Cooperative Interval Games

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 4: Cooperative Interval Games

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

Interval solutions for cooperative interval games

Big boss interval games

Handling interval solutions

References

Introduction

Introduction

This lecture is based on the papers
Cooperative interval games: a survey by Branzei et al., which was
published in Central European Journal of Operations Research
(CEJOR),
Set-valued solution concepts using interval-type payoﬀs for interval
games by Alparslan G¨k et al., which will appear in Journal of
o
Mathematical Economics (JME) and
Convex interval games by Alparslan G¨k, Branzei and Tijs, which
o
was published in Journal of Applied Mathematics and Decision
Sciences.

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

Cooperative game theory deals with coalitions who coordinate their
actions and pool their winnings.
The main problem: Dividing the rewards/costs among the
members of the formed coalition.
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.
In this research, rewards/costs taken into account are not random
variables, but just closed and bounded intervals of real numbers
with no probability distribution attached.

Introduction

Motivation

Idea of interval approach: In most economic and OR situations
rewards/costs are not precise.

Possible: Estimating the intervals to which rewards/costs belong.

Why cooperative interval games are important?
Useful for modeling real-life situations.

Aim: generalize and extend the classical theory to intervals and
apply it to economic situations, popular OR games.

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

Introduction

Classical cooperative games
A cooperative game < N, v >
N = {1, 2, ..., n}:set of players
v : 2N → R: characteristic function, v (∅) = 0
v (S): worth (or value) of coalition S.
x ∈ RN : payoﬀ vector
G N : class of all cooperative games with player set N
The core (Gillies (1959)) of a game < N, v > is the set

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

The idea: Giving every coalition S at least their worth v (S) so that
no coalition protests



A cooperative interval game is an ordered pair < N, w >,
where N is the set of players and w is the characteristic
function of the game.
N = {1, 2, ..., n}, w : 2N → I (R) is a map, assigning to each
coalition S ∈ 2N a closed interval, such that w (∅) = [0, 0].
w (S) = [w (S), w (S)]: worth (value) of S.
w (S): lower bound, w (S): upper bound
IG N :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)
< N, w1 − w2 > is defined by (w1 − w2 )(S) = w1 (S) − w2 (S)
with |w1 (S)| ≥ |w2 (S)|
Classical cooperative games associated with < N, w >:
Border games < N, w >, < N, w >
Length game < N, |w | >, where |w | (S) = w (S) − w (S) for
each S ∈ 2N .
w = w + |w |


Interval core
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

< 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.
For details on classical cooperative game theory we refer to
Branzei, Dimitrov and Tijs (2008).


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.


Generalization of Bondareva (1963) and Shapley (1967)
< 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.


The interval Weber Set
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 (N) = [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 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 big boss games.
BBIG N : the class of big boss interval games
marginal contribution of each player i ∈ N:
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))

big boss interval point: B(w ) = ([0, 0], . . . , [0, 0], w (N))
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 with one agent with a storage capacity and
other agents have goods to stored to generate benefits.
In classical cooperative game theory holding situations are
modelled by using big boss games.
We refer to Tijs, Meca and L´pez (2005).
o
We consider a holding situation with interval data and construct a
holding interval game which turns out to be a big boss interval
game.
Example 1: 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 1 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 ).
For more details see Alparslan G¨k, Branzei and Tijs (2010).
o


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 2:
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 ).
For more details see Branzei, Tijs and Alparslan G¨k (2010).
o

References

References
[1] 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).
[2] 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 (2009) DOI:
10.1115/2009/342089.
[3] Alparslan G¨k S.Z., Branzei R., 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.
[4] 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).

References

References
[5] 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.
[6] 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 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.
[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.

References

References
[9] Muto S., Nakayama M., Potters J. and Tijs S., On big boss
games, The Economic Studies Quarterly Vol.39, No. 4 (1988)
303-321.
[10] Shapley L.S., On balanced sets and cores, Naval Research
Logistics Quarterly 14 (1967) 453-460.
[11] 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.
[12] 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.
[13]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.

Cooperative Interval Games

Recomendados

Recomendados

Más contenido relacionado

Destacado

Destacado (6)

Más de SSA KPI

Más de SSA KPI (20)

Último

Último (20)

Cooperative Interval Games