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Methods for comparison of coalition influence on games in characteristic function form and their interrelationships
Applied Mathematics and Computation 217 (2010) 4047–4050
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Methods for comparison of coalition influence on games in characteristic function form and their interrelationships 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
i n f o
Keywords: Game theory Cooperative games Relations Influence Coalitions
a b s t r a c t This paper extends two existent methods, called the blockability relation and the viability relation, for simple games to compare influence of coalitions, to those for games in characteristic function form, and shows that the newly defined relations satisfy transitivity and completeness. It is shown in this paper that for every game in characteristic function form the blockability relation and the viability relation have a complementary interrelationship. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction One of the important issues with respect to games in characteristic function form with transferable utility is to know which coalitions will be formed in the games. Furnishing methods for comparison of coalition influence in such games contributes to resolve the issue, because coalitions with more influence tend to be formed more frequently. In the framework of simple games, which constitute a special class of games in characteristic function form, there are such methods to compare coalition influence as the desirability relation [1,5], the blockability relation [2,4], and the viability relation [3]. The desirability relation compares coalitions with respect to how much the coalitions are close to have enough power to completely control the decision of the situation. The blockability relation compares coalitions with respect to how much they can make other coalitions not have such power. The viability relation compares coalitions with respect to how robust they are over deviation of members. These relations are mathematically defined using the concepts of winning and losing. Because being winning and losing coalitions can be expressed by payoffs 1 and 0, respectively, analogous relations can be defined on games in characteristic function form. In this paper, the blockability relation and the viability relation are extended to those for games in characteristic function form, respectively. This paper gives propositions which verify that these relations for games in characteristic function form are certainly extensions of the corresponding relations for simple games, and provides some examples how the proposed relations work. Moreover, it is verified that the new relations satisfy completeness and transitivity. These properties are desirable, because if a relation satisfies completeness and transitivity, each coalition can be naturally assigned a real number which represents the relation and indicates how much the coalition influences the final decision. Finally, it is shown that the newly defined relations have a complementary interrelationship. The structure of this paper is as follows: the framework of games and the existent relations for simple games are presented in the next section. In Section 3, the definitions of the newly proposed relations for games in characteristic function form are provided, and the propositions mentioned above are given. In Section 4, the proposition about the complementary interrelationship of the relations is provided. The last section is devoted to concluding remarks.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (T. Inohara). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.10.012
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K. Kojima, T. Inohara / Applied Mathematics and Computation 217 (2010) 4047–4050
2. Framework In this section, the notation employed in this paper is introduced. Let N = {1, 2, . . . , n} be a set of n players. Each non-empty subset of N is called coalition, and a coalition S = {i1, i2, . . . , im} is often denoted by i1 i2 im for simplicity. A characteristic function v : 2N ! R such that v(;) = 0 assigns a real number to each coalition, where 2N and R denote the power set of N and the set of all real numbers, respectively. A pair (N, v) is said to be a game in characteristic function form with transferable utility, simply called a game in this paper. A game which satisfies the following four conditions is called a simple game: (i) v(S) 2 {0,1} for all S N; (ii) v(;) = 0; (iii) v(N) = 1; (iv) for S, T N if S T then v(S) 6 v(T). A coalition S such that v(S) = 1 is said to be a winning coalition. A coalition S such that v(S) = 0 is said to be a losing coalition. The next example of simple games is used throughout this paper. Example 1 (Simple games). Consider N = {1, 2, 3, 4} and v such that v(S) = 1 if S is 12, 123, 124, 234, or 1234, and v(S) = 0, otherwise. The pair (N, v) is a simple game. The blockability relation for simple games is defined as follows. Definition 1 (Blockability relations for simple games [2]). Consider a simple game (N, v). For coalitions S and S0 , S b S0 is defined as: for all winning coalition T, if TnS0 is a losing coalition, then TnS is also a losing coalition. b is called the blockability relation for (N, v). S b S0 expresses that if coalition S0 can make winning coalition T losing by deviation then coalition S can also make T losing by that. The next lemma is convenient to specify the blockability relation b between two coalitions. Lemma 1 [2]. Consider a simple game (N, v) and the blockability relation b for (N, v). Then, it is satisfied that for all coalitions S and S0 , S b S0 is equivalent to B(S) B(S0 ), where for S N, B(S) = {Tjv(T) = 1 and v(TnS) = 0}. The next example shows how Definition 1 and Lemma 1 work. Example 2. Consider the simple game in Example 1. Then, we have B(12) = {12, 123, 124, 234, 1234} and B(34) = {234}, because, for example, 234 2 B(34) since v(234) = 1 and v(234n34) = v(2) = 0. By Lemma 1, 12 b 34 holds, because B(12) B(34). That is, all winning coalitions become losing by the deviation of 12, while winning coalitions other than 234 do not become losing by the deviation of 34. The definition of viability relations for simple games is given. Definition 2 (Viability relations for simple games [3]). Consider a simple game (N, v). For coalitions S and S0 , S v S0 is defined as: for all coalition T 2 2N, if S0 nT is a winning coalition, then SnT is also a winning coalition. v is called the viability relation for (N, v). This relation says that if coalition S0 does not become losing by the deviation of T, then S does not become losing coalition by that, ether. The next lemma is useful for specifying the viability relation v for simple games. Lemma 2 [3]. Consider a simple game (N, v) and the blockability relation v for (N, v). Then, it is satisfied that for all coalitions S and S0 , S v S0 is equivalent to V(S) V(S0 ), where for S N, V(S) = {Tjv(SnT) = 1}. The next example shows how Definition 2 and Lemma 2 work. Example 3. Consider the simple game in Example 1. Then, we have V(1234) = {1, 3, 4, 34} and V(124) = {3, 4, 34}, because, for example, 34 2 V(124) since v(124n34) = v(12) = 1. By Lemma 2, 1234 v 124 holds because V(1234) V(124). 3. New relations for games in characteristic function form In this section, the relations for simple games are extended to those for games in characteristic function form, and some properties of them are verified. Then, propositions which imply that these relations for games in characteristic function form are indeed extensions of the corresponding relations for simple games are verified. The blockability relation for games in characteristic function form is defined as follows. Definition 3 (Blockability relations for games in characteristic function form). Consider a game (N, v). For a coalition T, let B*(T) P be UN v ðU n TÞ. For coalitions S and S0 , S B S0 is defined as B*(S) 6 B*(S0 ). B is called the blockability relation for (N, v). S B S0 expresses that coalition S can decrease the value of the characteristic function v by deviating from U more than coalition S0 can do. The next example shows how Definition 3 works.
K. Kojima, T. Inohara / Applied Mathematics and Computation 217 (2010) 4047–4050
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Example 4. Consider the simple game in Example 1. For coalitions 12 and 34, we have
B ð12Þ ¼
X
v ðU n 12Þ ¼ 4 ½v ð;Þ þ v ð3Þ þ v ð4Þ þ v ð34Þ ¼ 0;
UN
B ð34Þ ¼
X
v ðU0 n 34Þ ¼ 4 ½v ð;Þ þ v ð1Þ þ v ð2Þ þ v ð12Þ ¼ 4:
0
U N
By the definition of B, it holds that 12 B 34. Since simple games constitute a special class of games in characteristic function form, the blockability relation B for games in characteristic function form can be applied to every simple game. The next proposition shows that the blockability relation B which is applied to a simple game is implied by b. Proposition 1. For a simple game (N, v) and coalitions S1, S2 N, we have that if S1 b S2 then S1 B S2. Proof. Assume that S1 b S2. Then, by Lemma 1, we have B(S1) B(S2), which implies jB(S1)j P jB(S2)j. Elements of B(S) are winning coalitions which become losing by the deviation of S. Thus, we see that B*(S1) 6 B*(S2), which means S1 B S2. h The next proposition gives some properties which the blockability relation for games in characteristic function form satisfies. Proposition 2. The blockability relation B for a game (N, v) is transitive and complete. Proof. (Transitivity) If S1 B S2 and S2 B S3 for S1, S2, S3 N, then B*(S1) 6 B*(S2) and B*(S2) 6 B*(S3) hold. This implies that B*(S1) 6 B*(S3), which means S1 B S3. (Completeness) For S1, S2 N, B*(S1) and B*(S2) are real numbers. Hence, we have B*(S1) 6 B*(S2) or B*(S2) 6 B*(S1). This implies that S1 B S2 or S2 B S1. h The viability relation for games in characteristic function form is defined as follows. Definition 4 (Viability relations for games in characteristic function form). Consider a game (N, v). For a coalition T, let V*(T) be P V 0 V 0 0 UN v ðT n UÞ. For coalitions S and S , S S is defined as V*(S) P V*(S ). is called the viability relation for (N, v). V 0 S S expresses that coalition S can defend the value of the characteristic function from the deviation of U more than coalition S0 can do. The next example shows how Definition 4 works. Example 5. Consider the simple game in Example 1. For coalitions 124 and 234, we have
V ð124Þ ¼
X
v ð124 n UÞ ¼ 2
X VN
v ðU0 Þ ¼ 4;
U 0 124
UN
V ð234Þ ¼
X
v ð234 n VÞ ¼ 2
X
v ðV 0 Þ ¼ 2:
0
V 234
By the definition of V, it holds that 124 V 234. The next proposition shows that the viability relation V which is applied to a simple gam is implied by v. Proposition 3. For a simple game (N, v) and coalitions S1, S2 N, we have that if S1 v S2, then S1 V S2. Proof. Assume that S1 v S2. By Lemma 2, we have V(S1) V(S2), which implies jV(S1)j P jV(S2)j. Elements of V(S) are coalitions which cannot make S losing by deviation. Thus, we have that V*(S1) P V*(S2), which means S1 V S2. h The next proposition gives some properties which the viability relation for games in characteristic function form satisfies. Proposition 4. The viability relation V for a game (N, v) is transitive and complete. Proof. (Transitivity) If S1 V S2 and S2 V S3 for S1, S2, S3 N, then V*(S1) P V*(S2) and V*(S2) P V*(S3) hold. This implies that V*(S1) P V*(S3), which means S1 V S3. (Completeness) For S1, S2 N, V*(S1) and V*(S2) are real numbers. Hence, we have V*(S1) P V*(S2) or V*(S2) P V*(S1). This implies that S1 V S2 or S2 V S1. h 4. Complementary relationship between new relations This section shows a complementary interrelationship between the blockability relation and the viability relation for games in characteristic function form.
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Proposition 5. Consider a game (N, v). Let B and V be the blockability relation and the viability relation for (N, v), respectively. For S1, S2 N, we have that S1 B S2 if and only if NnS2 V NnS1. P P Proof. Assume that S1 B S2. Then we have B*(S1) 6 B*(S2), which means TN v ðT n S1 Þ 6 TN v ðT n S2 Þ. Since T N can be P P P expressed by NnU, if one takes NnT as U; TN v ðT n S1 Þ 6 TN v ðT n S2 Þ can be rewritten as v ððN n UÞ n S1 Þ 6 P PUN v ððN n UÞ n S Þ. For sets X, Y, and Z N, we generally have that (XnY)nZ = (XnZ)nY. Therefore, 2 UN v ððN n UÞ n S1 Þ 6 PUN P P UN v ððN n UÞ n S2 Þ can be rewritten UN v ððN n S1 Þ n UÞ 6 UN v ððN n S2 Þ n UÞ. By the definition of the viability relation, we have NnS2 V NnS1. h Proposition 5 shows that for every game in characteristic function form the blockability relation and the viability relation have a complementary interrelationship. 5. Concluding remarks This paper proposed two methods for games in characteristic function form to compare coalition influence on a game, and verified that the newly proposed methods are extensions of existent methods for simple games. Since the new relations satisfy completeness and transitivity, they can be employed for defining some indices which indicate the influence of coalitions in games. Moreover, in this paper, it was shown that the new methods have a complementary interrelationship. One of the important topics to be investigated in the future research opportunities is to provide some definition of coalition influence indices based on these relations. Such indices make it easier to compare the coalition influence and to know which coalitions are formed more frequently. References [1] E. Einy, The desirability relation of simple games, Mathematical Social Sciences 10 (1985) 155–168. [2] 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. [3] T. Kitamura, Comparison of coalition influence on group decision, Department of Value and Decision Science Master’s Thesis, Tokyo Institute of Technology (2008). [4] T. Kitamura, T. Inohara, A characterization of completeness of blockability relations with respect to unanimity, Applied Mathematics and Computation 197 (2) (2008) 715–718. [5] A.D. Taylor and W.S. Zwicker, Simple Games: Desirability Relation, Trading, Pseudoweightings, Princeton University Press, Princeton, New Jersey.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Methods for comparison of coalition influence on games in characteristic function form and their interrelationships 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
i n f o
Keywords: Game theory Cooperative games Relations Influence Coalitions
a b s t r a c t This paper extends two existent methods, called the blockability relation and the viability relation, for simple games to compare influence of coalitions, to those for games in characteristic function form, and shows that the newly defined relations satisfy transitivity and completeness. It is shown in this paper that for every game in characteristic function form the blockability relation and the viability relation have a complementary interrelationship. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction One of the important issues with respect to games in characteristic function form with transferable utility is to know which coalitions will be formed in the games. Furnishing methods for comparison of coalition influence in such games contributes to resolve the issue, because coalitions with more influence tend to be formed more frequently. In the framework of simple games, which constitute a special class of games in characteristic function form, there are such methods to compare coalition influence as the desirability relation [1,5], the blockability relation [2,4], and the viability relation [3]. The desirability relation compares coalitions with respect to how much the coalitions are close to have enough power to completely control the decision of the situation. The blockability relation compares coalitions with respect to how much they can make other coalitions not have such power. The viability relation compares coalitions with respect to how robust they are over deviation of members. These relations are mathematically defined using the concepts of winning and losing. Because being winning and losing coalitions can be expressed by payoffs 1 and 0, respectively, analogous relations can be defined on games in characteristic function form. In this paper, the blockability relation and the viability relation are extended to those for games in characteristic function form, respectively. This paper gives propositions which verify that these relations for games in characteristic function form are certainly extensions of the corresponding relations for simple games, and provides some examples how the proposed relations work. Moreover, it is verified that the new relations satisfy completeness and transitivity. These properties are desirable, because if a relation satisfies completeness and transitivity, each coalition can be naturally assigned a real number which represents the relation and indicates how much the coalition influences the final decision. Finally, it is shown that the newly defined relations have a complementary interrelationship. The structure of this paper is as follows: the framework of games and the existent relations for simple games are presented in the next section. In Section 3, the definitions of the newly proposed relations for games in characteristic function form are provided, and the propositions mentioned above are given. In Section 4, the proposition about the complementary interrelationship of the relations is provided. The last section is devoted to concluding remarks.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (T. Inohara). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.10.012
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K. Kojima, T. Inohara / Applied Mathematics and Computation 217 (2010) 4047–4050
2. Framework In this section, the notation employed in this paper is introduced. Let N = {1, 2, . . . , n} be a set of n players. Each non-empty subset of N is called coalition, and a coalition S = {i1, i2, . . . , im} is often denoted by i1 i2 im for simplicity. A characteristic function v : 2N ! R such that v(;) = 0 assigns a real number to each coalition, where 2N and R denote the power set of N and the set of all real numbers, respectively. A pair (N, v) is said to be a game in characteristic function form with transferable utility, simply called a game in this paper. A game which satisfies the following four conditions is called a simple game: (i) v(S) 2 {0,1} for all S N; (ii) v(;) = 0; (iii) v(N) = 1; (iv) for S, T N if S T then v(S) 6 v(T). A coalition S such that v(S) = 1 is said to be a winning coalition. A coalition S such that v(S) = 0 is said to be a losing coalition. The next example of simple games is used throughout this paper. Example 1 (Simple games). Consider N = {1, 2, 3, 4} and v such that v(S) = 1 if S is 12, 123, 124, 234, or 1234, and v(S) = 0, otherwise. The pair (N, v) is a simple game. The blockability relation for simple games is defined as follows. Definition 1 (Blockability relations for simple games [2]). Consider a simple game (N, v). For coalitions S and S0 , S b S0 is defined as: for all winning coalition T, if TnS0 is a losing coalition, then TnS is also a losing coalition. b is called the blockability relation for (N, v). S b S0 expresses that if coalition S0 can make winning coalition T losing by deviation then coalition S can also make T losing by that. The next lemma is convenient to specify the blockability relation b between two coalitions. Lemma 1 [2]. Consider a simple game (N, v) and the blockability relation b for (N, v). Then, it is satisfied that for all coalitions S and S0 , S b S0 is equivalent to B(S) B(S0 ), where for S N, B(S) = {Tjv(T) = 1 and v(TnS) = 0}. The next example shows how Definition 1 and Lemma 1 work. Example 2. Consider the simple game in Example 1. Then, we have B(12) = {12, 123, 124, 234, 1234} and B(34) = {234}, because, for example, 234 2 B(34) since v(234) = 1 and v(234n34) = v(2) = 0. By Lemma 1, 12 b 34 holds, because B(12) B(34). That is, all winning coalitions become losing by the deviation of 12, while winning coalitions other than 234 do not become losing by the deviation of 34. The definition of viability relations for simple games is given. Definition 2 (Viability relations for simple games [3]). Consider a simple game (N, v). For coalitions S and S0 , S v S0 is defined as: for all coalition T 2 2N, if S0 nT is a winning coalition, then SnT is also a winning coalition. v is called the viability relation for (N, v). This relation says that if coalition S0 does not become losing by the deviation of T, then S does not become losing coalition by that, ether. The next lemma is useful for specifying the viability relation v for simple games. Lemma 2 [3]. Consider a simple game (N, v) and the blockability relation v for (N, v). Then, it is satisfied that for all coalitions S and S0 , S v S0 is equivalent to V(S) V(S0 ), where for S N, V(S) = {Tjv(SnT) = 1}. The next example shows how Definition 2 and Lemma 2 work. Example 3. Consider the simple game in Example 1. Then, we have V(1234) = {1, 3, 4, 34} and V(124) = {3, 4, 34}, because, for example, 34 2 V(124) since v(124n34) = v(12) = 1. By Lemma 2, 1234 v 124 holds because V(1234) V(124). 3. New relations for games in characteristic function form In this section, the relations for simple games are extended to those for games in characteristic function form, and some properties of them are verified. Then, propositions which imply that these relations for games in characteristic function form are indeed extensions of the corresponding relations for simple games are verified. The blockability relation for games in characteristic function form is defined as follows. Definition 3 (Blockability relations for games in characteristic function form). Consider a game (N, v). For a coalition T, let B*(T) P be UN v ðU n TÞ. For coalitions S and S0 , S B S0 is defined as B*(S) 6 B*(S0 ). B is called the blockability relation for (N, v). S B S0 expresses that coalition S can decrease the value of the characteristic function v by deviating from U more than coalition S0 can do. The next example shows how Definition 3 works.
K. Kojima, T. Inohara / Applied Mathematics and Computation 217 (2010) 4047–4050
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Example 4. Consider the simple game in Example 1. For coalitions 12 and 34, we have
B ð12Þ ¼
X
v ðU n 12Þ ¼ 4 ½v ð;Þ þ v ð3Þ þ v ð4Þ þ v ð34Þ ¼ 0;
UN
B ð34Þ ¼
X
v ðU0 n 34Þ ¼ 4 ½v ð;Þ þ v ð1Þ þ v ð2Þ þ v ð12Þ ¼ 4:
0
U N
By the definition of B, it holds that 12 B 34. Since simple games constitute a special class of games in characteristic function form, the blockability relation B for games in characteristic function form can be applied to every simple game. The next proposition shows that the blockability relation B which is applied to a simple game is implied by b. Proposition 1. For a simple game (N, v) and coalitions S1, S2 N, we have that if S1 b S2 then S1 B S2. Proof. Assume that S1 b S2. Then, by Lemma 1, we have B(S1) B(S2), which implies jB(S1)j P jB(S2)j. Elements of B(S) are winning coalitions which become losing by the deviation of S. Thus, we see that B*(S1) 6 B*(S2), which means S1 B S2. h The next proposition gives some properties which the blockability relation for games in characteristic function form satisfies. Proposition 2. The blockability relation B for a game (N, v) is transitive and complete. Proof. (Transitivity) If S1 B S2 and S2 B S3 for S1, S2, S3 N, then B*(S1) 6 B*(S2) and B*(S2) 6 B*(S3) hold. This implies that B*(S1) 6 B*(S3), which means S1 B S3. (Completeness) For S1, S2 N, B*(S1) and B*(S2) are real numbers. Hence, we have B*(S1) 6 B*(S2) or B*(S2) 6 B*(S1). This implies that S1 B S2 or S2 B S1. h The viability relation for games in characteristic function form is defined as follows. Definition 4 (Viability relations for games in characteristic function form). Consider a game (N, v). For a coalition T, let V*(T) be P V 0 V 0 0 UN v ðT n UÞ. For coalitions S and S , S S is defined as V*(S) P V*(S ). is called the viability relation for (N, v). V 0 S S expresses that coalition S can defend the value of the characteristic function from the deviation of U more than coalition S0 can do. The next example shows how Definition 4 works. Example 5. Consider the simple game in Example 1. For coalitions 124 and 234, we have
V ð124Þ ¼
X
v ð124 n UÞ ¼ 2
X VN
v ðU0 Þ ¼ 4;
U 0 124
UN
V ð234Þ ¼
X
v ð234 n VÞ ¼ 2
X
v ðV 0 Þ ¼ 2:
0
V 234
By the definition of V, it holds that 124 V 234. The next proposition shows that the viability relation V which is applied to a simple gam is implied by v. Proposition 3. For a simple game (N, v) and coalitions S1, S2 N, we have that if S1 v S2, then S1 V S2. Proof. Assume that S1 v S2. By Lemma 2, we have V(S1) V(S2), which implies jV(S1)j P jV(S2)j. Elements of V(S) are coalitions which cannot make S losing by deviation. Thus, we have that V*(S1) P V*(S2), which means S1 V S2. h The next proposition gives some properties which the viability relation for games in characteristic function form satisfies. Proposition 4. The viability relation V for a game (N, v) is transitive and complete. Proof. (Transitivity) If S1 V S2 and S2 V S3 for S1, S2, S3 N, then V*(S1) P V*(S2) and V*(S2) P V*(S3) hold. This implies that V*(S1) P V*(S3), which means S1 V S3. (Completeness) For S1, S2 N, V*(S1) and V*(S2) are real numbers. Hence, we have V*(S1) P V*(S2) or V*(S2) P V*(S1). This implies that S1 V S2 or S2 V S1. h 4. Complementary relationship between new relations This section shows a complementary interrelationship between the blockability relation and the viability relation for games in characteristic function form.
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Proposition 5. Consider a game (N, v). Let B and V be the blockability relation and the viability relation for (N, v), respectively. For S1, S2 N, we have that S1 B S2 if and only if NnS2 V NnS1. P P Proof. Assume that S1 B S2. Then we have B*(S1) 6 B*(S2), which means TN v ðT n S1 Þ 6 TN v ðT n S2 Þ. Since T N can be P P P expressed by NnU, if one takes NnT as U; TN v ðT n S1 Þ 6 TN v ðT n S2 Þ can be rewritten as v ððN n UÞ n S1 Þ 6 P PUN v ððN n UÞ n S Þ. For sets X, Y, and Z N, we generally have that (XnY)nZ = (XnZ)nY. Therefore, 2 UN v ððN n UÞ n S1 Þ 6 PUN P P UN v ððN n UÞ n S2 Þ can be rewritten UN v ððN n S1 Þ n UÞ 6 UN v ððN n S2 Þ n UÞ. By the definition of the viability relation, we have NnS2 V NnS1. h Proposition 5 shows that for every game in characteristic function form the blockability relation and the viability relation have a complementary interrelationship. 5. Concluding remarks This paper proposed two methods for games in characteristic function form to compare coalition influence on a game, and verified that the newly proposed methods are extensions of existent methods for simple games. Since the new relations satisfy completeness and transitivity, they can be employed for defining some indices which indicate the influence of coalitions in games. Moreover, in this paper, it was shown that the new methods have a complementary interrelationship. One of the important topics to be investigated in the future research opportunities is to provide some definition of coalition influence indices based on these relations. Such indices make it easier to compare the coalition influence and to know which coalitions are formed more frequently. References [1] E. Einy, The desirability relation of simple games, Mathematical Social Sciences 10 (1985) 155–168. [2] 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. [3] T. Kitamura, Comparison of coalition influence on group decision, Department of Value and Decision Science Master’s Thesis, Tokyo Institute of Technology (2008). [4] T. Kitamura, T. Inohara, A characterization of completeness of blockability relations with respect to unanimity, Applied Mathematics and Computation 197 (2) (2008) 715–718. [5] A.D. Taylor and W.S. Zwicker, Simple Games: Desirability Relation, Trading, Pseudoweightings, Princeton University Press, Princeton, New Jersey.