Selection-based Approach to Cooperative Interval Games

Jan Bok and Milan Hlad

ık

Charles University in Prague, Faculty of Mathematics and Physics, Department of Applied Mathematics,

Malostransk

e n

est

ı 25, 11800, Prague, Czech Republic

Keywords:

Cooperative Game Theory, Interval Analysis, Core.

Abstract:

Cooperative interval games are a generalized model of cooperative games in which worth of every coalition

corresponds to a closed interval representing the possible outcomes of its cooperation. Selections are all

possible outcomes of the interval game with no additional uncertainty. We introduce new selection-based

classes of interval games and prove their characterizations and relations to existing classes based on the weakly

better operator. We show new results regarding the core and imputations. Then we introduce the deﬁnition of

strong imputation and strong core. We also examine a problem of equality of two different versions of core,

which is the main stability solution of cooperative games.

1 INTRODUCTION

Uncertainty and inaccurate data are issues occurring

very often in the real world situations. Therefore it is

important to be able to make decisions even when the

exact data are not available and only bounds on them

are known.

In classical cooperative game theory, every group

of players (coalition) knows precise reward for their

cooperation; in cooperative interval games, only the

worst and the best possible outcome are known. Such

situations can be naturally modeled with intervals en-

capsulating these outcomes.

Cooperation under interval uncertainty was ﬁrst

considered by Branzei, Dimitrov and Tijs in 2003 to

study bankruptcy situations (Branzei et al., 2004) and

later further extensively studied by G

ok in her PhD

thesis (Alparslan-G

ok, 2009) and other papers writ-

ten together with Branzei et al. (see the references

section of (Branzei et al., 2010) for more).

However, their approach is almost exclusively

aimed on interval solutions, that is on payoff distri-

butions consisting of intervals and thus containing

another uncertainty. This is in contrast with selec-

tions – possible outcomes of an interval game with

no additional uncertainty. Selection-based approach

was never systematically studied and not very much

is known. This paper is trying to ﬁx this and summa-

rizes our results regarding a selection-based approach

to interval games.

This paper has the following structure. Section

2 is a preliminary section which concisely presents

necessary deﬁnitions and facts on classical cooper-

ative games, interval analysis and cooperative inter-

val games. Section 3 is devoted to new selection-

based classes of interval games. We consequently

prove their characterizations and relations to existing

classes. Section 4 focuses on the so called core in-

cidence problem which asks under which conditions

are the selection core and the set of payoffs generated

by the interval core equal. In Section 5, deﬁnitions

of strong core and strong imputation are introduced

as new concepts. We show some remarks on strong

core, one of them being a characterization of games

with the strong imputation and strong core.

On Mathematical Notation

We will use ≤ relation on real vectors. For every x,y ∈

we write x ≤ y if x

≤ y

holds for every 1 ≤ i ≤ N.

We do not use symbol ⊂ in this paper. Instead, ⊆

and ( are used for subset and proper subset, respec-

tively, to avoid ambiguity.

2 PRELIMINARIES

2.1 Classical Cooperative Game Theory

Comprehensive sources on classical cooperative

game theory are for example (Branzei et al., 2000)

(Driessen, 1988) (Gilles, 2010) (Peleg and Sudh

olter,

Bok J. and Hladík M..

Selection-based Approach to Cooperative Interval Games.

DOI: 10.5220/0005221600340041

In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 34-41

ISBN: 978-989-758-075-8

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

2007). For more info about on applications, see

e.g. (Bilbao, 2000) (Curiel, 1997) (Lemaire, 1991).

Here we present only necessary background theory

for study of interval games. We examine games

with transferable utility (TU), therefore by coopera-

tive game we mean cooperative TU game.

Deﬁnition 2.1. (Cooperative Game) Cooperative

game is an ordered pair (N,v), where N =

{1,2,. .., n} is a set of players and v : 2

→ R is a

characteristic function of the cooperative game. We

further assume that v(

0) = 0.

The set of all cooperative games with player set N

is denoted by G

Subsets of N are called coalitions and N itself is

called a grand coalition.

Note 2.2. We often write v instead of (N, v), because

we can easily identify game with its characteristic

function without loss of generality.

To further analyze players’ gains, we will need a

payoff vector which can be interpreted as a proposed

distribution of reward between players.

Deﬁnition 2.3. (Payoff vector) Payoff vector for a co-

operative game (N,v) is a vector x ∈ R

with x

de-

noting reward given to ith player.

Deﬁnition 2.4. (Imputation) An imputation of

(N,v) ∈ G

is a vector x ∈ R

such that

∑

i∈N

v(N) and x

≥ v({i}) for every i ∈ N.

The set of all imputations of a given cooperative

game (N,v) is denoted by I(v).

Deﬁnition 2.5. (Core) The core of (N,v) ∈ G

is the

set

C(v) =

(

x ∈ I(v),

∑

i∈S

≥ v(S),∀S ⊆ N

)

There are many important classes of cooperative

games. Here we show the most important ones.

Deﬁnition 2.6. (Monotonic game) A game (N, v) is

monotonic if for every T ⊆ S ⊆ N we have

v(T ) ≤ v(S).

Informally, in monotonic games, bigger coalitions

are stronger.

Deﬁnition 2.7. (Superadditive game) A game (N, v)

is superadditive if for every S,T ⊆ N, S ∩ T =

0 we

have

v(T ) + v(S) ≤ v(S ∪ T ).

In a superadditive game, coalition has no incen-

tive to divide itself, since together, they will always

achieve at least as much as separated.

Superadditive game is not necessarily monotonic.

Conversely, monotonic game is not necessarily super-

additive. However, these classes have a nonempty in-

tersection. Check Caulier’s paper (Caulier, 2009) for

more details on relation of these two classes.

Deﬁnition 2.8. (Additive game) A game (N, v) is ad-

ditive if for every S,T ⊆ N, S ∩ T =

0 we have

v(T ) + v(S) = v(S ∪ T ).

Observe that additive games are superadditive as

well.

Another important type of game is a convex game.

Deﬁnition 2.9. (Convex game) A game (N,v) is con-

vex if its characteristic function is supermodular. The

characteristic function is supermodular if for every

S ⊆ T ⊆ N holds

v(T ) + v(S) ≤ v(S ∪ T ) + v(S ∩ T ).

Clearly, supermodularity implies superadditivity.

Convex games have many nice properties. we

show the most important one.

Theorem 2.10. (Shapley 1971 (Shapley, 1971)) If a

game (N,v) is convex, then its core is nonempty.

2.2 Interval Analysis

Deﬁnition 2.11. (Interval) The interval X is a set

X := [X,X] = {x ∈ R : X ≤ x ≤ X}.

With X being the lower bound and X being the upper

bound of the interval.

From now on, when we say an interval we mean a

closed interval. The set of all real intervals is denoted

by IR.

The following deﬁnition shows how to do a basic

arithmetics with intervals (Moore et al., 2009).

Deﬁnition 2.12. For every X ,Y,Z ∈ IR and 0 /∈ Z de-

ﬁne

X +Y := [X +Y ,X +Y ],

X −Y := [X −Y ,X −Y ],

X ·Y := [min S,max S], S = {XY ,XY ,XY ,XY },

X / Z := [min S,max S], S = {X/Z,X /Z,X /Z,X /Z}.

2.3 Cooperative Interval Games

Deﬁnition 2.13. (Cooperative Interval Game) A co-

operative game is an ordered pair (N, w), where N =

{1,2,. .., n} is a set of players and w : 2

→ IR is a

characteristic function of the cooperative game. We

further assume that w(

0) = [0,0].

The set of all interval cooperative games on player

set N is denoted by IG

Selection-basedApproachtoCooperativeIntervalGames

Note 2.14. We often write w(i) instead of w({i}).

Remark 2.15. Each cooperative interval game in

which the characteristic function maps to degenerate

intervals only can be associated with some classical

cooperative game. Converse holds as well.

Deﬁnition 2.16. (Border games) For every (N,w) ∈

, border games (N, w) ∈ G

(lower border game)

and (N,w) ∈ G

(upper border game) are given by

w(S) = w(S) and w(S) = w(S) for every S ∈ 2

Deﬁnition 2.17. (Length game) The length game of

(N,w) ∈ IG

is the game (N,|w|) ∈ G

with

|w|(S) = w(S) − w(S), ∀S ∈ 2

The basic notion of our approach will be a selec-

tion and consequently a selection imputation and a se-

lection core.

Deﬁnition 2.18. (Selection) A game (N, v) ∈ G

is a

selection of (N, w) ∈ IG

if for every S ∈ 2

we have

v(S) ∈ w(S). Set of all selections of (N,w) is denoted

by Sel(w).

Note that border games are particular examples of

selections.

Deﬁnition 2.19. (Interval selection imputation) The

set of interval selection imputations (or just selection

imputations) of (N,w) ∈ IG

is deﬁned as

S I (w) =

[



I(v) | v ∈ Sel(w)



Deﬁnition 2.20. (Interval selection core) Interval se-

lection core (or just selection core) of (N, w) ∈ IG

deﬁned as

S C (w) =

[



C(v) | v ∈ Sel(w)



ok (Alparslan-G

ok, 2009) choose an approach

using a weakly better operator. That was inspired by

(Puerto et al., 2008).

Deﬁnition 2.21. (Weakly better Operator ) Interval

I is weakly better than interval J (I  J) if I ≥ J and

I ≥ J. Furthermore, I  J if and only if I ≤ J and

I ≤ J. Interval I is better than J (I  J) if and only if

I  J and I 6= J.

Their deﬁnition of imputation and Core is as follows.

Deﬁnition 2.22. (Interval imputation) The set of in-

terval imputations of (N,w) ∈ IG

is deﬁned as

I (w) :=

,.. .,I

) ∈ IR

∑

i∈N

= w(N), I

 w(i), ∀i ∈ N

Deﬁnition 2.23. (Interval core) An interval core of

(N,w) ∈ IG

is deﬁned as

C (w) :=

,.. .,I

) ∈ I (w) |

∑

i∈S

 w(S), ∀S ∈ 2

\ {

Important difference between deﬁnitions of inter-

val and selection core and imputation is that selection

concepts yield a payoff vectors from R

, while I and

C yield vectors from IR

Note 2.24. (Notation) Throughout the papers on co-

operative interval games, notation, especially of core

and imputations, is not uniﬁed. It is therefore possible

to encounter different notation from ours.

Also, in these papers, selection core is called core

of interval game. We consider that confusing and that

is why do we use term selection core instead. The term

selection imputation is used because of its connection

with selection core.

The following classes of interval games have

been studied earlier (see e.g. (Alparslan-G

ok et al.,

2009b)).

Deﬁnition 2.25. (Size monotonicity) A game

(N,w) ∈ IG

is size monotonic if for every T ⊆ S ⊆ N

we have

|w|(T ) ≤ |w|(S).

That is when its length game is monotonic.

The class of size monotonic games on player set N

is denoted by SMIG

As we can see, size monotonic games capture sit-

uations in which an interval uncertainty grows with

the size of coalition.

Deﬁnition 2.26. (Superadditive interval game) A

game (N,w) ∈ IG

is superadditive interval game if

for every S,T ⊆ N, S ∩ T =

0 holds

w(T ) + w(S)  w(S ∪ T ),

and its length game is superadditive. We denote by

SIG

class of superadditive interval games on player

set N.

We should be careful with the following analogy

of convex game, since unlike for classical games, su-

permodularity is not the same as convexity.

Deﬁnition 2.27. (Supermodular interval game) An in-

terval game (N, v) is supermodular interval if for ev-

ery S ⊆ T ⊆ N holds

v(T ) + v(S)  v(S ∪ T ) + v(S ∩ T ).

We get immediately that interval game is super-

modular interval if and only if its border games are

convex.

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

Deﬁnition 2.28. (Convex interval game) An interval

game (N, v) is convex interval if its border games and

length game are convex.

We write CIG

for a set of convex interval games

on player set N.

Convex interval game is supermodular as well but

converse does not hold in general. See (Alparslan-

ok et al., 2009b) for characterizations of convex in-

terval games and discussion on their properties.

3 SELECTION-BASED CLASSES

OF INTERVAL GAMES

We will now introduce new classes of interval games

based on the properties of their selections. We think

that it is natural way to generalize special classes from

classical cooperative game theory. Consequently, we

show their characterizations and relation to classes

from preceding subsection.

Deﬁnition 3.1. (Selection Monotonic Interval Game)

An interval game (N, v) is selection monotonic if all

its selections are monotonic games. The class of such

games on player set N is denoted by SeMIG

Deﬁnition 3.2. (Selection Superadditive Interval

Game) An interval game (N,v) is selection superaddi-

tive if all its selections are superadditive games. The

class of such games on player set N is denoted by

SeSIG

Deﬁnition 3.3. (Selection Convex Interval Game) An

interval game (N,v) is selection convex if all its se-

lections are convex games. The class of such games

on player set N is denoted by SeCIG

We see that many properties persist. For example,

a selection convex game is a selection superadditive

as well. Selection monotonic and selection superaddi-

tive are not subset of each other but their intersection

is nonempty. Furthermore, selection core of selection

convex game is nonempty, which is an easy observa-

tion.

We will now show characterizations of these three

classes and consequently show their relations to exist-

ing classes presented in Subsection 2.3.

Proposition 3.4. An interval game (N,w) is selection

monotonic if and only if for every S,T ∈ 2

, S ( T

holds

w(S) ≤ w(T ).

Proof. For the “only if” part, suppose that (N,w) is a

selection monotonic and

w(S) > w(T ) for some S,T ∈

, S ( T . Then selection (N,v) with v(S) = w(S)

and v(T ) = w(T ) clearly violates monotonicity and

we arrive at a contradiction with assumptions.

Now for the “if” part. For any two subsets S, T

of N, one of the situations S ( T , T ( S or S = T

occur. For S = T , in every selection v, v(S) ≤ v(S)

holds. As for the other two situations, it is obvious

that monotonicity cannot be violated as well since

v(S) ≤ w(S) ≤ w(T ) ≤ v(T ).

Note 3.5. Notice the importance of using S ( T in

the formulation of Proposition 3.4. It is because us-

ing of S ⊆ T (thus allowing situation S = T ) would

imply w(S) ≤ w(S) for every S in selection monotonic

game which is obviously not true in general. In char-

acterizations of selection superadditive and selection

convex games, similar situation arises.

Proposition 3.6. An interval game (N,w) is selection

superadditive if and only if for every S,T ∈ 2

such

that S ∩ T =

0, S 6=

0, T 6=

0 holds

w(S) + w(T ) ≤ w(S ∪ T ).

Proof. Similar to proof of Proposition 3.4.

Proposition 3.7. An interval game (N,w) is selection

convex if and only if for every S,T ∈ 2

such that S 6⊆

T , T 6⊆ S, S 6=

0, T 6=

0 holds

w(S) + w(T ) ≤ w(S ∪ T ) + w(S ∩ T ).

Proof. Similar to proof of Proposition 3.4. Note that

we exclude cases S ⊆ T and S ⊆ T since w(S) +

w(T ) ≤ w(S) + w(T ) is too restrictive.

Now let us look on relation with existing classes

of interval games.

For selection monotonic and size monotonic

games, their relation is obvious. For nontrivial games

(that is games with the player set size greater than

one), a selection monotonic game is not necessarily

size monotonic. Converse is the same.

Proposition 3.8. For every player set N with |N| > 1,

the following assertions hold.

(i) SeSIG

6⊆ SIG

(ii) SIG

6⊆ SeSIG

(iii) SeSIG

∩ SIG

Proof. In (i), we can construct the counterexample in

the following way.

Let us construct game (N,w). For w(

0), interval is

given. Now for any nonempty coalition, set w(S) :=

[2|S| − 2,2|S| − 1]. For any S,T ∈ 2

with S and T

being nonempty, the following holds with the fact that

|S| + |T | = |S ∪ T | taken into account.

w(S) + w(T ) = (2|S| − 1) + (2|T | − 1)

= 2|S ∪ T | − 2

= w(S ∪ T )

Selection-basedApproachtoCooperativeIntervalGames

So (N,w) is selection superadditive by Proposition

3.6. Its length game, however, is not superadditive,

since for any two nonempty coalitions with the empty

intersection |w|(S) + |w|(T ) = 2 6≤ 1 = |w|(S ∪ T )

holds.

In (ii), we can construct the following counterex-

ample (N, w

). Set w

(S) = [0,|S|] for any nonempty

S. The lower border game is trivially superaddi-

tive. For the upper game, w

(S) +w

(T ) = |S| +|T | =

|S ∪ T | = w

(S ∪ T ) for any S,T with the empty inter-

section, so the upper game is superadditive. Observe

that the length game is the same as the upper border

game. This shows interval superadditivity.

However, (N,w

) is clearly not selection superad-

ditive because of nonzero upper bounds, zero lower

bounds of nonempty coalitions and the characteriza-

tion of SeSIG

taken into account.

(iii) Nonempty intersection can be argumented

easily by taking some superadditive game (N, c) ∈

. Then we can deﬁne corresponding game (N,d) ∈

with

d(S) = [c(S),c(S)], ∀S ∈ 2

Game (N,d) is selection superadditive since its only

selection is (N,c). And it is superadditive inter-

val game since border games are supermodular and

length game is |w|(S) = 0 for every coalition, which

trivially implies its superadditivity.

Proposition 3.9. For every player set N with |N| > 1,

following assertions hold.

(i) SeCIG

6⊆ CIG

(ii) CIG

6⊆ SeCIG

(iii) SeCIG

∩ CIG

Proof. For (i), take a game (N, w) assigning to each

nonempty coalition S interval [2

|S|

−2, 2

|S|

−1]. From

Proposition 3.7, we get that for inequalities which

must hold in order to meet necessary conditions of

game to be selection convex, |S| < |S ∪ T | and |T | <

|S ∪T | must hold. That gives the following inequality.

w(S) + w(T ) ≤ (2

|S∪T |−1

− 1) + (2

|S∪T |−1

− 1)

= 2

|S∪T |

− 2

= w(S ∪ T )

≤ w(S ∪ T ) + w(S ∩ T )

This concludes that (N, w) is selection convex. We see

that the border games and the length game are convex

too. To have a game so that it is selection convex

and not convex interval game, we can take (N, c) and

set c(S) := w(S) for S 6= N and v(N) := [w

(S),w(S)].

Now the game (N,c) is still selection convex, but its

length game is not convex, so (N,v) is not convex in-

terval game, which is what we wanted.

In (ii), we can take a game (N, w

) from the proof

of Proposition 3.8(ii). From the fact that |S| + |T | =

|S∪T |+|S∩T |, it is clear that w

is convex. The lower

border game is trivially convex and the length game is

the same as upper. However, for nonempty S,T ∈ 2

such that S 6⊆ T , T 6⊆ S, S 6=

0, T 6=

0, convex selection

games characterization is clearly violated.

As for (iii), we can use the same steps as in (iii) of

Proposition 3.8 or we can use a game (N,w) from (i)

of this theorem.

4 CORE COINCIDENCE

In G

ok’s PhD thesis (Alparslan-G

ok, 2009), the fol-

lowing topic is suggested: “A difﬁcult topic might be

to analyze under which conditions the set of payoff

vectors generated by the interval core of a coopera-

tive interval game coincides with the core of the game

in terms of selections of the interval game.”

We decided to examine this topic. We call it a

core coincidence problem. This subsection shows our

results.

Note 4.1. We remind the reader that whenever we

talk about relation and maximum, minimum, maxi-

mal, minimal vectors, we mean relation ≤ on real

vectors unless we say otherwise.

The main thing to notice is that while the interval

core gives us a set of interval vectors, selection core

gives us a set of real numbered vectors. To be able to

compare them, we need to assign to a set of interval

vectors a set of real vectors generated by these interval

vectors. That is exactly what the following function

gen does.

Deﬁnition 4.2. The function gen : 2

→ 2

maps

to every set of interval vectors a set of real vectors. It

is deﬁned as

gen(S) =

[

s∈S



,.. .,x

) | x

∈ s

, s ∈ IR



Core coincidence problem can be formulated as

this: What are the necessary and sufﬁcient condition

to satisfy gen(C (w)) = S C (w)?

The main results of this subsection are the two fol-

lowing theorems which give an answer to the afore-

mentioned question.

Note 4.3. In the following text, by mixed system we

mean a system of equalities and inequalities.

Proposition 4.4. For every interval game (M,w) we

have gen(C (w)) ⊆ S C (w).

Proof. For any x

∈ gen(C (w)), inequality v(N) ≤

∑

i∈N

≤ v(N) obviously holds. Furthermore, x

is in

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the core for any selection of the interval game (N,s)

with s given by

s(S) =

 

∑

i∈N

∑

i∈N



if S = N



w(S),min(

∑

i∈S

,w(S))



if otherwise.

Clearly, Sel(s) ⊆ Sel(w) and Sel(s) 6=

0. That con-

cludes gen(C (w)) ⊆ S C (w).

Theorem 4.5. (Core coincidence characterization)

For every interval game (N,w) we have gen(C (w)) =

S C (w) if and only if for every x ∈ S C (w) there exist

nonnegative vectors l

(x)

and u

(x)

such that

∑

i∈N

− l

(x)

) = w(N), (4.1)

∑

i∈N

+ u

(x)

) = w(N), (4.2)

∑

i∈S

− l

(x)

) ≥ w(S), ∀S ∈ 2

\ {

0}, (4.3)

∑

i∈S

+ u

(x)

) ≥ w(S), ∀S ∈ 2

\ {

0}. (4.4)

Proof. First, we observe that with Proposition 4.4

taken into account, we only need to take care of

gen(C (w)) ⊇ S C (w) to obtain equality.

For gen(C (w)) ⊇ S C (w), suppose we have some

x ∈ S C (w). For this vector, we need to ﬁnd some in-

terval X ∈ C (w) such that x ∈ gen(X). This is equiva-

lent to the task of ﬁnding two nonnegative vectors l

(x)

and u

(x)

such that

([x

−l

(x)

]),[x

−l

(x)

],.. .,[x

−l

(x)

]) ∈ C (w).

From the deﬁnition of interval core, we can see that

these two vectors have to satisfy exactly the mixed

system (4.1) − (4.4). That completes the proof.

Example 4.6. Consider an interval game with N =

{1,2} and w({1}) = w({2}) = [1, 3] and w(N) =

[1,4]. Then vector (2,2) lies in the core of the

selection with v({1}) = v({2}) = 2 and v(N) = 4.

However, to satisfy equation (4.1), we need to have

∑

i∈N

= 3 which means that either l

or l

has to be

greater than 1. That means we cannot satisfy (4.3)

and we conclude that gen(C (w)) 6= S C (w).

The following theorem shows that it sufﬁces to

check only minimal and maximal vectors of S C (w).

Theorem 4.7. For every interval game (N, w), if there

exist vectors q,r, x ∈ R

such that q,r ∈ gen(C (w))

and q

≤ x

≤ r

for every i ∈ N, then x ∈ gen(C (w)).

Proof. Let l

(r)

(q)

be the corresponding

vectors in sense of Theorem 4.5. We need to ﬁnd vec-

tors l

(x)

and u

(x)

satisfying (4.1) − (4.4) of Theorem

4.5.

Let’s deﬁne vectors dq,dr ∈ R

= x

− q

= r

− x

Finally, we deﬁne l

(x)

and u

(x)

in this way:

(x)

= dq

+ l

(q)

(x)

= dr

+ u

(r)

We need to check that we satisfy (4.1) − (4.4) for

x, l

(x)

and u

(x)

We will show (4.2) only, since remain-

ing ones can be done in a similar way.

∑

i∈N

− l

(x)

) =

∑

i∈N

− dq

− l

(q)

)

∑

i∈N

− x

+ q

− l

(q)

)

∑

i∈N

− l

(q)

)

= w(N).

For games with additive border games (see Deﬁ-

nition 2.8), we got a following result.

Theorem 4.8. For an interval game (N, w)

with additive border games, the payoff vector

(w(1),w(2),. .., w(n)) ∈ gen(C (w)).

Proof. First, let us look on an arbitrary additive game

(A,v

). From additivity condition and the fact that

we can write any subset of A as a union of one-

player sets we conclude that v

(A) =

i∈A

({i}) for

every coalition A. This implies that vector a with

= v

({i}) is in the core.

This argument can be applied to border games of

(N,w). The vector q ∈ R

with q

= w(i) is an ele-

ment of the core of (N,w) and an element of S C (w).

For the vector q we want to satisfy the mixed sys-

tem (4.1)-(4.4) of Theorem 4.5.

Take the vector l containing zeros only and the

vector u with u

= |w|(i). From the additivity, we get

that

∑

i∈N

− l

= w(N) and

∑

i∈N

+ u

= w(N).

Additivity further implies that inequalities (4.3)

and (4.4) hold for q, l and u. Therefore, q is an el-

ement of gen(C (w)).

Theorem 4.8 implies that for games with ad-

ditive border games, we need to check the exis-

tence of vectors l and u from (4.1) − (4.4) of The-

orem 4.5 for maximal vectors of S C only. That

follows from the fact that for any vector y ∈

S C (w) holds (w(1), w(2),.. .,w(n)) ≤ y. In other

words, (w(1),w(2), ... ,w(n)) is a minimum vector of

S C (w).

Selection-basedApproachtoCooperativeIntervalGames

5 STRONG IMPUTATION AND

CORE

In this subsection, our focus will be on a new concept

of strong imputation and strong core.

Deﬁnition 5.1. (Strong imputation) For a game

(N,w) ∈ IG

a strong imputation is a vector x ∈ R

such that x is an imputation for every selection of

(N,w).

Deﬁnition 5.2. (Strong core) For a game (N, w) ∈

the strong core is the union of vectors x ∈ R

such that x is an element of core of every selection of

(N,w).

Strong imputation and strong core can be consid-

ered as somewhat “universal” solutions. We show the

following three simple facts about the strong core.

Proposition 5.3. For every interval game with

nonempty strong core, w(N) is a degenerate interval.

Proof. Easily by the fact that an element c of strong

core must be efﬁcient for every selection and therefore

∑

i∈N

= w(N) = w(N).

This leads us to characterizing games with

nonempty strong core.

Proposition 5.4. (Strong core characterization) An

interval game (N,w) has a nonempty strong core if

and only if w(N) is a degenerate interval and the up-

per game w has a nonempty core.

Proof. Combination of Proposition 5.3 and the fact

that an element c of the strong core has to satisfy

∑

i∈S

≥ v(S), ∀v ∈ Sel(w), ∀S ∈ 2

0. We see

that this fact is equivalent to condition

∑

i∈S

≥

w(S), ∀S ∈ 2

0. Proving an equivalence is then

straightforward.

Strong core also has the following important prop-

erty.

Proposition 5.5. For every element c of the strong

core of (N,w), c ∈ gen(C (w)).

Proof. The vector c has to satisfy mixed system

(4.1)-(4.4) of Theorem 4.5 for some l, u ∈ IR

. We

show that l

= u

= 0 will do the thing.

Equations (4.1) and (4.2) are satisﬁed by taking

Proposition 5.3 into account. Inequalities (4.3) and

(4.4) are satisﬁed as the consequence of Proposition

5.4.

Note 5.6. The reason behind the using of name strong

core and strong imputation comes form the interval

linear algebra, where strong solutions of interval sys-

tem are a solutions for any realization (selection) of

interval matrices A and b in Ax = b.

Note 5.7. One could say why we did not introduce

a strong game as game in which each of its selec-

tion does have an nonempty core. This is because

such games are already deﬁned as strongly balanced

games (see e.g. (Alparslan-G

ok et al., 2008)).

6 CONCLUDING REMARKS

Selections of an interval game are very important,

since they do not contain any additional uncertainty.

On the top of that, selection-based classes and strong

core and imputation have crucial property that al-

though we deal with uncertain data, all possible out-

comes preserve important properties. In case of se-

lection classes it is preserving superadditivity, super-

modularity etc. In case of strong core it is an invariant

of having particular stable payoffs in each selection.

Furthermore, “weak” concepts like S C are important

as well since if S C is empty, no selection has stable

payoff.

The importance of studying selection-based

classes instead of the existing classes using  oper-

ator can be further illustrated by the following two

facts:

• Classes based on weakly better operator may con-

tain games with selections that do not have any link

with the properties of their border games and con-

sequently no link with the name of the class. For

example, superadditive interval games may contain

a selection that is not superadditive.

• Selection-based classes are not contained in corre-

sponding classes based on weakly better operator.

Therefore, the results on existing classes are not di-

rectly extendable to selection-based classes.

Our results provide an important tool for handling

cooperative situations involving interval uncertainty

which is a very common situation in various OR prob-

lems. Some of the speciﬁc applications of interval

games were already examined. See (Alparslan-G

et al., 2014) (Alparslan-G

ok et al., 2009a) (Alparslan-

ok et al., 2013) for applications to mountain sit-

uations, airport games and forest situations, respec-

tively. However, these papers do not use a selection-

based approach and therefore to study implications of

our approach to them can be an interesting theme for

another research.

To further study properties of selection-based

classes is a possible topic. Fruitful direction can be

an extension of the deﬁnition of stable set to interval

games in a selection way. For example, one could ex-

amine a union or an intersection of sets of stable sets

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

for each selection. Studying nucleolus and other con-

cepts in interval games context may be interesting as

well.

ACKNOWLEDGEMENTS

The authors were supported by the Czech Science

Foundation Grant P402/13-10660S.

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Selection-basedApproachtoCooperativeIntervalGames