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Axiomatizations of coalition aggregation functions
Mathematical Social Sciences 103 (2020) 69–75
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Axiomatizations of coalition aggregation functions Takaaki Abe School of Political Science and Economics, Waseda University, 1-6-1, Nishi-waseda, Shinjuku-ku, Tokyo 169-8050, Japan
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Article history: Received 21 July 2019 Received in revised form 10 November 2019 Accepted 3 December 2019 Available online 16 December 2019 Keywords: Axiomatization Coalition formation Coalition structure Monotonicity
a b s t r a c t We axiomatize Hart and Kurz’s (1983) two coalition aggregation functions known as the γ -function and the δ -function. A coalition aggregation function is a mapping that assigns a partition to each profile of coalitions selected by players. Through our axiomatization results, we observe that neither the γ function nor the δ -function satisfies monotonicity. We propose a monotonic function and axiomatically characterize it. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The main purpose of this paper is to answer the following question: what coalition structure should be assigned to a profile of coalitions selected by players? To clarify this question, we begin with a simple example. Suppose that there are three players, N = {1, 2, 3}. Every player chooses a coalition that she/he wants to form: for example, each student submits a list of students with whom he wants to share a room. We suppose that players 1 and 2 want to form the three-person coalition {1, 2, 3}, while player 3 wants to form the two-person coalition with player 2, namely, {2, 3}. Their choices are summarized as
σ = (123, 123, 23). For simplicity, we omit the parentheses of a coalition if it is not confusing: we write 123 instead of {1, 2, 3}. We call such a tuple σ a coalition profile. Now, what coalition structure is ‘‘optimal’’ for this coalition profile? The first attempts to address this question were made by Hart and Kurz (1983, 1984). They defined two aggregation rules known as the γ -function Bγ and the δ -function Bδ . These two rules exhibit two different views of coalition formation. On one hand, one might consider a coalition to be a result from a unanimous agreement among all its members. The γ -function formulates this view. Let S ⊆ N be a coalition of players. If all the players in S choose the same coalition S, then S is formed. However, if someone in S chooses a different coalition, the proposers of S are partitioned into singletons. In the three-person example above, neither 123 nor 23 obtains a unanimous agreement. Therefore, the γ -function assigns the three one-person coalitions to the coalition profile. On the other hand, one might consider that a coalition is the largest E-mail address: [email protected]. https://doi.org/10.1016/j.mathsocsci.2019.12.001 0165-4896/© 2019 Elsevier B.V. All rights reserved.
set of players who share the same choice. In this view, the choice of a coalition means the set of all players with whom he is willing to share the same coalition. The δ -function exhibits this view. In the same example, players 1 and 2 choose the same coalition 123. In other words, they accept all players. However, player 3 does not admit player 1. Although we will provide its formal definition in the next section, we can define the δ -function by simply grouping the players who choose the same coalition. Therefore, the δ -function assigns the two-person coalition to players 1 and 2. Player 3 is the only player who chooses coalition 23. Hence, the δ -function assigns his one-person coalition to player 3. In general, we call such a function that assigns a coalition structure to each coalition profile a coalition aggregation function. Our motivation for studying a coalition aggregation function is to associate players’ choices with coalition formation theory. Coalition formation theory has been mainly studied in the frameworks in which players do not select any action and/or choice, e.g., cooperative games and hedonic games. However, when we form a team or a group, our choices and decisions must implicitly/explicitly influence its result. Therefore, we attempt to analyze coalition formation in terms of players’ choices. However, besides the γ - and δ -functions, there are many coalition aggregation functions. How should we choose one coalition aggregation function? As Hart and Kurz (1983) state, ‘‘there is no universally correct answer to this problem’’. Nevertheless, once one chooses a certain function, she must be supposed to answer why she selects it. Our axiomatization results must be helpful to answer this question. As we will discuss in the following sections, the γ - and δ -functions do not satisfy a property called monotonicity. Monotonicity is a guarantee for players who have an acceptive attitude. To see this, we fix a player and the choices of the other players. Even if the player changes his choice to accept more players
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T. Abe / Mathematical Social Sciences 103 (2020) 69–75
Table 1 Axioms and CA-functions. Proposition Proposition Proposition Proposition
2.1 2.2 3.2 3.1
δ
B Bγ B≈ ̸ ∃B
UN
DA
PD
PD−
IR
DR+
MO
⊕ ⊕ ⊕ +
⊕ ⊕ −
+ + − +
+ + ⊕
⊕ − −
− ⊕ −
− − ⊕ +
by choosing a superset, the γ - and δ -functions may provide him with a smaller coalition. Monotonicity guarantees that his acceptive change does not end up such a contrary result: even if his invitation is declined, it does not break his original affiliation. After we axiomatize the γ -function and the δ -function in the next section, we propose a monotonic coalition aggregation function in Section 3. Its axiomatization is also provided. In Section 4, we summarize our results and offer some concluding remarks. Table 1 in Section 4 shows the axiomatic systems of the coalition aggregation functions we study in this paper. All the proofs and the independence of the axioms are provided in the appendix. 2. The γ -function and the δ-function Let N = {1, . . . , n} be a set of players. A coalition is a subset S ⊆ N. We denote the cardinality of coalition S by |S |. We use n to denote |N |. We assume n ≥ 3. For every i ∈ N, Ai is the set of coalitions that contain i, formally, Ai = {S ⊆ N |i ∈ S }. For example, let N = {1, 2, 3}. We have A1 = {1, 12, 13, 123}, A2 = {2, 12, 23, 123}, A3 = {3, 13, 23, 123}.
For every nonempty coalition S ⊆ N, let AS = ×i∈S Ai . We use σ to denote an element of AN . We call σ a coalition profile. We typically use P to denote a partition (or a coalition structure). For every coalition S ⊆ N, let Π (S) be the set of all partitions of S. Let Pi denote the coalition player i belongs to in partition P . A coalition aggregation function (CA-function) is a mapping that assigns a partition to each coalition profile, B : AN → Π (N). Since B(σ ) is a partition, let Bi (σ ) denote the coalition in partition B(σ ) that contains player i. Now, we introduce the two CA-functions proposed by Hart and Kurz (1983). We begin with the δ -function. The δ -function is given as Bδ (σ ) = {T ⊆ N |i, j ∈ T ⇐⇒ σi = σj }.
The δ -function focuses on the players who choose the same coalition. To see this, let N = {1, 2, 3, 4} and σ = (12, 12, 234, 234). Since players 1 and 2 choose the same coalition 12, they belong to coalition 12, namely, B1δ (σ ) = B2δ (σ ) = 12. Similarly, since players 3 and 4 choose the same coalition 234, they belong to coalition 34, namely, B3δ (σ ) = B4δ (σ ) = 34. Hence, the resulting coalition
structure is Bδ (σ ) = {12, 34}. In contrast, the γ -function requires players to make a unanimous agreement. Formally, Bγ (σ ) = {Tσi |i ∈ N }, where Tσi =
{
σi {i}
if σj = σi for every j ∈ σi , otherwise.
Consider the coalition profile σ = (12, 12, 234, 234) again. The list of coalitions proposed in σ is {12, 234}. Since all members in coalition 12 agree to form the coalition 12, coalition 12 is formed. However, for coalition 234, player 2 does not agree to join it. Therefore, coalition 234 lacks a unanimous agreement and is not formed. The γ -function does not allow the proposers of a failed coalition to form any coalition among themselves. Thus,
the resulting coalition structure is Bγ (σ ) = {12, 3, 4}. Note that a player who chooses his own one-person coalition σi = {i}, if any, achieves a unanimous agreement by himself. The two CA-functions for n = 3 are described in Tables Bδ and Bγ . For simplicity, we use symbols N, X, Y, Z, I to denote partitions as follows: N = {123}, X = {12, 3}, Y = {13, 2}, Z = {23, 1}, I = {1, 2, 3}. For example, Bδ (123, 123, 123) = {123}(= N) and Bδ (123, 123, 13) = {12, 3}(= X). We now introduce axioms to characterize these two CAfunctions. By axiomatizing a CA-function, we ‘‘decompose’’ a CA-function into three components: agreement, disagreement, and reaction to a deviation. In other words, we specify each CA-function in terms of these three properties. The first axiom describes a form of agreement among players. Axiom (Unanimity, UN). For any σ ∈ AN and any ∅ ̸ = S ⊆ N, if σj = S for every j ∈ S, then S ∈ B(σ ). This axiom states that if all members of a coalition agree on the formation of the coalition, then the coalition should be formed. For convenience, we call player i a unanimous member if σj = σi for every j ∈ σi (or equivalently, if there exists S ⊆ N such that i ∈ S and σj = S for every j ∈ S). All CA-functions that we discuss in this paper satisfy this axiom. Axiom (Disagreement, DA). For any σ ∈ AN and any i, j ∈ N, if σi ̸= σj , then Bi (σ ) ̸= Bj (σ ). We consider axiom DA to be the simplest form of disagreement. If player i’s choice is different from player j’s choice, then these two players become separate. For example, if player i invites player k, while player j does not, then their proposals clearly conflict over player k. A CA-function that obeys DA assigns different coalitions to such players. In other words, the coincidence between choices is a necessary condition for two players to belong to the same coalition. In this sense, this axiom might be thought of as the most strict form of disagreement. We will also introduce some weaker disagreement axioms later. Table UN and Table DA describe these axioms for n = 3. In the tables, for example, YI means that both {13, 2} ( = Y) and {1, 2, 3} ( = I) are possible and that the other coalition structures (N, X, and Z) are ruled out. In the same manner, ‘‘any’’ means that all partitions are possible. We now introduce an axiom that serves the third component of a CA-function, which describes the remaining players’ reaction to one’s deviation from an agreement. Axiom (Integrative Reaction, IR). For any σ ∈ AN , any i ∈ N, and any σi′ ∈ Ai with σi′ ̸ = σi , if |Bi (σ )| ≥ 3, then Bj (σi′ , σ−i ) = Bk (σi′ , σ−i ) for any j, k ∈ Bi (σ ) \ {i} with j ̸ = k. Let σ be a current coalition profile and B(σ ) be the corresponding coalition structure. In the coalition structure B(σ ), if |Bi (σ )| ≥ 3, then player i shares a coalition with at least two players, say j and k (hence, Bi (σ ) ⊇ {i, j, k}). Axiom IR guarantees that even after player i changes his choice from σi to new coalition σi′ , players j and k can keep their coalition without player i. In this sense, IR admits a partial agreement among players: even if a player splits off from the original agreement, the partial agreement of the remaining players is enough for them to sustain their own coalition. The following axiom is a variant of IR. Axiom (Disintegrative Reaction, DR). For any σ ∈ AN , any i ∈ N, and any σi′ ∈ Ai with σi′ ̸ = σi , if |Bi (σ )| ≥ 3, then Bj (σi′ , σ−i ) ̸ = Bk (σi′ , σ−i ) for any j, k ∈ Bi (σ ) \ {i} with j ̸ = k.
T. Abe / Mathematical Social Sciences 103 (2020) 69–75
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Table Bδ A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
123 N Z Z Z
123 12 Y X I I
123 N I I I
123 12 I X I I
23 Y I I I
23 I I I I
123 N any any ZI
123 12 any X any ZI
123 any ZI ZI ZI
123 12 23 YI YI XI I I I I I
23 any any any ZI
2 Y I I I
2 I I I I
2 YI YI YI I
2 YI I I I
123 X I Y I
13 12 23 I I X I Y Y I I
2 I I Y I
123 X I I I
23 12 23 I Z X Z I Z I Z
2 I I I I
123 X I I I
3 12 I X I I
23 I I I I
2 I I I I
123 I I Y I
Table Bγ 13 12 23 2 I I I X I I Y Y Y I I I
123 I I I I
23 12 23 I Z X Z I Z I Z
2 I I I I
123 I I I I
3 12 I X I I
23 I I I I
2 I I I I
123 any any Y ZI
Table UN 13 12 23 2 any any YI X any YI Y Y Y ZI ZI I
123 any any any ZI
23 12 23 any Z X Z any Z ZI Z
2 YI YI YI I
123 XI XI XI I
3 12 XI X XI I
23 XI XI XI I
2 I I I I
123 XI I YI I
Table DA 13 12 23 2 I I I XI I I YI YI YI I I I
123 XI I I I
23 12 23 I ZI XI ZI I ZI I ZI
2 I I I I
123 XI I I I
3 12 I XI I I
23 I I I I
2 I I I I
This axiom can be seen as the counterpart of IR: it no longer allows players j and k to stay in the same coalition after i’s deviation. Therefore, DR does not admit a partial agreement among the remaining players. More specifically, the coincidence of the choices among the remaining players does not bring about any coalition of the remaining players. In the same manner, we now introduce a coalitional version of DR. Axiom (Coalitional Disintegrative Reaction, DR+ ). For any σ ∈ AN , any ∅ ̸ = S ⊆ N, and any σS′ ∈ AS with σj′ ̸ = σj for all j ∈ S, if there exists T ∈ B(σ ) such that T ⊇ S and |T \ S | ≥ 2, then Bj (σS′ , σ−S ) ̸ = Bk (σS′ , σ−S ) for any j, k ∈ T \ S with j ̸ = k. In this axiom, in addition to individual deviations, coalitional deviations are also incorporated. Hence, DR+ implies DR. Note that for n = 3, DR+ is equivalent to DR.1 Given these four axioms, we obtain the following two characterizations. Proposition 2.1. it is Bδ .
CA-function B satisfies UN, DA, IR if and only if
1 Because of the constraints |B (σ )| ≥ 3 (in IR) and |T \ S | ≥ 2 (in DR+ ), the i restrictions of these axioms are modest: for n = 3, only if B(σ ) = {123} for some σ can these axioms restrict CA-functions. Moreover, as shown in Proposition 2.2, DR+ is needed instead of DR to characterize Bγ . To see this, we consider (a part of) the following CA-function for n = 4: B(1234, 1234, 1234, 1234) = {1234}, B(1, 1234, 1234, 1234) = {1, 2, 3, 4}, B(1, 2, 1234, 1234) = {1, 2, 34}. This function does not violate DR because the constraint required to player 2 is slack: player 2 belongs to his one-person coalition B2 (1, 1234, 1234, 1234) = {2}. In other words, this player 2 does not achieve the if-part of axiom DR because |{2}| < 3, even if he changes his choice 1234 to 2. However, the function above cannot be Bγ as B(1, 2, 1234, 1234) = {1, 2, 34} ̸ = {1, 2, 3, 4} = + Bγ (1, 2, 1234, 1234). Axiom DR works to rule out such a function.
Proposition 2.2. if it is Bγ .
CA-function B satisfies UN, DA, DR+ if and only
The proofs and the independence of the axioms are provided in the appendix. In view of the propositions, we can attribute the difference between the two CA-functions to the third component of the axiomatic systems, namely, IR or DR+ , which means that reaction to a deviation differentiates these CA-functions. Moreover, no CA-function satisfies both IR and DR simultaneously, while some functions satisfy neither IR nor DR. Such CA-functions should describe another type of reaction. The CA-function we propose in the next section is one of such functions. 3. Monotonicity In the previous section, we have axiomatized the γ - and δ -functions. However, because of IR and DR (or DR+ ), these two functions have an inflexible feature for one’s deviation. To see this, we consider the leftmost profiles in Tables Bγ and Bδ . The two functions assign (N, I, I, I) and (N, Z, Z, Z) to these profiles, respectively. We suppose that player 1 changes his choice from coalition 123 to another coalition. Then, the two assignments (N, I, I, I) and (N, Z, Z, Z) imply that the γ - and δ -functions do not take into account what coalition player 1 chooses as his new choice: the partitions of the non-deviating players are independent of the new choice of player 1. In this section, we introduce another CAfunction by using monotonicity that also incorporates the new choice of a deviating player. There are one similarity and two differences between monotonicity and the previous two axioms, IR and DR. The similarity is that a certain player changes his choice. Therefore, we consider
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T. Abe / Mathematical Social Sciences 103 (2020) 69–75
monotonicity to be an axiom that serves the third component of an axiomatic system. The first difference is that his new choice is now a superset of the first choice. The second difference is that monotonicity focuses on the player who changes a choice instead of the remaining players. Axiom (Monotonicity, MO). For any σ ∈ AN , any i ∈ N, and any σi′ ∈ Ai , if σi ⊆ σi′ , then Bi (σi , σ−i ) ⊆ Bi (σi′ , σ−i ). This axiom states that if a player changes his choice to a larger coalition in the sense of superset, then he should belong to a larger coalition or at least the same coalition. As mentioned in Section 1, we can consider this axiom to be a guarantee for a player who accepts more players: when a player changes his choice, if his new choice admits and allows more players to enter, then this friendly change does not give him a smaller coalition. In other words, even if his invitation is declined by some of the invited players, that failure does not break his original affiliation. Both the γ - and δ -functions violate monotonicity.2 In view of Propositions 2.1 and 2.2, one might consider that by simply replacing IR or DR+ with MO, we can obtain a CAfunction. However, no CA-function simultaneously satisfies these three axioms: UN DA, and MO. Worse, this negative result holds even for a weaker version of DA. To see this, we introduce the following notion and axiom. Players i and j are said to be a (direct) pair in σ if i ∈ σj and j ∈ σi : namely, if there is a mutual acceptance between the two players, we say that they are a pair. σ We denote a pair by i ∼ j. Axiom (Pairwise Disagreement, PD). For any σ ∈ AN and any σ i, j ∈ N, if not i ∼ j, then Bi (σ ) ̸ = Bj (σ ). This axiom requires that if at least one of the two players does not accept the other, then they are separate. Moreover, this axiom is equivalent to the following statement: if j ̸ ∈ σi then j ̸ ∈ Bi (σ ). Therefore, axiom PD interprets σi as player i’s claim that he does not want to share a coalition with the players in N \ σi . Table PD describes this axiom for n = 3. In view of Tables DA and PD, it may be clear that DA implies PD. The following result shows the impossibility result mentioned above. Proposition 3.1. and MO.
No CA-function simultaneously satisfies UN, PD,
Proposition 3.1 implies that PD is still restrictive if a CAfunction satisfies UN and MO. However, we can obtain a positive result by slightly relaxing PD. We say players i and j are an indirect pair in σ if there is a sequence of players k1 , . . . , kM such that σ k1 = i, kM = j, and km ∼ km+1 for m = 1, . . . , M − 1. We denote σ
an indirect pair by i ≈ j. Axiom (Weak Pairwise Disagreement, PD− ). For any σ ∈ AN and σ
any i, j ∈ N, if not i ≈ j, then Bi (σ ) ̸ = Bj (σ ). This axiom states that if player j has no connection with any player in the sequence of pairs that starts from player i, then the player j does not share the same coalition with the player i. Therefore, in order to join the group of player i, player j must obtain acceptance from at least one member of the group. Note that if i and j are a direct pair, then they are also an indirect pair. Hence, PD implies PD− , and we have DA ⇒ PD ⇒ PD− . 2 It is straightforward to confirm this fact. Table Bδ shows that we have Bδ (12, 23, 23) = {1, 23}( = Z). However, if player 3 changes his choice to 123, then Bδ (12, 23, 123) = {1, 2, 3}( = I). Therefore, B3δ (12, 23, 23) = 23 ̸ ⊆ 3 = B3δ (12, 23, 123). The same holds for Bγ .
Table PD− describes this axiom for n = 3. The following proposition states that PD− , together with UN and MO, specifies a unique CA-function. Proposition 3.2. CA-function B satisfies UN, PD− , MO if and only if it is B≈ , where σ
B≈ (σ ) = {T ⊆ N |i, j ∈ T ⇐⇒ i ≈ j}.
We call B≈ the pairwise function. The pairwise function for n = 3 is described in Table B≈ . Below, we provide three interpretations for this function.
• The pairwise function states that two players belong to the same coalition if and only if they are an indirect pair. We may interpret this definition as ‘‘the friend of your friend is your friend’’. Relaxing the disagreement axiom yields this friendly feature. • Different from the γ - and δ -functions, the pairwise function takes into account the coalition a deviating player chooses as his new choice. To see this, we focus on the leftmost profiles in Table B≈ , namely, (N, N, N, Z). The pairwise function assigns the two-person coalition to the remaining players if player 1 chooses his one-person coalition, while it assigns the three-person coalition to them if player 1 keeps a connection with either players 2 or 3. • Indirect pair ≈ is a binary relation. Moreover, it is an equivalence relation. In general, an equivalence relation divides a set of elements into some disjoint equivalence classes. In our model, indirect pair divides the player set into some disjoint coalitions, namely, a partition. Such a resulting partition is the partition B≈ (σ ). 4. Conclusion Table 1 summarizes our axiomatization results. In the table, symbol ⊕ means that the axiom is used for the axiomatization of the CA-function; + means that the CA-function satisfies the axiom; − means that the CA-function violates the axiom. In view of Table 1, one may consider that we can use PD or PD− instead of DA to axiomatize the γ - and δ -functions since both functions violate MO. However, if we employ PD or PD− , the axiomatic system admits some functions. For example, both B1 and B2 provided in the appendix satisfy UN, PD− , and either IR or DR+ , while the functions are neither Bγ nor Bδ , which violates the uniqueness of a CA-function. Moreover, in this paper, we considered UN to be an axiom that all CA-functions should satisfy. If we consider a weak form of unanimity, then the axiomatic system consisting of PD, MO, and the weak unanimity might specify another CA-function. We conclude this paper with discussing a construction of a non-cooperative game based on a CA-function. Once we select a coalition aggregation function and define players’ preferences over coalitions, we can obtain a non-cooperative game where every player simultaneously chooses a coalition to join.3 This allows us to apply non-cooperative game solutions to analyze coalition formation. Below, we elaborate the construction of a non-cooperative game based on a CA-function, which leads to our future research. We first fix the player set N and select a CAfunction B. A non-cooperative game is a triple of the player set, 3 Hence, by selecting a coalition aggregation function, we can formulate every hedonic game as a non-cooperative game. Hedonic games play a central role to analyze coalition formation and stable coalition structures. Hedonic games are introduced by Banerjee et al. (2001) and Bogomolnaia and Jackson (2002). Gallo and Inarra (2018) associate a hedonic game with a type of bankruptcy problem (d, E) with claims d = (d1 , . . . , dn ) and the endowment E to be divided among the agents and discuss coalition formation problem among the agents.
T. Abe / Mathematical Social Sciences 103 (2020) 69–75
A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
123 any XZI YZI ZI
123 12 XYI XI YI I
123 any any any ZI
123 12 any XI YI I
123 N N N Z
23 YZI ZI YZI ZI
23 any ZI any ZI
123 12 N X Y I
23 N Z N Z
2 YI I YI I
2 YI I YI I
2 Y I Y I
123 XYI XI YI I
Table PD 13 12 23 2 XYI YI YI XI I I YI YI YI I I I
123 any XI YI I
Table PD− 13 12 23 2 any YI YI XI I I YI YI YI I I I
123 N X Y I
Table 13 12 23 N Y X I Y Y I I
strategy sets, and a payoff function. We use the player set N as the player set of the non-cooperative game. We consider Ai to be player i’s strategy set: every player simultaneously chooses a coalition to join. Since a CA-function is independent of players’ preferences, we now extend our model to contain them. A CAfunction B associates a coalition profile with a partition, where a coalition profile can be seen as a strategy profile because Ai is player i’s strategy set for every i ∈ N, formally, B : ×i∈N Ai → Π (N). Therefore, to construct a payoff function, we need a function that connects a partition with a payoff profile. We denote such a function by φ : Π (N) → RN . Such a function φ is also studied by Hart and Kurz (1984), Casajus (2009), and Abe (2018). One can choose φ so that it represents a game-theoretic or economic situation of interest. For example, Hart and Kurz (1984) and Abe (2018) provide φ that represents a symmetric majority game, and Casajus (2009) formulates a gloves game. Finally, we combine φ and B and obtain a payoff function φ◦B. As a result, we obtain non-cooperative game (N , (Ai )i∈N , φ ◦ B). Our next step is to make a theoretical analysis for the generated non-cooperative game. We conjecture that there are some relationships between the axioms and solution concepts for a non-cooperative game in the generated game. Acknowledgments The author is grateful to two anonymous reviewers and the editor for their valuable comments and suggestions. The author thanks Yukihiko Funaki and Yukio Koriyama for helpful comments and gratefully acknowledges the financial support from JSPS, Japan Grant-in-Aid for Research Activity Start-up (No. 19K23206) and Waseda Univeristy, Japan Grant-in-Aid for Research Base Creation (2019C-486). Appendix. Proofs In this section, let σ
Di (σ ) := {j ∈ N |j ∼ i} for any i ∈ N and any σ ∈ AN . Note that i ∈ Di (σ ) for any i ∈ N and any σ ∈ AN .
123 XZI XZI ZI ZI
123 any any ZI ZI
23 12 XI XI I I
23 12 XI XI I I
73
23 ZI ZI ZI ZI
2 I I I I
23 ZI ZI ZI ZI
123 XI XI I I
2 I I I I
123 XI XI I I
3 12 XI XI I I
23 I I I I
3 12 XI XI I I
2 I I I I
23 I I I I
2 I I I I
B≈
2 Y I Y I
123 N N Z Z
23 12 23 X Z X Z I Z I Z
2 I I I I
123 X X I I
3 12 X X I I
23 I I I I
2 I I I I
Proof of Proposition 2.1 (Axiomatization of Bδ ). We begin with the if-part. UN: Assume that there is S ⊆ N such that σj = S for every j ∈ S and S ̸ ∈ Bδ (σ ). Since S ̸ ∈ Bδ (σ ), there are i ∈ S and k ∈ S such that Biδ (σ ) ̸ = Bkδ (σ ): players i and k belong to different coalitions, while σi = σk . This contradicts the definition of Bδ . DA: This immediately follows from the definition. IR: Let i ∈ N and j, k ∈ Biδ (σ ). They are three different players. Since i, j, k ∈ Biδ (σ ), we have σi = σj = σk . It follows from σj = σk that Bjδ (σi′ , σ−i ) = Bkδ (σi′ , σ−i ). We now show the only-if-part. Step 1. Let σ ∈ AN . For any nonempty S ⊆ N, define HSσ = {i ∈ N |σi = S }. We first prove that H σ := {HSσ |S ⊆ N , HSσ ̸ = ∅} is a partition of N. For any two different coalitions S , T ⊆ N, we have HSσ ∩ HTσ = ∅ because if there is a player i such that i ∈ HSσ ∩ HTσ then that ⋃ it holds ⋃ S =σ σi = T . This contradicts S ̸= T . Moreover, σ = N because if there is a player i ∈ N \ H (S) then H S ⊆N S ⊆N S
σi ̸= S for any S ⊆ N. This contradicts σi ∈ Ai ⊆ 2N . Hence, H σ is a partition of N. We now define K σ := {S ∈ H σ |σj = S for all j ∈ S } and Lσ := H σ \ K σ . Step 2. By UN, for each coalition S ∈ K σ , S ∈ B(σ ). Step 3. For each coalition T ∈ Lσ with |T | = 1, say {i} = T , we have σi ̸ = σj for any j ∈ N \ {i}. Hence, by DA, Bi (σ ) = {i}. Now, we fix a coalition T ∈ Lσ with |T | ≥ 2 and consider a player i ∈ T . Let R := σi . Note that it follows from T ∈ Lσ that σj = R for every j ∈ T , σj ̸= R for every j ∈ N \ T .
(1)
If R = T , then T ∈ K σ , which contradicts T ∈ Lσ = H σ \ K σ ; if R is a proper subset of T , then for some player k in T \ R, k ̸ ∈ R = σk , which contradicts σk ∈ Ak . Hence, we have R ⊋ T . It follows that |R| ≥ 3. For the strategy profile σ and the coalition R, define the following strategy profile σ 0 ∈ AN : for every j ∈ N,
σ = 0 j
{
R
σj
if j ∈ R, otherwise.
By UN, R ∈ B(σ 0 ). Let R \ T = {i1 , . . . , im } for some natural number m. Note that for every i ∈ R \ T , σi0 = R.
74
T. Abe / Mathematical Social Sciences 103 (2020) 69–75
Now, define σ 1 ∈ AN as
σj1 =
{
σj σj0
if j = i1 , otherwise.
In view of i1 ̸ ∈ T and (1), we have σi1 ̸ = R. By IR, for any j, k ∈ R \ {i1 }, we have Bj (σ 1 ) = Bk (σ 1 ), which implies Bj (σ 1 ) ⊇ R \ {i1 } for every j ∈ R \ {i1 }. Since σj1 ̸ = R for any j ∈ (N \ R) ∪ {i1 } and σj1 = R for any j ∈ R \ {i1 }, DA implies that Bj (σ 1 ) ⊆ R \ {i1 } for every j ∈ R \ {i1 }. Hence, we obtain Bj (σ 1 ) = R \ {i1 } for every j ∈ R \ {i1 }. In the same manner, given σ and σ 1 , we define σ 2 ∈ AN as
σj2 =
{
σj σj1
Proof of Proposition 2.2 (Axiomatization of Bγ ). We first prove the if-part. UN: This follows in the same manner as Bδ . DA: If γ γ σi ̸= σj , then Bi (σ ) = {i} ̸= {j} = Bj (σ ). DR+ : Let T ∈ Bδ (σ ). Let S ⊆ T , j ∈ T \ S, and k ∈ T \ S. Since σi′ ̸ = σi = T for every i ∈ S, γ γ we have Bj (σS′ , σ−S ) = {j} and Bk (σS′ , σ−S ) = {k}. We now show the only-if-part. Step 1 and Step 2 are the same as Proposition 2.1. Step 3. For any coalition T ∈ Lσ with |T | = 1, say {i} = T , we have σi ̸ = σj for any j ∈ N \ {i}. Hence, by DA, Bi (σ ) = {i}. Now, we fix a coalition T ∈ Lσ with |T | ≥ 2 and consider a player i ∈ T . Let R := σi . In the same manner as Proposition 2.1, we have R ⊋ T and |R| ≥ 3. For the given strategy profile σ , similarly define strategy profile σ 0 ∈ AN as
{
R
σj
if j ∈ R, otherwise.
By UN, R ∈ B(σ 0 ). Let R \ T = {i1 , . . . , im } for some natural number m. In view of T ∈ H σ , we have σj ̸ = R for every j ∈ N \ T . Hence, σj0 = R ̸= σj for every j ∈ R \ T . It follows from DR+ that Bi (σ ) ̸ = Bj (σ ) for any different i, j ∈ T .
(2)
Since σj ̸ = R for every j ∈ N \ T and σj = R for every j ∈ T , in view of DA, we have Bi (σ ) ̸ = Bj (σ ) for any i ∈ T and any j ∈ N \ T .
and obtain {k, i} ∈ B(σ k ) by UN. Now, let σ ∗ be
σj∗ =
{
N {j, i}
if j = i, otherwise.
From MO, it follows that
{1, i} ⊆ Bi (σ ∗ ), {2, i} ⊆ Bi (σ ∗ ), . . . , {n, i} ⊆ Bi (σ ∗ ).
if j = i2 , otherwise.
Similarly, by IR and DA, we obtain Bj (σ 2 ) = R \ {i1 , i2 } for every j ∈ R \ {i1 , i2 }. Repeating this procedure until im , we have σ m = σ and, hence, Bj (σ ) = Bj (σ m ) = R \ {i1 , . . . , im } for every j ∈ R \ {i1 , . . . , im }. As T = R \ {i1 , . . . , im }, we obtain Bj (σ ) = T for every j ∈ T . Hence, in view of Step 2, B(σ ) = K σ ∪ Lσ = H σ . Step 4. We show H σ = {S ⊆ N |i, j ∈ S ⇐⇒ σi = σj }. Let S ∈ H σ . There exists T ⊆ N such that for any j ∈ S, σj = T . Hence, it holds that i, j ∈ S ⇒ σi = σj . Now consider j ∈ N \ S. There exists T ′ ⊆ N such that T ′ ̸ = T and σj = T ′ . Hence, it follows that i ∈ S and j ̸ ∈ S ⇒ σi ̸ = σj . Thus S ∈ Bδ . Now let S ∈ Bδ . Let T := σj = σk for any j, k ∈ S. Then we have H σ (T ) = S and H σ (T ) ∈ H σ , which means S ∈ H σ . In view of Steps 3 and 4, B(σ ) = H σ = Bδ (σ ). This completes the proof. □
σj0 =
namely, σ 1 = ({1, i}, {2, i}, . . . , {i − 1, i}, {1, i}, {i + 1, i}, . . . , {n, i}). In view of UN, {1, i} ∈ B(σ 1 ). In the same manner, for k = 1, . . . , i − 1, i + 1, . . . , n, we define { {k, i} if j = i, σjk = {j, i} otherwise
(3)
Hence, Bi (σ ∗ ) = N. This holds for every i ∈ N, namely, Bi (σ ∗ ) = N for every i ∈ N. However, since it does not hold that 2 ∈ σ1∗ and 1 ∈ σ2∗ , players 1 and 2 are not a pair. Hence, in view of PD, B1 (σ ∗ ) ̸ = B2 (σ ∗ ). This is a contradiction. □ Proof of Proposition 3.2 (Axiomatization of B≈ ). We first show σ the if-part. UN: If σj = S for every j ∈ S, then i ∼ j for any i, j ∈ S, σ
which implies i ≈ j for any i, j ∈ S. Hence, S ∈ B≈ (σ ). PD− : This immediately follows from the definition. MO: Fix i ∈ N. In view σ
of the definition, Bi≈ (σ ) = {j ∈ N |i ≈ j}. Since σi′ ⊇ σi , we have σ
(σ ′ ,σ i
−i )
Bi≈ (σ ) = {j ∈ N |i ≈ j} ⊇ {j ∈ N |i ≈ j} = Bi≈ (σi′ , σ−i ). We now show the only-if-part. Let σ ∈ AN . Let K σ := {∅ ̸ = σ S ⊆ N |σ ⋃j = S for all j ∈ S }. In view ofσ UN, K ⊆ B(σ ). Let σ ˆ ˆ K := is not a unanimous S ∈K σ S. Every player i in N \ K member: there is a player j ∈ σi such that σj ̸ = σi . We partition ˆσ into {S1 , . . . , SM } for some natural number M by indirect N \K pairs: for each m = 1, . . . , M, σ
i ≈ j for any i, j ∈ Sm , σ
i ̸ ≈ j for any i ∈ Sm and any j ̸ ∈ Sm . We fix an arbitrary coalition S ∈ {S1 , . . . , SM }. From PD− , it follows that Bi (σ ) ⊆ S for every i ∈ S .
(4)
Below, we prove that Bi (σ ) ⊇ S for every i ∈ S. To this end, we first show the following claim. Claim. Let B be an CA-function and σ ∈ AN . If B satisfies UN and MO, then Bi (σ ) ⊇ Di (σ ) for every i ∈ N. Proof of Claim. Assume that there exists k ∈ Di (σ ) such that k ̸ ∈ Bi (σ ). Since k ∈ Di (σ ), we have k ∈ σi and i ∈ σk . We define σ ′ as follows: for every j ∈ N,
{ σ σj = j {i, k} ′
if j ∈ N \ {i, k}, otherwise (namely, j = i or k).
By UN, Bi (σ ′ ) = Bk (σ ′ ) = {i, k}. Note that σi ⊇ {i, k} = σi′ because i ∈ σi and k ∈ σi . Now define σ ∗ as follows: for every j ∈ N,
σj∗ =
{
σj {i, k}
if j ∈ N \ {k}, otherwise (namely, j = k).
From (2) and (3), it follows that Bi (σ ) ̸ = Bj (σ ) for any i ∈ T and any j ∈ N \ {i}. Hence, Bi (σ ) = {i} for each i ∈ T . This holds for every coalition T ∈ Lσ with |T |⋃≥ 2. Hence, we have σ }, where Lˆ σ = B(σ ) = K σ ∪ {{i}|i ∈ Lˆ T ∈Lσ T . Thus, B (σ ) = Bγ (σ ), which completes the proof. □
By MO, Bi (σ ∗ ) ⊇ Bi (σ ′ ) = {i, k}. Since k ∈ Bk (σ ∗ ), Bi (σ ∗ ) = Bk (σ ∗ ). Hence, Bk (σ ∗ ) ⊇ {i, k}. In a similar manner, by MO, Bk (σ ) ⊇ Bk (σ ∗ ) ⊇ {i, k}. Since i ∈ Bi (σ ), Bi (σ ) = Bk (σ ). Hence, Bi (σ ) ⊇ {i, k}. This contradicts k ̸ ∈ Bi (σ ). This completes the claim. //
Proof of Proposition 3.1 (No CA-function Simultaneously Satisfies UN, PD, MO). Let B satisfy UN, PD, and MO. Fix a player i ∈ N. Let σ 1 ∈ AN be
We now prove that Bi (σ ) ⊇ S for every i ∈ S. We first fix i ∈ S and j ∈ Di (σ ). We have {i, j} ⊆ Di (σ ) and {i, j} ⊆ Dj (σ ). The Claim above implies that
σj1 =
{
{1, i} {j, i}
if j = i, otherwise,
Bi (σ ) ⊇ Di (σ ) ⊇ {i, j}, and Bj (σ ) ⊇ Dj (σ ) ⊇ {i, j}.
T. Abe / Mathematical Social Sciences 103 (2020) 69–75
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B1 A3 A1 \ A2
123 12 13 1
123 N Z Z Z
123 12 Y X I I
123 N I I I
123 12 I X I I
23 Y I X Z
2 Y I Y I
123 X I Y I
13 12 23 I I X I Y Y I I
123 I I Y I
13 12 23 I I X I Y Y I I
2 Y I Y I
123 X I I Z
23 12 23 I Z X Z I Z I Z
2 I I I I
123 X I I I
3 12 I X I I
23 I I I I
2 I I I I
2 Y Y Y I
123 I I I Z
23 12 23 I Z X Z I Z I Z
2 I I I I
123 I I I I
3 12 I X I I
23 I I I I
2 I I I I
2 I I Y I
123 I I I I
23 12 23 I Z X Z I Z I Z
2 I I I I
123 I I I I
3 12 I X I I
23 I I I I
2 I I I I
2 Y I Y I
123 N N Z Z
23 12 23 X Z X Z I Z I Z
2 I I I I
123 X X I I
3 12 X X I I
23 I I I I
2 I I I I
B2 A3 A1 \ A2
123 12 13 1
23 I I X Z
2 I I Y I
B3 A3 A1 \ A2
123 12 13 1
123 N Z Z Z
123 12 Y X I I
123 N N N Z
123 12 N X N I
23 Y I X I
2 Y I I I
123 I I Y I
13 12 23 I I X I Y Y I I
123 N X Y I
13 12 23 N Y X I Y Y I I
B4 A3 A1 \ A2
123 12 13 1
23 N N N Z
2 Y I Y I
Hence, Bi (σ ) = Bj (σ ), which implies that Bi (σ ) = Bj (σ ) ⊇ (Di (σ ) ∪ Dj (σ )). We now consider the same j and fix k ∈ Di (σ ) ∪ Dj (σ ). In the same manner, we have Bj (σ ) = Bk (σ ) ⊇ (Dj (σ ) ∪ Dk (σ )). For the player j, we have Bi (σ ) = Bj (σ ) = Bk (σ ) ⊇ (Di (σ ) ∪ Dj (σ ) ∪ Dk (σ )). Applying this procedure to every i′ ∈ S ⋃ ′ and every j ∈ Di′ (σ ), we have Bi (σ ) ⊇ j∈S Dj (σ ) = S for every i ∈ S. Hence, we have Bi (σ ) ⊇ S for every i ∈ S. In view of (4), we have Bi (σ ) = S for every i ∈ S. Hence, we σ
ˆσ . For every i ∈ Kˆσ , have Bi (σ ) = {j ∈ N |i ≈ j} for every i ∈ N \ K σ
since i is a unanimous member, Bi (σ ) = σi = {j ∈ N |i ≈ j}. Thus, σ
B(σ ) = {T ⊆ N |i, j ∈ T ⇐⇒ i ≈ j}.
□
Independence Define Bs as follows: Bs (σ ) = {{i}|i ∈ N } for every σ ∈ AN . Functions B1 to B4 are provided in the following table, in which underlined partitions violate the corresponding axiom. Independence for Bδ • Bγ satisfies UN and DA, but violates IR. • B1 satisfies UN and IR, but violates DA. • Bs satisfies DA and IR, but violates UN.
Independence for Bγ
• Bδ satisfies UN and DA, but violates DR+ . • B2 satisfies UN and DR+ , but violates DA. • Bs satisfies DA and DR+ , but violates UN. Independence for B≈ • B3 satisfies UN and PD− , but violates MO. • B4 satisfies UN and MO, but violates PD− . • Bs satisfies PD− and MO, but violates UN. References Abe, T., 2018. Stable coalition structures in symmetric majority games: A coincidence between myopia and farsightedness. Theory Decis. 85, 353–374. Banerjee, S., Konishi, H., Sönmez, T., 2001. Core in a simple coalition structure game. Soc. Choice Welf. 18, 135–153. Bogomolnaia, A., Jackson, M.O., 2002. The stability of hedonic coalition structures. Games Econ. Behav. 38, 201–230. Casajus, A., 2009. Outside options, component efficiency, and stability. Games Econ. Behav. 65, 49–61. Gallo, O., Inarra, E., 2018. Rationing rules and stable coalition structures. Theor. Econ. 13, 933–950. Hart, S., Kurz, M., 1983. Endogenous formation of coalitions. Econometrica 51, 1047–1064. Hart, S., Kurz, M., 1984. Stable coalition structures. In: Holler, M.J. (Ed.), Coalitions and Collective Action, Vol. 51. pp. 235–258.
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Mathematical Social Sciences journal homepage: www.elsevier.com/locate/mss
Axiomatizations of coalition aggregation functions Takaaki Abe School of Political Science and Economics, Waseda University, 1-6-1, Nishi-waseda, Shinjuku-ku, Tokyo 169-8050, Japan
article
info
Article history: Received 21 July 2019 Received in revised form 10 November 2019 Accepted 3 December 2019 Available online 16 December 2019 Keywords: Axiomatization Coalition formation Coalition structure Monotonicity
a b s t r a c t We axiomatize Hart and Kurz’s (1983) two coalition aggregation functions known as the γ -function and the δ -function. A coalition aggregation function is a mapping that assigns a partition to each profile of coalitions selected by players. Through our axiomatization results, we observe that neither the γ function nor the δ -function satisfies monotonicity. We propose a monotonic function and axiomatically characterize it. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The main purpose of this paper is to answer the following question: what coalition structure should be assigned to a profile of coalitions selected by players? To clarify this question, we begin with a simple example. Suppose that there are three players, N = {1, 2, 3}. Every player chooses a coalition that she/he wants to form: for example, each student submits a list of students with whom he wants to share a room. We suppose that players 1 and 2 want to form the three-person coalition {1, 2, 3}, while player 3 wants to form the two-person coalition with player 2, namely, {2, 3}. Their choices are summarized as
σ = (123, 123, 23). For simplicity, we omit the parentheses of a coalition if it is not confusing: we write 123 instead of {1, 2, 3}. We call such a tuple σ a coalition profile. Now, what coalition structure is ‘‘optimal’’ for this coalition profile? The first attempts to address this question were made by Hart and Kurz (1983, 1984). They defined two aggregation rules known as the γ -function Bγ and the δ -function Bδ . These two rules exhibit two different views of coalition formation. On one hand, one might consider a coalition to be a result from a unanimous agreement among all its members. The γ -function formulates this view. Let S ⊆ N be a coalition of players. If all the players in S choose the same coalition S, then S is formed. However, if someone in S chooses a different coalition, the proposers of S are partitioned into singletons. In the three-person example above, neither 123 nor 23 obtains a unanimous agreement. Therefore, the γ -function assigns the three one-person coalitions to the coalition profile. On the other hand, one might consider that a coalition is the largest E-mail address: [email protected]. https://doi.org/10.1016/j.mathsocsci.2019.12.001 0165-4896/© 2019 Elsevier B.V. All rights reserved.
set of players who share the same choice. In this view, the choice of a coalition means the set of all players with whom he is willing to share the same coalition. The δ -function exhibits this view. In the same example, players 1 and 2 choose the same coalition 123. In other words, they accept all players. However, player 3 does not admit player 1. Although we will provide its formal definition in the next section, we can define the δ -function by simply grouping the players who choose the same coalition. Therefore, the δ -function assigns the two-person coalition to players 1 and 2. Player 3 is the only player who chooses coalition 23. Hence, the δ -function assigns his one-person coalition to player 3. In general, we call such a function that assigns a coalition structure to each coalition profile a coalition aggregation function. Our motivation for studying a coalition aggregation function is to associate players’ choices with coalition formation theory. Coalition formation theory has been mainly studied in the frameworks in which players do not select any action and/or choice, e.g., cooperative games and hedonic games. However, when we form a team or a group, our choices and decisions must implicitly/explicitly influence its result. Therefore, we attempt to analyze coalition formation in terms of players’ choices. However, besides the γ - and δ -functions, there are many coalition aggregation functions. How should we choose one coalition aggregation function? As Hart and Kurz (1983) state, ‘‘there is no universally correct answer to this problem’’. Nevertheless, once one chooses a certain function, she must be supposed to answer why she selects it. Our axiomatization results must be helpful to answer this question. As we will discuss in the following sections, the γ - and δ -functions do not satisfy a property called monotonicity. Monotonicity is a guarantee for players who have an acceptive attitude. To see this, we fix a player and the choices of the other players. Even if the player changes his choice to accept more players
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T. Abe / Mathematical Social Sciences 103 (2020) 69–75
Table 1 Axioms and CA-functions. Proposition Proposition Proposition Proposition
2.1 2.2 3.2 3.1
δ
B Bγ B≈ ̸ ∃B
UN
DA
PD
PD−
IR
DR+
MO
⊕ ⊕ ⊕ +
⊕ ⊕ −
+ + − +
+ + ⊕
⊕ − −
− ⊕ −
− − ⊕ +
by choosing a superset, the γ - and δ -functions may provide him with a smaller coalition. Monotonicity guarantees that his acceptive change does not end up such a contrary result: even if his invitation is declined, it does not break his original affiliation. After we axiomatize the γ -function and the δ -function in the next section, we propose a monotonic coalition aggregation function in Section 3. Its axiomatization is also provided. In Section 4, we summarize our results and offer some concluding remarks. Table 1 in Section 4 shows the axiomatic systems of the coalition aggregation functions we study in this paper. All the proofs and the independence of the axioms are provided in the appendix. 2. The γ -function and the δ-function Let N = {1, . . . , n} be a set of players. A coalition is a subset S ⊆ N. We denote the cardinality of coalition S by |S |. We use n to denote |N |. We assume n ≥ 3. For every i ∈ N, Ai is the set of coalitions that contain i, formally, Ai = {S ⊆ N |i ∈ S }. For example, let N = {1, 2, 3}. We have A1 = {1, 12, 13, 123}, A2 = {2, 12, 23, 123}, A3 = {3, 13, 23, 123}.
For every nonempty coalition S ⊆ N, let AS = ×i∈S Ai . We use σ to denote an element of AN . We call σ a coalition profile. We typically use P to denote a partition (or a coalition structure). For every coalition S ⊆ N, let Π (S) be the set of all partitions of S. Let Pi denote the coalition player i belongs to in partition P . A coalition aggregation function (CA-function) is a mapping that assigns a partition to each coalition profile, B : AN → Π (N). Since B(σ ) is a partition, let Bi (σ ) denote the coalition in partition B(σ ) that contains player i. Now, we introduce the two CA-functions proposed by Hart and Kurz (1983). We begin with the δ -function. The δ -function is given as Bδ (σ ) = {T ⊆ N |i, j ∈ T ⇐⇒ σi = σj }.
The δ -function focuses on the players who choose the same coalition. To see this, let N = {1, 2, 3, 4} and σ = (12, 12, 234, 234). Since players 1 and 2 choose the same coalition 12, they belong to coalition 12, namely, B1δ (σ ) = B2δ (σ ) = 12. Similarly, since players 3 and 4 choose the same coalition 234, they belong to coalition 34, namely, B3δ (σ ) = B4δ (σ ) = 34. Hence, the resulting coalition
structure is Bδ (σ ) = {12, 34}. In contrast, the γ -function requires players to make a unanimous agreement. Formally, Bγ (σ ) = {Tσi |i ∈ N }, where Tσi =
{
σi {i}
if σj = σi for every j ∈ σi , otherwise.
Consider the coalition profile σ = (12, 12, 234, 234) again. The list of coalitions proposed in σ is {12, 234}. Since all members in coalition 12 agree to form the coalition 12, coalition 12 is formed. However, for coalition 234, player 2 does not agree to join it. Therefore, coalition 234 lacks a unanimous agreement and is not formed. The γ -function does not allow the proposers of a failed coalition to form any coalition among themselves. Thus,
the resulting coalition structure is Bγ (σ ) = {12, 3, 4}. Note that a player who chooses his own one-person coalition σi = {i}, if any, achieves a unanimous agreement by himself. The two CA-functions for n = 3 are described in Tables Bδ and Bγ . For simplicity, we use symbols N, X, Y, Z, I to denote partitions as follows: N = {123}, X = {12, 3}, Y = {13, 2}, Z = {23, 1}, I = {1, 2, 3}. For example, Bδ (123, 123, 123) = {123}(= N) and Bδ (123, 123, 13) = {12, 3}(= X). We now introduce axioms to characterize these two CAfunctions. By axiomatizing a CA-function, we ‘‘decompose’’ a CA-function into three components: agreement, disagreement, and reaction to a deviation. In other words, we specify each CA-function in terms of these three properties. The first axiom describes a form of agreement among players. Axiom (Unanimity, UN). For any σ ∈ AN and any ∅ ̸ = S ⊆ N, if σj = S for every j ∈ S, then S ∈ B(σ ). This axiom states that if all members of a coalition agree on the formation of the coalition, then the coalition should be formed. For convenience, we call player i a unanimous member if σj = σi for every j ∈ σi (or equivalently, if there exists S ⊆ N such that i ∈ S and σj = S for every j ∈ S). All CA-functions that we discuss in this paper satisfy this axiom. Axiom (Disagreement, DA). For any σ ∈ AN and any i, j ∈ N, if σi ̸= σj , then Bi (σ ) ̸= Bj (σ ). We consider axiom DA to be the simplest form of disagreement. If player i’s choice is different from player j’s choice, then these two players become separate. For example, if player i invites player k, while player j does not, then their proposals clearly conflict over player k. A CA-function that obeys DA assigns different coalitions to such players. In other words, the coincidence between choices is a necessary condition for two players to belong to the same coalition. In this sense, this axiom might be thought of as the most strict form of disagreement. We will also introduce some weaker disagreement axioms later. Table UN and Table DA describe these axioms for n = 3. In the tables, for example, YI means that both {13, 2} ( = Y) and {1, 2, 3} ( = I) are possible and that the other coalition structures (N, X, and Z) are ruled out. In the same manner, ‘‘any’’ means that all partitions are possible. We now introduce an axiom that serves the third component of a CA-function, which describes the remaining players’ reaction to one’s deviation from an agreement. Axiom (Integrative Reaction, IR). For any σ ∈ AN , any i ∈ N, and any σi′ ∈ Ai with σi′ ̸ = σi , if |Bi (σ )| ≥ 3, then Bj (σi′ , σ−i ) = Bk (σi′ , σ−i ) for any j, k ∈ Bi (σ ) \ {i} with j ̸ = k. Let σ be a current coalition profile and B(σ ) be the corresponding coalition structure. In the coalition structure B(σ ), if |Bi (σ )| ≥ 3, then player i shares a coalition with at least two players, say j and k (hence, Bi (σ ) ⊇ {i, j, k}). Axiom IR guarantees that even after player i changes his choice from σi to new coalition σi′ , players j and k can keep their coalition without player i. In this sense, IR admits a partial agreement among players: even if a player splits off from the original agreement, the partial agreement of the remaining players is enough for them to sustain their own coalition. The following axiom is a variant of IR. Axiom (Disintegrative Reaction, DR). For any σ ∈ AN , any i ∈ N, and any σi′ ∈ Ai with σi′ ̸ = σi , if |Bi (σ )| ≥ 3, then Bj (σi′ , σ−i ) ̸ = Bk (σi′ , σ−i ) for any j, k ∈ Bi (σ ) \ {i} with j ̸ = k.
T. Abe / Mathematical Social Sciences 103 (2020) 69–75
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Table Bδ A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
123 N Z Z Z
123 12 Y X I I
123 N I I I
123 12 I X I I
23 Y I I I
23 I I I I
123 N any any ZI
123 12 any X any ZI
123 any ZI ZI ZI
123 12 23 YI YI XI I I I I I
23 any any any ZI
2 Y I I I
2 I I I I
2 YI YI YI I
2 YI I I I
123 X I Y I
13 12 23 I I X I Y Y I I
2 I I Y I
123 X I I I
23 12 23 I Z X Z I Z I Z
2 I I I I
123 X I I I
3 12 I X I I
23 I I I I
2 I I I I
123 I I Y I
Table Bγ 13 12 23 2 I I I X I I Y Y Y I I I
123 I I I I
23 12 23 I Z X Z I Z I Z
2 I I I I
123 I I I I
3 12 I X I I
23 I I I I
2 I I I I
123 any any Y ZI
Table UN 13 12 23 2 any any YI X any YI Y Y Y ZI ZI I
123 any any any ZI
23 12 23 any Z X Z any Z ZI Z
2 YI YI YI I
123 XI XI XI I
3 12 XI X XI I
23 XI XI XI I
2 I I I I
123 XI I YI I
Table DA 13 12 23 2 I I I XI I I YI YI YI I I I
123 XI I I I
23 12 23 I ZI XI ZI I ZI I ZI
2 I I I I
123 XI I I I
3 12 I XI I I
23 I I I I
2 I I I I
This axiom can be seen as the counterpart of IR: it no longer allows players j and k to stay in the same coalition after i’s deviation. Therefore, DR does not admit a partial agreement among the remaining players. More specifically, the coincidence of the choices among the remaining players does not bring about any coalition of the remaining players. In the same manner, we now introduce a coalitional version of DR. Axiom (Coalitional Disintegrative Reaction, DR+ ). For any σ ∈ AN , any ∅ ̸ = S ⊆ N, and any σS′ ∈ AS with σj′ ̸ = σj for all j ∈ S, if there exists T ∈ B(σ ) such that T ⊇ S and |T \ S | ≥ 2, then Bj (σS′ , σ−S ) ̸ = Bk (σS′ , σ−S ) for any j, k ∈ T \ S with j ̸ = k. In this axiom, in addition to individual deviations, coalitional deviations are also incorporated. Hence, DR+ implies DR. Note that for n = 3, DR+ is equivalent to DR.1 Given these four axioms, we obtain the following two characterizations. Proposition 2.1. it is Bδ .
CA-function B satisfies UN, DA, IR if and only if
1 Because of the constraints |B (σ )| ≥ 3 (in IR) and |T \ S | ≥ 2 (in DR+ ), the i restrictions of these axioms are modest: for n = 3, only if B(σ ) = {123} for some σ can these axioms restrict CA-functions. Moreover, as shown in Proposition 2.2, DR+ is needed instead of DR to characterize Bγ . To see this, we consider (a part of) the following CA-function for n = 4: B(1234, 1234, 1234, 1234) = {1234}, B(1, 1234, 1234, 1234) = {1, 2, 3, 4}, B(1, 2, 1234, 1234) = {1, 2, 34}. This function does not violate DR because the constraint required to player 2 is slack: player 2 belongs to his one-person coalition B2 (1, 1234, 1234, 1234) = {2}. In other words, this player 2 does not achieve the if-part of axiom DR because |{2}| < 3, even if he changes his choice 1234 to 2. However, the function above cannot be Bγ as B(1, 2, 1234, 1234) = {1, 2, 34} ̸ = {1, 2, 3, 4} = + Bγ (1, 2, 1234, 1234). Axiom DR works to rule out such a function.
Proposition 2.2. if it is Bγ .
CA-function B satisfies UN, DA, DR+ if and only
The proofs and the independence of the axioms are provided in the appendix. In view of the propositions, we can attribute the difference between the two CA-functions to the third component of the axiomatic systems, namely, IR or DR+ , which means that reaction to a deviation differentiates these CA-functions. Moreover, no CA-function satisfies both IR and DR simultaneously, while some functions satisfy neither IR nor DR. Such CA-functions should describe another type of reaction. The CA-function we propose in the next section is one of such functions. 3. Monotonicity In the previous section, we have axiomatized the γ - and δ -functions. However, because of IR and DR (or DR+ ), these two functions have an inflexible feature for one’s deviation. To see this, we consider the leftmost profiles in Tables Bγ and Bδ . The two functions assign (N, I, I, I) and (N, Z, Z, Z) to these profiles, respectively. We suppose that player 1 changes his choice from coalition 123 to another coalition. Then, the two assignments (N, I, I, I) and (N, Z, Z, Z) imply that the γ - and δ -functions do not take into account what coalition player 1 chooses as his new choice: the partitions of the non-deviating players are independent of the new choice of player 1. In this section, we introduce another CAfunction by using monotonicity that also incorporates the new choice of a deviating player. There are one similarity and two differences between monotonicity and the previous two axioms, IR and DR. The similarity is that a certain player changes his choice. Therefore, we consider
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T. Abe / Mathematical Social Sciences 103 (2020) 69–75
monotonicity to be an axiom that serves the third component of an axiomatic system. The first difference is that his new choice is now a superset of the first choice. The second difference is that monotonicity focuses on the player who changes a choice instead of the remaining players. Axiom (Monotonicity, MO). For any σ ∈ AN , any i ∈ N, and any σi′ ∈ Ai , if σi ⊆ σi′ , then Bi (σi , σ−i ) ⊆ Bi (σi′ , σ−i ). This axiom states that if a player changes his choice to a larger coalition in the sense of superset, then he should belong to a larger coalition or at least the same coalition. As mentioned in Section 1, we can consider this axiom to be a guarantee for a player who accepts more players: when a player changes his choice, if his new choice admits and allows more players to enter, then this friendly change does not give him a smaller coalition. In other words, even if his invitation is declined by some of the invited players, that failure does not break his original affiliation. Both the γ - and δ -functions violate monotonicity.2 In view of Propositions 2.1 and 2.2, one might consider that by simply replacing IR or DR+ with MO, we can obtain a CAfunction. However, no CA-function simultaneously satisfies these three axioms: UN DA, and MO. Worse, this negative result holds even for a weaker version of DA. To see this, we introduce the following notion and axiom. Players i and j are said to be a (direct) pair in σ if i ∈ σj and j ∈ σi : namely, if there is a mutual acceptance between the two players, we say that they are a pair. σ We denote a pair by i ∼ j. Axiom (Pairwise Disagreement, PD). For any σ ∈ AN and any σ i, j ∈ N, if not i ∼ j, then Bi (σ ) ̸ = Bj (σ ). This axiom requires that if at least one of the two players does not accept the other, then they are separate. Moreover, this axiom is equivalent to the following statement: if j ̸ ∈ σi then j ̸ ∈ Bi (σ ). Therefore, axiom PD interprets σi as player i’s claim that he does not want to share a coalition with the players in N \ σi . Table PD describes this axiom for n = 3. In view of Tables DA and PD, it may be clear that DA implies PD. The following result shows the impossibility result mentioned above. Proposition 3.1. and MO.
No CA-function simultaneously satisfies UN, PD,
Proposition 3.1 implies that PD is still restrictive if a CAfunction satisfies UN and MO. However, we can obtain a positive result by slightly relaxing PD. We say players i and j are an indirect pair in σ if there is a sequence of players k1 , . . . , kM such that σ k1 = i, kM = j, and km ∼ km+1 for m = 1, . . . , M − 1. We denote σ
an indirect pair by i ≈ j. Axiom (Weak Pairwise Disagreement, PD− ). For any σ ∈ AN and σ
any i, j ∈ N, if not i ≈ j, then Bi (σ ) ̸ = Bj (σ ). This axiom states that if player j has no connection with any player in the sequence of pairs that starts from player i, then the player j does not share the same coalition with the player i. Therefore, in order to join the group of player i, player j must obtain acceptance from at least one member of the group. Note that if i and j are a direct pair, then they are also an indirect pair. Hence, PD implies PD− , and we have DA ⇒ PD ⇒ PD− . 2 It is straightforward to confirm this fact. Table Bδ shows that we have Bδ (12, 23, 23) = {1, 23}( = Z). However, if player 3 changes his choice to 123, then Bδ (12, 23, 123) = {1, 2, 3}( = I). Therefore, B3δ (12, 23, 23) = 23 ̸ ⊆ 3 = B3δ (12, 23, 123). The same holds for Bγ .
Table PD− describes this axiom for n = 3. The following proposition states that PD− , together with UN and MO, specifies a unique CA-function. Proposition 3.2. CA-function B satisfies UN, PD− , MO if and only if it is B≈ , where σ
B≈ (σ ) = {T ⊆ N |i, j ∈ T ⇐⇒ i ≈ j}.
We call B≈ the pairwise function. The pairwise function for n = 3 is described in Table B≈ . Below, we provide three interpretations for this function.
• The pairwise function states that two players belong to the same coalition if and only if they are an indirect pair. We may interpret this definition as ‘‘the friend of your friend is your friend’’. Relaxing the disagreement axiom yields this friendly feature. • Different from the γ - and δ -functions, the pairwise function takes into account the coalition a deviating player chooses as his new choice. To see this, we focus on the leftmost profiles in Table B≈ , namely, (N, N, N, Z). The pairwise function assigns the two-person coalition to the remaining players if player 1 chooses his one-person coalition, while it assigns the three-person coalition to them if player 1 keeps a connection with either players 2 or 3. • Indirect pair ≈ is a binary relation. Moreover, it is an equivalence relation. In general, an equivalence relation divides a set of elements into some disjoint equivalence classes. In our model, indirect pair divides the player set into some disjoint coalitions, namely, a partition. Such a resulting partition is the partition B≈ (σ ). 4. Conclusion Table 1 summarizes our axiomatization results. In the table, symbol ⊕ means that the axiom is used for the axiomatization of the CA-function; + means that the CA-function satisfies the axiom; − means that the CA-function violates the axiom. In view of Table 1, one may consider that we can use PD or PD− instead of DA to axiomatize the γ - and δ -functions since both functions violate MO. However, if we employ PD or PD− , the axiomatic system admits some functions. For example, both B1 and B2 provided in the appendix satisfy UN, PD− , and either IR or DR+ , while the functions are neither Bγ nor Bδ , which violates the uniqueness of a CA-function. Moreover, in this paper, we considered UN to be an axiom that all CA-functions should satisfy. If we consider a weak form of unanimity, then the axiomatic system consisting of PD, MO, and the weak unanimity might specify another CA-function. We conclude this paper with discussing a construction of a non-cooperative game based on a CA-function. Once we select a coalition aggregation function and define players’ preferences over coalitions, we can obtain a non-cooperative game where every player simultaneously chooses a coalition to join.3 This allows us to apply non-cooperative game solutions to analyze coalition formation. Below, we elaborate the construction of a non-cooperative game based on a CA-function, which leads to our future research. We first fix the player set N and select a CAfunction B. A non-cooperative game is a triple of the player set, 3 Hence, by selecting a coalition aggregation function, we can formulate every hedonic game as a non-cooperative game. Hedonic games play a central role to analyze coalition formation and stable coalition structures. Hedonic games are introduced by Banerjee et al. (2001) and Bogomolnaia and Jackson (2002). Gallo and Inarra (2018) associate a hedonic game with a type of bankruptcy problem (d, E) with claims d = (d1 , . . . , dn ) and the endowment E to be divided among the agents and discuss coalition formation problem among the agents.
T. Abe / Mathematical Social Sciences 103 (2020) 69–75
A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
A3 A1 \ A2
123 12 13 1
123 any XZI YZI ZI
123 12 XYI XI YI I
123 any any any ZI
123 12 any XI YI I
123 N N N Z
23 YZI ZI YZI ZI
23 any ZI any ZI
123 12 N X Y I
23 N Z N Z
2 YI I YI I
2 YI I YI I
2 Y I Y I
123 XYI XI YI I
Table PD 13 12 23 2 XYI YI YI XI I I YI YI YI I I I
123 any XI YI I
Table PD− 13 12 23 2 any YI YI XI I I YI YI YI I I I
123 N X Y I
Table 13 12 23 N Y X I Y Y I I
strategy sets, and a payoff function. We use the player set N as the player set of the non-cooperative game. We consider Ai to be player i’s strategy set: every player simultaneously chooses a coalition to join. Since a CA-function is independent of players’ preferences, we now extend our model to contain them. A CAfunction B associates a coalition profile with a partition, where a coalition profile can be seen as a strategy profile because Ai is player i’s strategy set for every i ∈ N, formally, B : ×i∈N Ai → Π (N). Therefore, to construct a payoff function, we need a function that connects a partition with a payoff profile. We denote such a function by φ : Π (N) → RN . Such a function φ is also studied by Hart and Kurz (1984), Casajus (2009), and Abe (2018). One can choose φ so that it represents a game-theoretic or economic situation of interest. For example, Hart and Kurz (1984) and Abe (2018) provide φ that represents a symmetric majority game, and Casajus (2009) formulates a gloves game. Finally, we combine φ and B and obtain a payoff function φ◦B. As a result, we obtain non-cooperative game (N , (Ai )i∈N , φ ◦ B). Our next step is to make a theoretical analysis for the generated non-cooperative game. We conjecture that there are some relationships between the axioms and solution concepts for a non-cooperative game in the generated game. Acknowledgments The author is grateful to two anonymous reviewers and the editor for their valuable comments and suggestions. The author thanks Yukihiko Funaki and Yukio Koriyama for helpful comments and gratefully acknowledges the financial support from JSPS, Japan Grant-in-Aid for Research Activity Start-up (No. 19K23206) and Waseda Univeristy, Japan Grant-in-Aid for Research Base Creation (2019C-486). Appendix. Proofs In this section, let σ
Di (σ ) := {j ∈ N |j ∼ i} for any i ∈ N and any σ ∈ AN . Note that i ∈ Di (σ ) for any i ∈ N and any σ ∈ AN .
123 XZI XZI ZI ZI
123 any any ZI ZI
23 12 XI XI I I
23 12 XI XI I I
73
23 ZI ZI ZI ZI
2 I I I I
23 ZI ZI ZI ZI
123 XI XI I I
2 I I I I
123 XI XI I I
3 12 XI XI I I
23 I I I I
3 12 XI XI I I
2 I I I I
23 I I I I
2 I I I I
B≈
2 Y I Y I
123 N N Z Z
23 12 23 X Z X Z I Z I Z
2 I I I I
123 X X I I
3 12 X X I I
23 I I I I
2 I I I I
Proof of Proposition 2.1 (Axiomatization of Bδ ). We begin with the if-part. UN: Assume that there is S ⊆ N such that σj = S for every j ∈ S and S ̸ ∈ Bδ (σ ). Since S ̸ ∈ Bδ (σ ), there are i ∈ S and k ∈ S such that Biδ (σ ) ̸ = Bkδ (σ ): players i and k belong to different coalitions, while σi = σk . This contradicts the definition of Bδ . DA: This immediately follows from the definition. IR: Let i ∈ N and j, k ∈ Biδ (σ ). They are three different players. Since i, j, k ∈ Biδ (σ ), we have σi = σj = σk . It follows from σj = σk that Bjδ (σi′ , σ−i ) = Bkδ (σi′ , σ−i ). We now show the only-if-part. Step 1. Let σ ∈ AN . For any nonempty S ⊆ N, define HSσ = {i ∈ N |σi = S }. We first prove that H σ := {HSσ |S ⊆ N , HSσ ̸ = ∅} is a partition of N. For any two different coalitions S , T ⊆ N, we have HSσ ∩ HTσ = ∅ because if there is a player i such that i ∈ HSσ ∩ HTσ then that ⋃ it holds ⋃ S =σ σi = T . This contradicts S ̸= T . Moreover, σ = N because if there is a player i ∈ N \ H (S) then H S ⊆N S ⊆N S
σi ̸= S for any S ⊆ N. This contradicts σi ∈ Ai ⊆ 2N . Hence, H σ is a partition of N. We now define K σ := {S ∈ H σ |σj = S for all j ∈ S } and Lσ := H σ \ K σ . Step 2. By UN, for each coalition S ∈ K σ , S ∈ B(σ ). Step 3. For each coalition T ∈ Lσ with |T | = 1, say {i} = T , we have σi ̸ = σj for any j ∈ N \ {i}. Hence, by DA, Bi (σ ) = {i}. Now, we fix a coalition T ∈ Lσ with |T | ≥ 2 and consider a player i ∈ T . Let R := σi . Note that it follows from T ∈ Lσ that σj = R for every j ∈ T , σj ̸= R for every j ∈ N \ T .
(1)
If R = T , then T ∈ K σ , which contradicts T ∈ Lσ = H σ \ K σ ; if R is a proper subset of T , then for some player k in T \ R, k ̸ ∈ R = σk , which contradicts σk ∈ Ak . Hence, we have R ⊋ T . It follows that |R| ≥ 3. For the strategy profile σ and the coalition R, define the following strategy profile σ 0 ∈ AN : for every j ∈ N,
σ = 0 j
{
R
σj
if j ∈ R, otherwise.
By UN, R ∈ B(σ 0 ). Let R \ T = {i1 , . . . , im } for some natural number m. Note that for every i ∈ R \ T , σi0 = R.
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Now, define σ 1 ∈ AN as
σj1 =
{
σj σj0
if j = i1 , otherwise.
In view of i1 ̸ ∈ T and (1), we have σi1 ̸ = R. By IR, for any j, k ∈ R \ {i1 }, we have Bj (σ 1 ) = Bk (σ 1 ), which implies Bj (σ 1 ) ⊇ R \ {i1 } for every j ∈ R \ {i1 }. Since σj1 ̸ = R for any j ∈ (N \ R) ∪ {i1 } and σj1 = R for any j ∈ R \ {i1 }, DA implies that Bj (σ 1 ) ⊆ R \ {i1 } for every j ∈ R \ {i1 }. Hence, we obtain Bj (σ 1 ) = R \ {i1 } for every j ∈ R \ {i1 }. In the same manner, given σ and σ 1 , we define σ 2 ∈ AN as
σj2 =
{
σj σj1
Proof of Proposition 2.2 (Axiomatization of Bγ ). We first prove the if-part. UN: This follows in the same manner as Bδ . DA: If γ γ σi ̸= σj , then Bi (σ ) = {i} ̸= {j} = Bj (σ ). DR+ : Let T ∈ Bδ (σ ). Let S ⊆ T , j ∈ T \ S, and k ∈ T \ S. Since σi′ ̸ = σi = T for every i ∈ S, γ γ we have Bj (σS′ , σ−S ) = {j} and Bk (σS′ , σ−S ) = {k}. We now show the only-if-part. Step 1 and Step 2 are the same as Proposition 2.1. Step 3. For any coalition T ∈ Lσ with |T | = 1, say {i} = T , we have σi ̸ = σj for any j ∈ N \ {i}. Hence, by DA, Bi (σ ) = {i}. Now, we fix a coalition T ∈ Lσ with |T | ≥ 2 and consider a player i ∈ T . Let R := σi . In the same manner as Proposition 2.1, we have R ⊋ T and |R| ≥ 3. For the given strategy profile σ , similarly define strategy profile σ 0 ∈ AN as
{
R
σj
if j ∈ R, otherwise.
By UN, R ∈ B(σ 0 ). Let R \ T = {i1 , . . . , im } for some natural number m. In view of T ∈ H σ , we have σj ̸ = R for every j ∈ N \ T . Hence, σj0 = R ̸= σj for every j ∈ R \ T . It follows from DR+ that Bi (σ ) ̸ = Bj (σ ) for any different i, j ∈ T .
(2)
Since σj ̸ = R for every j ∈ N \ T and σj = R for every j ∈ T , in view of DA, we have Bi (σ ) ̸ = Bj (σ ) for any i ∈ T and any j ∈ N \ T .
and obtain {k, i} ∈ B(σ k ) by UN. Now, let σ ∗ be
σj∗ =
{
N {j, i}
if j = i, otherwise.
From MO, it follows that
{1, i} ⊆ Bi (σ ∗ ), {2, i} ⊆ Bi (σ ∗ ), . . . , {n, i} ⊆ Bi (σ ∗ ).
if j = i2 , otherwise.
Similarly, by IR and DA, we obtain Bj (σ 2 ) = R \ {i1 , i2 } for every j ∈ R \ {i1 , i2 }. Repeating this procedure until im , we have σ m = σ and, hence, Bj (σ ) = Bj (σ m ) = R \ {i1 , . . . , im } for every j ∈ R \ {i1 , . . . , im }. As T = R \ {i1 , . . . , im }, we obtain Bj (σ ) = T for every j ∈ T . Hence, in view of Step 2, B(σ ) = K σ ∪ Lσ = H σ . Step 4. We show H σ = {S ⊆ N |i, j ∈ S ⇐⇒ σi = σj }. Let S ∈ H σ . There exists T ⊆ N such that for any j ∈ S, σj = T . Hence, it holds that i, j ∈ S ⇒ σi = σj . Now consider j ∈ N \ S. There exists T ′ ⊆ N such that T ′ ̸ = T and σj = T ′ . Hence, it follows that i ∈ S and j ̸ ∈ S ⇒ σi ̸ = σj . Thus S ∈ Bδ . Now let S ∈ Bδ . Let T := σj = σk for any j, k ∈ S. Then we have H σ (T ) = S and H σ (T ) ∈ H σ , which means S ∈ H σ . In view of Steps 3 and 4, B(σ ) = H σ = Bδ (σ ). This completes the proof. □
σj0 =
namely, σ 1 = ({1, i}, {2, i}, . . . , {i − 1, i}, {1, i}, {i + 1, i}, . . . , {n, i}). In view of UN, {1, i} ∈ B(σ 1 ). In the same manner, for k = 1, . . . , i − 1, i + 1, . . . , n, we define { {k, i} if j = i, σjk = {j, i} otherwise
(3)
Hence, Bi (σ ∗ ) = N. This holds for every i ∈ N, namely, Bi (σ ∗ ) = N for every i ∈ N. However, since it does not hold that 2 ∈ σ1∗ and 1 ∈ σ2∗ , players 1 and 2 are not a pair. Hence, in view of PD, B1 (σ ∗ ) ̸ = B2 (σ ∗ ). This is a contradiction. □ Proof of Proposition 3.2 (Axiomatization of B≈ ). We first show σ the if-part. UN: If σj = S for every j ∈ S, then i ∼ j for any i, j ∈ S, σ
which implies i ≈ j for any i, j ∈ S. Hence, S ∈ B≈ (σ ). PD− : This immediately follows from the definition. MO: Fix i ∈ N. In view σ
of the definition, Bi≈ (σ ) = {j ∈ N |i ≈ j}. Since σi′ ⊇ σi , we have σ
(σ ′ ,σ i
−i )
Bi≈ (σ ) = {j ∈ N |i ≈ j} ⊇ {j ∈ N |i ≈ j} = Bi≈ (σi′ , σ−i ). We now show the only-if-part. Let σ ∈ AN . Let K σ := {∅ ̸ = σ S ⊆ N |σ ⋃j = S for all j ∈ S }. In view ofσ UN, K ⊆ B(σ ). Let σ ˆ ˆ K := is not a unanimous S ∈K σ S. Every player i in N \ K member: there is a player j ∈ σi such that σj ̸ = σi . We partition ˆσ into {S1 , . . . , SM } for some natural number M by indirect N \K pairs: for each m = 1, . . . , M, σ
i ≈ j for any i, j ∈ Sm , σ
i ̸ ≈ j for any i ∈ Sm and any j ̸ ∈ Sm . We fix an arbitrary coalition S ∈ {S1 , . . . , SM }. From PD− , it follows that Bi (σ ) ⊆ S for every i ∈ S .
(4)
Below, we prove that Bi (σ ) ⊇ S for every i ∈ S. To this end, we first show the following claim. Claim. Let B be an CA-function and σ ∈ AN . If B satisfies UN and MO, then Bi (σ ) ⊇ Di (σ ) for every i ∈ N. Proof of Claim. Assume that there exists k ∈ Di (σ ) such that k ̸ ∈ Bi (σ ). Since k ∈ Di (σ ), we have k ∈ σi and i ∈ σk . We define σ ′ as follows: for every j ∈ N,
{ σ σj = j {i, k} ′
if j ∈ N \ {i, k}, otherwise (namely, j = i or k).
By UN, Bi (σ ′ ) = Bk (σ ′ ) = {i, k}. Note that σi ⊇ {i, k} = σi′ because i ∈ σi and k ∈ σi . Now define σ ∗ as follows: for every j ∈ N,
σj∗ =
{
σj {i, k}
if j ∈ N \ {k}, otherwise (namely, j = k).
From (2) and (3), it follows that Bi (σ ) ̸ = Bj (σ ) for any i ∈ T and any j ∈ N \ {i}. Hence, Bi (σ ) = {i} for each i ∈ T . This holds for every coalition T ∈ Lσ with |T |⋃≥ 2. Hence, we have σ }, where Lˆ σ = B(σ ) = K σ ∪ {{i}|i ∈ Lˆ T ∈Lσ T . Thus, B (σ ) = Bγ (σ ), which completes the proof. □
By MO, Bi (σ ∗ ) ⊇ Bi (σ ′ ) = {i, k}. Since k ∈ Bk (σ ∗ ), Bi (σ ∗ ) = Bk (σ ∗ ). Hence, Bk (σ ∗ ) ⊇ {i, k}. In a similar manner, by MO, Bk (σ ) ⊇ Bk (σ ∗ ) ⊇ {i, k}. Since i ∈ Bi (σ ), Bi (σ ) = Bk (σ ). Hence, Bi (σ ) ⊇ {i, k}. This contradicts k ̸ ∈ Bi (σ ). This completes the claim. //
Proof of Proposition 3.1 (No CA-function Simultaneously Satisfies UN, PD, MO). Let B satisfy UN, PD, and MO. Fix a player i ∈ N. Let σ 1 ∈ AN be
We now prove that Bi (σ ) ⊇ S for every i ∈ S. We first fix i ∈ S and j ∈ Di (σ ). We have {i, j} ⊆ Di (σ ) and {i, j} ⊆ Dj (σ ). The Claim above implies that
σj1 =
{
{1, i} {j, i}
if j = i, otherwise,
Bi (σ ) ⊇ Di (σ ) ⊇ {i, j}, and Bj (σ ) ⊇ Dj (σ ) ⊇ {i, j}.
T. Abe / Mathematical Social Sciences 103 (2020) 69–75
75
B1 A3 A1 \ A2
123 12 13 1
123 N Z Z Z
123 12 Y X I I
123 N I I I
123 12 I X I I
23 Y I X Z
2 Y I Y I
123 X I Y I
13 12 23 I I X I Y Y I I
123 I I Y I
13 12 23 I I X I Y Y I I
2 Y I Y I
123 X I I Z
23 12 23 I Z X Z I Z I Z
2 I I I I
123 X I I I
3 12 I X I I
23 I I I I
2 I I I I
2 Y Y Y I
123 I I I Z
23 12 23 I Z X Z I Z I Z
2 I I I I
123 I I I I
3 12 I X I I
23 I I I I
2 I I I I
2 I I Y I
123 I I I I
23 12 23 I Z X Z I Z I Z
2 I I I I
123 I I I I
3 12 I X I I
23 I I I I
2 I I I I
2 Y I Y I
123 N N Z Z
23 12 23 X Z X Z I Z I Z
2 I I I I
123 X X I I
3 12 X X I I
23 I I I I
2 I I I I
B2 A3 A1 \ A2
123 12 13 1
23 I I X Z
2 I I Y I
B3 A3 A1 \ A2
123 12 13 1
123 N Z Z Z
123 12 Y X I I
123 N N N Z
123 12 N X N I
23 Y I X I
2 Y I I I
123 I I Y I
13 12 23 I I X I Y Y I I
123 N X Y I
13 12 23 N Y X I Y Y I I
B4 A3 A1 \ A2
123 12 13 1
23 N N N Z
2 Y I Y I
Hence, Bi (σ ) = Bj (σ ), which implies that Bi (σ ) = Bj (σ ) ⊇ (Di (σ ) ∪ Dj (σ )). We now consider the same j and fix k ∈ Di (σ ) ∪ Dj (σ ). In the same manner, we have Bj (σ ) = Bk (σ ) ⊇ (Dj (σ ) ∪ Dk (σ )). For the player j, we have Bi (σ ) = Bj (σ ) = Bk (σ ) ⊇ (Di (σ ) ∪ Dj (σ ) ∪ Dk (σ )). Applying this procedure to every i′ ∈ S ⋃ ′ and every j ∈ Di′ (σ ), we have Bi (σ ) ⊇ j∈S Dj (σ ) = S for every i ∈ S. Hence, we have Bi (σ ) ⊇ S for every i ∈ S. In view of (4), we have Bi (σ ) = S for every i ∈ S. Hence, we σ
ˆσ . For every i ∈ Kˆσ , have Bi (σ ) = {j ∈ N |i ≈ j} for every i ∈ N \ K σ
since i is a unanimous member, Bi (σ ) = σi = {j ∈ N |i ≈ j}. Thus, σ
B(σ ) = {T ⊆ N |i, j ∈ T ⇐⇒ i ≈ j}.
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Independence Define Bs as follows: Bs (σ ) = {{i}|i ∈ N } for every σ ∈ AN . Functions B1 to B4 are provided in the following table, in which underlined partitions violate the corresponding axiom. Independence for Bδ • Bγ satisfies UN and DA, but violates IR. • B1 satisfies UN and IR, but violates DA. • Bs satisfies DA and IR, but violates UN.
Independence for Bγ
• Bδ satisfies UN and DA, but violates DR+ . • B2 satisfies UN and DR+ , but violates DA. • Bs satisfies DA and DR+ , but violates UN. Independence for B≈ • B3 satisfies UN and PD− , but violates MO. • B4 satisfies UN and MO, but violates PD− . • Bs satisfies PD− and MO, but violates UN. References Abe, T., 2018. Stable coalition structures in symmetric majority games: A coincidence between myopia and farsightedness. Theory Decis. 85, 353–374. Banerjee, S., Konishi, H., Sönmez, T., 2001. Core in a simple coalition structure game. Soc. Choice Welf. 18, 135–153. Bogomolnaia, A., Jackson, M.O., 2002. The stability of hedonic coalition structures. Games Econ. Behav. 38, 201–230. Casajus, A., 2009. Outside options, component efficiency, and stability. Games Econ. Behav. 65, 49–61. Gallo, O., Inarra, E., 2018. Rationing rules and stable coalition structures. Theor. Econ. 13, 933–950. Hart, S., Kurz, M., 1983. Endogenous formation of coalitions. Econometrica 51, 1047–1064. Hart, S., Kurz, M., 1984. Stable coalition structures. In: Holler, M.J. (Ed.), Coalitions and Collective Action, Vol. 51. pp. 235–258.