We are concerned with the problem of core membership testing for hedonic coalition formation games, which is to decide whether a certain coalition structure belongs to the core of a given game. We show that this problem is co-NP complete when players' preferences are additive.additivity, coalition formation, core, co-NP completeness, hedonic games

Dinko Dimitrov

Shao Chin Sung

English

Research Papers in Economics

Institute of
Mathematical
Economics
Working Papers
374
November 2005
On Core Membership Testing for Hedonic
Coalition Formation Games
Shao Chin Sung and Dinko Dimitrov
IMW  Bielefeld University
Postfach 100131
33501 Bielefeld  Germany
email: imw@wiwi.uni-bielefeld.de
http://www.wiwi.uni-bielefeld.de/~imw/Papers/showpaper.php?374
ISSN: 0931-6558On core membership testing for
hedonic coalition formation games
Shao Chin Sung￿
Department of Industrial and Systems Engineering
Aoyama Gakuin University
5-10-1 Fuchinobe, Sagamihara City, Kanagawa, 229-8551, Japan
Email: son@ise.aoyama.ac.jp
Dinko Dimitrov
Institute of Mathematical Economics
Bielefeld University
P.O. Box 100131, 33501 Bielefeld, Germany
Email: d.dimitrov@wiwi.uni-bielefeld.de
November 21, 2005
Abstract
We are concerned with the problem of core membership testing
for hedonic coalition formation games, which is to decide whether a
certain coalition structure belongs to the core of a given game. We
show that this problem is co-NP complete when players￿preferences
are additive.
JEL Classi￿cation: C71, C63.
Keywords: additivity, coalition formation, core, co-NP completeness,
hedonic games.
￿Corresponding author.
11 Introduction
The study of computational complexity in hedonic coalition formation games,
or simply hedonic games, has a short history, although these issues in coopera-
tive and non-cooperative game theory are being gradually recognized. Maybe
the reason is that the formal model of a hedonic game was only recently in-
troduced (cf. Banerjee, Konishi, and S￿nmez (2001) and Bogomolnaia and
Jackson (2002)). This model consist of a ￿nite set of players and a preference
relation for each player de￿ned over the set of all coalitions containing the
corresponding player. The outcome of a hedonic game is a coalition structure
(i.e., a partition of the set of players into coalitions). A coalition structure
is called stable if there is no group of individuals who can all be better o⁄
by forming a new deviating coalition. The core of a hedonic game is the
collection of all stable coalition structures.
Computational complexity issues related to hedonic games in a general
setting are studied by Ballester (2004). As shown by this author, the prob-
lem to decide whether a given hedonic game has a nonempty core is NP-
complete1. CechlarovÆ and HajdukovÆ (2002, 2004) and Dimitrov, Borm,
Hendrickx, and Sung (2004) also elaborate on the computational complexity
of core related solution concepts for hedonic games but in a less general set-
ting, i.e., in games with some restrictions imposed on players￿preferences. In
particular, Dimitrov, Borm, Hendrickx, and Sung (2004) consider preference
pro￿les based on aversion to enemies that consitute a small subdomain of
the domain of additive preferences, and show that ￿nding a core member
for such games is NP-complete. The corresponding preference domains are
formally introduced in the next section.
1For an introduction to computational complexity, de￿nitions of NP, NP-complete,
NP-hard, and a catalog of NP-complete problems, we refer to Garey and Johnson (1979).
2In this note we consider the problem of core membership testing for he-
donic games. Given a hedonic game and a coalition structure, the problem of
core membership testing is to decide whether the coalition structure belongs
to the core of the game. We show that this problem is co-NP complete when
players￿preferences are additive. Indeed, the co-NP completeness is shown
by a reduction to hedonic games in which players￿preferences are based on
aversion to enemies. Hence, the preference domain based on aversion to en-
emies turns out to have a referential role with respect to this computational
complexity issue.
2 Preliminaries
Let N = f1;:::;ng be a ￿nite set of players. A coalition is a nonempty
subset of N. For each player i 2 N, we denote by Ai = fX ￿ N j i 2 Xg
the collection of all coalitions containing i. A collection ￿ of coalitions is
called a coalition structure if ￿ is a partition of N, i.e., all coalitions in ￿
are pairwise disjoint and
S
X2￿ X = N. We denote by CN the collection
of all coalition structures. For each coalition structure ￿ 2 CN and each
player i 2 N, we denote by ￿(i) the coalition in ￿ which contains i, i.e.,
￿(i) 2 ￿ \ Ai.
We assume that each player i 2 N is endowed with a preference ￿i
over Ai, i.e., a binary relation over Ai which is re￿ exive, complete, and
transitive. Moreover, we assume that the preference of each player i 2 N
over CN is purely hedonic, i.e., it is completely characterized by ￿i in such a
way that, for each ￿;￿0 2 CN, player i weakly prefers ￿ to ￿0 if and only if
￿(i) ￿i ￿0(i).
A hedonic game is a pair hN;￿i of a ￿nite set N of players and a pro￿le
￿= (￿1;:::;￿n) of players￿preferences. We denote by G the collection of
3all hedonic games. Let hN;￿i 2 G. We say that a coalition X is a deviation
from a coalition structure ￿ in hN;￿i if X ￿i ￿(i) for each i 2 X. We say
that a coalition structure ￿ is stable in hN;￿i if no deviation from ￿ exists,
i.e., for each coalition X, there exists i 2 X satisfying ￿(i) ￿i X. The core of
a hedonic game hN;￿i, denoted by ￿(N;￿), is the collection of all coalition
structures which are stable in hN;￿i.
Let ￿= (￿1;:::;￿n) be a preference pro￿le. We say that ￿ is additive
if, for each i 2 N, there exists a function vi : N ! R characterizing ￿i in
such a way that, for all X;Y 2 Ai,
￿ X ￿i Y if and only if
P
j2X vi(j) ￿
P
j2Y vi(j).
For simplicity, by vi(X) we denote
P
j2X vi(j) for each i 2 N and for
each X 2 Ai.
Given an additive preference pro￿le ￿ and any two players i;j 2 N, we
say that j is a friend (enemy) of i if and only if vi(j) > 0 (vi(j) < 0); if
vi(j) = 0, and we say that j is a neutral coalitional partner of i. Finally, we
say that a preference pro￿le ￿ is based on aversion to enemies if ￿ is additive,
and for each i 2 N, vi(￿) 2 f￿n;1g with vi(i) = 1. Hence, restricting players￿
preferences in such a way displays a situation in which each player i 2 N has
very strong enemies, very weak friends, and no neutral coalitional partners.
3 Core membership testing
In this section we study the problem of core membership testing formulated
as follows:
The Problem of Core Membership Testing (cmt)
Given: A hedonic game hN;￿i 2 G and a coalition structure ￿ 2 CN.
4Question: Is ￿ 2 ￿(N;￿)?
This problem belongs to the complexity class co-NP, i.e., the complexity
class containing the complements of the decision problems in the complexity
class NP. The complement of a decision problem is de￿ned as the problem
with the ￿YES￿and ￿NO￿answer reversed. The complement of the cmt
problem can then be described as follows. Given a hedonic game hN;￿i 2 G
and a coalition structure ￿ 2 CN, and ask whether ￿ 62 ￿(N;￿), i.e., whether
there is a deviation X from ￿ in hN;￿i. This problem, the complement of
cmt, belongs to NP, because in polynomial time of n one can
(1) guess non-deterministically a coalition X,
(2) test deterministically whether X is a deviation from ￿ in hN;￿i, and
(3) the answer is ￿YES￿if some coalition X is a deviation from ￿ in hN;￿i,
and otherwise ￿NO￿ .
Hence, the cmt problem belongs to co-NP.
Before we show that this problem is co-NP complete when players￿pref-
erence are additive, let us ￿rst recall some properties of hedonic games with
preference pro￿les based on aversion to enemies. For more details the reader
is referred to Dimitrov, Borm, Hendrickx, and Sung (2004). Let hN;￿i 2 G
be a hedonic game with preference pro￿le ￿ based on aversion to enemies.
It is known that
￿ the core ￿(N;￿) is always nonempty.
In order to describe the properties of core members, let us introduce some
terminology. Let H = (V;E) be a (undirected) graph, where V is the set
of vertices and E is the set of edges, i.e., each edge is a set consisting of
5two di⁄erent vertices from V . A clique X in H is a subset of V such that
fi;jg 2 E for each i;j 2 X with i 6= j.
Let HhN;￿i = (V;E) be a (undirected) graph with
￿ V = N, and
￿ E = ffi;jg ￿ V j i 6= j and vi(j) = vj(i) = 1g,
and let ￿ 2 CN be a core member, i.e., ￿ 2 ￿(N;￿). Then, it is known
that
￿ each X 2 ￿ is a clique in HhN;￿i.
Suppose X is not a clique in HhN;￿i for some X 2 ￿. Then, vi(j) = ￿n
for some i;j 2 X, which implies that vi(X) < vi(i). Hence, X ￿i fig, and
thus, fig is a deviation from ￿ in hN;￿i. Therefore ￿ 62 ￿(N;￿).
Moreover, it is known that
￿ at least one of the largest cliques in HhN;￿i belongs to ￿.
Suppose ￿ does not contain any of the largest cliques in HhN;￿i, and let
X be one of the largest cliques in HhN;￿i. Then, for each i 2 X, we have
vi(X) = jXj > j￿(i)j ￿ vi(￿(i)):
Hence, X is a deviation from ￿ in hN;￿i, i.e., ￿ = 2 ￿(N;￿). It follows from
this property that the problem of ￿nding a core member of a given hedonic
game, with preference pro￿le based on aversion to enemies, is at least as
hard as the problem for ￿nding a largest clique in a given graph, which is a
NP-hard optimization problem.
We are now ready to present our result.
6Theorem 1 The problem of core membership testing for hedonic games with
additive preference pro￿le is co-NP complete.
Proof. As already mentioned, the cmt problem belongs to co-NP. It su¢ ces
to show that this problem is co-NP hard. The co-NP hardness is shown
by a polynomial time reduction from a co-NP complete problem called the
clique problem, which is the complement of the clique problem. The
clique problem is de￿ned as follows:
Clique Problem (clique)
Given: A graph G = (V;E) and a positive integer 2 ￿ K ￿ jV j.
Question: Does G contain a clique of size K?
Let (G;K) be an instance of the clique problem, i.e., G = (V;E) is
a graph and K is a positive integer such that 2 ￿ K ￿ jV j. De￿ne a
hedonic game hN;￿i as follows. Take N = f1;2;:::;K ￿ 1g ￿ V to be the
set of players, and let n = jNj = (K ￿ 1)jV j. For each (k;s) 2 N, the
preference ￿(k;s) of player (k;s) is characterized by the function v(k;s), which
is de￿ned as follows: For each (‘;t) 2 N,
v(k;s)(‘;t) =
8
<
:
1 if s = t;
1 if k = ‘ and fs;tg 2 E;
￿n otherwise.
Observe that the transformation from (G;K) to the game hN;￿i can be
done in O(jV j4) time. Hence, it is a polynomial time reduction. Moreover,
observe that players￿preferences are, in fact, based on aversion to enemies.
Next, de￿ne ￿ = fXs j s 2 V g with Xs = f1;2;:::;K ￿1g￿fsg for each
s 2 V . Obviously ￿ is a partition of N, i.e., ￿ is a coalition structure. Also
notice that each Xs is a clique of size K ￿1 in HhN;￿i, because v(k;s)(‘;s) = 1
7for all k;‘ 2 f1;2;:::;K ￿ 1g and for each s 2 V . In the following we show
that ￿ 62 ￿(N;￿) if and only if G contains a clique of size K.
(1) Suppose that G contains a clique Y of size K. Then, for each k 2
f1;2;:::;K ￿ 1g, fkg ￿ Y is also a clique of size K in HhN;￿i. Hence, each
fkg ￿ Y is a deviation from ￿, and therefore ￿ = 2 ￿(N;￿).
(2) Suppose ￿ = 2 ￿(N;￿). Then there exists a deviation Z from ￿. Since
each Xs 2 ￿ is a clique of size K￿1 in HhN;￿i, Z must also be a clique of size
at least K in HhN;￿i in order to be a deviation from ￿. Let Z0 be a subset
of Z of size K. Since Z is a clique in HhN;￿i, Z0 is also a clique in HhN;￿i.
Then, by de￿nition, we have either k = k0 or s = s0 for all (k;s);(k0;s0) 2 Z0
with (k;s) 6= (k0;s0).
We show that k = k0 and s 6= s0 for all (k;s);(k0;s0) 2 Z0 with (k;s) 6=
(k0;s0), i.e., Z0 ￿ fkg ￿ V for some k 2 f1;2;:::;K ￿ 1g. Let (k;s) 2 Z0.
Then, there exists (k0;s0) 2 Z0 such that s0 6= s, because jZ0j = K > K ￿1 =
jXsj, and thus, we have k0 = k. We are done when K = 2. When K ￿ 3,
there exists (k00;s00) 2 Z0 such that (k00;s00) 6= (k;s) and (k00;s00) 6= (k0;s0).
When s00 = s, we have k00 6= k, and thus, we have k00 6= k = k0 and s00 = s 6= s0,
so that there is no edge between (k0;s0) and (k00;s00) in HhN;￿i, and Z0 cannot
be a clique in HhN;￿i. The same argument hold when s00 = s0. Hence, k = k0
and s 6= s0 for all (k;s);(k0;s0) 2 Z0 with (k;s) 6= (k0;s0). i.e., Z0 ￿ fkg ￿ V
for some k 2 f1;2;:::;K ￿ 1g.
Finally, since Z0 is a clique of size K in HhN;￿i and Z0 ￿ fkg ￿ V for
some k 2 f1;2;:::;K ￿ 1g, we have fs;tg 2 E for each (k;s);(k;t) 2 Z0.
Therefore, fs 2 V j (k;s) 2 Z0g is a clique of size K in G.
8References
[1] Ballester, C. (2004): NP-completeness in hedonic games, Games and
Economic Behavior 49, 1-30.
[2] Banerjee, S., H. Konishi, and T. S￿nmez (2001): Core in a simple coalition
formation game, Social Choice and Welfare 18, 135-153.
[3] Bogomolnaia, A. and M. O. Jackson (2002): The stability of hedonic
coalition structures, Games and Economic Behavior 38, 201-230.
[4] CechlarovÆ, K. and J. HajdukovÆ (2004): Stable partitions with W-
preferences, Discrete Applied Mathematics 138, 333-347.
[5] CechlarovÆ, K. and J. HajdukovÆ (2002): Computational complexity of
stable partitions with B-preferences, International Journal of Game The-
ory 31, 353-364.
[6] Dimitrov, D., P. Borm, R. Hendrickx, and S.-C. Sung (2004): Simple
priorities and core stability in hedonic games, Social Choice and Welfare,
forthcoming.
[7] Garey, M. and D. Johnson (1979): Computers and Intractability. A Guide
to the Theory of NP-Completeness, Freeman, New York.
9

On core membership testing for hedonic coalition formation games

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On core membership testing for hedonic coalition formation games

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