Noncooperative Formation of Coalitions in
Hedonic Games
Francis Bloch
E¤rosyni Diamantoudiy
February 14, 2006
Abstract
We study a bargaining procedure of coalition formation in the
class of hedonic games, where players’ preferences depend solely on
the coalition they belong to. We show that if a hedonic game is totally balanced then for every core element there exists a stationary
perfect equilibrium that supports it. The reverse is not true and it is
established by means of a counter example. We also identify a condition that guarantees both uniqueness of the core as well of stationary
perfect equilibria. In the other extreme, we identify a condition that
guarantees the emptiness of the core as well as the non-existence of
stationary perfect equilibria in the roommate problem. Moreover, we
show that if the game has an empty core then the stationary perfect
equilibrium, if it exists, will involve unacceptable o¤ers.
Keywords: Coalition formation, hedonic games, bargaining, marriage problem
JEL Classification Numbers: C78, C71
1
Introduction
Hedonic games represent situations where agents form groups in order to
enjoy the pleasure of each other’s company.1 In a hedonic game, players have
GREQAM, Université de la Méditerranée and Warwick University, [email protected]
y
Concordia University, [email protected]
1
The term "hedonic game" from the Greek root meaning "pleasure" has been coined
by Drèze and Greenberg (1980).
1
preferences over the set of coalitions they can form, and cannot make sidepayments to each other. This is a general class of games which encompasses
many matching problems (the marriage problem, the roommate problem,
the college admissions problem, etc.)2 . Hedonic games have recently been
studied by Banerjee et al. (2001), Bogomolnaia and Jackson (2001) and Papai
(2004) who propose di¤erent su¢ cient conditions for the nonemptiness and
uniqueness of the core while Diamantoudi and Xue (2003) analyze various
stability notions including the core.
Since Gale and Shapley (1962), it is known that the core is not empty in
the marriage problem. This has recently led researchers to propose noncooperative games implementing the core for matching problems where the core is
known to be nonempty.3 For example, in the marriage problem, consider the
following two-stage game which is adapted from Kamecke (1989). Men simultaneously announce the women they want to marry in the …rst stage, and
women simultaneously choose among the men who proposed to them in the
second stage. This game implements the core of the marriage problem. In a
variant of the game where men announce sequentially the women they want to
marry, and women then simultaneously choose among the men who proposed
to them, the unique subgame perfect equilibrium of the game gives rise to
the men-optimal stable matching. Alcalde (1996) proposes another sequential
game implementing stable marriages. Alcalde, Perez-Castrillo and RomeroMedina (1998) and Perez-Castrillo and Sotomayor (2002) study games inspired by Kamecke (1989) implementing the core of assignment games (where
players have the ability to make side-payments). Alcalde and Romero-Medina
(2000) and (2005) extend the game proposed by Kamecke (1989) to the college admissions problem, where one side of the market (the colleges) can be
matched to many agents on the other side.
From a market design point of view, sequential games implementing the
2
Hedonic games can also be viewed as a special class of NTU games where the characteristic function assigns a unique payo¤ pro…le to each coalition.
3
This literature is related to, but quite di¤erent from, an earlier literature looking
at the strategic properties of matching algorithms like the Gale-Shapley (1982) deferred
acceptance algorithm in the marriage problem. (See the book by Roth and Sotomayor
(1990) for an introduction to the literature on strategyproofness in matching models).
The main di¤erence is that in the earlier literature the market is centralized and preferences have to be truthfully reported, hence strategyproofness considers dominant strategy
implementation of the core of matching games. By contrast, the recent literature assumes
decentralized markets, considers sequential games, and uses the concept of subgame perfect
implementation.
2
core of matching problems can be of great practical use. From a positive
point of view, it is di¢ cult to see how these games would be played without
the intervention of a social planner designing the game. Our objective in
this paper is not to design a noncooperative game implementing the core,
but rather to study a “natural”procedure of coalition formation which tries
to mimic the way agents would form groups in an environment without a
social planner. The procedure we consider is inspired by Rubinstein (1982)’s
alternating-o¤ers bargaining game, and can be described as the following
game of courtship. (We use the metaphor of the marriage problem to describe
the game.) Men and women are ordered according to a …xed protocol. A
woman makes the …rst o¤er to a man who may accept or reject the proposal.
If the proposal is accepted, the couple is formed (for life), and another agent
is selected by the protocol to make the next o¤er. If the man rejects the
o¤er, he makes a counter-o¤er to a woman in the next period. This game of
courtship has an in…nite horizon, and we assume that agents have a preference
for marriages happening sooner rather than later, and that they obtain a zero
utility if the game continues forever4 .
The courtship model can be naturally extended to hedonic games where
the proposer can propose the formation of any coalition. Then, each (potential) coalition member, following the protocol order accepts or rejects the
proposal. In the event of unanimous acceptance the coalition forms and
withdraws from the game. In the event of a rejection, the rejector becomes
the next proposer. Our model is of course closely related to other models extending Rubinstein (1982)’s alternating-o¤ers game to coalitional bargaining.
Chatterjee et al. (1993) consider the same game as the one we study in the
context of games with transferable utility. They …nd that stationary perfect
equilibrium outcomes belong to the core only under very stringent conditions.
Bloch (1996) applies the same bargaining procedure to a context where players’ payo¤s depend on the entire coalition structure. While his analysis is
close to that of this paper, the focus is very di¤erent, as Bloch (1996) primarily studies the e¤ect of externalities across coalitions, and does not prove any
result relating the set of stationary perfect equilibrium outcomes to the core.
Ray and Vohra (1999) use the same bargaining game to study the formation
of coalitions in a context with externalities and side payments. Other pa4
It should be noted that the game of courtship we consider di¤ers signi…cantly from the
various dynamic approaches to the matching problem as we do not distinguish between
men and women in the protocol.
3
pers considering di¤erent procedures of coalitional bargaining include Okada
(1996), Seidmann and Winter (1998) and Maskin (2003).
Our analysis is divided into two parts. We …rst analyze hedonic games
with nonempty cores, and study whether the set of stationary perfect equilibrium outcomes of the game coincides with the core. Our …rst result shows
that any core outcome can be supported by a stationary perfect equilibrium.
However –and this is the central result of our analysis –the game of courtship
can produce unstable outcomes. We show this by counterexample, exhibiting
a stationary perfect equilibrium in a marriage game with six agents, where
the equilibrium outcome does not belong to the core. In order to guarantee
that all stationary perfect equilibrium outcomes belong to the core, we identify a strong su¢ cient condition (the "top coalition property" …rst introduced
by Banerjee et al. (2001)) under which the core of the hedonic game is singlevalued, and is the unique stationary perfect outcome of the courtship game.
In the second part of the analysis, we consider hedonic games with empty
cores. The question here is whether the game admits a stationary perfect
equilibrium, and if it does, what type of (unstable) outcome is reached by the
game of courtship. We identify a class of environments for which stationary
perfect equilibria fail to exist (roommate problems with odd top cycles), and
show that when a stationary perfect equilibrium exists, it must involve some
players making unacceptable o¤ers.
The rest of the paper is organized as follows. We present the model
(hedonic games and the bargaining procedure) in Section 2. Section 3 is
devoted to the analysis of games with nonempty cores. Section 4 contains
our results on games with empty cores, and Section 5 concludes.
2
2.1
The Model
Hedonic Games
We consider a set N of players, indexed by i = 1; 2; :::; n. A coalition S is
a nonempty subset of N . A coalition structure is a partition of the set of
players into disjoint coalitions. We let S(i; ) denote the coalition to which
player i belongs in coalition structure . In many problems of interest, the
set of admissible coalitions is a strict subset of the power set of N , and we
will let S denote the set of admissible coalitions. Finally, we denote by Si
the set of coalitions in S containing player i, Si = fS 2 Sji 2 Sg:
4
De…nition 2.1 A hedonic game H is de…ned by a set of admissible coalitions, S and preference orderings i over Si for all players i 2 N:5
The structure of hedonic games generalizes a number of matching problems considered in the literature. The marriage and roommate problems are
particular instances of hedonic games where the set of admissible coalitions
are pairs and singletons. One-to-many and many-to-many matching problems can also be represented as hedonic games, with appropriately de…ned
admissible coalitions. The general structure of hedonic games thus accommodates a large number of problems of group formation without transferable
utility. However, hedonic games do not allow for externalities across groups:
players’preferences are only de…ned over the set of coalitions they belong to,
and do not depend on the coalitions formed by other players.
The most natural concept of stability for hedonic games is the core –
simply de…ned in this context as the set of coalition structures such that no
group of players could be made better o¤ by forming a new group. Formally,
we de…ne:
De…nition 2.2 The core of a hedonic game H is the set of coalition structures
for which there does not exist a coalition T of players such that
T i S(i; ) for all i 2 .
In the particular context of one-to-one matching problems, coalition structures only contain pairs and singletons, and can be identi…ed with matchings.
It is easy to check that the core is then equivalent to the set of stable matchings (see, for example, Roth and Sotomayor, 1990).
2.2
A Noncooperative Model of Coalition Formation
We analyze a sequential noncooperative game of coalition formation inspired
by Rubinstein (1982)’s alternating-o¤ers bargaining game. More precisely,
the procedure we study is identical to the procedure studied by Bloch (1996)
in the context of games with externalities across coalitions. Variants of this
procedure, where players can choose how to divide the coalitional surplus,
have been analyzed by Chatterjee et al. (1993) in TU games and by Ray and
Vohra (1999) for games with externalities.
5
We only consider here hedonic games where players have strict (i.e. asymmetric)
preference orderings.
5
We suppose that players are ordered according to an exogenous rule of
order (or protocol) – which may depend on the set of active players in
the game. Initially, the …rst player according to the rule of order i proposes
to form a coalition to which she belongs, S. All players in S then respond
sequentially to the o¤er. If all players accept the o¤er, coalition S leaves
the game, and the set of active players becomes N nS. The …rst player in
N nS then makes a proposal, etc.. If one of the players rejects the o¤er, the
coalition is not formed and the set of active players remains the same. The
player who rejected the o¤er becomes the proposer at the next round.
We consider a game without discounting. However, in order to prevent
players from remaining in the game ad in…nitum, we suppose that continuous play (an outcome denoted ?) gives a lower payo¤ than any coalition,
S i ? for all i and all S 3 i. Furthermore, we suppose that players
have lexicographic preferences, and prefer an outcome to be reached earlier
than later, i.e. if S is obtained at two di¤erent dates t; t0 with t < t0 ; then
(S; t) i (S; t0 ):6
A state in the game is simply given by the set A of active players. (This
is the only payo¤-relevant information after any history of the game). A
strategy for player i speci…es either a coalition S 2 Si (when player i is a
proposer), or a response in fY; N g (when player i is a respondent) after every
history of the game. A strategy is stationary if proposals and responses only
depend on the state of the game, and not on the entire history.
De…nition 2.3 A stationary perfect equilibrium (SPE) is a vector of stationary strategies for all the players such that, after every history of the game,
all players choose an optimal action.
We do not elaborate here on the de…nition of SPE nor on the reasons
for which we restrict attention to stationary strategies. It is well known that
without this restriction, the set of equilibrium strategies in a sequential bargaining game can be very large (see, e.g. Sutton (1986) for a TU bargaining
game with three players). It is also well known that, if other players employ
stationary strategies, it is optimal for a player to use a stationary strategy.
Hence, stationary strategies are global best responses to each other.
We note that hedonic games only provide an ordinal structure of preferences over the outcomes of the sequential game of coalition formation. For
6
We abuse notations by using the same preferences to denote preferences over coalitions
in the hedonic game, and preferences over outcomes in the coalition formation game.
6
this reason, we consider a game tree without chance moves and only consider
deterministic outcomes of the game. Hence, we do not allow for a random
choice of proposers at any stage of the game, and restrict attention to pure
strategy equilibria.
3
Coalition Formation in Games with Nonempty Cores
We …rst consider the outcome of the sequential procedure of coalition formation in hedonic games with nonempty cores. Because the procedure we
consider is recursive, in order to solve the game, we need to assume that the
core of any restriction of the set of players is nonempty. In other words, we
de…ne "totally balanced" hedonic games as games for which any restriction
of the game to a subset of players has a nonempty core.
De…nition 3.1 The restriction of the hedonic game H to a set S N , H S
is de…ned by S S = fT 2 SjT
Sg. The sets SiS can easily be de…ned, and
S
i is the restriction of the preference orderings to Si .
De…nition 3.2 A hedonic game H is totally balanced if the core of H is
nonempty, and the core of any restriction H S of H is nonempty.
Of course, total balancedness is a very demanding condition. We note
however, that in the particular context of the marriage problem, stable
matchings exist for any set of players (Shapley and Gale (1962), and Roth
and Sotomayor (1990)). Hence, marriage games are totally balanced hedonic
games. Our …rst result shows that in this class of games, any core coalition structure can be supported by a stationary perfect equilibrium of the
sequential game of coalition formation.
Proposition 3.1 Let H be a totally balanced game. For any coalition structure in the core of H, there exists a stationary perfect equilibrium with
outcome :
Proof. We construct the equilibrium strategies. At the initial state, when
A = N , let every player i propose S(i; ) and accept any coalition T such
that T i S(i; ). At any state A that contains only blocks of the coalition
7
structure , let the players in A still propose S(i; ) and accept coalitions
such that T i S(i; ). (Note that the restriction of to the set A; A ,
necessarily belongs to the core of the restriction H A .) Now consider some
state A which is not on the equilibrium path and contains some coalition not
in , and apply the following recursive procedure. If jAj = n 1, select one
core coalition structure A in H A : If jAj = k < n 1, either A belongs to
0
the path of the formation of some coalition structure A de…ned earlier, in
which case players form the coalition structure A which is the restriction of
A0
to A, or one selects a core coalition structure, A in the core of H A :
We show that these strategies form a subgame perfect equilibrium of the
game. Consider a deviation by player i; who announces S 0 6= S(i; A ) at some
state, or rejects S(i; A ) and announces S 0 . First note that because players
prefer the outcome to be reached earlier, rejecting S(i; A ) to o¤er S(i; A )
next period cannot be an equilibrium strategy. Hence, after rejecting S(i; A ),
the player must announce S 0 6= S(i; A ). If some player j rejects the o¤er S 0 ,
player j forms the coalition S(j; A ). By construction, the coalition structure
A
will then be formed at the equilibrium of the game. Hence, player i
will still belong to coalition S(i; A ) and given that she prefers to form a
coalition earlier, this deviation cannot be pro…table. Hence we conclude that
the deviation can only be pro…table if all players in S 0 accept i0 s o¤er. Hence,
S 0 j S(j; A ) for all j 6= i. This contradicts the fact that A belongs to the
core of H A .
Proposition 3.1 shows that any core coalition structure can be reached as
a stationary perfect outcome of the game. It does not imply however that all
equilibrium outcomes are stable. It came to us as a surprise to realize that
the last statement is not true: there exist totally balanced hedonic games for
which the sequential process of coalition formation results in unstable equilibrium outcomes for some protocol. This is demonstrated by the following
example.
Example 3.1 Consider a marriage problem with three men and three women,
8
with the following preferences7 :
m1
m2
m3
w1
w2
w3
w2
w3
w1
w2
w3
w1
w3
w2
w1
m2
m3
m1
m3
m1
m2
m2
m1
m3
This marriage problem has a unique stable matching
= ((m1 ; w1 ); (m2; w3 ); (m3 ; w2 )):
However, we construct here an equilibrium which gives rise to the unstable
matching 0 = ((m1 ; w1 ); (m2; w2 ); (m3 ; w3 )) when player m1 makes the …rst
o¤er.
At the initial state, where A = N , let the following be the players’proposal and acceptance strategies:
Player:
m1
Proposal:
w1
Acceptance: w1 ; w2 ; w3
m2
m3
w1
w2
w3
w3
w2 ; w3
w2
w2 ; w3
m1
m 1 ; m2 ; m3
m3
m3
m2
m2
At states A 6= N , construct strategies as in the proof of Proposition
3.1 by selecting a stable matching at state A. For the particular states
A0 = fm2 ; m3 ; w2 ; w3 g and A00 = fm2 ; m3 ; w1 ; w2 ; w3 g which admit two sta0
ble matchings, pick the stable matchings A = ((m2 ; w2 ); (m3 ; w3 )) and
00
A
= ((w1 ); (m2 ; w2 ); (m3 ; w3 ))8 respectively: All other states admit only
one stable matching.
Given the proof of Proposition 3.1, we only need to check that the strategies form a subgame perfect equilibrium at the initial state. If man m1
deviates and proposes either to w2 or w3 , his o¤er is rejected, and either the
pair (m3 ; w2 ) or the pair (m2 ; w3 ) forms. In both cases, man m1 is matched to
7
We assume that staying single is the worst outcome for all players.
We note that we are free to choose the stable matching at this state. The construction
of the proof of Proposition 3.1 …xes the coalition structure only at states which are induced
by earlier states. The state A0 can only be induced by the states N and A00 . For A00 , we
select the same matching. State N is treated separately, and we do not require that players
follow up on the coalition structure targeted at state N .
8
9
w1 in the unique stable matching following the formation of either pair, and
man m1 prefers to accept w1 immediately. Similarly, woman w1 has no incentive to reject man m1 : Women w2 and w3 are matched to their top choice,
and hence have no incentive to deviate. If man m2 deviates and proposes to
w1 , his o¤er is accepted, and he is matched to w1 which is a worse outcome
than being married to woman w3 . If he proposes to w2 instead, his o¤er is
rejected, woman w2 marries man m3 and man m2 is married to woman w3
in the unique stable matching following the formation of the pair (m3 ; w2 ).
Hence man m2 has no pro…table deviation. A similar argument shows that
man m3 has no pro…table deviation.
Finally, we note that given these equilibrium strategies, if man m1 makes
the …rst o¤er, the outcome of the game is the unstable matching 0 .
A careful inspection of example 3.1 shows that it relies on two important features. First, there is a unique stable matching when all players
are active, but after the couple (m1 ; w1 ) has left the game, there are two
stable matchings: the men optimal matching ((m2 ; w2 ); (m3 ; w3 )) and the
women optimal matching ((m2 ; w3 ); (m2 ; w3 )). Second, at the initial state,
men m2 and m3 are unable to induce the subgame leading to the men optimal matching. In order to do so, they need to wait until the formation
of the pair (m1 ; w1 ). But they cannot induce players m1 and w1 to form a
match, because woman w1 always accepts their o¤ers. These two features
are required to construct a stationary perfect equilibrium which results in an
unstable matching. We also note that there also exist protocols for which
the stable marriage is obtained (for example if man m2 makes the …rst o¤er).
Whether one can construct examples where, for every protocol, there exists
an unstable equilibrium outcome, remains an open question.
Our next task is to characterize the set of hedonic games for which it is
impossible to construct stationary perfect equilibrium strategies resulting in
unstable outcomes. To this end, we will consider games which admit a unique
element in the core for every subgame. The following de…nition, taken from
Banerjee, Konishi and Sonmez (2001) will prove very useful.
De…nition 3.3 (Banerjee, Konishi, Sonmez (2001)) Given a non-empty set
of players V
N; a non-empty subset S
V is a top-coalition of V i¤
for any i 2 S and any T
V with i 2 T; we have S i T: A hedonic game
H satis…es the top-coalition property i¤ for any non-empty set of players
V
N; there exists a top-coalition S of V and every player i of V has a
feasible coalition Si 2 Si such that Si \ S 6= ?:
10
De…nition 3.3 states that, for all subsets of players, there exists a coalition
which is the top coalition for all its members. Furthermore, all players have a
feasible coalition which contains some member of the top coalition. Arguably,
this is a very restrictive condition. Banerjee, Konishi and Sonmez (2001) provide a number of applications where the top coalition property holds. They
note that the top coalition property implies the common ranking property
of Farrell and Scotchmer (1988). In the particular context of the marriage
problem, the condition will be satis…ed if all men agree on the ranking of
women and all women agree on the ranking of men. Alternatively, it will
also be satis…ed if preferences satisfy some "assortative property", such that
man mi ranks woman wi …rst (and then ranks all women according to the
minimal distance between his index and the index of the woman) and women
have a corresponding ranking. In the roommate problem, the top coalition
property corresponds to the
reducibility proposed by Alcalde (1995).
Banerjee, Konishi and Sonmez (2001) establish that if the hedonic game
satis…es the top coalition property, there exists a unique coalition structure
in the core. Note that the proof of their result is much simpler if the hedonic
game is a marriage game. Suppose that there are multiple stable matchings,
and let w and m be the women and men optimal matchings respectively,
with w 6= m . Let S = fmj w (m) 6= m (m)g [ fwj w (w) 6= m (w)g. When
the player set is S, it is clearly impossible to …nd a couple (m; w) who are
each other’s top choice.
We are now ready to prove one of the main results of the Section.
Proposition 3.2 Suppose that the hedonic game satis…es the top coalition
condition. Then the unique stable coalition structure
is the unique stationary perfect equilibrium outcome of the noncooperative game of coalition
formation for all protocols.
Proof. The proof is by induction on the number of players. If n = 1, the
result is obviously true. Suppose that the stable coalition structure is the
unique stationary perfect equilibrium outcome for any k < n, and consider
a hedonic game with n players.
Suppose that there exist two stationary perfect equilibrium outcomes
and 0 with 6= 0 and let be the unique stable coalition structure. Consider
the top coalition T . Clearly, the top coalition is formed along all equilibrium
paths, so that T 2 and T 2 0 . Furthermore, if any player in T is called to
11
make an o¤er, she will propose to form the top coalition T . Because and
0
are two di¤erent equilibrium outcomes, there must exist another coalition
T 0 which forms at the initial state, such that the coalition structure 0 arises
after the formation of T 0 . We now argue that T 0 2 . Because players in T 0
could have proposed to a player in T instead of forming the coalition T 0 we
either have T 0 = S(i; ) or T 0 i S(i; ) for all players i in T 0 . The latter
possibility contradicts the fact that is a stable coalition structure, so that
T 0 2 . Because T; T 0 both belong to and 0 , there must exist two di¤erent
stationary perfect outcomes in the hedonic game with player set N n(T [ T 0 ).
This last statement contradicts our induction hypothesis, and thus completes
the proof of the Theorem.
Thus, Theorem 3.2 identi…es a condition on hedonic games under which
all stationary perfect equilibrium outcomes are stable coalition structures.
This condition is su¢ cient, but by no means necessary.9
4
Coalition Formation in Games with Empty
Cores
When the core of the hedonic game is empty, existence of stationary perfect
equilibrium becomes an issue. The following example, taken from Bloch
(1996), shows that in fact stationary perfect equilibria may fail to exist. (A
similar example was discovered independently by Livshits (2002)).
Example 4.1 A roommate problem with cyclical preferences:
1
2
3
12 23 13
13 12 23
123 123 123
1
2
3
9
One can easily check that in a marriage problem with two men and two women, which
admits two stable matchings (the men optimal and women optimal) and thus does not
satisfy the top coalition property, the only stationary perfect equilibrium outcomes are
the two stable matchings.
12
In this Example, the bargaining game has no stationary perfect equilibrium. There cannot be an equilibrium where f123g is formed: any two-payer
coalition has an incentive to deviate. If the coalition f12g forms in equilibrium, players 2 and 3 have an incentive to deviate (by stationarity player
1 would reject 3’s o¤er to form f13g and hence player 3 would accept the
o¤er f23g). The same argument shows that no two-player coalition can form.
Obviously, the outcome cannot be that all players are singletons.
Example 4.1 can be extended to an arbitrary number of players, by using
the following property.
De…nition 4.1 Consider the roommate problem. A “top ring”is an ordered
subset of agents (x1 ; x2 ; :::; xk ); k 3; such that (subscript modulo k)
xi+1
xi
xi
1
xj for all j 6= i + 1; i
1:
Clearly, the "top ring" property is satis…ed only for very particular preference pro…les. It implies a cycle over each player’s peak. Interestingly enough
the type of “top ring”, odd or even, is critical in determining whether the
core is empty or not.
Proposition 4.1 If the “top ring” is odd, that is, k is odd, then the core of
the game is always empty. If the “top ring”is even then the core of the game
is always non-empty.
Proof. For the …rst part of the proposition observe that in every partition
there is at least one player left alone. Let this player be xi : Now consider
player xi 1 : Whether xi 1 is also alone or paired with someone else, he will
always want to form a blocking pair with xi since xi is his top choice. Similarly, xi prefers xi 1 to being alone as xi 1 is xi ’s second best choice. Hence
for every partition there exists a blocking pair.
For the second part of the proposition consider the partition where half
the players are paired with their top choices: (x1 ; x2 ); :::(xi ; xi+1 ); :::(xk 1 ; xk ):
Note that all the odd players x1 ; :::; xi+1 ; :::; xk 1 are matched with their top
choices, hence they cannot be part of a blocking pair. From the construction
of the ring it follows that all the even players x2 ; :::xi+1 ; :::xk are matched
with their second best choices and they would only form a blocking pair with
their top choice, which is an odd player. Hence no blocking pair can form
and the proposed partition is always in the core.
13
The “top ring” condition is very similar to Chung’s (2000) "ring" property. Chung (2000) considers similar cycles, but does not require that the
cycles operate over the players’peaks. Chung only requires that xi+1 xi xi 1
and does not place any restriction on the ranking of the other alternatives
xj ; j 6= i + 1; i 1: He shows that the absence of “odd rings” implies nonemptiness of the core but not the reverse. We show that if the preference
pro…le has an “odd ring” that moreover operates on the top alternatives,
then the core is empty. However, if preferences have an “odd ring” but an
“even top ring”the core will not be empty.
Our next Proposition shows that in roommate problems with an odd top
ring, stationary perfect equilibria do not exist.
Proposition 4.2 If there exist an odd-size subset of N; S; where players
’preferences pro…le have a “top ring” then the game does not admit a stationary perfect equilibrium.
Proof. Consider the subgame where only the subset S is left unmatched.
Suppose the protocol is following the natural ordering of the players -this
is not an essential assumption as the players are symmetric. Note that no
partition where x2 is matched with someone other than x1 or x3 can be
supported in equilibrium since x1 will bene…t by deviating and proposing to
x2 who will accept. Suppose that (x1 ; x2 ) is supported in equilibrium, this
implies that x3 must be matched in equilibrium with x4 ; else x2 would have
an incentive to reject x1 and propose to x3 who would accept. This argument
can continue until we are left with the following partition:
(x1 ; x2 ); :::(xi ; xi+1 ); :::(xk 2 ; xk 1 ); (xk ):
For such a partition to be supported in equilibrium it must be that xk rejects
xk 1 : For xk to reject xk 1 in equilibrium it must be that x1 accepts xk :
However, since x2 (by stationarity always) accepts x1 it cannot be the case
that x1 accepts xk -he has an incentive to reject and propose to x2 :
Now suppose that (x2 ; x3 ) is supported in equilibrium, than it must be
that x4 is matched with x5 , else x3 would reject x2 and propose to x4 who
would accept. This argument leads us to the following partition:
(xk ; x1 ); (x2 ; x3 ); :::(xi 1 ; xi ); :::(xk 3 ; xk 2 ); (xk 1 ):
Observe that both partitions consist of pairs where one of the partners is
with his top choice whereas the other is with his second best. The only
14
di¤erence is that in the …rst partition the odd players are peaked whereas in
the second the even players are peaked. Thus, a similar argument shows that
the second partition cannot be supported by a stationary perfect equilibrium
either, only now we start with xk 2 : For xk 2 to accept xk 3 it must be that
xk 1 rejects him. For that to happen it must be that xk accepts xk 1 . But
xk , by stationarity accepts x1:
Proposition 4.2 identi…es a class of hedonic games with empty cores for
which stationary perfect equilibria do not exist. This does not imply however
that stationary perfect equilibria always fail to exist when the core is empty.
First, we prove a simple lemma to show that when the core is empty, some
players must make an unacceptable o¤er.
Lemma 4.1 Let A be the set of players who make acceptable o¤ers at some
state A. Then the coalition structure implied by these o¤ers, A ,must belong
to the core of H A
Proof. Suppose to the contrary that there exists a coalition S in A such that
S i S(i; A ) for all i 2 S. Then clearly, any player i in S has a pro…table
deviation – namely to reject S(i; A ) and propose instead the formation of
coalition S:
Lemma 4.1 has an immediate Corollary:
Corollary 4.1 Suppose that the hedonic game H has an empty core. Then,
there is no stationary perfect equilibrium where all players make acceptable
o¤ers at the initial state.
Building on Corollary 4.1, we construct below an example where the
hedonic game has an empty core, but a stationary perfect equilibrium exists.
Naturally, this stationary perfect equilibrium involves some players making
unacceptable o¤ers.
Example 4.2 Suppose n = 6 and players have the following preferences over
admissible coalitions:
15
1
2
3
4
5
6
1245 1245 34
46
35
56
1236 1236 1236 1245 1245 1236
13
2
35
34
56
46
16
36
45
45
36
1
3
4
5
6
We …rst check that the core is empty. Coalition f1245g cannot be part
of a core coalition structure because f46g and f35g have an incentive to deviate. Similarly, f1236g cannot be part of a core coalition structure because
f34gand f56g have an incentive to deviate. The coalition structure cannot
contain f13g and f16g as players 3 and 6 have an incentive to deviate. Hence,
the only candidates for core coalition structures must have f1g and f2g as
elements and the other elements cannot be singletons. But if the structure
is f1j2j34j56g players f1245g have an incentive to deviate; if the structure
is f1j2j35j46g, players f1236g have an incentive to deviate, and if the structure is f1j2j36j45g, both coalitions f1236g and f1245g have an incentive to
deviate.
Furthermore, we can check that any proper subgame has a nonempty
core. (The list of stable coalition structures is given in the Appendix).
Next, consider the following strategies. At the initial stage, 1 proposes
f1g, 2 proposes f2g, 3 proposes f123g, 4 proposes f24g 5 proposes f25g and
6 proposes f123g. Hence players 3; 4; 5 and 6 all make unacceptable o¤ers
at the initial state. At the state where 1 has left the game and the active
set of players is S = f23456g, suppose that the protocol assigns player 2
to make the …rst o¤er, and let 2 propose to form f2g; 3 and 4 propose to
form f34g and 5 and 6 propose to form f56g. At the state where 2 has
left the game and the active set of players is S = f13456g, suppose that
the protocol assigns 3 to make the …rst o¤er. Let 1 propose to form f1g; 3
and 5 propose to form f35g, 4 and 6 propose to form f46g: When the set of
active players is f3456g; 3 and 4 propose to form f34g and 5 and 6 propose
to form f56g: These are the only sets of active players for which there exist
multiple core coalition structures. In any other state and o¤ the equilibrium
path, construct strategies where players agree to form the unique core stable
structure, as in Proposition 3.1.
16
By Proposition 3.1, these strategies form a subgame perfect equilibrium
in any continuation game o¤ the equilibrium path. Now consider the initial
state, where all players are active. First notice that players 3; 4; 5 and 6
obtain their optimal outcome. Players 3 and 6 make an o¤er that player 1
rejects at some equilibrium, so player 1 forms a singleton and players 3 and 6
form coalitions f34g and f56g. Similarly, by making an unacceptable o¤er to
player 2, players 4 and 5 induce the formation of their optimal groups, f46g
and f35g, because player 3 is the …rst proposer after 2 has left. Given this,
players 1 and 2 have no pro…table deviation: any o¤er to form a coalition that
they prefer to remaining as singletons involves the agreement of some player
in f3; 4; 5; 6g and will be rejected by some player. Next consider the state
when 1 has left the game. Players 2, 3 and 6 obtain their optimal outcome.
Players 4 and 5 have no incentive to deviate: if they make an unacceptable
o¤er to 2, they will remain matched to players 3 and 6 respectively after 2
has left the game. When player 2 has left the game, players 4 and 5 obtain
their optimal outcome. Any o¤er by player 1 to form f13g or f16g will be
rejected as players 3 and 6 prefer to remain single than to form a group with
1. Players 3 and 6 have no pro…table deviation either. If they propose to
4 or 5, their o¤er will be rejected and if they propose to 1, their o¤er will
be accepted, inducing a worse outcome than what they get in equilibrium.
Finally, it is clear that when 1 and 2 have left the game, the prescribed
strategies form a subgame perfect equilibrium.
5
Conclusion
In this paper we analyze a bargaining game of coalition formation where the
primitives are derived from a hedonic game, i.e., players’preferences depend
solely on the coalition they belong. We investigate the relationship between
the core of the hedonic game and the set of stationary perfect equilibria. In
the case of non-empty cores we obtain sharp results. If the hedonic game
is totally balanced then every core coalition structure can be supported by
stationary perfect equilibrium, while the reverse is not true. Moreover, we
establish that when the game satis…es the “top coalition” property, not only
does the core contain a unique element, but this element is supported by
the unique stationary perfect equilibrium. If the core is empty, matters are
more complicated. We are able to show that if a stationary equilibrium exists,
then it occurs with delay. Lastly, we identify the “odd top ring” condition
17
that guarantees both an empty core and non-existence of stationary perfect
equilibria.
Of course, we are aware that our results critically depend on the speci…c
extensive form that we have studied. While we believe that our courtship
game –a simple extension of Rubinstein (1982)’s alternative o¤ers protocol
–is a natural procedure to study, we would like to test the robustness of our
results to changes in the extensive form in future research. For instance, our
main counterexample, showing that unstable marriages can result in equilibrium, depends on the fact that players are forced to make o¤ers, and cannot
choose to pass their turn and let other players make o¤ers. The protocol also
plays a major role in this counterexample, as unstable marriages will only
result for some speci…c ordering of the players. This suggests that more work
is needed to identify precisely which elements of the bargaining procedure
are crucial to our results.
6
Appendix
Stable coalition structures for strict subsets of N in Example 4.2
If n = 5, the following are core structures:
12345
12346
12356
12456
13456
23456
1j2j34j5
1j2j3j46
1j2j35j6
1j2j4j56
1j34j56 and 1j35j46
2j34j56 and 2j35j46:
For n = 4; if S = f1245g or S = f1236g, then these are the core structures. Otherwise, if 1 or 2 belong to S, they remain as singletons in the core
structure. The following are core coalition structures:
1345
1346
1356
1456
and 2345
and 2346
and 2356
and 2456
12xy
3456
1j34j5 and 2j34j5
1j3j46 and 2j3j46
1j35j6 and 2j35j6
1j4j56 and 2j4j56
1j2jxy
34j56 or 35j46:
For n = 3, if 1 or 2 belong to S, they remain as singletons, and the other
two players form a pair. (If S contains both 1 and 2, the last player remains
18
as a singleton). If S only contains players in f3; 4; 5; 6g, the following are
core structures:
345 34j5
346 3j46
356 35j6
456 4j56
If n = 2, either S contains a pair in f3; 4; 5; 6g and this pair is a core
structure, or otherwise, the core structure is given by two singletons.
7
References
1. Alcalde, J. (1995) “Exchange-proofness or divorce-proofness? Stability in one-sided matching markets?," Economic Design 1, 275-287.
2. Alcalde, J. (1996) “Implementation of Stable Solutions to Marriage
Problems," Journal of Economic Theory 69, 240-254.
3. Alcalde, J. Perez-Castrillo, D. & Romero Medina A. (1998)
"Hiring Procedures to Implement Stable Allocations" Journal of Economic Theory 82, 469-480.
4. Alcalde, and Romero Medina A. (2000) "Simple Mechanisms
to Implement the Core of College Admissions Problems" Games and
Economic Behavior 31, 294-302..
5. Alcalde, and Romero Medina A. (2005) "Sequential Decisions in
the College Admissions Problem" Economics Letters 86, 153-158.
6. Banerjee, S., Konishi, H. & Sonmez, T. (2001) “Core in a Simple
Coalition Formation Game,”Social Choice and Welfare 18, 135-153.
7. Bloch, F. (1996) “Sequential Formation of Coalitions in Games with
Externalities and Fixed Payo¤ Division,”Games and Economic Behavior 14, 90-123.
8. Bogomolnaia, A. & Jackson M.O. (2002) “The Stability of Hedonic Coalition Structures,”Games and Economic Behavior 38, 201-230.
19
9. Chaterjee, K., Dutta, B., Ray, D., and Sengupta, K. (1993)
“A Noncooperative Theory of Coalitional Bargaining,”Review of Economic Studies 60, 463-477.
10. Chung, K.-S. (2000) “On the Existence of Stable Roommate Matchings,”Games and Economic Behavior 33, 206-230.
11. Diamantoudi, E. & Xue, L. (2003) “Farsighted Stability in Hedonic
Games,”Social Choice and Welfare 21, 39-61.
12. Farrell, J. and S. Scotchmer (1988) “Partnerships”, Quarterly
Journal of Economics,103, 279-297.
13. Gale, D. and Shapley, L. (1962) “College Admissions and the Stability of Marriage”American Mathematics Monthly 69, 9-15.
14. Kamecke, U. (1989) "Non-cooperative Matching Games", International Journal of Game Theory 18, 423-431.
15. Livshits, I. (2002) "On Non-existence of Pure Strategy Markov Perfect Equilibrium", Economics Letters 76, 393-96.
16. Maskin, E. (2003) “Bargaining, Coalitions and Externalities”, presidential address to the Econometric Society.
17. Okada, A. (1996) “A Noncooperative Coalitional Bargaining Game
with Random Proposers,”Games and Economic Behavior 16, 97-108.
18. Papai, S. (2004) “Unique Stability in Simple Coalition Formation
Games,”Games and Economic Behavior 48, 337-354.
19. Perez-Castrillo, D. and M. Sotomayor (2002) "A Simple Selling and Buying Procedure" Journal of Economic Theory 103, 461-74
20. Ray, D. and Vohra, R. (1999) “A Theory of Endogenous Coalition
Structures,”Games and Economic Behavior 26, 286-336.
21. Roth, A. and Sotomayor, M. (1990) “Two-Sided Matching: A
Study in Game Theoretic Modelling and Analysis”Cambridge University Press.
20
22. Rubinstein, A. (1982) “Perfect Equilibrium in A Bargaining Model,”
Econometrica 50, 97-109.
23. Seidmann, D, and Winter, E. (1998) “A Theory of Gradual Coalition Formation”Review of Economic Studies 65, 793-815.
24. Sutton, J. (1986) “Noncooperative Bargaining Theory: An Introduction,”Review of Economic Studies 53, 709-724.
21