Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value

European Journal of Operational Research 216 (2012) 638–646

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value Yoshio Kamijo a,⇑, Takumi Kongo b a b

Waseda Institute for Advanced Study, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan Faculty of Political Science & Economics, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan

a r t i c l e

i n f o

Article history: Received 28 January 2011 Accepted 10 August 2011 Available online 18 August 2011 Keywords: Game theory Axiomatization Shapley value Egalitarian value Solidarity value Banzhaf value

a b s t r a c t This study provides a unified axiomatic characterization method of one-point solutions for cooperative games with transferable utilities. Any one-point solution that satisfies efficiency, the balanced cycle contributions property (BCC), and the axioms related to invariance under a player deletion is characterized as a corollary of our general result. BCC is a weaker requirement than the well-known balanced contributions property. Any one-point solution that is both symmetric and linear satisfies BCC. The invariance axioms necessitate that the deletion of a specific player from games does not affect the other players’ payoffs, and this deletion is different with respect to solutions. As corollaries of the above characterization result, we are able to characterize the well-known one-point solutions, the Shapley, egalitarian, and solidarity values, in a unified manner. We also studied characterizations of an inefficient one-point solution, the Banzhaf value that is a well-known alternative to the Shapley value. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Consider a situation that is well described by the standard notion of a cooperative game with a transferable utility (TU game) and consider a one-point solution concept that prescribes how players divide the worth of their total cooperation among themselves. We deal with the setting of variable player sets, in particular, with how a player’s payoff in one TU game is related to that in another TU game (especially in the case of a subgame of the former game). More specifically, we explore the problem of when a player’s payoff is not affected by the elimination of some other player from the original situation. From cooperative game theory, we know that there are two relevant factors in considering this problem. Suppose N is an initial player set, v is a characteristic function of N, and k 2 N is a player who leaves the game. Then, one of the relevant factors is the difference in the worth of the grand coalition, i.e., the difference between the worth of the initial player set v(N) and that after player k leaves, v(Nn{k}). The other relevant factor is how the bargaining power of players in Nn{k} has been altered by the elimination of player k. For example, when player i makes a large contribution to the coalitions containing k and zero contribution to any other coalition, he is expected to lose considerable bargaining power after k is deleted. ⇑ Corresponding author. Tel.: +81 3 5286 2103. E-mail addresses: [email protected] (Y. Kamijo), kongo_takumi@toki. waseda.jp (T. Kongo). 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.08.011

A null player is a player who makes zero contribution to any other coalition. The worth of the total cooperation is unaltered by the elimination of a null player. In addition, most studies have assumed that the bargaining power of the remaining players is not affected by null player deletion. The original work of Shapley (1953) on the axiomatization of the Shapley value, by using the carrier axiom, which requires the same payoff for players before and after a null player’s elimination, was later carefully explored by Derks and Haller (1999), who provided a necessary and sufficient condition for a solution concept to satisfy the property. In this paper, we explore other types of invariance in the payoffs before and after the elimination of a player. Considering the two factors relevant to the payoff changes when a player is removed, we know that the invariance in the payoff occurs only when the following points are true about the deleted player:  in the original situation where the player was in set N, his payoff was only his marginal contribution, v(N)  v(Nn{k}), and  in the situation where the player is deleted, the absolute or relative importance or bargaining power of the remaining players that could have been counted by a solution concept were unaltered. The first condition is naturally obtained when we deal with a solution concept that distributes the worth of the total cooperation among the players. (i.e., when we focus on an ‘‘efficient’’ solution). The second condition implies that the invariance of the relative importance or bargaining power of players is strong enough to

Y. Kamijo, T. Kongo / European Journal of Operational Research 216 (2012) 638–646

result in the invariance of the payoffs of the remaining players. Thus, it is possible that a player other than a null player exists whose deletion does not affect the payoff of the remaining player. Here, we consider a player who makes a contribution to each coalition in the same manner. Thus, his deletion does not change the relative bargaining power of the other players. However, note that there is some arbitrariness in the meaning of ‘‘making a contribution to each coalition in the same manner.’’ To relieve this arbitrariness, we consider two versions of such a player and examine the invariance properties with respect to the elimination of each type of such players. One player is a proportional player, who makes a contribution to each coalition proportional to its worth and the size of the coalition, while the other player is a quasi-proportional player, who also makes a proportional contribution to each coalition but in a way that slightly differs from the manners in which the proportional player contributes. We first show that there exist solution concepts in a cooperative game that satisfy these two invariance properties. One invariance property is the invariance in payoffs from proportional player deletion, which is satisfied by the egalitarian value or the equal division value assigning an equal division of v(N) to each of the players in N. The second invariance property, i.e., the invariance in payoffs from quasi-proportional player deletion, is satisfied by the solidarity value introduced by Nowak and Radzik (1994), which is similar to the Shapley value but differs in using the average marginal contributions instead of the marginal contributions of a player.1 Interestingly, although the difference between these two invariance properties lies in the subtle difference in the meaning of ‘‘making a contribution to each coalition in the same manner,’’ the solution concepts that satisfy these invariance axioms are quite different. Next, we attempt to axiomatize these two solution concepts by using the corresponding invariance axioms. In the case of a null player, Derks and Haller (1999) did not succeed in axiomatizing solution concepts by using the invariance in payoff from the deletion of a null player. Recently, Kamijo and Kongo (2010) axiomatized the Shapley value through this property and their newly defined balanced contributions property. The original balanced contributions property (BC) of Myerson (1980) requires that for any two players, the claim of one player against another, measured by a solution concept, should be balanced with the counter claim from the second player against the first. In contrast, the balanced contributions property proposed by Kamijo and Kongo (2010), which is called the balanced cycle contributions property (BCC), does not require that claims between two players be balanced but rather necessitates that claims among all players should be balanced in a cyclical manner; i.e., for any order of players, the sum of the claims from each player against his successor is balanced with the sum of the claims from that player against his predecessor. Kamijo and Kongo (2010) show that the Shapley value is a unique one-point solution concept that is efficient and satisfies BCC and the invariance in payoff from the deletion of a null player. One merit of using BCC rather than BC is that while the Shapley value is a unique efficient solution concept satisfying BC, there are several solution concepts that satisfy BCC. In fact, we prove in this paper that any one-point solution that is both symmetric and linear satisfies BCC. Thus, both the egalitarian value and the solidarity value satisfy BCC. Moreover, we show that similar to the results of Kamijo and Kongo (2010), the egalitarian value is axiomatized by the efficiency, BCC, and the invariance from a proportional player deletion, and the solidarity value is axiomatized by the first two along with the invariance from a quasi-proportional player deletion. Thus, we provide new axiomatic foundations of the egalitar1 It is worth mentioning that in its recent paper Calvo (2008) found the so-called random removal value for NTU games, where the solidarity value turned out to be the resulting value for TU games.

639

ian value and the solidarity value, respectively.2 Furthermore, combined with the result of Kamijo and Kongo (2010), we can see that the difference among the three major one-point solution concepts—the Shapley value, the egalitarian value, and the solidarity value—lies in the selection of a player ‘‘whose deletion does not affect your payoff.’’3 An alternative to the Shapley value, the Banzhaf value (Banzhaf, 1965; Owen, 1975) is a well-known inefficient solution concept in TU games. A number of studies have compared axiomatizations between the Banzhaf and Shapley values (see, e.g., Lehrer, 1988; Haller, 1994; Feltkamp, 1995; Nowak, 1997; Nowak and Radzik, 2000; Alonso-Meijide et al., 2007; Casajus, 2011b). In most of their results, the Banzhaf value is characterized by replacing efficiency in the sets of axioms characterizing the Shapley value with 2-efficiency, which requires that a merger of two players into a single player does not benefit or harm the two players. A similar observation can be made for our axiomatization of the Shapley value. By replacing the efficiency with 2-efficiency and efficiency with respect to one-person games in the axiomatization of the Shapley value by Kamijo and Kongo (2010), the Banzhaf value is characterized. This result allows us to observe that the difference between the Shapley and the Banzhaf values arises from the difference between efficiency-related axioms. Moreover, we show that, in contrast with the axiomatization results for efficient solutions, there are no one-point solutions that satisfy 2-efficiency, efficiency with respect to one-person games, BCC, and the invariance from a proportional or quasi-proportional player deletion. The paper is organized as follows. In the next section, we provide the definitions and notation. In Section 3, we introduce the axioms related to the invariance under player deletion. In Section 4, we explain BCC introduced by Kamijo and Kongo (2010) and show that both the egalitarian and the solidarity values satisfy it. In Section 5, we present the general axiomatization results, including axiomatizations of the above two values. In Section 6, we present the axiomatization of the Banzhaf value and two impossibility results. Section 7 offers a conclusion.

2. Preliminaries Let N # N be a finite set of players and v : 2N ! R with v(;) = 0 be a characteristic function. A pair (N, v) is a cooperative game with transferable utility, or simply, a game. Let C be the set of all games and let jNj = n, where jj represents the cardinality of the set. A nonempty subset S # N is a coalition, and v(S) is the worth of the coalition. For simplicity, we will represent each singleton {i} # N as i when the possibility of confusion does not exist. A value or one-point solution concept on C is a function that associates each game (N, v) 2 C with an n-dimensional vector in RN . Let u be a value on C. Given a game (N, v) 2 C, two players i,j 2 N are symmetric if for any S # Nn{i, j}, v(S [ i) = v(S [ j). A value u is symmetric if for every i, j 2 N that are symmetric in (N, v) 2 C, ui(N, v) = uj(N, v). A value u is linear if for any real value a; b 2 R and any two games (N, v), (N, w) 2 C, au(N, v) + bu(N, w) = u(N, av + bw), where (N, av + bw) is defined as (av + bw)(S) = av(S) + bw(S) for any S # N. A value u is efficient if 2 In the literature, several axiomatizations of the egalitarian value are proposed (for the NTU game framework, see Kalai, 1977; Kalai and Samet, 1985; for the TU game framework, see van den Brink, 2007; for the axiomatization of a class of solutions to which the egalitarian value belongs, see van den Brink and Funaki, 2009). The solidarity value is axiomatized by Nowak and Radzik (1994), Casajus (2011a), and Driessen (2010). 3 These kinds of unified frameworks for the characterizations of solutions are also provided by Gómez-Rúa and Vidal-Puga (2010) for a broad class of values in TU games with coalition structures. The characterizations of several power indices for the simple games discussed in Lorenzo-Freire et al. (2007) can be seen as a unified framework for characterizations, as well.

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P

i2N ui ðN; v Þ ¼ v ðNÞ. In the first part of this paper, we will mainly consider three efficient values. The first of these efficient values is the Shapley value (Shapley, 1953). The Shapley value / for (N, v) is defined by

/i ðN; v Þ ¼

X ðn  sÞ!ðs  1Þ! ðv ðSÞ  v ðS n iÞÞ n! S # N;S3i

for each i 2 N, where s is the cardinality of the set S. Thus, the Shapley value of a player is the weighted average of his marginal contributions to coalitions. The second efficient value we consider is the egalitarian value EV, which is defined for i 2 N as follows:

EVi ðN; v Þ ¼

v ðNÞ n

:

For reasons that are clear from this definition, the egalitarian value is often called the equal-division value. To introduce the third efficient value, we first define the average marginal contributions of the members in coalition S: for each S # N,

Av ðSÞ ¼

1X ðv ðSÞ  v ðS n kÞÞ: s k2S

The solidarity value SV, proposed by Nowak and Radzik (1994), is defined by

X ðn  sÞ!ðs  1Þ! v A ðSÞ SVi ðN; v Þ ¼ n! S # N;S3i for i 2 N. Thus, the solidarity value of a player is the weighted average of Av(S) where the player belongs to S. In the literature, the difference among the values is commonly explained from the viewpoint of who obtains a zero payoff. A player i is a null player (NP) in game (N, v) if v(S [ i)  v(S) = 0 for any S # Nni. A player i 2 N is a nullifying player in (N, v) if v(S) = 0 for all S # N with i 2 S. A player i 2 N is an A-null player in (N, v) if Av(S) = 0 for all S # N with i 2 S. van den Brink (2007) provided the axiomatizations of both the Shapley value and the egalitarian value, and showed that the difference between the two values arises from the players who obtain zero payoffs, i.e., a null player for the Shapley value and a nullifying player for the egalitarian value. The comparison of the axiomatization of the Shapley value by Shapley (1953) with that of the solidarity value by Nowak and Radzik (1994) suggests that the difference between the two values is also from the players who obtain zero payoffs, i.e., a null player for the Shapley value and an A-null player for the solidarity value. A similar observation is obtained from the comparison of the axiomatization of the Shapley value by van den Brink (2001) with that of the solidarity value by Casajus (2011a).

Invariance from NP deletion: For any (N, v) 2 C, if k 2 N is an NP in (N, v), ui(N, v) = ui(Nnk, v) for any i 2 Nnk. In Derks and Haller (1999), this axiom is referred to as the Null Player Out property. Derks and Haller (1999) explore the class of solution concepts satisfying this property and provide a necessary and sufficient condition for a linear value to satisfy the property. It is easily shown that the Shapley value satisfies the invariance from NP deletion. Note that this property is different from the usual null player property that requires that the null players obtain zero payoffs. The other type of invariance from a player deletion that we are interested in is the invariance of the bargaining power of other players before and after a player’s deletion. We introduce two candidates for this type of player. A player k is a proportional player (PP) in (N, v) if v(k) = 0 and v ðS [ kÞ  v ðSÞ ¼ v ðSÞ for any S # Nnk, s S – ;. A player k is a quasi-proportional player (P’P) if ðSÞ v ðS [ kÞ  v ðSÞ ¼ vsþ1 for any S # Nnk. If k is a PP, then

v ðS [ kÞ v ðSÞ sþ1

¼

s

ð1Þ

:

Thus, PP makes zero contribution to any coalition if the contribution of a player is measured in per capita terms. On the other hand, if player k is P’P, the following holds for any S # Nnk:

v ðS [ kÞ sþ2

¼

v ðSÞ : sþ1

ð2Þ

Thus, as in the PP case, P’P makes zero contribution to any coalition if we measure the contribution of a player in per capita terms, but the worth of coalition S is now divided by s + 1 instead of being divided by s, which might reflect the existence of one additional player who is not described in the original TU game but who plays some special role in the underlying situation.4 The idea that the deletion of PP or P’P does not affect the bargaining power of the other players motivates the following two axioms. Invariance from PP deletion: For any (N, v) 2 C, if k 2 N is a PP in (N, v), ui(N, v) = ui(Nnk, v) for any i 2 Nnk. Invariance from P’P deletion: For any (N, v) 2 C, if k 2 N is a P’P in (N, v), ui(N, v) = ui(Nnk, v) for any i 2 Nnk. Interestingly, the egalitarian value and the solidarity value satisfy invariance from PP deletion and from P’P deletion, respectively. Lemma 1. (i) The egalitarian value satisfies the invariance from PP deletion. (ii) The solidarity value satisfies the invariance from P’P deletion. Proof. (i) Suppose player k is PP in (N, v). Then, for any i 2 Nnk,

3. Invariance from player deletion Let u be a value on C. Let us imagine how a payoff suggested by u will change if a certain player leaves the game and u is applied to a new situation. In particular, we are interested in the kinds of players that can be removed from the game without affecting the payoffs of the other players. These kinds of players are different from one solution to another. The literature provides a partial answer to the above question. Because a null player gives a zero contribution to every coalition, it is natural to require that a solution should assign the same payoffs to each player (other than a null player) when the null player is there and after the null player has left the game. This view yields the following axiom.

EVi ðN; v Þ ¼

v ðNÞ n

n

¼ n1

v ðN n kÞ v ðN n kÞ n

¼

n1

¼ EVi ðN n k; v Þ:

(ii) First, we show that for each (N, v) 2 C and for each i 2 N,

SVi ðN; v Þ ¼

v ðNÞ

where hðsÞ ¼

n

þ

X S(N;S3i

ðnsÞ!ðs1Þ! n!

hðsÞ

v ðSÞ X v ðSÞ  ; hðs þ 1Þ s þ 1 S # Nni sþ1

for s with 1 6 s 6 n,

4 For instance, the ring games obtained by n bidder collusions in a one-object auction situation are related to market situations that consist of n bidders and one seller (for details, see Branzei et al., 2009).

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By definition,

1X SVi ðN; v Þ ¼ hðtÞ ðv ðTÞ  v ðT n jÞÞ t j2T T # N;T3i X

v ðNÞ

¼

n

X

þ

X

hðtÞv ðTÞ 

T(N;T3i

hðtÞ

T # N;T3i

1X v ðT n jÞ t j2Tni

X

1  hðtÞ v ðT n iÞ: t T # N;T3i

ð3Þ

The third term of the last expression can be reduced to



X

hðtÞ

T # N;T3i

X ðn  s  1Þ!s! 1 1X ðn  sÞv ðSÞ v ðT n jÞ ¼  t j2Tni n! sþ1 S(N;S3i X ðn  sÞ!ðs  1Þ! s v ðSÞ n! sþ1 S(N;S3i X s hðsÞ v ðSÞ; ¼ s þ 1 S(N;S3i

¼

where the first equality is true because for a coalition S with i 2 S, v(S) appears in the summation n  s times and its coefficient is ðns1Þ!s! 1 . Thus, the sum of the second and the third terms in (3) n! sþ1 P 1 is reduced to S(N;S3i hðsÞ sþ1 v ðSÞ: The fourth term in (3) is reduced P 1 to  S # Nni hðs þ 1Þ sþ1 v ðSÞ, by setting S = Tni. This gives us the desired result. Suppose player k is P’P in (N, v). Then, for any i 2 Nnk, SVi ðN; v Þ  SVi ðN n k; v Þ ¼ 

v ðN n kÞ n1 X

þ

T(Nnk;T63i

¼

v ðNÞ



X T(Nnk;T3i

v ðNÞ n

þ

X

hðtÞ

T(N;T3i

v ðTÞ tþ1

X



T(N;T63i

hðt þ 1Þ

v ðTÞ tþ1

ðn  1  tÞ!ðt  1Þ! v ðTÞ ðn  1Þ! tþ1

ðn  t  2Þ!t! v ðTÞ ðn  1Þ! t þ 1

1!ðn  2Þ! 1 v ðN n kÞ v ðN n kÞ  n! n n1 X X v ðTÞ v ðS [ kÞ hðtÞ hðs þ 1Þ þ þ t þ 1 S(Nnk;S3i sþ2 T(Nnk;T3i n



þ

X T(Nnk;T3i



X

ðn  1  tÞ!ðt  1Þ! v ðTÞ ðn  1Þ! tþ1 hðt þ 1Þ

T(Nnk;T63i



X S(Nnk;S63i

hðs þ 2Þ

v ðTÞ tþ1 sþ2

X

The last equality arises from Eq. (2).

Balanced cycle contributions property (BCC): For any (N, v) 2 C and any order (i1, i2, . . . , in) on N, n  X



ui‘ ðN; v Þ  ui‘ ðN n i‘1 ; v Þ ¼

‘¼1

n  X



ui‘ ðN; v Þ  ui‘ ðN n i‘þ1 ; v Þ ;

‘¼1

where i0 = in and in+1 = i1. The Shapley value satisfies BCC because it satisfies BC, and thus, /i‘ ðN; v Þ  /i‘ ðN n i‘1 ; v Þ ¼ /i‘1 ðN; v Þ  /i‘1 ðN n i‘ ; v Þ always holds. P Because the term n‘¼1 ui‘ ðN; v Þ in the definition of BCC is common to both sides, the condition described in the axiom is reduced to n X ‘¼1

ui‘ ðN n i‘1 ; v Þ ¼

n X

ui‘ ðN n i‘þ1 ; v Þ;

‘¼1

where i0 = in and in+1 = i1. This is a more convenient representation of BCC. As such, henceforth we will use this representation of BCC. Note that in a two-person game ({i, j}, v), the conditions required by BCC are automatically satisfied because both the left- and righthand sides of the equations of the condition are ui(i, v) + uj(j, v). BCC requires that cycle contributions among all players should be balanced for any order of the set of all players and its inverse order. Similarly, we can consider the property that cycle contributions among all groups of three (or more) players should be balanced for any order of the group and its inverse order. A balance of contributions among any group of three players is defined as follows: BCC for three players: For any (N, v) 2 C and for any three-player coalition {i, j, k} # N,

v ðS [ kÞ

ðn  t  2Þ!t! v ðTÞ ðn  1Þ! t þ 1 T(Nnk;T63i   n þ 1 v ðNÞ v ðN n kÞ  ¼ n nþ1 n X ðn  s  1Þ!ðs  1Þ! s v ðS [ kÞ v ðSÞ   þ ðn  1Þ! n sþ2 sþ1 S(Nnk;S3i X ðn  s  2Þ!s! s þ 1 v ðS [ kÞ v ðSÞ   ¼ 0:  ðn  1Þ! n sþ2 sþ1 S(Nnk;S63i þ

where (Nnj, v) and (Nni, v) are the restrictions of (N, v) on Nnj and Nni, respectively. Myerson (1980) showed that the Shapley value satisfies BC and is the unique efficient value on C satisfying it. Kamijo and Kongo (2010) proposed a milder condition than BC that would be satisfied by several solutions in addition to the Shapley value. They introduced a balanced cycle contributions property (BCC), which does not require that claims between any two players be balanced but rather requires that claims among all players be balanced in a cyclical manner; i.e., for any order of players, the sum of the claims of each player against his successor is balanced with the sum of the claims of that player against his predecessor. Formally, BCC is defined as follows.

ui ðN n k; v Þ þ uj ðN n i; v Þ þ uk ðN n j; v Þ ¼ ui ðN n j; v Þ þ uj ðN n k; v Þ þ uk ðN n i; v Þ: Note that in the above property, there is no need to specify the order of the players i, j, k because the condition remains unchanged for any order and its inverse. It is straightforward that any value on C satisfying BC also satisfies BCC for three players. Thus, BCC for three players is a weaker property than BC. Kamijo and Kongo (2010) discuss the following relation between BCC and BCC for three players.

h

4. The balanced cycle contributions property Let u be a value on C. The balanced contributions property (BC) of Myerson (1980) requires that for any two players, the claim of one player against the other, which is measured by the contribution of the first player to the second given a certain solution concept, is balanced with the counter claim of the second against the first. Formally, for any (N, v) 2 C and for any {i, j} # N,

ui ðN; v Þ  ui ðN n j; v Þ ¼ uj ðN; v Þ  uj ðN n i; v Þ;

Proposition 1 (Kamijo and Kongo, 2010). BCC and BCC for three players are equivalent. From Myerson’s result on the axiomatization of the Shapley value, we know that the Shapley value is a unique efficient solution satisfying BC. In contrast, BCC (BCC for three players) is satisfied by several (efficient) solutions. In particular, below we show that any symmetric and linear value satisfies BCC for three players. To obtain this result, we first show several lemmas.  (Hammer Given a value u, its additive efficient normalization u and Holzman, 1992) is defined as follows. For any (N, v) 2 C and for any i 2 N,

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 i ðN; v Þ ¼ ui ðN; v Þ þ u



v ðNÞ 

P

i2N

ui ðN; v Þ

n

 :

 of a value u Lemma 5. If the additive efficient normalization u satisfies BCC for three players, then the original value u satisfies BCC for three players.

Obviously, the additive efficient normalization of any efficient value is the value itself. The following holds for the additive efficient normalization of any value. Lemma 2. If a value u is symmetric and linear, then the additive  is symmetric, linear, and efficient normalization of the value u efficient.

Proof. Obvious from the definitions.

h

By Proposition 1, Lemmas 4 and 5 hold when we replace BCC for three players with BCC. From Lemmas 2 through 5 and Proposition 1, we obtain the following. Theorem 1. If a value u is symmetric and linear, then it satisfies BCC.

Proof. We omit the proof as it is very obvious. h

Because both the egalitarian value and the solidarity value are symmetric and linear, they satisfy BCC.

The following lemma was proved by Ruiz et al. (1998). Lemma 3 (Lemma 9 in Ruiz et al., 1998). A value is symmetric, linear, and efficient if and only if there exists qns ðs ¼ 1; 2; . . . n  1Þ such that for any (N, v) 2 C and i 2 N,

ui ðN; v Þ ¼

v ðNÞ n

X

þ

qns

v ðSÞ s

S(N;S3i

X



qns

S(N:S63i

v ðSÞ : ns

ð4Þ

The following lemma describes the relationships between the above formula and BCC. Lemma 4. If a value is represented by Eq. (4), then it satisfies BCC for three players. Proof. Let (N, v) 2 C and let {i, j, k} # N. If n 6 2, a value u trivially satisfies BCC for three players. If n P 3, then,

ui ðN n k; v Þ þ uj ðN n i; v Þ þ uk ðN n j; v Þ ¼

v ðN n kÞ n1 þ

X

þ

X

qn1 s

v ðSÞ

X

v ðN n kÞ n1 

qn1 s

X

v ðSÞ s

X

qn1 s

X

qn1 s

S(Nnj;S3i;k

v ðSÞ s

v ðSÞ v ðN n iÞ þ n1s n1

v ðSÞ v ðN n jÞ þ n1s n1

qn1 s

v ðSÞ n1s X

þ

qn1 s

v ðSÞ s

S(Nnk;S3i;S63j

X v ðSÞ v ðN n iÞ v ðSÞ þ þ qn1 s n1s n1 s S(Nni;S3j;k

v ðSÞ s

S(Nni;S3j;S63k

þ

qn1 s

S(Nnj:S63k

qn1 s

qn1 s

S(Nnk:S63i

X



S(Nnk;S3i;j

qn1 s

X



S(Nni:S63j

X

þ

S(Nnk:S63i

þ

s X



s

S(Nnj;S3k

¼

v ðSÞ

S(Nnk;S3i

S(Nni;S3j

þ

qn1 s

v ðSÞ s



X

qn1 s

S(Nni:S63j

þ

X

v ðSÞ v ðN n jÞ þ n1s n1

qn1 s

S(Nnj;S3k;S63i

v ðSÞ s



X

qn1 s

S(Nnj:S63k

v ðSÞ n1s v ðN n jÞ X n1 v ðSÞ X n1 v ðSÞ v ðN n kÞ ¼ þ  þ qs qs n1 s n  1  s n1 S(Nnj;S3i S(Nnj:S63i 

þ

X

qn1 s

S(Nnk;S3j

þ

X S(Nni;S3k

v ðSÞ s

n1 v ðSÞ

qs

s



X

qn1 s

v ðSÞ v ðN n iÞ þ n1s n1

qn1 s

v ðSÞ n1s

S(Nnk:S63j



X S(Nni:S63k

¼ ui ðN n j; v Þ þ uj ðN n k; v Þ þ uk ðN n i; v Þ:



The next lemma describes an invariance of BCC with respect to additive efficient normalization.

5. Axiomatization of the egalitarian and solidarity values In this section, we provide a general axiomatization result by using the invariance axioms and BCC. The following axiom is basic: Efficiency (EF): For any ðN; v Þ 2 C;

P

i2N

ui ðN; v Þ ¼ v ðNÞ.

The Shapley value is axiomatized by the EF axiom and BC (Myerson, 1980). Kamijo and Kongo (2010) showed that the Shapley value is still axiomatized if we replace BC with BCC, which is weaker. However, we must add an additional axiom, the invariance from NP deletion, to axiomatize the Shapley value. Theorem 2 (Kamijo and Kongo, 2010). The Shapley value is the unique value on C that satisfies EF, BCC, and the invariance from NP deletion. One important remark is that the combination of BCC and the invariance from NP deletion does not imply BC. Thus, the result of Kamijo and Kongo (2010) is not a simple translation of the result of Myerson (1980). The literature contains no studies in which the egalitarian and solidarity values are axiomatized using the axiom related to BC. One reason is that BC is so strong that the Shapley value is the only efficient one-point solution that can satisfy BC. BCC is weaker than BC, and some efficient one-point solutions other than the Shapley value satisfy it. In fact, by Theorem 1, the egalitarian value and the solidarity value both satisfy BCC because both values are symmetric and linear. In the following, we give a useful theorem suggesting that there exists at most one solution that satisfies EF, BCC, and the invariance from a deletion of a certain type of players. We obtain the characterizations of the Shapley, egalitarian, and solidarity values as corollaries of the theorem. Let Q : C ! R with Q(;, v) = 0 for any (;, v) 2 C. Q is a function that associates each game (N, v) 2 C with a real number. Given a function Q and a game (N, v), player k 2 N is a Q-related player in (N, v) if v(S [ k) = Q(S, v) for any S # Nnk. Note that by the definition of Q, v(k) = 0 for any Q-related player k. The following is a generalized representation of the invariance axioms. Q-invariance (QI): If k 2 N is a Q-related player in (N, v), then for any i 2 Nnk, ui(N, v) = ui(Nnk, v). For example, NP, PP, P’P will be Q-related players if we choose the following functions as Q: for S – ;,  Q(1)(S, v) = v(S),  Q ð2Þ ðS; v Þ ¼ sþ1 v ðSÞ, s  Q ð3Þ ðS; v Þ ¼ sþ2 v ðSÞ. sþ1 The following general result is helpful to axiomatize some values in TU games.

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Theorem 3. For any function Q : C ! R with Q ð;; v Þ ¼ 0, there is at most one value satisfying EF, QI, and BCC.

Table 1 Independence of the axioms in Corollary 1. Values/properties

Proof. Let u be a value on C that satisfies EF, QI, and BCC. We show that the uniqueness of the value by using induction with respect to the number of players. When jNj = 1, by EF, ui(N, v) = v(i) for i 2 N. Assume that for any game with less than n players, u(N, v) is uniquely determined. We show that u(N, v) is uniquely determined when N = {1, 2, . . . , n}. Take any integer k with k R N. We extend (N, v) to (N0 , w) as follows:

EF

EV 2

8 < v ðNÞ  v ðN n iÞ ui ðN; v Þ ¼ jNnPj v ðNÞ : n 0 The Shapley value

if i 2 P if i ¼ minj2NnP j otherwise

BCC



+

+

+



+

+

+



P: the set of all proportional players, +: satisfies, : does not satisfy.

Table 2 Independence of the axioms in Corollary 2.

N0 ¼ N [ k

Values/properties

EF

BCC

and for any S # N0 ,

 wðSÞ ¼

SV 2

v ðSÞ

if k R S; Q ðS n k; v Þ if k 2 S:

ui ðN; v Þ ¼

(

v ðNÞ  v ðNPn iÞ ð1jP0 jÞv ðNÞþ

jNnP0 j

0

Invariance from PP deletion

The Shapley value

0

j2P 0

v ðNnjÞ

if i 2 P 0

Invariance from P’P deletion



+

+

+



+

+

+



otherwise

Clearly, k is a Q-related player in (N , w) and in (N nj, w) for any j 2 N. In addition, (N0 nk, w) = (N, v) and (N0 n{j, k}, w) = (Nnj, v) for any j 2 N. Consider an ordering (1, k, 2, . . . , n) on N0 . By BCC,

P : the set of all quasi-proportional players, +: satisfies, : does not satisfy.

u1 ðN0 n n; wÞ þ uk ðN0 n 1; wÞ þ    þ un ðN0 n ðn  1Þ; wÞ ¼ u1 ðN0 n k; wÞ þ uk ðN0 n 2; wÞ þ    þ un ðN 0 n 1; wÞ:

Theorem 2 is obtained as the corollary of Theorem 3 and the fact that the Shapley value satisfies the invariance from NP deletion, by letting Q(N, v) = Q(1)(N, v). Furthermore, the following characterization results can be easily obtained from Lemma 1 and Theorem 3.

ð5Þ

By EF and QI,

uk ðN0 n 1; wÞ ¼ wðN0 n 1Þ 

X

ui ðN0 n 1; wÞ

i2Nn1

¼ Q ðN n 1; v Þ 

X

ui ðN n 1; v Þ

i2Nn1

¼ Q ðN n 1; v Þ  v ðN n 1Þ: 0

Thus, uk(N n1, w) is uniquely determined. Similarly, uk(N0 n2, w) is also uniquely determined. By QI, ui(N0 nj, w) = ui(Nnj, v) for any j 2 N and any i 2 Nnj. Therefore, Eq. (5) is equivalent to

u1 ðN n n; v Þ þ Q ðN n 1; v Þ  v ðN n 1Þ þ u2 ðN; v Þ þ    þ un ðN n ðn  1Þ; v Þ ¼ u1 ðN; v Þ þ Q ðN n 2; v Þ  v ðN n 2Þ þ u2 ðN n 3; v Þ þ    þ un ðN n 1; v Þ; or

u1 ðN; v Þ  u2 ðN; v Þ ¼ QðN n 1; v Þ  v ðN n 1Þ  Q ðN n 2; v Þ þ v ðN n 2Þ  u2 ðN n 3; v Þ      un ðN n 1; v Þ þ u1 ðN n n; v Þ þ u3 ðN n 2; v Þ þ    þ un ðN n ðn  1Þ; v Þ: Let b1 denote the right-hand side of the above equation. By the induction hypothesis, b1 is uniquely determined. Applying a similar argument to the orderings (1, 2, k, 3, . . . , n), (1, 2, 3, k, 4, . . . , n), . . . , and (1, 2, . . . , n  1, k, n), we obtain the following (n  1) equations:

u1 ðN; v Þ  u2 ðN; v Þ ¼ b1 ; u2 ðN; v Þ  u3 ðN; v Þ ¼ b2 ; .. .

0

Corollary 1. Let Q(N, v) = Q(2) (N, v). Then, a Q-related player is a PP and QI coincides with the invariance from PP deletion. Since the egalitarian value satisfies EF, BCC, and the invariance from PP deletion, it is the unique value on C that satisfies the invariance from PP deletion, EF, and BCC. Corollary 2. Let Q(N, v) = Q(3)(N, v). Then, a Q-related player is a P’P and QI coincides with the invariance from P’P deletion. Since the solidarity value satisfies EF, BCC, and the invariance from P’P deletion, it is the unique value on C that satisfies the invariance from P’P deletion, EF, and BCC. For the independence of the axioms in Corollaries 1 and 2, see Tables 1 and 2 below. From Theorem 2 and Corollaries 1 and 2, we obtain the comparable axiomatizations of the Shapley value, the egalitarian value, and the solidarity value. These axiomatizations state that the difference between the three values lies in the difference between the players whose deletion does not affect the other players’ payoffs. To conclude this section, we discuss the invariance from nullifying player deletion defined as follows. Invariance from nullifying player deletion: For any (N, v) 2 C, if k 2 N is a nullifying player in (N, v), ui(N, v) = ui(Nnk, v) for any i 2 Nnk. A nullifying player will be a Q-related player if Q(4)(N, v) = 0. Then, the following is a natural question, is there a value that obeys EF, BCC, and QI with respect to Q = Q(4)?5 The following theorem gives a negative answer to this question. Theorem 4. There is no value on C that satisfies EF and the invariance from nullifying player deletion.

un1 ðN; v Þ  un ðN; v Þ ¼ bn1 : By EF,

u1 ðN; v Þ þ u2 ðN; v Þ þ    þ un ðN; v Þ ¼ v ðNÞ: Since these n equalities are linearly independent, u(N, v) is uniquely determined. h

Proof. We show this fact by a contradiction. Let u be a value on C. Take a three-person game ({i, j, ‘}, v) where players j, ‘ are nullifying players, i.e., v(S) = 0 if S 3 j or S 3 ‘. 5

This question was pointed out by one of the referees.

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By the invariance from nullifying player deletion and EF, for h = j, ‘,

ui ðfi; j; ‘g; v Þ ¼ ui ðfi; hg; v Þ ¼ ui ði; v Þ ¼ v ðiÞ:

ð6Þ

In addition, by the invariance from nullifying player deletion, EF, and Eq. (6), for h = j, ‘,

uh ðfi; j; ‘g; v Þ ¼ uh ðfi; hg; v Þ ¼ v ðfi; hgÞ  ui ðfi; hg; v Þ ¼ v ðiÞ: ð7Þ Generally, Eqs. (6) and (7) contradict EF, because ui({i, j, ‘}, v) + uj({i, j, ‘}, v) + u‘({i, j, ‘}, v) = v(i) – 0 = v({i, j, ‘}). h Remark 1. The function Q considered in the paper reminds the potential function studied by Hart and Mas-Colell (1989). Hence one can ask about the possible relations between them. Theorem 4 says that (in general) there is no such relations, because for any game the potential function always exists and is unique. In view of Theorem 4, also another open question arises: under what conditions on Q in Theorem 3, is there exactly one value satisfying EF, QI and BCC?

6. Axiomatization of the Banzhaf value

X

1 n1

2

S # N;S3i

v ðSÞ v ðS [ jÞ

if i R S;

Proof. First, we show that 2EF and the invariance from NP deletion imply the null player property. Given (N, v) 2 C, if k 2 N is a null player in (N, v), then

where the first equality follows from the invariance from NP deletion, the second equality follows from the fact that k is an NP, and the third equality follows from 2EF. Thus, uk(N, v) = 0. Let (N, v) 2 C and consider jNj P 2. Given a game (N, v) and k R N, the null-extended game (N0 , w) of a game (N, v) with respect to k is defined as follows:

wðSÞ ¼ v ðS n kÞ: Clearly, jN0 j P 3 and player k is an NP in game (N0 , w) and in any of its restriction (N00 , w) satisfying N00 # N0 and k 2 N00 . Take any i, j 2 N # N0 and consider the set {i, j, k}. By BCC for three players,

ui ðN 0 n k; wÞ þ uj ðN0 n i; wÞ þ uk ðN0 n j; wÞ ¼ ui ðN0 n j; wÞ þ uj ðN0 n k; wÞ þ uk ðN0 n i; wÞ:

ð8Þ

By the fact that 2EF and the invariance from NP deletion together imply the null player property, Eq. (8) is equivalent to BC of Myerson. h

Lemma 7. The Banzhaf value is characterized by BC of Myerson, 2EF, and EF1.

otherwise:

In the amalgamated game, players i and j act as if they are a single player. 2-efficiency (2EF): For any (N, v) 2 C and any {i, j} ui ðN; v Þ þ uj ðN; v Þ ¼ ui ðNij ; v ij Þ:

Lemma 6. 2EF, the invariance from NP deletion, and BCC (for three players) together imply BC of Myerson.

Next, we give a new characterization of the Banzhaf value using Myerson’s BC.

v ij : 2Nnj ! R, such that 

This property requires the usual efficiency for one-person games. It is straightforward that the Banzhaf value is efficient with respect to one-person games.6 We show the relationships between Myerson’s BC and our axioms.

and for any S # N0 ,

Nij ¼ N n j

v ij ðSÞ ¼

ui ði; v Þ ¼ v ðiÞ:

N0 ¼ N [ k

ðv ðSÞ  v ðS n iÞÞ:

Hence, like the Shapley value, the Banzhaf value is a weighted average of each player’s marginal contributions to all coalitions containing that player. However, the coefficient for each coalition is different from what is used in the definition of the Shapley value. The Banzhaf value is not efficient, but it does satisfy BC (see Sánchez, 1997). Because BCC is weaker than BC, the Banzhaf value also satisfies BCC. Furthermore, the Banzhaf value satisfies the invariance from NP deletion (see Nowak and Radzik, 2000; Casajus, 2011b). In the following, we describe two efficiency-related properties. Given a game (N, v) 2 C and any {i, j} # N, the amalgamated game (Nij, vij) is given by

and

Efficiency with respect to 1-person games (EF1): For any one-person game (i, v),

ui ðN; v Þ ¼ ui ðN n k; v Þ ¼ ui ðNik ; v ik Þ ¼ ui ðN; v Þ þ uk ðN; v Þ;

Next, we replace efficiency with other efficiency-related properties and characterize the Banzhaf value (Banzhaf, 1965; Owen, 1975), which is a well-known inefficient value on the class of all games. The Banzhaf value BV is defined as follows: for any (N, v) 2 C and for any i 2 N,

BV i ðN; v Þ ¼

(see, e.g., Lehrer, 1988; Haller, 1994; Nowak, 1997; Nowak and Radzik, 2000; Alonso-Meijide et al., 2007; Casajus, in press, 2011b). However, in our characterization by using the invariance from NP deletion and BCC, 2EF is not enough and another efficiency-related property is needed.

# N,

2EF requires that any merger of two players i and j into a single player i neither benefits nor harms the two players. Hence, it is a kind of collusion-proofness. Recently, collusion-proofness and similar properties have attracted the attention of those who study cooperative games in an open anonymous environment (see, e.g., Yokoo et al., 2005). The Banzhaf value satisfies 2EF. Instead of EF found in the characterizations of the Shapley value, a version of 2EF plays a key role in the characterizations of the Banzhaf value

Proof. Since we have already shown that the Banzhaf value satisfies these three properties, we need to only show that the value u satisfying the three properties is uniquely determined. The proof is by induction with respect to the number of players in the games. Let (N, v) 2 C and let u be a value on C that satisfies the three properties. In case jNj = 1, EF1 implies that ui(i, v) = v(i). Assume that u is uniquely determined when there are less than n players and consider the case jNj = n P 2. Fix i 2 N and consider j 2 Nni. By 2EF,

6 Algaba et al. (2004) use this axiom to characterize the Banzhaf value for the games on antimatroid.

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Applying BCC to the proportionally-extended game ({i, j, k}, w), we have

Table 3 Independence of the axioms in Theorem 5. Values/properties

2EF

EF1

BCC

Invariance from NP deletion

The Shapley value

 +

+ 

+ +

+ +

+

+



+

+

+

+



vÞ uðN; v Þ ¼ BV ðN; 2

ui ðN; v Þ ¼ ui ðN; v Þ ¼



0

if i 2 X

v ðNÞ otherwise: 2n1jXj v ðNÞ for any i2N 2n1

ui ðfi; jg; wÞ þ uj ðfj; kg; wÞ þ uk ðfi; kg; wÞ ¼ ui ðfi; kg; wÞ þ uj ðfi; jg; wÞ þ uk ðfj; kg; wÞ () ui ðfi; jg; v Þ  uj ðfi; jg; v Þ ¼ ui ðfi; kg; wÞ þ uk ðfj; kg; wÞ  uj ðfj; kg; wÞ  uk ðfi; kg; wÞ:

ð12Þ

By Eqs. (10)–(12),

X: the set of all null players in (N, v), +: satisfies, : does not satisfy.

ui ðfi; jg; v Þ  uj ðfi; jg; v Þ ¼ 2wðiÞ  2wðjÞ  wðfi; kgÞ þ wðfj; kgÞ ¼ 0:

ui ðN; v Þ þ uj ðN; v Þ ¼ ui ðNij ; v ij Þ:

ð13Þ

By BC of Myerson,

ui ðN; v Þ  uj ðN; v Þ ¼ ui ðN n j; v Þ  uj ðN n i; v Þ:

The last equation follows from Eq. (1). Combining Eqs. (9) and (13), we obtain

Thus,

ui ðfi; jg; v Þ ¼ uj ðfi; jg; v Þ ¼

1 ui ðN; v Þ ¼ ðui ðN ij ; v ij Þ þ ui ðN n j; v Þ  uj ðN n i; v ÞÞ: 2

We now produce a contradiction for the case of general three-person games. Namely, take any game (N, v) with N = {i, j, ‘}. By 2EF and Eq. (14), we get the following equalities:

By the induction assumption, the right-hand side is uniquely determined. Therefore, u is uniquely determined for any n-person game. h Lemmas 6 and 7 together imply the following new characterization results of the Banzhaf value using BCC. Theorem 5. The Banzhaf value is the unique value on C that satisfies 2EF, EF1, BCC, and the invariance from NP deletion.7 For the independence of 2EF, EF1, BCC, and the invariance from NP deletion, see Table 3. Unlike the general axiomatization results obtained in Section 5, the invariance from PP or P’P deletion does not characterize any value in conjunction with 2EF, EF1, and BCC. More precisely, no value satisfies the four axioms, as shown in the following. Theorem 6. (i) There is no value on C that satisfies 2EF, EF1, BCC, and the invariance from PP deletion. (ii) There is no value on C that satisfies 2EF, EF1, BCC, and the invariance from P’P deletion. Proof. We show (i) via a contradiction. Let u be a value on C satisfying all four properties mentioned in (i). First, consider a oneperson game. By EF1, u is uniquely determined as ui(i, v) = v(i). Next, consider an arbitrary two-person game ({i, j}, v). By 2EF and EF1,

ui ðfi; jg; v Þ þ uj ðfi; jg; v Þ ¼ ui ðfi; jgij ; v ij Þ ¼ v ij ðiÞ ¼ v ðfi; jgÞ:

ð9Þ

If j is a PP in ({i, j}, v), the invariance from PP deletion and EF1 imply

ui ðfi; jg; v Þ ¼ ui ði; v Þ ¼ v ðiÞ:

ð10Þ

v ðfi; jgÞ 2

ui ðN; v Þ þ uj ðN; v Þ ¼ ui ðNij ; v ij Þ ¼

ð14Þ

:

v ðNÞ

; 2 v ðNÞ ; uj ðN; v Þ þ u‘ ðN; v Þ ¼ uj ðN j‘ ; v j‘ Þ ¼ 2 v ðNÞ : u‘ ðN; v Þ þ ui ðN; v Þ ¼ u‘ ðN‘i ; v ‘i Þ ¼ 2 Hence,

ui ðN; v Þ ¼ uj ðN; v Þ ¼ u‘ ðN; v Þ ¼

v ðNÞ 4

ð15Þ

:

Consider the case where player ‘ is a PP. By Eq. (1), it follows that v ðfi; j; ‘gÞ ¼ 32 v ðfi; jgÞ. By the invariance from PP deletion and Eq. (14),

ui ðfi; j; ‘g; v Þ ¼ ui ðfi; jg; v Þ ¼

v ðfi; jgÞ 2

:

On the other hand, by Eq. (15),

ui ðfi; j; ‘g; v Þ ¼

v ðfi; j; ‘gÞ 4

¼

3 v ðfi; jgÞ: 8

This is a contradiction. A similar proof can be used to prove (ii). The main difference between the proofs for (i) and (ii) is that we use the quasiproportionally extended game defined below instead of a proportionally-extended game. Given a game (N, v), a quasiproportionally extended game with PP’ k R N is defined as (N [ k, w), where for any S # N [ k,

( wðSÞ ¼

0 jSjþ1 jSjjS\kj

if S ¼ k;

v ðS n kÞ

otherwise:



In addition, by Eqs. (9) and (10),

uj ðfi; jg; v Þ ¼ v ðfi; jgÞ  v ðiÞ:

ð11Þ

Given a two-person game ðfi; jg; v Þ, consider the following proportionally-extended game ({i, j, k}, w), where for any S # {i, j, k},

( wðSÞ ¼

0 jSj jSjjS\kj

if S ¼ k;

v ðS n kÞ

otherwise:

One can easily deduce that k is a PP in ({i, j, k}, w), ({i, k}, w), and ({j, k}, w). Furthermore, ðfi; jg; wÞ ¼ ðfi; jg; v Þ.

Note that our impossibility results in Theorem 6 are essential, i.e., there are one-point solutions that satisfy any three out of the four axioms. The egalitarian (or solidarity) value satisfies EF1, BCC, and the invariance from PP (P’P) deletion, but not 2EF. A value that always assigns zero for all players satisfies 2EF, BCC, and the invariance from PP (P’P) deletion, but not EF1. Let P(N, v) be the set of all PPs in (N, v), let P0 (N, v) be the set of all P’P in (N, v), let v ðNÞ 2 u1i ðN; v Þ be 0 if i 2 P(N, v) and 2n1jPðN; v Þj otherwise, and let ui ðN; v Þ v ðNÞ 1 2 be 0 if i 2 P0 (N, v) and n1jP 0 ðN;v Þj otherwise. The value u (u ) satisfies 2

2EF, EF1, and the invariance from PP (P’P) deletion, but not BCC. Let 7

Casajus (in press) characterizes the Banzhaf value by 2EF and the dummy player axiom, which is stronger than EF1.

ðNÞ u3i ðN; v Þ ¼ v2n1 for any i 2 N. This value satisfies 2EF, EF1, and BCC,

but not the invariance from PP or P’P deletion.

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Table 4 Summary of results.

Invariance from NP deletion Invariance from PP deletion Invariance from P’P deletion

EF & BCC

2EF & EF1 & BCC

The Shapley value (Kamijo and Kongo, 2010) The egalitarian value (Corollary 1) The solidarity value (Corollary 2)

The Banzhaf value (Theorem 5) No solution (Theorem 6 (i)) No solution (Theorem 6 (ii))

7. Concluding remarks In this paper, we have provided a general method of axiomatic characterization of one-point solutions for cooperative games with transferable utilities. Any one-point solution that is efficient, and that satisfies both the balanced cycle contributions property (symmetry and linearity together imply this property), and the axioms related to the invariance under a player deletion is characterized as a corollary of our general results. This is true, for instance, for various well-known one-point solutions. The Shapley value is characterized by focusing on the invariance from null player deletion, the egalitarian value is characterized by focusing on the invariance from the proportional player deletion, and the solidarity value is characterized by focusing on the invariance from quasi-proportional player deletion. Our result can be seen as a generalization of the result of Kamijo and Kongo (2010), and it captures the similarities and differences among the three values quite well. Furthermore, the center of imputation set (CIS) value (Driessen and Funaki, 1991) is also characterized in a similar manner.8 We also examined the characterizations of inefficient one-point solutions. Replacing efficiency with 2-efficiency and efficiency with respect to one-person games in the lists of axioms that Kamijo and Kongo (2010) used to characterize the Shapley value, we were able to characterize the Banzhaf value. However, we showed that no solution satisfies the two efficiency-related properties, the balanced cycle contributions property, and the invariance from proportional or quasi-proportional player deletion. The main results obtained in this paper are summarized in Table 4. Acknowledgments The authors thank Emilio Calvo, Jean Derks, and Yukihiko Funaki, who provided us with helpful comments and discussions. The authors are grateful to an associate editor and anonymous referees for their useful comments and suggestions. All remaining errors are, of course, the authors’. References Algaba, E., Bilbao, J.M., van den Brink, R., Jimenez-Losada, A., 2004. An axiomatization of the Banzhaf value for cooperative games on antimatroids. Mathematical Methods of Operations Research 59, 147–166. Alonso-Meijide, J., Carreras, F., Fiestras-Janeiro, M., Owen, G., 2007. A comparative axiomatic characterization of the Banzhaf–Owen coalitional value. Decision Support Systems 43, 701–712. Arin, J., Feltkamp, V., in press. Coalitional games: Monotonicity and Core. European Journal of Operational Research. doi:10.1016/j.ejor.2011.07.025.

8 For balanced TU games, core-selections are one of the areas of active research (e.g., Alexia value by Tijs et al. (2011), the core-center by González-Díaz and SánchezRodríguez (2009), and a monotonic core solution by Arin and Feltkamp (in press)). To extend our method to such non-linear solutions is a topic of interest.

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