This paper deals with the research area of cooperative interval games arising from airport situations with interval data. We also extend to airport interval games some results from classical theory.cooperative interval games;concave games;airport games;cost games;interval data

Alparslan-Gok, S.Z.

Brânzei, R.

Tijs, S.H.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
No. 2008–57 
 
  
COOPERATIVE INTERVAL GAMES ARISING FROM AIRPORT 
SITUATIONS WITH INTERVAL DATA 
 
 
By S.Z. Alparslan Gök, R. Brânzei, S.Tijs 
 
 
 
June 2008 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISSN 0924-7815 Cooperative interval games arising
from airport situations with interval
data
S.Z. Alparslan G¨ ok ∗† R. Branzei ‡ S. Tijs §
Abstract
This paper deals with the research area of cooperative interval
games arising from airport situations with interval data. We also
extend to airport interval games some results from classical theory.
JEL Classiﬁcation: C71
Keywords: cooperative interval games, concave games, airport
games, cost games, interval data
1 Introduction
In literature much attention is paid to airport situations and related games.
We refer here to Littlechild and Owen (1977), Littlechild and Thompson
(1977) and Tijs and Driessen (1986). In airport situations costs of the coali-
tions are considered. A cost game < N,c > is a cooperative game, where N
∗Institute of Applied Mathematics, Middle East Technical University, 06531 Ankara,
Turkey and S¨ uleyman Demirel University, Faculty of Arts and Sciences, Department of
Mathematics, 32 260 Isparta, Turkey, e-mail: alzeynep@metu.edu.tr
†This author acknowledges the support of TUBITAK (Turkish Scientiﬁc and Technical
Research Council).
‡Faculty of Computer Science, “Alexandru Ioan Cuza” University, Ia¸ si, Romania, e-
mail: branzeir@infoiasi.ro
§CentER and Department of Econometrics and OR, Tilburg University, Tilburg,
The Netherlands and Department of Mathematics, University of Genoa, Italy, e-mail:
S.H.Tijs@uvt.nl
1is the set of players, and c : 2N → R is a function assigning to each coali-
tion S ∈ 2N a real number in which c(S) is the cost of the coalition S with
c(∅) = 0. A game < N,c > is called concave (or submodular) if and only
if c(S ∪ T) + c(S ∩ T) ≤ c(S) + c(T) for all S,T ∈ 2N. It is well known
that airport games are concave. The economists Baker (1965) and Thomp-
son (1971) proposed an appealing rule now called the Baker-Thompson rule.
The idea is that only users of a piece of the runway pay for that piece and
they share the cost of it equally. The core C (Gillies (1959)) and the Shapley
value φ (Shapley (1953)) are central solution concepts deﬁned on the class
of classical cooperative games. The Shapley value is a core element on the
class of concave games. Littlechild and Owen (1973) showed that the Baker-
Thompson rule corresponds to the Shapley value. For a game < N,c > and
a coalition K ∈ 2N \ {∅} the dual K-unanimity game u∗
K is deﬁned by
u
∗
K(S) =
 
1, K ∩ S  = ∅
0, otherwise,
and the Shapley value φ(u∗
K) of the dual K-unanimity game u∗
K is deﬁned by
φi(u
∗
K) =
 
1/|K|, i ∈ K
0, i ∈ N \ K.
We recall that the decomposition of an airport game < N,c > using dual
unanimity games is given by c =
 m
k=1 tku∗
∪m
r=kNr, where tk is the extra cost to
extend a runway which is already suitable for landings of planes of type k−1
to the one which is suitable for landings of planes of type k, and N = ∪m
r=1Nr
is the set of all users of the runway.
In this paper we consider airport situations, where cost of pieces of the runway
are intervals. Then, we associate as in the classical case to such a situation an
interval cost game and extend to airport interval games the results presented
above.
The rest of the paper is organized as follows. We recall in Section 2 basic
notions and facts from interval calculus and the theory of cooperative interval
games. Section 3 is devoted to airport situations with interval data and
related airport interval games. We conclude in Section 4 with some ﬁnal
remarks on other economic and OR situations with interval data.
22 Preliminaries
We start with some preliminaries from interval calculus (Alparslan G¨ ok,
Branzei and Tijs (2008a)). We denote by I(R) the set of all closed inter-
vals in R, and by I(R)N the set of all n-dimensional vectors with elements
in I(R).
Let I,J ∈ I(R) with I =
 
I,I
 
, J =
 
J,J
 
, |I| = I − I and α ∈ R+. Then,
I + J =
 
I + J,I + J
 
; αI =
 
αI,αI
 
. The partial substraction operator
I − J is deﬁned, only if |I| ≥ |J|, by I − J =
 
I − J,I − J
 
. We say that
I is weakly better than J, which we denote by I < J, if and only if I ≥ J
and I ≥ J. We also use the reverse notation J 4 I, if and only if J ≤ I and
J ≤ I.
Now, we give basic deﬁnitions and some useful results of cooperative inter-
val cost games inspired by the theory of cooperative interval reward games
(Alparslan G¨ ok, Miquel and Tijs (2008), Alparslan G¨ ok, Branzei and Tijs
(2008b)).
An interval cost game is an ordered pair < N,d > where N = {1,2,...,n}
is the set of players, and d : 2N → I(R) is the characteristic function such
that d(∅) = [0,0]. For each S ∈ 2N, the worth set (or worth interval) d(S)
of the coalition S in the interval game < N,d > is of the form [d(S),d(S)],
where d(S) is the lower bound and d(S) is the upper bound of d(S). Some
classical cooperative cost games associated with an interval game < N,d >
will play a key role, namely the border games < N,d >, < N,d > and the
length game < N,|d| >, where |d|(S) = d(S) − d(S) for each S ∈ 2N.
Let < N,d1 > and < N,d2 > be interval cost games. We say that d1 4
d2 if d1(S) 4 d2(S) for each S ∈ 2N. We deﬁne < N,d1 + d2 > by
(d1 + d2)(S) = d1(S) + d2(S) for each S ∈ 2N. For < N,d1 > and < N,d2 >
with |d1(S)| ≥ |d2(S)| for each S ∈ 2N, < N,d1 − d2 > is deﬁned by
(d1 − d2)(S) = d1(S) − d2(S). Given < N,d > and λ ∈ R+ we deﬁne
< N,λd > by (λd)(S) = λ   d(S) for each S ∈ 2N.
Let < N,d > be an interval cost game. Then, the interval core C(d) is deﬁned
by
C(d) =
 
(I1,...,In) ∈ I(R)
N|
 
i∈N
Ii = d(N),
 
i∈S
Ii 4 d(S), for all S ∈ 2
N \ {∅}
 
.
A game < N,d > is called size monotonic if < N,|d| > is monotonic, i.e.
|d|(S) ≤ |d|(T) for all S,T ∈ 2N with S ⊂ T. We denote by SMIGN the
3class of size monotonic interval games with player set N.
An interval game < N,d > is called concave if < N,d > is submodular and
< N,|d| > is concave (or submodular), i.e.
d(S) + d(T) < d(S ∪ T) + d(S ∩ T), (1)
and |d|(S) + |d|(T) ≥ |d|(S ∪ T) + |d|(S ∩ T) for all S,T ∈ 2N.
In the following we give a characterization for concave interval games.
Proposition 2.1. Let < N,d > be a concave interval game. Then the
following assertions hold:
(i) A game < N,d > is submodular if and only if < N,d > and < N,d >
are concave (or submodular);
(ii) A game < N,d > is concave if and only if < N,|d| > and < N,d >,
< N,d > are concave (or submodular);
(iii) A game < N,d > is concave if and only if < N,d > and < N,|d| >
are concave (or submodular).
Proof. (i) This assertion follows from formula (1).
(ii) By deﬁnition < N,d > is concave if and only if < N,d > and
< N,|d| > are both submodular. By (i), < N,d > is submodular if
and only if its border games are concave (or submodular). Now, since
submodularity of < N,|d| > is the same with its concavity, we conclude
that < N,d > is concave if and only if
< N,d >, < N,d > and < N,|d| > are concave (or submodular).
(iii) This assertion follows easily from (ii) by noting that < N,|d| >,
< N,d > and < N,d > are concave (or submodular) if and only if
< N,|d| > and < N,d > are concave (or submodular) because
d = d + |d|.
Note that the fact that < N,|d| > is concave (or submodular) implies
that < N,|d| > is a monotonic game because for each S,T ∈ 2N with S ⊂ T
we have |d|(S)+|d|(T \S) ≥ |d|(T)+|d|(∅), and this implies that |d|(S) ≥
|d|(T). As a by-product we have each concave interval game < N,d > is size
monotonic.
4Denote by Π(N) the set of permutations σ : N → N. Let d ∈ SMIGN
and σ ∈ Π(N). The interval marginal vector of d with respect to σ, mσ(d),
corresponds to a situation, where the players enter a room one by one in the
order σ(1),σ(2),...,σ(n) and each player is given the marginal contribution
he/she creates by entering. We denote the set of predecessors of i in σ by
Pσ(i) = {r ∈ N|σ−1(r) < σ−1(i)}, where σ−1(i) denotes the entrance number
of player i, and deﬁne
m
σ
i (d) = d(Pσ(i) ∪ {i}) − d(Pσ(i)) for each i ∈ N.
The interval Shapley value Φ : SMIGN → I(R)N is deﬁned by Φ(d) =
1
n!
 
σ∈Π(N) mσ(d), for each w ∈ SMIGN. We notice that the interval Shap-
ley value Φ(d) is an interval core element on the class of concave interval
games.
3 Airport situations with interval data and
related games
Consider the aircraft fee problem of an airport with one runway. Suppose
that the planes which are to land are classiﬁed into m types. One can think
that the runway is divided into m consecutive pieces, namely P1,...,Pm, and
P1 is suﬃcient for planes of type 1, P1 and P2 together are suﬃcient for planes
of type 2, P1, P2 and P3 are together suﬃcient for planes of type 3 etc.. Let
Tj < [0,0] be the interval cost of piece Pj. Let Nj be the set of players who
own a plane of type j. Then, N = ∪m
j=1Nj is the set of all users of the runway.
Let nj denote the number of planes of type j. Let S ⊂ N be a coalition with
S∩Nj  = ∅ and S∩Nj+1 = ∅. Then, this coalition needs the pieces P1,...,Pj
of the runway. The interval cost of the used pieces of the runway is equal
to
 j
i=1 Ti. The characteristic function d of the corresponding cost game
< N,d > is given by d(∅) = [0,0] and d(S) =
 j
i=1 Ti if S ∩ Nj  = ∅ and
S ∩ Nj+1 = ∅ for each S ⊂ N.
To give a formal description of d we introduce interval games of the form
< N,Tc >, where < N,c > is the classical cooperative cost game and T ∈
I(R+). We denote by I(R+) the set of all closed intervals in R+.
Let T ∈ I(R+) and c : 2N → R. Then, the interval game < N,Tc > is
deﬁned by (Tc)(S) = c(S)T for each S ∈ 2N.
We notice that the Φ(Tc) for the interval game < N,Tc > is related with
5the Shapley value φ(c) of the classical game < N,c > as follows:
Φi(Tc) = φi(c)T for each i ∈ N. (2)
Now we give the description of the airport interval game as follows:
d =
m  
k=1
Tku
∗
∪m
r=kNr. (3)
In the following proposition we show that airport interval games are concave.
Proposition 3.1. 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 unanim-
ity games are concave (or submodular). By (3) we have, d =
 m
k=1 T ku∗
∪m
r=kNr
and |d| =
 m
k=1 |Tk|u∗
∪m
r=kNr are concave because T k ≥ 0 and |Tk| ≥ 0 for each
k. By Proposition 2.1 < N,d > is concave.
Next we propose an interval cost allocation rule β, which we call the
interval Baker-Thompson rule. For a given airport interval situation the
Baker-Thompson allocation for a player i of type j is as follows:
βi(d) =
j  
k=1
(
m  
r=k
nr)
−1Tk. (4)
Note that for the piece Pk of the runway the users are ∪m
r=kNr, i.e. there
are
 m
r=k nr users. So, (
 m
r=k nr)−1Tk is the equal cost share of each user.
This means that a player i of type j contributes to the cost of the pieces
P1,...,Pj.
We notice that it is helpful to implement the interval Baker-Thompson al-
location when the uncertainty regarding the costs of pieces of the runway is
removed. For details regarding the use of interval solutions for determining
the distribution of achieved common costs we refer the reader to Branzei,
Tijs and Alparslan G¨ ok (2008b).
In the following proposition we show that the interval Baker-Thompson al-
location rule coincides with the interval Shapley value of the corresponding
airport interval game.
6Proposition 3.2. Let < N,d > be an airport interval game with d as in
(3). Then, the interval allocation β(d) whose components are given by (4)
corresponds to Φ(d).
Proof. Notice that Φ(d) is additive. Then, for i ∈ Nj we have
Φi(d) = Φi(
m  
k=1
Tku
∗
∪m
r=kNr) =
m  
k=1
Φi(Tku
∗
∪m
r=kNr)
=
m  
k=1
φi(u
∗
∪m
r=kNr)Tk
=
j  
k=1
(
m  
r=k
nr)
−1Tk = βi(d),
where the second equality follows from the additivity of Φ(d) and the third
equality follows from (2).
Note that if we consider the special case N1 = {1},N2 = {2},...,Nn =
{n}. Then, β(d) = (
T1
n ,
T1
n +
T2
n−1,...,
T1
n +
T2
n−1 + ... + Tn
1 ). Here, each piece
of the runway is completely paid by the users and all users of the same piece
contribute equally. It is proved in Alparslan G¨ ok, Branzei and Tijs (2008b)
that the interval Shapley value is an interval core element for convex interval
games. Since the airport interval games are concave by Proposition 3.1 the
proof of the following proposition is straightforward.
Proposition 3.3. Let < N,d > be an airport interval game with d as in
(3). Then, Φ(d) ∈ C(d).
Example 3.1. Let < N,d > be a three-person airport interval game cor-
responding to the airport interval situations depicted in Figure 1. The in-
terval costs of the pieces are given by T1 = [30,45], T2 = [20,40] and
T3 = [100,120]. Then, d(∅) = [0,0], d(1) = [30,45], d(2) = d(1,2) = [50,85]
and d(3) = d(1,3) = d(2,3) = d(N) = [150,205]. Clearly, < N,d > and
< N,|d| > are submodular (or concave). So, < N,d > is concave by Propo-
sition 2.1. The following table shows the interval marginal vectors of the
game, where rows correspond to orderings of players and columns correspond
72 3 1
[100,120] [20,40] [30,45]
Figure 1: An airport situation with interval data
to players
123
132
213
231
312
321

     

[30,45] [20,40] [100,120]
[30,45] [0,0] [120,160]
[0,0] [50,85] [100,120]
[0,0] [50,85] [100,120]
[0,0] [0,0] [150,205]
[0,0] [0,0] [150,205]

     

.
Note that d = [30,45]u∗
{1,2,3} + [20,40]u∗
{2,3} + [100,120]u∗
{3} and
Φ(d) = ([10,15],[20,35],[120,155]) ∈ C(d).
Notice also that
Φ(d) = Φ(
3  
k=1
Tku
∗
∪3
r=kNr) = φ(u
∗
{1,2,3})T1 + φ(u
∗
{2,3})T2 + φ(u
∗
{3})T3
= (
1
3
,
1
3
,
1
3
)[30,45] + (0,
1
2
,
1
2
)[20,40] + (0,0,1)[100,120] = β(d).
4 Final remarks
In this paper we studied airport situations with interval data and related
games. Other economic and Operations research situations with interval
8data and related interval games have been also studied. We refer here to
Branzei, Tijs and Alparslan G¨ ok (2008a), Branzei and Dall’Aglio (2008a,b)
for bankruptcy situations, Alparslan G¨ ok et al. (2008) for sequencing situa-
tions and Moretti et al. (2008) for minimum cost spanning tree situations.
Weber, Alparslan G¨ ok and S¨ oyler (2007) and Weber et al. (2007) considered
environmental and gene-network problems with interval uncertainty. It is a
topic for further research to associate cooperative interval games with such
situations.
Other OR situations and combinatorial optimization problems with interval
data among which ﬂow situations, linear production situations and holding
situations can give rise to interesting interval games. The existing literature
on related classical games can be an inspiration source for further research
(Borm, Hamers and Hendrickx (2001), Curiel (1997), Kalai and Zemel (1982),
Owen (1975), Tijs, Meca and L´ opez (2005).
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Cooperative Interval Games Arising from Airport Situations with Interval Data

Cooperative Interval Games Arising from Airport Situations with Interval Data

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