The Subcoalition Perfect Core of Cooperative Games

Transcription

1 The Subcoalition Perfect Core of Cooperative Games J. Drechsel and A. Kimms 1 August 2008 Address of correspondence: Julia Drechsel and Prof. Dr. Alf Kimms Lehrstuhl für Logistik und Verkehrsbetriebslehre Mercator School of Management University of Duisburg Essen Lotharstr Duisburg Germany julia.drechsel@uni due.de alf.kimms@uni due.de URL: due.de/log/ 1 Corresponding Author

2 The Subcoalition Perfect Core of Cooperative Games Abstract The core is a set valued solution concept for cooperative games. In situations where the characteristic function is not monotone the classical definition may not be sufficient. Hence, we propose a subset of the core that is called subcoalition perfect core. It will be proven that the subcoalition perfect core coincides with the set of non negative core allocations. Furthermore, a general mathematical programming algorithm is provided which may be applied in many applications to compute an element in the subcoalition perfect core. In addition, we discuss an application where the characteristic function is not monotone and perform a computational study. Keywords: cooperative game theory, core, mathematical programming, lot sizing JEL classification: C63, C71, D24, M11 1 Definition of the Subcoalition Perfect Core 1.1 Review of the Core A cooperative game (with transferable utilities) is defined by a pair (N, c) where N is an index set of players and c : 2 N IR is a characteristic function which assigns to every coalition S 2 N a value c(s) (with c( ) = 0). Let us assume that c is a cost function ( the lower the better ), but in settings where c is a benefit function the following material can straightforwardly be adapted. A solution of the cooperative game (with transferable utilities) is a distribution of the cost shares, i.e. a cost allocation π i IR for every player i N. One of the most prominent solution concepts for cooperative games is the core (Gillies, 1959) which is the set of cost allocations (vectors of size N ) which are efficient and rational (stable). Mathematically, the core of a game (N, c) is defined to be: C(N, c) = {π IR N i N π i = c(n) and i S π i c(s) for all S N, S }. If c is subadditive, i.e. c(s 1 )+c(s 2 ) c(s 1 S 2 ) for all S 1, S 2 N with S 1 S 2 =, there is an incentive for the players to form the grand coalition which has to be seen with care as we will point out below. The core defining inequality guarantees that no coalition S has to carry a higher cost in the grand coalition than in a standalone situation (stability condition). Of particular importance for us is the following Lemma 1: Let (N, c) be a cooperative game. If c is monotone, i.e. c(s) S Ŝ N, all cost assignments in the core are non negative. c(ŝ) for all Proof: π i = c(n) j N\{i} π j c(n) c(n\{i}) 0 for all i N. Note that subadditivity of c does not imply monotonicity of c. 1

3 1.2 Motivation for a Core Variant Assume that c is not monotone (but it may be subadditive). In such a case there exists a coalition S and a supercoalition Ŝ S (Ŝ N) such that c(ŝ) < c(s). For Ŝ N this means that the grand coalition and an arbitrary core cost allocation π C(N, c) may not be the best choice for S (even if c is subadditive). S may prefer the game (Ŝ, c) over the game (N, c). In other words, S may try to push Ŝ\S to form a coalition Ŝ instead of the grand coalition. This can be interpreted as a force of anarchy in the game and the cost allocation π C(N, c) may not be considered as stable any longer if c(ŝ) < i S π i c(s) is true. Such a force of anarchy may be unwanted and core cost allocations may be considered as feasible only if no coalition S has to carry a higher cost in the grand coalition than in any other supercoalition Ŝ of S. Mathematically, this can be defined as follows: C SP (N, c) = {π IR N i N π i = c(n) and i S (1) π i c(ŝ) for all S N, S and S Ŝ N}. An equivalent formulation is C SP (N, c) = {π IR N i N π i = c(n) and i S π i min c(ŝ) for all S N, S }. S Ŝ N We call C SP (N, c) the subcoalition perfect core. It is easy to see that C SP (N, c) C(N, c) holds in general, and that C SP (N, c) = C(N, c) is true if c is monotone. 1.3 Proof of a Characteristic Property We will now prove that the subcoalition perfect core equals the set of non negative core allocations C + (N, c) = {π C(N, c) π i 0 for all i N}. Theorem 1: Let (N, c) be a cooperative game. The subcoalition perfect core coincides with the set of non negative core allocations, i.e. C + (N, c) = C SP (N, c). Proof: If c is monotone the proof follows immediately from Lemma 1 and the fact that c(s) = min S Ŝ N c(ŝ) for all coalitions S and so we have C+ (N, c) = C(N, c) = C SP (N, c) if c is monotone. What remains to show is the case where c is not monotone. C + (N, c) C SP (N, c) : Let the set of non negative core allocations be non empty, because the proof is trivial otherwise. Assume there exists a non negative core allocation π C + (N, c) which is not in the subcoalition perfect core, i.e. two coalitions S Ŝ N exist with c(ŝ) < i S π i c(s). The chosen core allocation π fulfills π i + π i = π i c(ŝ) < π i. i S i Ŝ\S i Ŝ i S 2

4 Thus, π i < 0 for at least one player i Ŝ\S. Mathematically, this finding can be expressed by the following implication: S N : S Ŝ N : i S π i > c(ŝ) i N : π i < 0 This is a contradiction to the assumption π C + (N, c) and by contraposition we get i N : π i 0 S N : S Ŝ N : i S π i c(ŝ). Hence, π C SP (N, c) must hold. C + (N, c) C SP (N, c) : Let the set of subcoalition perfect core allocations be non empty, because the proof is trivial otherwise. Assume there exists a subcoalition perfect core allocation π C SP (N, c) with π i < 0 for at least one player i N, i.e. π C + (N, c). The chosen cost allocation is efficient and so we have c(n) = j N π j. Because of π i < 0 we get j N π j < j N\{i} π j. But π is an element from the subcoalition perfect core and therefore j N\{i} π j c(n) must hold which leads to the conclusion that c(n) < c(n). Hence, π C + (N, c) must be true. In summary we have C + (N, c) = C SP (N, c). The remaining text is organized as follows: Section 2 describes a general procedure to compute a subcoalition perfect core cost allocation which can be applied in many situations where the characteristic function is defined by an optimization problem. In Section 3 an application is described where the characteristic function of the cooperative game is not monotone. The details of the procedure from Section 2 will also be revealed to show how the algorithm can be applied for a particular problem. The results of a computational study are provided in Section 4 and concluding remarks in Section 5 finish the paper. The paper makes the following contributions: It contributes to the theory of cooperative games by introducing a new solution concept, the subcoalition perfect core. It contributes to practical problem solving by providing a rather general algorithm to compute a subcoalition perfect core allocation. 2 Computing a Subcoalition Perfect Core Cost Allocation Drechsel and Kimms (2007) have proposed a mathematical programming procedure which can be used in very general settings to compute a core allocation for cooperative games. If the core is empty, the procedure detects emptiness. This procedure can also be used in our context with a slight modification so that a subcoalition perfect core allocation is determined. To be self contained, we briefly repeat the algorithm here. The definition of the (subcoalition perfect) core specifies a constraint satisfaction problem where the 3

5 number of constraints is in the order of 2 N. Hence, Drechsel and Kimms (2007) suggested to run a row generation procedure. The master problem is a linear program of the following form where S is a (small) subset of coalitions for which the (subcoalition perfect) core defining inequality is explicitly taken into account. Master problem MP (S): min v subject to π i = c(n) i N π i v c(s) i S π i 0 v 0 S S i N It should be emphasized that without Theorem 1 the master problem would not have the non negativity condition for the cost allocation. The iterative row generation procedure can be outlined as follows: 1. Define a small initial set S, e.g. S = {{1},..., { N }}. 2. Solve the linear program MP (S) optimally. 3. If v > 0 then stop. The game instance has an empty (subcoalition perfect) core. 4. Otherwise, find a coalition S S (S ) such that i S π i > c(s ). An optimization subproblem SP (π) which searches for a coalition S that violates the (subcoalition perfect) core defining inequality most ( i S π i c(s ) is maximized) can, for instance, be defined. 5. If no such coalition S can be found (SP (π) has a non positive optimum objective function value) then stop. The current values π i define a (subcoalition perfect) core allocation. 6. Otherwise, update S = S {S } and return to Step 2. Step 4 of the algorithm is the only problem specific part in this approach. In Section 3 we will provide the details of the subproblem to be solved for a specific application where the characteristic function is not monotone. 3 An Application: Cooperative Lot Sizing with Capacity Constraints and Transshipment Opportunities Roughly speaking, lot sizing is a problem which deals with the question of the right quantity when placing an order or producing inhouse. Several ordering/lot sizing games 4

6 have been discussed in the literature before and they can be classified by the underlying ordering/lot sizing problem: The economic order quantity (EOQ) model, see Meca et al. (2004) and Dror and Hartman (2007), the economic production quantity (EPQ) model, see Meca et al. (2003), Meca (2007), and Meca et al. (2007), the Wagner Whitin model (some authors refer to Wagner Whitin based games as setup inventory (SI) games and to the special case where setup costs are zero as production inventory (PI) games), see Chen and Zhang (2006a), Drechsel and Kimms (2007), Guardiola et al. (2006), Guardiola et al. (2007), and van den Heuvel et al. (2007), the capacitated lot sizing problem, see Drechsel and Kimms (2008), and the newsvendor model, see Chen and Zhang (2006b), Dror et al. (2007), Hartmann et al. (2000), Hartman and Dror (2005), Müller et al. (2002), Özen et al. (2008), and Slikker et al. (2001). We refer to Meca and Timmer (2008) for a recent and very comprehensive survey on ordering/lot sizing models with multiple players. 3.1 The Cooperative CLSP Game A scenario where a cooperative game occurs and the characteristic function is not monotone is described by Drechsel and Kimms (2008). They consider a cooperative lot sizing problem where players may produce items in order to meet some known (or estimated) demand. The problem considered by Drechsel and Kimms (2008) is an extension of the well known capacited lot sizing problem (CLSP, see e.g. Billington et al., 1983). Due to scarce capacities it may not be possible for some players to meet their own demand and so they need to buy items from others. Another incentive to buy from other players may stem from lower costs. So, players may ship items among each other which costs money. The classical CLSP is an N P hard optimization problem where one single decision maker i has to determine a production plan for T periods of time. Several items are to be produced which share a common resource with availability R it. Let K be the index set of the items under consideration and d ikt be the dynamic demand of item k in period t that is faced by the decision maker i. Let b ikt be the decision maker s production coefficient for item k in period t, i.e. the number of resource units required to produce one unit of product k by i in t. Given these parameters, the decision variables q ikt can be used to represent the production plan which is to be computed where q ikt is the quantity of item k that is produced by i in period t. Let the parameter c P ikt be the unit production cost coefficient. The objective is to find a production plan with minimum costs where a tradeoff exists between setup costs (a fixed setup cost c S ikt is incurred whenever i produces a positive quantity of product k in period t and a binary decision variable x ikt can be used to indicate whether or not a setup takes place) and holding costs (items can be produced and kept on stock until needed for each unit of k on stock at the end of period t a holding cost c H ikt is charged and a decision variable I ikt can be used to represent 5

7 the number of items on stock I ik0 is a parameter and stands for the initial inventory of player i regarding item k). The focus of our attention is a cooperative CLSP setting. To the best of our knowledge there is only one single paper on this problem by Sambasivan and Yahya (2005) who were motivated by a real world problem in a large U.S. company manufacturing steel rolled products, but these authors do not discuss the problem of allocating costs which is our concern. The idea behind a cooperative CLSP is that several players work together in the sense that players cannot only produce to meet own demand, but also to fulfill other players demand. Let N be the index set of all players under consideration. Such a situation seems to be senseful, if the capacity of some players is insufficient to fulfill their demand such that other players can help out. It seems to be senseful as well, if some players have lower costs than others. This setting however means that items must be transported among the players. Let c T ijkt be the per unit transportation cost for shipping product k from player i to player j in period t and a ijkt the transported quantity. If a set S N of players forms a coalition the cooperative optimization problem is the following: c(s) = min ( T c S iktx ikt + c P iktq ikt + c H ikti ikt + ) c T ijkta ijkt (2) i S k K t=1 j S s.t. I ikt = I ik,t 1 + q ikt + a jikt d ikt a ijkt i S; k K; t = 1,..., T (3) j S j S b ikt q ikt R it i S; t = 1,..., T (4) k K q ikt M kt x ikt i S; k K; t = 1,..., T (5) q ikt, I ikt 0 i S; k K; t = 1,..., T (6) a ijkt 0 i, j S; k K; t = 1,..., T (7) x ikt {0, 1} i S; k K; t = 1,..., T (8) The objective (2) is minimizing total costs. Inventory balance constraints (3) take into account that transshipments to or from a player see the decision variables defined in (7) can take place. The underlying assumption is that such transshipments happen within a cooperation only. (4) are the capacity constraints. Due to (5) production cannot be done without a setup. M kt = i S T τ=t d ikτ is a sufficiently large number. It should be noted that extensions like setup times, backlogging, overtime, limited stocking capacity, multi level product structures, limited transportation capacity or fixed transportation costs can easily be added to the model. Drechsel and Kimms (2008) present heuristics to solve the optimization problem (2) (8). They also discuss the problem of finding core cost allocations which may be considered as fair the minmax core is introduced in their work. The issue of subcoalition perfect core cost allocations has not been treated. Note that c( ) = 0 by construction and c(s) 0. To be well defined let us assume c(s) =, if the optimization problem has no feasible solution for the coalition S. The characteristic function c may not be monotone, because adding players to a coalition may indeed decrease total costs of the enlarged coalition, if the additional players can offer additional capacity for a low cost, for instance. To prove the non monotonicity of c consider a very simple example with two players ( N = 2), one product ( K = 1), 6

8 one period (T = 1), a very simple demand pattern d 111 = 1, d 211 = 0, and a very simple capacity pattern R 11 = R 21 = 1 with b 111 = b 211 = 1. Assume that player 2 has lower setup cost coefficients than player 1, i.e. c S 211 < c S 111. For the sake of simplicity assume c P 211 = c P 111 = 0. Furthermore assume c T 2111 = 0. All other parameters may have arbitrary non negative values. The interpretation is that player 2 owns unused resources which can cheaply be provided to other player 1. In this example, we have c({1}) = c S 111 (and c({2}) = 0). For the grand coalition we get c({1, 2}) = c S 211 < c S 111 = c({1}) which proves the non monotonicity of c in this example. 3.2 The Problem Specific Subproblem If we want to apply the procedure described in Section 2 to the cooperative CLSP game, we need to specify the details of Step 4, the subproblem, of that procedure. The following mixed integer programming formulation defines the problem to solve. In our implementation we have used standard software (CPLEX) to solve the subproblem optimally. Subproblem SP (π): max o + i N π i z i (9) subject to o = i N k K t=1 ( T c S iktx ikt + c P iktq ikt + c H ikti ikt + ) c T ijkta ijkt j N I ikt = I ik,t 1 + q ikt + a jikt d ikt z i a ijkt i N; k K; t = 1,..., T (11) j N j N b ikt q ikt R it i N; t = 1,..., T (12) k K q ikt M kt x ikt i N; k K; t = 1,..., T (13) x ikt z i i N; k K; t = 1,..., T (14) a ijkt M kt z i i, j N; k K; t = 1,..., T (15) q ikt, I ikt 0 i N; k K; t = 1,..., T (16) a ijkt 0 i, j N; k K; t = 1,..., T (17) x ikt {0, 1} i N; k K; t = 1,..., T (18) o 0 (19) z i {0, 1} i N (20) The decision variable o (19) is defined to be the value c(s ) for a certain coalition S due to (10). The decision variable z i see (20) indicates which players from N are in that coalition S where z i = 1 means i S (z i = 0 stands for i S ). The constraints (11), (12), and (13) and the decision variables (16), (17), and (18) stem from the CLSP model formulation (2) (8). To make sure that only those players who are selected by the z i variables affect the solution, we have (11), (14), and (15). If the optimum objective function value of this subproblem is positive then the (subcoalition perfect) core defining (10) 7

9 constraint for the coalition S will be added to the master problem. Otherwise, the whole procedure will terminate and π is a (subcoalition perfect) core allocation. Once more it should be emphasized that Theorem 1 is very important here. Without Theorem 1 we would have no non negativity condition for the π i values in the master problem formulation M P (S) (which is no problem). But the more severe consequence is that the subproblem SP (π) would require a different model formulation: max o + i N π i z i (21) subject to (10), (12), (13), (16), (17), (18), (19), and I ikt = I ik,t 1 + q ikt + a jikt d ikt z i a ijkt i N; k K; t = 1,..., T (22) j N j N x ikt z i i N; k K; t = 1,..., T (23) a ijkt M kt z i i, j N; k K; t = 1,..., T (24) z i z i i N (25) z i {0, 1} i N (26) z i {0, 1} i N (27) Two sets of binary decision variables would be needed. Having the definition (1) of the subcoalition perfect core in mind, the set of variables z i identifies the coalition S and the set of variables z i identifies the coalition Ŝ in the subcoalition perfect core defining constraint i S π i c(ŝ). S Ŝ is assured via (25). This problem formulation requires more run time to be solved than (9) (20) and therefore Theorem 1 is not only a theoretical insight, but it helps to reduce the run time effort for computing a (subcoalition perfect) core allocation. 4 Computational Study We have implemented the proposed procedure using the software package AMPL/CPLEX running on hardware equipped with an AMD Athlon 64X2 Dual Core Processor with 2.41 GHz and 1.96 GB RAM. We have used random instances of the cooperative CLSP game where integral random numbers with uniform distribution were drawn from the following intervals: c S ikt [0; 200], cp ikt [0; 15], ch ikt [0; 10], ct ijkt [0; 15], b ikt = t {1,...,t} b ikt d ikt t {1,...,t 1} R it }; k K b iktd max ]. b ik [1; 5], and R it [max{0; k K The (integral) demand was randomly chosen as d ikt [0; d max ] with d max = 20, but for one player, say player N, we defined d N kt = 1 which made the occurrence of a non monotone characteristic function very probable (all our random instances with 10 or more players turned out to be non monotone and most of the instances with a smaller number of players are non monotone as well). In all of our tests we used T = 6. The number of players N was systematically varied, N {3, 5, 10, 15}, in combination with K = 3 products. For N = 3 we additionally tested K = 1. For each parameter combination 15 random instances were generated. Results in terms of the average number of iterations can be seen in Table 1. The average run time performance measured in CPU seconds is provided in Table 2. As we can see in Table 3, the time requirement per iteration is larger 8

10 if we use the subproblem formulation (21) (27) instead of (9) (20). On the other hand, the use of (21) (27) leads to a lower average number of iterations. N / K 3/1 3/3 5/3 10/3 15/3 (9) (20) (21) (27) Table 1: Average Number of Iterations for Cooperative CLSP Game Instances N / K 3/1 3/3 5/3 10/3 15/3 (9) (20) (21) (27) Table 2: Average Run Time (CPU Seconds) for Cooperative CLSP Game Instances N / K 3/1 3/3 5/3 10/3 15/3 (9) (20) (21) (27) Table 3: Average Ratio of Run Time (CPU-Seconds) and Number of Iterations for Cooperative CLSP Game Instances 5 Conclusions In this paper we have discussed cooperative games where the characteristic function is not monotone. We revealed that in such settings a force of anarchy exists which means that a coalition may exist which has an incentive not to work in the grand coalition if the core is used as a solution concept. As an alternative we have introduced the subcoalition perfect core which is a subset of the core. If we use the subcoalition perfect core as a solution concept the force of anarchy phenomenon vanishes and a selected cost allocation can be considered as more stable than an arbitrary core allocation. In addition to that we have proven that the subcoalition perfect core coincides with the set of non negative core allocations. To make the subcoalition perfect core of practical use as a new solution concept, we have also discussed a very general mathematical programming approach to compute a subcoalition perfect core allocation. The approach is general, because only a subproblem of the approach is problem specific and that subproblem indeed can be an arbitrary optimization problem. This algorithm is only a slight modification of a procedure proposed by Drechsel and Kimms (2007) to compute a core allocation. It became possible to use this algorithm, because we have proven in this paper that the subcoalition perfect core equals the set of non negative core allocations. Without this proof, the subproblem in the Drechsel and Kimms (2007) procedure requires to be changed to make the subproblem 9

11 much more harder to be solved which limits the applicability. Fortunately, our findings helped to avoid this additional complexity. To demonstrate the applicability of this subcoalition perfect core routine, we have used a cooperative capacitated lot sizing game which was introduced by Drechsel and Kimms (2008). The characteristic function of this game is not monotone in general which makes it a good candidate for the proposed approach. A computational study showed that games with up to 15 players can be solved in less than two minutes of CPU time. Given that finding a core allocation is an N P hard problem and that for 15 players the number of (subcoalition perfect) core defining inequalities to be taken into account is in the order of 2 15, we believe that this paper makes a contribution to theory by introducing a new solution concept as well as a contribution to practical problem solving by providing a mathematical programming algorithm. References [1] Billington, P.J., McCLain, J.O., Thomas, L.J., (1983), Mathematical Programming Approaches to Capacity Constrained MRP Systems: Review, Formulation and Problem Reduction, Management Science, Vol. 29, [2] Chen, X., Zhang, J., (2006a), Duality Approaches to Economic Lot Sizing Games, Working Paper, University of Illinois at Urbana Champaign [3] Chen, X., Zhang, J., (2006b), A Stochastic Programming Duality Approach to Inventory Centralization Games, Working Paper, University of Illinois at Urbana Champaign [4] Drechsel, J., Kimms, A., (2007), Computing Core Allocations in Cooperative Games with an Application to Procurement Problems, Working Paper, University of Duisburg Essen [5] Drechsel, J., Kimms, A., (2008), Solutions and Fair Cost Allocations for Cooperative Lot Sizing with Transshipments and Scarce Capacities, Working Paper, University of Duisburg Essen, in preparation [6] Dror, M., Guardiola, L.A., Meca, A., Puerto, J., (2007), Dynamic Realization Games in Newsvendor Inventory Centralization, International Journal of Game Theory, to appear [7] Dror, M., Hartman, B.C., (2007), Shipment Consolidation: Who Pays for It and How Much?, Management Science, Vol. 53, pp [8] Gillies, D.B., (1959), Solutions to General Non Zero Sum Games, in: Tucker, A.W., Luce, R.D., (eds.), Contributions to the Theory of Games IV, Princeton, Princeton University Press, pp [9] Guardiola, L.A., Meca, A., Puerto, J., (2006), Coordination in Periodic Review Inventory Situations, Working Paper, Universidad Miguel Hernández de Elche 10

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