Hedonic Coalition Formation and Individual Preferences
Szilvia Pápai∗
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 Drèze 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 Sönmez (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, Pápai (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. Cechlárová 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 .
3
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, S 0 , S̄ ∈ Πi such that S ⊂ S̄, S 0 ⊂ S̄, SPi S 0 Ri S̄, and S 0 Pi {i}, we have S
T 0
S Ri S 0 .
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.
We say that a preference domain R̄N ⊆ RN satisfies the Inclusion Property if for all
R ∈ R̄N and for all i ∈ N, Ri satisfies the Inclusion Property.
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 S̄ ⊇
S
S 0
S , S̄Ri S or S̄Ri S 0 .
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 S̄ = {i} then {i} is
the first ranked coalition, (b) if S̄ 6= {i} then S̄ is the first ranked coalition, all coalitions
5
ranked above {i} are supersets of S̄, and for all coalitions S, S 0 that are ranked above {i}
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 Cechlárová 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.
A preference domain R̄N ⊆ RN is anonymous if it satisfies the following two conditions:
(1) Permutation: Let fij : N → N be a bijection such that fij (i) = j. Then, for all i ∈ N
and Ri ∈ R̄i , and for all j ∈ N \ {i}, there exists Rj ∈ R̄j such that for all S, S 0 ∈ Πi , SPi S 0
implies that fij (S)Pj fij (S 0 ), where fij (S) =
S
l∈S
fij (l).1
(2) Dummy: Let i ∈ N, Ri ∈ R̄i and j ∈ N \ {i}. Then there exists a “dummy player”
1
When we define fij later in the proofs, we will use the fij (S) = S 0 notation.
6
l ∈ N \{i, j} that “mimics” j, i.e., there exists R̃i ∈ R̄i such that for all S with SPi {i}, l 6∈ S,
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
get R̃i . We can also interpret these players with their dummy replicas as “split” players,
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 Sönmez, 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
R̄N is rich if, for all i ∈ N, for all S ∈ Πi , there exists Ri ∈ R̄i such that Ri ranks S first.
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.
A preference domain R̄N guarantees the existence of a stable coalition structure
if there exists a stable coalition structure at all R ∈ R̄N .
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).
For all S̄ ∈ Π, a coalition S ⊂ S̄ has a full support above S̄ with respect to R if
for all i ∈ S, SPi {i} and SPi S̄.
Let RN ∈ R̄N be an anonymous and rich preference domain that satisfies the inclusion
Property. Fix R ∈ R̄N . Then for all i ∈ N, Ri satisfies the Inclusion Property, and we
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
Π̂N ⊆ ΠN be such that ΠˆN = {S ∈ ΠN | there is no S̃ ⊂ S for some i ∈ S̃, S̃Pi S}. For all
i ∈ N , let ΠiN = {S ∈ Π̂N = {S ∈ Π̂N |i ∈ S}. Let N̄ = {i ∈ N |Π̂iN 6= ∅}, |N̄ | = n̄, and
number players so that N̄ = {1, . . . , n̄}. Note that since ΠN 6= ∅, N̄ 6= ∅. For all i ∈ N̄ , let
Π̂iN̄ = Π̂iN , and let Π̂N = Π̂N̄ . For all i ∈ N̄ , there exists a unique coalition S such that for
all S 0 ∈ Π̂N \ {S}, S ⊂ S 0 .
We can construct S as follows. Let Π̂N be the set of coalitions that have a full support
above N . Let Π̂iN = {S ∈ ΠN |S ∈ Πi }. Fix S ∈ Π̂iN and let Π̂N̄ := Π̂iN̄ \ {S}.
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
Let S := S
(S
T 0
S ) ∈ Π̂N̄ , by the Inclusion Property.
T 0
T
ˆ i , and go to Step 3. Otherwise there exists S̄ ⊂
S if (S S 0 ) ∈ |pi
N
T 0
i
T 0
S ) ∈ Π̂N such that for some j ∈ S̄, S̄Pj S
8
S . Then let S := S̄. Go to Step 3.
Step 2 Let Π̄iN̄ := Π̄N̄ \ {S 0 }. Go to Step 4.
Step 3 Let Π̄iN̄ := Π̄N̄ \ {S 0 , S
T 0
S }. Go to Step 4.
Step 4 If Π̄iN̄ = ∅, stop. Otherwise, go to Step 1.
Let S i := S. Next, we will show that {S 1 , . . . , S n̄ } is a partition of N̄ . Fix i, j ∈ N such
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.
Fix l ∈ S i
T j
S (note: l = i, j is possible). Then S i ∈ Π̂lN̄ and S j ∈ Π̂lN̄ . Thus,
T
by the construction of Π̂lN̄ , S j Pl (S i S j ). Given that S i , S j is an independent pair, this
i
j
iT j
contradicts the Inclusion Property. Therefore, for all i, j ∈ N̄ , if S 6= S then S
S = ∅.
Then, given that for all i ∈ N̄ , i ∈ S i , {S 1 , . . . , S N } is a partition of N̄ .
Next, we will show that there exists a stable coalition structure for N̄ . If {S 1 , . . . , S N̄ }
is stable for N̄ then we are done. Assume, thus, that it is not stable. Then there exists
a blocking coalition S ⊆ N̄ at R, i.e., for all i ∈ S, SPi S i . First note that there is no
i ∈ N̄ such that S ⊂ S i , given the construction of Π̂N . Next, suppose that S
some i ∈ N , and S, S i is an independent pair. Fix l ∈ S
the Inclusion Property implies that (S
T i
S =
6 ∅ for
T i
S . Then, given that SPl S i ,
T i
S )Pl S i , which contradicts the construction of Π̂N .
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∈N̄ \S and S, and apply a similar argument to it as we did
above to {S 1 , . . . , S n̄ }. Given that N̄ is finite, this proves that there exists a stable coalition
¯ ⊆ N \ N̄ ,
structure for N̄ . Now take N \ N̄ , and use the same argument as above to find N̄
for which there exists a stable coalition structure.
S ¯
¯ , respectively, add up
Consider N̄ N̄
. If the stable coalition structures for N̄ and N̄
S ¯
S ¯
to a stable coalition structure for N̄ N̄
, then we can proceed to N \ (N̄ N̄
). Suppose
that’s not the case. Then there exists a blocking coalition S 0 at R. Let i ∈ S 0 \ N̄ , and
i . Then,
let σN denote the stable coalition structure that we identified for N̄ . Let S = σN̄
9
since S 0 Pi SPi N , the Inclusion Property implies that if S, S 0 is an independent pair then
(S
T 0
T
S )Pi S. Since this is true for all i ∈ S S 0 , the construction of Π̂N is contradicted.
Since there exists j ∈ N \ N̄ (which means that j 6∈ S) such that j ∈ S 0 , we have S ⊂ S 0 .
i .
Note, moreover, that there exists S̃ ⊆ N̄ such that S N̄ = S̃ and S̃ = i∈S̃ σN̄
¯ = S̃, and repeat the argument in the above paragraph for N̄ S N̄
¯
Now we can let N̄
¯ in each
as many times as necessary. Each blocking coalition contains some player from N̄
T
S
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
N̄
S ¯
stable coalition structure for N̄ N̄
.
Finally, we can repeat the above steps to identify a stable coalition structure in this
S ¯ S
manner for N̄ N̄
. . . = 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
domain R̄N which guarantees the existence of a stable coalition structure, but does not
satisfy the Inclusion Property. Then there exist i ∈ N and R̄ ∈ R̄N such that R̄i does not
satisfy the Inclusion Property. This means that there exist coalitions S, S 0 , S̄ ∈ Πi such
that S ⊆ S̄, S 0 ⊆ S̄, S P̄i S 0 Ri S̄, S 0 P̄i {i}, and S 0 P̄i S
Suppose S ⊂ S 0 . Then S
S
T 0
S.
T 0
S = S, and S 0 P̄i S is a contradiction. Suppose S 0 ⊂ S. Then
T 0
S = S 0 , and S 0 P̄i S 0 is a contradiction. Hence, S, S 0 is an independent pair.
10
Step 2: Given that R̄N is anonymous, we can use (2) Dummy to find Ri ∈ Ri with the
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 ⊂ S̄ 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
let l ∈ S 0 \ S. Let S̃ = {j, l}
S
S̄ \ (S
S 0
S ). If |S| =
6 |S̃|, use (2) Dummy to expand S, S 0 , S̃,
as necessary, in order to get the following:
(i) |S| = |S 0 | = |S̃|,
(ii) S
T 0
S is unchanged,
T 0T
(iii) S
S
S̃ = ∅,
T
T
T
(iv) |S S 0 | = |S S̃| = |S 0 S̃|.
S
0S
0
(v) |S \ (S
S̃)| = |S \ (S
S̃)| = |S̃ \ (S
S 0
T
S )| = |S S 0 |.
Step 3: Next, we will define R−i ∈ R̄N −i , using (1) Permutation in the definition of
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 .
1. Let µS̄ : S̄ → S̄ be a cycle permutation of S̄ such that µS̄ (i) = j, µS̄(S
T 0
S) =
T
S 0
S
S
S ), µS̄ (S̃ \ (S S 0 )) = S̃ 0 \ (S S̃), µS̄ (S 0 \ S S̃) =
S
T
T
T
T
S \ (S 0 S̃), µS̄ (S \ S 0 S̃) = S 0 S̃ 0 , and µS̄ (S 0 S̃) = S S̃ 0 . Note that µS̄ (S) =
S
T
S̃, µS̄ (S
T
S̃) = S̃ \ (S
S̃, µS¯0 (S) = S, and µS̄ (S̄) = S̄. For all h ∈ S̄, let fij (h) = µS̄ (h), and for all h ∈ N \ S̄,
let fij (h) = h. Let Rj ∈ R̄j such that Rj = fij (Ri ). Thus, S̃Pj SPj S̄.
2. Let µ0S̄ : S̄ → S̄ be a cycle permutation of S̄ such that µ0S̄ (j) = l, µ0 S̄(S
S
0T
S̃, µ0S̄ (S
0T
S̃) = S 0 \ (S
S
S̃), µ0S̄ (S 0 \ (S
11
S
S̃)) = S \ (S
0S
S̃), µ0S̄ (S \ S
T
S̃) =
0T
S̃) =
S̃ \ (S
S 0
T
T
T
T
S ), µ0S̄ (S̃ \ S S 0 ) = S S 0 , and µ0S̄ (S S 0 ) = S S̃. Note that µ0S̄ (S̃) =
S 0 , µ0S̄ (S) = tildeS, and µ0S̄ (S̄) = S̄. For all h ∈ S̄, let fjl (h) = µ0S̄ (h), and for all
h ∈ N \ S̄, let fjl (h) = h. Let Rl ∈ R̄l such that Rl = fjl (Rj ). Thus, S 0 Pj S̃Pj S̄.
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 ∈ R̄ such that Ri0 = fii0 (Ri ). Note
that SPi0 S 0 Pi0 S̄.
4. Let j 0 ∈ S
of S
T
T
0
S̃ such that j 0 6= j. Let µjS T S̃ : S
T
0
S̃ such that µjS T S̃ (j) = (j 0 ). For all h ∈ S
all h ∈ N \ (S
T
S̃ → S
T
T
S̃ be a cycle permutation
0
S̃, let fjj 0 (h) = µjS T S̃ (h), and for
0
S̃), let fjj 0 (h) = h. Let Rj 0 ∈ R̄j such that Rj 0 = fjj 0 (Rj ). Note that
SPj 0 S̃Pj 0 S̄.
5. Let l0 ∈ S 0
of S
0T
T
0
S̃ such that l0 6= l. Let µjS 0 T S̃ : S 0
l0
T
S̃ → S 0
S̃ such that µS 0 T S̃ (l) = (l0 ). For all h ∈ S
for all h ∈ N \ (S 0
T
0T
T
S̃ be a cycle permutation
0
S̃, let fll0 (h) = µlS 0 T S̃ (h), and
0
S̃), let fll0 (h) = h. Let Rl0 ∈ R̄l such that Rl0 = fll0 (Rl ). Note
that S 0 Pl0 S̃Pl0 S̄.
6. Given that R̄N is rich, we will now define preferences for those players that are exactly
in one of S, S 0 , and S̃. For all i0 ∈ S \ (S 0
j 0 ∈ S̃ \ (S
S
S̃), let Ri0 rank S first. Similarly, for all
S 0
S
S ), let Rj 0 rank S 0 first, and for or all l0 ∈ S 0 \ (S S̃), let Rl0 rank S 0
first.
7. Finally, given that R̄N is rich, for all h ∈ N \ S̄, let Rh rank {h} first.
Points 1-7. completely define R ∈ R̄N .
Step 4: We will show that there is no stable coalition structure at R. First note that all
stable coalition structures for R have to contain [N \ S̄], since otherwise there would be some
h ∈ N \ S̄ such that {h} blocks the coalition structure at R. Let S1 , S2 ∈ {S, S 0 , S̃}such
12
that S1 6= S2 . Suppose that there exist T ∈ σS̄ 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
that T 6= S̄, T 6= S1
T
S2 , fhh0 (T ) 6= T . Therefore, there exists T 0 6= T, T 0 6= S1 , S2 , T 0 ⊆ S̄
such that |T 0 | = |T |, T 0 Ph0 {h}, and fhh0 (T 0 ) = T . However, given the construction of Ri
in Step 2, and given the definition of R ∈ R̄N in Step 3, such a T 0 does not exist. This is
a contradiction, which implies that there is no T as described above. Therefore, the only
individually rational coalitions at R are S, S 0 , S̃, S̄, S
T 0 T
T
S , S S̃, and S 0 S̃.
S
Now suppose that there exists a partition of S̄, σS̄ , such that σ = σS̄
[N \ S̄] is a stable
coalition structure at R. Then, if σS̄ = {S̄}, then coalitions S, S 0 ,, and S̃ block σS̄ at R. If
T
T
S
T
T 0 S
S } [N \(S S 0 )] then S̃ blocks σ at R. Similarly, if σS̄ = {S S̃} [N \(S S̃)]
T
S
T
then S 0 blocks σ at R, and if σS̄ = {S 0 S̃} [N \ (S 0 S̃)] then S blocks σ at R. Finally,
σS̄ = {S
if S ∈ σS̄ then S̃ blocks, if S̃ ∈ σS̄ then S 0 blocks, and if S 0 ∈ σS̄ then S blocks σS̄ at R.
Therefore, there is no stable coalition structure at R ∈ R̄N . This is a contradiction, since
R̄N guarantees the existence of a stable coalition structure.
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