Hedonic Coalition Formation and Individual Preferences Szilvia P´apai∗ January 2011

Abstract We examine properties of individual preferences for which there exists a stable hedonic coalition structure, and prove that an anonymous and rich preference domain guarantees the existence of a stable coalition structure if and only if it satisfies the Inclusion Property. The Inclusion Property requires that if a player ranks two nonsingleton coalitions above any superset of their union then the intersection of these two coalitions should be ranked above at least one of these two coalitions. Keywords: coalition formation, matching, stability, core

1

Introduction

We examine the hedonic coalition formation problem, which got its name from Dr`eze and Greenberg (1980), who referred to the players’ subjective ranking of coalitions, based on the identity of the coalition’s members, as hedonic preferences. Contrary to this origin of the name of the hedonic coalition formation model, which is based on direct preferences over coalitions (rather than on coalitional values that are divided up among the members), hedonic coalition formation may be interpreted as one based on objective preferences over coalitions. Given that players only care about the coalition they join, and that each player is ∗

Department of Economics, Concordia University and CIREQ; e-mail: [email protected]

1

exactly in one coalition (which may consist of one player only), it is a simple model of coalition formation that generalizes the well-known two-sided matching models (in particular, the marriage model), whose ample literature is surveyed by Roth and Sotomayor (1990). Given strict preferences, if the coalition structure is stable (in the core) then there is no coalition of players that would strictly prefer to form this coalition to staying in their respective coalitions in the coalition structure. It is well known that the core of the hedonic coalition formation problem may be empty, or in other words, there may not exist a stable hedonic coalition structure for a given coalition formation problem. This is demonstrated by Banerjee, Konishi, and S¨ onmez (2001), and Bogomolnaia and Jackson (2002), which offer sufficient conditions for the existence of stable coalition structures, in terms of restrictions on preference profiles. In addition, P´apai (2004) gives a characterization of the existence of a unique stable coalition structure in terms of restrictions on coalitions. In this paper we focus on restrictions on individual preferences, rather than restrictions on the preference profile or on feasible coalitions. We introduce a property of individual preferences called Inclusion Property, and prove that under the assumptions of anonymity and richness of the preference domain this property is the only one that guarantees the existence of a stable coalition structure. Cechl´arov´a and Romero-Medina (2001), Alcalde and Romero-Medina (2006), Alcalde and Revilla (2004), and Dimitrov et al. (2004) propose sufficient conditions for the existence of a stable hedonic coalition structure using individual preference restrictions. We identify two sufficient conditions for the existence of stable coalition structures, Intersection Restriction and Union Restriction, both of which imply the Inclusion property, and compare these properties to the conditions proposed in the above papers. We show that, when comparable, their sufficient conditions are stronger than one or the other of our two properties. Note, furthermore, that in contrast to these previous papers, which give sufficient conditions, we provide a characterization. Given how demanding these properties are, our result reveals that ensuring the existence of a stable coalition structure, by anonymously restricting the available individual preferences, is difficult. 2

2

Hedonic Coalition Formation

There is a finite set of players N = {1, . . . , n}, and a set of preferences Ri for each player i ∈ N . We will refer to each nonempty subset of N as a coalition, and let Π = {S ⊆ N |S 6= ∅} denote the set of coalitions. Furthermore, let Πi = {S ⊆ N |i ∈ S} denote the set of coalitions that player i is a member of. Player i’s preferences Ri ∈ Ri strictly order the elements of Πi . Preferences are strict, complete, and transitive. We will write SPi S 0 to indicate strict preferences, and SRi S 0 to indicate that either SPi S 0 or S = S 0 . We assume that players only care about the coalition they join. A (hedonic) coalition formation problem is defined by a pair (N, R), where R = (R1 , . . . , Rn ) is a preference profile in RN and RN = R1 × . . . × Rn is the Cartesian product of Ri ’s. Throughout this paper N is fixed, and thus a coalition formation problem is simply defined by a preference profile R ∈ RN . Given R ∈ RN and S ∈ Π, we denote (Ri )i∈S by RS . We also write R−i to denote RN \{i} , and R−S to denote RN \S . A coalition structure is a set of coalitions {S1 , . . . , Sk }, with n ≥ k ≥ 1, which is a partition of N (i.e.,

Sk

t=1 St

= N , where all St are pairwise disjoint). For all coalitions

S ⊆ N , let [S] = {{i} : i ∈ S} denote the set of singletons for the members of S. Let Σ denote the set of all coalition structures, and let σ ∈ Σ denote a coalition structure. Moreover, let σi be the coalition in σ that contains i. Given N and R ∈ RN , a coalition S ⊆ N blocks σ ∈ Σ at R if for all i ∈ S, SPi σi . A coalition structure σ is stable at R if there is no coalition that blocks σ at R. Note that stable coalition structures are core coalition structures, since the two notions are identical in this context. A property of individual preferences is said to guarantee the existence of a stable coalition structure if there exists a stable coalition structure for all coalition formation problems R ∈ RN such that, for all i ∈ N, Ri satisfies the property in question. A coalition structure σ is Pareto efficient at R if there is no σ 0 ∈ Σ, σ 0 6= σ, such that for all players i ∈ N with σi0 6= σi , σi0 Pi σi .

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A coalition S ∈ Π is individually rational at R if for all i ∈ S, SPi {i}. For all R ∈ RN , let ΠIR (R) = {S ∈ Π|for all i ∈ S, SPi {i}} be the set of individually rational coalitions at R. We note that if σ is stable at R then σ ⊆ ΠIR (R) [N ]. For all R ∈ RN , let RIR denote the corresponding preference S

profile in which rankings of coalitions in ΠIR (R) are preserved, while all coalitions not in ΠIR (R) are ranked below {i} for each player i. Furthermore, let R : S denote preference profile R restricted to players in coalition S such that coalitions containing players not in S are deleted.

3

The Inclusion Property

We are now ready to introduce the main property of individual preferences that is used in the characterization result. Inclusion Property Preferences Ri satisfy the Inclusion Property if for all coalitions ¯ S 0 ⊂ S, ¯ SPi S 0 Ri S, ¯ and S 0 Pi {i}, we have S T S 0 Ri S 0 . S, S 0 , S¯ ∈ Πi such that S ⊂ S, This property says that if two non-singleton coalitions have a superset of their union ranked below both of them, then their intersection is ranked above at least on of them. ¯ N ⊆ RN satisfies the Inclusion Property if for all We say that a preference domain R ¯ N and for all i ∈ N, Ri satisfies the Inclusion Property. R∈R The following two conditions, called Intersection Restriction and Union Restriction, both imply the Inclusion Property. Intersection Restriction: For all coalitions S, S 0 such that SPi {i} and S 0 Pi {i}, (S or (S

T 0 S )Ri S

T 0 S )Ri S 0 .

Intersection Restriction requires that for any two coalitions preferred to staying alone, 4

their intersection should be ranked above at least one of the two coalitions. Union Restriction: For all coalitions S, S 0 such that SPi {i} and S 0 Pi {i}, for all S¯ ⊇ S

S 0 ¯ ¯ iS0. S , SRi S or SR

Union Restriction requires that for any two coalitions preferred to staying alone, their union, and each superset of their union, should be ranked above at least one of the two coalitions. Alcalde and Romero-Medina (2006) proposes four sufficient properties of individual preferences. Union Responsiveness Condition: Ri satisfies the Union Responsiveness Condition if for all coalitions S, S 0 such that S 0 ⊂ S and S 0 6= Chi (S), SPi S 0 , where Chi (S) denotes player i’s first ranked choice among coalitions only containing members of S. Both the Union Responsiveness Condition and Union Restriction imply that the 1st or 2nd ranked coalition is the grand coalition. The Union Responsiveness Condition is more demanding than Union Restriction, however, because it has requirements for all supersets, not just for unions. Singularity: for each coalition S, SPi {i} implies that S = Chi (N ). This property is very restrictive, since it implies that the 1st or 2nd ranked coalition has to be {i}. Therefore, Singularity implies both Intersection Restriction and Union Restriction. Intersection Responsiveness Condition: for all coalitions S, S 0 , SPi S 0 implies that (S

T 0 S )Ri S 0 .

The Intersection Responsiveness Condition is very closely related to Intersection Restriction, but it is stronger, since it has requirements for coalitions ranked below {i}, unlike Intersection Restriction. Essentiality: there exists an essential coalition S for i such that (a)if S¯ = {i} then {i} is the first ranked coalition, (b) if S¯ = 6 {i} then S¯ is the first ranked coalition, all coalitions 5

¯ and for all coalitions S, S 0 that are ranked above {i} ranked above {i} are supersets of S, such that S ⊂ S 0 , SPi S 0 . This property, which guarantees the existence of a unique stable coalition structure, implies Intersection Restriction. This can be seen since for any two coalitions ranked above staying alone, their intersection is a subset (not necessarily strict) of both coalitions, and Essentiality requires that the intersection coalition, if distinct from both, be ranked higher than both. A somewhat weaker version of Essentiality, Top Responsiveness is used in Alcalde and Revilla (2004), which also implies Intersection Restriction, but allows for multiple stable coalition structures. Other sufficient properties of individual preferences were proposed by Cechl´arov´a and Romero-Medina (2001), and Dimitrov et al. (2004). The first one assumes that preferences are based on the most preferred and least preferred member of a coalition, while the latter bases the evaluation of coalitions on a the existence of a set of “friends,” and introduces two properties in which preferences are entirely determined by this set. All of these properties require lots of indifferences, and thus are not directly comparable to our properties in the context where only strict preferences are considered.

4

The Characterization Result

We make two assumptions on the preference domain, anonymity and richness. ¯ N ⊆ RN is anonymous if it satisfies the following two conditions: A preference domain R (1) Permutation: Let fij : N → N be a bijection such that fij (i) = j. Then, for all i ∈ N ¯ i , and for all j ∈ N \ {i}, there exists Rj ∈ R ¯ j such that for all S, S 0 ∈ Πi , SPi S 0 and Ri ∈ R implies that fij (S)Pj fij (S 0 ), where fij (S) =

S

l∈S

fij (l).1

¯ i and j ∈ N \ {i}. Then there exists a “dummy player” (2) Dummy: Let i ∈ N, Ri ∈ R 1

When we define fij later in the proofs, we will use the fij (S) = S 0 notation.

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˜i ∈ R ¯ i such that for all S with SPi {i}, l 6∈ S, l ∈ N \{i, j} that “mimics” j, i.e., there exists R and for all S such that SPi {i} and j ∈ S, S is replaced in Ri by S 0 = S {l}, in order to S

˜ i . We can also interpret these players with their dummy replicas as “split” players, get R i.e., players that are split into more than one player. Since this assumption may appear, at first, restrictive, we will argue that without such an assumption it does not make much sense to talk about individual preference restrictions, as opposed to preference profile restrictions. Firstly, we note that any preference profile restrictions may be expressed as restrictions on individual preferences if anonymity is not required. For example, having a top coalition S (Banerjee, Konishi, and S¨onmez, 2001) for N at preference profile R, which is obviously a preference profile restriction, can be expressed in terms of individual preference restrictions as follows. There exists a coalitions S, such that for all i ∈ S, Ri ∈ (S). Clearly, these individual preference restrictions are not anonymous. Note also that all individual preference restrictions in the papers mentioned earlier are anonymous. Secondly, we can easily show that without the anonymity assumption the existence of a stable coalition structure does not have any implication for a particular player’s individual preferences. Fix i ∈ N , and let R−i be such that for all j ∈ N \ {i}, Rj ∈ (N \ {i}). Then, for an arbitrary Ri , the coalition structure σ = {N \ {i}, {i}} is stable. Thus, we need the assumption of anonymity in order to establish the necessity of any property of individual preferences for the existence of a stable coalition structure. Our second assumption is richness of the preference domain. The preference domain ¯ N is rich if, for all i ∈ N, for all S ∈ Πi , there exists Ri ∈ R ¯ i such that Ri ranks S first. R This is a standard assumption and it is satisfied by all reasonable domain restrictions, while it rules out some trivial domains that contain very few preferences. ¯ N guarantees the existence of a stable coalition structure A preference domain R ¯N . if there exists a stable coalition structure at all R ∈ R We state our main result below. 7

Theorem: An anonymous and rich preference domain guarantees the existence of a stable coalition structure if and only if it satisfies the Inclusion Property. Proof: Sufficiency: We will show that if an anonymous and rich preference domain satisfies the Inclusion Property then it guarantees the existence of a stable coalition structure. We will need the following definitions. A coalition S is a top coalition of R if for all i ∈ S, Ri ∈ (S). ¯ with respect to R if For all S¯ ∈ Π, a coalition S ⊂ S¯ has a full support above S ¯ for all i ∈ S, SPi {i} and SPi S. ¯ N be an anonymous and rich preference domain that satisfies the inclusion Let RN ∈ R ¯ N . Then for all i ∈ N, Ri satisfies the Inclusion Property, and we Property. Fix R ∈ R need to show that there exists a stable coalition structure at R. Let ΠN be the set of coalitions that have a full support above N . If ΠN = ∅ then {N } is a stable coalition structure and we are done. Assume, therefore, that ΠN 6= ∅. Let ˆ N ⊆ ΠN be such that ΠˆN = {S ∈ ΠN | there is no S˜ ⊂ S for some i ∈ S, ˜ SP ˜ i S}. For all Π ˆ N = {S ∈ Π ˆ N |i ∈ S}. Let N ¯ = {i ∈ N |Π ˆ i 6= ∅}, |N ¯| = n i ∈ N , let ΠiN = {S ∈ Π ¯ , and N ¯ = {1, . . . , n ¯ = ¯ , let number players so that N ¯ }. Note that since ΠN 6= ∅, N 6 ∅. For all i ∈ N ˆ i¯ = Π ˆ i , and let Π ˆN = Π ˆ ¯ . For all i ∈ N ¯ , there exists a unique coalition S such that for Π N N N ˆ N \ {S}, S ⊂ S 0 . all S 0 ∈ Π ˆ N be the set of coalitions that have a full support We can construct S as follows. Let Π ˆ i = {S ∈ ΠN |S ∈ Πi }. Fix S ∈ Π ˆ i and let Π ˆ ¯ := Π ˆ i¯ \ {S}. above N . Let Π N N N N Step 1 Fix S 0 ∈ ΠiN . If S ⊂ S 0 , go to Step 2. If S 0 ⊂ S, let S := S 0 . Go to Step 2. If S, S 0 is an independent pair, note that (S

T 0 ˆ ¯ , by the Inclusion Property. S)∈Π N

T 0 T ˆ i , and go to Step 3. Otherwise there exists S¯ ⊂ S if (S S 0 ) ∈ |pi N T 0 ˆ i such that for some j ∈ S, ¯ SP ¯ j S T S 0 . Then let S := S. ¯ Go to Step 3. (S S ) ∈ Π N

Let S := S

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¯ i¯ := Π ¯ ¯ \ {S 0 }. Go to Step 4. Step 2 Let Π N N ¯ i¯ := Π ¯ ¯ \ {S 0 , S T S 0 }. Go to Step 4. Step 3 Let Π N N ¯ i¯ = ∅, stop. Otherwise, go to Step 1. Step 4 If Π N ¯ . Fix i, j ∈ N such Let S i := S. Next, we will show that {S 1 , . . . , S n¯ } is a partition of N that S i 6= S j . Suppose S i

T j S = 6 ∅. Since both S i and S j are in ΠN , S i 6⊂ S j and S j 6⊂ S i .

Thus, S i , S j is an independent pair. T j ˆ l¯ and S j ∈ Π ˆ l¯ . Thus, S (note: l = i, j is possible). Then S i ∈ Π N N T j l j i i j ˆ by the construction of ΠN¯ , S Pl (S S ). Given that S , S is an independent pair, this i j iT j ¯

Fix l ∈ S i

contradicts the Inclusion Property. Therefore, for all i, j ∈ N , if S 6= S then S

S = ∅.

¯ , i ∈ S i , {S 1 , . . . , S N } is a partition of N ¯. Then, given that for all i ∈ N ¯ . If {S 1 , . . . , S N¯ } Next, we will show that there exists a stable coalition structure for N ¯ then we are done. Assume, thus, that it is not stable. Then there exists is stable for N ¯ at R, i.e., for all i ∈ S, SPi S i . First note that there is no a blocking coalition S ⊆ N ¯ such that S ⊂ S i , given the construction of Π ˆ N . Next, suppose that S T S i 6= ∅ for i∈N some i ∈ N , and S, S i is an independent pair. Fix l ∈ S the Inclusion Property implies that (S

T i S . Then, given that SPl S i ,

T i ˆN. S )Pl S i , which contradicts the construction of Π S i

Therefore, the blocking coalition S is such that S =

i∈S

S , where |S| ≥ 2. Now consider

the partition consisting of {S i }i∈N¯ \S and S, and apply a similar argument to it as we did ¯ is finite, this proves that there exists a stable coalition above to {S 1 , . . . , S n¯ }. Given that N ¯ ⊆ N \N ¯ . Now take N \ N ¯ , and use the same argument as above to find N ¯, structure for N for which there exists a stable coalition structure. ¯ . If the stable coalition structures for N ¯ , respectively, add up ¯ SN ¯ and N Consider N ¯ , then we can proceed to N \ (N ¯ ). Suppose ¯ SN ¯ SN to a stable coalition structure for N ¯ , and that’s not the case. Then there exists a blocking coalition S 0 at R. Let i ∈ S 0 \ N ¯ . Let S = σ i¯ . Then, let σN denote the stable coalition structure that we identified for N N

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since S 0 Pi SPi N , the Inclusion Property implies that if S, S 0 is an independent pair then (S

T 0 T ˆ N is contradicted. S )Pi S. Since this is true for all i ∈ S S 0 , the construction of Π

¯ (which means that j 6∈ S) such that j ∈ S 0 , we have S ⊂ S 0 . Since there exists j ∈ N \ N ¯ such that S T N ¯ = S˜ and S˜ = S ˜ σ i¯ . Note, moreover, that there exists S˜ ⊆ N i∈S N ¯ ¯ ˜ ¯ SN Now we can let N = S, and repeat the argument in the above paragraph for N ¯ in each as many times as necessary. Each blocking coalition contains some player from N round, and the blocking coalition S 0 is ranked higher in each round by the same player in ¯ . Thus, this procedure ends in a finite number of steps. This means that there exists a N ¯. ¯ SN stable coalition structure for N Finally, we can repeat the above steps to identify a stable coalition structure in this ¯ S . . . = N. ¯ SN manner for N Necessity: We will show that if an anonymous and rich preference domain guarantees the existence of a stable coalition structure then it satisfies the Inclusion Property. We will call two coalitions S, S 0 an independent pair if S 6⊂ S 0 , and S 0 6⊂ S. Note first that if the Inclusion Property is satisfied by some player’s preferences, then any preference obtained from these preferences using anonymity also satisfies the Inclusion Property. This implies that there exist anonymous preference domains that satisfy the Inclusion Property. Step 1: Suppose, by contradiction, that there exists an anonymous and rich preference ¯ N which guarantees the existence of a stable coalition structure, but does not domain R ¯∈R ¯ N such that R ¯ i does not satisfy the Inclusion Property. Then there exist i ∈ N and R satisfy the Inclusion Property. This means that there exist coalitions S, S 0 , S¯ ∈ Πi such ¯ S 0 ⊆ S, ¯ S P¯i S 0 Ri S, ¯ S 0 P¯i {i}, and S 0 P¯i S T S 0 . that S ⊆ S, Suppose S ⊂ S 0 . Then S S

T 0 S = S, and S 0 P¯i S is a contradiction. Suppose S 0 ⊂ S. Then

T 0 S = S 0 , and S 0 P¯i S 0 is a contradiction. Hence, S, S 0 is an independent pair.

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¯ N is anonymous, we can use (2) Dummy to find Ri ∈ Ri with the Step 2: Given that R following properties. For simplicity, we will use the same coalition names as before for the coalitions that will be transformed by the (2) Dummy assumption. If there exist T, T 0 ⊂ S¯ such that T, T 0 6= S, S 0 , T Pi S 0 , T 0 Pi {i}, and |T | = |T 0 |, then note first that T, T 0 is an independent pair. Then there exists j ∈ T 0 \ T , and we can use (2) Dummy in the definition of anonymity to make |T 0 | = |T | + 1. If we keep iterating this step, we will eventually not have coalitions T, T 0 as specified above. Let |S| = |S 0 | such that |S \ S 0 | ≥ 2, which implies that |S 0 \ S| ≥ 2. Let j ∈ S \ S 0 and S S ˜ use (2) Dummy to expand S, S 0 , S, ˜ let l ∈ S 0 \ S. Let S˜ = {j, l} S¯ \ (S S 0 ). If |S| = 6 |S|,

as necessary, in order to get the following: ˜ (i) |S| = |S 0 | = |S|, (ii) S

T 0 S is unchanged, T 0T ˜

(iii) S

S

S = ∅,

T T˜ T˜ (iv) |S S 0 | = |S S| = |S 0 S|. S˜ S˜ S T (v) |S \ (S 0 S)| = |S 0 \ (S S)| = |S˜ \ (S S 0 )| = |S S 0 |.

¯ N −i , using (1) Permutation in the definition of Step 3: Next, we will define R−i ∈ R anonymity. We will need the following definition. µ : S → S is a cycle permutation function if it is a bijection such that S = {i1 , . . . , ik } and µ(i1 ) = i2 , µ(i2 ) = i3 , . . . , µ(ik ) = i1 . ¯ T S0) = 1. Let µS¯ : S¯ → S¯ be a cycle permutation of S¯ such that µS¯ (i) = j, µS(S T˜ T˜ S S S˜ T˜ S, µS¯ (S S) = S˜ \ (S S 0 ), µS¯ (S˜ \ (S S 0 )) = S˜0 \ (S S), µS¯ (S 0 \ S S) = S T T T T ˜ µ ¯ (S \ S 0 S) ˜ = S 0 S˜0 , and µ ¯ (S 0 S) ˜ = S S˜0 . Note that µ ¯ (S) = S \ (S 0 S),

S

S

S

S

˜ µ ¯0 (S) = S, and µ ¯ (S) ¯ = S. ¯ For all h ∈ S, ¯ let fij (h) = µ ¯ (h), and for all h ∈ N \ S, ¯ S, S S S ˜ j SPj S. ¯ ¯ j such that Rj = fij (Ri ). Thus, SP let fij (h) = h. Let Rj ∈ R ¯ T S) ˜ = 2. Let µ0S¯ : S¯ → S¯ be a cycle permutation of S¯ such that µ0S¯ (j) = l, µ0 S(S S0

T˜ 0 T˜ S˜ S˜ S˜ T˜ S, µS¯ (S 0 S) = S 0 \ (S S), µ0S¯ (S 0 \ (S S)) = S \ (S 0 S), µ0S¯ (S \ S 0 S) =

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S T T T T˜ ˜ = S˜ \ (S S 0 ), µ0S¯ (S˜ \ S S 0 ) = S S 0 , and µ0S¯ (S S 0 ) = S S. Note that µ0S¯ (S)

¯ = S. ¯ For all h ∈ S, ¯ let fjl (h) = µ0¯ (h), and for all S 0 , µ0S¯ (S) = tildeS, and µ0S¯ (S) S ¯ let fjl (h) = h. Let Rl ∈ R ¯ l such that Rl = fjl (Rj ). Thus, S 0 Pj SP ˜ j S. ¯ h ∈ N \ S, T 0 T T 0 S such that i0 = 6 i. Let µiS T S 0 : S S 0 → S S 0 be a cycle permutation T T 0 0 of S S 0 such that µiS T S (i) = (i0 ). For all h ∈ S S 0 , let fii0 (h) = µiS T S 0 (h), and T 0 ¯ i0

3. Let i0 ∈ S

for all h ∈ N \ (S

S ), let fii0 (h) = h. Let Ri0 ∈ R such that Ri0 = fii0 (Ri ). Note

¯ that SPi0 S 0 Pi0 S. 0 T˜ T T S such that j 0 = 6 j. Let µjS T S˜ : S S˜ → S S˜ be a cycle permutation 0 0 T˜ T let fjj 0 (h) = µjS T S˜ (h), and for of S S˜ such that µjS T S˜ (j) = (j 0 ). For all h ∈ S S, T˜ ¯ j0

4. Let j 0 ∈ S

all h ∈ N \ (S

S), let fjj 0 (h) = h. Let Rj 0 ∈ R such that Rj 0 = fjj 0 (Rj ). Note that

˜ j 0 S. ¯ SPj 0 SP 0 T T T˜ S such that l0 = 6 l. Let µjS 0 T S˜ : S 0 S˜ → S 0 S˜ be a cycle permutation T T˜ 0 0 of S 0 S˜ such that µSl 0 T S˜ (l) = (l0 ). For all h ∈ S 0 S, let fll0 (h) = µlS 0 T S˜ (h), and 0T ˜ ¯ l0

5. Let l0 ∈ S 0

for all h ∈ N \ (S

S), let fll0 (h) = h. Let Rl0 ∈ R such that Rl0 = fll0 (Rl ). Note

¯ ˜ l0 S. that S 0 Pl0 SP ¯ N is rich, we will now define preferences for those players that are exactly 6. Given that R ˜ For all i0 ∈ S \ (S 0 S S), ˜ let Ri0 rank S first. Similarly, for all in one of S, S 0 , and S. S S˜ j 0 ∈ S˜ \ (S S 0 ), let Rj 0 rank S 0 first, and for or all l0 ∈ S 0 \ (S S), let Rl0 rank S 0

first. ¯ N is rich, for all h ∈ N \ S, ¯ let Rh rank {h} first. 7. Finally, given that R ¯N . Points 1-7. completely define R ∈ R Step 4: We will show that there is no stable coalition structure at R. First note that all ¯ since otherwise there would be some stable coalition structures for R have to contain [N \ S], ˜ h ∈ N \ S¯ such that {h} blocks the coalition structure at R. Let S1 , S2 ∈ {S, S 0 , S}such 12

that S1 6= S2 . Suppose that there exist T ∈ σS¯ and h ∈ T such that T S1 , S2 , h ∈ S1

T

T

S1

T

S2 6= ∅, T 6=

S2 . Moreover, assuming without loss of generality that S1 Ph S2 , we have

T Ph S2 . Note that then |T | ≥ 2, and there exists h0 ∈ T, h0 6= h. Then, since T ∈ σ and σ is a stable coalition structure at R, T Ph0 {h0 }. Thus, by the construction of R, and given ¯ T 6= S1 T S2 , fhh0 (T ) 6= T . Therefore, there exists T 0 6= T, T 0 6= S1 , S2 , T 0 ⊆ S¯ that T 6= S, such that |T 0 | = |T |, T 0 Ph0 {h}, and fhh0 (T 0 ) = T . However, given the construction of Ri ¯ N in Step 3, such a T 0 does not exist. This is in Step 2, and given the definition of R ∈ R a contradiction, which implies that there is no T as described above. Therefore, the only ˜ S, ¯ S individually rational coalitions at R are S, S 0 , S,

T 0 T˜ T˜ S , S S, and S 0 S. ¯ is a stable ¯ σ ¯ , such that σ = σ ¯ S[N \ S] Now suppose that there exists a partition of S, S S

¯ then coalitions S, S 0 ,, and S˜ block σ ¯ at R. If coalition structure at R. Then, if σS¯ = {S}, S T T˜ S T˜ T 0 S S } [N \(S S 0 )] then S˜ blocks σ at R. Similarly, if σS¯ = {S S} [N \(S S)] T˜ S T˜ then S 0 blocks σ at R, and if σS¯ = {S 0 S} [N \ (S 0 S)] then S blocks σ at R. Finally,

σS¯ = {S

if S ∈ σS¯ then S˜ blocks, if S˜ ∈ σS¯ then S 0 blocks, and if S 0 ∈ σS¯ then S blocks σS¯ at R. ¯ N . This is a contradiction, since Therefore, there is no stable coalition structure at R ∈ R ¯ N guarantees the existence of a stable coalition structure. R

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