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Core, least core and nucleolus for multiple scenario cooperative games
European Journal of Operational Research 164 (2005) 225–238 www.elsevier.com/locate/dsw
Decision Aiding
Core, least core and nucleolus for multiple scenario cooperative games q M.A. Hinojosa a b
a,1
, A.M. M armol
b,*,1
, L.C. Thomas
c
Departamento de Economıa y Empresa, Universidad Pablo de Olavide, Ctra. de Utrera Km. 1, 41012 Sevilla, Spain Departamento de Economıa Aplicada III, Universidad de Sevilla, Avda. Ramon y Cajal n°. 1, 41018 Sevilla, Spain c School of Management, University of Southampton, Highfield, Southampton 5017 1BJ, UK Received 25 September 2002; accepted 3 September 2003 Available online 5 March 2004
Abstract Multiple scenario cooperative games model situations where the worth of the coalitions is valued in different scenarios simultaneously or under different states of nature. In this paper we analyze solution concepts for this class of games keeping the multidimensional nature of the characteristic function. We obtain extensions of the notions of core, least core and nucleolus, and explore the relationship among these solution concepts. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Game theory; Cooperative games; Multiple scenario; Lexicographical solutions
1. Introduction Multiple scenario cooperative games is a particular class of cooperative vector valued transferable utility (TU) games, where a single resource is allocated, but the effectiveness of the allocations for each coalition is measured by a vector instead of by a scalar. These games arise in managerial and public decision making when a bundle of goods has to be allocated among a set of individuals, taking into account the worth or values of the coalitions under different scenarios. The different scenarios can be interpreted as the different states of the world in presence of uncertainty. When the agents cannot assign (objective or subjective) probabilities to the occurrence of different states of the world, standard expected utility arguments cannot be applied. In this situations the values of the coalitions are represented by a vector containing all the possible results.
q This research was conducted while A.M. Marmol and M.A. Hinojosa were visiting the School of Management, University of Southampton (UK). * Corresponding author. Tel.: +34-95-455-7554; fax: +34-95-455-1667. E-mail address: [email protected] (A.M. Marmol). 1 The research of these authors is partially supported by the Spanish Ministry of Science and Technology projects BFM2002-11282E and BEC2003-03111.
0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.09.028
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Some classes of vector valued cooperative games have been addressed in J€ ornsten et al. (1995) and Fern andez et al. (2002b). Voorneveld and Van den Nouweland (1998) introduced a general model of cooperative multicriteria game with public and private criteria which includes the so-called multicommodity games as a particular case. Related work on multicommodity games and its applications can be seen in Derks and Tijs (1986), Nouweland et al. (1989), Nishizaki and Sakawa (2001), and Fernandez et al. (2002a, in press). Multiple scenario cooperative games differ from conventional cooperative games in the dimension of the coalition values, and they also differ from cooperative multicriteria games in that a single good instead of a set of different commodities has to be allocated among the players. Formally, in multiple scenario cooperative games an allocation is represented by a vector whereas in multicriteria games it is represented by a matrix. One example would be the allocation of the total cost of construction of the distribution system of a certain good (electricity, water, etc.) from a common supplier to different costumers (players). The allocation must be fair, in the sense that each group of costumers (coalitions) will not have to pay more than the cost of constructing their own distribution system. In a multiple scenario framework, if there is uncertainty about the costs associated to the coalitions, the aim becomes to identify allocations of costs that are satisfactory under all scenarios. A second application is where the players are specialists in investment funds with expertise in investing in different areas. They can decide to form coalitions to pool expertise investing the coalition total, and the allocation of the funds to the players has to be done taking into account the values of the coalitions under future economic scenarios. The class of multiple scenario cooperative games has been considered recently in Nishizaki and Sakawa (2001), that have presented extensions of the notions of least core and nucleolus. Their approach consists of measuring the effectiveness of an allocation by a scalar, that can be either an aggregation by weighting coefficients of the vector valued excess function, or an aggregation by minimal components of the excess function. They also consider the solution obtained when fixing m 1 levels and trying to maximize the other one. The solutions obtained by these procedures strongly depend on the method chosen to scalarize the problem. The objective of this paper is to propose and analyze different solution concepts for the class of multiple scenario cooperative games, and to provide procedures to obtain the solutions. We concentrate on core concepts. The idea underlying core allocations is to provide incentives for cooperation excluding those unstable outcomes in which some coalitions are able to reach better results for all their members. In the case of a multiple scenario cooperative game the vector valued characteristic function induces different core concepts depending on the preference structure that the players are willing to accept. However, as in conventional games, these core concepts do not provide a single solution for the multiple scenario game. In fact, there can be too many outcomes in these solution sets but also they can be empty for some games. Therefore, in this paper we also explore other solution concepts based in a criterion of equity: the least-core and the nucleolus. We keep the multidimensional nature of the characteristic function to obtain extensions of the notions of e-core, least core, and nucleolus. It is important to point out that these solutions concepts can be characterized as the solutions of multicriteria linear problems and therefore computed with the available specific software. In Section 2, we present the model and the notation used in the paper. Solutions related to the classical concept of core are addressed in Section 3. In Section 4 we extend the concept of least core to the case of multiple scenario games and the allocations in the least core are characterized as the solutions of a multicriteria linear problem. Section 5 presents the generalized nucleolus and describe a procedure to construct it. In Section 6 an example is discussed, and Section 7 is devoted to the conclusions.
2. Multiple scenario cooperative games A multiple scenario cooperative game with transferable utility is a triplet ðN ; v; EÞ where N ¼ f1; 2; . . . ; ng is the set of players (a nonempty subset S N of the player set is called a coalition), E is
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the total amount of the issue that has to be allocated and v is a vector valued function, the characteristic function of the game, which associates any coalition S N with vðSÞ 2 Rm , that is the worth of coalition S. The components of vðSÞ, vj ðSÞ, j ¼ 1; . . . ; m, represent the worth of S in the m scenarios, that is, vj ðSÞ can be interpreted as the amount of issue that coalition S could obtain by itself under scenario j. If the multiple scenario game is played an interesting question which arises is how the total amount of issue, E, should be allocated among the various players taking into account the worth of the coalitions under all scenarios. As in conventional games, an allocation is a payoff vector, X 2 Rn , whose components P represent the payoff for each player, X is the payoff for the ith player. Therefore X P 0, 8i 2 N , and i i i2N Xi ¼ E holds. P The sum X ðSÞ ¼ i2S Xi is the overall payoff obtained by coalition S, thus X ðN Þ ¼ E. We will denote the set of allocations of the game by I ðEÞ. A solution concept for cooperative games associates a set (possibly empty) of allocations with each game. We will be interested in valuing the differences between what the coalitions obtain with a certain allocation, and their values in the game. For each allocation and each coalition, we define the vector valued excess function eðX ; SÞ whose components are ej ðX ; SÞ ¼ vj ðSÞ X ðSÞ, j ¼ 1; . . . ; m. This function measures the excesses of coalition S under allocation X in every scenario and will play a central role in the definition of the solutions concepts for these games. In what follows, we will denote by aj , j ¼ 1; . . . ; m, the components of vector a 2 Rm , and for a; b 2 Rm , we will use the following notation: a 5 b if aj 6 bj , 8j ¼ 1; . . . ; m; a 6 b if a 5 b and a 6¼ b; ab (a£b) if a 5 b (a 6 b) is not true.
3. Core solutions When extending the core concept of the conventional Game Theory to vector valued games, different concepts are obtained depending on the ordering considered in the payoff space. In this section we define some of these solution concepts for multiple scenario cooperative games and explore the relationships among them. Firstly we consider allocations in which no coalition has any incentive to deviate irrespective of the scenario and define the concept of preference core. Definition 3.1. The preference core of the multiple PCðN ; v; EÞ ¼ fX 2 I ðEÞ j eðX ; SÞ 5 0; 8S N ; S 6¼ ;g.
scenario game ðN ; v; EÞ is defined as
Notice that the preference core is the intersection of the cores of each scalar scenario game, ðN ; vj Þ where v ðN Þ ¼ E, 8j ¼ 1; . . . ; m, induced by the multiple scenario game. The next result characterizes the preference core of the multiple scenario game as the core of the scalar game obtained by valuing each coalition at the most optimistic scenario. Consider the scalar game ðN ; vmax Þ, whose characteristic function is defined as vmax ðSÞ ¼ Maxj¼1;...;m vj ðSÞ, 8S N , S 6¼ ;. The core of this game is CðN ; vmax Þ ¼ fX 2 I ðEÞ j vmax ðSÞ X ðSÞ 6 0; 8S N ; S 6¼ ;g. It is straightforward to prove that the preference core of the multiple scenario cooperative game coincides with the core of the game ðN ; vmax Þ. j
Proposition 3.1. PCðN ; v; EÞ ¼ CðN ; vmax Þ. As a consequence the preference core of the game is not empty if and only if the game ðN ; vmax Þ is balanced. Next we define a weaker core concept for multiple scenario cooperative games, the dominance core. To be in the dominance core it suffices that the allocation gives a nonpositive excess for each coalition, that is, for each coalition there is at least one scenario in which it has no reason to deviate.
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Definition 3.2. The dominance core of the multiple scenario game ðN ; v; EÞ is defined as DCðN ; v; EÞ ¼ fX 2 I ðEÞ j 8S N ; S 6¼ ;; eðX ; SÞ 0g. The dominance core is contained in the core of the scalar game obtained when the worth of each coalition is valued at the most pessimistic scenario, that is DCðN ; v; EÞ CðN ; vmin Þ, where the scalar game ðN ; vmin Þ is defined by vmin ðSÞ ¼ Minj¼1;...;m vj ðSÞ, 8S N , S 6¼ ;. It is worth noting that the inclusion relation DCðN ; v; EÞ CðN ; vmin Þ can be strict. To analyze the difference between these two sets of allocations, let us denote by T the set of coalitions whose value in the multiple scenario game is not the same in every scenario, i.e., T ¼ fS N ; vj ðSÞ > vmin ðSÞ; for some j ¼ 1; . . . ; mg. The dominance core of the multiple scenario game, and the core of the game ðN ; vmin Þ differ in those allocations in CðN ; vmin Þ that give to a coalition, S 2 T , exactly its value in the worst scenario. Formally the result is: Proposition 3.2 CðN ; vmin Þ ¼ DCðN ; v; EÞ [ fX 2 CðN ; vmin Þ j 9S 2 T ; X ðSÞ ¼ vmin ðSÞg:
Proof. It is easy to see that DCðN ; v; EÞ is included in CðN ; vmin Þ. In order to prove the other inclusion, suppose that X 2 CðN ; vmin Þ and X 62 DCðN ; v; EÞ, then 9S N , vðSÞ X ðSÞ P 0. This implies that vmin ðSÞ X ðSÞ P 0 and also that vj ðSÞ > vmin ðSÞ, for some j ¼ 1; . . . ; m, thus S 2 T . But, as X 2 CðN ; vmin Þ, vmin ðSÞ X ðSÞ 6 0, and it follows that vmin ðSÞ ¼ X ðSÞ for some S 2 T . h The dominance core is usually too wide as a set of solutions, and often the preference core is a too restrictive set or even empty. In practice, it may be interesting not only to look for a nonpositive excess, but to establish certain bounds on the excess function in order to obtain solutions that achieve a better performance. This is the idea underlying the next solution concept, that the extension of the notion of e-core. Let p 2 Rm , p ¼ ðp1 ; . . . ; pm Þ, be a vector of admissible excess levels. Definition 3.3. The preference p-core for the multiple scenario game ðN ; v; EÞ, PpCðN ; v; EÞ, is the set of allocations PpCðN ; v; EÞ ¼ fX 2 I ðEÞ j eðX ; SÞ 5 p; 8S N ; S 6¼ ;g:
Definition 3.4 The dominance p-core for the multiple scenario game ðN ; v; EÞ, DpCðN ; v; EÞ, is the set of allocations DpCðN ; v; EÞ ¼ fX 2 I ðEÞ j eðX ; SÞ p; 8S N ; S 6¼ ;g:
For p ¼ 0, PpCðN ; v; EÞ ¼ PCðN ; v; EÞ and DpCðN ; v; EÞ ¼ DCðN ; v; EÞ. For p P 0, the concept of preference p-core is a relaxation of the notion of preference core, in that it allows the excess function to be positive if maintained below the levels established for each criteria, and the dominance p-core is also a relaxation of the notion of dominance core. On the other hand, for p 6 0 the concept of preference p-core becomes more restrictive than the preference core, requiring that the maximum excesses attained are below zero, and the concept of dominance p-core is more restrictive than the dominance core.
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The following relationships between these concepts are easy to prove. Proposition 3.3. If p; q 2 Rm , p 5 q, then the following inclusion relations hold: PpCðN ; v; EÞ PqCðN ; v; EÞ; DpCðN ; v; EÞ DqCðN ; v; EÞ:
When the admissible level vectors are incomparable the inclusions are established in the following result. Proposition 3.4. If p; q 2 Rm , p q and q p then PpCðN ; v; EÞ DqCðN ; v; EÞ and PqCðN ; v; EÞ DpCðN ; v; EÞ. Proof. If p q and q p, there exists j ¼ 1; . . . ; m such that pj < qj . Let X 2 PpCðN ; v; EÞ, eðX ; SÞ 5 p, 8S N , and therefore eðX ; SÞ q, 8S N . It follows that X 2 DqCðN ; v; EÞ. Analogously the other inclusion holds. h Decreasing the admissible excess levels, p, yields to allocations that are more satisfactory for all the coalitions in the preference p-core, PpCðN ; v; EÞ, but the set can be empty. The objective now is to find feasible allocations that reach the smallest maximal excess level as possible. This lead us to define the generalized least core as a solution concept for multiple scenario cooperative games.
4. The generalized least core Definition 4.1. The maximum excess vector for allocation X 2 I ðEÞ, is the vector pðX Þ 2 Rm whose components are pj ðX Þ ¼ Max ej ðX ; SÞ, j ¼ 1; . . . ; m. SN
The maximum excess vector is a vector valued measure of the worst performance of an allocation in the set of all the coalitions. Indeed, the allocation is valued by the worst value attained in each scenario. Notice that the values pðX Þ 2 Rm may not be attained simultaneously, because different components can be obtained from different coalitions. Nevertheless, the vector is an upper bound for the values of the excess function. In order to formalize the idea of minimizing the worst performance we consider those allocations that minimize the maximum excess vector in the multicriteria sense, arriving to the natural extension of the concept of least core. Definition 4.2. p 2 Rm is a nondominated maximum excess vector for the multiple scenario game ðN ; v; EÞ, if 9X 2 I ðEÞ such that p ¼ pðX Þ, and 9 = Y 2 I ðEÞ, such that pðY Þ 6 p. We will denote the set of nondominated maximum excess vectors by P1 . Definition 4.3. The generalized least core of the multiple scenario game ðN ; v; EÞ, GLCðN ; v; EÞ, is the set of allocations, X 2 I ðEÞ, whose maximum excess vectors are nondominated. Allocations in the generalized least core are those whose vector of maximum excess is Pareto optimal or nondominated, in the sense that no other allocation has a better maximum excess vector.
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The following result establishes that given a certain nondominated maximum excess vector, the set of allocations where this level is attained coincides with the corresponding preference p-core. Proposition 4.1. If p 2 Rm is a nondominated maximum excess vector for the multiple scenario game ðN ; v; EÞ, then fX 2 I ðEÞ j pðX Þ ¼ pg ¼ PpCðN ; v; EÞ:
Proof. Recall that PpCðN ; v; EÞ ¼ fX 2 I ðEÞ j eðX ; SÞ 5 p; 8S N ; S 6¼ ;g. It is easy to see that if X 2 fX 2 I ðEÞ j pðX Þ ¼ pg, then X 2 PpCðN ; v; EÞ. On the other hand, let p 2 P1 , and X 2 PpCðN ; v; EÞ. Since eðX ; SÞ 5 p, 8S N , MaxSN ej ðX ; SÞ 6 pj , 8j ¼ 1; . . . ; m, and pðX Þ 5 p. If pðX Þ ¼ q 6 p, then p 62 P1 . It follows that pðX Þ ¼ p. h As a consequence of this result, the generalized least core can be represented as the union of the preferences p-cores associated to nondominated maximum excess vectors, that is, [ GLCðN ; v; EÞ ¼ PpCðN ; v; EÞ: p2P1
In order to characterize the allocations in the generalized least core, consider the following multicriteria linear programming problem associated to the game ðN ; v; EÞ, that we will denote by (GMLP)1 Min
ðp1 ; p2 ; . . . ; pm Þ
s:t::
vj ðSÞ X ðSÞ 6 pj ;
8S N ; 8j ¼ 1; . . . ; m;
X ðN Þ ¼ E; X P 0: Proposition 4.2. Let ðX ; pÞ be a nondominated solution of (GMLP)1 , then X 2 GLCðN ; v; EÞ and p is its maximum excess vector. Conversely, if X 2 GLCðN ; v; EÞ and p is its maximum excess vector, then ðX ; pÞ is a nondominated solution of (GMLP)1 . Proof. It follows from Definition 4.3 that the allocations in the generalized least core are the Pareto optimal or nondominated solutions of the following multicriteria nonlinear problem: Min
ðp1 ðX Þ; p2 ðX Þ; . . . ; pm ðX ÞÞ
s:t::
X ðN Þ ¼ E; X P 0;
which is equivalent to the linear multicriteria problem (GMLP)1 . h This result allows the calculation of the generalized least core by solving a linear multicriteria problem with m objectives and m ð2n 1Þ constraints. Although the dimensionality of the problem can be high, appropriate software, such as ADBASE (Steuer, 1995), is currently available to solve it. The whole set of nondominated solutions of (GMLP)1 , in general is not a convex set, and therefore the generalized least core is not necessarily convex, as can be seen in the example presented in Section 6. Nevertheless, for every p 2 P1 , the corresponding preference p-core, PpCðN ; v; EÞ, is a polyhedral set whose
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extreme points are obtained when solving the problem (GMLP)1 . Furthermore, as a consequence of the connectedness of the solutions of multicriteria linear problems, the generalized least core is a connected set. The generalized least core may include too many vectors to be useful in the practice of decision-making. In the next section we refine this solution concept by applying recursively the same idea of obtaining the allocations associated to vectors of maximum excess that are nondominated.
5. The generalized nucleolus In this section we establish the extension of the concept of nucleolus for multiple scenario games. The nucleolus was first proposed as a solution for scalar cooperative games in Schmeidler (1969), and its properties have been extensively explored thereafter. A good revision is in Maschler (1992). It has been considered as a crucial solution of the game in many applications specially in relation to cost allocations problems and bankruptcy situations (see, for instance, Littlechild and Thompson, 1977; Young, 1985; Aumann and Maschler, 1985; Gow and Thomas, 1998). The idea underlying this concept is the selection of the allocations in the least core for those who are better in a lexicographical sense. In the scalar case it is possible to define a lexicographical order among the allocations and therefore determine an unique best solution, the nucleolus. In the case of a vector valued measure of the allocations we deal with the binary relation that we will define next. For each allocation of the game, X , we construct an m ð2n 2Þ matrix, pðX Þ. The jth row of matrix pðX Þ, pj ðX Þ, j ¼ 1; . . . ; m, contains the values that the excess function attains in the jth scenario in every coalition under allocation X , arranged in order of decreasing magnitude. Matrix pðX Þ is a matrix-valued measure of the performance of allocation X with respect to all the coalitions and in every scenario. Notice that the first column of pðX Þ, p1 ðX Þ is the vector of maximum excess for allocation X , pðX Þ. The second column, p2 ðX Þ is the vector of second biggest values of the excess that can be attained with allocation X in every scenario. In general, pk ðX Þ represents the vector of values in the kth place. Definition 5.1. Allocation X is lexicographically better than allocation Y , in the multiple scenario game ðN ; v; EÞ, (X 6 lex Y ), if pk ðX Þ 6 pk ðY Þ for the first column, k, in which matrix pðX Þ and matrix pðY Þ are different. Notice that this binary relation does not define a complete order in the space of allocations, in fact, when for column k, pk ðX Þ£pk ðY Þ and pk ðY Þ£pk ðX Þ, the allocations are incomparable. In the particular case n where m ¼ 1, this relation coincides with the lexicographical order in R2 2 . Definition 5.2. Allocation X is lexicographically nondominated in the multiple scenario game ðN ; v; EÞ, if there is no other allocation Y such that Y 6 lex X . This dominance relation permits to make a selection among the allocations and to define the solution concept that extends the notion of nucleolus to multiple scenario games. Definition 5.3. The generalized nucleolus for the multiple scenario game ðN ; v; EÞ, GN ðN ; v; EÞ, is the set of allocations, X 2 I ðEÞ, that are lexicographically nondominated. The following result establishes that when for some level p 2 Rm the preference p-core, PpCðN ; v; EÞ, is nonempty, it includes some of the allocations in the generalized nucleolus.
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Proposition 5.1. If PpCðN ; v; EÞ 6¼ ;, then PpCðN ; v; EÞ \ GN ðN ; v; EÞ 6¼ ;. Proof. Let PpCðN ; v; EÞ be nonempty. If p 2 P1 , from Proposition 4.1, the preference p-core, PpCðN ; v; EÞ, coincides with the set of allocations whose maximum excess vector is exactly p. As p is nondominated, there exists at least a lexicographically nondominated allocation in PpCðN ; v; EÞ. If p 62 P1 , there exists q 2 P1 such that q 6 p. It follows from Proposition 3.3 that PqCðN ; v; EÞ PpCðN ; v; EÞ, and the lexicographically nondominated allocations in PqCðN ; v; EÞ are included in the nucleolus. h As an immediate consequence, we obtain the extension of the known result in scalar cooperative games. Corollary 5.1. If the preference core, PCðN ; v; EÞ is nonempty, the nucleolus has nonempty intersection with the preference core. It is worth noting that, in general, the generalized nucleolus is not included in the preference core. Nevertheless, as a consequence of Proposition 3.3, the following result holds. Proposition 5.2. If q 6 p, 8q 2 P1 , then GN ðN ; v; EÞ PpCðN ; v; EÞ. To construct the generalized nucleolus, once the generalized least core is obtained, look at the nondominated levels p 2 P1 , and if PpCðN ; v; EÞ is a singleton, it is contained in the generalized nucleolus. In other case the lex-nondominated allocations in PpCðN ; v; EÞ have to be obtained. Each one of the sets PpCðN ; v; EÞ is a polyhedral set. Let its extreme points be X 1 ; . . . ; X rðpÞ . We will prove that, when p 2 P1 , for every scenario j ¼ 1; . . . ; m, there is at least a coalition that attains the maximum excess level given by vector p for each X 2 PpCðN ; v; EÞ. Proposition 5.3. If p 2 P1 , then 8j ¼ 1; . . . ; m, fS N j S 6¼ ;; ej ðX ; SÞ ¼ pj ; 8X 2 PpCðN ; v; EÞg 6¼ ;:
Proof. Consider p 2 P1 . For j ¼ 1; . . . ; m, let sðjÞ be defined as follows: sðjÞ ¼ fS N j vj ðSÞ X k ðSÞ ¼ pj ; 8k ¼ 1; . . . ; rðpÞg: We are going to prove that sðjÞ 6¼ ;, 8j ¼ 1; 2; . . . ; m. Assume that sðjÞ ¼ ;. Then, for any coalition S, there exists an extreme point X l , such that vj ðSÞ X l ðSÞ < pj . Define the following allocation: Y ¼
rðpÞ 1 X X k 2 PpCðN ; v; EÞ: rðpÞ k¼1
Now, for any coalition, S N , and for any scenario j, vj ðSÞ Y j ðSÞ ¼
rðpÞ rðpÞ rðpÞ 1 X j 1 X 1 X j v ðSÞ X k ðSÞ ¼ ðv ðSÞ X k ðSÞÞ < pj ; rðpÞ k¼1 rðpÞ k¼1 rðpÞ k¼1
and this contradicts the fact that p is a nondominated vector. Notice that if a coalition attains a level in the extreme points of PpCðN ; v; EÞ, it also attains this level in all the allocations of PpCðN ; v; EÞ. h
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It follows from the result proved in Proposition 5.3 that when the set PpCðN ; v; EÞ is obtained from the (GMLP)1 , at least m constraints are active in any of the allocations X 2 PpCðN ; v; EÞ, what makes it possible to find the allocations in PpCðN ; v; EÞ with nondominated maximum second levels. In order to do this, for each j ¼ 1; . . . ; m, we select a coalition T ðjÞ 2 sðjÞ ¼ fS N j vj ðSÞ X k ðSÞ ¼ pj ; 8k ¼ 1; . . . ; rðpÞg, and consider the following multicriteria problem, (GMLP)2 , where we replace the corresponding m inequality constraints with the equality consisting of the level pj that the coalition attains in scenario j: Min
ðq1 ; q2 ; . . . ; qm Þ
s:t::
vj ðT ðjÞÞ X ðT ðjÞÞ ¼ pj ; j
j
v ðSÞ X ðSÞ 6 q ;
8j ¼ 1; . . . ; m;
8j ¼ 1; . . . ; m; 8S N ; S 6¼ T ðjÞ;
X ðN Þ ¼ E; X 2 PpCðN ; v; EÞ: The following proposition characterizes those allocations in the generalized least core whose vectors of second maximum levels are nondominated. Proposition 5.4. Let p 2 P1 . If ðX ; qÞ is a nondominated solution of (GMLP)2 , then p ¼ p1 ðX Þ and q ¼ p2 ðX Þ. Conversely, if X 2 PpCðN ; v; EÞ and p2 ðX Þ is such that, 9 = Y 2 PpCðN ; v; EÞ, p2 ðY Þ 6 p2 ðX Þ, then ðX ; qÞ, where q ¼ p2 ðX Þ, is a nondominated solution of problem (GMLP)2 . We will denote by P2 the set of nondominated vectors, q 2 Rm , obtained when solving (GMLP)2 . If there is a unique allocation associated to q 2 P2 , the allocation is in the generalized nucleolus of the game, in other case, the same procedure can be applied to obtain the allocations with nondominated p3 ðX Þ, until the solution is unique. To solve problem (GMLP)2 , notice that the nondominated extreme solutions of (GMLP)1 associated to p provide the set of extreme points of PpCðN ; v; EÞ, therefore this set can be represented as ( ) X
PpCðN ; v; EÞ ¼ X 2 I ðEÞ j X ¼ Lp k; ki ¼ 1; k P 0 ; i¼1;...;rðpÞ
where Lp is a matrix whose rðpÞ columns are the extreme points of PpCðN ; v; EÞ. If we substitute X ¼ Lp k the problem (GMLP)2 is transformed to Min
ðq1 ; q2 ; . . . ; qm Þ
s:t::
vj ðT ðjÞÞ MðT ðjÞÞk ¼ pj ; j
j
v ðSÞ MðSÞk 6 q ;
8j ¼ 1; . . . ; m
8j ¼ 1; . . . ; m; 8S N ; S 6¼ T ðjÞ;
t
e k ¼ 1; k P 0; where M is a ð2n 2Þ rðpÞ matrix whose columns represent the overall payoff obtained by the ð2n 2Þ coalitions with the extreme points of PpCðN ; v; EÞ. MðSÞ stands for the row in M associated to coalition S. Notice that although we have added the condition X 2 PpCðN ; v; EÞ to the original problem (GMLP)1 , the number of constraints has not increased, and each of the constraints still corresponds to a certain coalition and a certain scenario. The recursive procedure to obtain allocations in the generalized nucleolus in a multiple scenario cooperative game can be described as follows.
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In a first step we obtain a nondominated maximum excess vector, p, by solving problem (GMLP)1 . If the preference p-core is a singleton, PpCðN ; v; EÞ ¼ fX g, then allocation X is lexicographically nondominated, and therefore it is in the generalized nucleolus. In other case, the extreme points of PpCðN ; v; EÞ are provided as part of the solution of (GMLP)1 . In a second step, by solving (GMLP)2 we obtain the allocations in PpCðN ; v; EÞ whose second maximum excess levels are nondominated. In general, in step k, a vector of nondominated kth maximum excess level is obtained. The procedure finishes when in a certain step the set of allocations associated to the corresponding nondominated maximum excess level has a unique element. In any step a transformation of the type X ¼ Lp k can be performed once the extreme points of the corresponding set are known, and at least m inequality constraints become active. Since n independent equality constraints determine uniquely the solution, the procedure will finish in at most n steps. The generalized nucleolus is a subset of the generalized least core from where it may be possible to select a suitable allocation. The selection of this single allocation can be done once the whole set is obtained, or it can also be done by choosing one of the nondominated maximum excess vectors in each step. The characterization of the successive maximum excess vectors as the solutions of multicriteria linear programming problems, established in Propositions 4.2 and 5.4, suggests the possibility of incorporating additional information about the probability of occurrence of the different scenarios to the model. A methodology to incorporate information about the importance of the criteria in multicriteria linear problems based in the extreme points of the information sets is proposed in Marmol et al. (1998). In M armol et al. (2002) a procedure to sequentially incorporate preference information to the decision problem is provided. It is possible to apply this methodology to the process of obtaining allocations in the generalized nucleolus of the multiple scenario cooperative game in order to direct the search for a final consensus allocation. Alternatively, other decision criteria (such as min–max, Laplace criterion, etc.) can be considered to obtain nondominated solutions of the problems in each step.
6. Example The following example illustrates the procedure to obtain the preference and the dominance cores, the generalized least core and the generalized nucleolus of a multiple scenario game. The example is taken from Nishizaki and Sakawa (2001). Consider a three person cooperative game with three scenarios. For all the scenarios, the values of oneperson coalitions are zero, and the value of the grand coalition is one, because the game is zero–one normalized. The values of the two person coalitions, are shown in the following table. S
Scenario I
Scenario II
Scenario III
f1; 2g f1; 3g f2; 3g
0.3 0.5 0.4
0.3 0.3 0.7
0.1 0.5 0.7
The preference core and the dominance cores of this multiple scenario game have been obtained as shown in Propositions 3.1 and 3.2, respectively, PCðN ; v; EÞ ¼ fX 2 R3 j X P 0; X1 þ X2 þ X3 ¼ 1; X1 þ X2 P 0:3; X1 þ X3 P 0:5; X2 þ X3 P 0:7g; DCðN ; v; EÞ ¼ fX 2 R3 j X P 0; X1 þ X2 þ X3 ¼ 1; X1 þ X2 > 0:1; X1 þ X3 > 0:3; X2 þ X3 > 0:4g: Notice that the dominance core is a subset of the core of the pessimistic game, that does not include all the allocations in the frontier, as is established in Proposition 3.2. In Fig. 1 we represent the preference core and the dominance core of the game on the simplex fX 2 R3 j X P 0; X1 þ X2 þ X3 ¼ 1g.
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(0, 1, 0)
PC
DC
(0, 0, 1)
(1, 0, 0)
Fig. 1. Preference core and dominance core.
To construct the generalized least core for this game, using ADBASE to solve the associated (GMLP)1 , we obtain four extreme nondominated solutions: ðX1 ; p1 Þ ¼ ð0:15; 0:35; 0:50; 0:15; 0:15; 0:15Þ; ðX2 ; p2 Þ ¼ ð0:15; 0:30; 0:55; 0:15; 0:15; 0:15Þ; ðX3 ; p3 Þ ¼ ð0:20; 0:30; 0:50; 0:20; 0:10; 0:10Þ; ðX4 ; p4 Þ ¼ ð0:30; 0:25; 0:45; 0:25; 0; 0Þ: p1 , p2 and p3 determine a nondominated facet, and P the segment P joining p3 and p4 is also a nondominated edge. If we denote by F1 ¼ fX 2 R3 j X P 0; X ¼ 3i¼1 ki Xi ; 3i¼1 ki ¼ 1; k P 0g and by F2 ¼ fX 2 R3 j X P 0; X ¼ k1 X3 þ k2 X4 ; k P 0; k1 þ k2 ¼ 1g, the generalized least core of the game is the union of these sets of allocations: GLCðN ; v; EÞ ¼ F1 [ F2 : Fig. 2 is graphical representation of the generalized least core. We denote by P1i , i ¼ 1; 2, the setPof vectors Pof maximum excess associated with3 the allocations in Fi , i ¼ 1; 2, that is, P11 ¼ fp 2 R3 j p ¼ 3i¼1 ki pi ; 3i¼1 ki ¼ 1; k P 0g and P12 ¼ fp 2 S R j p ¼ k1 p3 þ k2 p4 ; k P 0; k1 þ k ¼ 1g. Then the sets F and F can be viewed as F ¼ and 1 2 1 p2P11 PpCðN ; v; EÞ S 2 F2 ¼ p2P1 PpCðN ; v; EÞ. 2
X1 α
F1
X2
α'
X3
F2 X4
Fig. 2. Generalized least core.
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In order to obtain the generalized nucleolus, we analyze the sets F1 and F2 . There is a unique allocation associated to the levels p 2 P12 , therefore for each p 2 P12 , the set of allocations PpCðN ; v; EÞ is a singleton and is contained in the generalized nucleolus. The extremes ðX1 ; p1 Þ and ðX2 ; p2 Þ have the same maximum excess vector. In fact all the nondominated levels p 2 P11 , p 6¼ p3 , are not attained in an unique allocation. To see this, notice that as p1 ¼ p2 , the set P11 can be represented as P11 ¼ fp 2 R3 j p ¼ ap1 þ ð1 aÞp3 ; 0 < a 6 1g, that is, each p 2 P11 has associated an unique a, 0 < a 6 1, and the set of allocations where pa is attained can be represented as Ppa CðN ; v; EÞ ¼ fX 2 R3 j X ¼ k½aX1 þ ð1 aÞX3 þ ð1 kÞ½aX2 þ ð1 aÞX3 ; 0 6 k 6 1g, in words, Ppa CðN ; v; EÞ is the segment joining the allocations that attain level pa , in the edges X1 X3 and X2 X3 (see Fig. 2). Next step is to find the allocations whose second maximum excess level is minimal in each of the sets Ppa CðN ; v; EÞ. To illustrate the procedure we show how it is done in Pð0:15;0:15;0:15Þ CðN ; v; EÞ. The extreme points of this set are allocations X1 and X2 . To select the allocations of Pð0:15;0:15;0:15Þ C that are in the generalized nucleolus, as vj ðf1gÞ Xi ðf1gÞ ¼ 0:15, j ¼ 1; 2; 3, i ¼ 1; 2, we choose coalition T ðjÞ ¼ f1g, 8j ¼ 1; 2; 3, and solve problem (GMLP)2 , where we replace with the equality constraints the three constraints associated with coalition f1g: Min
ðq1 ; q2 ; . . . ; qm Þ
s:t::
vj ðSÞ X ðSÞ 6 qj ; j ¼ 1; . . . ; m; S N ; S 6¼ f1g; vj ðf1gÞ X ðf1gÞ ¼ 0:15; j ¼ 1; . . . ; m; X ðN Þ ¼ E; X 2 Pð0:15;0:15;0:15Þ C:
Problem (GMLP)2 has a unique optimal solution, X ¼ ð0:15; 0:325; 0:525Þ and q ¼ ð0:175; 0:15; 0:15Þ. Therefore this allocation is in the generalized nucleolus with levels p1 ðX Þ ¼ ð0:15; 0:15; 0:15Þ and p2 ðX Þ ¼ ð0:175; 0:15; 0:15Þ. For the other nondominated levels pa , 0 < a 6 1, solving the corresponding (GMLP)2 , we obtain unique allocations Xa ¼ ð0:2 0:05a; 0:3 þ 0:025a; 0:5 þ 0:025aÞ with vectors of levels qa ¼ ð0:2 0:025a; 0:1 þ 0:05a; 0:1 þ 0:05aÞ.
(0 , 1, 0)
X1 GLC X2
GN X3
DC X4
PC
(0 , 0, 1)
(1 , 0, 0)
Fig. 3. Preference core, dominance core, generalized least core and generalized nucleolus.
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The set of allocations obtained, together with the allocations of F2 constitute the generalized nucleolus of the game. It is worth noting that these allocation are the mid-points of the segments Ppa CðN ; v; EÞ and they are on the line determined by the segment F2 . Therefore the generalized nucleolus is the segment joining allocations X4 and X5 (see Fig. 3). Fig. 3 is a graphical representation of the solution sets of the game on the simplex fX 2 R3 j X P 0; X1 þ X2 þ X3 ¼ 1g. Notice that, in this game, allocations in the nucleolus and in the least core are contained in the preference core, as a consequence of Proposition 5.2.
7. Conclusions The recent development of multicriteria analysis techniques allows us to address vector valued games, in particular multiple scenario cooperative games. For this class of games we have established extensions of some of the most usual solution concepts of cooperative games. In relation to the core, the vector valued characteristic function leads to different concepts depending on the preference structure that the players are willing to accept. The natural extension of the concept of least core consists of the set of allocations whose vectors of maximum excess are nondominated, and these allocations can be computed by solving a multicriteria linear programming problem. A lexicographical dominance relation plays the role of the lexicographic ordering in the scalar case, and provides the basis to define the generalized nucleolus. We have shown that this set of allocations can be characterized as the set of nondominated solutions of a sequence of multicriteria linear problems. It is worth to point out that the choice of a unique allocation in the nucleolus involves the selection of a nondominated maximum excess vector in each step, this suggests the possibility of developing procedures to include additional information about the scenarios in the decision process.
References Aumann, R.J., Maschler, M., 1985. Game theoretic analysis of a bankruptcy problem from the Talmud. Journal of Economic Theory 36, 195–213. Derks, J.M., Tijs, S.H., 1986. Stable outcome for multi-commodity flow games. Methods of Operations Research 55, 493–504. Fern andez, F.R., Hinojosa, M.A., Marmol, A.M., Puerto, J., 2002a. Solution concepts in multiple criteria linear production games. In: Trzaskalik, T., Michnik, J. (Eds.), Multiple Objective and Goal Programming. Recent Developments. Advances in Soft Computing. Physica-Verlag, pp. 257–271. Fern andez, F.R., Hinojosa, M., Puerto, J., 2002b. Core solutions in vector-valued games. Journal of Optimization Theory and Applications 112, 331–360. Fern andez, F.R., Hinojosa, M., Puerto, J., in press. Set valued TU-games. European Journal of Operational Research. Gow, S.H., Thomas, L.C., 1998. Interchange fees for bank ATM networks. Naval Research Logistic 45, 407–417. J€ ornsten, K., Lind, M., Tind, J., 1995. Stable payment schemes of TU-games with multiple criteria. Revised version of Publication 93, 3, Department of Operations Research, University of Aarhus, Arhus, Denmark. Littlechild, S.C., Thompson, G.F., 1977. Aircraft landing fees: A game theory approach. The Bell Journal of Economics 8, 186–204. M armol, A.M., Puerto, J., Fernandez, F.R., 1998. The use of partial information on weights in multicriteria decision problems. Journal of Multi-criteria Decision Analysis 7, 322–329. M armol, A.M., Puerto, J., Fernandez, F.R., 2002. Sequential incorporation of imprecise information in multiple criteria decision processes. European Journal of Operational Research 137, 123–133. Maschler, M., 1992. The bargaining set, kernel and nucleolus. In: Aumann, R.J., Hart, S. (Eds.), Handbook of Game Theory with Economics Applications, vol. I. Elsevier Science Publishers, Amsterdam. Nishizaki, I., Sakawa, M., 2001. Fuzzy and Multiobjective Games for Conflict Resolution. Physica-Verlag. Nouweland, A., van den Aarts, H., Borm, P., 1989. Multi-commodity games. Methods of Operations Research 63, 329–338. Schmeidler, D., 1969. The nucleolus of a characteristic function game. SIAM Journal of Applied Mathematics 16, 1163–1170.
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Steuer, R.E., 1995. Manual for the ADBASE. Multiple objective linear programming package. University of Georgia, Athens, GA. Voorneveld, M., Van den Nouweland, A., 1998. Cooperative multicriteria games with public and private criteria: An investigation of core concepts. CentER Discussion Paper Series 1998, 62, Tilburg University. Young, H.P., 1985. Cost allocation. In: Young, H.P. (Ed.), Fair Allocation Proceedings of Symposia in Applied Mathematics, vol. 33. American Mathematical Society, Providence, RI, pp. 69–94.
Decision Aiding
Core, least core and nucleolus for multiple scenario cooperative games q M.A. Hinojosa a b
a,1
, A.M. M armol
b,*,1
, L.C. Thomas
c
Departamento de Economıa y Empresa, Universidad Pablo de Olavide, Ctra. de Utrera Km. 1, 41012 Sevilla, Spain Departamento de Economıa Aplicada III, Universidad de Sevilla, Avda. Ramon y Cajal n°. 1, 41018 Sevilla, Spain c School of Management, University of Southampton, Highfield, Southampton 5017 1BJ, UK Received 25 September 2002; accepted 3 September 2003 Available online 5 March 2004
Abstract Multiple scenario cooperative games model situations where the worth of the coalitions is valued in different scenarios simultaneously or under different states of nature. In this paper we analyze solution concepts for this class of games keeping the multidimensional nature of the characteristic function. We obtain extensions of the notions of core, least core and nucleolus, and explore the relationship among these solution concepts. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Game theory; Cooperative games; Multiple scenario; Lexicographical solutions
1. Introduction Multiple scenario cooperative games is a particular class of cooperative vector valued transferable utility (TU) games, where a single resource is allocated, but the effectiveness of the allocations for each coalition is measured by a vector instead of by a scalar. These games arise in managerial and public decision making when a bundle of goods has to be allocated among a set of individuals, taking into account the worth or values of the coalitions under different scenarios. The different scenarios can be interpreted as the different states of the world in presence of uncertainty. When the agents cannot assign (objective or subjective) probabilities to the occurrence of different states of the world, standard expected utility arguments cannot be applied. In this situations the values of the coalitions are represented by a vector containing all the possible results.
q This research was conducted while A.M. Marmol and M.A. Hinojosa were visiting the School of Management, University of Southampton (UK). * Corresponding author. Tel.: +34-95-455-7554; fax: +34-95-455-1667. E-mail address: [email protected] (A.M. Marmol). 1 The research of these authors is partially supported by the Spanish Ministry of Science and Technology projects BFM2002-11282E and BEC2003-03111.
0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2003.09.028
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Some classes of vector valued cooperative games have been addressed in J€ ornsten et al. (1995) and Fern andez et al. (2002b). Voorneveld and Van den Nouweland (1998) introduced a general model of cooperative multicriteria game with public and private criteria which includes the so-called multicommodity games as a particular case. Related work on multicommodity games and its applications can be seen in Derks and Tijs (1986), Nouweland et al. (1989), Nishizaki and Sakawa (2001), and Fernandez et al. (2002a, in press). Multiple scenario cooperative games differ from conventional cooperative games in the dimension of the coalition values, and they also differ from cooperative multicriteria games in that a single good instead of a set of different commodities has to be allocated among the players. Formally, in multiple scenario cooperative games an allocation is represented by a vector whereas in multicriteria games it is represented by a matrix. One example would be the allocation of the total cost of construction of the distribution system of a certain good (electricity, water, etc.) from a common supplier to different costumers (players). The allocation must be fair, in the sense that each group of costumers (coalitions) will not have to pay more than the cost of constructing their own distribution system. In a multiple scenario framework, if there is uncertainty about the costs associated to the coalitions, the aim becomes to identify allocations of costs that are satisfactory under all scenarios. A second application is where the players are specialists in investment funds with expertise in investing in different areas. They can decide to form coalitions to pool expertise investing the coalition total, and the allocation of the funds to the players has to be done taking into account the values of the coalitions under future economic scenarios. The class of multiple scenario cooperative games has been considered recently in Nishizaki and Sakawa (2001), that have presented extensions of the notions of least core and nucleolus. Their approach consists of measuring the effectiveness of an allocation by a scalar, that can be either an aggregation by weighting coefficients of the vector valued excess function, or an aggregation by minimal components of the excess function. They also consider the solution obtained when fixing m 1 levels and trying to maximize the other one. The solutions obtained by these procedures strongly depend on the method chosen to scalarize the problem. The objective of this paper is to propose and analyze different solution concepts for the class of multiple scenario cooperative games, and to provide procedures to obtain the solutions. We concentrate on core concepts. The idea underlying core allocations is to provide incentives for cooperation excluding those unstable outcomes in which some coalitions are able to reach better results for all their members. In the case of a multiple scenario cooperative game the vector valued characteristic function induces different core concepts depending on the preference structure that the players are willing to accept. However, as in conventional games, these core concepts do not provide a single solution for the multiple scenario game. In fact, there can be too many outcomes in these solution sets but also they can be empty for some games. Therefore, in this paper we also explore other solution concepts based in a criterion of equity: the least-core and the nucleolus. We keep the multidimensional nature of the characteristic function to obtain extensions of the notions of e-core, least core, and nucleolus. It is important to point out that these solutions concepts can be characterized as the solutions of multicriteria linear problems and therefore computed with the available specific software. In Section 2, we present the model and the notation used in the paper. Solutions related to the classical concept of core are addressed in Section 3. In Section 4 we extend the concept of least core to the case of multiple scenario games and the allocations in the least core are characterized as the solutions of a multicriteria linear problem. Section 5 presents the generalized nucleolus and describe a procedure to construct it. In Section 6 an example is discussed, and Section 7 is devoted to the conclusions.
2. Multiple scenario cooperative games A multiple scenario cooperative game with transferable utility is a triplet ðN ; v; EÞ where N ¼ f1; 2; . . . ; ng is the set of players (a nonempty subset S N of the player set is called a coalition), E is
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the total amount of the issue that has to be allocated and v is a vector valued function, the characteristic function of the game, which associates any coalition S N with vðSÞ 2 Rm , that is the worth of coalition S. The components of vðSÞ, vj ðSÞ, j ¼ 1; . . . ; m, represent the worth of S in the m scenarios, that is, vj ðSÞ can be interpreted as the amount of issue that coalition S could obtain by itself under scenario j. If the multiple scenario game is played an interesting question which arises is how the total amount of issue, E, should be allocated among the various players taking into account the worth of the coalitions under all scenarios. As in conventional games, an allocation is a payoff vector, X 2 Rn , whose components P represent the payoff for each player, X is the payoff for the ith player. Therefore X P 0, 8i 2 N , and i i i2N Xi ¼ E holds. P The sum X ðSÞ ¼ i2S Xi is the overall payoff obtained by coalition S, thus X ðN Þ ¼ E. We will denote the set of allocations of the game by I ðEÞ. A solution concept for cooperative games associates a set (possibly empty) of allocations with each game. We will be interested in valuing the differences between what the coalitions obtain with a certain allocation, and their values in the game. For each allocation and each coalition, we define the vector valued excess function eðX ; SÞ whose components are ej ðX ; SÞ ¼ vj ðSÞ X ðSÞ, j ¼ 1; . . . ; m. This function measures the excesses of coalition S under allocation X in every scenario and will play a central role in the definition of the solutions concepts for these games. In what follows, we will denote by aj , j ¼ 1; . . . ; m, the components of vector a 2 Rm , and for a; b 2 Rm , we will use the following notation: a 5 b if aj 6 bj , 8j ¼ 1; . . . ; m; a 6 b if a 5 b and a 6¼ b; ab (a£b) if a 5 b (a 6 b) is not true.
3. Core solutions When extending the core concept of the conventional Game Theory to vector valued games, different concepts are obtained depending on the ordering considered in the payoff space. In this section we define some of these solution concepts for multiple scenario cooperative games and explore the relationships among them. Firstly we consider allocations in which no coalition has any incentive to deviate irrespective of the scenario and define the concept of preference core. Definition 3.1. The preference core of the multiple PCðN ; v; EÞ ¼ fX 2 I ðEÞ j eðX ; SÞ 5 0; 8S N ; S 6¼ ;g.
scenario game ðN ; v; EÞ is defined as
Notice that the preference core is the intersection of the cores of each scalar scenario game, ðN ; vj Þ where v ðN Þ ¼ E, 8j ¼ 1; . . . ; m, induced by the multiple scenario game. The next result characterizes the preference core of the multiple scenario game as the core of the scalar game obtained by valuing each coalition at the most optimistic scenario. Consider the scalar game ðN ; vmax Þ, whose characteristic function is defined as vmax ðSÞ ¼ Maxj¼1;...;m vj ðSÞ, 8S N , S 6¼ ;. The core of this game is CðN ; vmax Þ ¼ fX 2 I ðEÞ j vmax ðSÞ X ðSÞ 6 0; 8S N ; S 6¼ ;g. It is straightforward to prove that the preference core of the multiple scenario cooperative game coincides with the core of the game ðN ; vmax Þ. j
Proposition 3.1. PCðN ; v; EÞ ¼ CðN ; vmax Þ. As a consequence the preference core of the game is not empty if and only if the game ðN ; vmax Þ is balanced. Next we define a weaker core concept for multiple scenario cooperative games, the dominance core. To be in the dominance core it suffices that the allocation gives a nonpositive excess for each coalition, that is, for each coalition there is at least one scenario in which it has no reason to deviate.
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Definition 3.2. The dominance core of the multiple scenario game ðN ; v; EÞ is defined as DCðN ; v; EÞ ¼ fX 2 I ðEÞ j 8S N ; S 6¼ ;; eðX ; SÞ 0g. The dominance core is contained in the core of the scalar game obtained when the worth of each coalition is valued at the most pessimistic scenario, that is DCðN ; v; EÞ CðN ; vmin Þ, where the scalar game ðN ; vmin Þ is defined by vmin ðSÞ ¼ Minj¼1;...;m vj ðSÞ, 8S N , S 6¼ ;. It is worth noting that the inclusion relation DCðN ; v; EÞ CðN ; vmin Þ can be strict. To analyze the difference between these two sets of allocations, let us denote by T the set of coalitions whose value in the multiple scenario game is not the same in every scenario, i.e., T ¼ fS N ; vj ðSÞ > vmin ðSÞ; for some j ¼ 1; . . . ; mg. The dominance core of the multiple scenario game, and the core of the game ðN ; vmin Þ differ in those allocations in CðN ; vmin Þ that give to a coalition, S 2 T , exactly its value in the worst scenario. Formally the result is: Proposition 3.2 CðN ; vmin Þ ¼ DCðN ; v; EÞ [ fX 2 CðN ; vmin Þ j 9S 2 T ; X ðSÞ ¼ vmin ðSÞg:
Proof. It is easy to see that DCðN ; v; EÞ is included in CðN ; vmin Þ. In order to prove the other inclusion, suppose that X 2 CðN ; vmin Þ and X 62 DCðN ; v; EÞ, then 9S N , vðSÞ X ðSÞ P 0. This implies that vmin ðSÞ X ðSÞ P 0 and also that vj ðSÞ > vmin ðSÞ, for some j ¼ 1; . . . ; m, thus S 2 T . But, as X 2 CðN ; vmin Þ, vmin ðSÞ X ðSÞ 6 0, and it follows that vmin ðSÞ ¼ X ðSÞ for some S 2 T . h The dominance core is usually too wide as a set of solutions, and often the preference core is a too restrictive set or even empty. In practice, it may be interesting not only to look for a nonpositive excess, but to establish certain bounds on the excess function in order to obtain solutions that achieve a better performance. This is the idea underlying the next solution concept, that the extension of the notion of e-core. Let p 2 Rm , p ¼ ðp1 ; . . . ; pm Þ, be a vector of admissible excess levels. Definition 3.3. The preference p-core for the multiple scenario game ðN ; v; EÞ, PpCðN ; v; EÞ, is the set of allocations PpCðN ; v; EÞ ¼ fX 2 I ðEÞ j eðX ; SÞ 5 p; 8S N ; S 6¼ ;g:
Definition 3.4 The dominance p-core for the multiple scenario game ðN ; v; EÞ, DpCðN ; v; EÞ, is the set of allocations DpCðN ; v; EÞ ¼ fX 2 I ðEÞ j eðX ; SÞ p; 8S N ; S 6¼ ;g:
For p ¼ 0, PpCðN ; v; EÞ ¼ PCðN ; v; EÞ and DpCðN ; v; EÞ ¼ DCðN ; v; EÞ. For p P 0, the concept of preference p-core is a relaxation of the notion of preference core, in that it allows the excess function to be positive if maintained below the levels established for each criteria, and the dominance p-core is also a relaxation of the notion of dominance core. On the other hand, for p 6 0 the concept of preference p-core becomes more restrictive than the preference core, requiring that the maximum excesses attained are below zero, and the concept of dominance p-core is more restrictive than the dominance core.
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The following relationships between these concepts are easy to prove. Proposition 3.3. If p; q 2 Rm , p 5 q, then the following inclusion relations hold: PpCðN ; v; EÞ PqCðN ; v; EÞ; DpCðN ; v; EÞ DqCðN ; v; EÞ:
When the admissible level vectors are incomparable the inclusions are established in the following result. Proposition 3.4. If p; q 2 Rm , p q and q p then PpCðN ; v; EÞ DqCðN ; v; EÞ and PqCðN ; v; EÞ DpCðN ; v; EÞ. Proof. If p q and q p, there exists j ¼ 1; . . . ; m such that pj < qj . Let X 2 PpCðN ; v; EÞ, eðX ; SÞ 5 p, 8S N , and therefore eðX ; SÞ q, 8S N . It follows that X 2 DqCðN ; v; EÞ. Analogously the other inclusion holds. h Decreasing the admissible excess levels, p, yields to allocations that are more satisfactory for all the coalitions in the preference p-core, PpCðN ; v; EÞ, but the set can be empty. The objective now is to find feasible allocations that reach the smallest maximal excess level as possible. This lead us to define the generalized least core as a solution concept for multiple scenario cooperative games.
4. The generalized least core Definition 4.1. The maximum excess vector for allocation X 2 I ðEÞ, is the vector pðX Þ 2 Rm whose components are pj ðX Þ ¼ Max ej ðX ; SÞ, j ¼ 1; . . . ; m. SN
The maximum excess vector is a vector valued measure of the worst performance of an allocation in the set of all the coalitions. Indeed, the allocation is valued by the worst value attained in each scenario. Notice that the values pðX Þ 2 Rm may not be attained simultaneously, because different components can be obtained from different coalitions. Nevertheless, the vector is an upper bound for the values of the excess function. In order to formalize the idea of minimizing the worst performance we consider those allocations that minimize the maximum excess vector in the multicriteria sense, arriving to the natural extension of the concept of least core. Definition 4.2. p 2 Rm is a nondominated maximum excess vector for the multiple scenario game ðN ; v; EÞ, if 9X 2 I ðEÞ such that p ¼ pðX Þ, and 9 = Y 2 I ðEÞ, such that pðY Þ 6 p. We will denote the set of nondominated maximum excess vectors by P1 . Definition 4.3. The generalized least core of the multiple scenario game ðN ; v; EÞ, GLCðN ; v; EÞ, is the set of allocations, X 2 I ðEÞ, whose maximum excess vectors are nondominated. Allocations in the generalized least core are those whose vector of maximum excess is Pareto optimal or nondominated, in the sense that no other allocation has a better maximum excess vector.
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The following result establishes that given a certain nondominated maximum excess vector, the set of allocations where this level is attained coincides with the corresponding preference p-core. Proposition 4.1. If p 2 Rm is a nondominated maximum excess vector for the multiple scenario game ðN ; v; EÞ, then fX 2 I ðEÞ j pðX Þ ¼ pg ¼ PpCðN ; v; EÞ:
Proof. Recall that PpCðN ; v; EÞ ¼ fX 2 I ðEÞ j eðX ; SÞ 5 p; 8S N ; S 6¼ ;g. It is easy to see that if X 2 fX 2 I ðEÞ j pðX Þ ¼ pg, then X 2 PpCðN ; v; EÞ. On the other hand, let p 2 P1 , and X 2 PpCðN ; v; EÞ. Since eðX ; SÞ 5 p, 8S N , MaxSN ej ðX ; SÞ 6 pj , 8j ¼ 1; . . . ; m, and pðX Þ 5 p. If pðX Þ ¼ q 6 p, then p 62 P1 . It follows that pðX Þ ¼ p. h As a consequence of this result, the generalized least core can be represented as the union of the preferences p-cores associated to nondominated maximum excess vectors, that is, [ GLCðN ; v; EÞ ¼ PpCðN ; v; EÞ: p2P1
In order to characterize the allocations in the generalized least core, consider the following multicriteria linear programming problem associated to the game ðN ; v; EÞ, that we will denote by (GMLP)1 Min
ðp1 ; p2 ; . . . ; pm Þ
s:t::
vj ðSÞ X ðSÞ 6 pj ;
8S N ; 8j ¼ 1; . . . ; m;
X ðN Þ ¼ E; X P 0: Proposition 4.2. Let ðX ; pÞ be a nondominated solution of (GMLP)1 , then X 2 GLCðN ; v; EÞ and p is its maximum excess vector. Conversely, if X 2 GLCðN ; v; EÞ and p is its maximum excess vector, then ðX ; pÞ is a nondominated solution of (GMLP)1 . Proof. It follows from Definition 4.3 that the allocations in the generalized least core are the Pareto optimal or nondominated solutions of the following multicriteria nonlinear problem: Min
ðp1 ðX Þ; p2 ðX Þ; . . . ; pm ðX ÞÞ
s:t::
X ðN Þ ¼ E; X P 0;
which is equivalent to the linear multicriteria problem (GMLP)1 . h This result allows the calculation of the generalized least core by solving a linear multicriteria problem with m objectives and m ð2n 1Þ constraints. Although the dimensionality of the problem can be high, appropriate software, such as ADBASE (Steuer, 1995), is currently available to solve it. The whole set of nondominated solutions of (GMLP)1 , in general is not a convex set, and therefore the generalized least core is not necessarily convex, as can be seen in the example presented in Section 6. Nevertheless, for every p 2 P1 , the corresponding preference p-core, PpCðN ; v; EÞ, is a polyhedral set whose
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extreme points are obtained when solving the problem (GMLP)1 . Furthermore, as a consequence of the connectedness of the solutions of multicriteria linear problems, the generalized least core is a connected set. The generalized least core may include too many vectors to be useful in the practice of decision-making. In the next section we refine this solution concept by applying recursively the same idea of obtaining the allocations associated to vectors of maximum excess that are nondominated.
5. The generalized nucleolus In this section we establish the extension of the concept of nucleolus for multiple scenario games. The nucleolus was first proposed as a solution for scalar cooperative games in Schmeidler (1969), and its properties have been extensively explored thereafter. A good revision is in Maschler (1992). It has been considered as a crucial solution of the game in many applications specially in relation to cost allocations problems and bankruptcy situations (see, for instance, Littlechild and Thompson, 1977; Young, 1985; Aumann and Maschler, 1985; Gow and Thomas, 1998). The idea underlying this concept is the selection of the allocations in the least core for those who are better in a lexicographical sense. In the scalar case it is possible to define a lexicographical order among the allocations and therefore determine an unique best solution, the nucleolus. In the case of a vector valued measure of the allocations we deal with the binary relation that we will define next. For each allocation of the game, X , we construct an m ð2n 2Þ matrix, pðX Þ. The jth row of matrix pðX Þ, pj ðX Þ, j ¼ 1; . . . ; m, contains the values that the excess function attains in the jth scenario in every coalition under allocation X , arranged in order of decreasing magnitude. Matrix pðX Þ is a matrix-valued measure of the performance of allocation X with respect to all the coalitions and in every scenario. Notice that the first column of pðX Þ, p1 ðX Þ is the vector of maximum excess for allocation X , pðX Þ. The second column, p2 ðX Þ is the vector of second biggest values of the excess that can be attained with allocation X in every scenario. In general, pk ðX Þ represents the vector of values in the kth place. Definition 5.1. Allocation X is lexicographically better than allocation Y , in the multiple scenario game ðN ; v; EÞ, (X 6 lex Y ), if pk ðX Þ 6 pk ðY Þ for the first column, k, in which matrix pðX Þ and matrix pðY Þ are different. Notice that this binary relation does not define a complete order in the space of allocations, in fact, when for column k, pk ðX Þ£pk ðY Þ and pk ðY Þ£pk ðX Þ, the allocations are incomparable. In the particular case n where m ¼ 1, this relation coincides with the lexicographical order in R2 2 . Definition 5.2. Allocation X is lexicographically nondominated in the multiple scenario game ðN ; v; EÞ, if there is no other allocation Y such that Y 6 lex X . This dominance relation permits to make a selection among the allocations and to define the solution concept that extends the notion of nucleolus to multiple scenario games. Definition 5.3. The generalized nucleolus for the multiple scenario game ðN ; v; EÞ, GN ðN ; v; EÞ, is the set of allocations, X 2 I ðEÞ, that are lexicographically nondominated. The following result establishes that when for some level p 2 Rm the preference p-core, PpCðN ; v; EÞ, is nonempty, it includes some of the allocations in the generalized nucleolus.
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Proposition 5.1. If PpCðN ; v; EÞ 6¼ ;, then PpCðN ; v; EÞ \ GN ðN ; v; EÞ 6¼ ;. Proof. Let PpCðN ; v; EÞ be nonempty. If p 2 P1 , from Proposition 4.1, the preference p-core, PpCðN ; v; EÞ, coincides with the set of allocations whose maximum excess vector is exactly p. As p is nondominated, there exists at least a lexicographically nondominated allocation in PpCðN ; v; EÞ. If p 62 P1 , there exists q 2 P1 such that q 6 p. It follows from Proposition 3.3 that PqCðN ; v; EÞ PpCðN ; v; EÞ, and the lexicographically nondominated allocations in PqCðN ; v; EÞ are included in the nucleolus. h As an immediate consequence, we obtain the extension of the known result in scalar cooperative games. Corollary 5.1. If the preference core, PCðN ; v; EÞ is nonempty, the nucleolus has nonempty intersection with the preference core. It is worth noting that, in general, the generalized nucleolus is not included in the preference core. Nevertheless, as a consequence of Proposition 3.3, the following result holds. Proposition 5.2. If q 6 p, 8q 2 P1 , then GN ðN ; v; EÞ PpCðN ; v; EÞ. To construct the generalized nucleolus, once the generalized least core is obtained, look at the nondominated levels p 2 P1 , and if PpCðN ; v; EÞ is a singleton, it is contained in the generalized nucleolus. In other case the lex-nondominated allocations in PpCðN ; v; EÞ have to be obtained. Each one of the sets PpCðN ; v; EÞ is a polyhedral set. Let its extreme points be X 1 ; . . . ; X rðpÞ . We will prove that, when p 2 P1 , for every scenario j ¼ 1; . . . ; m, there is at least a coalition that attains the maximum excess level given by vector p for each X 2 PpCðN ; v; EÞ. Proposition 5.3. If p 2 P1 , then 8j ¼ 1; . . . ; m, fS N j S 6¼ ;; ej ðX ; SÞ ¼ pj ; 8X 2 PpCðN ; v; EÞg 6¼ ;:
Proof. Consider p 2 P1 . For j ¼ 1; . . . ; m, let sðjÞ be defined as follows: sðjÞ ¼ fS N j vj ðSÞ X k ðSÞ ¼ pj ; 8k ¼ 1; . . . ; rðpÞg: We are going to prove that sðjÞ 6¼ ;, 8j ¼ 1; 2; . . . ; m. Assume that sðjÞ ¼ ;. Then, for any coalition S, there exists an extreme point X l , such that vj ðSÞ X l ðSÞ < pj . Define the following allocation: Y ¼
rðpÞ 1 X X k 2 PpCðN ; v; EÞ: rðpÞ k¼1
Now, for any coalition, S N , and for any scenario j, vj ðSÞ Y j ðSÞ ¼
rðpÞ rðpÞ rðpÞ 1 X j 1 X 1 X j v ðSÞ X k ðSÞ ¼ ðv ðSÞ X k ðSÞÞ < pj ; rðpÞ k¼1 rðpÞ k¼1 rðpÞ k¼1
and this contradicts the fact that p is a nondominated vector. Notice that if a coalition attains a level in the extreme points of PpCðN ; v; EÞ, it also attains this level in all the allocations of PpCðN ; v; EÞ. h
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It follows from the result proved in Proposition 5.3 that when the set PpCðN ; v; EÞ is obtained from the (GMLP)1 , at least m constraints are active in any of the allocations X 2 PpCðN ; v; EÞ, what makes it possible to find the allocations in PpCðN ; v; EÞ with nondominated maximum second levels. In order to do this, for each j ¼ 1; . . . ; m, we select a coalition T ðjÞ 2 sðjÞ ¼ fS N j vj ðSÞ X k ðSÞ ¼ pj ; 8k ¼ 1; . . . ; rðpÞg, and consider the following multicriteria problem, (GMLP)2 , where we replace the corresponding m inequality constraints with the equality consisting of the level pj that the coalition attains in scenario j: Min
ðq1 ; q2 ; . . . ; qm Þ
s:t::
vj ðT ðjÞÞ X ðT ðjÞÞ ¼ pj ; j
j
v ðSÞ X ðSÞ 6 q ;
8j ¼ 1; . . . ; m;
8j ¼ 1; . . . ; m; 8S N ; S 6¼ T ðjÞ;
X ðN Þ ¼ E; X 2 PpCðN ; v; EÞ: The following proposition characterizes those allocations in the generalized least core whose vectors of second maximum levels are nondominated. Proposition 5.4. Let p 2 P1 . If ðX ; qÞ is a nondominated solution of (GMLP)2 , then p ¼ p1 ðX Þ and q ¼ p2 ðX Þ. Conversely, if X 2 PpCðN ; v; EÞ and p2 ðX Þ is such that, 9 = Y 2 PpCðN ; v; EÞ, p2 ðY Þ 6 p2 ðX Þ, then ðX ; qÞ, where q ¼ p2 ðX Þ, is a nondominated solution of problem (GMLP)2 . We will denote by P2 the set of nondominated vectors, q 2 Rm , obtained when solving (GMLP)2 . If there is a unique allocation associated to q 2 P2 , the allocation is in the generalized nucleolus of the game, in other case, the same procedure can be applied to obtain the allocations with nondominated p3 ðX Þ, until the solution is unique. To solve problem (GMLP)2 , notice that the nondominated extreme solutions of (GMLP)1 associated to p provide the set of extreme points of PpCðN ; v; EÞ, therefore this set can be represented as ( ) X
PpCðN ; v; EÞ ¼ X 2 I ðEÞ j X ¼ Lp k; ki ¼ 1; k P 0 ; i¼1;...;rðpÞ
where Lp is a matrix whose rðpÞ columns are the extreme points of PpCðN ; v; EÞ. If we substitute X ¼ Lp k the problem (GMLP)2 is transformed to Min
ðq1 ; q2 ; . . . ; qm Þ
s:t::
vj ðT ðjÞÞ MðT ðjÞÞk ¼ pj ; j
j
v ðSÞ MðSÞk 6 q ;
8j ¼ 1; . . . ; m
8j ¼ 1; . . . ; m; 8S N ; S 6¼ T ðjÞ;
t
e k ¼ 1; k P 0; where M is a ð2n 2Þ rðpÞ matrix whose columns represent the overall payoff obtained by the ð2n 2Þ coalitions with the extreme points of PpCðN ; v; EÞ. MðSÞ stands for the row in M associated to coalition S. Notice that although we have added the condition X 2 PpCðN ; v; EÞ to the original problem (GMLP)1 , the number of constraints has not increased, and each of the constraints still corresponds to a certain coalition and a certain scenario. The recursive procedure to obtain allocations in the generalized nucleolus in a multiple scenario cooperative game can be described as follows.
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In a first step we obtain a nondominated maximum excess vector, p, by solving problem (GMLP)1 . If the preference p-core is a singleton, PpCðN ; v; EÞ ¼ fX g, then allocation X is lexicographically nondominated, and therefore it is in the generalized nucleolus. In other case, the extreme points of PpCðN ; v; EÞ are provided as part of the solution of (GMLP)1 . In a second step, by solving (GMLP)2 we obtain the allocations in PpCðN ; v; EÞ whose second maximum excess levels are nondominated. In general, in step k, a vector of nondominated kth maximum excess level is obtained. The procedure finishes when in a certain step the set of allocations associated to the corresponding nondominated maximum excess level has a unique element. In any step a transformation of the type X ¼ Lp k can be performed once the extreme points of the corresponding set are known, and at least m inequality constraints become active. Since n independent equality constraints determine uniquely the solution, the procedure will finish in at most n steps. The generalized nucleolus is a subset of the generalized least core from where it may be possible to select a suitable allocation. The selection of this single allocation can be done once the whole set is obtained, or it can also be done by choosing one of the nondominated maximum excess vectors in each step. The characterization of the successive maximum excess vectors as the solutions of multicriteria linear programming problems, established in Propositions 4.2 and 5.4, suggests the possibility of incorporating additional information about the probability of occurrence of the different scenarios to the model. A methodology to incorporate information about the importance of the criteria in multicriteria linear problems based in the extreme points of the information sets is proposed in Marmol et al. (1998). In M armol et al. (2002) a procedure to sequentially incorporate preference information to the decision problem is provided. It is possible to apply this methodology to the process of obtaining allocations in the generalized nucleolus of the multiple scenario cooperative game in order to direct the search for a final consensus allocation. Alternatively, other decision criteria (such as min–max, Laplace criterion, etc.) can be considered to obtain nondominated solutions of the problems in each step.
6. Example The following example illustrates the procedure to obtain the preference and the dominance cores, the generalized least core and the generalized nucleolus of a multiple scenario game. The example is taken from Nishizaki and Sakawa (2001). Consider a three person cooperative game with three scenarios. For all the scenarios, the values of oneperson coalitions are zero, and the value of the grand coalition is one, because the game is zero–one normalized. The values of the two person coalitions, are shown in the following table. S
Scenario I
Scenario II
Scenario III
f1; 2g f1; 3g f2; 3g
0.3 0.5 0.4
0.3 0.3 0.7
0.1 0.5 0.7
The preference core and the dominance cores of this multiple scenario game have been obtained as shown in Propositions 3.1 and 3.2, respectively, PCðN ; v; EÞ ¼ fX 2 R3 j X P 0; X1 þ X2 þ X3 ¼ 1; X1 þ X2 P 0:3; X1 þ X3 P 0:5; X2 þ X3 P 0:7g; DCðN ; v; EÞ ¼ fX 2 R3 j X P 0; X1 þ X2 þ X3 ¼ 1; X1 þ X2 > 0:1; X1 þ X3 > 0:3; X2 þ X3 > 0:4g: Notice that the dominance core is a subset of the core of the pessimistic game, that does not include all the allocations in the frontier, as is established in Proposition 3.2. In Fig. 1 we represent the preference core and the dominance core of the game on the simplex fX 2 R3 j X P 0; X1 þ X2 þ X3 ¼ 1g.
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(0, 1, 0)
PC
DC
(0, 0, 1)
(1, 0, 0)
Fig. 1. Preference core and dominance core.
To construct the generalized least core for this game, using ADBASE to solve the associated (GMLP)1 , we obtain four extreme nondominated solutions: ðX1 ; p1 Þ ¼ ð0:15; 0:35; 0:50; 0:15; 0:15; 0:15Þ; ðX2 ; p2 Þ ¼ ð0:15; 0:30; 0:55; 0:15; 0:15; 0:15Þ; ðX3 ; p3 Þ ¼ ð0:20; 0:30; 0:50; 0:20; 0:10; 0:10Þ; ðX4 ; p4 Þ ¼ ð0:30; 0:25; 0:45; 0:25; 0; 0Þ: p1 , p2 and p3 determine a nondominated facet, and P the segment P joining p3 and p4 is also a nondominated edge. If we denote by F1 ¼ fX 2 R3 j X P 0; X ¼ 3i¼1 ki Xi ; 3i¼1 ki ¼ 1; k P 0g and by F2 ¼ fX 2 R3 j X P 0; X ¼ k1 X3 þ k2 X4 ; k P 0; k1 þ k2 ¼ 1g, the generalized least core of the game is the union of these sets of allocations: GLCðN ; v; EÞ ¼ F1 [ F2 : Fig. 2 is graphical representation of the generalized least core. We denote by P1i , i ¼ 1; 2, the setPof vectors Pof maximum excess associated with3 the allocations in Fi , i ¼ 1; 2, that is, P11 ¼ fp 2 R3 j p ¼ 3i¼1 ki pi ; 3i¼1 ki ¼ 1; k P 0g and P12 ¼ fp 2 S R j p ¼ k1 p3 þ k2 p4 ; k P 0; k1 þ k ¼ 1g. Then the sets F and F can be viewed as F ¼ and 1 2 1 p2P11 PpCðN ; v; EÞ S 2 F2 ¼ p2P1 PpCðN ; v; EÞ. 2
X1 α
F1
X2
α'
X3
F2 X4
Fig. 2. Generalized least core.
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In order to obtain the generalized nucleolus, we analyze the sets F1 and F2 . There is a unique allocation associated to the levels p 2 P12 , therefore for each p 2 P12 , the set of allocations PpCðN ; v; EÞ is a singleton and is contained in the generalized nucleolus. The extremes ðX1 ; p1 Þ and ðX2 ; p2 Þ have the same maximum excess vector. In fact all the nondominated levels p 2 P11 , p 6¼ p3 , are not attained in an unique allocation. To see this, notice that as p1 ¼ p2 , the set P11 can be represented as P11 ¼ fp 2 R3 j p ¼ ap1 þ ð1 aÞp3 ; 0 < a 6 1g, that is, each p 2 P11 has associated an unique a, 0 < a 6 1, and the set of allocations where pa is attained can be represented as Ppa CðN ; v; EÞ ¼ fX 2 R3 j X ¼ k½aX1 þ ð1 aÞX3 þ ð1 kÞ½aX2 þ ð1 aÞX3 ; 0 6 k 6 1g, in words, Ppa CðN ; v; EÞ is the segment joining the allocations that attain level pa , in the edges X1 X3 and X2 X3 (see Fig. 2). Next step is to find the allocations whose second maximum excess level is minimal in each of the sets Ppa CðN ; v; EÞ. To illustrate the procedure we show how it is done in Pð0:15;0:15;0:15Þ CðN ; v; EÞ. The extreme points of this set are allocations X1 and X2 . To select the allocations of Pð0:15;0:15;0:15Þ C that are in the generalized nucleolus, as vj ðf1gÞ Xi ðf1gÞ ¼ 0:15, j ¼ 1; 2; 3, i ¼ 1; 2, we choose coalition T ðjÞ ¼ f1g, 8j ¼ 1; 2; 3, and solve problem (GMLP)2 , where we replace with the equality constraints the three constraints associated with coalition f1g: Min
ðq1 ; q2 ; . . . ; qm Þ
s:t::
vj ðSÞ X ðSÞ 6 qj ; j ¼ 1; . . . ; m; S N ; S 6¼ f1g; vj ðf1gÞ X ðf1gÞ ¼ 0:15; j ¼ 1; . . . ; m; X ðN Þ ¼ E; X 2 Pð0:15;0:15;0:15Þ C:
Problem (GMLP)2 has a unique optimal solution, X ¼ ð0:15; 0:325; 0:525Þ and q ¼ ð0:175; 0:15; 0:15Þ. Therefore this allocation is in the generalized nucleolus with levels p1 ðX Þ ¼ ð0:15; 0:15; 0:15Þ and p2 ðX Þ ¼ ð0:175; 0:15; 0:15Þ. For the other nondominated levels pa , 0 < a 6 1, solving the corresponding (GMLP)2 , we obtain unique allocations Xa ¼ ð0:2 0:05a; 0:3 þ 0:025a; 0:5 þ 0:025aÞ with vectors of levels qa ¼ ð0:2 0:025a; 0:1 þ 0:05a; 0:1 þ 0:05aÞ.
(0 , 1, 0)
X1 GLC X2
GN X3
DC X4
PC
(0 , 0, 1)
(1 , 0, 0)
Fig. 3. Preference core, dominance core, generalized least core and generalized nucleolus.
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The set of allocations obtained, together with the allocations of F2 constitute the generalized nucleolus of the game. It is worth noting that these allocation are the mid-points of the segments Ppa CðN ; v; EÞ and they are on the line determined by the segment F2 . Therefore the generalized nucleolus is the segment joining allocations X4 and X5 (see Fig. 3). Fig. 3 is a graphical representation of the solution sets of the game on the simplex fX 2 R3 j X P 0; X1 þ X2 þ X3 ¼ 1g. Notice that, in this game, allocations in the nucleolus and in the least core are contained in the preference core, as a consequence of Proposition 5.2.
7. Conclusions The recent development of multicriteria analysis techniques allows us to address vector valued games, in particular multiple scenario cooperative games. For this class of games we have established extensions of some of the most usual solution concepts of cooperative games. In relation to the core, the vector valued characteristic function leads to different concepts depending on the preference structure that the players are willing to accept. The natural extension of the concept of least core consists of the set of allocations whose vectors of maximum excess are nondominated, and these allocations can be computed by solving a multicriteria linear programming problem. A lexicographical dominance relation plays the role of the lexicographic ordering in the scalar case, and provides the basis to define the generalized nucleolus. We have shown that this set of allocations can be characterized as the set of nondominated solutions of a sequence of multicriteria linear problems. It is worth to point out that the choice of a unique allocation in the nucleolus involves the selection of a nondominated maximum excess vector in each step, this suggests the possibility of developing procedures to include additional information about the scenarios in the decision process.
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