Partially ordered cooperative games: extended core and Shapley value

Ann Oper Res (2008) 158: 143–159 DOI 10.1007/s10479-007-0242-9 Partially ordered cooperative games: extended core and Shapley value J. Puerto ·F.R. Fernández ·Y. Hinojosa Published online: 27 September 2007 © Springer Science+Business Media, LLC 2007 Abstract In this paper we analyze cooperative games whose characteristic function takes values in a partially ordered linear space. Thus, the classical solution concepts in cooperative game theory have to be revisited and redefined: the core concept, Shapley–Bondareva theorem and the Shapley value are extended for this class of games. The classes of standard, vector-valued and stochastic cooperative games among others are particular cases of this general theory. Keywords Cooperative games ·Core ·Shapley value ·Partial order 1 Introduction Game theorists have tried over years to keep connected the development of Game Theory to actual applications. This orientation is very important from an economic point of view and also from the practitioners point of view. Nevertheless, nowadays game theory is not only an economic tool but also an interesting mathematical discipline. In this regard, games themselves are mathematical objects worth to be investigated. One may argue that in mathematical-economics the stress is put in the applicability however there are many aspects within the Theory of Games that remain open and beg for further analysis. The research of the authors is partially supported by Spanish DGICYT grant numbers MTM2004-0909, HA2003-0121, HI2003-0189, MTM2007-67433-C02-01, P06-FQM-01366. J. Puerto ( )·F.R. Fernández Departamento de Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Sevilla, 41012 Sevilla, Spain e-mail: [email protected] F.R. Fernández e-mail: [email protected] Y. Hinojosa Departamento de Economía Aplicada I, Facultad CCEE, Universidad de Sevilla, Sevilla, Spain e-mail: [email protected] 144 Ann Oper Res (2008) 158: 143–159 One of these situations can be identified in the allocation of divisible entities among the agents operating in an optimization problem. One of the branches of the theory of games, namely TU-cooperative game theory, has covered partially this field. Indeed, all this theory is founded in that either worth or cost will be allocated to the players. This means that any object to be allocated is given a value through a “utility” and only this virtual utility can be decomposed and finally allocated. One can think of situations, as for instance the following story, that fits bad to the paradigm above. Three guys are in the middle of the desert and have in common one orange, one grapefruit and one watermelon. Each one of them must go in a different direction and needs the fruits as future refreshment. It is clearly not optimal to cut the fruits in pieces since conservation would be impossible. Moreover, here the monetary value is of no interest. This is of course, also, the case of heritages of goods having familiar or subjective interest. It is clear to us that considering the problem of how to allocate mathematical objects as a whole requires a titanic effort (as considered as a single project). Therefore, one can try to proceed by stages. In a first stage we restrict ourselves to the problem of how to allocate elements of a partially ordered linear space. The standard analysis of cooperative TU-games assumes that the payoff of any coalition is valued by a real number. Here we replace this assumption allowing the payoffs to be elements of any partially ordered linear space. Different extensions of cooperative games have being the games with a continuous of player by Aumann and Shapley (1974)orthe fuzzy coalition theory by Aubin (1987). In recent years another productive line of research has been to impose different structures on the set of coalitions for the players in the games. This analysis gives rise to the so called cooperative games on combinatorial structures (see Bilbao 2000). Our analysis is completely different instead of imposing conditions on the argument of the characteristic function we extend the nature of the payoffs. Particular instances of this model have been already considered in (Fernández et al. 2002a; Granot 1977; Nishizaki and Sakawa 2001; Suijs 2000; Suijs et al. 1998,1999; Timmer 2001). Our goal is to extend the solution concepts of the classical cooperative game theory to this new class of games. Specifically, we study the two most widely used solution concepts within the theory of cooperative games: (1) the core set (set solution), and; (2) the Shapley value. The partially ordered cooperative game theory includes as a particular instance the standard TU-games, as well as some other classes of games such as vector-valued and stochastic games. The results in this paper are summarized in the following: (1) introduction of a new class of cooperative games whose characteristic function ranges on any linear space, (2) definition of different core concepts according to the domination relationship defined on the space, (3) characterization of non-emptiness of the core; and, (4) extension of the Shapley value to this class of games and its characterization by potentials and axiomatically. The paper is organized as follows. In Sect. 2we present the basic concepts and definitions concerning partially ordered cooperative games. Section 3contains the results concerning the core. We present the notion of core set and its characterization in terms of coalitional dominance. We also prove an analogous to Bondareva–Shapley Theorem that holds for partially ordered cooperative games using a general form of duality by Jahn (1983). Section 4 introduces the extended Shapley value. It also characterizes this value using extended potentials and a set of axioms. The paper ends with the references cited in the text. Ann Oper Res (2008) 158: 143–159 145 2 Basic concepts Let ℵbe a linear space over the real field. A partially ordered cooperative game (N, v) is asetN={1,2,...,n}of players and a map v:2N∪{∅}→ℵon the set 2Nof all subsets of Nsuch that v(∅)=ℵ(ℵis the null vector in the space ℵ). The elements of the set N are called players and the function vis the characteristic function of the game. The function v(S) is the worth of the coalition S. We denote by POn(ℵ)the family of all the partially ordered cooperative games defined on the space ℵ. We assume that there exists a partial order (reflexivity and transitivity) defined on the set ℵ(see Roubens and Vincke 1985). We represent by the corresponding strict partial order and by ∼the indifference relationship. Associated with there is another binary reflexive relation defined by: XYiff not(Y X), ∀X= Y, that is important to be considered in our analysis. We require to this partial order a natural densedness condition: for any X= Y∈ℵ,XY⇒∃Z∈ℵ,XZY. (1) It is worth noting that the well-known scalar, vector-valued (Fernández et al. 2002a)and stochastic cooperative games (Fernández et al. 2002b; Suijs et al. 1999) are particular cases of this formulation, just considering ℵ=Rwith the ≥order over the reals, Rnwith the component-wise order and L1(R)with the stochastic dominance order, respectively. The cases above are examples that can be found in the literature of game theory although one can think of many other interesting structures where this approach can be applied. If players agree on cooperation then an interesting question which arises is how the worth v(N) should be allocated among the various players. The natural extension of the idea of allocation (or preimputation) used in scalar games to the partially ordered cooperative games consists of using an allocation X=(X1,...,X n) where Xi∈ℵ,i=1,...,n, stands for the payoff of the i-th player. We assume that the worth of v(N) must be allocated to the players in N. This is the well-known efficiency principle. Therefore, in the rest of the paper we will consider only allocations that satisfy n  i=1 Xi∼v(N). The set of allocations of a partially ordered cooperative game (N, v) is denoted by I∗(N, v). Formally, I∗(N, v) =(X1,...,X n)∈ℵ n: n  i=1 Xi∼v(N). Among all the allocations of the game (N, v) ∈POn(ℵ)we are interested in those which cannot be dominated by the worth given to the coalitions. Thus, some kind of ordering concept is necessary to perform these comparisons. 146 Ann Oper Res (2008) 158: 143–159 3 Core solutions The minimum requirement imposed on allocations so that players do not refuse them is the following: each individual player igets a payoff Xibeing not worse than the worth v(i) given by the characteristic function of the game. The set of all the allocations that fulfill this property, I(N,v), is called imputation set of the game. I(N,v)={X∈I∗(N, v) :Xiv(i) ∀i}. Keeping tracks of the development followed in the standard theory the next step is to impose the collective rationality to those imputations proposed as good allocations. This idea was first suggested by (Gillies 1959) and later formalized (for scalar games) under the name of core of the game. Definition 3.1 The core of the partially ordered cooperative game (N, v) ∈POn(ℵ)is defined as the set of allocations such that XS:= i∈SXiis as least as preferred as v(S),for every coalition Sand it is denoted by core(N, v;)={X∈I∗(N, v)/XSv(S) ∀S⊂N}. Notice that ⊂stands for strict inclusion, while ⊆will be used in the paper for the regular inclusion. Example 3.1 Let us consider the partially ordered lineal space of continuous functions X:[0,2]−→Rsuch that X(t) =I[0,1](t)f1(t) +I[1,2](t)f2(t) where fii=1,2areaffine functions, IA(t) =1ift∈A, 0otherwise; and the partial order given for any X, Y in this space by XYif X(t) ≥Y(t) for any t∈[0,2]. We consider the three-person game (N, v) whose characteristic function is given by: S{1} {2} {3} {1,2} {1,3} {2,3} {1,2,3} f1(t) 23+t1+3t6+t3+3t4+5t10 +2t f2(t) 3/2+t/27/2+t/27/2+t/27 5+t17/2+t/212 It is clear that the allocation given by X1X2X3 f1(t) 35−t2+3t f2(t) 7/2−t/27/2+t/25 belongs to core(N, v, ). In order to characterize the core we need to introduce the coalitional dominance induced by the relationship .LetX, Y ∈I∗(N, v) and S⊆Na coalition. Ydominates Xthrough S according to and we will denote Y S domXif YSXSand v(S)YS. This concept leads us to consider the notion of non-dominated imputation by allocations. Ann Oper Res (2008) 158: 143–159 147 Definition 3.2 An imputation X∈I(N,v) of the game (N, v) is non-dominated by allocations if for any coalition S⊆Nit does not exist an allocation Y∈I∗(N, v) such that Y S domX. This set is given by: NDIA(N, v, )={X∈I(N,v)/S⊆N,Y ∈I∗(N, v), Y ∼ X:Y S domX} This set exhibits a close relation with the concept of core given in Definition 3.1. Theorem 3.1 The following relationship NDIA(N, v;)=core(N, v;)holds. Proof Let us assume that X∈ NDIA(N, v;)then it must exist a coalition S⊂Nand Y∈I∗(N, v) such that: v(S) YSXS. This implies that XSv(S) does not hold and hence X∈ core(N, v;). Conversely, let X∈ core(N, v;). Then, it exists S⊆Nsuch that XSv(S) does not hold. Therefore, v(S) XS. Now, we can apply the densedness property (1)anditmust exists Y∈ℵsatisfying: v(S) YXS, and hence X∈ NDIA(N, v;). Our next result is a first sufficient condition for non-emptiness of the core. Let ube a function u:ℵ→Rsatisfying for any X1,X 2∈ℵ,X1X2⇒u(X1)≥u(X2).(uagrees with the partial order .) We define the set C(N,vu)={X∈I∗(N, v) :u(v(S)) ≤u(XS), ∀S⊆N}. It is worth noting that this set may be used as solution concept if players agree on allocating the worth in the game through a utility function u. Lemma 3.1 For any uthat agrees with the partial order ,the relationship core(N, v, )⊆C(N,vu), holds. Proof Let us assume that X∈core(N, v, )but X∈ C(N,vu). Then, it must exist a coalition Ssuch that u(XS) < u(v(S)). However, this is not possible because uagrees with the partial order . We can give alternative conditions on the non-emptiness of the core. Let us assume that the partial order is defined by the family of functions U. That means that XY⇔ u(X) ≥u(Y ), ∀u∈U. Then, we can establish the following theorem. Theorem 3.2 It holds that: core(N, v, )= u∈U C(N,vu). 148 Ann Oper Res (2008) 158: 143–159 Proof The inclusion core(N, v, )⊆u∈UC(N,vu)is clear by the definition of C(N,vu) and Lemma 3.1. Then, let us assume that X∈u∈UC(N,vu)and X∈ core(N, v, ). Thus, it must exist S⊆Nsuch that not(XSv(S)). This is equivalent to that it exists ¯u∈Usuch that ¯u(v(S)) > ¯u(XS). However, this means that X∈ u∈UC(N,vu). Let us consider a utility function uon ℵ. Associated with uwe define the following scalar cooperative game (N, vu)where the characteristic function is given by vu(S) =u(v(S)) for any S⊆N. Theorem 3.2 is particularly important when the cone that characterizes the partial order is finitely generated. (XY⇔uj(X) ≥uj(Y ), j =1,...,k.) Then we get the following lemma. (Recall that the core in a scalar cooperative game is not empty if and only if the game is balanced. See Owen (1995).) Lemma 3.2 core(N, v, )=∅if the scalar game (N, vuj)is not balanced for some j= 1,...,k. Notice that the very important case of a partial order defined by individual utilities, described above, fits into this category. It is worth noting that for the core to be empty it suffices that a game (N, vui)has empty core, although some other games (N, vuj), j = imay have nonempty core. Example 3.2 (Vector-valued games, Fernández et al. 2002a) Let us consider the space (Rk,≥), where for any x,y ∈Rk,x≥ymeans xi≥yifor i=1,...,k.Thegame(N, v) whose characteristic function vis defined: v:2N→Rk,v(∅)=0 is called vector-valued game. In this case the partial order is generated by the utility functions ui(x) =xi,i=1,...,k. Therefore, by Theorem 3.2 core(N, v, ≥)=k i=1C(N,vui). In this particular case, any element X∈core(N, v, ≥)is a k×nmatrix whose i-th row Xiis a core allocation of the scalar game (N, vui). 3.1 A necessary and sufficient condition for non-emptiness of the core In the following we address a characterization of core(N, v, )similar to the one known as Bondareva–Shapley Theorem. First of all, we would like to recall the idea behind that theorem. The theorem states a primal minimization problem whose feasible set defines the imputations in the core while the objective function is just the value obtained by the grand coalition with a given imputation. To this problem (primal feasibility problem) it is associated a dual maximization problem. The feasible set of this problem is taken as a definition of balancedness. Proving strong duality between these two problems allows to characterize the non-emptiness of the core as soon as the dual problem is bounded from above. This is essentially what is done by Bondareva–Shapley theorem. This analysis is not only privative of the family of games defined over the real line. In general, every time that we are able to define the core of a game as the feasible set of a primal minimization problem and strong duality is proven with respect to a dual maximization problem we can do a similar argument and a necessary and sufficient condition for the core is generated. This is the argument in our approach. In order to be able to prove such characterization we assume that is induced by a convex cone Dℵ,thatisXY⇔X−Y∈Dℵ. We also consider the space Zbeing the 2n−2-fold Cartesian product of the linear space ℵ, i.e. Z=ℵ 2n−2. The space Zis partially Ann Oper Res (2008) 158: 143–159 149 ordered by the convex cone DZ=(Dℵ)2n−2. (The reader may notice that this is not a restriction because all the interesting cases fall into the considered case.) Let ℵ∗and Z∗be the topological dual of the spaces ℵand Z.ForanyY∈ℵand ˆ Y∈ℵ ∗we denote by ˆ Y,Y the pairing between the elements of the primal and the dual spaces, e.g. the action of the continuous linear functional ˆ Yon Y. Therefore, ˆ Y,Y= ˆ Y(Y). (The analogous definition holds for the pairing between Zand Z∗). The ordering cone of the topological dual space ℵ∗is given by: Dℵ∗:= { ˆ Y∈ℵ ∗:ˆ Y,Y≥0,∀Y∈Dℵ} and the quasi-interior of Dℵ∗is given by: D# ℵ∗:= { ˆ Y∈ℵ ∗:ˆ Y,Y>0,∀Y∈Dℵ\{ℵ}}, where ℵdenotes the zero of the space ℵ. Let us define two linear mappings from ℵninto ℵand Z, respectively, as follows: C:ℵ n−→ ℵ X=(X1,...,X n)−→ C(X) = n  i=1 Xi A:ℵ n−→ Z X=(X1,...,X n)−→ A(X) = i∈S XiS⊂N . We denote by ∗applied to an operator its adjoint operator, namely C∗,A∗and T∗denote the adjoint mappings of C,Aand T, respectively. Finally, let L(Z, ℵ)be the linear space over Rof continuous linear mappings from Zinto ℵ. Definition 3.3 The partially ordered cooperative game (N, v, )is balanced if for any ˆ Y∈ D# ℵ∗there exists T∈L(Z, ℵ)such that: 1. (C −TA) ∗(ˆ Y)∈(Dℵ∗)n, 2. T∗(ˆ Y)∈DZ∗, 3. v(N) T ((v(S))S⊂N)and does not exist Tsuch that T((v(S))S⊂N)T ((v(S))S⊂N) We note in passing that the above balancedness condition is similar to the one in the scalar case and reduces to the usual one when we consider scalar cooperative games. It states that it must exist a maximal (non-dominated by the partial order )elementTbeing inferior that v(N). It is based on a feasibility condition on a dual problem that will appear in the proof of the characterization theorem. The reader may notice that there exist in the literature some other extensions of the concept of balancedness to class of games without side payment as for instance in (Billera 1970; Kannai 1992; Keiding and Thorlund-Petersen 1987; Shapley 1973) and the more recent by (Predtetchinski and Herings 2004)wherea necessary and sufficient condition on the non-emptiness of the core of a cooperative game without side payment is given. Let us consider the following set FA={X∈ℵ n:A(X) −(v(S)S⊂N)∈DZ,X∈(Dℵ)n}. 150 Ann Oper Res (2008) 158: 143–159 First of all, we assume that (v(S))S⊂N= Z. Moreover, we impose that our continuous linear map Averifies the Slater type stability condition (see Jahn 1983, p. 346), i.e. there exists (X1,...,X n)∈ℵ nsuch that A(X) −(v(S))S⊂N∈ ◦ DZ, the topological interior of DZ. These are stability conditions to ensure a certain type of duality defined on the game payoff function. We assume further that {C(X) :X∈FA}+Dℵis convex with non-empty algebraic interior. (2) This hypothesis ensures that any minimal element of the set {C(X) :X∈FA}is properly minimal (see Jahn 1984). It is worth noting that our problem always fulfills condition (2)in finite dimension spaces. Thus, in the scalar case it always holds without explicitly imposed. Theorem 3.3 The game (N, v, )is balanced if and only if core(N, v, )=∅. Proof Let us consider the following problem: (P ):“min” C(X) s.t.:A(X) −(v(S))S⊂N∈DZ, X∈(Dℵ)n where “min” must be understood in the sense of minimal points in the order . Under our hypothesis on stability of the map Awe can apply the dual by (Jahn 1983) which for problem (P) turns out to be: (D):“max” T ((v(S))S⊂N) s.t.:(C −TA) ∗(ˆ Y)∈(Dℵ∗)n, T∗(ˆ Y)∈DZ∗, T∈L(Z, ℵ), ˆ Y∈D# ℵ∗. These two dual problems satisfy that any maximal solution of (D) corresponds to a properly minimal element of (P ) and conversely any properly minimal element of (P ) is a maximal element of (D) by (2). Let us assume that (N, v, )is balanced with T. This element corresponds to a maximal solution of (D). Then, by duality there exists a feasible solution ˆ Xto (P ) such that it is properly minimal and C( ˆ X) =T ((v(S))S⊂N). Therefore, we have v(N) −C( ˆ X) ℵ;and there exists a core allocation in core(N, v, ). Conversely, assume w.l.o.g. that v(N) is a minimal element of (P ). Since we have imposed the condition (2)on(P ) any minimal element is properly minimal. Then, by duality there exists a maximal element ˆ Tof (D) such that v(N) =T ((v(S))S⊂N). This implies that the game is balanced.  The application of this characterization to the well-known componentwise partial order of Rmis natural. The reader should notice that for m=1 it reduces to the standard case of Rwith the natural order, and therefore the definition of balancedness will give us the classic one for real valued cooperative games. Example 3.3 (Balancedness of vector-valued games with the componentwise order of Rm) The elements in the case of ℵ=Rmwith the componentwise order given by the cone Dℵ=Rm +are the following. The dual space ℵ∗=Rm, and the dual cone Dℵ∗=Rm +.This Ann Oper Res (2008) 158: 143–159 151 implies that Z=Rm(2n−2).ThemapC:R(m·n)×1→Rm×1is given by C=(cj i)i=1...m j=1...m×n where cj i=1ifj=(k −1)m +i, for k=1,2,...n, 0otherwise. The map A:R(m·n)×1−→ R(m(2n−2))×1is given by A=(AS)t S⊂N(tdenotes the transpose), where for any S⊂N,AS=(aj i,S)i=1...m j=1...m·n ,and aj i,S =1ifj=(k −1)m +i, for k=1,2,...n, and k∈S 0otherwise. Notice that since we consider the componentwise order in Rm, in the adapted problem (P ) (see page 150), there are mconstraints per each coalition S⊂N\∅. Therefore, the dimension of Ais ((2n−2)·m) ×(m ·n). Finally, Tis an element of L(Z, Rm), the space of continuous linear maps from Zto Rm.ThismeansthatTis a matrix of dimensions m×[(2n−2)·m]. To simplify, we write T=(T1,...,T m)twhere Ti=(T S i)S⊂N∈R1×[(2n−2)m],andTS i=(tS,j i)j=1...m ∈R1×m. In order to compute G:= C−TA, we must compute the matrix E:= T×A= (T1A,T2A,...,T mA)t,being TiA= S⊂N TS iAS∈R1×(mn). Notice that the row vector TiA:= (ej i)j=1...m is given by: ej i= S⊂N m  r=1 tS,r iaj r,S ,for j=1...mn. According to the definition of aj r,S we obtain that aj r,S =1iffj=(k −1)m +r, for some k=1,2,...,n, and k∈S, and thus ej i= S⊂N k∈S tS,(k−1)m+i i. Hence, we obtain G:= (C −TA):= (gj i)=(cj i−ej i)i=1...m j=1...m×n . Here the adjoint G∗of Gis the transpose matrix, e.g. G∗=Gt. Now we are in conditions to write down the balancedness conditions. For the sake of simplicity we first write conditions 2, and 3, and then condition 1. (2) For any ˆ Y∈Rm +\{0},Ttˆ Y≥0. This is equivalent to tS,j i≥0foranyi, j and S⊂N. (3) For any i=1,...,mit must hold vi(N) ≥ S⊂N n  k=1 tS,(k−1)m+i ivi(S). 158 Ann Oper Res (2008) 158: 143–159 Remark 4.3 We assume that the games are normalized with respect to the cone cone(B), induced by the positive linear combinations of the elements of the basis of ℵ. This means that v({i})∈−cone(B)for any i∈N. Notice that it does not mean loss of generality since we can translate all the payoffs by a fix vector maintaining the ordering relationships among them. Theorem 4.3 The extended Shapley value is the unique value defined on all the partially ordered cooperative games satisfying Axioms A1, A2, and A3. Proof It is clear that if Sis a carrier for wpj Sthen any superset Tsuch that S⊆Tis also a carrier. Thus, by axiom A1 we get that i∈Tϕi[wpj S]=pjfor any S⊆T. This fact together with Remark 4.3 implies that ϕi[wpj S]=ℵfor i∈ S. Now, by using A2 and the linearity implied by A3we can deduce that for any c>0 ϕi[cwpj S]=cpj sif i∈S, ℵotherwise. Let pi S=(ℵ,..., S pi,...,ℵ)t∈ℵ 2n−1, it is clear that since Bis a basis of ℵthen ¯ B=(pi S)i∈I,S⊆Nis a basis of ℵ2n−1. For any game (N, v) ∈POn(ℵ),let(dpi S)i∈I,S⊆Nbe the unique set of scalar in the representation of vin the basis ¯ B.Thisis, v= pi∈B S⊆N dpi Spi S. Let U⊂Nbe any coalition, then v(U) = pi T⊆U dpi Tpi= pi T⊆U S⊆U S⊇T (−1)s−tdpi Tpi, = pi S⊆U T⊆S (−1)s−tdpi Tpi let cpi S= T⊆S (−1)s−tdpi T= pi S⊆U cpi Spi= pi S⊆N cpi Swpi S(U). Therefore, using axiom A3we have ϕj[v]= pi S⊆N cpi Sϕj[wpi S]= pi S⊆N j∈S cpi S pi s = pi S⊆N j∈S  T⊆S (−1)s−tdpi Tpi s= pi T⊆N S⊆N T∪{j}⊆S (−1)s−tdpi T spi Ann Oper Res (2008) 158: 143–159 159 let γj(T ) = S⊆N T∪{j}⊆S (−1)s−t1 s= pi pi T⊆N j∈T γj(T )[dpi T−dpi T\{j}] since γj(T ) =(t −1)!(n −t)! n!= T⊆N j∈T (t −1)!(n −t)! n! pi [dpi T−dpi T\{j}]pi = T⊆N j∈T (t −1)!(n −t)! n!(v(T ) −v(T \{j})). Therefore, the extended Shapley value is uniquely characterized by axioms A1, A2, and A3. References Aubin, J. P. (1987). Optima and equilibria: an introduction to nonlinear analysis. Heidelberg: Springer. Aumann, R. J., & Shapley, L. S. (1974). Values of non-atomic games. Princeton: Princeton University Press. Bilbao, M. (2000). Cooperative games on combinatorial structures.Theory and decision library, series C. Dordrecht: Kluwer Academic. Billera, L. J. (1970). Some theorems on the core of an n-person game without side payment. SIAM Journal of Applied Mathematics,18, 567–579. Fernández, F. R., Hinojosa, M. A., & Puerto, J. (2002a). Core solutions of vector-valued games. Journal of Optimization Theory and Applications,112, 331–360. Fernández, F. R., Puerto, J., & Zafra, M. J. (2002b). Cores of Stochastic cooperative games. International Game Theory Review,4(3), 265–280. Gillies, D. B. (1959). Solutions to general non-zero-sum Games. In D. Luce & A. W. Tucker (Eds.), Annals of Mathematics Studies: Vol. 40.Contributions to the theory of games IV (pp. 47–85). Princeton: Princeton University Press. Granot, D. (1977). Cooperative games in stochastic characteristic function form. Management Science,23, 621–630. Hart, S., & Mas-Colell, A. (1989). Potential, value and consistency. Econometrica,57, 589–614. Jahn, J. (1983). Duality in vector optimization. Mathematical Programming,25, 343–355. Jahn, J. (1984). Scalarization in vector optimization. Mathematical Programming,29, 203–218. Kannai, Y. (1992). The core and balancedness. In R. J. Aumann & S. Hart (Eds.), Handbook of game theory with economic applications (pp. 355–395). Amsterdam: North-Holland. Keiding, H., & Thorlund-Petersen, L. (1987). The core of a cooperative game without side payments. Journal of Optimization Theory and Applications,54, 273–288. Nishizaki, I., & Sakawa, M. (2001). Fuzzy and multiobjective games for conflict resolution. Heidelberg: Physica. Owen, G. (1995). Game theory. New York: Academic Press. Predtetchinski, A., & Herings, P. J.-J. (2004). A necessary and sufficient condition for non-emptiness of the Core of a non-transferable utility game. Journal of Economic Theory,116, 84–92. Roubens, M., & Vincke, P. (1985). Lectures notes in economic and mathematical systems.Preference modelling. Berlin: Springer. Suijs, J. (2000). Cooperative decision-making under risk. Boston: Kluwer Academic. Suijs, J., Borm, P., & De Waegenaere, A. (1998). Stochastic cooperative games in insurance. Insurance,22, 209–228. Suijs, J., Borm, P., De Waegenaere, A., & Tijs, S. (1999). Cooperative games with stochastic payoffs. European Journal of Operational Research,113, 193–205. Shapley, L. S. (1953). A value for n-persongames.InH.W.Kuhn&A.W.Tucker(Eds.),Annals of mathematics studies: Vol. 28.Contributions to the theory of games II (pp. 307–317). Princeton: Princeton University Press. Shapley, L. R. (1973). On balanced games without side payments. In T. C. Hu & S. M. Robinson (Eds.), Mathematical programming (pp. 261–290). New York: Academic Press. Timmer, J. (2001). Cooperative behaviour, uncertainty and operations research. Center Dissertation Series. Tilburg. Reproducedwithpermissionofthecopyrightowner.Furtherreproductionprohibitedwithoutpermission.