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How to Handle Interval Solutions for Cooperative Interval Games
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AACIMP 2011 Summer School. Operational Research Stream. Lecture by Sırma Zeynep Alparslan Gok.

AACIMP 2011 Summer School. Operational Research Stream. Lecture by Sırma Zeynep Alparslan Gok.

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    How to Handle Interval Solutions for Cooperative Interval Games How to Handle Interval Solutions for Cooperative Interval Games Presentation Transcript

    • 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
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011Outline Introduction Allocation rules The one-stage procedure The multi-stage procedure Final remarks References
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionIntroduction 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionMotivation 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionRelated 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionCooperative 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
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionInterval 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)
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionInterval 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).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Allocation rulesAllocation 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)).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Allocation rulesAllocation 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 ).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Allocation rulesBankruptcy 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Allocation rulesBankruptcy 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 for each bankruptcy problem (E , d) and all i ∈ N.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Allocation rulesBankruptcy 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 for each bankruptcy problem (E , d) and all 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Allocation rulesBankruptcy 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The one-stage procedureThe one-stage procedure 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)
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The one-stage procedureThe one-stage procedure 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The one-stage procedure 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}.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The one-stage procedureExample 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
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The one-stage procedureExample 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
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The one-stage procedureRemark 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).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The one-stage procedureRemark 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The multi-stage procedureThe multi-stage procedure 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 ) .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The multi-stage procedureThe multi-stage procedure 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)
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The multi-stage procedureThe multi-stage procedure 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The multi-stage procedureThe multi-stage procedure 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}
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The multi-stage procedure 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}.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The multi-stage procedureRemarks 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The multi-stage procedureRemarks 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 .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The multi-stage procedureExample 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 3Stage 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
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 The multi-stage procedureExample continuedStage 2. The amount R (2) − R (1) = 5 is distributed over agents in N 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 8Stage 3. The amount R (3) − R (2) = 15 is distributed over agents in N 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
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final remarksFinal 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final remarksFinal 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
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final remarksFinal 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final remarksFinal 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [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).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [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).