- Email: [email protected]
A note on the core of voting games
Journal of Mathematical Economics 33 Ž2000. 367–372 www.elsevier.comrlocaterjmateco
A note on the core of voting games Nicolas Gabriel Andjiga
a,b
, Boniface Mbih
b,)
a
b
CREAM, Ecole Normale Superieure de Yaounde, ´ ´ Yaounde, ´ Cameroon CREME, Faculte de Sciences Economiques et de Gestion, UniÕersite´ de Caen, 14023 Caen, Cedex 14, France
Received 17 April 1997; received in revised form 7 September 1998; accepted 20 February 1999
Abstract In this paper, we give a definition of the dominance relation which is slightly different from the usual dominance relation, in order to explicitly take into consideration the possibility for a voting game Ža simple game. to be non-monotonic. In this context, we obtain a characterization of core stable voting games which is a generalization of Nakamura’s theorem. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: C71; D71 Keywords: Dominance; Core stability; Voting game
1. Introduction It seems natural to consider that when some group of individuals has the power to enforce a decision, a larger group containing the former also has this same power. This idea has been formalized in cooperative game theory Žand more precisely in the literature on voting games. as a monotonicity property. However, this assumption, it seems to us, is not always realistic and indeed one can think of
)
Corresponding author. [email protected]
Tel.:
q33-231-5654-25;
Fax:
q33-131-9361-94;
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 0 3 - 8
E-mail:
368
N.G. Andjiga, B. Mbih r Journal of Mathematical Economics 33 (2000) 367–372
situations in which the introduction of an additional individual within a group reduces the power of this group. These situations can happen in at least two circumstances, illustrated by the two following examples. The first case arises when, for example, in a town council, members of moderate parties forming a majority systematically give up supporting any motion in favour of which an extreme-right party, the minority, expresses its preference; such a motion may be suspect to the public opinion. One illustration of the second case is given by the well-known no show and monotonicity paradoxes in social choice theory, where, roughly speaking, the winner of an election is displaced after one or more voters change their preferences in a way favourable to him, without any change in the relative rankings of the other candidates. Plurality rule with runoff and Hare system of preferential voting are the usual examples of non-monotonic rules Žsee Section 8.6 of Brams and Fishburn, 1982 and Moulin, 1988, p. 235.. In the context of voting games Žalso called simple games . one natural way of deriving a social comparison of the alternatives is the use of the dominance relation Žsee Section 2.; and many results have been obtained concerning the existence—whatever the configuration of individual preferences—of some social decision upon which no coalition can improve in the process of choosing a social outcome; this is the important problem of the non-emptiness of the core, or core stability. The central result on this topic is the theorem of Nakamura Ž1979., which provides a necessary and sufficient condition for a simple game to be core stable. From a conceptual viewpoint, this analysis has been extended in several ways. For example, Andjiga and Moulen Ž1988. consider non-neutral voting games, in which a coalition may have the power to decide over a pair Ž a,b . of alternatives, but not over some other pair Ž c,d .; they provide a characterization of core stability for this class of voting games, when indifference is ruled out from individual preferences. Truchon Ž1995. extends this latter result to the case of individual preferences admitting indifference; he also provides sufficient conditions for the core stability of possibly non-symmetric Žanonymous. voting games, among other results. From a technical viewpoint, one can note that the non-emptiness of the core is always obtained in terms of the acyclicity of the dominance relation; and it is known from Blair and Pollak Ž1982. that acyclicity requires the presence of vetoers over a critical number of pairs of alternatives when the number of alternatives is greater than the number of individuals Žsee also Banks, 1995.. A more complete list of references on this theme can be found in Truchon’s paper. The goal of this paper is also to extend Nakamura’s analysis, though in a different way; we introduce a version of the dominance relation explicitly taking into account the possibility of non-monotonicity over the subsets of individuals in the society, and we show that a generalization of Nakamura’s theorem can be obtained in this context. The paper is organized as follows: Section 2 introduces the basic notation and definitions; Section 3 is devoted to the presentation of results and proofs, and Section 4 concludes the paper.
N.G. Andjiga, B. Mbih r Journal of Mathematical Economics 33 (2000) 367–372
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2. Notation and definitions Let N be a finite set of n individuals, the set of players, who are choosing among a finite set A of alternatiÕes. We assume that every player has a preference relation R i , which is a linear order over A, i.e., a complete, antisymmetric and transitive binary relation. L will denote the set of all linear orders over A, and LN, the n-fold Cartesian product of L, will be the set of all profiles over A. We assume that the distribution of Ždecision-.power among the players is specified W , A., where W , the set of winning by a Õoting game ŽVG. denoted G s Ž N,W coalitions of G, is a non-empty family of non-empty subsets of N. W , A. is said to be monotonic if for all S,T ; N, A voting game G s Ž N,W w S g W and S ; T x T g W . The words ‘simple game’ are sometimes used for monotonic voting games Žsee, for example, Peleg, 1984.. In this paper, we are interested in voting games that are possibly non-monotonic. We start by defining two notions of dominance for VGs.
´
W , A. be a VG, and let x, y g A and R N g LN . Definition 1. Let G s Ž N,W Ži. x dominates y Žwith respect to the pair Ž G, R N .., written xDŽ G, R N . y, if there exists some S g W such that for all i g S, xR i y; Žii. x w-dominates y, written xD w Ž G, R N . y, if i g N: xR i y4 g W ; Žiii. the core Žresp. the w-core . of Ž G, R N . is the set C Ž G, R N . Žresp. C w Ž G, R N .. of undominated Žresp. w-undominated. alternatives in A, with respect to Ž G, R N .; Živ. G is stable Žresp. w-stable. if for all R N g LN , C Ž G, R N . / B Žresp. C Ž G, R N . / B.. One can note that: Ža. in the above definition, point Ži. is the usual notion of dominance for simple games Žsee Nakamura, 1979.; Žb. if G is a monotonic simple game, then the notions of dominance and w-dominance are identical; Žc. let W , A. be a VG and let G m s Ž N,W W m , A. be the monotonic coÕer of G G s Ž N,W m where W s T ; N:S g W for some S ; T 4 . Clearly G m is monotonic, and the dominance relation of G equals the w-dominance relation of G m . Note that Nakamura’s definition of a simple game does not include monotonicity, but since he uses the dominance relation associated with G, his result bears on the monotonic cover of G. Now, our main problem in this paper is to find the precise conditions under which a VG is w-stable, i.e., the conditions under which its w-core is non-empty for every profile R N of preferences. Let us first give an example to illustrate the difference between the two notions of stability given in Definition 1. Example 1. Let N s 1,2,34 and A s x, y, z 4 ; suppose individual preferences are as follows: xR1 yR1 z, yR 2 zR 2 x and zR 3 xR 3 y; suppose G is a VG such that W s 24 , 34 , 2,344 . Then C Ž R N . s B; further, G is clearly not monotonic, and as the reader can easily check, the only w-undominated alternative is z; hence C w Ž R N . s z 4.
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N.G. Andjiga, B. Mbih r Journal of Mathematical Economics 33 (2000) 367–372
This example shows that in the general case the dominance and the w-dominance Žrelations. of a voting game will be distinct; furthermore, G is w-stable Žthe alternative ranked last in player 1’s preference relation is always in the core., but not stable. Let us now turn to our main result.
3. Core stability In this section, we prove the main result of the paper: the characterization of w-stable VGs. To start, let us introduce the following notation: V Ž G . s s ; W :F S j :S j g s 4 s B4 and V w Ž G . s s ; W :F S j :S j g s 4 s B and j S j :S j g s 4 s N 4 . We then give the following definition. W , A. be a VG. Definition 2. Let G s Ž N,W Ži. The Nakamura’s number of G, denoted n Ž G ., is given by n Ž G . s q` if V Ž G . s B, and n Ž G . s min s g V ŽG. < s < otherwise. Žii. The w-number of G, denoted n w Ž G . is given by n w Ž G . s q` if V w Ž G . s B, and n w Ž G . s min s g V w ŽG. < s < otherwise. From the above definition we now recall Nakamura’s theorem before giving our own results. W , A. be a simple game. Then G is Theorem 1 (Nakamura, 1979). Let G s Ž N,W stable if and only if < A < - n Ž G .. We first write the following result. W , A. be a VG. If G is monotonic, then n Ž G . s n w Ž G .. Theorem 2. Let G s Ž N,W Proof. First, it is clear from the definitions of V w Ž G . and V Ž G . that V w Ž G . : V Ž G .; hence n Ž G . F n w Ž G .. Now, let s s S1 , . . . ,Sm 4 be a subset of W such that n Ž G . s m. If j S j :S j g s 4 s N, then n w Ž G . s n Ž G .. If j S j :S j g s 4 ; N, consider the sets S and s X such that S s N y j S j :S j g s 4 and s X s S1X s S1 j S,SX2 s S2 , . . . ,SXm s Sm 4 . By monotonicity, S1X g W . Then F SXj :SXj g s X 4 s B and j SXj :SXj g s X 4 s N, and again n w Ž G . s n Ž G .. Q.E.D. Note in particular that for non-symmetric majority games G we will in general have n Ž G . - n w Ž G . as shown in the following example. Example 2. Let G be a majority game with n s 11, in which the winning coalitions are the following: every coalition with at least nine individuals, and S1 s 1,2,3,4,5,64 , S2 s 1,2,3,7,8,94 , S3 s 4,5,6,7,8,94 , S4 s 2,4,5,7,8,10,11 4 . The reader can check that the only three winning coalitions with an empty
N.G. Andjiga, B. Mbih r Journal of Mathematical Economics 33 (2000) 367–372
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intersection are S1 , S2 , and S3 , whereas S1 j S2 j S3 / N and S1 j S2 j S3 j S4 s N; thus n Ž G . s 3 - n w Ž G . s 4. The next result is useful for the characterization of w-stable VGs. Lemma 1. The two following statements are true. Ž i . Given a subset B s a1 , . . . ,a m 4 of A and a profile R N g LN , define Ž S j . js1, . . . , m by S j s i g N:a j R i a jq14 , where a mq1 s a1 , for all j s 1, . . . ,m. Then F S j , j s 1, . . . ,m4 s B and j S j , j s 1, . . . ,m4 s N. Ž ii . Conversely, given a collection Ž S j . js1, . . . , m of non-empty subsets of N satisfying F S j , j s 1, . . . ,m4 s B and j S j , j s 1, . . . ,m4 s N and m F < A <, then for every subset B s a1 , . . . ,a m 4 of A, there exists R N g LN such that for all j s 1, . . . ,m, S j s i g N:a j R i a jq14 , where a mq1 s a1. Proof. Let us first show that Ži. is true. Sm F S j :j / m4 s B because if i g F S j s 1, . . . ,m4 then R i will contain a cycle; and j S j s 1, . . . ,m4 s N because if there exists some i such that i f j S j s 1, . . . ,m4 then R i will contain the opposite cycle. We now show that Žii. is true. Let M s 1, . . . ,m4 be an index set. Since F S j : j g M 4 s B and j S j : j g M 4 s N, for every i in N, we must have B ; M Ž i . ; M, where M Ž i . s j g M:i g S j 4 . So, for every i g N, RŽ i . s Ž x j , x jq1 .: j g M Ž i .4 j Ž x jq1 , x j .: j f M Ž i .4 has no cycle and is thus contained in a linear order R i. Finally, by construction, profile R N s Ž R 1, . . . , R n . satisfies the required condition. Q.E.D. W , A. be a VG. Then G is w-stable if and only if Theorem 3. Let G s Ž N,W < A < - n w Ž G .. Proof. Sufficiency: we have to show that if < A < - n w Ž G ., then G is w-stable; suppose on the contrary that G is not w-stable, then there exists some profile R N g LN such that C w Ž G, R N . s B. Since A is finite, there exists some subset B s a1 , . . . ,a m 4 of A such that a j D w Ž G, R N . a jq1 for every j s 1, . . . ,m, with a mq 1 s a1. By the definition of w-dominance, we must have the following: for all j s 1, . . . ,m, S j s i g N:a j R i a jq14 g W ; and by Lemma 1, F S j , j s 1, . . . ,m4 s B and j S j , j s 1, . . . ,m4 s N; thus n w Ž G . F m; and since m s < B <, we must have n w Ž G . F < A <. Necessity: we have to show that if G is w-stable, then < A < - n w Ž G .; suppose on the contrary that n w Ž G . F < A <; further suppose m s n w Ž G . and let Ž S j ., j s 1, . . . ,m be a collection of winning coalitions satisfying F S j , j s 1, . . . ,m4 s B and j S j , j s 1, . . . ,m4 s N. Since m s n w Ž G . F < A <, we can partition A in m non-empty subsets denoted Bj , j s 1, . . . ,m. Now, let RŽ i . s Ž x, y .: x g Bj and y g Bjq1 if i g S j 4 j Ž y, x .: x g Bj and y g Bjq1 if i f S j 4 . Then RŽ i . is contained in a linear order R i and by Lemma 1 Žii., there exists R N g LN such that C w Ž G, R N . s B. Q.E.D.
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4. Concluding remarks To end this paper, we shall say a few words about weak orders. More precisely, one interesting question is how the results in this paper are modified if instead of assuming individual preferences to be strict orders Ži.e., linear orders., we assume that they are weak orders Ži.e., complete and transitive binary relations over the set of alternatives.. Indeed, the weak order hypothesis in place of strict orders leads to only slight modifications in the definitions of dominance and stability; and we then speak of w-stability. However, we can make the following remarks: Ži. if G is monotonic, then the dominance and the w-dominance are equivalent; Žii. G is stable if and only if G is w-stable Žsee Peleg, 1984.; Žiii. If G is w-w-stable, then G is w-stable. But the converse is not true; and the reader can check that the game defined in Example 1 is w-stable but not w-w-stable.
Acknowledgements We would like to thank Joel ¨ Moulen for helpful remarks and stimulating discussions. Further insightful comments and suggestions are due to an anonymous referee.
References Andjiga, N.G., Moulen, J., 1988. Binary games in constitutional form and collective choice. Mathematical Social Sciences 16, 189–201. Banks, J.S., 1995. Acyclic social choice from finite sets. Social Choice and Welfare 12, 293–310. Blair, D.H., Pollak, R.A., 1982. Acyclic collective choice rules. Econometrica 50, 931–993. Brams, S.J., Fishburn, P.C., 1982. Approval Voting. Birkhauser, Boston. ¨ Moulin, H., 1988. Axioms of Cooperative Decision-Making. Cambridge Univ. Press, Cambridge. Nakamura, K., 1979. The vetoers in a simple game with ordinal preferences. International Journal of Game Theory 8, 55–61. Peleg, B., 1984. Game-Theoretic Analysis of Voting in Committees. Cambridge Univ. Press, Cambridge. Truchon, M., 1995. Voting games and collective choice rules. Mathematical Social Sciences 29, 165–179.
A note on the core of voting games Nicolas Gabriel Andjiga
a,b
, Boniface Mbih
b,)
a
b
CREAM, Ecole Normale Superieure de Yaounde, ´ ´ Yaounde, ´ Cameroon CREME, Faculte de Sciences Economiques et de Gestion, UniÕersite´ de Caen, 14023 Caen, Cedex 14, France
Received 17 April 1997; received in revised form 7 September 1998; accepted 20 February 1999
Abstract In this paper, we give a definition of the dominance relation which is slightly different from the usual dominance relation, in order to explicitly take into consideration the possibility for a voting game Ža simple game. to be non-monotonic. In this context, we obtain a characterization of core stable voting games which is a generalization of Nakamura’s theorem. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: C71; D71 Keywords: Dominance; Core stability; Voting game
1. Introduction It seems natural to consider that when some group of individuals has the power to enforce a decision, a larger group containing the former also has this same power. This idea has been formalized in cooperative game theory Žand more precisely in the literature on voting games. as a monotonicity property. However, this assumption, it seems to us, is not always realistic and indeed one can think of
)
Corresponding author. [email protected]
Tel.:
q33-231-5654-25;
Fax:
q33-131-9361-94;
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 0 3 - 8
E-mail:
368
N.G. Andjiga, B. Mbih r Journal of Mathematical Economics 33 (2000) 367–372
situations in which the introduction of an additional individual within a group reduces the power of this group. These situations can happen in at least two circumstances, illustrated by the two following examples. The first case arises when, for example, in a town council, members of moderate parties forming a majority systematically give up supporting any motion in favour of which an extreme-right party, the minority, expresses its preference; such a motion may be suspect to the public opinion. One illustration of the second case is given by the well-known no show and monotonicity paradoxes in social choice theory, where, roughly speaking, the winner of an election is displaced after one or more voters change their preferences in a way favourable to him, without any change in the relative rankings of the other candidates. Plurality rule with runoff and Hare system of preferential voting are the usual examples of non-monotonic rules Žsee Section 8.6 of Brams and Fishburn, 1982 and Moulin, 1988, p. 235.. In the context of voting games Žalso called simple games . one natural way of deriving a social comparison of the alternatives is the use of the dominance relation Žsee Section 2.; and many results have been obtained concerning the existence—whatever the configuration of individual preferences—of some social decision upon which no coalition can improve in the process of choosing a social outcome; this is the important problem of the non-emptiness of the core, or core stability. The central result on this topic is the theorem of Nakamura Ž1979., which provides a necessary and sufficient condition for a simple game to be core stable. From a conceptual viewpoint, this analysis has been extended in several ways. For example, Andjiga and Moulen Ž1988. consider non-neutral voting games, in which a coalition may have the power to decide over a pair Ž a,b . of alternatives, but not over some other pair Ž c,d .; they provide a characterization of core stability for this class of voting games, when indifference is ruled out from individual preferences. Truchon Ž1995. extends this latter result to the case of individual preferences admitting indifference; he also provides sufficient conditions for the core stability of possibly non-symmetric Žanonymous. voting games, among other results. From a technical viewpoint, one can note that the non-emptiness of the core is always obtained in terms of the acyclicity of the dominance relation; and it is known from Blair and Pollak Ž1982. that acyclicity requires the presence of vetoers over a critical number of pairs of alternatives when the number of alternatives is greater than the number of individuals Žsee also Banks, 1995.. A more complete list of references on this theme can be found in Truchon’s paper. The goal of this paper is also to extend Nakamura’s analysis, though in a different way; we introduce a version of the dominance relation explicitly taking into account the possibility of non-monotonicity over the subsets of individuals in the society, and we show that a generalization of Nakamura’s theorem can be obtained in this context. The paper is organized as follows: Section 2 introduces the basic notation and definitions; Section 3 is devoted to the presentation of results and proofs, and Section 4 concludes the paper.
N.G. Andjiga, B. Mbih r Journal of Mathematical Economics 33 (2000) 367–372
369
2. Notation and definitions Let N be a finite set of n individuals, the set of players, who are choosing among a finite set A of alternatiÕes. We assume that every player has a preference relation R i , which is a linear order over A, i.e., a complete, antisymmetric and transitive binary relation. L will denote the set of all linear orders over A, and LN, the n-fold Cartesian product of L, will be the set of all profiles over A. We assume that the distribution of Ždecision-.power among the players is specified W , A., where W , the set of winning by a Õoting game ŽVG. denoted G s Ž N,W coalitions of G, is a non-empty family of non-empty subsets of N. W , A. is said to be monotonic if for all S,T ; N, A voting game G s Ž N,W w S g W and S ; T x T g W . The words ‘simple game’ are sometimes used for monotonic voting games Žsee, for example, Peleg, 1984.. In this paper, we are interested in voting games that are possibly non-monotonic. We start by defining two notions of dominance for VGs.
´
W , A. be a VG, and let x, y g A and R N g LN . Definition 1. Let G s Ž N,W Ži. x dominates y Žwith respect to the pair Ž G, R N .., written xDŽ G, R N . y, if there exists some S g W such that for all i g S, xR i y; Žii. x w-dominates y, written xD w Ž G, R N . y, if i g N: xR i y4 g W ; Žiii. the core Žresp. the w-core . of Ž G, R N . is the set C Ž G, R N . Žresp. C w Ž G, R N .. of undominated Žresp. w-undominated. alternatives in A, with respect to Ž G, R N .; Živ. G is stable Žresp. w-stable. if for all R N g LN , C Ž G, R N . / B Žresp. C Ž G, R N . / B.. One can note that: Ža. in the above definition, point Ži. is the usual notion of dominance for simple games Žsee Nakamura, 1979.; Žb. if G is a monotonic simple game, then the notions of dominance and w-dominance are identical; Žc. let W , A. be a VG and let G m s Ž N,W W m , A. be the monotonic coÕer of G G s Ž N,W m where W s T ; N:S g W for some S ; T 4 . Clearly G m is monotonic, and the dominance relation of G equals the w-dominance relation of G m . Note that Nakamura’s definition of a simple game does not include monotonicity, but since he uses the dominance relation associated with G, his result bears on the monotonic cover of G. Now, our main problem in this paper is to find the precise conditions under which a VG is w-stable, i.e., the conditions under which its w-core is non-empty for every profile R N of preferences. Let us first give an example to illustrate the difference between the two notions of stability given in Definition 1. Example 1. Let N s 1,2,34 and A s x, y, z 4 ; suppose individual preferences are as follows: xR1 yR1 z, yR 2 zR 2 x and zR 3 xR 3 y; suppose G is a VG such that W s 24 , 34 , 2,344 . Then C Ž R N . s B; further, G is clearly not monotonic, and as the reader can easily check, the only w-undominated alternative is z; hence C w Ž R N . s z 4.
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N.G. Andjiga, B. Mbih r Journal of Mathematical Economics 33 (2000) 367–372
This example shows that in the general case the dominance and the w-dominance Žrelations. of a voting game will be distinct; furthermore, G is w-stable Žthe alternative ranked last in player 1’s preference relation is always in the core., but not stable. Let us now turn to our main result.
3. Core stability In this section, we prove the main result of the paper: the characterization of w-stable VGs. To start, let us introduce the following notation: V Ž G . s s ; W :F S j :S j g s 4 s B4 and V w Ž G . s s ; W :F S j :S j g s 4 s B and j S j :S j g s 4 s N 4 . We then give the following definition. W , A. be a VG. Definition 2. Let G s Ž N,W Ži. The Nakamura’s number of G, denoted n Ž G ., is given by n Ž G . s q` if V Ž G . s B, and n Ž G . s min s g V ŽG. < s < otherwise. Žii. The w-number of G, denoted n w Ž G . is given by n w Ž G . s q` if V w Ž G . s B, and n w Ž G . s min s g V w ŽG. < s < otherwise. From the above definition we now recall Nakamura’s theorem before giving our own results. W , A. be a simple game. Then G is Theorem 1 (Nakamura, 1979). Let G s Ž N,W stable if and only if < A < - n Ž G .. We first write the following result. W , A. be a VG. If G is monotonic, then n Ž G . s n w Ž G .. Theorem 2. Let G s Ž N,W Proof. First, it is clear from the definitions of V w Ž G . and V Ž G . that V w Ž G . : V Ž G .; hence n Ž G . F n w Ž G .. Now, let s s S1 , . . . ,Sm 4 be a subset of W such that n Ž G . s m. If j S j :S j g s 4 s N, then n w Ž G . s n Ž G .. If j S j :S j g s 4 ; N, consider the sets S and s X such that S s N y j S j :S j g s 4 and s X s S1X s S1 j S,SX2 s S2 , . . . ,SXm s Sm 4 . By monotonicity, S1X g W . Then F SXj :SXj g s X 4 s B and j SXj :SXj g s X 4 s N, and again n w Ž G . s n Ž G .. Q.E.D. Note in particular that for non-symmetric majority games G we will in general have n Ž G . - n w Ž G . as shown in the following example. Example 2. Let G be a majority game with n s 11, in which the winning coalitions are the following: every coalition with at least nine individuals, and S1 s 1,2,3,4,5,64 , S2 s 1,2,3,7,8,94 , S3 s 4,5,6,7,8,94 , S4 s 2,4,5,7,8,10,11 4 . The reader can check that the only three winning coalitions with an empty
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intersection are S1 , S2 , and S3 , whereas S1 j S2 j S3 / N and S1 j S2 j S3 j S4 s N; thus n Ž G . s 3 - n w Ž G . s 4. The next result is useful for the characterization of w-stable VGs. Lemma 1. The two following statements are true. Ž i . Given a subset B s a1 , . . . ,a m 4 of A and a profile R N g LN , define Ž S j . js1, . . . , m by S j s i g N:a j R i a jq14 , where a mq1 s a1 , for all j s 1, . . . ,m. Then F S j , j s 1, . . . ,m4 s B and j S j , j s 1, . . . ,m4 s N. Ž ii . Conversely, given a collection Ž S j . js1, . . . , m of non-empty subsets of N satisfying F S j , j s 1, . . . ,m4 s B and j S j , j s 1, . . . ,m4 s N and m F < A <, then for every subset B s a1 , . . . ,a m 4 of A, there exists R N g LN such that for all j s 1, . . . ,m, S j s i g N:a j R i a jq14 , where a mq1 s a1. Proof. Let us first show that Ži. is true. Sm F S j :j / m4 s B because if i g F S j s 1, . . . ,m4 then R i will contain a cycle; and j S j s 1, . . . ,m4 s N because if there exists some i such that i f j S j s 1, . . . ,m4 then R i will contain the opposite cycle. We now show that Žii. is true. Let M s 1, . . . ,m4 be an index set. Since F S j : j g M 4 s B and j S j : j g M 4 s N, for every i in N, we must have B ; M Ž i . ; M, where M Ž i . s j g M:i g S j 4 . So, for every i g N, RŽ i . s Ž x j , x jq1 .: j g M Ž i .4 j Ž x jq1 , x j .: j f M Ž i .4 has no cycle and is thus contained in a linear order R i. Finally, by construction, profile R N s Ž R 1, . . . , R n . satisfies the required condition. Q.E.D. W , A. be a VG. Then G is w-stable if and only if Theorem 3. Let G s Ž N,W < A < - n w Ž G .. Proof. Sufficiency: we have to show that if < A < - n w Ž G ., then G is w-stable; suppose on the contrary that G is not w-stable, then there exists some profile R N g LN such that C w Ž G, R N . s B. Since A is finite, there exists some subset B s a1 , . . . ,a m 4 of A such that a j D w Ž G, R N . a jq1 for every j s 1, . . . ,m, with a mq 1 s a1. By the definition of w-dominance, we must have the following: for all j s 1, . . . ,m, S j s i g N:a j R i a jq14 g W ; and by Lemma 1, F S j , j s 1, . . . ,m4 s B and j S j , j s 1, . . . ,m4 s N; thus n w Ž G . F m; and since m s < B <, we must have n w Ž G . F < A <. Necessity: we have to show that if G is w-stable, then < A < - n w Ž G .; suppose on the contrary that n w Ž G . F < A <; further suppose m s n w Ž G . and let Ž S j ., j s 1, . . . ,m be a collection of winning coalitions satisfying F S j , j s 1, . . . ,m4 s B and j S j , j s 1, . . . ,m4 s N. Since m s n w Ž G . F < A <, we can partition A in m non-empty subsets denoted Bj , j s 1, . . . ,m. Now, let RŽ i . s Ž x, y .: x g Bj and y g Bjq1 if i g S j 4 j Ž y, x .: x g Bj and y g Bjq1 if i f S j 4 . Then RŽ i . is contained in a linear order R i and by Lemma 1 Žii., there exists R N g LN such that C w Ž G, R N . s B. Q.E.D.
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4. Concluding remarks To end this paper, we shall say a few words about weak orders. More precisely, one interesting question is how the results in this paper are modified if instead of assuming individual preferences to be strict orders Ži.e., linear orders., we assume that they are weak orders Ži.e., complete and transitive binary relations over the set of alternatives.. Indeed, the weak order hypothesis in place of strict orders leads to only slight modifications in the definitions of dominance and stability; and we then speak of w-stability. However, we can make the following remarks: Ži. if G is monotonic, then the dominance and the w-dominance are equivalent; Žii. G is stable if and only if G is w-stable Žsee Peleg, 1984.; Žiii. If G is w-w-stable, then G is w-stable. But the converse is not true; and the reader can check that the game defined in Example 1 is w-stable but not w-w-stable.
Acknowledgements We would like to thank Joel ¨ Moulen for helpful remarks and stimulating discussions. Further insightful comments and suggestions are due to an anonymous referee.
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