Coalitional Games with Interval-Type Payoffs: A Survey

6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011

Cooperative Game Theory. Operations Research
Games. Applications to Interval Games
Lecture 6: Coalitional Games with Interval-Type Payoﬀs: A
Survey

Sırma Zeynep Alparslan G¨k
o
S¨leyman Demirel University
u
Faculty of Arts and Sciences
Department of Mathematics
Isparta, Turkey
email:zeynepalparslan@yahoo.com

August 13-16, 2011


Outline

Introduction

Cooperative interval games

Classes of cooperative interval games

Economic and OR situations with interval data

Handling interval solutions

Final remarks and outlook

References

Introduction

Introduction

This lecture is based on the book
Cooperative Interval Games: Theory and Applications
by Alparslan G¨k published by
o
Lambert Academic Publishing (LAP).
For more information please see:
http://www.morebooks.de/store/gb/book/cooperative-
interval-games/isbn/978-3-8383-3430-1
The book is the PhD thesis of Alparslan G¨k entitled
o


from Middle East Technical University, Ankara-Turkey.

Introduction

Motivation

Game theory:
Mathematical theory dealing with models of conﬂict and
cooperation.
Many interactions with economics and with other areas such
as Operations Research (OR) and social sciences.
Tries to come up with fair divisions.
A young ﬁeld of study:
The start is considered to be the book Theory of Games and
Economic Behaviour by von Neumann and Morgernstern
(1944).
Two parts: non-cooperative and cooperative.

Introduction

Motivation continued...

Cooperative game theory deals with coalitions which
coordinate their actions and pool their winnings.
The main problem: How to divide the rewards or costs among
the members of the formed coalition?
Generally, the situations are considered from a deterministic
point of view.
Basic models in which probability and stochastic theory play a
role are: chance-constrained games and cooperative games
with stochastic/random payoﬀs.

Introduction

Motivation continued...
Idea of interval approach:
In most economic and OR situations rewards/costs are not
precise.
Possible to estimate the intervals to which rewards/costs
belong.
Why cooperative interval games are important?
Useful for modeling real-life situations.
Aim: generalize the classical theory to intervals and apply it to
economic situations and OR situations.
In this study, rewards/costs taken into account are not
random variables, but just closed and bounded intervals of
real numbers with no probability distribution attached.

Introduction

Interval calculus

I (R): the set of all closed and bounded intervals in R
I , J ∈ I (R), I = I , I , J = J, J , |I | = I − I , α ∈ R+
addition: I + J = I + J, I + J
multiplication: αI = αI , αI
subtraction: deﬁned only if |I | ≥ |J|
I − J = I − J, I − J
weakly better than: I J if and only if I ≥ J and I ≥ J
I J if and only if I ≤ J and I ≤ J
better than: I J if and only if I J and I = J
I J if and only if I J and I = J


Classical cooperative games versus cooperative interval
games
< N, v >, N := {1, 2, ..., n}: set of players
v : 2N → R: characteristic function, v (∅) = 0
v (S): worth (or value) of coalition S
G N : the class of all coalitional games with player set N
< N, w >, N: set of players
w : 2N → I (R): characteristic function, w (∅) = [0, 0]
w (S) = [w (S), w (S)]: worth (value) of S
IG N :the class of all interval games with player set N
Example (LLR-game): Let < N, w > be an interval game with
w ({1, 3}) = w ({2, 3}) = w (N) = J [0, 0] and w (S) = [0, 0]
otherwise.


Arithmetic of interval games

w1 , w2 ∈ IG N , λ ∈ R+ , for each S ∈ 2N
w1 w2 if w1 (S) w2 (S)
< N, w1 + w2 > is defined by (w1 + w2 )(S) = w1 (S) + w2 (S).
< N, λw > is defined by (λw )(S) = λ · w (S).
Let w1 , w2 ∈ IG N such that |w1 (S)| ≥ |w2 (S)| for each
S ∈ 2N . Then < N, w1 − w2 > is defined by
(w1 − w2 )(S) = w1 (S) − w2 (S).
Classical cooperative games associated with < N, w >
Border games: < N, w > and < N, w >
Length game: < N, |w | >, where |w | (S) = w (S) − w (S) for
each S ∈ 2N .


Preliminaries on classical cooperative games
< N, v > is called a balanced game if for each balanced map
λ : 2N {∅} → R+ we have

λ(S)v (S) ≤ v (N).
S∈2N {∅}

The core (Gillies (1959)) C (v ) of v ∈ G N is deﬁned by

C (v ) = x ∈ RN | xi = v (N); xi ≥ v (S), ∀S ∈ 2N .
i∈N i∈S

Theorem (Bondareva (1963), Shapley (1967)): Let < N, v > be an
n-person game. Then, the following two assertions are equivalent:
(i) C (v ) = ∅.
(ii) < N, v > is a balanced game.


Selection-based solution concepts

Let < N, w > be an interval game.
v is called a selection of w if v (S) ∈ w (S) for each S ∈ 2N .
Sel(w ): the set of selections of w

The core set of an interval game < N, w > is deﬁned by

C (w ) := ∪ {C (v )|v ∈ Sel(w )} .


Selection-based solution concepts
An interval game < N, w > is strongly balanced if for each
balanced map λ it holds that
λ(S)w (S) ≤ w (N).
S∈2N {∅}

Proposition: Let < N, w > be an interval game. Then, the
following three statements 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.
Proof: (i) ⇔ (ii) follows from Bondareva-Shapley theorem.
(i) ⇔ (iii) follows using w (N) ≤ v (N) ≤ w (N) and
λ(S)w (S) ≤ λ(S)v (S) ≤ λ(S)w (S)
S∈2N {∅} S∈2N {∅} S∈2N {∅}

for each balanced map λ.


Interval solution concepts
I (R)N : set of all n-dimensional vectors with elements in I (R).
The interval imputation set:

I(w ) = (I1 , . . . , In ) ∈ I (R)N | Ii = w (N), Ii w (i), ∀i ∈ N .
i∈N
The interval core:

C(w ) = (I1 , . . . , In ) ∈ I(w )| Ii w (S), ∀S ∈ 2N {∅} .
i∈S
Example (LLR-game) continuation:

C(w ) = (I1 , I2 , I3 )| Ii = J, Ii w (S) ,
i∈N i∈S
C(w ) = {([0, 0], [0, 0], J)} .


Classical cooperative games (Part I in Branzei, Dimitrov
and Tijs (2008))

< N, v > is convex if and only if the supermodularity condition

v (S ∪ T ) + v (S ∩ T ) ≥ v (S) + v (T )

for each S, T ∈ 2N holds.
< N, v > is concave if and only if the submodularity condition

v (S ∪ T ) + v (S ∩ T ) ≤ v (S) + v (T )

for each S, T ∈ 2N holds.


Convex and concave interval games

< N, w > is supermodular if

w (S) + w (T ) w (S ∪ T ) + w (S ∩ T ) for all S, T ∈ 2N .

< N, w > is convex if w ∈ IG N is supermodular and
|w | ∈ G N is supermodular (or convex).
< N, w > is submodular if

w (S) + w (T ) w (S ∪ T ) + w (S ∩ T ) for all S, T ∈ 2N .

< N, w > is concave if w ∈ IG N is submodular and |w | ∈ G N
is submodular (or concave).


Illustrative examples

Example 1: Let < N, w > be the two-person interval game with
w (∅) = [0, 0], w ({1}) = w ({2}) = [0, 1] and w (N) = [3, 4].
Here, < N, w > is supermodular and the border games are convex,
but |w | ({1}) + |w | ({2}) = 2 > 1 = |w | (N) + |w | (∅).
Hence, < N, w > is not convex.
Example 2: Let < N, w > be the three-person interval game with
w ({i}) = [1, 1] for each i ∈ N,
w (N) = w ({1, 3}) = w ({1, 2}) = w ({2, 3}) = [2, 2] and
w (∅) = [0, 0].
Here, < N, w > is not convex, but < N, |w | > is supermodular,
since |w | (S) = 0, for each S ∈ 2N .


Example (unanimity interval games):
Let J ∈ I (R) such that J [0, 0] and let T ∈ 2N {∅}. The
unanimity interval game based on T is deﬁned for each S ∈ 2N by

J, T ⊂S
uT ,J (S) =
[0, 0] , otherwise.

< N, |uT ,J | > is supermodular, < N, uT ,J > is supermodular:

uT ,J (A ∪ B) uT ,J (A ∩ B) uT ,J (A) uT ,J (B)
T ⊂ A, T ⊂B J J J J
T ⊂ A, T ⊂B J [0, 0] J [0, 0]
T ⊂ A, T ⊂B J [0, 0] [0, 0] J
T ⊂ A, T ⊂B J or [0, 0] [0, 0] [0, 0] [0, 0].


Size monotonic interval games

< N, w > is size monotonic if < N, |w | > is monotonic, i.e.,
|w | (S) ≤ |w | (T ) for all S, T ∈ 2N with S ⊂ T .
SMIG N : the class of size monotonic interval games with
player set N.
For size monotonic games, w (T ) − w (S) is deﬁned for all
S, T ∈ 2N with S ⊂ T .
CIG N : the class of convex interval games with player set N.
CIG N ⊂ SMIG N because < N, |w | > is supermodular implies
that < N, |w | > is monotonic.


I-balanced interval games
< N, w > is I-balanced if for each balanced map λ

λS w (S) w (N).
S∈2N {∅}

IBIG N : class of interval balanced games with player set N.
CIG N ⊂ IBIG N
CIG N ⊂ (SMIG N ∩ IBIG N )

Theorem: Let w ∈ IG N . Then the following two assertions are
equivalent:
(i) C(w ) = ∅.
(ii) The game w is I-balanced.


Solution concepts for cooperative interval games
Π(N): set of permutations, σ : N → N, of N
Pσ (i) = r ∈ N|σ −1 (r ) < σ −1 (i) : set of predecessors of i in σ
The interval marginal vector mσ (w ) of w ∈ SMIG N w.r.t. σ:

miσ (w ) = w (Pσ (i) ∪ {i}) − w (Pσ (i))

for each i ∈ N.
Interval Weber set W : SMIG N I (R)N :

W(w ) = conv {mσ (w )|σ ∈ Π(N)} .

Example: N = {1, 2}, w ({1}) = [1, 3], w ({2}) = [0, 0] and
w ({1, 2}) = [2, 3 1 ]. This game is not size monotonic.
2
m(12) (w )is not deﬁned.
w (N) − w ({1}) = [1, 1 ]: undeﬁned since |w (N)| < |w ({1})|.
2


The interval Shapley value
The interval Shapley value Φ : SMIG N → I (R)N :

1
Φ(w ) = mσ (w ), for each w ∈ SMIG N .
n!
σ∈Π(N)

Example: N = {1, 2}, w ({1}) = [0, 1], w ({2}) = [0, 2],
w (N) = [4, 8].

1
Φ(w ) = (m(12) (w ) + m(21) (w ));
2
1
Φ(w ) = ((w ({1}), w (N) − w ({1})) + (w (N) − w ({2}), w ({2}))) ;
2
1 1 1
Φ(w ) = (([0, 1], [4, 7]) + ([4, 6], [0, 2])) = ([2, 3 ], [2, 4 ]).
2 2 2


Properties of solution concepts

W(w ) ⊂ C(w ), ∀w ∈ CIG N and W(w ) = C(w ) is possible.
Example: N = {1, 2}, w ({1}) = w ({2}) = [0, 1] and
w (N) = [2, 4] (convex).
W(w ) = conv m(1,2) (w ), m(2,1) (w )
m(1,2) (w ) = ([0, 1], [2, 4] − [0, 1]) = ([0, 1], [2, 3])
m(2,1) (w ) = ([2, 3], [0, 1]])
m(1,2) (w ) and m(2,1) (w ) belong to C(w ).
([ 2 , 1 4 ], [1 1 , 2 4 ]) ∈ C(w )
1 3
2
1

no α ∈ [0, 1] exists satisfying
αm(1,2) (w ) + (1 − α)m(2,1) (w ) = ([ 1 , 1 4 ], [1 1 , 2 1 ]).
2
3
2 4
Φ(w ) ∈ W(w ) for each w ∈ SMIG N .
Φ(w ) ∈ C(w ) for each w ∈ CIG N .


The square operator
Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) with a ≤ b.
Then, we denote by a b the vector

a b := ([a1 , b1 ] , . . . , [an , bn ]) ∈ I (R)N

generated by the pair (a, b) ∈ RN × RN .
Let A, B ⊂ RN . Then, we denote by A B the subset of
I (R)N deﬁned by

A B := {a b|a ∈ A, b ∈ B, a ≤ b} .

For a multi-solution F : G N RN we deﬁne
F : IG N I (R)N by F = F(w ) F(w ) for each w ∈ IG N .


Square solutions and related results
C (w ) = C (w ) C (w ) for each w ∈ IG N .
Example: N = {1, 2}, w ({1}) = [0, 1], w ({2}) = [0, 2],
w (N) = [4, 8].
1 1
(2, 2) ∈ C (w ), (3 , 4 ) ∈ C (w ).
2 2
1 1 1 1
(2, 2) (3 , 4 ) = ([2, 3 ], [2, 4 ]) ∈ C (w ) C (w ).
2 2 2 2

C(w ) = C (w ) for each w ∈ IBIG N .
W (w ) = W (w ) W (w ) for each w ∈ IG N .
C(w ) ⊂ W (w ) for each w ∈ IG N .
C (w ) = W (w ) for each w ∈ CIG N .
W(w ) ⊂ W (w ) for each w ∈ CIG N .


Classical cooperative games
Theorem (Shapley (1971) and Shapley-Weber-Ichiishi (1981,
1988)):
Let v ∈ G N . The following ﬁve assertions are equivalent:
(i) < N, v > is convex.
(ii) For all S1 , S2 , U ∈ 2N with S1 ⊂ S2 ⊂ N U

v (S1 ∪ U) − v (S1 ) ≤ v (S2 ∪ U) − v (S2 ).

(iii) For all S1 , S2 ∈ 2N and i ∈ N such that S1 ⊂ S2 ⊂ N {i}

v (S1 ∪ {i}) − v (S1 ) ≤ v (S2 ∪ {i}) − v (S2 ).

(iv) Each marginal vector mσ (v ) of the game v with respect to
the permutation σ belongs to the core C (v ).
(v) W (v ) = C (v ).


Basic characterizations for convex interval games

Theorem:
Let w ∈ IG N be such that |w | ∈ G N is supermodular. Then, the
following three assertions are equivalent:
(i) w ∈ IG N is convex.
(ii) For all S1 , S2 , U ∈ 2N with S1 ⊂ S2 ⊂ N U

w (S1 ∪ U) − w (S1 ) w (S2 ∪ U) − w (S2 ).

(iii) For all S1 , S2 ∈ 2N and i ∈ N such that S1 ⊂ S2 ⊂ N {i}

w (S1 ∪ {i}) − w (S1 ) w (S2 ∪ {i}) − w (S2 ).


Basic characterizations of convex interval games

Proposition:
Let w ∈ IG N . Then the following assertions hold:
(i) A game < N, w > is supermodular if and only if its border
games < N, w > and < N, w > are convex.
(ii) A game < N, w > is convex if and only if its length game
< N, |w | > and its border games < N, w >, < N, w > are
convex.
(iii) A game < N, w > is convex if and only if its border game
< N, w > and the game < N, w − w > are convex.



Theorem: Let w ∈ IBIG N . Then, the following assertions are
equivalent:
(i) w is convex.
(ii) |w | is supermodular and C(w ) = W (w ).
Proof: By (ii) of Proposition, w is convex if and only if |w | , w and
w are convex. Clearly, the convexity of |w | is equivalent with its
supermodularity.
Further, w and w are convex if and only if W (w ) = C (w ) and
W (w ) = C (w ).
These equalities are equivalent with W (w ) = C (w ). Finally,
since w is I-balanced by hypothesis, we have C(w ) = W (w ).



Theorem: Let w ∈ IG N . Then, the following assertions are
equivalent:
(i) w is convex.
(ii) |w | is supermodular and mσ (w ) ∈ C(w ) for all σ ∈ Π(N).

Proposition: Let w ∈ CIG N . Then, W(w ) ⊂ C(w ).
Proof: By the above theorem we have mσ (w ) ∈ C(w ) for each
σ ∈ Π(N). Now, we use the convexity of C(w ).


Population interval monotonic allocation schemes (pmias)
(inspired by Sprumont (1990))
For a game w ∈ IG N and a coalition S ∈ 2N {∅}, the interval
subgame with player set T is the game wT deﬁned by
wT (S) := w (S) for all S ∈ 2T .
T IBIG N : class of totally I-balanced interval games (interval
games whose all subgames are I-balanced) with player set N.
We say that for a game w ∈ T IBIG N a scheme
A = (AiS )i∈S,S∈2N {∅} with AiS ∈ I (R)N is a pmias of w if
(i) i∈S AiS = w (S) for all S ∈ 2N {∅},
(ii) AiS AiT for all S, T ∈ 2N {∅} with S ⊂ T and for each
i ∈ S.


Population interval monotonic allocation schemes

A pmias allocates a larger payoﬀ to each player as the coalitions
grow larger.
In order to take the possibility of partial cooperation a pmias
speciﬁes not only how to allocate w (N) but also how to allocate
w (S) of every coalition S ∈ 2N {∅}.

We say that for a game w ∈ CIG N an imputation
I = (I1 , . . . , In ) ∈ I(w ) is pmias extendable if there
exists a pmas A = (AiS )i∈S,S∈2N {∅} such that
AiN = Ii for each i ∈ N.

Theorem: Let w ∈ CIG N . Then each element I of W(w ) is
extendable to a pmias of w .


Population interval monotonic allocation schemes
Example: Let w ∈ CIG N with w (∅) = [0, 0],
w ({1}) = w ({2}) = w ({3}) = [0, 0],
w ({1, 2}) = w ({1, 3}) = w ({2, 3}) = [2, 4] and w (N) = [9, 15]. It
is easy to check that the interval Shapley value for this game
generates the pmias depicted as
 

 1 2 3 

N 
 [3, 5] [3, 5] [3, 5] 

{1, 2} 
 [1, 2] [1, 2] ∗ 

{1, 3} 
 [1, 2] ∗ [1, 2] .

{2, 3} 
 ∗ [1, 2] [1, 2] 

{1} 
 [0, 0] ∗ ∗ 

{2}  ∗ [0, 0] ∗ 
{3} ∗ ∗ [0, 0]


Classical big boss games versus big boss interval games
Classical big boss games (Muto et al. (1988), Tijs (1990)):
< N, v > is a big boss game with n as big boss if
(i) v ∈ G N is monotonic, i.e.,
v (S) ≤ v (T ) if for each S, T ∈ 2N with S ⊂ T ;
(ii) v (S) = 0 if n ∈ S;
/
(iii) v (N) − v (S) ≥ i∈NS (v (N) − v (N {i}))
for all S, T with n ∈ S ⊂ N.
Big boss interval games:
< N, w > is a big boss interval game if < N, w > and
< N, w − w > are classical (total) big boss games.
BBIG N : the class of big boss interval games.
Marginal contribution of each player i ∈ N to the grand coalition:
Mi (w ) := w (N) − w (N {i}).


Properties of big boss interval games
Theorem: Let w ∈ SMIG N . Then, the following conditions are
equivalent:
(i) w ∈ BBIG N .
(ii) < N, w > satisﬁes
(a) Veto power property:
w (S) = [0, 0] for each S ∈ 2N with n ∈ S.
/
(b) Monotonicity property:
w (S) w (T ) for each S, T ∈ 2N with n ∈ S ⊂ T .
(c) Union property:

w (N) − w (S) (w (N) − w (N {i}))
i∈NS

for all S with n ∈ S ⊂ N.


T -value (inspired by Tijs(1981))

the big boss interval point: B(w ) := ([0, 0], . . . , [0, 0], w (N));
the union interval point:
n−1
U(w ) := (M1 (w ), . . . , Mn−1 (w ), w (N) − Mi (w )).
i=1

The T -value T : BBIG N → I (R)N is deﬁned by

1
T (w ) := (U(w ) + B(w )).
2


Holding situations with interval data
Holding situations: one agent has a storage capacity and other
agents have goods to store to generate benefits.
In classical cooperative game theory, holding situations are
modeled by using big boss games (Tijs, Meca and L´pez (2005)).
o
For a holding situation with interval data one can construct a
holding interval game which turns out to be a big boss interval
game.

Example: Player 3 is the owner of a holding house which has
capacity for one container. Players 1 and 2 have each one
container which they want to store. If player 1 is allowed to store
his/her container, then the benefit belongs to [10, 30] and if player
2 is allowed to store his/her container, then the benefit belongs to
[50, 70].


Example continues ...

The situation described corresponds to an interval game as follows:

The interval game < N, w > with N = {1, 2, 3} and
w (S) = [0, 0] if 3 ∈ S, w (∅) = w ({3}) = [0, 0],
/
w ({1, 3}) = [10, 30] and w (N) = w ({2, 3}) = [50, 70] is a big
boss interval game with player 3 as big boss.
B(w ) = ([0, 0], [0, 0], [50, 70]) and
U(w ) = ([0, 0], [40, 40], [10, 30]) are the elements of the
interval core.
T (w ) = ([0, 0], [20, 20], [30, 50]) ∈ C(w ).


Bi-monotonic interval allocation schemes (inspired by
Branzei, Tijs and Timmer (2001))

Pn : the set {S ⊂ N|n ∈ S} of all coalitions containing the big
boss.
Take a game w ∈ BBIG N with n as a big boss.
We call a scheme B := (BiS )i∈S,S∈Pn an (interval) allocation
scheme for w if (BiS )i∈S is an interval core element of the
subgame < S, w > for each coalition S ∈ Pn . Such an allocation
scheme B = (BiS )i∈S,S∈Pn is called a bi-monotonic (interval)
allocation scheme (bi-mias) for w if for all S, T ∈ Pn with S ⊂ T
we have BiS BiT for all i ∈ S {n}, and BnS BnT .
Remark: In a bi-mias the big boss is weakly better oﬀ in larger
coalitions, while the other players are weakly worse oﬀ.


Bi-monotonic interval allocation schemes
We say that for a game w ∈ BBIG N with n as a big boss, an
imputation I = (I1 , . . . , In ) ∈ I(w ) is bi-mias extendable if
there exists a bi-mas B = (BiS )i∈S,S∈Pn such that BiN = Ii for
each i ∈ N.
Theorem: Let w ∈ BBIG N with n as a big boss and let I ∈ C(w ).
Then, I is bi-mias extendable.
Example continues: The T -value generates a bi-mias represented
by the following matrix:
 
1 2 3
N  [0, 0] [20, 20] [30, 50] 
 
{1, 3}  [5, 15]
 ∗ [5, 15]  .

{2, 3}  ∗ [25, 35] [25, 35] 
{3} ∗ ∗ [0, 0]


Airport situations with interval data
In airport situations, the costs of the coalitions are considered
(Driessen (1988)):
One runway and m types of planes (P1 , . . . , Pm pieces of the
runway: P1 for type 1, P1 and P2 for type 2, etc.).
Tj [0, 0]: the interval cost of piece Pj .
Nj : the set of players who own a plane of type j.
nj : the number of (owners of) planes of type j.
< N, d > is given by
N = ∪m Nj : the set of all users of the runway;
j=1
d(∅) = [0, 0], d(S) = j Ti
i=1
if S ∩ Nj = ∅, S ∩ Nk = ∅ for all j + 1 ≤ k ≤ m.
S needs the pieces P1 , . . . , Pj of the runway. The interval cost of
the used pieces of the runway is j Ti . i=1



m ∗
Formally, d = k=1 Tk u∪m Nr , where
r =k

∗ 1, K ∩ S = ∅
uK (S) :=
0, otherwise.

Interval Baker-Thompson allocation for a player i of type j:
j m
γi := ( nr )−1 Tk .
k=1 r =k

Proposition: Interval Baker-Thompson allocation agrees with the
interval Shapley value Φ(d).


Proposition: Let < N, d > be an airport interval game. Then,
< N, d > is concave.

Proof: It is well known that non-negative multiples of classical dual
unanimity games are concave (or submodular). By formal
deﬁnition of d the classical games d = m T k uk,m and
k=1
∗
m ∗
|d| = k=1 |Tk | uk,m are concave because T k ≥ 0 and |Tk | ≥ 0
for each k, implying that < N, d > is concave.

Proposition: Let (N, (Tk )k=1,...,m ) be an airport situation with
interval data and < N, d > be the related airport interval game.
Then, the interval Baker-Thompson rule applied to this airport
situation provides an allocation which belongs to C(d).


Example:
< N, d > airport interval game interval costs: T1 = [4, 6],
T2 = [1, 8],
d(∅) = [0, 0], d(1) = [4, 6], d(2) = d(1, 2) = [4, 6] + [1, 8] = [5, 14],
∗ ∗
d = [4, 6]u{1,2} + [1, 8]u{2} ,
Φ(d) = ( 1 ([4, 6] + [0, 0]), 2 ([1, 8] + [5, 14])) = ([2, 3], [3, 11]),
2
1
1 1
γ = ( 2 [4, 6], 2 [4, 6] + [1, 8]) = ([2, 3], [3, 11]) ∈ C(d).


Sequencing situations with interval data
Sequencing situations with one queue of players, each with one
job, in front of a machine order. Each player must have his/her job
processed on this machine, and for each player there is a cost
according to the time he/she spent in the system (Curiel, Pederzoli
and Tijs (1989)).

A one-machine sequencing interval situation is described as a
4-tuple (N, σ0 , α, p),
σ0 : a permutation deﬁning the initial order of the jobs
α = ([αi , αi ])i∈N ∈ I (R+ )N , p = ([p i , p i ])i∈N ∈ I (R+ )N : vectors
of intervals with αi , αi representing the minimal and maximal
unitary cost of the job of i, respectively, p i , p i being the minimal
and maximal processing time of the job of i, respectively.



To handle such sequencing situations, we propose to use either
the approach based on urgency indices or the approach based
on relaxation indices. This requires to be able to compute
α p p
either ui = p i , αi (for each i ∈ N) or ri = αi , αii (for each
p i i i
i ∈ N), and such intervals should be pair-wise disjoint.
Interval calculus: Let I , J ∈ I (R+ ).
We deﬁne · : I (R+ ) × I (R+ ) → I (R+ ) by I · J := [I J, I J].
Let Q := (I , J) ∈ I (R+ ) × I (R+ {0}) | I J ≤ I J .
I I I
We deﬁne ÷ : Q → I (R+ ) by J := [ J , J ] for all (I , J) ∈ Q.


Example (a): Consider the two-agent situation with
p1 = [1, 4], p2 = [6, 8], α1 = [5, 25], α2 = [10, 30]. We can compute
4
5
u1 = 5, 25 , u2 = 3 , 15 and use them to reorder the jobs as the
4
intervals are disjoint.

Example (b): Consider the two-agent situation with
p1 = [1, 3], p2 = [4, 6], α1 = [5, 6], α2 = [11, 12]. Here, we can
compute r1 = 5 , 1 , r2 = 11 , 1 , but we cannot reorder the jobs
1
2
4
2
as the intervals are not disjoint.

Example (c): Consider the two-agent situation with
p1 = [1, 3], p2 = [5, 8], α1 = [5, 6], α2 = [10, 30]. Now, r1 is defined
but r2 is undefined. On the other hand, u1 is undefined and u2 is
defined, so no comparison is possible; consequently, the reordering
cannot take place.


Let i, j ∈ N. We deﬁne the interval gain of the switch of jobs i and
j by
αj pi − αi pj , if jobs i and j switch
Gij :=
[0,0], otherwise.
The sequencing interval game:

w := Gij u[i,j] .
i,j∈N:i<j

Gij ∈ I (R) for all switching jobs i, j ∈ N and
u[i,j] is the unanimity game deﬁned as:

1, if {i, i + 1, ..., j − 1, j} ⊂ S
u[i,j] (S) :=
0, otherwise.


The interval equal gain splitting rule is deﬁned by
1
IEGSi (N, σ0 , α, p) = 2 Gij + 12 Gij , for each
j∈N:i<j j∈N:i>j
i ∈ N.
Proposition: Let < N, w > be a sequencing interval game. Then,
1
i) IEGS(N, σ0 , α, p) = 2 (m(1,2...,n) (w ) + m(n,n−1,...,1) (w )).
ii) IEGS(N, σ0 , α, p) ∈ C(w ).
Proposition: Let < N, w > be a sequencing interval game. Then,
< N, w > is convex.
Example: Consider the interval situation with N = {1, 2},
σ0 = {1, 2}, p = (2, 3) and α = ([2, 4], [12, 21]).
The urgency indices are u1 = [1, 2] and u2 = [4, 7], so that the two
jobs may be switched.
We have:
G12 = [18, 30], IEGS(N, σ0 , α, p) = ([9, 15], [9, 15]) ∈ C(w ).


Bankruptcy situations with interval data
In a classical bankruptcy situation, a certain amount of moneyd
has to be divided among some people, N = {1, . . . , n}, who have
individual claims ci , i ∈ N on the estate, and the total claim is
weakly larger than the estate. The corresponding bankruptcy game
vE ,d : vE ,d (S) = (E − i∈NS di )+ for each S ∈ 2N , where
x+ = max {0, x} (Aumann and Maschler (1985)).
A bankruptcy interval situation with a ﬁxed set of claimants
N = {1, 2, . . . , n} is a pair (E , d) ∈ I (R) × I (R)N , where
E = [E , E ] [0, 0] is the estate to be divided and d is the
vector of interval claims with the i-th coordinate di = [d i , d i ]
(i ∈ N), such that [0, 0] d1 d2 . . . dn and
E < n di.
i=1
BRI N : the family of bankruptcy interval situations with set of
claimants N.


We deﬁne a subclass of BRI N , denoted by SBRI N , consisting of all
bankruptcy interval situations such that

|d(N S)| ≤ |E | for each S ∈ 2N with d(N S) ≤ E .

We call a bankruptcy interval situation in SBRI N a strong
bankruptcy interval situation. With each (E , d) ∈ SBRI N we
associate a cooperative interval game < N, wE ,d >, deﬁned by

wE ,d (S) := [vE ,d (S), vE ,d (S)] for each S ⊂ N.

SBRIG N : the family of all bankruptcy interval games wE ,d
with (E , d) ∈ SBRI N .



Example: Let (E , d) be a two-person bankruptcy situation.
We suppose that the claims of the players are closed intervals
d1 = [70, 70] and d2 = [80, 80], respectively,
and the estate is E = [100, 140].
Then, the corresponding game < N, wE ,d > is given by

wE ,d (∅) = [0, 0], wE ,d (1) = [20, 60], wE ,d (2) = [30, 70]

and wE ,d (1, 2) = [100, 140].

This game is supermodular, but is not convex because
|wE ,d | ∈ G N is not convex.


How to use interval games and their solutions in
interactive situations
Stage 1 (before cooperation starts):
with N = {1, 2, . . . , n} set of participants with interval data ⇒
interval game < N, w > and interval solutions ⇒ agreement for
cooperation based on an interval solution ψ and signing a binding
contract (specifying how the achieved outcome by the grand
coalition should be divided consistently with Ji = ψi (w ) for each
i ∈ N.

Stage 2 (after the joint enterprise is carried out):
The achieved reward R ∈ w (N) is known; apply the agreed upon
protocol speciﬁed in the binding contract to determine the
individual shares xi ∈ Ji .
Natural candidates for rules used in protocols are bankruptcy rules.



Example:
w (1) = [0, 2], w (2) = [0, 1] and w (1, 2) = [4, 8].
1 1
Φ(w ) = ([2, 4 2 ], [2, 3 2 ]). R = 6 ∈ [4, 8]; choose proportional rule
(PROP) deﬁned by

di
PROPi (E , d) := E
j∈N dj

for each bankruptcy problem (E , d) and all i ∈ N.
(Φ1 (w ), Φ2 (w )) +
PROP(R − Φ1 (w ) − Φ2 (w ); Φ1 (w ) − Φ1 (w ), Φ2 (w ) − Φ2 (w ))
1 1
= (2, 2) + PROP(6 − 2 − 2; (2 2 , 1 2 ))
1 3
= (3 4 , 2 4 ).


Conclusion and future work

The State-of-the-art of interval game literature:
Branzei R., Dimitrov D. and Tijs S., “Shapley-like values for
interval bankruptcy games”, Economics Bulletin 3 (2003) 1-8.
Alparslan G¨k S.Z., Branzei R., Fragnelli V. and Tijs S.,
o
“Sequencing interval situations and related games”, to appear
in Central European Journal of Operations Research (CEJOR).
Alparslan G¨k S.Z., Branzei O., Branzei R. and Tijs S.,
o
“Set-valued solution concepts using interval-type payoﬀs for
interval games”, to appear in Journal of Mathematical
Economics (JME).


Alparslan G¨k S.Z., Branzei R. and Tijs S., “Convex interval
o
games”, Journal of Applied Mathematics and Decision
Sciences, Vol. 2009, Article ID 342089, 14 pages (2009a)
DOI: 10.1115/2009/342089.
Alparslan G¨k S.Z., Branzei R. and Tijs S., “Big boss interval
o
games”, International Journal of Uncertainty, Fuzziness and
Knowledge-Based Systems (IJUFKS), Vol. 19, no:1 (2011)
pp.135-149.
Branzei R. and Alparslan G¨k S.Z., “Bankruptcy problems
o
with interval uncertainty”, Economics Bulletin, Vol. 3, no. 56
(2008) pp. 1-10.


Branzei R., Mallozzi L. and Tijs S., “Peer group situations
and games with interval uncertainty”, International Journal of
Mathematics, Game Theory, and Algebra, Vol.19, Issues 5-6
(2010).
Branzei R., Tijs S. and Alparslan G¨k S.Z., “Some
o
characterizations of convex interval games”, AUCO Czech
Economic Review, Vol. 2, no.3 (2008) 219-226.
Branzei R., Tijs S. and Alparslan G¨k S.Z., “How to handle
o
interval solutions for cooperative interval games”,
International Journal of Uncertainty, Fuzziness and
Knowledge-based Systems, Vol.18, Issue 2, (2010) 123-132.
Branzei R., Branzei O., Alparslan G¨k S.Z., Tijs S.,
o
“Cooperative interval games: a survey”, Central European
Journal of Operations Research (CEJOR), Vol.18, no.3
(2010) 397-411.


Moretti S., Alparslan G¨k S.Z., Branzei R. and Tijs S.,
o
“Connection situations under uncertainty and cost monotonic
solutions”, Computers and Operations Research, Vol.38, Issue
11 (2011) pp.1638-1645.
Branzei R., Alparslan Gk S.Z. and Branzei O., “On the
Convexity of Interval Dominance Cores”, to appear in Central
European Journal of Operations Research (CEJOR), DOI:
10.1007/s10100-010-0141-z.
Alparslan G¨k S.Z., Branzei R. and Tijs S., “Airport interval
o
games and their Shapley value”, Operations Research and
Decisions, Issue 2 (2009).
Alparslan G¨k S.Z., Miquel S. and Tijs S., “Cooperation
o
under interval uncertainty”, Mathematical Methods of
Operations Research, Vol. 69, no.1 (2009) 99-109.


Alparslan G¨k S.Z., “Cooperative interval games”, PhD
o
Dissertation Thesis, Institute of Applied Mathematics, Middle
East Technical University, Ankara-Turkey (2009).
Alparslan G¨k S.Z., Branzei R. and Tijs S., “The interval
o
Shapley value: an axiomatization”, Central European Journal
of Operations Research (CEJOR), Vol.18, Issue 2 (2010) pp.
131-140.
Future work:
Promising topic (interesting open problems exist to be
generalized in the theory of cooperative interval games).
Other OR situations and combinatorial optimization problems
with interval data can be modeled by using cooperative
interval games, e.g., ﬂow, linear production, holding
situations, ﬁnancial and energy markets.

References

References
[1]Alparslan G¨k S.Z., Cooperative Interval Games: Theory and
o
Applications, Lambert Academic Publishing (LAP), Germany
(2010) ISBN:978-3-8383-3430-1.
[2]Aumann R. and Maschler M., Game theoretic analysis of a
bankruptcy problem from the Talmud, Journal of Economic
Theory 36 (1985) 195-213.
[3] Bondareva O.N., Certain applications of the methods of linear
programming to the theory of cooperative games, Problemly
Kibernetiki 10 (1963) 119-139 (in Russian).
[4] Branzei R., Dimitrov D. and Tijs S., Models in Cooperative
Game Theory, Springer, Game Theory and Mathematical Methods
(2008).
[5] Branzei R., Tijs S. and Timmer J., Information collecting
situations and bi-monotonic allocation schemes, Mathematical
Methods of Operations Research 54 (2001) 303-313.

References

References

[6]Curiel I., Pederzoli G. and Tijs S., Sequencing games, European
Journal of Operational Research 40 (1989) 344-351.
[7]Driessen T., Cooperative Games, Solutions and Applications,
Kluwer Academic Publishers (1988).
[8] Gillies D. B., Solutions to general non-zero-sum games. In:
Tucker, A.W. and Luce, R.D. (Eds.), Contributions to the theory
of games IV, Annals of Mathematical Studies 40. Princeton
University Press, Princeton (1959) pp. 47-85.
[9]Ichiishi T., Super-modularity: applications to convex games and
to the greedy algorithm for LP, Journal of Economic Theory 25
(1981) 283-286.

References

References
[10] Muto S., Nakayama M., Potters J. and Tijs S., On big boss
games, The Economic Studies Quarterly Vol.39, No. 4 (1988)
303-321.
[11]Shapley L.S., On balanced sets and cores, Naval Research
Logistics Quarterly 14 (1967) 453-460.
[12] Shapley L.S., Cores of convex games, International Journal of
Game Theory 1 (1971) 11-26.
[13] Sprumont Y., Population Monotonic Allocation Schemes for
Cooperative Games with Transferable Utility, Games and Economic
Behavior 2 (1990) 378-394.
[14] Tijs S., Bounds for the core and the τ -value, In: Moeschlin
O., Pallaschke D. (eds.), Game Theory and Mathematical
Economics, North Holland, Amsterdam (1981) pp. 123-132.

References

References

[15] Tijs S., Big boss games, clan games and information market
games. In:Ichiishi T., Neyman A., Tauman Y. (eds.), Game Theory
and Applications. Academic Press, San Diego (1990) pp.410-412.
[16]Tijs S., Meca A. and L´pez M.A., Beneﬁt sharing in holding
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situations, European Journal of Operational Research 162 (1)
(2005) 251-269.
[17] von Neumann, J. and Morgernstern, O., Theory of Games and
Economic Behaviour, Princeton: Princeton University Press
(1944).
[18] Weber R., Probabilistic values for games, in Roth A.E. (Ed.),
The Shapley Value: Essays in Honour of Lloyd S. Shapley,
Cambridge University Press, Cambridge (1988) 101-119.

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