Set-valued TU-games

UNCORRECTED PROOF 2Set-valued TU-games 3F.R. Fern aandez a , M.A. Hinojosa b , J. Puerto a,* 4 a Facultad de Mathematicas Dept., Departamento de Estad ııstica e Investigacioon Operativa, Universidad de Sevilla, 541012 Sevilla, Spain 6 b Departamento de Econom ııa y Empresa, Universidad Pablo de Olavide, Sevilla, Spain 7Received 6 May 2002; accepted 3 March 2003 8Abstract 9The goal of this paper is to explore solution concepts for set-valued TU-games. Several stability conditions can be 10 defined since one can have various interpretations of an improvement within the multicriteria framework. We present 11 two different core solution concepts and explore the relationships among them. These concepts generalize the classic 12 core solution for scalar games and can be considered under different preference structures. We give characterizations for 13 the non-emptiness of these core sets and apply the results to four multiobjective operational research games. 14 2003 Published by Elsevier B.V. 15 Keywords: Multiobjective analysis; Game theory; Core 16 1. Introduction 17 It is currently accepted that real-world decision processes are multivalued. This assertion means that 18 decision-making is actually based on several (more than one) criteria. Obviously, using several criteria 19 implies the non-existence of a total order among the evaluation of the different alternatives. Thus, regarding 20 the scalar case, where all the optimal decisions share the same evaluation, in multicriteria decision-making 21 the above property does not make sense. In the latter case, the decision-maker may accept many different 22 alternatives provided that their evaluations are non-dominated componentwise. 23 Modelling conflict situations where several criteria must be considered simultaneously leads in a natural 24 way to multiobjective game theory (see e.g. Bergstresser and Yu, 1977; Blackwell, 1956; Hwang and Lin, 25 1987; Shapley, 1959). In this framework the evaluation given to the alternatives considered by the agents is 26 not a unique value but a set of non-dominated vectors (see Fern aandez et al., 1998; Fern aandez and Puerto, 27 1996; Puerto and Fern aandez, 1995). 28 The discussion above leads us to consider the class of the multiobjective cooperative TU-games. Within 29 this class any coalition Sof player is given a characteristic set of vectors. These vectors represent the non- * Corresponding author. Tel.: +954557940; fax: 954622800. E-mail addresses: [email protected] (F.R. Fern aandez), [email protected] (M.A. Hinojosa), [email protected] (J. Puerto). 0377-2217/03/$ - see front matter 2003 Published by Elsevier B.V. doi:10.1016/S0377-2217(03)00398-9 European Journal of Operational Research xxx (2003) xxx–xxx www.elsevier.com/locate/dsw EOR 5770 No. of Pages 15, DTD = 4.3.1 7 July 2003 Disk used ARTICLE IN PRESS UNCORRECTED PROOF 30 dominated payoffs that the members of a coalition can ensure by themselves. Notice that different from the 31 classic scalar case, in this framework, coalitions may support any of their admissible payoffs in their 32 characteristic set of vectors. Hence, in this class of TU-games one looks not only for fair allocations of the 33 grand coalitionÕs payoffs but for which of the grand coalitionÕs payoffs the above question can be answered 34 in an affirmative way. 35 When the characteristic set of vectors are singletons, we obtain the class of vector-valued games (see 36 Fern aandez et al., 2002). In addition, if the number of criteria considered by the agents is only one we obtain 37 the standard theory of cooperative TU-games. 38 It is also worth noting that with this class we can model any game whose characteristic set of vectors is 39 given implicitly as the set of non-dominated vectors of a multiobjective program. In particular Operation 40 Research games (see Borm et al., 2001) may be analyzed within this new framework when more than one 41 objective is simultaneously considered in the optimization process. Examples are multiobjective flow games, 42 multiobjective minimum spanning tree games, multiobjective combinatorial optimization games, etc. 43 In order to illustrate the discussion above, we describe in detail three different classes of set-valued TU- 44 games: the multiobjective linear production game, the multiobjective continuous single facility location 45 game and the multiobjective minimum cost spanning tree game. It is worth noting that the two former 46 games come from a continuous multiobjective OR problem (the scalar version of these games were in- 47 troduced by Owen (1975) and Puerto et al. (2001), respectively) while the latter does from a combinatorial 48 one (the scalar version of this game was introduced by Bird, 1976). 49 1.1. The multiobjective linear production game 50 Consider the multiobjective linear production problem: ½Pv-max Cx s:t::x2FðPÞ:¼fx2Rp:Ax 6b;xP0g; 52 where C2Rkpis the matrix whose rows represent the kdifferent objectives of the problem; A2Rmpis the 53 technological matrix; b2Rmis the resource vector; xis the production vector and FðPÞis the decision set 54 for the problem [P]. 55 The solution concept for this problem is the set of efficient solutions: EðPÞ¼fx2Rp:9=y2FðPÞverifying Cy PCx;Cy 6¼ Cxg 57 and the set of values of the efficient solutions is: ZðPÞ¼fzðxÞ:zðxÞ¼Cx;x2EðPÞg: 59 This model can be considered as a game when the pool of resources is controlled by ndifferent agents 60 (players). Let us assume that player iholds a resource vector bi¼ðbi 1;bi 2;...;bi mÞt,i¼1;2;...;n. Thus, if 61 coalition Sof players is to form it controls a bundle of resources bðSÞ¼Pi2Sbi. This vector of resources 62 makes possible for the coalition Sto produce goods according to the following linear production problem: ½PSv-max Cx s:t::x2FðPSÞ:¼fx2Rp:Ax 6bðSÞ;xP0g: 64 Finding the set of efficient solutions EðPSÞof this problem, coalition Sobtains payoff vectors in the set 65 ZðPSÞ¼fz2Rk:z¼Cx;x2EðPSÞg. 66 This framework leads naturally to introduce the multiobjective linear production game with nplayers 67 (agents) and where each coalition, S, can guarantee vectors in ZðPSÞ. 2F.R. Fern aandez et al. / European Journal of Operational Research xxx (2003) xxx–xxx EOR 5770 No. of Pages 15, DTD = 4.3.1 7 July 2003 Disk used ARTICLE IN PRESS UNCORRECTED PROOF 68 1.2. The multiobjective continuous single facility location game 69 A continuous single facility location problem is a set of nusers of a certain facility, placed in ndifferent 70 points in the space Rmwith mP1. The problem consists of finding a location for the facility which min- 71 imizes the transportation cost (which depends on the distances from the users to the facility) plus the setup 72 cost. Formally, a continuous single facility location problem is a 4-tuplet ðN;U;d;KÞwhere: 73 •N¼fa1;...;angis a set of ndifferent points in Rm(with nP2). 74 •U:Rn!Ris a lower semicontinuous globalizing function satisfying that: (1) Uis definite, i.e. UðxÞ¼0if 75 and only if x¼0; (2) Uis monotone, i.e. UðxÞ6UðyÞwhenever x6y. 76 •d:RmRm!Ris a measure of distance, satisfying that, for every r;s2Rm,dðr;sÞ¼fðkrskÞ, where 77 fis a lower semicontinuous, non-decreasing and non-negative map from Rto Rwith fð0Þ¼0, and kk is 78 a norm on Rm. 79 •Kis the setup cost. This cost is independent of the number of users and of the location of the facility; it is 80 mostly installation cost. 81 Solving the continuous single facility location problem ðN;U;d;KÞfor SNmeans to find an  xx 2Rm 82 minimizing UðdSðxÞÞ, where dSðxÞis the vector in Rnwhose ith component is equal to dðx;aiÞif ai2S, and 83 equal to zero otherwise. We denote LðSÞ¼minx2RmUðdSðxÞÞ. We impose to simplify the analysis that the 84 setup cost must be greater than or equal to the total transportation cost, i.e. KPLðNÞ. 85 This is the classical version of the continuous single facility location problem. Here we consider a natural 86 variant of this problem in which the users in Nare interested not only in finding an optimal location of the 87 facility, but also in sharing the corresponding total costs. 88 Therefore we can associate with ðN;U;d;KÞa cost TU-game ðN;vÞwhose characteristic function vis 89 defined, for every SN¼fa1;...;ang,by: vðSÞ¼ KþLðSÞif S6¼;; 0ifS¼;:  91 Every cost TU-game defined in this way is what we call a continuous single facility location game. If several 92 (more than one) globalizing functions Uj,j¼1;...;kare simultaneously considered then we get a set- 93 valued TU-game. It is worth noting that in this situation LðSÞ¼v-minx2RmðU1ðdSðxÞÞ;...;UkðdSðxÞÞÞ. Thus 94 the set-valued TU game ðN;VÞis given by VðSÞ¼KþLðSÞfor any SN, and Vð;Þ ¼ f0g. 95 1.3. The multiobjective minimum spanning tree game 96 Consider a set of Nusers of some good that is supplied by a common supplier 0 (N0¼N[f0g). There is 97 a multiobjective cost associated to the distribution system that has to be divided among the users. This 98 situation can be formulated as a set-valued game with Nplayers and a characteristic function that asso- 99 ciates to each coalition Sa set VðSÞthat represent the Pareto-minimum cost of constructing a distribution 100 system among the users in Sfrom the source 0. 101 Let G¼ðN0;EÞbe the complete graph with set of nodes N0and set of edges (links) denoted by E. There is 102 a vector of costs associated with the use of each link. Let eij ¼eji ¼ðeij 1;eij 2;...;eij kÞdenote the vector-valued 103 cost of using the link fi;jg2E. A tree is a connected graph which contains no cycles. A Pareto-minimum 104 cost spanning tree for a given connected graph, with costs on the edges, is a spanning tree which has Pareto- 105 minimum costs among all spanning trees (see Ehrgott, 2000). 106 A Pareto-minimum cost spanning tree game, associated to the complete graph G¼ðN0;EÞ, is a pair 107 ðN;VÞwhere Nis the set of player and Vis the characteristic function defined by: F.R. Fern aandez et al. / European Journal of Operational Research xxx (2003) xxx–xxx 3 EOR 5770 No. of Pages 15, DTD = 4.3.1 7 July 2003 Disk used ARTICLE IN PRESS UNCORRECTED PROOF 108 1. Vð;Þ ¼ f0g. 109 2. For each non-empty coalition SN, VðSÞ¼ vmin Pfi;jg2ETS0 eij TS0:spanning tree ; 111 where ETS0is the set of edges of the spanning tree, TS0, that contains S0¼S[f0g;andv-min stands for 112 Pareto-minimization. 113 Remark that the resulting spanning tree TS0must contain S0but it may also contain some additional nodes. 114 To analyze multiobjective games we extend the classical individual and collective rationality principles 115 using two different orderings in the payoff space. The first one corresponds with a compromise attitude 116 towards negotiation where coalitions admit payoffs that are not worse in all the components than any 117 payoffs that they can ensure by themselves. The second one, is a more restrictive ordering that only accept 118 payoffs that get more in all the components than all payoffs that they can guarantee by themselves. Similar 119 approaches to these two analysis have been done in Fern aandez et al. (2002), J€ oornsten et al. (1995) and 120 Nouweland et al. (1989) and an application can be seen in Fern aandez et al. (2001). 121 The paper is organized as follows. In the second section we introduce the definition of set-valued TU- 122 game and the concept of allocation for those games. Moreover, we analyze two different domination re- 123 lationships that extend the classic domination concept in the scalar case. In Section 3, we introduce the non- 124 dominated allocations sets, NDA sets, and we show the relationship with the core concepts. In Section 2 we 125 study existence theorems for these solution concepts. All the results are illustrated with three different 126 classes of games. 127 2. Basic concepts 128 A set-valued TU-game is a pair ðN;VÞ, where N¼f1;2;...;ngis the set of players and Vis a function 129 which assigns to each coalition SNa compact subset VðSÞof Rk, the characteristic set of coalition S, such 130 that Vð;Þ ¼ 0. 131 Vectors in VðSÞrepresent the worths that the members of coalition Scan guarantee by themselves. 132 Notice that the characteristic function in these games are set-to-set maps instead of the usual set-to-point 133 maps. 134 We denote by GVthe family of all the set-valued TU-games, by Gvthe class of vector-valued TU-games 135 and by gvthe family of all the scalar TU-games. 136 Example 2.1. Consider the following two-objective linear production problem with three decision makers 137 (players) in which the matrix that represents the two objectives is C¼24 1:51  139 and the technological matrix is A¼177 488  t : 141 The resource vectors for each player are b1¼ð14;14;13Þt,b2¼ð18;9;22Þtand b3¼ð11;18;22Þt. Then, 142 the characteristic sets for every coalition S(SN) are VðSÞ¼ZðPSÞ¼convðzS 1;zS 2Þ(convðAÞmeans the 4F.R. Fern aandez et al. / European Journal of Operational Research xxx (2003) xxx–xxx EOR 5770 No. of Pages 15, DTD = 4.3.1 7 July 2003 Disk used ARTICLE IN PRESS UNCORRECTED PROOF 143 convex hull of the set A): 145145 Example 2.2. Let N¼fa1;a2;a3gbe a set of players located at the points 0, 1, 2 on the real line and assume 146 that 0 <e. We consider two globalizing functions U1,U2given by: U1ðdNðxÞÞ ¼ 1 2 ejx0jþ 1 4 þejx1jþ1 4jx2j; U2ðdNðxÞÞ ¼ 1 4jx0jþ 1 4 ejx1jþð 1 2þeÞjx2j: 148 The multiobjective continuous single facility location game is given by the characteristic set 149 VðSÞ¼KþLðSÞ, for any SNwhere: 151151 153153 The reader may notice that LðSÞare the non-dominated values of the corresponding bicriteria location 154 problems, i.e. LðSÞ¼vminðU1ðdSðxÞÞ;U2ðdSðxÞÞÞ. 155 Example 2.3. Consider the complete graph below. 2 157157 The bi-criteria Pareto-minimum cost spanning tree game associated to the graph is: S{1} {2} {3} {1,2} {1,3} {2,3} N zS 1(6.5,1.625) (9,2.225) (8.25,3.89) (16.75,4,68) (15,5.41) (17.38,6.27) (25,8.58) zS 2(3.71,2.78) (9,2.25) (8,4) (15.14,5.35) (11.28,6.96) (17.38,6.27) (28.14,9.36) S {1}, {2}, {3} {1,2} {1,3} LðSÞ0 0  1 4þe 1 4e  1 4e  xþ1 2 1 4e  xþ1þ2e  for all x2½0;2 S{2,3} N ðSÞexþ1 4e 1 42e  xþ3 4þ3e  for all x2½1;21 2xþ1 4e2exþ3 4þ3e  for all x2½1;2 S{1} {2} {3} {2,3} {1,2} {1,3} N VðSÞ1 3  1 2  1 5  2 3  3 5  ;2 6  2 6  ;4 5  F.R. Fern aandez et al. / European Journal of Operational Research xxx (2003) xxx–xxx 5 EOR 5770 No. of Pages 15, DTD = 4.3.1 7 July 2003 Disk used ARTICLE IN PRESS UNCORRECTED PROOF 159159 If a set-valued TU-game is played then an interesting question is how an achievable vector vN2VðNÞ 160 should be divided among the players. It is worth noting that this is the same situation that appears in scalar 161 TU-games, where the worth of vðNÞ2Rhas to be allocated among the players. Nevertheless, in the set- 162 valued case there are many elements that can be considered to be divided among the players. 163 The extension of the idea of allocation used in scalar games to set-valued TU-games consists of using a 164 payoff matrix (an element of Rkn) whose rows are allocations of the criteria. Since the payoffs are vectors, 165 the allocations in these games are matrices Xwith krows (criteria) and ncolumns (players). The ith column, 166 Xi, in matrix Xrepresents the payoffs of ith player for each criteria; therefore Xi¼ðxi 1;xi 2;...;xi kÞtare the 167 payoffs for player i. The jth row, Xj, in matrix Xis an allocation among the players of the total amount 168 obtained in each criteria; Xj¼ðx1 j;x2 j;...;xn jÞare the payoffs corresponding to criteria jfor each player. The 169 sum XS¼Pi2SXiis the overall payoff obtained by coalition S. 170 Matrix Xis an allocation of the game ðN;VÞ2GVif XN¼Pi2NXi2VðNÞ. The set of the allocations of 171 the game is denoted by IðN;VÞ. 172 3. Dominance and core concepts 173 An important point in the development of set-valued TU-games is the use of the new orderings defined in 174 the set of allocations. To this end, we must replace the complete order ‘‘ 6’’ in R, for the comparison 175 between allocations and the characteristic sets, by the considered orderings in Rk, that is, ‘‘be better or equal 176 componentwise’’, denoted by ‘‘=’’, and ‘‘not be worse’’, denoted by ‘‘i’’. 177 To simplify the presentation in the following, XSiVðSÞmeans XSivS8vS2VðSÞ, that is, there does not 178 exist vS2VðSÞsuch that XS5vS,XS6¼ vS. Analogously XS=VðSÞmeans XS=vS8vS2VðSÞ, that is, 179 XS jPvS j8j¼1;2;...;k;8vS2VðSÞ. 180 These orderings, above defined, lead us to two different core concepts in set-valued TU-games. When the 181 ordering is defined as ‘‘i’’, we have the following definition of core: 182 Definition 3.1. The dominance core of a game ðN;VÞ2GVis the set of allocations, X2IðN;VÞ, such that 183 XSiVðSÞ8SN. We will denote this set as CðN;V;iÞ. 184 Nevertheless, it may happen that in some situations the preference structure assumed by the agents is 185 stronger, and coalitions only accept allocations if they get more than the worth given by the characteristic 186 set. This assumption modifies the rationale of the decision process under the game and, therefore, the core 187 concept will be modified accordingly. Proceeding similarly, we introduce now the concept of core with 188 respect to the strong ordering, that we will call the preference core. 189 Definition 3.2. The preference core of a game ðN;VÞ2GVis the set of allocations, X2IðN;VÞ, such that 190 XS=VðSÞ8SN. We will denote this set as CðN;v;=Þ. 191 The preference core is always included in the dominance core. Thus, it may happen that the former set is 192 empty while the latter set is not. Nevertheless, if the preference core is non-empty then the players will only 193 agree on allocations within this set because all the players will be better off without assuming any com- 194 promise. Therefore, this solution concept must be considered in any set-valued game provided that we are 195 given tools to check whether it is non-empty. 196 The dominance core defined above coincides with the set of stable outcomes (SO) introduced by van den 197 Nouweland et al. (1989). Thus, our treatment is similar to that of these authors although our character- 6F.R. Fern aandez et al. / European Journal of Operational Research xxx (2003) xxx–xxx EOR 5770 No. of Pages 15, DTD = 4.3.1 7 July 2003 Disk used ARTICLE IN PRESS UNCORRECTED PROOF 198 ization is different. In addition, we characterize the preference core, a concept not considered in the above 199 mentioned paper. 200 Example 3.1. Let us assume a production situation where three agents can produce, using three different 201 technologies A, B, C, two types of goods. The characteristic set of any coalition Sis given by the production 202 levels of each good using the existing technologies, i.e. VðSÞis a set of three vectors (technologies) with two 203 components each one (goods). The following table defines the characteristic set-valued map of the game 204 ðN;VÞ. 206206 If the agents decide to cooperate and to produce with the technology A they must allocate the vector 207 of goods (5,4), the allocation X¼221 112  209 is in the preference core, while Y¼13=25=2 3=23=21  211 is in the dominance core and not in the preference core since Yf1;2g¼ð5=2;3Þjð3;1Þ, the third element of 212 the characteristic set Vðf1;2gÞ. 213 Imputations in the core (any of them) will be acceptable if no coalition can argue against its allocated 214 amount XS. To this end, we use the following dominance concepts, where Rk =stands for fx2Rk:xP0g. 215 Definition 3.3. Let us consider two matrices X,Y2Rknand a coalition S2N. 216 1. Ydominates Xthrough Saccording to i, and we will denote Ydom S iX, if: (a) YSiXS,YS6¼ XS, 218 (b) YS2VðSÞRk =. 219 2. Ydominates Xthrough Saccording to =, and we will denote Ydom S PX, if: (a) YS=XS,YS6¼ XS, 221 (b) YS2VðSÞRk =. 222 In scalar TU-games the set of non-dominated imputations has been widely considered (see Driessen, 1988 223 and the references therein). Nevertheless, in set-valued TU-games the concept which plays the important 224 role is the NDA set. These sets are defined by: 225 1. NDAðN;V;iÞ¼fX2IðN;VÞsuch that 9=SN,Y2IðN;VÞ,Ydom S iXg, 226 2. NDAðN;V;=Þ¼fX2IðN;VÞsuch that 9=S2N,Y2IðN;VÞ,Ydom S PXg. S{1} {2} {3} {1,2} {1,3} {2,3} N A(1/2,1) (5/2,3/2) (1,2) (5,4) B(1,1/2) (2,2) (2,1) (6,3) C(4/5,3/4) (3,1) (3/2,4/3) (3,6) F.R. Fern aandez et al. / European Journal of Operational Research xxx (2003) xxx–xxx 7 EOR 5770 No. of Pages 15, DTD = 4.3.1 7 July 2003 Disk used ARTICLE IN PRESS UNCORRECTED PROOF 227 Our following result proves that both core sets are sets of non-dominated allocations. 228 Theorem 3.1. The core sets hold the following properties: 229 1. CðN;V;=Þ¼NDAðN;V;iÞ, 230 2. CðN;V;iÞ¼NDAðN;V;=Þ. 231 Proof. We only prove 1. the proof of 2. being similar. 232 1. )Suppose that X2CðN;V;=Þand that X62 NDAðN;V;iÞ. Then there exists SNand 233 Y2IðN;VÞ, such that Ydom S iX, that is, YSiXS,YS6¼ XSand YS2VðSÞRk =, but it is not possible 234 because XS=VðSÞ. 235 (Suppose that X2NDAðN;V;iÞand that X62 CðN;V;=Þ. Then, there exists SNand vS2VðSÞ, 236 such that XSis not better componentwise then vS, that is, vSiXS. Now let us construct an allocation, Y,of 237 vSas follows: Yi¼ vS jSj8i2S; 08i62 S:  239 Allocation Yof vS2VðSÞdominates allocation Xthrough coalition Saccording to ibecause YS¼vSiXS 240 and YS2VðSÞ. Hence, it contradicts that X2NDAðN;V;iÞ. 241 4. Existence theorems 242 Once, we have defined the two core concepts and their relationships it is important to give conditions 243 that ensure non-emptiness of these cores. 244 4.1. Dominance core 245 For each scalarized vector k2K, K¼k2Rk;kj (>0;j¼1;...;ksuch that X k j¼1 kj¼1) 247 and any game ðN;VÞ2GV, we define the scalar game ðN;vkÞ2gvas: vkð;Þ ¼ 0;vkðSÞ¼ max vS2VðSÞRk = ktvS;8SN;S6¼;:ð1Þ 249 Using the game defined in (1) we establish a sufficient condition for the non-emptiness of the dominance 250 core. 251 Theorem 4.1. The core CðN;V;iÞof the game ðN;VÞ2GVis non-empty if there exists ^ kk 2Ksuch that the 252 scalar game ðN;v^ kkÞ2gvis balanced and it satisfies v^ kkðNÞ 6¼ 0. 253 Proof. Let it ^ kk be a weight in Ksuch that the scalar game ðN;v^ kkÞ2gv, defined in (1), is balanced and verify 254 v^ kkðNÞ 6¼ 0. Consider zS2ðVðSÞRk =Þsuch that ^ kktzS¼v^ kkðSÞ8SN. Notice that zS2VðSÞ, otherwise it is 255 possible to find another vector vS2VðSÞsuch that zS5vS,zS6¼ vS, and then ^ kktzS<^ kktvS. By Bondareva and 256 Shapley theorem (see Bondareva, 1963) there exists an allocation x2CðN;v^ kkÞ. 8F.R. Fern aandez et al. / European Journal of Operational Research xxx (2003) xxx–xxx EOR 5770 No. of Pages 15, DTD = 4.3.1 7 July 2003 Disk used ARTICLE IN PRESS UNCORRECTED PROOF 257 Now consider the matrix X2Rknwhose columns are: Xi¼xi v^ kkðNÞzN8i2N: 259 Since v^ kkðNÞ 6¼ 0, we prove that X2CðN;V;iÞ. Indeed, XN¼X n i¼1 xi v^ kkðNÞzN¼zN 261 and then X2IðN;VÞ. Assume that X62 CðN;V;iÞ. Then, there exists a coalition SNand a vector 262 wS2VðSÞsuch that XS6wS,XS6¼ wS, that is, ^ kktXS<^ kktwS. Then: max vS2VðSÞRk = ^ kktvSP^ kktwS>^ kktXS¼X i2S ^ kktXi¼Pi2Sxi v^ kkðNÞ^ kktzN¼xSPv^ kkðSÞ¼ max vS2VðSÞRk = ^ kktvS: 264 This is a contradiction.  265 This results is useful in finding elements in the dominance core of different set-valued games. 266 4.1.1. Multiobjective linear programming games 267 The set-valued characteristic function is usually defined through the set of non-dominated values of a 268 multiobjective programming problem. A particular case of these games are the Multiobjective Linear 269 Production Games. These games are characterized because the objective functions of the multiobjective 270 program are linear. In this situation we can obtain an allocation of the dominance core for any 271 z¼Cx 2VðNÞ. Indeed, given z¼Cx2VðNÞ, it is well-known that there exists a vector of weights ^ kk 2Rk, 272 ^ kk >0, such that xis the solution of the scalar problem: ½PNð^ kkÞ max ^ kktCx s:t::x2FðPNÞ: 274 Let ube an optimal solution of the dual problem of ½PNð^ kkÞ. The matrix X¼ðX1;X2;...;XnÞwhose 275 columns are Xi¼ðubi=^ kktzÞzbelongs to the dominance core. This follows from Theorem 4.1. Notice that 276 Xis an allocation of z. 277 We note in passing that the choice of z2VðNÞcan be done taking a weighting vector k>0. Procedures 278 guiding the agents to the choice of weighting vectors are described in Marmol et al. (2002) and the ref- 279 erences therein. 280 Example 2.1 (continued).Let us take ^ kk ¼ð0:8;0:2Þ. The problem PNð^ kkÞis: max 1:9x1þ3:4x2 s:a::x1þ8x2643;7x1þ4x2641;7x1þ8x2657;x1;x2P0: 282 An optimal solution of PNð^ kkÞis x1¼ð2:^ 33;5:08^ 33Þwith objective value z1¼ð25;8:58^ 33Þ. An the optimal so- 283 lution of the dual of PNð^ kkÞis u¼ð0:179167;0;0:245833Þ. The allocation in the dominance core obtained by 284 the above method, is: X¼6:567 9:938 8:495 2:254 3:412 2:917  : 286 4.1.2. Multiobjective continuous single facility location games 287 For this class of games we can provide a method to construct allocations in the dominance core. The 288 approach consists of applying Theorem 4.1 transforming the multiobjective game into a scalar continuous F.R. Fern aandez et al. / European Journal of Operational Research xxx (2003) xxx–xxx 9 EOR 5770 No. of Pages 15, DTD = 4.3.1 7 July 2003 Disk used ARTICLE IN PRESS