- Email: [email protected]
An acyclic relation for comparison of bargaining powers of coalitions and its interrelationship with bargaining set
Applied Mathematics and Computation 215 (2010) 3665–3668
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An acyclic relation for comparison of bargaining powers of coalitions and its interrelationship with bargaining set Kentaro Kojima, Takehiro Inohara * Department of Value and Decision Science, Graduate School of Social and Decision Science and Technology, Tokyo Institute of Technology, W9-38, 2-12-1 O-okayama Meguro, Tokyo 152-8552, Japan
a r t i c l e Keywords: Game theory Cooperative games Relations Bargaining Coalitions Power
i n f o
a b s t r a c t This paper proposes a method to compare bargaining power of coalitions within the framework of games in coalition form with transferable utility. The method is expressed by a relation on the set of all coalitions in a game, the relation which is defined based on the players’ bargaining power. It is shown in this paper that the newly defined relation satisfies acyclicity. Also, it is verified in this paper that the set of all individually rational payoff configurations under which all coalitions have the equal bargaining power coincides with the bargaining set. Some examples demonstrate how the newly proposed method works. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Cooperative games in which all players can bargain together with perfect information are introduced by Aumann and Maschler [1]. One of the fundamental questions regarding such cooperative games is what coalitions will form in the games. Bargaining power of coalitions in the games is helpful to get close to the answer of this question. In the literature, players in a game are compared regarding their bargaining power using a relation, which is derived by the concepts of objection and counter-objection [5,6], on the set of all players. In this paper a method is proposed for the comparison of bargaining power of coalitions by extending the relation to the one on the set of all coalitions. Some examples are provided to demonstrate how the newly proposed method works. Some types of relations have been proposed to compare coalitions with respect to their power within the framework of simple games, and such properties of the relations as completeness and transitivity have been investigated [2–4]. These properties are quite desirable for providing definitions of power indices, but often too demanding. Simple games, moreover, are appropriate to treat such decision making situations as voting committees, but not suitable for the analysis of situations of bargaining for profit allocation, the situation which are often expressed by games in coalition form. In this paper, acyclicity of the relation proposed for the comparison of bargaining power of coalitions is examined within the framework of games in coalition form with transferable utility (called games hereafter). Games are more general than simple games in the sense that games include simple games as special cases. Acyclicity is one of the important properties of relations, because one can find the maximal coalitions from all coalitions with respect to the relations. In this paper a proposition shows that the newly proposed relation on the set of coalitions satisfies acyclicity. Another proposition shows a relationship between the proposed relation and the bargaining set [1]. More specifically, it is verified that the set of all individually rational payoff configurations under which all coalitions have the equal bargaining power coincides with the bargaining set.
* Corresponding author. E-mail address: [email protected] (T. Inohara). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.11.004
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The structure of this paper is as follows: the framework of games and that of bargaining among players in the games are presented in the next section. In Section 3, the definition of the newly proposed relation on the set of all coalitions in games are provided, and the propositions mentioned above are given. The last section is devoted for concluding remarks. 2. Preliminaries In this section, the notation employed in this paper is introduced, and such concepts as games, payoff configurations, objections, and counter-objections are defined. These definitions are based on [5,6]. Let N ¼ f1; 2; . . . ; ng be a set of n players. Each non-empty subset of N is called a coalition, and a coalition S ¼ fi1 ; i2 ; . . . ; im g is often denoted by i1 i2 ; . . . ; im for simplicity. A coalition structure P of N is defined as a partition of N, which is defined as a 0 family fT 1 ; T 2 ; . . . ; T m g of pairwise disjoint (that is, T j \ T j0 ¼ ; if j – j ) non-empty coalitions T j ðj ¼ 1; 2; . . . ; mÞ whose union T is N. [m j¼1 j A characteristic function v : 2N n f;g ! R assigns a real number to each coalition, where 2N and R denotes the power set of N and the set of all real numbers, respectively. The pair ðN; v Þ is said to be a game. A payoff configuration for ðN; v Þ is a pair P ðx; PÞ of an n-vector x ¼ ðx1 ; x2 ; . . . ; xn Þ 2 Rn and a coalition structure P ¼ fT 1 ; T 2 ; . . . ; T m g of N satisfying i2T j xi ¼ v ðT j Þ for j ¼ 1; 2; . . . ; m. If a payoff configuration ðx; PÞ for ðN; v Þ satisfies xi P v ðfigÞ for all i 2 N; ðx; PÞ is said to be individually rational. An individually rational payoff configuration is often abbreviated by an i.r.p.c. The following definitions of objections, counter-objections, a relation on players, and acyclicity of relations are based on [5,6]. Definition 1 (Objections). Consider a game ðN; v Þ, and let ðx; PÞ be an i.r.p.c. for ðN; v Þ. Let, moreover, h and k be two distinct players in coalition T 2 P. An objection of k against h in ðx; PÞ is such an i.r.p.c. ðy; P0 Þ for ðN; v Þ that there exists T 0 2 P0 such that k 2 T 0 ; h R T 0 , and yi > xi for all i 2 T 0 . An objection of k against h expresses the situation that player k is insisting that player h does not have to be a member of k’s coalition, because k can form another coalition T 0 , in which h is not contained, such that the payoff yi of each member i of the new coalition T 0 will be more than xi . Definition 2 (Counter-objections). Consider a game ðN; v Þ, and let ðx; PÞ be an i.r.p.c. for ðN; v Þ. Let, moreover, h and k be two distinct players in coalition T 2 P. Suppose an objection ðy; P0 Þ of k against h, where T 0 2 P0 satisfies that k 2 T 0 ; h R T 0 , and yi > xi for all i 2 T 0 . Then, a counter-objection of h against k with respect to the objection ðy; P0 Þ is such an i.r.p.c. ðz; P00 Þ that there exists T 00 2 P00 such that h 2 T 00 , k R T 00 ; zi P xi for all i 2 T 00 , and zi P yi for all i 2 T 0 \ T 00 . A counter-objection of h with respect to the objection of k to form the coalition T 0 2 P0 , in which h is not contained, weakens the power of the objection, because h can form the coalition T 00 2 P00 , in which k is not contained and each member obtains equal or more payoff than in the case he/she participates in the original coalition T 2 P or in the coalition T 0 proposed in the objection of k. The next gives an example of objections and counter-objections. Example 1. Consider a game ðN; v Þ such that N ¼ f1; 2; 3g, v ð1Þ ¼ v ð2Þ ¼ v ð3Þ ¼ 0, v ð12Þ ¼ v ð13Þ ¼ v ð123Þ ¼ 100, and v ð23Þ ¼ 50. Then, consider the i.r.p.c. ðx; PÞ ¼ ðð75; 25; 0Þ; f12; 3gÞ. In this case, player 2 has an objection ðy; P0 Þ ¼ ðð0; 26; 24Þ; f1; 23gÞ against player 1, and player 1 has a counter-objection ðz; P00 Þ ¼ ðð76; 0; 24Þ; f13; 2gÞ with respect to the objection ðy; P0 Þ of player 2. A relation on the set N of all players can be defined based on the concepts of objections and counter-objections. Definition 3 (Relation on players in ðx; PÞ). Consider a game ðN; v Þ, and let ðx; PÞ be an i.r.p.c. for ðN; v Þ. Suppose two players h and k in N. Then, player k is said to be stronger than player h (or, equivalently, player h is weaker than player k) in ðx; PÞ, if and only if k has an objection against h, but h does not have any counter-objections with respect to the objection, denoted by k h. k is said to be equal to h, denoted by k h, if and only if neither k h nor h k hold. We see, Definition 1 of objections, that if k h, then k and h are elements of the same coalition in P. In other words, one has neither k h nor h k, if k and h belong to different coalitions in P. That is, the relation is, in general, a partial relation. The next gives a numerical example of the relation on the set N of all players. Example 2. Consider the game ðN; v Þ in Example 1, and suppose the i.r.p.c. ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ. The i.r.p.c. ðy; P0 Þ ¼ ðð0; 21; 29Þ; f1; 23gÞ is an objection of player 2 against player 1. Player 1, however, does not have any counterobjection ðz; P00 Þ with respect to this objection ðy; P0 Þ, because player 1 cannot obtain 80 if he/she offers 29 or more to player 3. Hence, we have that 2 1 in ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ. We have, in Example 1, that 2 1, but we never have that 1 2. This fact is guaranteed by the acyclicity of the relation . Acyclicity of relations is defined as follows: Definition 4 (Acyclicity of relations). Consider a game ðN; v Þ and the relation on the set N of players in ðx; PÞ. The relation is said to be acyclic, if and only if there do not exist such players 1; 2; . . . ; t that 1 2 t 1.
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Under the acyclicity of a relation on the set N of all players, one can find the maximal players from N with respect to the relation. The next lemma verifies that the relation defined in Definition 3 is acyclic. Lemma 1. Let ðx; PÞ be an i.r.p.c. for a game ðN; v Þ, then the relation on the set N of all players is acyclic. Proof. See [5].
h
This lemma implies, in particular, that the relation is asymmetric, that is, for i and j in N, if i j, then j i is not true. As defined in Definition 3, for h and k in N; k h denotes that neither k h nor h k hold. Using this relation on N, Aumann and Maschler [1] defines the concept of M-stability of i.r.p.c.s. for a game ðN; v Þ. Definition 5 (M-stability of i.r.p.c.s. [1]). Consider a game ðN; v Þ. An i.r.p.c. ðx; PÞ for ðN; v Þ is said to be M-stable, if and only if for all i and j in N; i j holds. Then, for a game ðN; v Þ, the set of all M-stable i.r.p.c.s. ðx; PÞ for ðN; v Þ is called the bargaining set of ðN; v Þ. 3. Relations on coalitions This section proposes a definition of a relation on the set of all coalitions in a game. An example demonstrates how the newly proposed relation works, and a theorem shows that the proposed relation is acyclic. Definition 6 (Relation on coalitions in ðx; PÞ). Consider a game ðN; v Þ, and let ðx; PÞ be an i.r.p.c. for ðN; v Þ. Suppose two coalitions S1 and S2 in N. Then, coalition S1 is said to be stronger than coalition S2 (or, equivalently, coalition S2 is weaker than coalition S1 ) in ðx; PÞ, denoted by S1 S2 , if and only if 1. for each i 2 S1 , there exists j 2 S2 such that i j, and 2. for each i 2 S1 and each j 2 S2 , it is not satisfied that j i. Then, S1 is said to be equal to S2 , denoted by S1 S2 , if and only if neither S1 S2 nor S2 S1 hold. Note that in Definition 6, S1 and S2 can be arbitrary non-empty subsets of N, and in particular, it is not assumed that S1 or S are coalitions in the coalition structure P. We see, from Definition 6 and the comments just after Definition 3, that if S1 S2 in ðx; PÞ and S1 \ T – ; for some T 2 P, then S2 \ T – ;. The following two numerical examples show how the newly proposed relation on the set of all coalitions works. Example 3 demonstrates that if a player i 2 N is identified with a one-player coalition fig in N, then the newly proposed relation on coalitions reserves the relation on players. 2
Example 3. In Example 2, we see that 2 1 in ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ in the game ðN; v Þ given in Example 1. We also see, by Lemma 1, that 1 2 does not hold in ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ. Therefore, it holds f2g f1g in ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ. Example 4 demonstrates how the newly proposed relation on the set of all coalitions works for comparing coalitions with two or more members. Example 4. Consider the game ðN; v Þ such that N ¼ f1; 2; 3; 4g, v ð1Þ ¼ v ð2Þ ¼ v ð3Þ ¼ v ð4Þ ¼ 0, v ð12Þ ¼ v ð13Þ ¼ v ð123Þ ¼ v ð134Þ ¼ v ð124Þ ¼ 80, v ð14Þ ¼ v ð23Þ ¼ v ð24Þ ¼ v ð34Þ ¼ 65, v ð234Þ ¼ 75, and v ð1234Þ ¼ 120. Let us compare two coalitions, 12 and 34, in the i.r.p.c. ðx; PÞ ¼ ðð30; 30; 30; 30Þ; f1234gÞ. The i.r.p.c. ðy; P0 Þ ¼ ðð40; 40; 0; 0Þ; f12; 3; 4gÞ is an objection of player 1 against player 3, and player 3 does not have any counter-objections ðz; P00 Þ against 1 with respect to this objection. Thus, we have 1 3, and thus, 3 1 is not satisfied by the asymmetry of . Similarly, the i.r.p.c. ðy; P0 Þ ¼ ðð40; 40; 0; 0Þ; f12; 3; 4gÞ is an objection of player 2 against player 4, and player 4 does not have any counter-objections ðz; P00 Þ against 2 with respect to this objection. So, we have 2 4, and thus, 4 2 is not satisfied by the asymmetry of . Player 1 has a counter-objection against player 4 with respect to each objection ðy; P0 Þ of player 4 against player 1, that is, 4 1 is not satisfied. Similarly, player 2 has a counter-objection ðz; P00 Þ against player 3 with respect to each objection ðy; P0 Þ of player 3 against player 2, that is, 3 2 is not satisfied. Therefore, since we have 1 3, ‘‘not 3 1,” 2 4, ‘‘not 4 2,” ‘‘not 4 1,” and ‘‘not 3 2,” we have 12 34 in ðx; PÞ ¼ ðð30; 30; 30; 30Þ; f1234gÞ. The next theorem verifies that the relation defined in Definition 6 is acyclic. Theorem 1. Let ðx; PÞ be an i.r.p.c. for a game ðN; v Þ. Then, the relation on the set of all coalitions is acyclic. Proof. Assume that coalitions S1 ; S2 ; . . . ; St in N satisfies that S1 S2 St S1 . Then, for each u ðu ¼ 1; 2; . . . ; t 1Þ u uþ1 u uþ1 2 Suþ1 such that k k . This implies that there exists a sequence of players and each k 2 Su , there exists a player k 1 2 t tþ1 k ; k ; . . . ; k ; k ; . . . such that
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2
t
tþ1
k k k k 1
tþ1
2tþ1
2
2t
2tþ1
k k tþ2
2tþ2
3t
k ; t
2t
3t
where k ; k ; k ; . . . 2 S1 ; k ; k ; k ; . . . 2 S2 ; . . ., and k ; k ; k ; . . . 2 St . v w Since the set N of all player is finite, one can find v and w such that w > v and k ¼ k , that is, the sub-sequence v
k k
v þ1
w1
k
w
k ¼k
v
of the above sequence is cyclic, but this contradicts Lemma 1. Hence, the relation on the set of all coalitions is acyclic.
h
By Theorem 1, it is verified that one can find the maximal coalitions from all coalitions with respect to the newly proposed relation on the set of all coalitions. More, as in the case of the relation on the set of all players, the relation on the set of all coalitions is asymmetric, that is, for coalitions S1 and S2 in N, if S1 S2 , then S2 S1 is not true. The next example shows that the relation on the set of all coalitions is not necessarily transitive. Example 5 [5]. Consider the game ðN; v Þ such that N ¼ f1; 2; 3; 4; 5g; v ð1Þ ¼ v ð2Þ ¼ v ð3Þ ¼ 0; v ð12Þ ¼ v ð13Þ ¼ v ð123Þ ¼ 30; v ð14Þ ¼ 40; v ð35Þ ¼ 20; v ð245Þ ¼ 30, and for B N; v ðBÞ ¼ 0, otherwise. We see f1g f2g; f2g f3g and f1g f3g in ðx; PÞ ¼ ðð10; 10; 10; 0; 0Þ; f123; 4; 5gÞ. Theorem 2 verifies that the set of all i.r.p.c.s. under which all coalitions have the equal bargaining power coincides with the bargaining set. Theorem 2. Let ðN; v Þ be a game. Then, for each i.r.p.c. ðx; PÞ for ðN; v Þ, we have that ðx; PÞ is M-stable () S1 S2 in ðx; PÞ for all coalitions S1 and S2 in N. Proof. Assume that ðx; PÞ is M-stable. Then, for all players i1 and i2 in N, we have i1 i2 by Definition 5. Therefore, we have that for all coalitions S1 and S2 ; S1 S2 by Definition 6. If S1 S2 for all coalitions S1 and S2 , then considering all one-player coalitions fi1 g and fi2 g in N, we have i1 i2 for all players i1 and i2 in N, which means that ðx; PÞ is M-stable. h 4. Conclusions In this paper, a relation on the set of all coalitions in a game in coalition form with transferable utility was newly proposed in Definition 6, and the fact that the newly proposed relation satisfies acyclicity was verified in Theorem 1. More, Theorem 2 showed that the set of all i.r.p.c.s. under which all coalitions have the equivalent bargaining power coincides with the bargaining set, which is originally defined by Aumann and Maschler [1]. The proposed relation compares coalitions with respect to the abilities of making objections, and gives some insights on which coalitions may deviate from a coalition. Relationships between the proposed relation on the set of all coalitions in a game in coalition form and solution concepts for the game should be examined in future research opportunities, because solution concepts for games can be derived from relations on the set of all coalitions as in [4]. Acknowledgment The authors would like to express their appreciation to the Editor-in-Chief, Dr. Melvin Scott, and anonymous referees who kindly provided helpful suggestions which improved the quality of their paper. A part of this work is supported by KAKENHI (19310097). References [1] R.J. Aumann, M. Maschler, The bargaining set for cooperative games, Annals of Mathematics Studies, vol. 52, Princeton University Press, Princeton, NJ, 1964. [2] E. Einy, The desirability relation of simple games, Mathematical Social Sciences 10 (1985) 155–168. [3] K. Ishikawa, T. Inohara, A method to compare influence of coalitions on group decision other than desirability relation, Applied Mathematics and Computation 188-1 (2007) 838–849. [4] T. Kitamura, T. Inohara, Comparison of coalition influence on group decision, Master’s Thesis, Department of Value and Decision Science, Tokyo Institute of Technology, 2008. [5] M. Davis, M. Mashler, Existance of stable payoff configurations for cooperative games essays in mathematical economics, in: Martin Shubik (Ed.), Honor of Oskar Morgenstern, Princeton University Press, 1962, pp. 39–52. [6] G. Owen, Game Theory, third ed., 1995, pp. 313–319.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An acyclic relation for comparison of bargaining powers of coalitions and its interrelationship with bargaining set Kentaro Kojima, Takehiro Inohara * Department of Value and Decision Science, Graduate School of Social and Decision Science and Technology, Tokyo Institute of Technology, W9-38, 2-12-1 O-okayama Meguro, Tokyo 152-8552, Japan
a r t i c l e Keywords: Game theory Cooperative games Relations Bargaining Coalitions Power
i n f o
a b s t r a c t This paper proposes a method to compare bargaining power of coalitions within the framework of games in coalition form with transferable utility. The method is expressed by a relation on the set of all coalitions in a game, the relation which is defined based on the players’ bargaining power. It is shown in this paper that the newly defined relation satisfies acyclicity. Also, it is verified in this paper that the set of all individually rational payoff configurations under which all coalitions have the equal bargaining power coincides with the bargaining set. Some examples demonstrate how the newly proposed method works. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Cooperative games in which all players can bargain together with perfect information are introduced by Aumann and Maschler [1]. One of the fundamental questions regarding such cooperative games is what coalitions will form in the games. Bargaining power of coalitions in the games is helpful to get close to the answer of this question. In the literature, players in a game are compared regarding their bargaining power using a relation, which is derived by the concepts of objection and counter-objection [5,6], on the set of all players. In this paper a method is proposed for the comparison of bargaining power of coalitions by extending the relation to the one on the set of all coalitions. Some examples are provided to demonstrate how the newly proposed method works. Some types of relations have been proposed to compare coalitions with respect to their power within the framework of simple games, and such properties of the relations as completeness and transitivity have been investigated [2–4]. These properties are quite desirable for providing definitions of power indices, but often too demanding. Simple games, moreover, are appropriate to treat such decision making situations as voting committees, but not suitable for the analysis of situations of bargaining for profit allocation, the situation which are often expressed by games in coalition form. In this paper, acyclicity of the relation proposed for the comparison of bargaining power of coalitions is examined within the framework of games in coalition form with transferable utility (called games hereafter). Games are more general than simple games in the sense that games include simple games as special cases. Acyclicity is one of the important properties of relations, because one can find the maximal coalitions from all coalitions with respect to the relations. In this paper a proposition shows that the newly proposed relation on the set of coalitions satisfies acyclicity. Another proposition shows a relationship between the proposed relation and the bargaining set [1]. More specifically, it is verified that the set of all individually rational payoff configurations under which all coalitions have the equal bargaining power coincides with the bargaining set.
* Corresponding author. E-mail address: [email protected] (T. Inohara). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.11.004
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The structure of this paper is as follows: the framework of games and that of bargaining among players in the games are presented in the next section. In Section 3, the definition of the newly proposed relation on the set of all coalitions in games are provided, and the propositions mentioned above are given. The last section is devoted for concluding remarks. 2. Preliminaries In this section, the notation employed in this paper is introduced, and such concepts as games, payoff configurations, objections, and counter-objections are defined. These definitions are based on [5,6]. Let N ¼ f1; 2; . . . ; ng be a set of n players. Each non-empty subset of N is called a coalition, and a coalition S ¼ fi1 ; i2 ; . . . ; im g is often denoted by i1 i2 ; . . . ; im for simplicity. A coalition structure P of N is defined as a partition of N, which is defined as a 0 family fT 1 ; T 2 ; . . . ; T m g of pairwise disjoint (that is, T j \ T j0 ¼ ; if j – j ) non-empty coalitions T j ðj ¼ 1; 2; . . . ; mÞ whose union T is N. [m j¼1 j A characteristic function v : 2N n f;g ! R assigns a real number to each coalition, where 2N and R denotes the power set of N and the set of all real numbers, respectively. The pair ðN; v Þ is said to be a game. A payoff configuration for ðN; v Þ is a pair P ðx; PÞ of an n-vector x ¼ ðx1 ; x2 ; . . . ; xn Þ 2 Rn and a coalition structure P ¼ fT 1 ; T 2 ; . . . ; T m g of N satisfying i2T j xi ¼ v ðT j Þ for j ¼ 1; 2; . . . ; m. If a payoff configuration ðx; PÞ for ðN; v Þ satisfies xi P v ðfigÞ for all i 2 N; ðx; PÞ is said to be individually rational. An individually rational payoff configuration is often abbreviated by an i.r.p.c. The following definitions of objections, counter-objections, a relation on players, and acyclicity of relations are based on [5,6]. Definition 1 (Objections). Consider a game ðN; v Þ, and let ðx; PÞ be an i.r.p.c. for ðN; v Þ. Let, moreover, h and k be two distinct players in coalition T 2 P. An objection of k against h in ðx; PÞ is such an i.r.p.c. ðy; P0 Þ for ðN; v Þ that there exists T 0 2 P0 such that k 2 T 0 ; h R T 0 , and yi > xi for all i 2 T 0 . An objection of k against h expresses the situation that player k is insisting that player h does not have to be a member of k’s coalition, because k can form another coalition T 0 , in which h is not contained, such that the payoff yi of each member i of the new coalition T 0 will be more than xi . Definition 2 (Counter-objections). Consider a game ðN; v Þ, and let ðx; PÞ be an i.r.p.c. for ðN; v Þ. Let, moreover, h and k be two distinct players in coalition T 2 P. Suppose an objection ðy; P0 Þ of k against h, where T 0 2 P0 satisfies that k 2 T 0 ; h R T 0 , and yi > xi for all i 2 T 0 . Then, a counter-objection of h against k with respect to the objection ðy; P0 Þ is such an i.r.p.c. ðz; P00 Þ that there exists T 00 2 P00 such that h 2 T 00 , k R T 00 ; zi P xi for all i 2 T 00 , and zi P yi for all i 2 T 0 \ T 00 . A counter-objection of h with respect to the objection of k to form the coalition T 0 2 P0 , in which h is not contained, weakens the power of the objection, because h can form the coalition T 00 2 P00 , in which k is not contained and each member obtains equal or more payoff than in the case he/she participates in the original coalition T 2 P or in the coalition T 0 proposed in the objection of k. The next gives an example of objections and counter-objections. Example 1. Consider a game ðN; v Þ such that N ¼ f1; 2; 3g, v ð1Þ ¼ v ð2Þ ¼ v ð3Þ ¼ 0, v ð12Þ ¼ v ð13Þ ¼ v ð123Þ ¼ 100, and v ð23Þ ¼ 50. Then, consider the i.r.p.c. ðx; PÞ ¼ ðð75; 25; 0Þ; f12; 3gÞ. In this case, player 2 has an objection ðy; P0 Þ ¼ ðð0; 26; 24Þ; f1; 23gÞ against player 1, and player 1 has a counter-objection ðz; P00 Þ ¼ ðð76; 0; 24Þ; f13; 2gÞ with respect to the objection ðy; P0 Þ of player 2. A relation on the set N of all players can be defined based on the concepts of objections and counter-objections. Definition 3 (Relation on players in ðx; PÞ). Consider a game ðN; v Þ, and let ðx; PÞ be an i.r.p.c. for ðN; v Þ. Suppose two players h and k in N. Then, player k is said to be stronger than player h (or, equivalently, player h is weaker than player k) in ðx; PÞ, if and only if k has an objection against h, but h does not have any counter-objections with respect to the objection, denoted by k h. k is said to be equal to h, denoted by k h, if and only if neither k h nor h k hold. We see, Definition 1 of objections, that if k h, then k and h are elements of the same coalition in P. In other words, one has neither k h nor h k, if k and h belong to different coalitions in P. That is, the relation is, in general, a partial relation. The next gives a numerical example of the relation on the set N of all players. Example 2. Consider the game ðN; v Þ in Example 1, and suppose the i.r.p.c. ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ. The i.r.p.c. ðy; P0 Þ ¼ ðð0; 21; 29Þ; f1; 23gÞ is an objection of player 2 against player 1. Player 1, however, does not have any counterobjection ðz; P00 Þ with respect to this objection ðy; P0 Þ, because player 1 cannot obtain 80 if he/she offers 29 or more to player 3. Hence, we have that 2 1 in ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ. We have, in Example 1, that 2 1, but we never have that 1 2. This fact is guaranteed by the acyclicity of the relation . Acyclicity of relations is defined as follows: Definition 4 (Acyclicity of relations). Consider a game ðN; v Þ and the relation on the set N of players in ðx; PÞ. The relation is said to be acyclic, if and only if there do not exist such players 1; 2; . . . ; t that 1 2 t 1.
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Under the acyclicity of a relation on the set N of all players, one can find the maximal players from N with respect to the relation. The next lemma verifies that the relation defined in Definition 3 is acyclic. Lemma 1. Let ðx; PÞ be an i.r.p.c. for a game ðN; v Þ, then the relation on the set N of all players is acyclic. Proof. See [5].
h
This lemma implies, in particular, that the relation is asymmetric, that is, for i and j in N, if i j, then j i is not true. As defined in Definition 3, for h and k in N; k h denotes that neither k h nor h k hold. Using this relation on N, Aumann and Maschler [1] defines the concept of M-stability of i.r.p.c.s. for a game ðN; v Þ. Definition 5 (M-stability of i.r.p.c.s. [1]). Consider a game ðN; v Þ. An i.r.p.c. ðx; PÞ for ðN; v Þ is said to be M-stable, if and only if for all i and j in N; i j holds. Then, for a game ðN; v Þ, the set of all M-stable i.r.p.c.s. ðx; PÞ for ðN; v Þ is called the bargaining set of ðN; v Þ. 3. Relations on coalitions This section proposes a definition of a relation on the set of all coalitions in a game. An example demonstrates how the newly proposed relation works, and a theorem shows that the proposed relation is acyclic. Definition 6 (Relation on coalitions in ðx; PÞ). Consider a game ðN; v Þ, and let ðx; PÞ be an i.r.p.c. for ðN; v Þ. Suppose two coalitions S1 and S2 in N. Then, coalition S1 is said to be stronger than coalition S2 (or, equivalently, coalition S2 is weaker than coalition S1 ) in ðx; PÞ, denoted by S1 S2 , if and only if 1. for each i 2 S1 , there exists j 2 S2 such that i j, and 2. for each i 2 S1 and each j 2 S2 , it is not satisfied that j i. Then, S1 is said to be equal to S2 , denoted by S1 S2 , if and only if neither S1 S2 nor S2 S1 hold. Note that in Definition 6, S1 and S2 can be arbitrary non-empty subsets of N, and in particular, it is not assumed that S1 or S are coalitions in the coalition structure P. We see, from Definition 6 and the comments just after Definition 3, that if S1 S2 in ðx; PÞ and S1 \ T – ; for some T 2 P, then S2 \ T – ;. The following two numerical examples show how the newly proposed relation on the set of all coalitions works. Example 3 demonstrates that if a player i 2 N is identified with a one-player coalition fig in N, then the newly proposed relation on coalitions reserves the relation on players. 2
Example 3. In Example 2, we see that 2 1 in ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ in the game ðN; v Þ given in Example 1. We also see, by Lemma 1, that 1 2 does not hold in ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ. Therefore, it holds f2g f1g in ðx; PÞ ¼ ðð80; 20; 0Þ; f12; 3gÞ. Example 4 demonstrates how the newly proposed relation on the set of all coalitions works for comparing coalitions with two or more members. Example 4. Consider the game ðN; v Þ such that N ¼ f1; 2; 3; 4g, v ð1Þ ¼ v ð2Þ ¼ v ð3Þ ¼ v ð4Þ ¼ 0, v ð12Þ ¼ v ð13Þ ¼ v ð123Þ ¼ v ð134Þ ¼ v ð124Þ ¼ 80, v ð14Þ ¼ v ð23Þ ¼ v ð24Þ ¼ v ð34Þ ¼ 65, v ð234Þ ¼ 75, and v ð1234Þ ¼ 120. Let us compare two coalitions, 12 and 34, in the i.r.p.c. ðx; PÞ ¼ ðð30; 30; 30; 30Þ; f1234gÞ. The i.r.p.c. ðy; P0 Þ ¼ ðð40; 40; 0; 0Þ; f12; 3; 4gÞ is an objection of player 1 against player 3, and player 3 does not have any counter-objections ðz; P00 Þ against 1 with respect to this objection. Thus, we have 1 3, and thus, 3 1 is not satisfied by the asymmetry of . Similarly, the i.r.p.c. ðy; P0 Þ ¼ ðð40; 40; 0; 0Þ; f12; 3; 4gÞ is an objection of player 2 against player 4, and player 4 does not have any counter-objections ðz; P00 Þ against 2 with respect to this objection. So, we have 2 4, and thus, 4 2 is not satisfied by the asymmetry of . Player 1 has a counter-objection against player 4 with respect to each objection ðy; P0 Þ of player 4 against player 1, that is, 4 1 is not satisfied. Similarly, player 2 has a counter-objection ðz; P00 Þ against player 3 with respect to each objection ðy; P0 Þ of player 3 against player 2, that is, 3 2 is not satisfied. Therefore, since we have 1 3, ‘‘not 3 1,” 2 4, ‘‘not 4 2,” ‘‘not 4 1,” and ‘‘not 3 2,” we have 12 34 in ðx; PÞ ¼ ðð30; 30; 30; 30Þ; f1234gÞ. The next theorem verifies that the relation defined in Definition 6 is acyclic. Theorem 1. Let ðx; PÞ be an i.r.p.c. for a game ðN; v Þ. Then, the relation on the set of all coalitions is acyclic. Proof. Assume that coalitions S1 ; S2 ; . . . ; St in N satisfies that S1 S2 St S1 . Then, for each u ðu ¼ 1; 2; . . . ; t 1Þ u uþ1 u uþ1 2 Suþ1 such that k k . This implies that there exists a sequence of players and each k 2 Su , there exists a player k 1 2 t tþ1 k ; k ; . . . ; k ; k ; . . . such that
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2
t
tþ1
k k k k 1
tþ1
2tþ1
2
2t
2tþ1
k k tþ2
2tþ2
3t
k ; t
2t
3t
where k ; k ; k ; . . . 2 S1 ; k ; k ; k ; . . . 2 S2 ; . . ., and k ; k ; k ; . . . 2 St . v w Since the set N of all player is finite, one can find v and w such that w > v and k ¼ k , that is, the sub-sequence v
k k
v þ1
w1
k
w
k ¼k
v
of the above sequence is cyclic, but this contradicts Lemma 1. Hence, the relation on the set of all coalitions is acyclic.
h
By Theorem 1, it is verified that one can find the maximal coalitions from all coalitions with respect to the newly proposed relation on the set of all coalitions. More, as in the case of the relation on the set of all players, the relation on the set of all coalitions is asymmetric, that is, for coalitions S1 and S2 in N, if S1 S2 , then S2 S1 is not true. The next example shows that the relation on the set of all coalitions is not necessarily transitive. Example 5 [5]. Consider the game ðN; v Þ such that N ¼ f1; 2; 3; 4; 5g; v ð1Þ ¼ v ð2Þ ¼ v ð3Þ ¼ 0; v ð12Þ ¼ v ð13Þ ¼ v ð123Þ ¼ 30; v ð14Þ ¼ 40; v ð35Þ ¼ 20; v ð245Þ ¼ 30, and for B N; v ðBÞ ¼ 0, otherwise. We see f1g f2g; f2g f3g and f1g f3g in ðx; PÞ ¼ ðð10; 10; 10; 0; 0Þ; f123; 4; 5gÞ. Theorem 2 verifies that the set of all i.r.p.c.s. under which all coalitions have the equal bargaining power coincides with the bargaining set. Theorem 2. Let ðN; v Þ be a game. Then, for each i.r.p.c. ðx; PÞ for ðN; v Þ, we have that ðx; PÞ is M-stable () S1 S2 in ðx; PÞ for all coalitions S1 and S2 in N. Proof. Assume that ðx; PÞ is M-stable. Then, for all players i1 and i2 in N, we have i1 i2 by Definition 5. Therefore, we have that for all coalitions S1 and S2 ; S1 S2 by Definition 6. If S1 S2 for all coalitions S1 and S2 , then considering all one-player coalitions fi1 g and fi2 g in N, we have i1 i2 for all players i1 and i2 in N, which means that ðx; PÞ is M-stable. h 4. Conclusions In this paper, a relation on the set of all coalitions in a game in coalition form with transferable utility was newly proposed in Definition 6, and the fact that the newly proposed relation satisfies acyclicity was verified in Theorem 1. More, Theorem 2 showed that the set of all i.r.p.c.s. under which all coalitions have the equivalent bargaining power coincides with the bargaining set, which is originally defined by Aumann and Maschler [1]. The proposed relation compares coalitions with respect to the abilities of making objections, and gives some insights on which coalitions may deviate from a coalition. Relationships between the proposed relation on the set of all coalitions in a game in coalition form and solution concepts for the game should be examined in future research opportunities, because solution concepts for games can be derived from relations on the set of all coalitions as in [4]. Acknowledgment The authors would like to express their appreciation to the Editor-in-Chief, Dr. Melvin Scott, and anonymous referees who kindly provided helpful suggestions which improved the quality of their paper. A part of this work is supported by KAKENHI (19310097). References [1] R.J. Aumann, M. Maschler, The bargaining set for cooperative games, Annals of Mathematics Studies, vol. 52, Princeton University Press, Princeton, NJ, 1964. [2] E. Einy, The desirability relation of simple games, Mathematical Social Sciences 10 (1985) 155–168. [3] K. Ishikawa, T. Inohara, A method to compare influence of coalitions on group decision other than desirability relation, Applied Mathematics and Computation 188-1 (2007) 838–849. [4] T. Kitamura, T. Inohara, Comparison of coalition influence on group decision, Master’s Thesis, Department of Value and Decision Science, Tokyo Institute of Technology, 2008. [5] M. Davis, M. Mashler, Existance of stable payoff configurations for cooperative games essays in mathematical economics, in: Martin Shubik (Ed.), Honor of Oskar Morgenstern, Princeton University Press, 1962, pp. 39–52. [6] G. Owen, Game Theory, third ed., 1995, pp. 313–319.