Avoiding unfairness of Owen allocations in linear production processes

Document downloaded from: This paper must be cited as: The final publication is available at Copyright Additional Information http://dx.doi.org/10.1016/j.ejor.2012.01.013 http://hdl.handle.net/10251/56504 Elsevier Perea Rojas Marcos, F.; Puerto Albandoz, J.; Fernández García, FR. (2012). Avoiding unfairness of Owen allocations in linear production processes. European Journal of Operational Research. 220(1):125-131. doi:10.1016/j.ejor.2012.01.013. Avoiding unfairness of Owen allocations in linear production processes Federico Perea1(Corresponding Author), Justo Puerto2, Francisco R. Fern´andez2 email: p[email protected]v.es,{puerto,fernande}@us.es, telephone and fax: +34 963877490 / +34 963877499 1Universitat Polit`ecnica de Val`encia (Spain) 2Universidad de Sevilla (Spain) Abstract This paper deals with cooperation situations in linear production problems in which a set of goods are to be produced from a set of resources so that a certain benefit function is maximized, assuming that resources not used in the production plan have no value by themselves. The Owen set is a well-know solution rule for the class of linear production processes. Despite their stability properties, Owen allocations might give null payoff to players that are necessary for optimal production plans. This paper shows that, in general, the aforementioned drawback cannot be avoided allowing only allocations within the core of the cooperative game associated to the original linear production process. In this paper a new solution set named EOwen is introduced. For any player whose resources are needed in at least one optimal production plan, the EOwen set contains at least one allocation that assigns a strictly positive payoff to such player. Keywords: Cooperative games, Linear production games, Allocations. Preprint submitted to EJOR September 23, 2011 1. Introduction A benefit cooperative game is a pair (N, v), where N={1,2, ..., n}is the set of players and v: 2N→Ris the characteristic function assigning to every coalition S⊂Nthe maximum benefit that the cooperation between players in Swould yield. For a complete introduction on cooperative game theory see for instance Owen (1995) or Forg´o et al. (1999). Assuming that the game is superadditive, that is v(S) + v(T)≤v(S∪T),∀S, T ⊂N, cooperation among all players is beneficial and, therefore, the grand coalition Nis to form. One of the main questions in cooperative game theory is how to distribute the benefit obtained by the grand coalition Namong the players. An allocation is a vector α∈Rn, such that αiis the payoff of player iand Pn i=1 αi=v(N). One well-accepted way of allocating v(N) among the players is to find allocations in the core. The core of a game (N, v), denoted by Core(N, v), is the set of allocations satisfying that no coalition of players can obtain a better payoff by acting separately from the rest of players. That is, Core(N, v) = {α∈Rn:v(S)≤α(S)∀S⊂N, v(N) = α(N)},(1) where α(S) = Pi∈Sαi,∀S⊂N. In principle, the core has at least two problems: the core of a game might be empty, that is, there are games for which no core allocations exist, and finding a core allocation might be a NPhard problem. Along the years, many other allocation rules have appeared in the literature. One of the most used allocation rules is the Shapley value, 2 which has attracted a lot of interest for its many applications, see Moretti and Patrone (2008). A linear production problem is a situation in which certain goods that can be sold in a market are to be produced from a set of available distinct resources. An implicit feature of the linear production problems we deal with in this paper is that the resources not used in the production plan have no value at all. Situations like this may arise when the resources are perishable and, if not used in the next production plan, they are wasted. Another example of this situation is found in some industries in developed countries that give their excesses to underdeveloped countries, to charity organizations, or even to other companies within the same area as long as they are not competing ones. This is beneficial for both parties: the donor party gets rid of excesses which, if not used, must be eliminated at certain cost, and the receiving party only has to pay for the shipping costs, which is usually cheaper than having to buy the material. In this paper we study a new set of allocations for linear production processes (LP processes for short), which arise when a bunch of players N= {1,...,n}with conflicting objectives control the resources of a LP problem. A cooperative game, called LP game, can be associated to each LP process. (Note that different LP processes may generate the same LP game.) An early reference to LP games can be found in Owen (1975). LP games are totally balanced games, so every subgame of a LP game has a non-empty core. By solving the dual problem of the underlying linear production problem we can obtain a set of allocations for LP processes known as Owen allocations (see Owen (1975)), which has been well-studied in the literature. One of its 3 main properties is that Owen allocations are always core allocations, and are easily computed. More recently, Van Gellekom et al. (2000) provided an axiomatic characterization of this solution set. In this paper we show that, despite their stability properties, Owen allocations do not always yield a fair distribution of the benefit obtained. For instance, a player whose resources are necessary for any optimal plan may receive a null payoff from Owen allocations. Such drawback is discussed in this paper, and an alternative allocation set is proposed. Since the pioneering work by Owen, several generalizations of LP games have appeared in the literature. Dubey and Shapley (1984) study a game in which players have partial control over the constraints of a general mathematical programming problem. Granot (1986) introduces another generalization in which the resources owned by a coalition are not restricted to be the sum of the resources of players in the coalition. Curiel et al. (1989) introduce LP games with committee control, obtaining results on the balancedness of these games, whose core has been more recently studied by Molina and Tejada (2004). The goal of this paper is to introduce a new set of allocations for linear production processes that avoid some of the aforementioned drawbacks of the Owen set. To this end, the rest of the paper is structured as follows. Section 2 gives a short introduction to LP processes and a motivation of the studied problem. Some definitions and technical results are given in Section 3. The allocation set proposed in this paper is introduced and analyzed in Section 4. An axiomatic characterization and some of its properties are given, as well as a discussion about the impossibility of finding core allocations that avoid 4 the unfairness problem of the Owen allocations we address in this paper. 2. Linear Production processes A LP problem is a situation in which there is a finite set of resources R={1,2,...,r}and from those resources a set P={1,2,...,p}of consumption goods can be produced. The production technologies are given by a matrix A∈Rr×p, where Aij ≥0 denotes the amount of resource inecessary to produce one unit of product j,∀i= 1, . . . , r, j = 1,...,p. It is also assumed that the demand of every product is large enough to sell all produced products, the unitary market price of product jbeing cj≥0. The objective of a LP problem is to decide how much of each product should be produced so that the general benefit is maximized. Assume now that a group of players N={1,...,n}control the resources R={1,2,...,r}, that is, player kowns Bik ≥0 units of resource i,k= 1, . . . , n, i = 1,...,r. Therefore, let B= (Bik)r×nbe the resource-player matrix. Let b∈Rrbe the resource vector, that is b=BeN, where eS∈Rn satisfying (eS)k= 1 if k∈S, and zero otherwise for all S⊆N. In other words, biis the total amount of resource iowned by the grand coalition, that is, bi=Pn k=1 Bik ∀i∈R. Thus, the maximum profit that can be made by the cooperation of all players is the value of problem PN: max cx s.t. Ax ≤b x≥0 (PN), min yb s.t. yA ≥c y≥0 (DN),(2) where DNis the dual problem of PN(see Bazaraa et al. (1990) for a description of duality theory in linear programming). It is easy to check that, 5 although players can try to produce separately, it is always more profitable to join their resources since the benefit they obtain this way is at least as high as the sum of the possible coalitions’ profits separately. For a coalition S⊂N, we define its characteristic function, v(S), via the optimal value of problem PS: max cx s.t. Ax ≤BeS x≥0 (PS), min yBeS s.t. yA ≥c y≥0 (DS),(3) where DSis the dual of PS. Problem PSis feasible and bounded for all possible coalitions if BeS>0, c≥0 and ∀j:cj>0 there is at least one resource i∈Rwith Aij >0. Each triple (A, B, c) satisfying the conditions above will be called in the following, according to Van Gellekom et al. (2000), a linear production process. Let Ldenote the class of LP processes. From the definition of the characteristic function vone can associate to each LP process a cooperative game (N, v). The reader may note that the same LP game can originate from different LP processes. Now a natural question arises: how to divide the profit made by the grand coalition among the players. Let us introduce some notation that will be useful in the rest of the paper. Let (A, B, c)∈ L. The feasible regions of problems PNand DN, see (2), are denoted by Fmax(A, B, c) := {x∈Rp +:Ax ≤b}, Fmin(A, B, c) := {y∈Rn +:yA ≥c}, (4) 6 respectively. The optimal values of problems PNand DNare denoted by vmax(A, B, c) := max{cx :x∈Fmax(A, B, c)}, vmin(A, B, c) := min{yb :y∈Fmin(A, B, c)}, (5) respectively, and the set of optimal solutions to PNand DNby Omax(A, B, c) := {x∈Fmax(A, B, c) : cx =vmax(A, B, c)}, Omin(A, B, c) := {y∈Fmin(A, B, c) : yb =vmin(A, B, c)}. (6) Asolution rule ϕon Lis a map assigning to every LP process (A, B, c)∈ La set Γ ⊂Rnsuch that Pi∈Nγi=vmax(A, B, c) for all γ∈Γ. Each member of this set is an allocation. A well-known solution rule for cooperative games is the core, see (1). One well-accepted solution rule specific for LP processes is the Owen set, defined from optimal solutions to the dual problem DN. Definition 1. Let (A, B, c)∈ L. The Owen set of (A, B, c)is Owen(A, B, c) := {yB :y∈Omin(A, B, c)}.(7) Owen (1975) proved that Owen(A, B, c)⊆Core(A, B, c) for every (A, B, c)∈ L. That is, Owen allocations are stable in the sense that no group of players can obtain a better payoff by acting separately. Despite these good properties, they should not be considered as ideal allocations. See the following example. Example 1. Consider the 3-player game (A, B, c)∈ L where A=         1 0 1 1 0 1 1 2         , B =         101 040 100 050         , c = 1 2  . 7 The corresponding dual problem D(N)is min 2y1+ 4y2+y3+5y4 s.t. y1+y2+y4≥1 y2+y3+2y4≥2 y1, y2, y3, y4≥0. (8) The characteristic function of the associated game is v({i}) = v({1,3}) = 0∀i= 1,2,3,v({1,2}) = 3, v({2,3}) = 1,v({1,2,3}) = 4. It can be checked that Omin(A, B, c) = {(1,0,2,0)}and, therefore, Owen(A, B, c) = {(1,0,2,0)B}={(3,0,1)}. This allocation is in the core of the game but, is it a “fair” allocation? Note that player 2 receives nothing but, without his resources, the optimal production plan cannot be achieved. So, the Owen allocation gives a null payoff to a player whose resources are necessary for the optimal production plan. What happened in Example 1 is a general drawback of the Owen set in LP processes. This is a consequence of the complementary slackness theorem, see Bazaraa et al. (1990), which says that if there is some surplus of resource iin an optimal solution x∗∈Omax(A, B, c) (meaning (Ax∗)i< bi), then y∗ i= 0 ∀y∗∈Omin(A, B, c). This condition for optimality conveys a simple economic principle: if there is a positive slack in a constrained primal resource, i.e. there are leftovers, then the additional quantities of that resource must have no value (shadow prices are zero). This means that only players owning resources that generate no surplus have the chance of receiving a strictly positive payoff from Owen allocations, which are based on shadow prices. This fact could make players get rid of their surpluses so that the 8 Proposition 1. Let b B∈[Bx∗, B]for some reduced matrix Bx∗associated to an optimal production plan x∗. Then we have that: Owen(A, B, c)⊆Owen(A, b B, c)⊆Owen(A, Bx∗, c). Proof. Let α∈Owen(A, B, c)⇒ ∃ y∈Omin(A, B, c) : α=yB. Let us see that y∈Omin(A, b B, c). min yb s.t. yA ≥c y≥0 (D), min yb b s.t. yA ≥c y≥0 (b D). Since y∈Omin(A, B, c), by definition one has that yis optimal for problem D. Therefore, yis feasible for problem b D. Besides, as we proved in Lemma 1, part 4, yb =yb b. Thus y∈Omin(A, b B, c). Applying again the complementary slackness theorem, one has that: αk= n X i=1 yiBik =X i:yi6=0 yiBik =X i:yi6=0 yib Bik ⇒α=yb B. Therefore we have proven that α∈Owen(A, b B, c), and as a consequence Owen(A, B, c)⊆Owen(A, b B, c). Owen(A, b B, c)⊆Owen(A, Bx∗, c) can be proven analogously.  An immediate corollary to the previous result states that the name Extended Owen set is meaningful, as EOwen contains the Owen set. Corollary 1. Owen(A, B, c)⊆EOwen(A, B, c)for all (A, B, c)∈ L. The following example proves that the inclusions in Proposition 1 and Corollary 1 may be strict. 15 Example 2. Take the LP process from Example 1. One can see that Omax(A, B, c) = {x∗= (2,1)}. Therefore, Bx∗(in this case unique) and a choice of b Bare: Bx∗=         1 0 1 0 3 0 1 0 0 0 4 0         ,b B=         1 0 1 0 3 0 1 0 0 0 5 0         . Then, we obtain that problem DN(Bx∗)and that corresponding to b B,DN(b B), are, respectively: min 2y1+ 3y2+y3+4y4 s.t. y1+y2+y4≥1 y2+y3+2y4≥2 y1, y2, y3, y4≥0 DN(Bx∗), min 2y1+ 3y2+y3+5y4 s.t. y1+y2+y4≥1 y2+y3+2y4≥2 y1, y2, y3, y4≥0 DN(b B). (16) From the solutions to these problems, which are the convex hulls of {(1,0,2,0), (0,1,1,0),(0,0,0,1)}and {(1,0,2,0),(0,1,1,0)}, respectively, one can see that Owen(A, Bx∗, c)(which in this example coincides with EOwen(A, B, c)) is the convex hull of {(3,0,1),(1,3,0),(0,4,0)}and Owen(A, b B, c)is the convex hull of {(3,0,1),(1,3,0)}. This way we prove that the relations described in Proposition 1 are strict. Besides, note that there are allocations in EOwen(A, B, c)that give player 2 a strictly positive payoff, unlike Owen(A, B, c). Note as well that EOwen(A, B, c)⊂Core(A, B, c). 16 From their definitions, it is easy to prove that for reduced LP processes, as introduced in Definition 2, the EOwen set and the Owen set coincide. Analogously, one can state that in LP processes where all the resources are completely used, the EOwen set coincides with the Owen set. The following proposition proves that allocations in the EOwen set distribute exactly v(N) among the players. Proposition 2. Let (A, B, c)∈ L and let γ∈EOwen(A, B, c). Then γis efficient. Proof. Let γ∈EOwen(A, B, c). Then there exists x∗∈Omax(A, B, c), Bx∗∈ B(A, B, x∗) and by∈Omin(A, Bx∗, c) such that γk=Pr i=1 byiBx∗ ik ∀k= 1,...,n. Therefore γ(N) = n X k=1 γk= n X k=1 r X i=1 byiBx∗ ik = r X i=1 byi n X k=1 Bx∗ ik = r X i=1 byibx∗ i=bybx∗.(17) Since by∈Omin(A, Bx∗, c) and x∗∈Omax(A, B, c), we know that bybx∗=cx∗= v(N). This concludes that γ(N) = v(N).  Another interesting property states that, for all players whose resources are necessary to produce the maximum benefit v(N) in some optimal production plan, there exists an allocation in EOwen that assigns them a strictly positive payoff. Theorem 1. Let (A, B, c)∈ L and x∗∈Omax(A, B, c), and let k∈Nbe a player such that some of the resources that he owns are needed for the optimal production plan x∗to be developed. Then there exists α∈EOwen(A, B, c) such that αk>0. 17 Proof. Let x∗∈Omax(A, B, c), and Bx∗∈ B(A, B, x∗). By the strict complementary slackness theorem, if the slack in the ith constraint of problem PN(Bx∗) is zero, then there exists a solution yto DN(Bx∗) such that yi>0 (see Theorem 10.7 in Vanderbei (1997)). Assuming that the units of the ith resource owned by player kare needed for the optimal production plan x∗, it is easy to see that Bx∗ ik >0. Then, the payoff of player kfrom the EOwen allocation αBx∗=yBx∗is, at least, yiBx∗ ik >0.  Note that this proposition allows us to state that the EOwen set always overcomes the unfairness problem illustrated in Example 1. Let us now introduce the property of upper limit inclusion (ULI), which will be useful for our characterization of EOwen. Property 1 (ULI). A solution rule ϕsatisfies ULI if for every (A, B, c)∈ L, every x∗∈Omax(A, B, c), and every matrix B′such that B′∈[Bx∗, B]for every Bx∗∈ B(A, B, x∗), we have that ϕ(A, B′, c)⊆ϕ(A, B, c). The following result proves that EOwen satisfies this property. Proposition 3. EOwen satisfies ULI. Proof. Let (A, B, c)∈ L. Consider x∗∈Omax(A, B, c), and let B′∈ [Bx∗, B] for every Bx∗reduced matrix associated to x∗. Similarly as we proved in Proposition 1, it can be seen that Omax(A, B′, c)⊆Omax(A, B, c). Besides, since we proved in Lemma 2 that B(A, B, x∗)⊇ B(A, B′, x∗), we 18 have EOwen(A, B, c) = Sx∗∈Omax(A,B,c)SBx∗∈B(A,B,x∗)Owen(A, Bx∗, c) ⊇Sx∗∈Omax(A,B′,c)SBx∗∈B(A,B,x∗)Owen(A, Bx∗, c) ⊇Sx∗∈Omax(A,B′,c)SBx∗∈B(A,B′,x∗)Owen(A, Bx∗, c) =EOwen(A, B′, c).  Now we are ready to give a characterization of the EOwen solution rule for linear production processes. Theorem 2. Let ϕbe a solution rule over L.ϕsatisfies ULI, coincides with the Owen set over LP processes without leftovers and is minimal if and only if ϕ≡EOwen. Proof. •Clearly EOwen coincides with the Owen set in LP processes without leftovers, and as proven in Proposition 3, EOwen satisfies ULI. Let us see that EOwen is minimal. For this purpose, let ϕbe a solution set that coincides with the Owen set in LP processes without leftovers and satisfies ULI, and let (A, B, c)∈ L. Therefore, Owen(A, Bx∗, c) = ϕ(A, Bx∗, c)⊆ϕ(A, B, c) for all x∗∈Omax(A, B, c) and all Bx∗∈ B(A, B, x∗). Hence EOwen(A, B, c) = Sx∗∈Omax(A,B,c)SBx∗∈B(A,B,x∗)Owen(A, Bx∗, c) =Sx∗∈Omax(A,B,c)SBx∗∈B(A,B,x∗)ϕ(A, Bx∗, c) ⊆ϕ(A, B, c) which proves that EOwen is minimal. 19 •Let ϕbe a solution rule over Lsatisfying the hypotheses of the theorem. Therefore ϕ(A, Bx∗, c) = Owen(A, Bx∗, c)∀x∗∈Omax(A, B, c) and for every Bx∗∈ B(A, B, x∗), because (A, Bx∗, c) has no leftovers. Now, since ϕsatisfies ULI we have that ϕ(A, Bx∗, c)⊆ϕ(A, B, c), therefore EOwen(A, B, c) = [ x∗∈Omax(A,B,c)[ Bx∗∈B(A,B,x∗) ϕ(A, Bx∗, c)⊆ϕ(A, B, c). From the minimality of ϕ, and since EOwen satisfies the hypotheses of the theorem, ϕ(A, B, c)⊆EOwen(A, B, c). Thus, EOwen(A, B, c) = ϕ(A, B, c).  Let us now introduce the following increasing monotonicity (IM) property that will lead us to another characterization of EOwen set: Property 2 (IM). A solution rule ϕsatisfies IM if for every (A, B, c)∈ L, every x∗∈Omax(A, B, c), and every B1, B2such that B1∈[Bx∗, B]for each Bx∗∈ B(A, B, x∗)and B2∈[B1, B], we have that ϕ(A, B1, c)⊆ϕ(A, B2, c). The following result states that increasing monotonicity is equivalent to upper limit inclusion. Lemma 4. Let ϕbe a solution rule over L. Then ϕsatisfies ULI if and only if ϕsatisfies IM. Proof. Trivially, if ϕsatisfies IM then it satisfies ULI too. Conversely, let (A, B, c)∈ L, and x∗∈Omax(A, B, c). Moreover, let B1∈[Bx∗, B] for each Bx∗∈ B(A, B, x∗) and B2∈[B1, B]. It is straightforward that 20 x∗∈Omax(A, B2, c). Then, since B1∈[Bx∗, B2] and ϕsatisfies IM, we have that ϕ(A, B1, c)⊆ϕ(A, B2, c), which concludes the proof.  From the equivalency between increasing monotonicity and upper limit inclusion, the following alternative characterization of EOwen trivially follows from Theorem 2. Corollary 2. Let ϕbe a solution rule over L.ϕsatisfies IM, coincides with the Owen set over LP processes without leftovers and is minimal if and only if ϕ≡EOwen. We finish this section by studying the relation between EOwen, the Owen set, and the core of the original game. It is obvious that Core(A, B, c)⊆ Core(A, Bx∗, c) for every optimal production plan x∗and every reduced matrix Bx∗associated to it, since vBx∗ (S)≤v(S) and vBx∗ (N) = v(N), where vBx∗ denotes the characteristic function of the corresponding reduced game. The following example shows that this relation may be strict. Example 3. Consider the LP process (A, B, c)where A=      1 0 1 1 0 1     , B =      1 0 1 3 1 0     , c = 2 1  . It can be seen that Omax(A, B, c) = {x∗= (1,1)}, and the characteristic function of the associated game is v({1}) = 2, v({2}) = 0, v({1,2}) = 3. There is a surplus of resource 2. One reduced matrix (in which both players 21 drop half of the units of resource 2 they had) is Bx∗=      1 0 0.5 1.5 1 0     , and vx∗({1}) = 1, vx∗({2}) = 0, v({1,2}) = 3. Therefore, Core(A, B, c) Core(A, Bx∗, c). Unfortunately, not all EOwen allocations are core allocations. However, it is not always possible to find allocations that avoid the unfairness drawback of the Owen set mentioned in this paper and remain in the core of the original game at the same time. Therefore, one has to look for some compromise between null-payoff to absolutely necessary players and un-stability. The following example illustrates the above statements. Example 4. Consider the LP process (A, B, c)with the following data, A=      1 0 1 1 0 1     , B =      100 010 001     , c = 1 1  . The characteristic function of the associated game is v({1,2}) = v({2,3}) = v({1,2,3}) = 1, and zero for any other coalition. Therefore, the core of this game consists of the singleton {(0,1,0)}. Note as well that without the resources of players 1 and 3, player 2 gets nothing, but it is not possible to give a positive payoff to players 1 and 3 with a core allocations. Let us calculate the EOwen set for this example. One can see that the extreme optimal production plans are x1= (1,0) and x2= (0,1), and that 22 B(A, B, x1)and B(A, B, x2)consist only of one matrix each (named B1and B2, respectively). B1has a diagonal equal to (1,1,0) and B2has a diagonal equal to (0,1,1). All non-diagonal entries are null for both matrices. It is easy to see that Omin(A, B1, c)is the union over t1≥1and t2≥0of the convex hulls of {(1,0, t1),(0,1, t2)}, and that Omin(A, B2, c)is the union over t1≥1and t2≥0of the convex hulls of {(t1,0,1),(t2,1,0)}. Therefore, Owen(A, B1, c)is the convex hull of {(1,0,0),(0,1,0)}and Owen(A, B2, c) is the convex hull of {(0,0,1),(0,1,0)}. For every non-extreme optimal solution xa= (a, 1−a) (a∈(0,1)),B(A, B, xa)consists only of one matrix Ba, in which the diagonal is (a, 1,1−a), and the rest is zero. The corresponding Omin(A, Ba, c)is the convex hull of {(0,1,0),(1,0,1)}, and therefore the Owen set of the corresponding reduced LP process is {(0,1,0),(a, 0,1−a)}. Therefore, EOwen(A, B, c)is the convex hull of {(1,0,0),(0,1,0),(0,0,1)} (all players can obtain a positive payoff from allocations in this set). To summarize, the Owen set, the EOwen set and the core of the original game have a relationship as shown in Figure 1. &% '$ &% '$ Owen EOwen Core(A, B, c) Figure 1: General relation between EOwen,Core and Owen . 23 Conclusions In this work we have introduced the EOwen set, a new solution rule on the class of linear production processes which overcomes certain drawbacks of the well-known Owen set, in the sense that one can always find an allocation that gives a strictly positive payoff to players whose resources are needed for (at least) one optimal production plan. Some examples in the paper show that Owen allocations do not satisfy this property. EOwen is defined as the union, over all possible optimal production plans and all possible reduced matrices, of the Owen sets over the corresponding reduced LP processes, in which players get rid of the leftovers in their resources according to a reduced matrix. Several theoretical properties and an axiomatic characterization of this new solution rule are given. By means of an example we also prove that, in general, it is not possible to find allocations that give non-null payoffs to players that are necessary in order to achieve the optimal value of v(N) by restricting to the core. We note that two types of players have been involved in this paper: group T1, consisting of players such that some of the resources they own are needed for (at least) one optimal production plan; and group T2, consisting of players whose resources are never completely used in any of the optimal production plans. Theorem 1 ensures that players in T1 can always find an allocation in EOwen that gives them a strictly positive payoff. Examples have shown that players in T2 may receive only zero payoffs from Owen allocations. Players in both T1 and T2 are of special interest, since they receive strictly positive payoffs from EOwen allocations, and may only receive zero payoffs from Owen allocations (see for instance examples 1 and 2). 24