V (K) = V ({i}) if K K. In particular, V ({i}) = V ({i}) for any i I. i K 7

8 If the conditions of the theorem hold for a GCF V then the conditions of Theorem 4 hold for the corresponding GCF V, too, because for any balanced weighting w : 2 I Z + we can substitute any coalition K K by the set of the singletons {{i} i K} and obtain a balanced weighting w : K [I] Z +. Thus, C(V ) by Theorem 4. On the other hand, C(V ) C(V, K), because the GCFs V and V enable the coalitions of K with the same power. To any CF v we can assign a GCF V v defined by for all coalitions K I. V 0 v (K) = {x IR K x K = i K x i v(k)} Lemma 2 For any CF v the corresponding GCF V v is compact and monotone, and it is superadditive iff CF v is superadditive. Furthermore, C(V v, K) = C(v, K) for any K. Proof. It is straightforward. To any GCF V let us assign the truncated GCF V defined by V 0 (K) = {x V 0 (K) y V 0 (K) for which y i x i, y i v i i I} for all coalitions K I. Equivalently, V P O(K) = V P O (K) {x IR K x i v i i K}. In other words, V P O (K) consists of all the vectors of V P O(K) acceptable for every player i K. Obviously, the truncated GCF V is compact if V is compact. Let us note also that every allocation x V (K) \ V (K) is not acceptable for at least one player i K. Example 7 K = {(1, 2), (1, 3), (2, 3, 4)} is partitionable, because for K there is no balanced weighting at all, while K [I] is not partitionable. Indeed, K = {(1, 2), (1, 3), (2, 3, 4), (4)} is a minimal balanced subfamily of K [I], and obviously K is not a partition. (In general, when we add the singletons many new balanced weightings can appear.) Let us observe that, according to Theorem 2, K is v-stable. E.g. if { 1 if S K for some K K v(s) =, 0 otherwise, then v being superadditive, C(v, K ) follows, and indeed x = (0, 1, 1, 1) C(v, K ). Let us consider the CF v K and the GCF V = V vk. Then etc., while V P O (1, 2) = {(x 1, x 2 ) x 1 + x 2 = 2} V P O (1, 3) = {(x 1, x 3 ) x 1 + x 3 = 2} V P O (2, 3, 4) = {(x 2, x 3, x 4 ) x 2 + x 3 + x 4 = 3} V P O (1, 2, 3, 4) = {(x 1, x 2, x 3, x 4 ) x 1 + x 2 + x 3 + x 4 = 3} 8

9 etc. V P O (1, 2) = {(x 1, x 2 ) x 1 + x 2 = 2, x 1 0, x 2 0} V P O (1, 3) = {(x 1, x 3 ) x 1 + x 3 = 2, x 1 0, x 3 0} V P O (2, 3, 4) = {(x 2, x 3, x 4 ) x 2 + x 3 + x 4 = 3, x 2 0, x 3 0, x 4 1} V P O (1, 2, 3, 4) = {(x 1, x 2, x 3, x 4 ) x 1 + x 2 + x 3 + x 4 = 3, x 1 0, x 2 0, x 3 0, x 4 1 }, Finally, to a GCF V let us assign another GCF V defined by V P O (K) = INT (n, V P O (K)), where INT (n, X) denotes the set of vectors in X the coordinates of which are all multiple of 1/n. It is easy to check that V is finitely generated. Theorem 6 (Gurvich and Vasin (1978)) A family of coalitions K is V-stable iff the extended family K [I] is partitionable. For the case K [I] the same result was obtained independently by Kaneko and Wooders (1982). Here we prove a stronger claim. Theorem 7. If K [I] is partitionable then the K-core is not empty for any compact superadditive GCF. Furthermore, if K [I] is not partitionable then there exists a superadditive finitely generated GCF such that its K-core is empty. Proof. For the first claim, let us apply Theorem 5. If w : K [I] Z + is a partition then (4.1) is implied by the superadditivity (P7). For the second claim, let us suppose that K [I] is not partitionable, and let us consider a minimal balanced weighting w : K [I] Z + which is not a partition. Let us consider its support K w = {K K [I] w(k) 0}, the CF v Kw and the GCF V vkw. According to Lemma 1, v Kw is superadditive and C(v Kw, K w ) =. Thus, V vkw is superadditive, compact, and C(V vkw, K w ) =, by Lemma 2. Unfortunately, we cannot just finish our proof yet, since K w can contain singletons which are not in K, and in principle, it could happen that the K w -core is empty while the (K w \ [I])-core is not. Let us consider the truncated GCF V v Kw. It is not difficult to check that it is monotone, superadditive, compact, and by the definition, any Pareto optimal allocation is acceptable for all the singletons from K. Thus indeed C(V v Kw, K w \ [I]) =, implying that C(V v Kw, K) =, since K w \ [I] K. Therefore the family K is not V-stable. Since the GCF V v Kw is not finitely generated, let us use instead the GCF V v Kw. Clearly, it is superadditive and by the definition, it is finitely generated. Let us prove that C(V v Kw, K w \[I]) =. Let us recall that we denote by n = I the number of players, and by m the multiplicity of the balanced family K w. Furthermore, for x = (x 1,..., x n ) IR n, let x K = i K x i for all K I, and in particular, let x I = i I x i. Finally, let A = {x = (x 1,..., x n ) IR n x I v Kw (I)} denote the set of outcomes. Let us introduce further δ i = 1 x i for i I, and let δ K = K x K for K I. In particular, δ I = n x I. Let us note that since K w is not partitionable, v Kw (I) n 1, and thus δ I = n x I 1 follows for every x A. Furthermore, since w is a balanced weighting of K w of multiplicity m, we have K K w w(k) K = mn, and thus w(k)δ(k) = mn w(k)x K = m(n x I ) m. K K w K K w 9

10 We claim that for any allocation x A there exists a coalition K = K x K w such that δ Kx K x /n. Indeed, if δ K < K /n for all K K w then we get a contradiction by m w(k)δ K < (1/n) w(k) K = m. K K w K K w Let us now prove that an allocation x A is not acceptable for the coalition K x with respect to V v Kw. As we have shown, we have x K n 1 n K x = n 1 n v K w (K x ). Let us consider the vector x obtained from x by incrementing each of its components x i, i K x by a strictly positive value to the nearest multiple of 1/n (if x i itself is a multiple of 1/n then we still add 1/n.) Then we still have x K K x = v Kw (K x ), and thus x belongs to V v Kw, and it is strictly and unanimously preferred to x by K. 5 Game forms and effectivity functions As before, let I denote the set of players and let A denote the set of outcomes. Subsets K I and B A denote, respectively, coalitions of players and blocks of outcomes. An effectivity functions (EFF) is defined as a mapping E : 2 I 2 A {0, 1}. In other words, an EFF is a Boolean function on the set of variables I A. We can as well treat it as a subset of 2 I 2 A, i.e. as the set of pairs (K, B) for which E(K, B) = 1. The notation E(K, B) = 1 (resp. E(K, B) = 0) is read as coalition K is effective (resp. is not effective) for block B and means that K is (resp. is not) able to guarantee that an outcome from B will be realized. An EFF is called K-monotone if E(K, B) = 1 and K K implies E(K, B) = 1. Analogously, it is called B-monotone if E(K, B) = 1 and B B implies E(K, B ) = 1. Finally, an EFF E is called monotone if it is both K- and B-monotone, i.e. if An EFF is called superadditive if E(K, B) = 1, K K B B E(K, B ) = 1. (E(K 1, B 1 ) = 1, E(K 2, B 2 ) = 1, K 1 K 2 = ) (E(K 1 K 2, B 1 B 2 ) = 1. The so called boundary conditions are defined as E(I, B) = E(K, A) = 1 for any B and for any K, E(, B) = E(K, ) = 0 for any B, and for any K. In particular, E(, A) = E(I, ) = 0. Let X i denote the (finite) set of strategies of player i I, and let X = i I X i. Let g : X A be a mapping, which assigns an outcome a A to any strategy-vector (x i i I). This mapping g is not supposed to be injective, i.e. the same outcome can be assigned to different strategy-vectors. 10

11 The mapping g (or sometimes the quadruple < I, A, X, g >) is called a game form. Thus, a game form is something like a game in normal form but without payoff, which is not yet specified. To any game form g we can assign an EFF E g defined by E g (K, B) = 1 ( x K g(x K, x I\K ) B x I\K )) for all K I, B A, where x K = {x i i K} and x I\K = {x i i K} are the strategy-vectors of the coalitions K and I \ K, respectively. The right hand part means that coalition K is able to guarantee that the resulting outcome belong to block B independently on the actions of the complementary coalition I \ K. An effectivity function E is called playable if it is realized by a game form, i.e. if there exists a game form g such that E = E g. Theorem 8 (Moulin-Peleg (1982)) An EFF is playable iff it is monotone, superadditive and satisfies the boundary conditions. 6 Normal form games and g-stability Let us call a mapping u : I A IR a utility function. The real number u(i, a) is interpreted as the profit of player i I in case outcome a A is realized. Let us denote by P R(K, a, u) the set of all those outcomes that coalition K strictly and unanimously prefers to a with respect to the utility function u, i.e. P R(K, a, u) = {a A u(i, a ) > u(i, a) i K}. Note that a P R(K, a, u). Given an EFF E and a utility function u, we say that the coalition K rejects an outcome a if E(K, P R(K, a, u)) = 1, i.e. if K is able to guarantee an outcome, better for all of its participants than a. The core C(E, u) is defined as the set of not rejected outcomes, i.e. C(E, u) = {a A E(K, P R(K, a, u)) = 0 K I}. Given E, u and a family of coalitions K, the K-core C(E, u, K) is defined as the set of outcomes not rejected by the coalitions from K, i.e. C(E, u, K) = {a A E(K, P R(K, a, u)) = 0 K K}. A game in normal form is defined as a pair < g, u >. If it is given then the corresponding core C(E g, u) and K-core C(E g, u, K) are determined, too. A family of coalitions K is called g-stable if the K-core is not empty for any game in normal form, or in other words, if C(E g, u, K) for any utility function u and for any game form g. 7 V-stability, g-stability and normality are equivalent In this section we allow the set of outcomes A and the sets of strategies X i, i I to be infinite, but the set of players I is still supposed to be finite. Given an EFF E and a utility function u : I A IR, let us define a CF v E,u and a GCF V E,u by 11

12 and v E,u (K) = sup B A,E(K,B)=1 inf a B u(i, a), i K V E,u (K) = U(i, a), B A,E(K,B)=1 a B i K where U(i, a) = {x IR I x i u(i, a)}. In other words, the CF v E,u and the GCF V E,u for any K I are, respectively, the sum and the sets of allocations which coalition K is able to guarantee in the game < I, A, E, u >, given in effectivity function form. Remark 1 The core and the K-core of such a game is defined by the corresponding GCF, i.e. C(E, u, K) = C(V E,u, K). The core and K-core of a normal form game < g, u > can be defined equivalently via the corresponding EFF, i.e. C(g, u, K) = C(E g, u, K). Thus, going from (g, u) through (E g, u) to V Eg,u we can lose some information about the game, but not about its core. However, if we substitute GCF V by CF v then this is not true anymore. Remark 2 If case of an infinite EFF E we shall always restrict ourself by a utility function u such that the CF v E,u and the GCF V E,u are well defined, i.e. finite and non-empty, respectively.. The following claims are straightforward. Lemma 3 Let E be an arbitrary EFF and u be a utility function. Then the following hold. (c1) If E is superadditive then both v E,u and V E,u are superadditive. (c2) If E is monotone then the GCF V E,u is monotone. (c3) If E is finite then the GCF V E,u is well defined and finitely generated. (c4) The GCF V E,u has the properties P1,P2,P3. Theorem 9 (von Neumann and Morgenstern (1944), 157.2,3) Every superadditive CF v is realizable by a finite game < g, u > in normal form, i.e. v = v E,u for some playable EFF E = E g and utility function u. In fact a much stronger claim holds. For any superadditive v there is a game such that any coalition K I is able to guarantee not only the sum v(k), but any allocation of this sum between its participants, as well. Of course, this game (or more exactly the strategy set X) is not finite. Furthermore, the inverse claims to (c2), (c3), and (c4) hold, too. Theorem 10 (Gurvich and Vasin (1978a)) Every superadditive GCF V is realizable by a game < g, u > in normal form (which may have a continuous set of strategies), i.e. V = V E,u for some playable EFF E = E g (which may have a continuous set of outcomes) and utility u. Theorem 11 Every finitely generated superadditive GCF V is realizable by a finite game < g, u > in normal form, i.e. V = V E,u for some finite playable EFF E = E g and utility u. 12

13 We provide here simpler proofs based on the theorem by Moulin and Peleg (1982). Proof of Theorem 10. Given a GCF V, let us define an EFF E and a utility u such that V E,u = V. The set of players I is already given. Let A = V (I) and let E be defined by E(K, B) = 1 iff B {x V (I) x 0 V 0 (K) : x i = x 0 i i K} or in other words, a coalition K is able to force any allocation from V 0 (K), but it cannot at all influence the allocations for players not in K. Let u be defined by u i (x) = x i for all x A. It is not difficult to check that E is monotone if V is monotone, E is superadditive if V is superadditive, and E always satisfies the boundary conditions, with maybe the exception of E(K, ) 0 for some K. Moreover, if V is monotone then E(K, ) = 0 for all K I. Let us recall also that, according to Proposition 2, any superadditive V is monotone. Thus, according to Theorem 8, V is realizable by a (continuous) playable EFF and a utility function. Proof of Theorem 11. Given a finitely generated GCF V, let us define an EFF E and a utility function u such that V E,u = V. The set of players I is already given. For any K I let us consider the finite set of generators B(K) V 0 (K), and define the set of outcomes as their union, i.e. A = K I B(K). For any outcome b B(K) let us introduce the set of outcomes B b which consists of all the outcomes a B(K ) such that K K and x i (a) x i (b) for all i K. In other words, B b consists of the outcome b and all the generators majorizing b. Now let us define an EFF E by E(K, B) = 1 iff b B(K) B b B. for any K except for K = I and K =. In other words, K is effective for any set of outcomes which consists of a generator from B(K) and all its majorizing generators, and of course, K is effective also for any set of outcomes containing one of the above mentioned ones. Let us define now E(I, B) = 1 B, E(I, ) = 0, and E(, B) = 0 B. By this definition, the EFF E is finite, B-monotone and satisfies the boundary conditions. Clearly, E is also K-monotone if V is monotone, and superadditive if V is superadditive (according to Proposition 2, any superadditive V is monotone.) Let us define a utility function u by setting u i (x) = x i for all x A. Thus, we obtain V E,u = V. Therefore, according to Theorem 8, any finitely generated superadditive GCF is realizable by a finite playable EFF and a utility function. Remark 3 Let us note, that Theorem 11 is a rather strong generalization of Theorem 9. However the proof given above is quite short, due to the use of the Moulin-Peleg theorem. For completeness, let us provide here a short proof for characterizing g-stable families as normal hypergraphs. (For other proofs, and further definitions see e.g. Le Breton, Owen and Weber (1992), Boros and Gurvich (1994), and Kuipers (1994).) Theorem 12 A family of coalitions K is g-stable iff the hypergraph K is normal. 13

14 Proof. By Theorems 7 and 11, the family K is g-stable if and only if K [I] is partitionable. It is immediate to see that this latter is equivalent with saying that the system of linear equations K i K K w(k) 1 for all i I, 0 w(k) 1 for all K K has integral extremal solutions (i.e. all solutions are convex combinations of integral solutions, see also Kaneko and Wooders (1982).) This integrality is exactly what was characterized by Lovász (1972a), as K being a normal hypergraph, or in other words, for which the collections C i = {K K i K}, i I form the family of cliques of a perfect graph with vertex set K. Acknowledgement. The second author thanks Salvador Barbera for the helpful discussions. References [1] R. Aharoni, A. Hajnal, E.C. Milner (1994), Interval covers of a linearly ordered set, Manuscript. [2] C. Berge (1961a), Farbung von Graphen, deren samtliche bzw deren ungerade Kreise starr sind, Wiss. Z. Martin Luter Univ. Halle Wittenbog, Math. Nat. Reihe 10 (1961) [3] C. Berge (1961b), Sur une conjecture relative au probleme des codes optimaux, Comm. 13eme Assemble Generale de l URSI, Tokyo, [4] C. Berge (1970), Graphes et Hypergraphes, (Dunod, Paris, 1970). English translation: Graphs and Hypergraphs, (North-Holland, Amsterdam, 1973). [5] C. Berge (1984), Minimax theorems for normal hypergraphs and balanced hypergraphs - a survey, Annals of Discrete Mathematics, 21 (1984) [6] C. Berge and P. Duchet (1983) (Seminaire MSH, Paris, 1983 Jan). [7] C. Berge and P. Duchet (1986), Solvability of perfect graphs, Proceedings of the Burnside- Raspai meeting (Barbados, 1986), (McGill Univ., Montreal, 1987). [8] O.N. Bondareva (1962), The core of N person game, Vestnik Leningrad University, 17, N 13, (1962) (in Russian). [9] O.N. Bondareva (1963), Some applications of linear programming methods to the theory of cooperative games, Problemy Kibernetiki, 10 (1963) (in Russian). [10] E. Boros and V. Gurvich (1994), Perfect graphs are kernel-solvable, Rutcor Research Report , Discrete Mathematics (1996), 159, p [11] V. Danilov and A. Sotskov (1991), The social choice mechanizms, ( Nauka, Moscow, 1991) 175 p. (in Russian). [12] V. Danilov and A. Sotskov (1992), Generalized convexity: some fixed point theorems and their applications, CEPREMAP, Preprint N 9213,

15 [13] G. Demange (1990). Intermediate preferences and stable coalition structures. Laboratorie d Econometrie de l Ecole Polytechnique. Preprint. [14] Fan Ky (1956), On systems of linear inequalities, in Linear inequalities and related systems, ed by H. Kuhn and A. Tucker (Princeton Univ. Press, Princeton, 1956). [15] J. Farkas (1902), Über die Theorie der Einfachen Ungleichungen, J. Reine und Angewandte Math. 124 (1902), [16] J. Greenberg (1993), Coalition structures, in Handbook of game theory with economic applications, ed R.Aumann and S.Hart (Elsevier North Holland, NY, 1993). [17] J. Greenberg and S. Weber (1986), Strong Tiebout equilibrium under restricted preferences domain, J. Econ. Theory 38 (1986) [18] V.A. Gurvich and A.A. Vasin (1977), Reconcilable sets of coalitions, In Questions of applied math. (Sibirian Energetic Inst. Irkutsk, 1977) (in Russian). Autor s abstract in English. Math. Rev. 81b: [19] V.A. Gurvich and A.A. Vasin (1978a), Reconcilable sets of coalitions for normal form games. In Numerical methods in optimization theory (Appl. math.), (Sibirian Energetic Inst. Irkutsk, 1978) (in Russian). Autor s abstract in English. Math. Rev. 81j: [20] V.A. Gurvich and A.A.Vasin (1978b), On Reconcilable sets of coalitions, 3-rd USSR Conference on Operation Research, Gorky, 1978, Abstract, (in Russian). [21] V.A. Gurvich and I.S. Menschikov (1989), Institutions of agreement, In Knowledge, News in the life, science and technics, serie Mathematics and Cybernetics. N 6, subsection 12.3, (1989) 42 p. (in Russian). [22] R. Aharoni and R. Holzman (1995), Fractional kernels in digraphs. Manuscript (1995), Dep. of Math., Technion, Haifa, To appear in J. Combinatorial Theory. [23] H. Keiding (1985), Necessary and sufficient conditions for stability of effectivity functions, International Journal of Game Theory, 14 (1985) [24] M. Kaneko and M.H. Wooders (1982), Cores of partitioning games, Mathematical Social Sciences, 3 (1982) [25] J. Kuipers (1994), Combinatorial methods in cooperative game theory, Ph.D. Dissertation, Maastricht University, [26] M. Le Breton, G. Owen and S. Weber (1992), Strongly balanced cooperative games, International Journal of Game Theory, 21 (1992) [27] L. Lovász (1972a), Normal hypergraphs and the weak perfect graph conjecture, Discrete Math., 2 N 3, (1972) [28] L. Lovász (1972b), A characterization of perfect graphs, J. Combinatorial Theory, B13, N 2 (1972)

16 [29] H. Moulin and B. Peleg (1982), Cores of effectivity functions and implementation theory, J. of Mathematical Economics, 10 (1982) [30] H. Moulin (1983), The strategy of social choice, (North-Holland, Amsterdam, 1983). [31] K. Nakamura (1975), The core of a simple game with ordinal preferences, Int. J. of game theory, 4, (1975) [32] J. von Neumann and O. Morgenstern (1944), Theory of games and economic behaviour, subsection 157.2,3 (Princeton, Princeton University Press, 1944). [33] B. Peleg (1965), An inductive method for constructing minimal balanced collections of finite sets, Naval Research Logistics Quarterly, 12 (1965) [34] T. Quint (1991), Necessary and sufficient conditions for balancedness in partitioning games, Mathematical Social Sciences, 22 (1991) [35] T. Quint (1994), Restricted houswapping games, Mimeo, Yale University, [36] H. Scarf (1967), The core of n person game, Econometrica 35, N 1 (1967) [37] L.S. Shapley (1965), On balanced sets and cores, RAND corporation, RM-4601-PR, [38] L.S. Shapley (1967), On balanced sets and cores, Naval Research Logistics Quarterly, 14, (1967) [39] L.S. Shapley and M. Shubik (1972), The assignment game. I: The core, International Journal of Game Theory, 1 (1972) [40] A.A. Vasin (1983), The models for the processes with few paticipants, section 2, (Moscow University Publishers, 1983) 84 p. (in Russian). 16