ABSTRACT In this paper, we study the computational complexity of solution concepts in the context of coalitional games. Firstly, we distinguish two different kinds of core, the undominated core and excess core, and investigate the difference and relationship between them. Secondly, we thoroughly investigate the computational complexity of undominated core and three farsighted solution concepts-farsighted core, farsighted stable set and largest consistent set

Chongjun Wang

Jun Wu

Junyuan Xie

Meilin Liu

Yusen Zhan

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On the Complexity of Undominated Core and Farsighted
Solution Concepts in Coalitional Games
(Extended Abstract)
Yusen Zhan1,2, Jun Wu1,3
∗
, Chongjun Wang1,2, Meilin Liu4,5, Junyuan Xie1,2
1State Key Laboratory for Novel Software Technology at Nanjing University, Nanjing 210023, China
2Department of Computer Science and Technology, Nanjing University,Nanjing 210023, China
3School of Computer and Information Engineering, Hohai University, Nanjing 210098, China
4Department of Computer Science and Engineering, Wright State University, Dayton, OH 45435, USA
yusen.zhan@iip.nju.edu.cn, {iip,chjwang,jyxie}@nju.edu.cn, 5meilin.liu@wright.edu
ABSTRACT
In this paper, we study the computational complexity of
solution concepts in the context of coalitional games. First-
ly, we distinguish two different kinds of core, the undom-
inated core and excess core, and investigate the difference
and relationship between them. Secondly, we thoroughly
investigate the computational complexity of undominated
core and three farsighted solution concepts—farsighted core,
farsighted stable set and largest consistent set.
Categories and Subject Descriptors
F.2 [Theory of Computation]: Analysis of Algorithms
and Problem Complexity; I.2.11 [Distributed Artificial
Intelligence]: Multiagent systems; J.4 [Computer Ap-
plications]: Social and Behavioral Sciences - Economics
General Terms
Economics, Theory
Keywords
Game theory (cooperative and non-cooperative), teamwork,
coalition formation, coordination, computational complexity
1. FARSIGHTED COALITIONAL GAMES
Recent years, coalitional games or cooperative games are
more and more influential among multiagent system commu-
nity due to their ability to capture the cooperative behaviors
among players or agents. A coalitional game is defined by
a set of players N = {1, . . . , n} and a characteristic func-
tion v : 2N → R where v(∅) = 0 and an imputation for
a coalitional game G is an n-vector x ∈ Rn satisfying (a)∑
i∈N xi = v(N), (b) xi ≥ v({i}) for all i ∈ N . And I(G)
denotes the set of imputations of game G. A fundamental
problem for coalitional games is to characterize certain sub-
sets of imputations, which are payoff distributions among
∗Jun Wu is the corresponding author.
Appears in: Proceedings of the 12th International Confer-
ence on Autonomous Agents and Multiagent Systems (AA-
MAS 2013), Ito, Jonker, Gini, and Shehory (eds.), May, 6– 10, 2013,
Saint Paul, Minnesota, USA.
Copyright c© 2013, International Foundation for Autonomous Agents and
Multiagent Systems (www.ifaamas.org). All rights reserved.
players, in terms of several interesting properties. Tradi-
tionally, these concepts have been studied in economics and
mathematics to shed light on the explanation of various e-
conomical intuition of fairness and stability, and also to pro-
vide a strict and profound mathematical analysis. The well-
known solution concepts include the Shapley value, the core,
the stable set and so on.
However, the (undominated) core and the stable set have
been criticized by Harsanyi[5] and Chwe[2]. They pointed
out that these two solution concepts are too myopic to cap-
ture the farsight behaviors of players or coalitions in real
economical environment. To overcome the myopia in the
core and the stable set, Harsanyi proposed a model with a
chairman to control the coalition formation process. Chwe
proposed a similar model and formalized a binary relation
called indirect dominance, which is slightly different from
the one discussed by Harsanyi, to capture the farsightedness
of coalitions. When indirect dominance is incorporated into
the undominated core and stable set, we get new solution
concepts. These new solution concepts are called the far-
sighted core and (vN-M) farsighted stable set, respectively.
Furthermore, Chwe defined another solution concept–largest
consistent set to conclude more outcomes. Béal et al.[1]
systematically studied these solution concepts in coalitional
games. Their results lay the foundation of our work.
2. THE CORES
There are two kinds of cores in the literature. The undom-
inated core, is defined by (direct) dominance( for x, y ∈ I(G).
And let S ⊆ N be a coalition, x directly dominates y via S,
denoted as y ≺S x, if (a) x(S) ≤ v(S), (b) yi < xi for all
i ∈ S), was originally introduced by Gillies[4]. The undom-
inated core can be expressed as {x ∈ I(G)|∀S ⊆ N, y ∈
I(G) s.t. x ≺S y}. However, the excess core, which is de-
fined by excess, i.e., e(S, x) = v(S) − x(S), is the common
one. Also it can be characterized by {x ∈ I(G)|e(S, x) =
v(S)− x(S) ≤ 0, ∀S ⊆ N}. We illustrate the difference and
relationship between the undominated core and the excess
core. Owen[6] proved that if the coalitional game is superad-
ditive, then the undominated core coincides with the excess
core. We show that in the general setting, the undominated
core is strictly larger than the excess one. Figure 1 summa-
rizes our major results.
1177
Excess Core 
Undominated Core 
All Imputations 
Figure 1: The undominated core and excess core
ing machine, we prove a necessary and sufficient condition
to determine that whether an imputation is not in the un-
dominated core in the full version of the paper. With this
condition, we could explicitly exam the stability of an im-
putation without considering the complicated direct domi-
nance relation.
3. FARSIGHTED SOLUTION CONCEPTS
Chwe[2] extended the direct dominance into undirect dom-
inance. For x, y ∈ I(G), let S ⊆ N be a coalition, we say
that x indirectly dominates y, denoted as y  x, if there
exist a finite sequence of imputations y = x1, . . . , xm = x
and a finite sequence of coalitions S1, . . . , Sm−1 such that
(a)xj+1(Sj) ≤ v(Sj), (b)xji < xi for all i ∈ Sj , where
j = 1, . . . ,m − 1. Based on this binary relation, the far-
sighted core can be defined as {x ∈ I(G)|∀S ⊆ N, y ∈
I(G) s.t. x  y}, the farsighted stable sets can be charac-
terized by {K ⊆ I(G)|∀x, y ∈ K, neither x  y nor y 
x,∀y ∈ I(G) \K, x ∈ K s.t. y  x}.
Furthermore, Chwe[2] considered the farsighted stable set
is too exclusive, since it may exclude some outcomes that
may not be consistent, but rational and foresight. To over-
came this, Chew suggested a new solution concept to capture
more inclusive farsighted stability: the consistent set.
Definition 1. A set K ⊆ I(G) is consistent if:
1. ∀x ∈ K, ∀z ∈ I(G) and ∀S ∈ 2N \ ∅ such that z(S) ≤
v(S), there exists y ∈ K, where y = z or z  y, such
that x ⊀S y;
2. ∀x ∈ I(G) \ K, these exist z ∈ I(G), S ∈ 2N \ ∅ with
z(S) ≤ v(S), such that ∀y ∈ K, where y = z or z  y,
and it holds that xi < yi for all i ∈ S.
Although the consistent set may not be unique, Chew([2])
proved that there exists a unique largest consistent set, i.e.,
it contains all the tother consistent sets. Both Chew[2] and
Béal[1] showed that the largest consistent set is not empty.
4. COMPUTATIONAL MODEL
To study the computational complexity of solution con-
cepts, we adopt the oracle setting to represent the coali-
tional games. In this setting, we consider the characteristic
function v as an oracle, i.e., given a coalition, it outputs the
Table 1: Summary of resultsSolution Concept
Problem
Membership Emptiness
Undominated Core co-NP-c NP-hard1
Farsighted Core co-NP-c NP-c
Farsighted Stable Set DP-c Nonempty
Largest Consistent Set Undecidable Nonempty
1The emptiness problem for games, whose corresponding
(0,1)-reduced games v(S) ≤ 1, is NP-complete. For the
general games, we conjecture that it is in ΣP2 .
value in polynomial time w.r.t. the size of the game repre-
sentation. Moreover, we assume that players are included in
the game representation, i.e., the size of the representation
of a coalition S is less or equal to the size of the whole game
representation.
5. COMPUTATIONAL RESULTS
Finally, based on the work of Béal et al.[1] and oracle
representation scheme in coalitional games, we thoroughly
investigate the computational issues of undominated core,
farsighted core, (vN-M) farsighted stable set and largest con-
sistent set. Moreover, in the literature, researchers mostly
discuss the following two problems[3]: (1) Given a game and
an imputation, deciding whether the imputation belongs to a
certain solution concept; (2) Given a game, deciding whether
a specific solution concept is empty. We call the first and
second the membership problem and emptiness problem, re-
spectively. We study the membership problems of the un-
dominated core, farsighted core, farsighted stable sets and
largest consistent set. For the emptiness problems, we only
investigate the undominated core and farsighted core since
the farsighted stable sets and largest consistent set are never
empty in general coalitional games[1]. Table 1 presented the
summary of our major results in this paper.
6. ACKNOWLEDGMENTS
This paper is supported by the National Program on Key
Basic Research Project (973 Program)(Grant No. CB505300),
the National Natural Science Foundation of China (Grant
No. 61021062, 61105069), the Science and Technology Sup-
port Foundation of Jiangsu Province(Grant No. BE2011171,
BE2012161), and the Scientific Research Foundation of Grad-
uate School of Nanjing University(Grant No. 2011CL07).
7. REFERENCES
[1] S. Béal, J. Durieu, and P. Solal. Farsighted coalitional
stability in tu-games. Mathematical Social Sciences,
56(3):303–313, 2008.
[2] M. Chwe. Farsighted coalitional stability. Journal of
Economic Theory, 63(2):299–325, 1994.
[3] X. Deng and C. Papadimitriou. On the complexity of
cooperative solution concepts. Mathematics of
Operations Research, 19(2):257–266, 1994.
[4] D. Gillies. Solutions to general non-zero-sum games.
Contributions to the Theory of Games, 4:47–85, 1959.
[5] J. Harsanyi. An equilibrium-point interpretation of
stable sets and a proposed alternative definition.
Management science, 20(11):1472–1495, 1974.
[6] G. Owen. Game theory, 3rd. Academic Press, 1995.
Moreover, to provide an polynomial certificate for the Tur-
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On the complexity of undominated core and farsighted solution concepts in coalition games

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On the complexity of undominated core and farsighted solution concepts in coalition games

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