Economic and Operations Research Situations with Interval Data

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 5: Economic and Operations Research Situations with
Interval Data

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 situations with interval data

Operations Research situations with interval data

References

Introduction

Introduction

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

Introduction

Introduction
This lecture is also based on the papers
Big boss interval games by Alparslan G¨k, Branzei and Tijs
o
which was published in International Journal of Uncertainty,
Fuzziness and Knowledge-Based Systems (IJUFKS),
Airport interval games and their Shapley value by Alparslan
G¨k, Branzei and Tijs which was published in Operations
o
Research and Decisions,
Bankruptcy problems with interval uncertainty by Branzei and
Alparslan G¨k which was published in Economics Bulletin and
o
Sequencing interval situations and related games by Alparslan
G¨k et al. which will appear in Central European Journal of
o
Operations Research (CEJOR).

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.


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).


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)} .


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 (1, 2) = [4, 8].

1
Φ(w ) = (m(12) (w ) + m(21) (w ));
2
1
Φ(w ) = ((w (1), w (1, 2) − w (1)) + (w (1, 2) − 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


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 ).


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

is the dual unanimity game.
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 money E
has to be divided among some people, N = {1, . . . , n}, who have
individual claims di , 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.

Note that (∗) implies vE ,d (S) ≤ vE ,d (S) for each S ∈ 2N .
SBRIG N : the family of all bankruptcy interval games wE ,d
with (E , d) ∈ SBRI N .


We notice that wE ,d ∈ SBRIG N is supermodular because vE ,d and
vE ,d ∈ G N are convex. The following example illustrates that
wE ,d ∈ SBRIG N is supermodular but not necessarily convex.
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.


Minimum cost spanning tree situations with interval data

There are also interesting results for interval extension of minimim
spanning tree situations. For further details please see the paper
Connection situations under uncertainty and cost monotonic
solutions by Moretti et al. which was published in Computers
and Operations Research.

References

References

[1]Alparslan G¨k S.Z., Cooperative interval games, PhD
o
Dissertation Thesis, Institute of Applied Mathematics, Middle East
Technical University, Ankara-Turkey (2009).
[2]Alparslan G¨k S.Z., Cooperative Interval Games: Theory and
o
Applications, Lambert Academic Publishing (LAP), Germany
(2010) ISBN:978-3-8383-3430-1.
[3]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).
[4]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).

References

References
[5]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.
[6]Aumann R. and Maschler M., Game theoretic analysis of a
bankruptcy problem from the Talmud, Journal of Economic
Theory 36 (1985) 195-213.
[7]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).
[8] Branzei R. and Alparslan G¨k S.Z., Bankruptcy problems with
o
interval uncertainty, Economics Bulletin, Vol. 3, no. 56 (2008) pp.
1-10.

References

References
[9]Branzei R., Dimitrov D. and Tijs S., Models in Cooperative
Game Theory, Springer, Game Theory and Mathematical Methods
(2008).
[10]Curiel I., Pederzoli G. and Tijs S., Sequencing games,
European Journal of Operational Research 40 (1989) 344-351.
[11]Driessen T., Cooperative Games, Solutions and Applications,
Kluwer Academic Publishers (1988).
[12]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.
[13] 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.

References

References

[14] Muto S., Nakayama M., Potters J. and Tijs S., On big boss
games, The Economic Studies Quarterly Vol.39, No. 4 (1988)
303-321.
[15]Shapley L.S., On balanced sets and cores, Naval Research
Logistics Quarterly 14 (1967) 453-460.
[16] 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.
[17] 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.

References

References

[18]Tijs S., Meca A. and L´pez M.A., Beneﬁt sharing in holding
o
situations, European Journal of Operational Research 162 (1)
(2005) 251-269.
[19] von Neumann, J. and Morgernstern, O., Theory of Games and
Economic Behaviour, Princeton: Princeton University Press
(1944).

Economic and Operations Research Situations with Interval Data

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