How to Handle Interval Solutions for 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 7: How to Handle Interval Solutions for 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

Allocation rules

The one-stage procedure

The multi-stage procedure

Final remarks

References

Introduction

Introduction

This lecture is based on the paper

How to handle interval solutions for cooperative interval games by
Branzei, Tijs and Alparslan G¨k,
o

which was published in

International Journal of Uncertainty, Fuzziness and
Knowledge-based Systems.

Introduction

Motivation

Uncertainty accompanies almost every situation in our lives
and it inﬂuences our decisions.
On many occasions uncertainty is so severe that we can only
predict some upper and lower bounds for the outcome of our
(collaborative) actions, i.e., payoﬀs lie in some intervals.
Cooperative interval games have been proved useful for
solving reward/cost sharing problems in situations with
interval data in a cooperative environment (see Branzei et al.
(2010) for a survey).
A natural way to incorporate the uncertainty of coalition
values into the solution of such reward/cost sharing problems
is by using interval solution concepts.

Introduction

Related literature
Many papers appeared on modeling economic and Operational
Research situations with interval data by using game theory, in
particular cooperative interval games, as a tool.
Branzei, Dimitrov and Tijs (2003), Alparslan G¨k, Miquel and
o
Tijs (2009), Alparslan G¨k (2009), Branzei et al. (2010),
o
Branzei, Mallozzi and Tijs (2010), Yanovskaya, Branzei and
Tijs (2010).
Kimms and Drechsel (2009).
Bauso and Timmer (2009) introduce dynamics into the theory
of cooperative interval games, whereas Mallozzi, Scalzo and
Tijs (2011) extend some results from the theory of
cooperative interval games by considering coalition values
given by means of fuzzy intervals.

Introduction

Cooperative interval games

< N, w >, N = {1, 2, . . . , n}: set of players
w : 2N → I (R): characteristic function, w (∅) = [0, 0]
w (S) = [w (S), w (S)]: worth (value) of S
w (S) : the lower bound, w (S): the upper bound of the interval
w (S)
I (R): the set of all closed and bounded intervals in R
I (R)N : set of all n-dimensional vectors with elements in I (R)
IG N : the class of all interval games with player set N

Introduction

Interval solution concepts
An interval solution concept on IG N is a map assigning to each
interval game w ∈ IG N a set of n-dimensional vectors whose
components belong to 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

The interval Shapley value Φ : SMIG N → I (R)N :
1
Φ(w ) = mσ (w ), for each w ∈ SMIG N .
n!
σ∈Π(N)

Introduction

Interval solution concepts
The payoff vectors x = (x1 , x2 , . . . , xn ) ∈ RN from the classical
cooperative transferable utility (TU) game theory are replaced by
n-dimensional vectors (J1 , . . . , Jn ) ∈ I (R)N , where Ji = [J i , J i ],
i ∈ N.
The players’ agreement on a particular interval allocation
(J1 , . . . , Jn ) based on an interval solution concept merely says that
the payoff xi that player i will receive when the outcome of the
grand coalition is known belongs to the interval Ji .
A procedure to transform an interval allocation
J = (J1 , . . . , Jn ) ∈ I (R)N into a payoff vector
x = (x1 , . . . , xn ) ∈ RN is therefore a basic ingredient of contracts
that people or businesses have to sign when they cannot estimate
with certainty the attainable coalition payoff(s).

Allocation rules

Allocation rules
Let N be a set of players that consider cooperation under interval
uncertainty of coalition values, i.e. knowing what each group S of
players (coalition) can obtain between two bounds, w (S) and
w (S), via cooperation.
If the players use cooperative game theory as a tool, they can
choose an interval solution concept, say the value-type solution Ψ,
that associates with the related cooperative interval game
< N, w > the interval allocation Ψ(w ) = (J1 , . . . , Jn ) which
guarantees for each player i ∈ N a final payoff within the interval
Ji = [J i , J i ] when the value of the grand coalition is known.
Clearly, w (N) = i∈N J i and w (N) = i∈N J i . For each i ∈ N
the interval [J i , J i ] can be seen as the interval claim of i on the
realization R of the payoff for the grand coalition N
(w (N) ≤ R ≤ w (N)).

Allocation rules

Allocation rules
One should determine payoffs xi ∈ [J i , J i ], i ∈ N (the feasibility
condition) such that i∈N xi = R (the efficiency condition).
Notice that in the case R = w (N) the payoff vector x equals
(J 1 , . . . , J n ), in the case R = w (N) we have x = (J 1 , . . . , J n ), but
in the case w (N) < R < w (N) there are infinitely many ways to
determine allocations (x1 , . . . , xn ) satisfying both the efficiency and
the feasibility conditions.
In the last case, we need suitable allocation rules to determine fair
allocations (x1 , . . . , xn ) of R satisfying the above conditions.
As players prefer as large payoffs as possible and the amount R to
be divided between them is smaller than i∈N J i , the players are
facing a bankruptcy-like situation, implying that bankruptcy rules
are good candidates for transforming an interval allocation
(J1 , . . . , Jn ) into a payoff vector (x1 , . . . , xn ).

Allocation rules

Bankruptcy rules

A bankruptcy situation with set of claimants N is a pair (E , d),
where E ≥ 0 is the estate to be divided and d ∈ RN is the vector
+
of claims such that i∈N di ≥ E . We denote by BR N the set of
bankruptcy situations with player set N.
A bankruptcy rule is a function f : BR N → RN which assigns to
each bankruptcy situation (E , d) ∈ BR N a payoﬀ vector
f (E , d) ∈ RN such that 0 ≤ f (E , d) ≤ d (reasonability) and
i∈N fi (E , d) = E (eﬃciency).
We only use three bankruptcy rules: the proportional rule (PROP),
the constrained equal awards (CEA) rule and the constrained equal
losses (CEL) rule.

Allocation rules

Bankruptcy rules
The rule PROP is deﬁned by
di
PROPi (E , d) = E
j∈N dj

for each bankruptcy problem (E , d) and all i ∈ N.
The rule CEA is deﬁned by

CEAi (E , d) = min {di , α} ,

where α is determined by

CEAi (E , d) = E ,
i∈N


Allocation rules

Bankruptcy rules
The rule CEL is deﬁned by

CELi (E , d) = max {di − β, 0} ,

where β is determined by

CELi (E , d) = E ,
i∈N

We introduce the notation F = {CEA, CEL, PROP} and let
f ∈ F. The choice of one speciﬁc f ∈ F in a certain bankruptcy
situation is based on the preference of the players involved in that
situation; other bankruptcy rules could be also considered as
elements of a larger F.

Allocation rules

Bankruptcy rules
When the value of the grand coalition becomes known in multiple
stages, i.e., updated estimates of the outcome of cooperation
within the grand coalition are considered during an allocation
process, more general division problems than bankruptcy problems
may arise.
We present the rights-egalitarian (f RE ) rule deﬁned by
1
fi RE (E , d) = di + n (E − i∈N di ), for each division problem (E , d)
and all i ∈ N.
The rights-egalitarian rule divides equally among the agents the
diﬀerence between the total claim D = i∈N di and the available
amount E , being suitable for all circumstances of division
problems; in particular, the amount to be divided can be either
positive or negative, the vector of claims d = (d1 , . . . , dn ) may
have negative components, and the amount to be divided may
exceed or fall short of the total claim D.


Let (J1 , . . . , Jn ) be an interval allocation, with Ji = [J i , J i ], i ∈ N,
satisfying i∈N J i = w (N) and i∈N J i = w (N), and let R be
the realization of w (N).
One can write R and J i , i ∈ N, as:

R = w (N) + (R − w (N)), (1)

J i = J i + (J i − J i ), (2)
implying that the problem (R − w (N), (J i − J i )i∈N ) is a bankruptcy
problem. Since R is the realization of w (N), one can expect that

w (N) ≤ R ≤ w (N). (3)


Next we describe and illustrate a simple (one-stage) procedure to
transform an interval allocation (J1 , . . . , Jn ) ∈ I (R)N into a payoff
vector x = (x1 , . . . , xn ) ∈ RN which satisfies

J i ≤ xi ≤ J i for each i ∈ N; (4)

xi = R. (5)
i∈N

The one-stage procedure (in the case when the value of the grand
coalition becomes known at once) uses as input data an interval
allocation (J1 , . . . , Jn ), the realized value of the grand coalition, R,
and function(s) specifying the division rule(s) for distributing the
amount R over the players. It determines for each player i, i ∈ N,
a payoff xi ∈ R such that J i ≤ xi ≤ J i , and i∈N xi = R.


Procedure One-Stage;
Input data: n, (Ji )i=1,n , R;
function f ;
begin
compute w (N) w (N) = i∈N J i ;
for i = 1 to n do
di = J i − J i
{endfor}
for i = 1 to n do
pi = fi (R − w (N), (di )i=1,n )
{endfor}
for i = 1 to n do
xi := J i + pi
{endfor}
Output data: x = (x1 , . . . , xn );
{end procedure}.


Example

Let < N, w > be the three-person interval game with
w (S) = [0, 0] if 3 ∈ S, w (∅) = w (3) = [0, 0], w (1, 3) = [20, 30]
/
and w (N) = w (2, 3) = [50, 90]. We assume that the realization of
w (N) is R = 60 and consider that cooperation within the grand
coalition was settled based on the use of the interval Shapley
value. Then, Φ(w ) = ([3 1 , 5], [18 1 , 35], [28 1 , 50]).
3 3 3
We determine individual uncertainty-free shares distributing the
amount R − w (N) = 10 among the three agents. Note that we
deal here with a classical bankruptcy problem (E , d) with E = 10,
d = (1 2 , 16 3 , 21 2 ).
3
2
3


Example continued

Using the one-stage procedure three times with PROP, CEA and
CEL in the role of f , respectively, we have

f PROP(E , d) CEA(E , d) CEL(E , d)
1 .
p ( 12 , 4 1 , 5 12 ) (1 2 , 4 1 , 4 1 ) (0, 2 2 , 7 2 )
5
6
5
3 6 6
1

Then, we obtain x as (3 1 , 18 1 , 28 1 ) + f (10, (1 2 , 16 3 , 21 3 )),
3 3 3 3
2 2

f ∈ F, shown in the next table.

f PROP(E , d) CEA(E , d) CEL(E , d)
.
x (3 4 , 22 2 , 33 4 ) (5, 22 2 , 32 2 ) (3 3 , 20 5 , 35 5 )
3 1 3 1 1 1
6 6


Remark
First, since R satisfies (3), one idea is to determine λ ∈ [0, 1] such
that
R = λw (N) + (1 − λ)w (N), (6)
and give to each i ∈ N the payoff

xi = λJ i + (1 − λ)J i . (7)

Note that J i ≤ xi ≤ J i and

xi = λ J i + (1 − λ) J i = λw (N) + (1 − λ)w (N) = R.
i∈N i∈N i∈N

So, x satisfies conditions (4) and (5).


Remark continued

Now, we notice that we can also write x = J + (1 − λ)(J − J).
So, the payoﬀ for player i ∈ N can be obtained in the following
manner: ﬁrst each player i ∈ N is allocated the amount J i ; second,
the amount R − i∈N J i is distributed over the players
proportionally with J i − J i , i ∈ N, which is equivalent with using
the bankruptcy rule PROP for a bankruptcy problem (E , d), where
the estate E equals R − i∈N J i and the claims di are equal to
J i − J i for each i ∈ N.



The multi-stage procedure (in the case when the value of the
grand coalition becomes known in multiple stages, say T ) uses as
input data an interval allocation (J1 , . . . , Jn ), a related sequence of
observed outcomes for the grand coalition, R (1) , . . . , R (T ) , and
function(s) specifying the division rule(s) for distributing the
amount R (t) − R (t−1) over the players at stage t, t = 1, . . . , T . It
determines for each player i ∈ N a payoﬀ xi ∈ R such that
J i ≤ xi ≤ J i , and i∈N xi = R (T ) .



In this section we introduce some dynamics in allocation processes
for procedures to transform an interval allocation
(J1 , . . . , Jn ) ∈ I (R)N into a payoﬀ vector x ∈ RN satisfying
conditions (4) and (5).
We assume that a ﬁnite sequence of updated estimates of the
outcome of the grand coalition, R (t) with t ∈ {1, 2, . . . , T }, is
available because the value of the grand coalition is known in
multiple stages, where

w (N) ≤ R (1) ≤ R (2) ≤ . . . ≤ R (T ) ≤ w (N). (8)



At any stage t ∈ {1, 2, . . . , T } a budget of ﬁxed size, R (t) − R (t−1) ,
where R (0) = w (N), is distributed among the players.
The decision as which portion of the budget each player will
receive at that stage depends on the historical allocation and is
speciﬁed by a predetermined allocation rule. As allocation rules at
each stage we consider either a bankruptcy rule f (in the case
when a bankruptcy problem arises) or a general division rule (for
example f RE ) otherwise.


Procedure Multi-Stage;
Input data: n, (Ji )i=1,n , T , (R (j) )j=1,T ;
function f , g ;
compute w (N) w (N) = i∈N J i ;
begin
R (0) := w (N);
for i = 1 to n do
di = J i − J i ; spi := 0
{endfor}
for t = 1 to T do
begin
D := 0;
for i = 1 to n do
D := D + di
{endfor}


if D > R (j) − R (j−1)
then for i = 1 to n do pi = fi (R (j) − R (j−1) , (di )i=1,n ) {endfor}
else for i = 1 to n do pi = gi (R (j) − R (j−1) , (di )i=1,n ) {endfor}
{endif}
for i = 1 to n do
di := di − pi ;
spi := spi + pi
{endfor}
{end}
{endfor}
for i = 1 to n do
xi := J i + spi
{endfor}
Output data: x = (x1 , . . . , xn );
{end procedure}.


Remarks

We notice that the One-Stage procedure appears as a special case
of the Multi-Stage procedure where T = 1. At each stage
t ∈ {1, . . . , T } of the allocation process the fixed amount
R (t) − R (t−1) , where R (t) is the estimate of the payoff for the
grand coalition at stage t, with R (0) = w (N) is distributed among
the players’ by taking into account the players’ updated claims at
the previous stage, di , i ∈ N, to determine the payoff portions, pi ,
i ∈ N.


Remarks continued

The calculation of the individual payoff portions is done using the
specified bankruptcy rule f when we deal with a bankruptcy
problem, i.e. when the total claim D is greater than R (j) − R (j−1)
(and all the individual claims are nonnegative).
These payoff portions are used further to update both the
aggregate portions spi and the individual claims di , i ∈ N.
Notice that under the assumption (8) our procedure assures that
all the individual claims are nonnegative as far as we apply a
bankruptcy rule f . However, the condition D > R (j) − R (j−1) may
be not satisfied requiring the use of a general division rule g like
f RE .


Example

Consider the interval game and the interval Shapley value as in the
previous example. But, suppose there are 3 updated estimates of
the realization of the payoﬀ for the grand coalition:
R (1) = 60; R (2) = 65 and R (3) = 80.
We have R (0) = 50; d = (1 2 , 16 2 , 21 2 ); sp = (0, 0, 0);
3 3 3
Stage 1. The amount R (1) − R (0) = 10 is distributed over agents in N
according to the claims d = (1 2 , 16 2 , 21 3 ). Note that
3 3
2

D = 40 > 10, so the bankruptcy rule PROP can be applied at
this stage yielding p = ( 12 , 4 1 , 5 12 ). Clearly,
5
6
5
5 1 5
sp = ( 12 , 4 6 , 5 12 ). The vector of claims becomes
d = (1 4 , 12 2 , 16 1 ).
1 1
4


Example continued

according to d = (1 1 , 12 1 , 16 1 ). Note that D = 30 > 5, so
4 2 4
the bankruptcy rule PROP can be applied yielding
p = ( 24 , 2 12 , 2 17 ). Then the adjusted vector of claims is
5 1
24
d = (1 24 , 10 12 , 13 13 ) and sp equals now ( 5 , 6 1 , 8 1 ).
1 5
24 8 4 8
according to d = (1 24 , 10 12 , 13 13 ). Since D = 24 5 > 15, we
1 5
24 6
can apply the bankruptcy rule PROP obtaining
p = ( 5 , 6 4 , 8 8 ). Then we obtain sp = (1 4 , 12 1 , 16 4 ) (No
8
1 1 1
2
1

claims are further needed because T = 3).
Finally, x = (3 3 + 1 1 , 18 1 + 12 2 , 28 1 + 16 4 ) = (4 12 , 30 5 , 44 12 ).
1
4 3
1
3
1 7
6
7

Final remarks

Final remarks
In collaborative situations with interval data, to settle cooperation
within the grand coalition using the cooperative game theory as a
tool, the players should jointly choose:
(i) An interval solution concept, for example a value-type interval
solution Ψ, that captures the interval uncertainty with regard
to the coalition values under the form of an interval allocation,
say J = (J1 , . . . , Jn ), where Ji = Ψi (w ) for all i ∈ N;
(ii) A procedure, specifying the allocation process and the
allocation rule(s) to be used during the allocation process, in
order to transform the interval allocation (J1 , . . . , Jn ) into a
payoﬀ vector (x1 , . . . , xn ) ∈ RN such that J i ≤ xi ≤ J i for
each i ∈ N and i∈N xi = R, where R is the revenue for the
grand coalition at the end of cooperation.

Final remarks

Final remarks

The two procedures presented transform an interval allocation into
a payoﬀ vector, under the assumption that only the uncertainty
with regard to the value of the grand coalition has been resolved.
In both procedures the vector of computed payoﬀ shares belongs to
the core1 C (v ) of a selection2 < N, v > of the interval game
< N, w >.

1
The core of a cooperative transferable utility game was introduced by
Gillies (1959).
2
Let < N, w > be an interval game; then v : 2N → R is called a selection of
w if v (S) ∈ w (S) for each S ∈ 2N (Alparslan G¨k, Miquel and Tijs (2009)).
o

Final remarks

Final remarks

In the sequel, we discuss two cases where besides the realization of
w (N) also the realizations of w (S) for some or all S ⊂ N are
known.
First, suppose that the uncertainty on all outcomes is resolved,
implying that a selection of the initial interval game is available.
Then, we can use for this selection a suitable classical solution (for
example the classical solution corresponding to the interval solution
Ψ) to determine a posteriori uncertainty-free individual shares.

Final remarks

Final remarks
Secondly, suppose that only the uncertainty on some coalition
values (including the payoff for the grand coalition) was resolved.
In such situations, we propose to adjust the initial interval
allocation (J1 , . . . , Jn ) using the same interval solution concept Ψ
which generated it, but for the interval game < N, w > where
w (S) = [RS , RS ] for all S ⊂ N whose worth realizations RS are
known, w (N) = R, w (∅) = [0, 0], and w (S) = w (S) otherwise.
Then, the obtained interval allocation for the game < N, w > will
be transformed into an allocation x = (x1 , . . . , xn ) ∈ R N of R
using our procedures.
Finally, an alternative approach for designing one-stage procedures
is to use taxation rules instead of bankruptcy rules by handing out
first J i and then taking away with the aid of a taxation rule the
deficit T = i∈N J i − R based on di = J i − J i for each i ∈ N.

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]Alparslan G¨k S.Z., “Cooperative interval games”, PhD
o
Dissertation Thesis, Institute of Applied Mathematics, Middle East
Technical University (2009).
[3]Alparslan G¨k S.Z., Miquel S. and Tijs S., “Cooperation under
o
interval uncertainty”, Mathematical Methods of Operations
Research, Vol. 69, no.1 (2009) 99-109.
[4] Bauso D. and Timmer J.B., “Robust Dynamic Cooperative
Games”, International Journal of Game Theory, Vol. 38, no. 1
(2009) 23-36.

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., Tijs S., Alparslan G¨k S.Z., “How to handle interval
o
solutions for cooperative interval games”, International Journal of
Uncertainty, Fuzziness and Knowledge-based Systems, Vol.18,
Issue 2 (2010) 123-132.
[7] Branzei R., Dimitrov D. and Tijs S., “Shapley-like values for
interval bankruptcy games”, Economics Bulletin Vol. 3 (2003) 1-8.
[8]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).

References

References
[9] Gillies D.B., Solutions to general non-zero-sum games. In:
Tucker, A.W. and Luce, R.D. (Eds.), Contributions to theory of
games IV, Annals of Mathematical Studies, Vol. 40. Princeton
University Press, Princeton (1959) pp. 47-85.
[10] Kimms A and Drechsel J., “Cost sharing under uncertainty: an
algorithmic approach to cooperative interval-valued games”, BuR -
Business Research, Vol. 2 (urn:nbn:de:0009-20-21721) (2009).
[11] Mallozzi L., Scalzo V. and Tijs S., “Fuzzy interval cooperative
games”, Fuzzy Sets and Systems, Vol. 165 (2011) pp.98-105.
[12] Yanovskaya E., Branzei R. and Tijs S., “Monotonicity
Properties of Interval Solutions and the Dutta-Ray Solution for
Convex Interval Games”, Chapter 16 in “Collective Decision
Making: Views from Social Choice and Game Theory”, series
Theory and Decision Library C, Springer Verlag Berlin/ Heidelberg
(2010).

How to Handle Interval Solutions for Cooperative Interval Games

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