Journal of Mathematical Economics 47 (2011) 621–626
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Journal of Mathematical Economics
journal homepage: www.elsevier.com/locate/jmateco
Set-valued solution concepts using interval-type payoffs for interval games
S.Z. Alparslan Gök a,∗ , O. Branzei b , R. Branzei c , S. Tijs d
a
Süleyman Demirel University, Faculty of Arts and Sciences, Department of Mathematics, 32 260 Isparta, Turkey
b
Richard Yvey School of Business, University of Western Ontario, N6A 3K7, London Ontario, Canada
c
Faculty of Computer Science, ‘‘Alexandru Ioan Cuza’’ University, 700483, Iaşi, Romania
d
CentER and Department of Econometrics and OR, Tilburg University, 90153, The Netherlands
article
info
Article history:
Received 1 March 2010
Received in revised form
24 July 2011
Accepted 29 August 2011
Available online 6 September 2011
Keywords:
Cooperative games
Interval games
The core
The dominance core
Stable sets
abstract
Uncertainty is a daily presence in the real world. It affects our decision making and may have influence
on cooperation. Often uncertainty is so severe that we can only predict some upper and lower bounds
for the outcome of our actions, i.e., payoffs lie in some intervals. A suitable game theoretic model to
support decision making in collaborative situations with interval data is that of cooperative interval
games. Solution concepts that associate with each cooperative interval game sets of interval allocations
with appealing properties provide a natural way to capture the uncertainty of coalition values into the
players’ payoffs. In this paper, some set-valued solution concepts using interval payoffs, namely the
interval core, the interval dominance core and the interval stable sets for cooperative interval games, are
introduced and studied. The main results contained in the paper are a necessary and sufficient condition
for the non-emptiness of the interval core of a cooperative interval game and the relations between the
interval core, the interval dominance core and the interval stable sets of such a game.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
Set-valued solution concepts like cores and stable sets are
widely used within the framework of games in characteristic
function form, i.e., games described by a map v that attaches
to each nonempty subset S of players a real value v(S ). The
characteristic function v is deterministic; there is no uncertainty
involved. However, uncertainty affects our decision making
activities on a daily basis. To incorporate uncertainty in cooperative
game theory is motivated by the need to handle uncertain
outcomes in collaborative situations. There are many sources of
uncertainty in the real world. We refer here to technological
and market uncertainty, noise in observation and experimental
design, incomplete information and vagueness in decision making.
On many occasions uncertainty is so severe that we can only
predict some upper and lower bounds for the outcome of our
actions, i.e., payoffs lie in some intervals. A suitable game theoretic
model to support decision making in collaborative situations with
interval data is that of cooperative interval games. In this model,
interval uncertainty affects coalition values, i.e., for each nonempty
coalition S the realized value belongs to an interval of real numbers
∗
Corresponding author.
E-mail addresses: [email protected] (S.Z. Alparslan Gök),
[email protected] (O. Branzei), [email protected] (R. Branzei),
[email protected] (S. Tijs).
0304-4068/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmateco.2011.08.008
instead of being sharply defined. The theory of cooperative interval
games has been recently born (Branzei et al., 2003). Next we supply
both empirical and theoretical background and motivation for the
study of cooperative interval games.
There are a handful of promising theoretically and practicallyrelevant insights for dealing with interval uncertainty in organizational phenomena. Gray (2000) and Loveridge (2000) proposed
network-based collaboration as a solution to interval uncertainty
dilemmas: organizations can deliberately mitigate uncertainty—
in part by disaggregating uncertain costs and rewards into subdomains where lower and upper boundaries can be more precisely
forecasted. The larger issue of whether and why individuals and
organizations choose to cooperate (or not) when faced with interval uncertainty data on outcomes or costs has generated a productive line of research in recent years. This question is particularly
timely and relevant for explaining the growing reliance on cooperation for bringing about radical innovation. For radical innovation
there is robust theoretical and empirical evidence that cooperation
can yield superior returns to innovation by organizations alone—
in homophilic (Ahuja et al., 2009) as well as heterogeneous constellations (Branzei, 2005). This makes the model of cooperative
interval games promising for solving sharing problems related to
radical innovation. Lavie et al. (2007) explain that partners join in
voluntarily as they face technological and market uncertainty. They
examine the distribution of benefits to partners in multipartner
alliances by concentrating on the dynamics of partner entry and
involvement (rather than the post-factum benefits which each
622
S.Z. Alparslan Gök et al. / Journal of Mathematical Economics 47 (2011) 621–626
partner may eventually attain). Their evaluation of how partners
come to these benefits during the course of the alliance suggests
the relevance of modelling interval-type uncertainty.
Interval uncertainty is the simplest and the most natural type
of uncertainty which may influence cooperation because lower
and upper bounds for future outcomes or costs of cooperation
can always be estimated based on available economic data. Differently, stochastic uncertainty and fuzzy uncertainty, which have already been considered within cooperative game theory, make use
of more sophisticated information which can be difficult to obtain or to argue. Basic models of cooperative games which consider
stochastic uncertainty are cooperative games in stochastic characteristic function (Granot, 1977; Suijs et al., 1999) and cooperative
games with random payoffs (Timmer, 2001; Timmer et al., 2005).
Recently a general game-theoretic model – the model of partially
ordered cooperative games (Puerto et al., 2008) – has been introduced which allows the payoff of any coalition to be an element
of any partially ordered linear space. This model catches as particular instances cooperative stochastic games, games with random
payoffs, cooperative interval games, and also cooperative vectorvalued games1 (Fernández et al., 2002). For partially ordered cooperative games the extended core, the nondominated core and
the Shapley value are introduced and studied. These solution concepts associate with each partially ordered cooperative game particular payoff vectors whose components belong to the same linear
space as the coalition values. Set-valued solutions for cooperative
interval games introduced and studied in our paper are in the same
spirit, i.e., the components of the payoff vectors generated by them
are intervals.
This paper considers the model of cooperative interval games as
a distinct one within cooperative game theory. To construct such a
game one observes a lower and an upper bound of the considered
coalitions or, more generally, the characteristic function intervalvalues may result from solving general optimization problems.
This is very important, for example from a computational and
algorithmic viewpoint. We extend the results on cooperative
interval games in Branzei et al. (2003) and Alparslan Gök et al.
(2009b) to specify the interval core, the interval dominance core
and the interval stable sets, which capture the interval-uncertainty
of coalition values within players’ payoffs. Interval payoff vectors
obtained by using interval solutions inform each player about what
he/she/it2 might expect to receive – between two bounds – as a
result of cooperation within the grand coalition. This information
helps individuals or organizations to choose to cooperate (or not)
when faced with interval uncertainty data on outcomes, and is also
useful to determine sharp payoffs for the cooperating players when
the uncertainty on the worth of the grand coalition is resolved
(Branzei et al., 2010b).
The paper is organized as follows. In Section 2 we recall basic
notions and facts from the theory of cooperative interval games.
In Section 3 we introduce the interval core and prove that a
cooperative interval game has a nonempty interval core if and only
if the game is I-balanced. Importantly, the connection between
the interval core and the core of an interval game is discussed
and relations between different existing notions of balancedness
in cooperative interval game theory are established. In Section 4
the classical dominance relation is extended to the interval setting
1 Cooperative vector-valued games arise naturally from collaborative situations
where players face a reward/cost sharing problem according to a finite set of
criteria, and for each criterion sharp values for each coalition can be evaluated.
Mathematically, cooperative interval games can be looked at as special cooperative
vector-valued games in the case only two criteria – one pessimistic and the other
one optimistic – are used to predict the outcomes of coalitions.
2 A player could be an organization/unit/project team.
and used to define the interval dominance core and interval stable
sets. Relations between the interval core, the interval dominance
core and the interval stable sets are studied. In Section 5 we offer a
summary, suggest some topics for further research and discuss our
original contribution to the theory of cooperative games.
2. Preliminaries
A cooperative game in coalitional form is an ordered pair ⟨N , v⟩,
where N = {1, . . . , n} (the set of players) and v : 2N → R is a
map, assigning to each coalition S ∈ 2N a real number, such that
v(∅) = 0. This function v is called the characteristic function of
the game and v(S ) is called the worth (or value) of coalition S. Often
we identify a game ⟨N , v⟩ with its characteristic function v . The set
of coalitional games with player set N is denoted by GN . We refer
the reader to Tijs (2003) and Part I in Branzei et al. (2008a) for an
introduction to classical cooperative game theory.
Let I (R) be the set of all closed intervals in R. A cooperative
interval game (in coalitional form) is an ordered pair ⟨N , w⟩ where
N = {1, . . . , n} is the set of players, and w : 2N → I (R) is the
characteristic function with w(∅) = [0, 0], which assigns to each
coalition S ∈ 2N a closed and bounded interval [w(S ), w(S )]. A
classical cooperative game ⟨N , v⟩ can be identified with ⟨N , w⟩,
where w(S ) = [v(S ), v(S )] for each S ∈ 2N . The family of
all interval games with player set N is denoted by IGN . In the
following, we recall some definitions and results from Alparslan
Gök et al. (2009b), where the focus is on balancedness of n-person
cooperative interval games and cores for two-person cooperative
interval games.
Let ⟨N , w⟩ be an interval game; then v : 2N → R is called a
selection of w if v(S ) ∈ w(S ) for each S ∈ 2N . The set Sel(w) of
selections of w plays a key role in defining the imputation set and
the core of a cooperative interval game. Thus, the imputation set
I (w) of ⟨N , w⟩ is defined by I (w) = ∪ {I (v)|v ∈ Sel(w)}, and the
core C (w) of ⟨N , w⟩ is defined by C (w) = ∪ {C (v)|v ∈ Sel(w)}.
Clearly, C (w) ̸= ∅ if and only if there exists a v ∈ Sel(w) with
C (v) ̸= ∅.
An interval game ⟨N , w⟩ is∑
called strongly balanced if for each
balanced map λ it holds that
S ∈2N \{∅} λ(S )w(S ) ≤ w(N ). Recall
N
that
∑ a map λ :S 2 \N{∅} → SR+ is called a balanced map if
= e . Here, e is the characteristic vector for
S ∈2N \{∅} λ(S )e
coalition S with
eSi

=
1
0
if i ∈ S
if i ∈ N \ S .
Proposition 1 in Alparslan Gök et al. (2009b) specifies:
Let ⟨N , w⟩ be an interval game. Then, the following three
assertions are equivalent:
(i) For each v ∈ Sel(w) the game ⟨N , v⟩ is balanced.
(ii) For each v ∈ Sel(w), C (v) ̸= ∅.
(iii) The interval game ⟨N , w⟩ is strongly balanced.
From Proposition 1 in Alparslan Gök et al. (2009b) it follows that
C (w) ̸= ∅ for a strongly balanced game ⟨N , w⟩, since for all v ∈
Sel(w), C (v) ̸= ∅.
An interval game ⟨N , w⟩ is called
∑strongly unbalanced if there
exists a balanced map λ such that
S ∈2N \{∅} λ(S )w(S ) > w(N ).
Then, C (v) = ∅ for all v ∈ Sel(w), which implies that C (w) = ∅.
If all the worth intervals of an interval game ⟨N , w⟩ are degenerate intervals then strong balancedness corresponds to balancedness and strong unbalancedness corresponds to unbalancedness
for classical cooperative
games

 ⟨N , v⟩.
Let I = I , I and J = J , J be two intervals. We say that I is
weakly better than J, which we denote by I < J, if and only if I ≥ J
and I ≥ J. Note that in the case I < J, then for each x ∈ J there
S.Z. Alparslan Gök et al. / Journal of Mathematical Economics 47 (2011) 621–626
exists y ∈ I such that x ≤ y and for each y ∈ I there exists x ∈ J
such that x ≤ y. We say that I is better than J, which we denote by
I ≻ J, if and only if I < J and I ̸= J. We also use the reverse notation
I 4 J, if and only if I ≤ J and I ≤ J and the notation I ≺ J, if and
only if I 4 J and I ̸= J.


The sets I (R) and D+ = (x, y) ∈ R2 |y ≥ x ⊂ R2 coincide,
and the partial order < on I (R) with the usual partial order ≥ on
R × R, implying that cooperative interval games are particular
vector-valued games and, consequently, special partially ordered
cooperative games. The relatively new theory of partially ordered
cooperative games is thus the theoretical framework for set-valued
solution concepts using interval payoffs.
Further, we use the notation I (R+ ) for the set of all closed
nonnegative intervals in R. In this paper, n-tuples of intervals I =
(I1 , . . . , In ) where Ii ∈ I (R) for each i ∈ N, will play a key role.
For further use we denote by I (R)N the set of all n-dimensional
vectors whose components are elements in I (R). Let Ii = [I i , I i ] be
the interval payoff of player i, and let I = (I1 , . . . , In ) be an
∑interval
payoff vector. Then, according to Moore (1995), we have i∈S Ii =

∑
∑
N
i∈S I i ,
i∈S I i ∈ I (R) for each S ∈ 2 \ {∅}. Next, we define
interval solution concepts for cooperative interval games w ∈ IGN .
Instead of w({i}), w({i, j}), etc., we often write w(i), w(i, j), etc.
3. The interval core
The interval imputation set I(w) of the interval game w , is
defined by

I(w) =
−
(I1 , . . . , In ) ∈ I (R)N |
Ii = w(N ), w(i) 4 Ii ,
i∈N

for all i ∈ N .
We note
∑ that i∈N Ii = w(N ) is equivalent with i∈N I i = w(N )
and i∈N I i = w(N ), and w(i) 4 Ii is equivalent with w(i) ≤ I i and
∑
w(i) ≤ I i , for each i ∈ N. Furthermore, i∈N Ii = w(N ) implies
for
∑ all i ∈ N and for all t ∈ w(N ) there exists xi ∈ Ii such that
i∈N xi = t. Notice that the interval uncertainty of coalition values
propagates into the interval uncertainty of individual payoffs and
we obtain interval payoff vectors as building blocks of interval
solutions. The interval imputation set consists of those interval
payoff vectors which assure the distribution of the uncertain worth
of the grand coalition such that each player can expect a weakly
better interval payoff than what he/she can expect on his/her own.
The interval core C (w) of the interval game w is defined by
∑
∑

C (w) =
(I1 , . . . , In ) ∈ I (R)N |
−
Ii = w(N ),
i∈N

−
Ii < w(S ), ∀S ∈ 2 \ {∅} .
N
i∈S
The interval core consists of those interval payoff vectors which
assure the distribution of the uncertain worth of the grand
coalition such that each coalition of players can expect a weakly
better interval payoff than what that group can expect on its own,
implying
that no coalition has any incentives∑to split off. Here,
∑
I
=
w(N ) is the efficiency condition and i∈S Ii < w(S ), S ∈
i
i∈N
2N \ {∅}, are the stability conditions of the interval payoff vectors.
Clearly, C (w) ⊂ I(w) for each w ∈ IGN . Notice that for two-person
cooperative interval games the interval imputation set coincides
with the interval core.
623
Example 3.1. Consider the auction interval game3 given by w({1})
= [14, 28], w({1, 2}) = [34, 68], w({1, 2, 3}) = [50, 100], and
w(S ) = [0, 0], for any other coalition
S.One
easily
check that
 16can


70
68
,
belongs
to the
the interval allocation [27, 54], 53
,
,
3
3
3
3
interval core of the game.
If the worth of the grand coalition is given by a degenerate
interval then the elements of the interval core are tuples of
degenerate intervals. Under this assumption, the necessary and
sufficient condition for the nonemptiness of the interval core is the
balancedness of the upper game.
Remark 3.1. The interval core of a cooperative interval game can
be obtained as a particular instance of the (extended) core of a
partially ordered cooperative game (Definition 3.1, p. 146 in Puerto
et al., 2008) in the case the characteristic function takes values in
the cone (not a linear space) I (R) endowed with the partial order
<. We notice that one can define an indifference relation ∼ on I (R)
as follows: I ∼ J iff I < J and J < I, which is equivalent with I ∼ J
if and only if I = J.
Some basic properties of the interval core are straightforward
extensions of the corresponding properties of the core of
traditional cooperative games (Gillies, 1959). Specifically, the
interval core correspondence C : IGN I (R)N is a superadditive
map; for each w ∈ IGN the set C (w) is a convex set, and the interval
core is relatively invariant with respect to strategic equivalence,
i.e. for all w, a ∈ IGN , and for each k > 0 we have
∑C (kw + a) =
kC (w) + C (a), where ⟨N , a⟩ is defined by a(S ) =
i∈S a({i}).
An interval game w ∈ IGN is called I∑
-balanced4 if for each
balanced map λ : 2N \ {∅} → R+ we have S ∈2N \{∅} λ(S )w(S ) 4
w(N ).
In the classical theory of cooperative games it is proved by Bondareva (1963) and Shapley (1967) that a game v ∈ GN is balanced
if and only if C (v) is nonempty. In Theorem 3.15 we extend this
result to cooperative interval games.
Theorem 3.1. Let w ∈ IGN . Then the following two assertions are
equivalent:
(i) C (w) ̸= ∅;
(ii) The game w is I-balanced.
In the following we discuss the connection between the interval
core C (w) and the core C (w) of an interval game ⟨N , w⟩, and
establish relations between different notions of balancedness for
interval games.
The elements of the sets C (w) and C (w) are of different type,
implying that we cannot compare the sets with respect to the
inclusion relation. Specifically, the elements of C (w) are vectors
x ∈ RN , whereas the elements of C (w) are vectors I ∈ I (R)N . But,
if all the worth intervals of the interval game ⟨N , w⟩ are degenerate
intervals then the interval core C (w) corresponds naturally to the
core C (w), since ([a1 , a1 ], . . . , [an , an ]) belongs to the interval core
C (w) if and only if (a1 , . . . , an ) is in the core C (w) for each ai ∈
R and i ∈ N. Furthermore, we notice that the interval core of
3 Interval games arising from second price sealed bid auctions with one object
where the bidders are facing interval uncertainty are defined from interval
valuations in Branzei et al. (2010a).
4 The same terminology ‘‘I-balancedness’’ had previously been used by Iehlé
(2007) in the context of hedonic games, but the concept is different from the one
developed in our paper.
5 This theorem is a corollary of Theorem 3.3 in Puerto et al. (2008). A
proof of Theorem 3.1 using the duality theory from linear programming theory
(see Theorem 1.34 in Branzei et al. (2008a)) is available from the authors upon
request.
624
S.Z. Alparslan Gök et al. / Journal of Mathematical Economics 47 (2011) 621–626
n-person cooperative interval games can generate via selections
(x1 , . . . , xn ) ∈ (I1 , . . . , In ) ∈ C (w), a set which has the same type
of elements as C (w). The two sets do not coincide for arbitrary
cooperative interval games, but they coincide in the case all the
coalitional worth values are degenerate intervals.
Proposition 3.1. Let w ∈ IGN . If the interval core C (w) is nonempty
then the core C (w) is nonempty.
Proof. Take (I1 , . . . , In ) ∈ C (w). Then, i∈N Ii = w(N ), implying
∑
∑
∑
that
w(N ), and i∈S Ii < w(S ),
i∈N I i =∑w(N ) and
i∈N I i =
∑
implying that
i∈S I i ≥ w(S ) and
i∈S I i ≥ w(S ). Let ⟨N , v⟩ be
the selection
of w with v(S ) =
∑
∑w(S ), v(N ) = w(N ) and let xi = I i .
Then,
i∈S xi ≥ w(S ) and
i∈N xi = w(N ) which shows that
C (w) ̸= ∅ and C (w) ̸= ∅ implying that C (w) is nonempty. ∑
Example 3.2 illustrates that the sufficient condition for the nonemptiness of the core of an interval cooperative game provided in
Proposition 3.1 is not a necessary one.
Example 3.2. Let ⟨N , w⟩ be a two-person interval game with
w(1, 2) = [6, 8], w(1) = [2, 4], w(2) = [5, 6] and w(∅) = [0, 0].
The interval game is not I-balanced since w(1) + w(2) ≻ w(1, 2).
According to Theorem 3.1 we conclude that C (w) = ∅. But,
C (w) ̸= ∅ since C (v) ̸= ∅ for some selections v ∈ Sel(w).
In the following proposition a relation between balancedness
(in terms of selections) and I-balancedness is given.
Proposition 3.2. Let ⟨N , w⟩ be a strongly balanced interval game;
then ⟨N , w⟩ is I-balanced.
−
S ∈2N \{∅}
So,
∑
S ∈2N \{∅}
λ(S )w(S ) ≥
−
λ(S )w(S ).
S ∈2N \{∅}
λ(S )w(S ) 4 w(N ). Hence, ⟨N , w⟩ is I-balanced.
Note that the converse of Proposition 3.2 is not true since there
exists v ∈ Sel(w) with C (v) ̸= ∅, implying that the core C (w) is
nonempty, but the interval core may be empty as we learn from
Example 3.2.
Furthermore, we note that if all the worth intervals of
the interval game ⟨N , w⟩ are degenerate intervals, then strong
balancedness and I-balancedness of the game also correspond to
the classical balancedness.
We end this section by pointing out that studying the interval
core on its own (instead of looking at it as the extended core
for partially ordered games on a particular domain) has two
additional practical advantages. First, the conditions guaranteeing
the nonemptiness of the interval core of an interval game are
simple. Second, there is an easy way to compute interval core
elements using elegant software (Kimms and Drechsel, 2009). Both
issues are important from the practitioners’ point of view.
4. The interval dominance core and stable sets
Let w ∈ IGN , I = (I1 , . . . , In ), J = (J1 , . . . , Jn ) ∈ I(w) and
S ∈ 2N \ {∅}. We say that I dominates J via coalition S, and denote
it by IdomS J, if
(i) I∑
i ≻ Ji for all i ∈ S,
(ii)
i∈S Ii 4 w(S ).
For S ∈ 2 \ {∅} we denote by D(S ) the set of those elements of
I(w) which are dominated via S. For I , J ∈ I(w), we say that I
dominates J and denote it by I dom J if there is an S ∈ 2N \ {∅}
such that IdomS J. I is called undominated if there exist no J and no
coalition S such that JdomS I.
The interval dominance core DC (w) of an interval game w ∈ IGN
consists of all undominated elements in I(w).
N
(i) The definition of the imputation set for a partially ordered
cooperative game is based on a binary reflexive relation
defined with the aid of the partial order under consideration,
whereas the definition of the interval imputation set for a
cooperative interval game is based on the partial order <.
(ii) The used dominance relations are different. Specifically, the
dominance relation in Puerto et al. (2008) is a coalitional
dominance while the definition of our dominance relation is
a straightforward generalization of the dominance relation in
classical cooperative game theory.
For w ∈ IGN a subset A of I(w) is an interval stable set if the
following conditions hold:6
(i) (Internal stability) There do not exist I , J ∈ A such that I dom J.
(ii) (External stability) For each I ̸∈ A there exists J ∈ A such that J
dom I.
Next, we study relations between the interval core, the interval
dominance core and the interval stable sets for cooperative interval
games.
Proof. Take a balanced map λ : 2N \ {∅} → R+ . Then
w(N ) ≥ w(N ) ≥
Remark 4.1. For partially ordered cooperative games, Puerto et al.
(2008) introduced the nondominated core (Definition 3.2, p. 147)
and proved (Theorem 3.1) that the (extended) core and the
nondominated core coincide for each partially ordered game.
Importantly, the interval dominance core of a cooperative interval
game cannot be obtained from the nondominated core of a partially
ordered cooperative game in the case the characteristic function
takes values in I (R), although both the cores are defined in the
same spirit (i.e. by extending the definition of the dominance core –
also called the undominated core – of a classical cooperative game).
There are two reasons for this:
Theorem 4.1. Let w ∈ IGN and let A be a stable set of w . Then,
C (w) ⊂ DC (w) ⊂ A.
Proof. In order to show that C (w) ⊂ DC (w) let us assume that
there is I ∈ C (w) such that I ̸∈ DC (w). Then, there exist a
J ∈ I(w) and∑
a coalition S ∈ 2N \ {∅} such that JdomS I. Thus,
I (S ) ≺ J (S ) =
i∈S Ji 4 w(S ) and Ji ≻ Ii for all i ∈ S implying that
I ̸∈ C (w). To prove next that DC (w) ⊂ A it is sufficient to show
I(w) \ A ⊂ I(w) \ DC (w). Take I ∈ I(w) \ A. Given the external
stability of A there is a J ∈ A with JdomI. Because the elements in
DC (w) are not dominated, we obtain I ̸∈ DC (w), or equivalently,
I ∈ I(w) \ DC (w). The inclusions stated in the previous theorem may be strict. The
following example, inspired by Tijs (2003), illustrates that the
inclusion of C (w) in DC (w) might be strict.
Example 4.1. Let ⟨N , w⟩ be the three-person interval game with
w(1, 2) = [2, 2], w(N ) = [1, 1] and w(S ) = [0, 0] if S ̸=
{{1, 2} , N }. Then, C (w) = ∅ because the game is not I-balanced
(note that w(1, 2)+w(3) ≻ w(N )). Further, D(S ) = ∅ if S ̸= {1, 2}
and D({1, 2}) = {I ∈ I(w)|I3 ≻ [0, 0]}. The elements I ∈ I(w)
which are undominated satisfy I3 = [0, 0]. Since the interval
dominance core is the set of undominated elements in I(w), the
interval dominance core of this game is nonempty.
Next, we introduce unanimity interval games, show that such
games are I-balanced games, give an explicit description of the
interval core for such games and prove that the interval core and
the interval dominance core coincide on the class of unanimity
6 The classical notion of stable sets was introduced by von Neumann and
Morgenstern (1944).
S.Z. Alparslan Gök et al. / Journal of Mathematical Economics 47 (2011) 621–626
interval games. Let J ∈ I (R+ ) and let T ∈ 2N \ {∅}. The unanimity
interval game based on J and T is defined by
uT , J ( S ) =
J,
[0, 0],

T ⊂S
otherwise,
for each S ∈ 2N . We notice that the interval core of the unanimity
interval game based on the degenerate interval J = [1, 1] corresponds to the core of the unanimity game in the traditional case.
Recall that for T ∈ 2N \ {∅} the traditional unanimity game based
on T , ⟨N , uT ⟩, is defined by
uT (S ) =

1,
0,
T ⊂S
otherwise,
for each S ∈ 2N \ {∅}. The core
the unanimity game
 C (uT ) of∑
⟨N , uT ⟩ is given by C (N , uT ) = x ∈ RN | ni=1 xi = 1, xi = 0

for i ∈ N \ T .
The next proposition gives a description of the interval core of a
unanimity interval game and shows that on the class of unanimity
games the interval core and the interval dominance core coincide.
We define K as follows:

K=
(I1 , . . . , In ) ∈ I (R)N |
−
Ii = J ,
i∈N

I i ≥ 0 for all i ∈ N , Ii = [0, 0] for i ∈ N \ T .
Proposition 4.1. Let ⟨N , uT ,J ⟩ be the unanimity interval game based
on coalition T and the interval payoff J < [0, 0]. Then, DC (uT ,J ) =
C (uT ,J ) = K .
Proof. First, we prove that the interval core of uT ,J can be described
as the set K . In order to show that C (uT ,J ) ⊂ K , let (I1 , . . . , In ) ∈
C (uT ,J ). Clearly, for each i ∈ N we have Ii < u∑
T ,J ({i}) and uT ,J ({i}) <
≥ 0 for all i ∈ N. Furthermore, i∈N Ii = uT ,J (N ) = J.
[0, 0]. So, I i∑
Since also
i∈T Ii < J, we conclude that Ii = 0 for i ∈ N \ T .
So, (I1 , . . . , In ) ∈ K . In order to show that K ⊂ C (uT ,J ), let
(I1 , . . . ,∑
In ) ∈ K . So, I i ≥ 0 for all i ∈ N , Ii = [0, 0] if i ∈
N \ T , i∈N Ii = J. Then (I1 , . . . , In ) ∈ C (uT ,J ), because it also
holds:
(i) ∑i∈S Ii < [∑
0, 0] = uT ,J (S ) if T \ S ̸= ∅,
(ii)
i∈S Ii =
i∈N Ii = uT ,J (N ) = J = uT ,J (S ) if T ⊂ S.
∑
Next, we prove that C (uT ,J ) = DC (uT ,J ). Note first that
C (uT ,J ) ⊂ DC (uT ,J ) by Theorem 4.1. We only have to prove that
DC (uT ,J ) ⊂ C (uT ,J ) or we need to show that for each I ̸∈ C (uT ,J )
we have I ̸∈ DC (uT ,J ). Take I ̸∈ C (uT ,J ). Then, there is a k ∈ N \ T
with Ik ̸= [0, 0]. Then, I ′ domT I, where Ii′ = [0, 0] for i ∈ N \ T and
Ii′ = Ii + |T1| Ik for i ∈ T . So, I ̸∈ DC (uT ,J ). The interval core might coincide with the interval dominance core
also for games which are not unanimity interval games as the
following example illustrates.
Example 4.2. Consider the game w described in Example 3.1. We
will show that DC (w) = C (w). Take I = (I1 , I2 , I3 ) ∈ I(w). Note
that if I1 ̸= [0, 0] then ([0, 0], I2 + 21 I1 , I3 + 12 I1 )dom{2,3} (I1 , I2 , I3 ).
So, I ̸∈ DC (w). Similarly, if I2 ̸= [0, 0] then I ̸∈ DC (w). Hence,
DC (w) ⊂ {([0, 0], [0, 0], J )} = C (w) by Example 3.1. On the
other hand we know, in view of Theorem 4.1, that C (w) ⊂ DC (w).
So, we conclude that DC (w) = C (w).
Further analysis of the equality between the interval dominance
core and the interval core for cooperative interval games can be
found in Branzei et al. (in press).
625
5. Conclusion and outlook
This paper deals with cooperative games whose characteristic
functions are interval-valued, i.e. the worth of a coalition is
not a real number, but a compact interval of real numbers.
This class of cooperative games has been applied to different
economic and operations research problems such as bankruptcy
situations, airport situations, sequencing situations and cost
allocation problems arising from connection situations.
A game-theoretic model which at first sight seems to be similar
to the model of cooperative interval games is that of cooperative
games with upper bounds (Carpente et al., 2010). Such games arise
from situations in which one can provide a sharp worth for each
coalition of players, and additionally, for some coalitions, a sharp
upper limit for the obtainable coalition’s worth. In other words,
the model of cooperative games with upper bounds is a hybrid
of a classical cooperative game and a partially defined classical
cooperative game with the same set of players. Hence, although
cooperative games with upper bounds can be seen as associating
intervals with (some) coalitions of players, as cooperative interval
games, they also deserve theoretical investigations on their own.
A cooperative interval game can be studied and used to solve
reward/cost sharing situations with interval data either through
its selections ⟨N , v⟩, where v is a characteristic function and
v(S ) ∈ [w(S ), w(S )], or using interval calculus. The first approach
was used in Alparslan Gök et al. (2009b) to introduce the core
C (w) of a cooperative interval game ⟨N , w⟩. The study in this
paper follows the second approach to introduce the interval core
C (w), the interval dominance core DC (w), and stable sets. A
worthwhile topic for further research might be to analyze under
which conditions the set of payoff vectors generated by the interval
core of a cooperative interval game coincides with the core of the
game using selections of the interval game. Studying stable sets
of a cooperative interval game in terms of selections of the game
can offer a further valuable extension of the theory of cooperative
interval games.
The central set-valued solution concept in this paper is the
interval core, and Theorem 3.1 is the most interesting result
regarding this solution concept. Two classes of interval games,
namely convex games and big boss interval games, have a nonempty interval core. These classes of interval games have been
recently developed by Alparslan Gök et al. (2009a, 2011). Another
interesting class of interval games, less developed so far, is that
of exact interval games. Exact interval games have been recently
introduced in Branzei et al. (2008b), who used them to characterize
convex interval games. A cooperative interval game is exact if for
each S ∈ 2N :
(i) there exists I = (I1 , . . . , In ) ∈ C (w)
∑ such that i∈S Ii = w(S );
(ii) there exists x ∈ C (|w|) such that i∈S xi = |w| (S ).
∑
Theorem 2 in Branzei et al. (2008b) proves the coincidence
between the classes of totally exact interval games and convex
interval games, based on a previous result by Biswas et al. (1999).
Future research can explore further characterizations of exact
interval games, especially whether (or not) results by Csóka et al.
(forthcoming) and Schmeidler (1972) can be extended beyond
exact TU-games.7
A final note about the novelty and originality of the solution
concepts for cooperative interval games introduced and studied in
this paper and the obtained results is in order. Remark 3.1 discusses
the relation between the interval core for cooperative interval
games and the extended core for partially ordered cooperative
games, and the implications upon our results in Section 3.
7 We thank one of the referees for suggesting this extension.
626
S.Z. Alparslan Gök et al. / Journal of Mathematical Economics 47 (2011) 621–626
Importantly, the connection between the interval core C (w), and
the core C (w) of an interval game ⟨N , w⟩ and the established
relations between different notions of balancedness are issues not
yet considered in the theory of partially ordered games. Remark 4.1
makes clear that the interval dominance core is different from the
nondominated core on the domain of cooperative interval games.
Furthermore, Remark 4.1 implies that our results regarding the
interval dominance core are complementary with the results on
the nondominated core for partially ordered cooperative games,
and the notion of interval stable set as well as the related results in
Theorem 4.1 are new.
Acknowledgments
The first author acknowledges the support of TUBITAK (Turkish
Scientific and Technical Research Council). Financial support from
the Government of Spain and FEDER under project MTM200806778-C02-01 is gratefully acknowledged by the third and
the fourth authors. The authors gratefully acknowledge two
anonymous reviewers.
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