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 ResearchGames. Applications to Interval Games

Lecture 7: How to Handle Interval Solutions for CooperativeInterval Games

Sırma Zeynep Alparslan GokSuleyman Demirel University

Faculty of Arts and Sciences

Department of Mathematics

Isparta, Turkey

email:[email protected]

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 byBranzei, Tijs and Alparslan Gok,

which was published in

International Journal of Uncertainty, Fuzziness andKnowledge-based Systems.


Introduction

Motivation

I Uncertainty accompanies almost every situation in our livesand it influences our decisions.

I On many occasions uncertainty is so severe that we can onlypredict some upper and lower bounds for the outcome of our(collaborative) actions, i.e., payoffs lie in some intervals.

I Cooperative interval games have been proved useful forsolving reward/cost sharing problems in situations withinterval data in a cooperative environment (see Branzei et al.(2010) for a survey).

I A natural way to incorporate the uncertainty of coalitionvalues into the solution of such reward/cost sharing problemsis by using interval solution concepts.


Introduction

Related literature

Many papers appeared on modeling economic and OperationalResearch situations with interval data by using game theory, inparticular cooperative interval games, as a tool.

I Branzei, Dimitrov and Tijs (2003), Alparslan Gok, Miquel andTijs (2009), Alparslan Gok (2009), Branzei et al. (2010),Branzei, Mallozzi and Tijs (2010), Yanovskaya, Branzei andTijs (2010).

I Kimms and Drechsel (2009).

I Bauso and Timmer (2009) introduce dynamics into the theoryof cooperative interval games, whereas Mallozzi, Scalzo andTijs (2011) extend some results from the theory ofcooperative interval games by considering coalition valuesgiven by means of fuzzy intervals.


Introduction

Cooperative interval games

< N,w >, N = {1, 2, . . . , n}: set of playersw : 2N → I (R): characteristic function, w(∅) = [0, 0]w(S) = [w(S),w(S)]: worth (value) of Sw(S) : the lower bound, w(S): the upper bound of the intervalw(S)

I I (R): the set of all closed and bounded intervals in RI I (R)N : set of all n-dimensional vectors with elements in I (R)

I IGN : the class of all interval games with player set N


Introduction

Interval solution conceptsAn interval solution concept on IGN is a map assigning to eachinterval game w ∈ IGN a set of n-dimensional vectors whosecomponents belong to I (R).The interval imputation set:

I(w) =

{(I1, . . . , In) ∈ I (R)N |

∑i∈N

Ii = w(N), Ii < w(i), ∀i ∈ N

}.

The interval core:

C(w) =

{(I1, . . . , In) ∈ I(w)|

∑i∈S

Ii < w(S), ∀S ∈ 2N \ {∅}

}.

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

Φ(w) =1

n!

∑σ∈Π(N)

mσ(w), for each w ∈ SMIGN .


Introduction

Interval solution concepts

The payoff vectors x = (x1, x2, . . . , xn) ∈ RN from the classicalcooperative transferable utility (TU) game theory are replaced byn-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 thatthe payoff xi that player i will receive when the outcome of thegrand coalition is known belongs to the interval Ji .A procedure to transform an interval allocationJ = (J1, . . . , Jn) ∈ I (R)N into a payoff vectorx = (x1, . . . , xn) ∈ RN is therefore a basic ingredient of contractsthat people or businesses have to sign when they cannot estimatewith certainty the attainable coalition payoff(s).


Allocation rules

Allocation rules

Let N be a set of players that consider cooperation under intervaluncertainty of coalition values, i.e. knowing what each group S ofplayers (coalition) can obtain between two bounds, w(S) andw(S), via cooperation.If the players use cooperative game theory as a tool, they canchoose 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) whichguarantees for each player i ∈ N a final payoff within the intervalJi = [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 therealization 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 feasibilitycondition) such that

∑i∈N xi = R (the efficiency condition).

Notice that in the case R = w(N) the payoff vector x equals(J1, . . . , Jn), in the case R = w(N) we have x = (J1, . . . , Jn), butin the case w(N) < R < w(N) there are infinitely many ways todetermine allocations (x1, . . . , xn) satisfying both the efficiency andthe feasibility conditions.In the last case, we need suitable allocation rules to determine fairallocations (x1, . . . , xn) of R satisfying the above conditions.As players prefer as large payoffs as possible and the amount R tobe divided between them is smaller than

∑i∈N J i , the players are

facing a bankruptcy-like situation, implying that bankruptcy rulesare 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 vectorof claims such that

∑i∈N di ≥ E . We denote by BRN the set of

bankruptcy situations with player set N.A bankruptcy rule is a function f : BRN → RN which assigns toeach bankruptcy situation (E , d) ∈ BRN a payoff vectorf (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 equallosses (CEL) rule.


Allocation rules

Bankruptcy rules

The rule PROP is defined by

PROPi (E , d) =di∑j∈N dj

E

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∑i∈N

CEAi (E , d) = E ,

for each bankruptcy problem (E , d) and all i ∈ N.


Allocation rules

Bankruptcy rules

The rule CEL is defined by

CELi (E , d) = max {di − β, 0} ,

where β is determined by∑i∈N

CELi (E , d) = E ,

for each bankruptcy problem (E , d) and all i ∈ N.We introduce the notation F = {CEA,CEL,PROP} and letf ∈ F . The choice of one specific f ∈ F in a certain bankruptcysituation is based on the preference of the players involved in thatsituation; other bankruptcy rules could be also considered aselements of a larger F .


Allocation rules

Bankruptcy rulesWhen the value of the grand coalition becomes known in multiplestages, i.e., updated estimates of the outcome of cooperationwithin the grand coalition are considered during an allocationprocess, more general division problems than bankruptcy problemsmay arise.We present the rights-egalitarian (f RE ) rule defined byf REi (E , d) = di + 1

n (E −∑

i∈N di ), for each division problem (E , d)and all i ∈ N.The rights-egalitarian rule divides equally among the agents thedifference between the total claim D =

∑i∈N di and the available

amount E , being suitable for all circumstances of divisionproblems; in particular, the amount to be divided can be eitherpositive or negative, the vector of claims d = (d1, . . . , dn) mayhave negative components, and the amount to be divided mayexceed 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 bankruptcyproblem. 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 totransform an interval allocation (J1, . . . , Jn) ∈ I (R)N into a payoffvector x = (x1, . . . , xn) ∈ RN which satisfies

J i ≤ xi ≤ J i for each i ∈ N; (4)∑i∈N

xi = R. (5)

The one-stage procedure (in the case when the value of the grandcoalition becomes known at once) uses as input data an intervalallocation (J1, . . . , Jn), the realized value of the grand coalition, R,and function(s) specifying the division rule(s) for distributing theamount 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 ;begincompute w(N)

{w(N) =

∑i∈N J i

};

for i = 1 to n dodi = J i − J i{endfor}for i = 1 to n dopi = fi (R − w(N), (di )i=1,n){endfor}for i = 1 to n doxi := J i + pi{endfor}Output data: x = (x1, . . . , xn);{end procedure}.



Example

Let < N,w > be the three-person interval game withw(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 ofw(N) is R = 60 and consider that cooperation within the grandcoalition was settled based on the use of the interval Shapleyvalue. Then, Φ(w) = ([3 1

3 , 5], [18 13 , 35], [28 1

3 , 50]).We determine individual uncertainty-free shares distributing theamount R − w(N) = 10 among the three agents. Note that wedeal here with a classical bankruptcy problem (E , d) with E = 10,d = (1 2

3 , 16 23 , 21 2

3 ).



Example continued

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

f PROP(E , d) CEA(E , d) CEL(E , d)

p ( 512 , 4

16 , 5

512 ) (1 2

3 , 416 , 4

16 ) (0, 2 1

2 , 712 )

.

Then, we obtain x as (3 13 , 18 1

3 , 28 13 ) + f (10, (1 2

3 , 16 23 , 21 2

3 )),f ∈ F , shown in the next table.

f PROP(E , d) CEA(E , d) CEL(E , d)

x (3 34 , 22 1

2 , 33 34 ) (5, 22 1

2 , 32 12 ) (3 1

3 , 20 56 , 35 5

6 ).



Remark

First, since R satisfies (3), one idea is to determine λ ∈ [0, 1] suchthat

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∑i∈N

xi = λ∑i∈N

J i + (1− λ)∑i∈N

J i = λw(N) + (1− λ)w(N) = R.

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 followingmanner: 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 usingthe bankruptcy rule PROP for a bankruptcy problem (E , d), wherethe 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 thegrand coalition becomes known in multiple stages, say T ) uses asinput data an interval allocation (J1, . . . , Jn), a related sequence ofobserved outcomes for the grand coalition, R(1), . . . ,R(T ), andfunction(s) specifying the division rule(s) for distributing theamount R(t) − R(t−1) over the players at stage t, t = 1, . . . ,T . Itdetermines for each player i ∈ N a payoff xi ∈ R such thatJ i ≤ xi ≤ J i , and

∑i∈N xi = R(T ).




In this section we introduce some dynamics in allocation processesfor procedures to transform an interval allocation(J1, . . . , Jn) ∈ I (R)N into a payoff vector x ∈ RN satisfyingconditions (4) and (5).We assume that a finite sequence of updated estimates of theoutcome of the grand coalition, R(t) with t ∈ {1, 2, . . . ,T}, isavailable because the value of the grand coalition is known inmultiple 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 willreceive at that stage depends on the historical allocation and isspecified by a predetermined allocation rule. As allocation rules ateach stage we consider either a bankruptcy rule f (in the casewhen a bankruptcy problem arises) or a general division rule (forexample f RE ) otherwise.



The multi-stage procedureProcedure 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

};

beginR(0) := w(N);for i = 1 to n dodi = J i − J i ; spi := 0{endfor}for t = 1 to T dobeginD := 0;for i = 1 to n doD := 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 dodi := di − pi ;spi := spi + pi{endfor}{end}{endfor}for i = 1 to n doxi := J i + spi{endfor}Output data: x = (x1, . . . , xn);{end procedure}.



Remarks

We notice that the One-Stage procedure appears as a special caseof the Multi-Stage procedure where T = 1. At each staget ∈ {1, . . . ,T} of the allocation process the fixed amountR(t) − R(t−1), where R(t) is the estimate of the payoff for thegrand coalition at stage t, with R(0) = w(N) is distributed amongthe players’ by taking into account the players’ updated claims atthe 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 thespecified bankruptcy rule f when we deal with a bankruptcyproblem, 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 theaggregate portions spi and the individual claims di , i ∈ N.Notice that under the assumption (8) our procedure assures thatall the individual claims are nonnegative as far as we apply abankruptcy rule f . However, the condition D > R(j) − R(j−1) maybe not satisfied requiring the use of a general division rule g likef RE .



Example

Consider the interval game and the interval Shapley value as in theprevious example. But, suppose there are 3 updated estimates ofthe 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

3 , 16 23 , 21 2

3 ); sp = (0, 0, 0);

Stage 1. The amount R(1) − R(0) = 10 is distributed over agents in Naccording to the claims d = (1 2

3 , 16 23 , 21 2

3 ). Note thatD = 40 > 10, so the bankruptcy rule PROP can be applied atthis stage yielding p = ( 5

12 , 416 , 5

512 ). Clearly,

sp = ( 512 , 4

16 , 5

512 ). The vector of claims becomes

d = (1 14 , 12 1

2 , 16 14 ).



Example continued

Stage 2. The amount R(2) − R(1) = 5 is distributed over agents in Naccording to d = (1 1

4 , 12 12 , 16 1

4 ). Note that D = 30 > 5, sothe bankruptcy rule PROP can be applied yieldingp = ( 5

24 , 21

12 , 21724 ). Then the adjusted vector of claims is

d = (1 124 , 10 5

12 , 13 1324 ) and sp equals now ( 5

8 , 614 , 8

18 ).

Stage 3. The amount R(3) − R(2) = 15 is distributed over agents in Naccording to d = (1 1

24 , 10 512 , 13 13

24 ). Since D = 24 56 > 15, we

can apply the bankruptcy rule PROP obtainingp = ( 5

8 , 614 , 8

18 ). Then we obtain sp = (1 1

4 , 12 12 , 16 1

4 ) (Noclaims are further needed because T = 3).

Finally, x = (3 13 + 1 1

4 , 18 13 + 12 1

2 , 28 13 + 16 1

4 ) = (4 712 , 30 5

6 , 44 712 ).


Final remarks

Final remarks

In collaborative situations with interval data, to settle cooperationwithin the grand coalition using the cooperative game theory as atool, the players should jointly choose:

(i) An interval solution concept, for example a value-type intervalsolution Ψ, that captures the interval uncertainty with regardto 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 theallocation rule(s) to be used during the allocation process, inorder to transform the interval allocation (J1, . . . , Jn) into apayoff vector (x1, . . . , xn) ∈ RN such that J i ≤ xi ≤ J i foreach 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 intoa payoff vector, under the assumption that only the uncertaintywith regard to the value of the grand coalition has been resolved.In both procedures the vector of computed payoff shares belongs tothe core1 C (v) of a selection2 < N, v > of the interval game< N,w >.

1The core of a cooperative transferable utility game was introduced byGillies (1959).

2Let < N,w > be an interval game; then v : 2N → R is called a selection ofw if v(S) ∈ w(S) for each S ∈ 2N (Alparslan Gok, Miquel and Tijs (2009)).


Final remarks

Final remarks

In the sequel, we discuss two cases where besides the realization ofw(N) also the realizations of w(S) for some or all S ⊂ N areknown.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 (forexample the classical solution corresponding to the interval solutionΨ) to determine a posteriori uncertainty-free individual shares.


Final remarks

Final remarks

Secondly, suppose that only the uncertainty on some coalitionvalues (including the payoff for the grand coalition) was resolved.In such situations, we propose to adjust the initial intervalallocation (J1, . . . , Jn) using the same interval solution concept Ψwhich 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′

= (x′1, . . . , x

′n) ∈ RN of R

using our procedures.Finally, an alternative approach for designing one-stage proceduresis to use taxation rules instead of bankruptcy rules by handing outfirst J i and then taking away with the aid of a taxation rule thedeficit T =

∑i∈N J i − R based on di = J i − J i for each i ∈ N.


References

References

[1]Alparslan Gok S.Z., “Cooperative Interval Games: Theory andApplications”, Lambert Academic Publishing (LAP), Germany(2010) ISBN:978-3-8383-3430-1.[2]Alparslan Gok S.Z., “Cooperative interval games”, PhDDissertation Thesis, Institute of Applied Mathematics, Middle EastTechnical University (2009).[3]Alparslan Gok S.Z., Miquel S. and Tijs S., “Cooperation underinterval uncertainty”, Mathematical Methods of OperationsResearch, Vol. 69, no.1 (2009) 99-109.[4] Bauso D. and Timmer J.B., “Robust Dynamic CooperativeGames”, International Journal of Game Theory, Vol. 38, no. 1(2009) 23-36.


References

References

[5] Branzei R., Branzei O., Alparslan Gok S.Z., Tijs S.,“Cooperative interval games: a survey”, Central European Journalof Operations Research (CEJOR), Vol.18, no.3 (2010) 397-411.[6]Branzei R., Tijs S., Alparslan Gok S.Z., “How to handle intervalsolutions for cooperative interval games”, International Journal ofUncertainty, Fuzziness and Knowledge-based Systems, Vol.18,Issue 2 (2010) 123-132.[7] Branzei R., Dimitrov D. and Tijs S., “Shapley-like values forinterval bankruptcy games”, Economics Bulletin Vol. 3 (2003) 1-8.[8]Branzei R., Mallozzi L. and Tijs S., “Peer group situations andgames with interval uncertainty”, International Journal ofMathematics, 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 ofgames IV, Annals of Mathematical Studies, Vol. 40. PrincetonUniversity Press, Princeton (1959) pp. 47-85.[10] Kimms A and Drechsel J., “Cost sharing under uncertainty: analgorithmic 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 cooperativegames”, Fuzzy Sets and Systems, Vol. 165 (2011) pp.98-105.[12] Yanovskaya E., Branzei R. and Tijs S., “MonotonicityProperties of Interval Solutions and the Dutta-Ray Solution forConvex Interval Games”, Chapter 16 in “Collective DecisionMaking: Views from Social Choice and Game Theory”, seriesTheory and Decision Library C, Springer Verlag Berlin/ Heidelberg(2010).

How to Handle Interval Solutions for Cooperative Interval Games

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