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On the core of cost-revenue games: Minimum cost spanning tree games with revenues
European Journal of Operational Research 237 (2014) 606–616
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On the core of cost-revenue games: Minimum cost spanning tree games with revenues q Arantza Estévez-Fernández a,⇑, Hans Reijnierse b a b
Tinbergen Institute and Department of Econometrics & OR, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands CentER and Department of Econometrics & OR, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
a r t i c l e
i n f o
Article history: Received 21 November 2012 Accepted 24 January 2014 Available online 4 February 2014 Keywords: Game theory Cost-revenue allocation problem Cooperative game Core Minimum cost spanning tree problem
a b s t r a c t In this paper, we analyze cost sharing problems arising from a general service by explicitly taking into account the generated revenues. To this cost-revenue sharing problem, we associate a cooperative game with transferable utility, called cost-revenue game. By considering cooperation among the agents using the general service, the value of a coalition is defined as the maximum net revenues that the coalition may obtain by means of cooperation. As a result, a coalition may profit from not allowing all its members to get the service that generates the revenues. We focus on the study of the core of cost-revenue games. Under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members, it is shown that a cost-revenue game has a nonempty core for any vector of revenues if, and only if, the dual game of the cost game has a large core. Using this result, we investigate minimum cost spanning tree games with revenues. We show that if every connection cost can take only two values (low or high cost), then, the corresponding minimum cost spanning tree game with revenues has a nonempty core. Furthermore, we provide an example of a minimum cost spanning tree game with revenues with an empty core where every connection cost can take only one of three values (low, medium, or high cost). Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we study cost sharing problems arising from a general service by explicitly taking into account the revenues generated by this service. In addition, we apply our findings to minimum cost spanning tree (mcst) problems with revenues. The study of cost sharing problems combined with revenues has been considered in the literature before. Littlechild and Owen (1976) initiated this stream of literature within the framework of airport problems, where they provide an algorithm to compute the nucleolus of airport games. Brânzei, Iñarra, Tijs, and Zarzuelo (2006) is a recent follow-up to Littlechild and Owen (1976). Suijs, Borm, Hamers, Quant, and Koster (2005) analyze the allocation of costs and revenues within a public network communication structure within the context of cooperative games with transferable utility. Meertens and Potters (2006) provide an algorithm to
q The authors thank two referees for their valuable suggestions for improvement. Special thanks go to Peter Borm and Ruud Hendrickx for their helpful comments in a previous version of this paper. ⇑ Corresponding author. Tel.: +31 20 5982294; fax: +31 20 5986020. E-mail addresses: [email protected] (A. Estévez-Fernández), [email protected] (H. Reijnierse).
http://dx.doi.org/10.1016/j.ejor.2014.01.056 0377-2217/Ó 2014 Elsevier B.V. All rights reserved.
compute the nucleolus of fixed tree games with revenues. Estévez-Fernández, Borm, Meertens, and Reijnierse (2009) focus on the analysis of the core of routing games with revenues. Here, we consider the situation in which a group of agents may cooperate to reduce the (possibly nonlinear) costs associated with the performance of a service. Each agent has some fixed revenue from the performance of the service. The question is how to share the net revenues obtained from cooperation among the agents. To solve this allocation problem, we use cooperative game theoretic techniques. By considering cooperation among the agents, we define a cost-revenue game where the value of a coalition is defined as the maximum net revenues that the coalition can obtain by means of cooperation. Note that with this definition, a coalition may profit from not allowing all its members to get the service that generates the revenues. In order to provide an allocation of the cost-revenue sharing problem at hand, we focus on the core of the cost-revenue game. The core of a game is the set of allocations of the total net revenues obtained by the cooperation of all agents that are stable in the sense that no group of agents has an incentive to deviate. In this paper, we characterize the class of cost-revenue games with a nonempty core under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members. It turns out that
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a cost-revenue game has a nonempty core for any vector of revenues that guarantees full service to all members of the grand coalition if, and only if, the dual game of the corresponding cost game has a large core. We apply the general results found for cost-revenue games to the study of mcst allocation problems with revenues. In an mcst allocation problem, a group of customers needs to be connected to a common information source; the customer needs a physical connection to the source (for instance, the cable for TV transmission) and this connection may be done either directly, or indirectly through other costumers. The cheapest way to connect all customers to the source is through a spanning tree that can be constructed by means of cooperation among the customers. The mcst allocation problem is, then, how to share the connection costs among the agents. Claus and Kleitman (1973) introduced this problem, which has been extensively analyzed in the literature. Granot and Huberman (1981) define mcst games and show that mcst games have a nonempty core. Granot and Huberman (1984) define allocation rules for mcst allocation problems which lead to elements of the core of the corresponding mcst games and show that, for some instances, some of these rules lead to the nucleolus of the associated game. Granot and Maschler (1998) generalize mcst games by studying spanning network games. In a spanning network situation, costs are associated to both edges and vertices and the costs may be negative (reflecting rewards). They provide sufficient conditions for non-emptiness of the core of spanning network games and characterize the set of essential coalitions. Besides, they study the computational complexity of the core and the nucleolus of this class of games, and the decomposition of the core and the nucleolus for some special cases. Kuipers, Solymosi, and Aarts (2000) provide a method to compute the nucleolus of an mcst game in polynomial time. Norde, Moretti, and Tijs (2004) show that mcst games posses population monotonic allocation schemes and provide an algorithm to find such a scheme. Tijs, Brânzei, Moretti, and Norde (2006) analyze a class of rules for mcst problems that are cost monotonic and induce monotonic allocation schemes. For a more extensive overview on the literature of mcst games, we refer to Moretti (2008). When the revenues obtained by the customers by being connected to the source are taking into account, we face an mcst allocation problem with revenues. In this paper, we define mcst games with revenues and we provide an example of an mcst game with revenues where it is profitable that all members of the grand coalition connect to the source and whose core is empty. In this example, the connection costs between two distinct players can only take one of three values: high, medium, or low cost. We show that if the connection costs can only take one of two values (high or low cost), then, the associated mcst game with revenues has a nonempty core under the assumption that it is profitable for the grand coalition to have all its members connected to the source. The remainder of the paper is organized as follows. Section 2 provides the basic definitions and terminology of graph theory and cooperative games used in this paper. In Section 3, we analyze cost-revenue games. Section 4 analyzes mcst games with revenues. We conclude with some further remarks in Section 5.
fv l ; v lþ1 g 2 E. We denote by Vðpathðv ; wÞÞ the set of nodes of pathðv ; wÞ and by Eðpathðv ; wÞÞ the set of edges of pathðv ; wÞ. Forand Eðpathðv ; wÞÞ ¼ mally, Vðpathðv ; wÞÞ ¼ fv 0 ; v 1 ; . . . ; v p g ffv 0 ; v 1 g; . . . ; fv p1 ; v p gg. A path v 0 ; . . . ; v p is called simple if all nodes are distinct. A cycle, j, is a sequence of nodes v 0 ; v 1 ; . . . ; v p with v 0 ¼ v p ; p P 3; v 0 ; v 1 ; . . . ; v p1 distinct nodes and, for each l 2 f0; . . . ; p 1g; fv l ; v lþ1 g 2 E. We denote by VðjÞ the set of nodes of j and by EðjÞ the set of edges of j. Formally, VðjÞ ¼ fv 0 ; v 1 ; . . . ; v p1 g and EðjÞ ¼ ffv 0 ; v 1 g; . . . ; fv p1 ; v p gg. Given a graph C, we denote by CðCÞ the set of cycles of C. A subset V 0 of V is called connected if, for every v ; w 2 V 0 , there is a path from v to w using only nodes of V 0 ; we denote by CðCÞ the set of all connected subsets of V, formally, CðCÞ ¼ fV 0 # V : V 0 is connectedg. A subset U of V is called maximally connected or a component if U is connected and, for any w 2 V n U; U [ fwg is not connected. Given V 0 # V, we denote by V 0jC the set of components of V 0 with respect to C. A graph C ¼ ðV; EÞ is a forest if it has no cycles and it is a tree if it is a connected forest. A rooted tree with root v 0 2 V is a tree where node v 0 is singled out. 2.2. Cooperative games A cooperative (transferable utility) game in characteristic function form is an ordered pair ðN; v Þ where N is a finite set of players and v : 2N ! R is the characteristic function satisfying v ð;Þ ¼ 0. In general, v ðSÞ represents the joint payoff that this coalition can obtain when its members decide to cooperate and is called the value of coalition S. A cooperative game can reflect costs or rewards. A game reflecting costs is denoted by a mapping c, while a game reflecting rewards is denoted by a mapping v. The following properties and definitions apply to games reflecting rewards. The central question within a cooperative framework is how to share among the players the revenues v ðNÞ that are obtained by means of cooperation. One of the most studied solution concepts in cooperative game theory is the core of a game, first introduced in Gillies (1953). The core of a game ðN; v Þ; Coreðv Þ, is the set of efficient allocations of v ðNÞ (exactly v ðNÞ is shared among the players) that are coalitionally rational (to which no coalition can reasonably object). Formally,1
Coreðv Þ ¼ x 2 RN : xðNÞ ¼ v ðNÞ; xðSÞ P v ðSÞ for all S N : The upper core of a game ðN; v Þ; Uðv Þ, is the set of not necessarily efficient allocations that are coalitionally rational. Formally,
Uðv Þ ¼ x 2 RN : xðSÞ P v ðSÞ for all S # N : Largeness of the core was first introduced in Sharkey (1982). The core of ðN; v Þ is said to be large if, for every x 2 Uðv Þ, there is a y 2 Coreðv Þ with yi 6 xi for each i 2 N. The class of games with a large core has been characterized in Estévez-Fernández (2012) by means of minimal cover inequalities. Given a finite set N, a collection C 2N n f;; Ng is called a minimal cover of N if the following two conditions are satisfied: (i) [S2C S ¼ N; (ii) [S2CnfS0 g S – N for every S0 2 C.
2. Preliminaries 2.1. Graph theory A graph C is a tuple ðV; EÞ in which V is the finite set of nodes and E is the set of edges. A subgraph ðV 0 ; E0 Þ of C is a graph with V 0 # V and E0 ¼ ffv ; wg 2 E : v ; w 2 V 0 g; we denote CjV 0 ¼ ðV 0 ; E0 Þ. A path from node v to w; pathðv ; wÞ, is a sequence of nodes v 0 ; v 1 ; . . . ; v p with v 0 ¼ v ; v p ¼ w and, for each l 2 f0; . . . ; p 1g;
By condition (i), it follows that C covers N and by condition (ii), we have that no element of C is superfluous. We denote by MCðNÞ the set of minimal covers of N. Let PðNÞ denote the set of partitions of N. Note that PðNÞ n ffNgg # MCðNÞ. 1
Here and further, for any vector x 2 RN , we denote xðSÞ :¼
P
i2S
xi .
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Given a finite set N and C 2 MCðNÞ, for every i 2 N, we denote by nðC; iÞ the number of times that i is ‘‘extra covered’’ in C; formally, nðC; iÞ ¼ jfS 2 C : i 2 Sgj 1. We denote by EðCÞ the set of elements of N that are ‘‘extra covered’’ by C; formally, EðCÞ ¼ fi 2 N : nðC; iÞ > 0g. EðCÞ Given C 2 MCðNÞ with EðCÞ – ;, a vector l 2 R2þ nf;g is said to be a vector of minimal cover balanced weights (mc-balanced vector) for C if
X
lS e S ¼
S22EðCÞ nf;g
X
nðC; iÞefig
ð2:1Þ
i2EðCÞ
S2C
X
lS eS
ð2:2Þ
S22EðCÞ nf;g
for every l 2 MðCÞ. Note that if EðCÞ ¼ ;, then, C is a non-trivial parP tition of N and S2C eS ¼ eN . Given a game ðN; v Þ and a minimal cover C 2 MCðNÞ, the minimal cover inequality associated with C is
8 < X
X S2C
v ðSÞ 6 v ðNÞ þ lmax 2MðCÞ:
9 = lS v ðSÞ : ;
ð2:3Þ
S22EðCÞ nf;g
Theorem 2.1 Estévez-Fernández, 2012. A game ðN; v Þ has a large core if, and only if, all minimal cover inequalities are satisfied. Macula (1995) provides the number of minimal covers of a set N k Pn 1 Pak 2 k 1 m!Sðn; mÞ where with jNj ¼ n, which is k¼2 k! m¼k mk k
ak ¼ minfn; 2 1g and Sðn; mÞ is the Stirling number of the second Pm
m n kind, which is given by Sðn; mÞ ¼ l . However, l¼1 ð1Þ l in some situations, analyzing largeness of the core of a game may be easier. Let ðN; v Þ be a game, a coalition S # N is called essential if, for every non-trivial partition hS1 ; . . . ; St i of P S; v ðSÞ > tl¼1 v ðSl Þ. We denote by Essðv Þ the set of essential coalitions of ðN; v Þ. Notice that only essential coalitions are needed for the description of both the core and the upper core of a game. Therefore, following the comments in the Final Remarks of Estévez-Fernández (2012), we can rewrite Theorem 2.1 as follows: 1 m!
ml
A game ðN; v Þ has a large core if, and only if,
X S2C
LðcÞ ¼ x 2 RN : xðSÞ 6 cðSÞ for all S # N : Following the definition of largeness of the core for reward games, we say that the cost game ðN; cÞ has a large anti core if, for any x 2 LðcÞ, there is a y 2 ACoreðcÞ with xi 6 yi for each i 2 N. The dual game of ðN; cÞ; ðN; c Þ, is defined, for every S # N, by
c ðSÞ ¼ cðNÞ cðN n SÞ: It is easily seen that ACoreðcÞ ¼ Coreðc Þ.
with eR 2 RN defined by eRi ¼ 1 if i 2 R and eRi ¼ 0 if i R R for every R 2 2N n f;g. We denote by MðCÞ the set of mc-balanced vectors for C. Trivially, MðCÞ – ; if EðCÞ – ;. It follows that
X eS ¼ eN þ
The lower core of a cost game ðN; cÞ; Lðv Þ, is the set of not necessarily efficient allocations that are coalitionally rational. Formally,
8 < X
v ðSÞ 6 v ðNÞ þ lmax 2MðCÞ:
S22
EðCÞ
lS v ðSÞ
nf;g
9 = ;
for every C 2 MCðNÞ such that C # Essðv Þ. A game ðN; v Þ is said to be exact (Schmeidler, 1972) if, for every S N, there exists x 2 Coreðv Þ such that xðSÞ ¼ v ðSÞ. Given a forest C ¼ ðN; EÞ, the game ðN; v Þ is said Ptto be forest essential with respect to2 C if v ðNÞ ¼ maxhS1 ;...;St i2PðNÞ l¼1 v ðSl Þ and all essential coalitions are connected with respect to C, (Essðv Þ # CðCÞ). If C is a tree, we say that ðN; v Þ is tree-essential with respect to C. Let ðN; cÞ be a cost game. The anti core of ðN; cÞ; ACoreðcÞ, is the set of efficient allocations of cðNÞ to which no coalition can reasonably object. Formally,
ACoreðcÞ ¼ x 2 RN : xðNÞ ¼ cðNÞ; xðSÞ 6 cðSÞ for all S N :
3. Cost-revenue games In this section, we analyze situations in which a group of agents cooperate in order to obtain a higher joint revenue. Each time a coalition decides to cooperate, an additive reward is obtained and a (possibly nonadditive) cost is generated. The problem at hand is how to share the net revenue that is obtained by the grand coalition among the agents. We define cost-revenue games and concentrate on the study of their cores. It turns out that a costrevenue game has a nonempty core for any vector of revenues that provides maximal cooperation if, and only if, the dual game of the cost game has a large core. Let ðN; cÞ be a cost game. Consider the case in which, next to the costs that a coalition must confront, the coalition also obtains revenues when it decides to cooperate. Let b 2 RNþ be the vector of revenues, where bi is the revenue that player i generates if i gets the service under consideration. Then, the total revenue that a coalition S # N can obtain by cooperation is
pb ðSÞ ¼ bðSÞ cðSÞ: Notice that, due to the revenue structure of the game, it may be more profitable for coalition S not to form as a whole. Given a cost game ðN; cÞ and a vector of revenues b 2 RNþ , we define the costrevenue game, ðN; v b Þ, for every S # N, by
v b ðSÞ ¼ max fpb ðRÞg: R#S The following example illustrates the computation of costrevenues games and shows that full participation of the grand coalition (v b ðNÞ ¼ pb ðNÞ) is relevant in order to have non-emptiness of the core of the corresponding cost-revenue game. In order to exclude situations like this example we will introduce later on Assumption 3.1. Example 3.1. Consider the six players cost game ðN; cÞ with characteristic function defined by if jSj ¼ 1; cðSÞ ¼
if jSj ¼ 2;
if jSj ¼ 3;
if jSj ¼ 4; if jSj ¼ 5;
2 if S f1;2;3g;
1 8 > <4 cðSÞ ¼ 3 > : 2 8 > <6 cðSÞ ¼ 4 > : 3 5 cðSÞ ¼ 4 cðSÞ ¼ 5;
otherwise; if S f1;2;3g; if S 2 ff1;6g;f2;5g;f3;4gg; otherwise; if S ¼ f1;2;3g; if S 2 ff1;2;5g;f1;2;6g;f1;3;4g;f1;3;6g;f2;3;4g;f2;3;5gg; otherwise; if S 2 ff1;2;3;4g;f1;2;3;5g;f1;2;3;6gg; otherwise;
if S ¼ N; cðSÞ ¼ 6:
Let b ¼ ð2; 2; 2; 0; 0; 0Þ. The corresponding cost-revenue-game is given by 2 In fact, a forest-essential game with respect to C is a game with the total dependency property (see Kuipers et al. (2000)) and where the grand coalition is stable (see Derks & Kuipers (1997)).
v b ðSÞ ¼
1 if f1; 2; 4g # S; or f1; 3; 5g # S; or f2; 3; 6g # S; 0
otherwise:
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A. Estévez-Fernández, H. Reijnierse / European Journal of Operational Research 237 (2014) 606–616 Table 1 Coalitional values of the cost game, dual game, additive revenues, and cost-revenue game in Example 3.2. S
f1g
f2g
f3g
f4g
f5g
f1; 2g
f1; 3g
f1; 4g
f1; 5g
f2; 3g
f2; 4g
f2; 5g
f3; 4g
f3; 5g
f4; 5g
cðSÞ c ðSÞ bðSÞ v b ðSÞ
1 0 1 0
1 0 1 0
1 0 1 0
2 0 1 0
1 0 0 0
2 0 2 0
1 1 2 1
2 1 2 0
2 0 1 0
1 1 2 1
2 1 2 0
2 0 1 0
3 0 2 0
1 1 1 0
2 1 1 0
S
f1; 2; 3g
f1; 2; 4g
f1; 2; 5g
f1; 3; 4g
f1; 3; 5g
f1; 4; 5g
f2; 3; 4g
f2; 3; 5g
f2; 4; 5g
f3; 4; 5g
cðSÞ c ðSÞ bðSÞ v b ðSÞ
2 1 3 1
2 2 3 1
3 0 2 0
3 1 3 1
2 1 2 1
2 2 2 0
3 1 3 1
2 1 2 1
2 2 2 0
3 1 2 0
S
f1; 2; 3; 4g
f1; 2; 3; 5g
f1; 2; 4; 5g
f1; 3; 4; 5g
f2; 3; 4; 5g
N
cðSÞ c ðSÞ bðSÞ v b ðSÞ
3 2 4 1
3 1 3 1
3 2 3 1
3 2 3 1
3 2 3 1
3 3 4 1
Note that v b ðNÞ ¼ 1 > 0 ¼ bðNÞ cðNÞ. Below, we explain how to compute the value of coalition f1; 2; 4g.
v b ðf1;2;4gÞ ¼ maxfpb ðf;gÞ; pb ðf1gÞ; pb ðf2gÞ; pb ðf4gÞ; pb ðf1;2gÞ; pb ðf1;4gÞ; pb ðf2;4gÞ; pb ðf1;2;4gÞg ¼ maxf0;2 2;2 2;0 1;4 4;2 2;2 2;4 3g ¼ maxf0;0;0;1;0;0;0;1g ¼ 1: Note that ðN; cÞ has a nonempty anti core since ð1; 1; 1; 1; 1; 1Þ 2 ACoreðcÞ. However, ðN; v b Þ has an empty core. To show this, suppose that x 2 Coreðv b Þ, then, 1 6 x1 þ x2 þ x4 , 1 6 x1 þ x3 þ x5 , 1 6 x2 þ x3 þ x6 , 0 6 x4 þ x5 þ x6 . By adding all these equations, we obtain 3 6 2xðNÞ ¼ 2v b ðNÞ ¼ 2, establishing a contradiction. Here, the first equality follows from x 2 Coreðv b Þ. As we have seen in the example above, when not all members of the grand coalition get the service that generates the revenues, the core of the cost-revenue game may be empty. For that reason, we assume3 from here that it is profitable to form the grand coalition. Assumption 3.1.
v b ðNÞ ¼ pb ðNÞ ¼ bðNÞ cðNÞ:
ð3:1Þ
Example 3.2. Consider the five players exact cost game ðN; cÞ with5 ACoreðcÞ ¼ convfð0; 0; 1; 2; 0Þ, ð1; 1; 0; 0; 1Þg, and let b ¼ ð1; 1; 1; 1; 0Þ. In Table 1, we provide the coalitional values of ðN; cÞ; ðN; c Þ; ðN; bÞ and ðN; v b Þ. Note that bðSÞ P c ðSÞ for every S # N. However, ðN; v b Þ has an empty core. To show this, suppose that x 2 Coreðv b Þ, then, 1 6 x1 þ x2 þ x3 þ x4 , 1 6 x1 þ x2 þ x3 þ x5 , 1 6 x1 þ x2 þ x4 þ x5 , 1 6 x1 þ x3 þ x4 þ x5 , 1 6 x2 þ x3 þ x4 þ x5 . By adding all these equations, we obtain 5 6 4xðNÞ ¼ 4v b ðNÞ ¼ 4, establishing a contradiction. Here, the first equality follows from x 2 Coreðv b Þ. The first result of this section states that if the core of a costrevenue game is nonempty, then, its core can be expressed in terms of the vector of revenues and the anti core of the cost game. Lemma 3.1. Let ðN; cÞ be a cost game, let b 2 RN þ be a vector of revenues satisfying Assumption 3.1, and let ðN; v b Þ be the corresponding cost-revenue game. Then,
Coreðv b Þ ¼ fb y : y 2 ACoreðcÞ with y 6 bg: Proof. We first show ‘‘ # ’’. If Coreðv b Þ ¼ ;, we are done. Let x 2 Coreðv b Þ and let y ¼ b x, we have to show that (i) y 2 ACoreðcÞ and (ii) y 6 b.
Taking into account that pb ðNÞ ¼ maxR # N fbðRÞ cðRÞg, it follows that Assumption 3.1 is equivalent to
bðSÞ P c ðSÞ for every S # N:
(i) Let S N. We have
bðSÞ cðSÞ 6 maxfbðRÞ cðRÞg ¼ v b ðSÞ 6 xðSÞ;
ð3:2Þ
R#S
This is because v b ðNÞ ¼ bðNÞ cðNÞ is equivalent to bðNÞ cðNÞ P bðRÞ cðRÞ for every R N, which, taking R ¼ N n S, reduces to Eq. (3.2). Assumption 3.1 may seem somehow restrictive. It ensures that the grand coalition will effectively form without leaving any player behind. The following example shows that non-emptiness of the anti core of the cost game is not sufficient to have non-emptiness of the core of the corresponding cost-revenue game, even when the cost game is exact.4 Therefore, revenues have a real impact in the structure of the core of cost-revenue games. It further shows why the game v b of Example 3.2 has an empty core; there are no elements of AcoreðcÞ that are dominated by b.
where the last inequality follows because x 2 Coreðv b Þ. Therefore,
yðSÞ ¼ bðSÞ xðSÞ 6 cðSÞ: bðNÞ cðNÞ ¼ v b ðNÞ ¼ xðNÞ;
where the first equality follows by Assumption 3.1 and the second one because x 2 Coreðv b Þ. This gives
yðNÞ ¼ bðNÞ xðNÞ ¼ cðNÞ:
ð3:4Þ
By Eqs. (3.3) and (3.4), we have that y 2 ACoreðcÞ. (ii) Core elements are in particular imputations, so for every i 2 N,
xi P v b ðfigÞ P 0 3
This assumption is however not mandatory for having a nonempty core. Think of a situation with v b ðNÞ ¼ 0 . 4 Similarly as in (reward) games, a cost game is exact if, for every S N, there is an x 2 ACoreðcÞ such that xðSÞ ¼ cðSÞ.
ð3:3Þ
Furthermore,
and, therefore, yi ¼ bi xi 6 bi for every i 2 N. 5
By conv we refer to the convex hull.
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Next, we show ‘‘’’. If there does not exist a y 2 ACoreðcÞ satisfying y 6 b, then, we are done. Let x ¼ b y with y 2 ACoreðcÞ and y 6 b. We have
xðNÞ ¼ bðNÞ yðNÞ ¼ bðNÞ cðNÞ ¼ v b ðNÞ;
ð3:5Þ
where the second equality follows from y 2 ACoreðcÞ and the third one is a direct consequence of Assumption 3.1. Furthermore, for every S N, let RS # S be such that v b ðSÞ ¼ pb ðRS Þ ¼ bðRS Þ cðRS Þ. We have that
xðSÞ P xðRS Þ ¼ bðRS Þ yðRS Þ P bðRS Þ cðRS Þ ¼ v b ðSÞ;
ð3:6Þ
where the first inequality is a direct consequence of x ¼ b y P 0 and RS # S, and the second inequality follows by y 2 ACoreðcÞ. Eqs. (3.5) and (3.6) imply that x 2 Coreðv b Þ. h The following result characterizes the class of cost games in which, for any vector of revenues satisfying Assumption 3.1, it follows that the corresponding cost-revenue game has a nonempty core. Theorem 3.2. Let ðN; cÞ be a cost game. Then, ðN; v b Þ has a nonempty core for any b 2 RN þ satisfying Assumption 3.1 if, and only if, ðN; c Þ has a large core.
between two different villages can only take one of, at most, two possible values (low and high cost). We call such games 2-mcst games with revenues. Second, we answer this question by showing that, under Assumption 3.1, every 2-mcst game with revenues has a nonempty core. Let N ¼ f1; 2; . . . ; ng denote the set of villages that have to be connected to a source. Let N 0 ¼ N [ f0g, where 0 denotes the source. Let C ¼ ðcij Þ be an N 0 N 0 -matrix, where cij P 0, for every i; j 2 N 0 , represents the costs to go from village i to village j. Since the cost matrix is nonnegative, it turns out that the graph that connects all villages to the source with a minimum cost is a tree. Throughout this article we assume that:
ðiÞ cii ¼ 0 for all i 2 N0 ; ðiiÞ cij ¼ cji
for all i; j 2 N0
ðsymmetryÞ:
The network ðN 0 ; CÞ is usually represented by the complete graph on N 0 with costs cij on the edge between nodes i and j. Given a network ðN 0 ; CÞ and a subset S # N, we denote S0 :¼ S [ f0g. A tree ðV; EÞ is called a spanning tree of S if V ¼ S0 . Let TðSÞ denote the set of all spanning trees of S and, for C ¼ ðS0 ; EÞ 2 TðSÞ, let cðC; SÞ denote the cost associated with the tree, formally,
cðC; SÞ ¼
X
cij :
fi;jg2E
Proof. We first show the ‘‘if’’ part. Let ðN; c Þ have a large core. We have to show that ðN; v b Þ has a nonempty core for any b 2 RNþ satisfying Assumption 3.1. Such a b satisfies bðSÞ P c ðSÞ for every S # N and, therefore, b 2 Uðc Þ. By largeness of the core of ðN; c Þ, there is a yb 2 Coreðc Þ ¼ ACoreðcÞ with b P yb . By Lemma 3.1, it follows that b yb 2 Coreðv b Þ. In order to show the ‘‘only if’’ part, let ðN; cÞ be such that ðN; v b Þ has a nonempty core for every b 2 RN þ satisfying Assumption 3.1. We have to show that ðN; c Þ has a large core. Let b 2 Uðc Þ. Then, b satisfies Assumption 3.1 since bðSÞ P c ðSÞ for every S # N. Choose any x 2 Coreðv b Þ. First, we have that b P b x since x P 0 by non-negativity of ðN; v b Þ; second, b x 2 ACoreðcÞ ¼ Coreðc Þ by Lemma 3.1. Therefore, ðN; c Þ has a large core. h The following example illustrates that largeness of the core of ðN; c Þ and largeness of the anti core of ðN; cÞ can be unrelated.
By associating each village in N with a player, an mcst game, ðN; cÞ, is defined, for every S # N, by
cðSÞ :¼ min fcðC; SÞg: C2TðSÞ
We add revenues to the situation by assuming that if village i 2 N is connected to the source, a revenue bi P 0 is obtained. An mcst problem with revenues is denoted by a tuple ðN 0 ; C; bÞ and the corresponding mcst game with revenues by ðN; v b Þ. Note that, under Assumption 3.1, we have v b ðNÞ ¼ bðNÞ cðNÞ. Example 4.1. Consider the mcst problem with revenues represented in Fig. 1, in which the numbers at the edges represent the connection costs and the boldfaced numbers at the nodes represent the revenues. The value of coalition f1; 2g is computed as follows:
v b ðf1; 2gÞ ¼ max f0; b1 cðf1gÞ; b2 cðf2gÞ; b1 þ b2 cðf1; 2gÞg Example 3.3. Consider the cost game ðN; cÞ given in Table 2. It can be easily seen that Coreðc Þ ¼ ACoreðcÞ ¼ fð1; 1; 1; 1Þg. Since, for any x 2 Uðc Þ; xi P 1 for every i 2 N, the core of ðN; c Þ is large. On the other hand, it can be checked that x ¼ ð1; 1:1; 0:9; 0:9Þ 2 LðcÞ. Therefore, the anti core of ðN; cÞ is not large since x2 ¼ 1:1 > 1 ¼ y2 with y ¼ ð1; 1; 1; 1Þ and, hence, there is no y 2 ACoreðcÞ ¼ fð1; 1; 1; 1Þg with xi 6 yi for every i 2 N. 4. Minimum cost spanning tree games with revenues In this section, we study minimum cost spanning tree (mcst) games with revenues. First, we provide an example of an mcst game with revenues that has an empty core. In this example, the connection cost between two different villages can only take one out of three possible values (low, medium, and high cost). Therefore, the arising question is whether we can find an mcst game with revenues with an empty core in which the connection cost
¼ max f0; 6 5; 2 10; 6 þ 2 10g ¼ max f0; 1; 8; 2g ¼ 1: The associated mcst game with revenues is given in Table 3. It can be checked that Coreðv b Þ ¼ convfð3; 0; 4Þ; ð1; 2; 4Þ; ð1; 0; 6Þg. In general, mcst games with revenues satisfying Assumption 3.1 can have an empty core as the following example illustrates. Example 4.2. Consider the mcst network N ¼ f1; 2; 3; 4; 5; 6; 7g and cost matrix defined by
8 0 > > > < 3 cij ¼ > >4 > : 5
ðN 0 ; CÞ
with
if i ¼ j; if fi;jg 2 ff0;1g;f1;2g;f1;3g;f1;4g;f2;7g;f3;6g;f4;5gg; if fi;jg 2 ff0;2g;f0;3g;f0;4g;f2;5g;f3;7g;f4;6gg; otherwise:
Table 2 Coalitional values of the cost game and dual game in Example 3.3. S
f1g
f2g
f3g
f4g
f1; 2g
f1; 3g
f1; 4g
f2; 3g
f2; 4g
f3; 4g
f1; 2; 3g
f1; 2; 4g
f1; 3; 4g
f2; 3; 4g
N
cðSÞ c ðSÞ
2 1
2 1
1 1
1 1
4 2
2 2
2 2
2 2
2 2
2 0
3 3
3 3
3 2
3 2
4 4
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611
is a node connected to all other nodes (the graph is a star), then, there are 2n1 þ n 1; n ¼ jNj, connected coalitions. In this case, the number of minimal covers using only essential coalitions exceeds k Pn1 1 Pak 2 k1 m!Sðn1;mÞ, in which a ¼ minfn1;2k 1g k k¼2 k! m¼k mk Pm n1 ml m 1 and Sðn 1; mÞ ¼ m! l . l¼1 ð1Þ l In order to describe the collection of essential coalitions with respect to ðN; c Þ, we introduce some extra notation. Given a 2-mcst network ðN 0 ; CÞ, we define the k1 -graph Ck1 by N 0 ; Ek1 ,
Fig. 1. The mcst problem with revenues in Example 4.1.
where
Ek1 ¼ fi; jg N0 : cij ¼ k1 :
Table 3 Coalitional values of the mcst game with revenues in Example 4.1.
Let F k1 # Ek1 be the set of edges that belong to any mcst for N, i.e.,
S
f1g
f2g
f3g
f1; 2g
f1; 3g
f2; 3g
N
v b ðSÞ
1
0
4
1
5
4
7
Note that ðN; c Þ does not have a large core by considering the minimal cover C ¼ ff1; 2; 4; 5g; f1; 2; 3; 7g; f1; 3; 4; 6gg. Since c ðf1; 2; 4; 5gÞ ¼ c ðf1; 2; 3; 7gÞ ¼ c ðf1; 3; 4; 6gÞ ¼ 10; c ðNÞ ¼ 21, the inequality
c ðf1; 2; 4; 5gÞ þ c ðf1; 2; 3; 7gÞ þ c ðf1; 3; 4; 6gÞ ¼ 30 > 21 þ 8 9 8 = < X ¼ c ðNÞ þ max lS c ðSÞ ; ; l2MðCÞ: EðCÞ S22
nf;g
shows the non-largeness of the core of c . As an optimal solution to ^ can be chothe involved maximization problem above, the vector l ^ S ¼ 1 if S 2 ff1; 2; 3g; f1; 4gg and l ^ S ¼ 0 othersen to be defined by l wise. It can be checked that b ¼ ð1; 2; 2; 2; 5; 5; 5Þ satisfies Assumption 3.1 and ðN; v b Þ has an empty core since v b ðf4gÞ ¼ v b ðf1; 2gÞ ¼ v b ðf1; 3gÞ ¼ 0; v b ðf2; 5; 7gÞ ¼ v b ðf3; 6; 7gÞ ¼ v b ðf4; 5; 6gÞ ¼ v b ðNÞ ¼ 1 and, therefore, if x 2 Coreðv b Þ, then, 0 6 x4 ; 0 6 x1 þ x2 ; 0 6 x1 þ x3 ; 1 6 x2 þ x5 þ x7 ; 1 6 x3 þ x6 þ x7 ; 1 6 P x4 þ x5 þ x6 and 7i¼1 xi ¼ 1. By adding the inequalities, we obtain P7 3 6 2 i¼1 xi ¼ 2, establishing a contradiction. Every connection cost in the mcst network of Example 4.2 can take one of three given values: 3; 4, or 5. We now will analyze the case in which connection costs between two different villages in an mcst network can take, at most, two values: low and high cost. It will turn out that, under Assumption 3.1, the corresponding mcst game with revenues always has a nonempty core. We refer to this type of mcst network as a 2-mcst network. Formally, ðN 0 ; CÞ is a 2-mcst network if, for every i; j 2 N 0 ; i – j; cij 2 fk1 ; k2 g with 0 6 k1 6 k2 . We will show non-emptiness of the core of 2-mcst games with revenues under Assumption 3.1. Their duals will play a crucial role. Therefore, we first provide some results on the dual game of a 2mcst game. The first result states that the dual game of an mcst b for the game is tree-essential with respect to any optimal tree C grand coalition. It is based on results from Kuipers (1994) (see Theorem 4.4 in Kuipers (1994) and comments below) and the proof is, therefore, omitted. Theorem 4.1 Kuipers, 1994. Let ðN 0 ; CÞ be an mcst network and let b be an optimal mcst for N. ðN; cÞ be the associated mcst game. Let C Then, the associated dual game ðN; c Þ is forest-essential with respect b jN . to C This result served as the base of a polynomial time algorithm by Kuipers (1994) for computing the nucleolus of mcst games. Unfortunately, it does not limit the number of essential coalitions to polyb jN is such that there nomially bounded proportions. For instance, if C
F k1 ¼
8 < E k1
if Ck1 is a forest; : E n [j2CðCk1 Þ EðjÞ if Ck1 is not a forest: k1
With minor abuse of language, we write essential coalition when we refer to an essential coalition with respect to ðN; c Þ. By Theorem 4.1, all essential coalitions are connected in every mcst for N. Since F k1 is contained in any mcst for N, every connected coalition in F k1 is also connected in any mcst for N. We will now analyze those coalitions that are connected in every mcst for N and that have a nonempty intersection with [j2CðCk1 Þ VðjÞ. Lemma 4.2. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let S # N with S 2 CðCk1 Þ. If there exists j 2 CðCk1 Þ with 1 < jS \ VðjÞj < b jN Þ for at least one mcst C b for N. jVðjÞj, then, S R Cð C Proof. Let j 2 CðCk1 Þ with 1 < jS \ VðjÞj < jVðjÞj. Let i; j 2 S \ VðjÞ and let l 2 VðjÞ n S. Then, there are two disjoint paths in j connecting i and j, one of them visiting l. Fix an edge e on the path that does b not visit l. Using Kruskal’s algorithm, we can find an mcst C containing all edges of j with the exception of e. The only path b Þ visits village l. connecting i and j that uses only edges in Eð C b. h Therefore, S is not connected in C As an immediate consequence of Theorem 4.1 and Lemma 4.2, we have the following corollary: if an essential coalition has a nonempty intersection with the set of nodes of a cycle of the k1 -graph, then, the intersection is either a singleton, or the set of nodes of the cycle. Corollary 4.3. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, let ðN; cÞ be the associated 2-mcst game, and let S 2 Essðc Þ. If there exists
j 2 CðCk1 Þ with S \ VðjÞ – ;, then, either S \ VðjÞ ¼ VðjÞ, or S \ VðjÞ ¼ fig for some i 2 S. Given a 2-mcst network ðN 0 ; CÞ with k1 -graph Ck1 and 2-mcst game ðN; cÞ, as a result of Corollary 4.3, we have that if two cycles
j1 ; j2 2 CðCk1 Þ share two nodes, then, an essential coalition, S, of the dual game satisfies that either jS \ ðVðj1 Þ [ Vðj2 ÞÞj 6 1, or Vðj1 Þ [ Vðj2 Þ # S. Following this, we define an equivalence relation in order to describe the set of essential coalitions of the dual game of a 2-mcst game. Let C ¼ ðN 0 ; EÞ be a graph and let j1 ; j2 2 CðCÞ. We say that j1 and j2 are related, j1 Rj2 , if jVðj1 Þ \ Vðj2 Þj P 2. The binary relation R is reflexive and symmetric, but does not need to be transitive. Therefore, we consider the transitive closure of R; Rþ , which is an equivalence relation. We denote by ½j the equivalence class of j 2 CðCÞ with respect to Rþ and by ½CðCÞ the set of equivalence
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Fig. 2. Example of the numbering of C in the proof of Theorem 4.6.
classes in C, so ½CðCÞ ¼ f½j : j 2 CðCÞg. Given j 2 CðCÞ, we denote ~ Þ and Eð½jÞ ¼ [j~ 2½j Eðj ~ Þ. Vð½jÞ ¼ [j~ 2½j Vðj Let ðN 0 ; CÞ be a 2-mcst network. We define EðN 0 ; CÞ by
l1 l2 lr lr X X X X 1 c ðT l Þ ¼ c ðT l Þ þ c ðT l Þ þ þ c ðT l Þ ¼ c [ll¼1 Tl l¼1
l¼1
n
EðN0 ; CÞ ¼ S 2 CðCk1 Þ : jS \ Vð½jÞj 2 f0; 1; jVð½jÞjg o for every ½j 2 ½CðCk1 Þ :
þ
l¼l1 þ1
l¼lr1 þ1
r c [ll¼1 Tl \ T lþ1 þ þ c [ll¼l T l þ1 r1
l1 1 X l¼1
þ
The following result is a direct consequence of Theorem 4.1 and Corollary 4.3 and the proof is, therefore, omitted.
lr 1 X
c
r X s [ll¼lr1 þ1 Tl \ T lþ1 ¼ c [ll¼l Tl s1 þ1 s¼1
l¼lr1 þ1
þ Corollary 4.4. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let ðN; cÞ be the associated 2-mcst game. Then, Essðc Þ # EðN 0 ; CÞ. Another crucial result states that if two essential coalitions have a nonempty intersection, then, the sum of their dual values equals the sum of the dual values of their union and intersection. Due to the technicality of the proof, it is postponed to Appendix A.
r X
ls 1 X
s¼1 l¼ls1 þ1
þ
ls 1 r X X s¼1 l¼ls1 þ1
þ max
8 <
l2MðCÞ:
c [ll¼ls1 þ1 Tl \ T lþ1 6 c ðNÞ c
[ll¼ls1 þ1 Tl \ T lþ1 6 c ðNÞ
X
9 = lT c ðTÞ ; ;
T22EðCÞ nf;g
Lemma 4.5. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let ðN; cÞ be the associated 2-mcst game. Then,
in which the second equality is a consequence of successively applying Lemma 4.5 and the first inequality is valid because n o 1 r [ll¼1 T l ; . . . ; [ll¼l T l is a partition of N and ðN; c Þ is forest-essenr1 þ1
c ðT 1 Þ þ c ðT 2 Þ ¼ c ðT 1 \ T 2 Þ þ c ðT 1 [ T 2 Þ
tial with respect to any mcst C for N by Theorem 4.1. Finally, the last inequality follows because l defined by lT ¼ 1 for every T ¼ [ll¼l þ1 Tl \ T lþ1 with s 2 f1; . . . ; rg and l 2 fls1 þ 1; . . . ;
for every T 1 ; T 2 2 EðN 0 ; CÞ with T 1 \ T 2 – ;. We can finally provide the main result of this Section.
s1
ls 1g and Theorem 4.6. Every 2-mcst game Assumption 3.1 has a nonempty core.
with
revenues
Proof. Let ðN 0 ; CÞ be a 2-mcst network and let ðN; cÞ be the associated 2-mcst game. Let b 2 RNþ be a vector of revenues satisfying Assumption 3.1. By Theorem 3.2, it suffices to show that ðN; c Þ has a large core. By Theorem 2.1, we have to show that, for all C 2 MCðNÞ,
8 < X c ðTÞ 6 c ðNÞ þ max l2MðCÞ: T2C
X
lT ¼ 0 otherwise, is an mc-balanced vector for C.
h
satisfying
9 = lT c ðTÞ : ;
T22EðCÞ nf;g
As already mentioned in Section 2, we can restrict our analysis to minimal covers whose elements are essential coalitions of ðN; c Þ. Therefore, fix C 2 MCðNÞ such that T 2 Essðc Þ for every T 2 C. By Corollary 4.4, T 2 EðN 0 ; CÞ for every T 2 C. Let C ¼ fT 1 ; . . . ; T l1 ; . . . ; T lr1 ; . . . ; T lr g be ordered as follows: 1 1 1 r T 1 \ T 2 – ;; . . . ; [ll¼1 T l \ T l1 – ;; [ll¼1 T l \ [ll¼l T l ¼ ; and for 1 þ1 s 1 T l \ T ls – ; every s 2 f2; . . . ; rg, T ls1 þ1 \ T ls1 þ2 – ;; . . . ; [ll¼l s1 þ1 s r and [ll¼l T l \ [ll¼l T l ¼ ; (see Fig. 2). Note that s þ1 s1 þ1 n o 1 r is a partition of N. We have T l ; . . . ; [ll¼l T [ll¼1 l þ1 r1
It remains an open and very interesting question whether it is possible to verify Assumption 3.1 for an mcst game with revenues in polynomial time. To conclude this Section, note that the cost game in Example 3.1 is a 2-mcst game with k1 ¼ 1; k2 ¼ 2, and C defined by cij ¼ 1 if fi; jg 2 ff0; 4g; f0; 5g; f0; 6g; f1; 4g; f1; 5g; f2; 4g; f2; 6g; f3; 5g; f3; 6gg and cij ¼ 2 otherwise. Therefore, Assumption 3.1 is, indeed, necessary in Theorem 4.6. 5. Further remarks Littlechild and Owen (1976) were the first to analyze cost problems arising from a general service facility by taking into account the revenues that the service generates. They restricted their study to the framework of airport problems, with a more recent follow up by Brânzei et al. (2006). Meertens and Potters (2006) consider fixed tree games with revenues. In both settings, the underlying cost game is concave6 and, therefore, its dual game is convex7 and has a large core (see Sharkey, 1982). 6 A cost game ðN; cÞ is concave if, for every i 2 N and every S T # N n fig; cðS [ figÞ cðSÞ P cðT [ figÞ cðTÞ. 7 A game ðN; v Þ is convex if, for every i 2 N and every S T # N n fig; v ðS [ figÞ v ðSÞ 6 v ðT [ figÞ v ðTÞ.
613
A. Estévez-Fernández, H. Reijnierse / European Journal of Operational Research 237 (2014) 606–616 Table 4 Coalitional values of the routing game, associated dual game, additive revenues, and routing game with revenues in Example 5.1. S
f1g
f2g
f3g
f4g
f5g
f1; 2g
f1; 3g
f1; 4g
f1; 5g
f2; 3g
f2; 4g
f2; 5g
f3; 4g
f3; 5g
f4; 5g
cp^ ðSÞ cp^ ðSÞ bðSÞ v p^ b ðSÞ
4 3 10 6
2 0 10 8
20 10 10.5 0
4 16 10 6
20 10 10 0
5 4 20 15
24 13 20.5 6
5 13 20 15
24 13 20 6
21 13 20.5 8
5 16 20 15
22 10 20 8
14 3 20.5 6.5
40 20 20.5 0
14 4 20 6
S
f1; 2; 3g
f1; 2; 4g
f1; 2; 5g
f1; 3; 4g
f1; 3; 5g
f1; 4; 5g
f2; 3; 4g
f2; 3; 5g
f2; 4; 5g
f3; 4; 5g
cp^ ðSÞ cp^ ðSÞ bðSÞ v p^ b ðSÞ
24 14 30.5 15
8 12 30 22
25 14 30 15
18 6 30.5 15
44 23 30.5 6
15 7 30 15
15 4 30.5 15.5
41 23 30.5 8
15 4 30 15
24 23 30.5 6.5
S
f1; 2; 3; 4g
f1; 2; 3; 5g
f1; 2; 4; 5g
f1; 3; 4; 5g
f2; 3; 4; 5g
N
cp^ ðSÞ cp^ ðSÞ bðSÞ v p^ b ðSÞ
18 8 40.5 22.5
44 24 40.5 15
18 8 40 22
28 26 40.5 15
25 24 40.5 15.5
28 28 50.5 22.5
Granot and Maschler (1998) consider spanning network situations. These situations generalize mcst situations in the sense that they allow for ‘Steiner-points’ (vertices at which no players are resided), costs can be negative and can be located at vertices, and coalitions are allowed to use arcs and vertices not adjacent to their residences (making the associated game monotone). Unlike in our setting, a coalition must connect each of its members to the root. They provide conditions under which the core of the associated game is nonempty (Theorem 4.13 in their paper). Their condition that the grand coalition connects all vertices to the root is similar to our Assumption 3.1, however their assumption that each edge (including its end point) used by the grand coalition has nonnegative costs quite differs from our situation. The nature of our model implies that each branch of an optimal tree has nonnegative benefits. This makes both approaches difficult to compare. Estévez-Fernández et al. (2009) consider routing games with revenues. Routing games are well-known and broadly studied in the literature. A routing game is similar to a traveling salesman game (see Potters, Curiel, & Tijs, 1992), except that a proper coalition cannot choose the order of the visits of the salesman, but must use the order induced by an optimal order to visit all cities. EstévezFernández et al. (2009) show that routing games with revenues have a nonempty core under the assumption that cooperation among the members of the grand coalition grants that the optimal tour will go through all cities. Example 5.1 illustrates that the dual game of a routing game does not always have a large core. This
might seem in contradiction with Theorem 3.2 in this article. In Example 5.1, we provide an explanation for this phenomenon. Example 5.1. Consider the routing problem with revenues ðN 0 ; C; bÞ with N ¼ f1; 2; 3; 4; 5g,
;
^ : 0 1 2 3 4 b ¼ ð10; 10; 10:5; 10; 10Þ, and optimal tour p 5 0. In Table 4, we provide the coalitional values of the routing game ðN; cp^ Þ, the corresponding dual game ðN; cp^ Þ; ðN; bÞ and ðN; v p^ b Þ. Note that bðSÞ P cp^ ðSÞ for every S # N. Still, ðN; v p^ b Þ has an empty core. To show this, suppose that x 2 Coreðv p^ b Þ; then, 15 6 x1 þ x4 ; 8 6 x2 , and 0 6 x3 þ x5 . By adding all these equations P we obtain 23 6 5i¼1 xi ¼ v p^ b ðNÞ ¼ 22:5 establishing a contradiction. Therefore, Coreðv p^ b Þ ¼ ;, and, consequently, ðN; cp^ Þ does not have a large core.
Table 5 Coalitional values of the traveling salesman game, associated dual game, additive revenues, and net revenues in Example 5.1. S
f1g
f2g
f3g
f4g
f5g
f1; 2g
f1; 3g
f1; 4g
f1; 5g
f2; 3g
f2; 4g
f2; 5g
f3; 4g
f3; 5g
f4; 5g
cts ðSÞ cts ðSÞ bðSÞ bðSÞ cts ðSÞ
4 3 10 6
2 0 10 8
20 11 10.5 9.5
4 16 10 6
20 12 10 10
5 4 20 15
24 13 20.5 3.5
5 13 20 15
24 13 20 4
21 13 20.5 0.5
5 16 20 15
22 13 20 2
14 3 20.5 6.5
40 20 20.5 19.5
14 4 20 6
S
f1; 2; 3g
f1; 2; 4g
f1; 2; 5g
f1; 3; 4g
f1; 3; 5g
f1; 4; 5g
f2; 3; 4g
f2; 3; 5g
f2; 4; 5g
f3; 4; 5g
cts ðSÞ cts ðSÞ bðSÞ bðSÞ cts ðSÞ
24 14 30.5 6.5
8 12 30 22
25 14 30 5
15 6 30.5 15.5
44 23 30.5 13.5
15 7 30 15
15 4 30.5 15.5
41 23 30.5 10.5
15 4 30 15
24 23 30.5 6.5
S
f1; 2; 3; 4g
f1; 2; 3; 5g
f1; 2; 4; 5g
f1; 3; 4; 5g
f2; 3; 4; 5g
N
cts ðSÞ cts ðSÞ bðSÞ bðSÞ cts ðSÞ
16 8 40.5 24.5
44 24 40.5 3.5
17 8 40 23
28 26 40.5 12.5
25 24 40.5 15.5
28 28 50.5 22.5
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A. Estévez-Fernández, H. Reijnierse / European Journal of Operational Research 237 (2014) 606–616
Hence, Assumption 3.1 is insufficient to ensure non-emptiness of the core for routing games with revenues. Estévez-Fernández et al. (2009) show that non-emptiness of the core is guaranteed if in the corresponding traveling salesman problem with revenues, it is optimal for N to visit all cities and to visit them in increasing order. This boils down to bðSÞ P cts ðSÞ for every S # N, where ðN; cts Þ denotes the traveling salesman game corresponding to distance matrix C. In Table 5, we provide the coalitional values of salesman game ðN; cts Þ, the corresponding dual game the traveling
N; cts , and ðN; bÞ. Note that without a prescribed order it is more profitable to only visit cities f1; 2; 3; 4g (in order 0–2–1–4–3–0) than to visit all cities in N. Besides, note that b is not an element of the upper core of ðN; cts Þ since b3 ¼ 10:5 < 11 ¼ cts ðf3gÞ and b5 ¼ 10 < 12 ¼ cts ðf5gÞ. Note that Example 5.1 illustrates that Assumption 3.1 is not a mandatory condition for non-emptiness of the core of cost games with revenues. Appendix A In this appendix, we show the proof of Lemma 4.5 in Section 4. Before giving the proof, we provide three auxiliary lemmas. Lemma A.1. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let ðN; cÞ be the corresponding 2-mcst game. Then,
o X n k j 2 N0 n T : fi; jg 2 F 1 i2T n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ T ¼ fig 1
c ðTÞ ¼ jTjk1 ðk2 k1 Þ
for every T 2 EðN 0 ; CÞ. Proof. By the definition of EðN 0 ; CÞ; T 2 CðCk1 Þ. Let U 2 N 0 k be the jC 1 component of Ck1 with T # U. We have c ðTÞ ¼ cðNÞ cðN n TÞ ¼ k1 jNj k1 jN 0jCk1 j 1 þ k2 jN 0jCk1 j 1 k1 jN n T j k1 ðjðN 0 n TÞjCk1 j 1Þ þ k2 ðjðN 0 n TÞjCk1 j 1Þ ¼ k1 jTj ðk2 k1 Þ jðN 0 n TÞjCk1 j jN 0jCk1 j ¼ k1 jTj ðk2 k1 Þ jðU n TÞjCk1 j 1 ¼ k1 jTj ðk2 k1 Þ ! o n o X n k1 k1 j 2 U n T : fi; jg 2 F þ ½j 2 ½CðC Þ : Vð½jÞ \ T ¼ fig 1 i2T
o X n k ¼ k1 jTj ðk2 k1 Þ j 2 N 0 n T : fi; jg 2 F 1 i2T n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ T ¼ fig 1 ;
in which the fourth equality is a direct consequence of T # U with U 2 N0jCk1 ; the fifth equality follows because the elements of ðU n TÞjCk1 are those components that we obtain when the edges between T and U n T are broken; the final equality is a direct consequence of T # U and fi; jg # U for every fi; jg 2 Ek1 with i 2 T. h Lemma A.2. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j. For any disjoint pair T 1 ; T 2 2 EðN 0 ; CÞ of essential coalitions we have (i) If jVð½jÞ \ T 1 j ¼ jVð½jÞ \ T 2 j ¼ 1 for some ½j 2 ½CðCk1 Þ, then, Vð½jÞ \ T 1 ¼ Vð½jÞ \ T 2 ; (ii) T 1 [ T 2 ; T 1 \ T 2 2 EðN 0 ; CÞ.
Proof. (i) On the contrary, suppose that Vð½jÞ \ T 1 ¼ fi1 g and Vð½jÞ \ T 2 ¼ fi2 g with i1 – i2 . Since jVð½jÞ \ T 1 j ¼ 1 and jVð½jÞ \ T 2 j ¼ 1, we have i1 2 T 1 n T 2 and i2 2 T 2 n T 1 . Since Vð½jÞ 2 CðCk1 Þ, there exists a path connecting i1 and i2 ; pathði1 ; i2 Þ, using only edges in Eð½jÞ. Let j 2 T 1 \ T 2 ; by connectivity of T 1 , there exists a path, pathði1 ; jÞ, connecting i1 and j in Ck1 using nodes of T 1 . Because jVð½jÞ \ T 1 j ¼ 1, no edge of pathði1 ; jÞ belongs to Eð½jÞ. Analogously, there exists a path, pathðj; i2 Þ, connecting j and i2 in Ck1 using nodes of T 2 and such that no edge in the path belongs to Eð½jÞ. Combining pathði1 ; jÞ and pathðj; i2 Þ, we can construct a path connecting i1 and i2 ; pathði1 ; i2 Þ , without using any edge in Eð½jÞ. Combining the paths 2 CðCk1 Þ pathði1 ; i2 Þ and pathði1 ; i2 Þ , there exists a cycle j Þ and ½j – ½j . This establishes a contradicwith i1 ; i2 2 Vð½j . tion since i1 ; i2 2 Vð½jÞ and, therefore, ½j ¼ ½j (ii) First, note that T 1 [ T 2 2 CðCk1 Þ since T 1 ; T 2 2 CðCk1 Þ with T 1 \ T 2 – ;. Moreover, for every ½j 2 ½CðCk1 Þ, we have that a. if Vð½jÞ # T 1 (analogously, Vð½jÞ # T 2 ), then, Vð½jÞ # T 1 [ T 2 ; b. if Vð½jÞ \ T 1 ¼ fig for some i 2 T 1 and jVð½jÞ \ T 2 j ¼ 1 (analogously, Vð½jÞ \ T 2 ¼ fig for some i 2 T 2 and jVð½jÞ \ T 1 j ¼ 1), then, Vð½jÞ \ T 2 ¼ fig by (i) of this lemma and Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig; c. if Vð½jÞ \ T 1 ¼ fig for some i 2 T 1 and Vð½jÞ \ T 2 ¼ ; (analogously, Vð½jÞ \ T 2 ¼ fig for some i 2 T 2 and Vð½jÞ \ T 1 ¼ ;), then, Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig; d. if Vð½jÞ \ T 1 ¼ ; and Vð½jÞ \ T 2 ¼ ;, then, Vð½jÞ\ ðT 1 [ T 2 Þ ¼ ;. Therefore, T 1 [ T 2 2 EðN 0 ; CÞ. Second, note that the intersection of two connected sets is connected and, therefore, T 1 \ T 2 2 CðCk1 Þ. Moreover, for every ½j 2 ½CðCk1 Þ, we have e. if Vð½jÞ \ T 1 ¼ ; (analogously, Vð½jÞ \ T 2 ¼ ;), then, Vð½jÞ \ ðT 1 \ T 2 Þ ¼ ;; f. if Vð½jÞ \ T 1 ¼ fig for some i 2 T 1 and jVð½jÞ \ T 2 j ¼ 1 (analogously, Vð½jÞ \ T 2 ¼ fig for some i 2 T 2 and jVð½jÞ \ T 1 j ¼ 1), then, Vð½jÞ \ T 2 ¼ fig by (i) of this lemma and Vð½jÞ \ ðT 1 \ T 2 Þ ¼ fig; g. if Vð½jÞ \ T 1 ¼ fig for some i 2 T 1 and Vð½jÞ # T 2 (analogously, Vð½jÞ \ T 2 ¼ fig for some i 2 T 2 and Vð½jÞ # T 1 ), then, Vð½jÞ \ ðT 1 \ T 2 Þ ¼ fig; h. if Vð½jÞ # T 1 and Vð½jÞ # T 2 , then, Vð½jÞ # T 1 \ T 2 . Therefore, T 1 \ T 2 2 EðN 0 ; CÞ.
h
Lemma A.3. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let T 1 ; T 2 2 EðN 0 ; CÞ with T 1 \ T 2 – ;. Then, (i)
P
i2T 1 nT 2 jfj k1
2 T 2 n T 1 : fi; jg 2 F k1 gj þ
P
i2T 2 nT 1 jfj
2 T1 n T2 :
fi; jg 2 F gj ¼ 0; P P k1 (ii) i2T 1 jfj 2 N 0 n T 1 : fi; jg 2 F gj þ i2T 2 jfj 2 N 0 n T 2 : fi; jg P P k1 2 F gj ¼ i2T 1 [T 2 jfj 2 N 0 n ðT 1 [ T 2 Þ : fi; jg 2 F k1 gj þ i2T 1 \ (iii)
T 2 jfj 2 N n ðT 1 \ T 2 Þ : fi; jg 2 F k1 gj; o P P n 0 k1 i2T 1 ½j 2 ½CðC Þ : Vð½jÞ \ T 1 ¼ fig þ i2T 2 jf½j 2 n P k1 ½CðC Þ : Vð½jÞ \ T 2 ¼ figgj ¼ i2T 1 [T 2 ½j 2 ½CðCk1 Þ : n P Vð½jÞ \ ðT 1 [ T 2 Þ ¼ figgj þ i2T 1 \T 2 ½j 2 ½CðCk1 Þ : Vð½jÞ\ ðT 1 \ T 2 Þ ¼ figgj.
A. Estévez-Fernández, H. Reijnierse / European Journal of Operational Research 237 (2014) 606–616
Proof. First, note that T 1 [ T 2 ; T 1 \ T 2 2 EðN 0 ; CÞ by Lemma A.2 (ii). (i) On the contrary, suppose that X X jfj 2 T 2 n T 1 : fi; jg 2 F k1 gj þ jfj 2 T 1 n T 2 : fi;jg 2 F k1 gj – 0: i2T 1 nT 2
i2T 2 nT 1
Then, there exist i1 2 T 1 n T 2 and i2 2 T 2 n T 1 with fi1 ; i2 g 2 F k1 . By definition of F k1 ; fi1 ; i2 g R [j2CðCk1 Þ EðjÞ. Let j 2 T 1 \ T 2 ; by connectivity of T 1 , there exists a path, pathði1 ; jÞ, connecting i1 and j in Ck1 using nodes of T 1 . Analogously, there exists a path, pathðj; i2 Þ, connecting j and i2 in Ck1 using nodes of T 2 . By concatenating pathði1 ; jÞ and pathðj; i2 Þ, we construct a path, pathði1 ; i2 Þ, connecting i1 and i2 without using edge fi1 ; i2 g. As a result, there exists a cycle j 2 CðCk1 Þ with fi1 ; i2 g 2 EðjÞ, establishing a contradiction. o X n o X n k k (ii) j 2 N0 n T 1 : fi;jg 2 F 1 þ j 2 N0 n T 2 : fi;jg 2 F 1 i2T 1
i2T 2
o o X n X n k k j 2 N0 n T 1 : fi;jg 2 F 1 þ j 2 N0 n T 1 : fi;jg 2 F 1
¼
i2T 1 nT 2
þ
i2T 1 \T 2
o o X n X n k k j 2 N0 n T 2 : fi;jg 2 F 1 þ j 2 N0 n T 2 : fi;jg 2 F 1
i2T 2 nT 1
i2T 1 \T 2
o n o X n k k ¼ j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 T 2 n T 1 : fi; jg 2 F 1 i2T 1 nT 2
þ
o n o X n k k j 2 N0 n ðT 1 [ T 2 Þ : fi; jg 2 F 1 þ j 2 T 2 n T 1 : fi; jg 2 F 1
i2T 1 \T 2
b. ½j 2 ½CðCk1 ÞðT 1 Þ and ½j R ½CðCk1 ÞðT 2 Þ. In this case, we have to distinguish between two new cases: Vð½jÞ \ T 2 ¼ ; and Vð½jÞ \ T 2 ¼ Vð½jÞ. b.1. Vð½jÞ \ T 2 ¼ ;. In this case, ½j 2 ½CðCk1 ÞðT 1 [ T 2 Þ and ½j R ½CðCk1 ÞðT 1 \ T 2 Þ. b.2. Vð½jÞ \ T 2 ¼ Vð½jÞ. In this case, ½j R ½CðCk1 Þ ðT 1 [ T 2 Þ and ½j 2 ½CðCk1 ÞðT 1 \ T 2 Þ. c. ½j R ½CðCk1 ÞðT 1 Þ and ½j 2 ½CðCk1 ÞðT 2 Þ. Again we distinguish between two new cases: Vð½jÞ \ T 1 ¼ ; and Vð½jÞ \ T 1 ¼ Vð½jÞ. c.1. Vð½jÞ \ T 1 ¼ ;. In this case, ½j 2 ½CðCk1 ÞðT 1 [ T 2 Þ and ½j R ½CðCk1 ÞðT 1 \ T 2 Þ. c.2. Vð½jÞ \ T 1 ¼ Vð½jÞ. In this case, ½j R ½CðCk1 Þ ðT 1 [ T 2 Þ and ½j 2 ½CðCk1 ÞðT 1 \ T 2 Þ. d. ½j R ½CðCk1 ÞðT 1 Þ and ½j R ½CðCk1 ÞðT 2 Þ. In this case, ½j R ½CðCk1 ÞðT 1 [ T 2 Þ and ½j R ½CðCk1 ÞðT 1 \ T 2 Þ. Following these cases, we have j½CðCk1 ÞðT 1 Þjþ j½CðCk1 ÞðT 2 Þj ¼ j½CðCk1 ÞðT 1 [ T 2 Þj þ j½CðCk1 ÞðT 1 \ T 2 Þj and, therefore,
o X n k ½j 2 ½CðC 1 Þ : Vð½jÞ \ T 1 ¼ fig i2T 1
o n o X n k k þ j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 T 1 n T 2 : fi;jg 2 F 1
þ
n o ¼ ½j 2 ½CðCk1 Þ : jVð½jÞ \ T 1 j ¼ 1 n o þ ½j 2 ½CðCk1 Þ : jVð½jÞ \ T 2 j ¼ 1 n o ¼ ½j 2 ½CðCk1 Þ : jVð½jÞ \ ðT 1 [ T 2 Þj ¼ 1 n o þ ½j 2 ½CðCk1 Þ : jVð½jÞ \ ðT 1 \ T 2 Þj ¼ 1 o X n k ¼ ½j 2 ½CðC 1 Þ : Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig
o n o X n k k j 2 N0 n ðT 1 [ T 2 Þ : fi; jg 2 F 1 þ j 2 T 1 n T 2 : fi; jg 2 F 1
i2T 1 \T 2
o o X n X n k k j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 T 2 n T 1 : fi;jg 2 F 1
¼
i2T 1 [T 2
þ
i2T 1 nT 2
o o X n X n k k j 2 T 2 n T 1 : fi;jg 2 F 1 þ j 2 T 1 n T 2 : fi;jg 2 F 1
i2T 1 \T 2
þ
i2T 2 nT 1
o n o X n k k j 2 N0 n ðT 1 [ T 2 Þ : fi; jg 2 F 1 þ j 2 T 1 n T 2 : fi; jg 2 F 1
i2T 1 \T 2
¼
o o X n X n k k j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 T 2 n T 1 : fi; jg 2 F 1
i2T 1 [T 2
i2T 1 [T 2
i2T 1 \T 2
o n o X n k k þ j 2 N0 n ðT 1 [ T 2 Þ : fi; jg 2 F 1 þ j 2 T 1 n T 2 : fi; jg 2 F 1
þ
i2T 1 \T 2
¼
o X n k ½j 2 ½CðC 1 Þ : Vð½jÞ \ ðT 1 \ T 2 Þ ¼ fig :
i2T 1 \T 2
o o X n X n k k j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 N0 n ðT 1 \ T 2 Þ : fi; jg 2 F 1 ;
i2T 1 [T 2
o X n k ½j 2 ½CðC 1 Þ : Vð½jÞ \ T 2 ¼ fig i2T 2
i2T 2 nT 1
þ
615
i2T 1 \T 2
in which the first equality follows from T 1 ¼ ðT 1 n T 2 Þ [ ðT 1 \ T 2 Þ and T 2 ¼ ðT 2 n T 1 Þ [ ðT 1 \ T 2 Þ. The second equality is a direct consequence of N 0 n T 1 ¼ ðN0 n ðT 1 [ T 2 ÞÞ [ ðT 2 n T 1 Þ and N 0 n T 2 ¼ ðN0 n ðT 1 [ T 2 ÞÞ [ ðT 1 n T 2 Þ. The third equality follows since T 1 [ T 2 ¼ ðT 1 n T 2 Þ [ ðT 2 n T 1 Þ [ ðT 1 \ T 2 Þ. The fourth equality is a consequence of (i) of this Lemma. The fifth equality follows because N 0 n ðT 1 \ T 2 Þ ¼ ðT 2 n T 1 Þ [ ðN 0 n ðT 1 [ T 2 ÞÞ [ ðT 1 n T 2 Þ. (iii) By definition of EðN 0 ; CÞ, if R 2 EðN 0 ; CÞ and ½j 2 ½CðCk1 Þ with Vð½jÞ \ R – ;, then, either Vð½jÞ \ R ¼ Vð½jÞ, or Vð½jÞ \ R ¼ fig for some i 2 R. As a result,
o Xn j ½j 2 ½CðCk1 Þ : Vð½jÞ \ R ¼ fig j i2R
n o ¼ j ½j 2 ½CðCk1 Þ : jVð½jÞ \ Rj ¼ 1 j:
n o For R # N, we denote ½CðCk1 ÞðRÞ ¼ ½j 2 ½CðCk1 Þ : jVð½jÞ \ Rj ¼ 1 . k1 k1 Next, we compare the sets ½CðC ÞðT 1 Þ and ½CðC ÞðT 2 Þ and the sets ½CðCk1 ÞðT 1 [ T 2 Þ and ½CðCk1 ÞðT 1 \ T 2 Þ. For this, we distinguish several cases: a. ½j 2 ½CðCk1 ÞðT 1 Þ and ½j 2 ½CðCk1 ÞðT 2 Þ. By Lemma A.2 (i), Vð½jÞ \ T 1 ¼ Vð½jÞ \ T 2 ¼ fig for some i 2 T 1 \ T 2 implying Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig and Vð½jÞ \ ðT 1 \ T 2 Þ ¼ fig. Therefore, ½j 2 ½CðCk1 ÞðT 1 [ T 2 Þ and ½j 2 ½CðCk1 Þ ðT 1 \ T 2 Þ.
Proof of Lemma 4.5. By Lemma A.2 (ii), we have that T 1 [ T 2 ; T 1 \ T 2 2 EðN 0 ; CÞ. This gives c ðT 1 Þ þ c ðT 2 Þ ¼ k1 jT 1 j ðk2 k1 Þ
o X n k j 2 N0 n T 1 : fi;jg 2 F 1
i2T 1 n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ T 1 ¼ fig 1 þ k1 jT 2 j ðk2 k1 Þ ! o n o X n k k j 2 N 0 n T 2 : fi; jg 2 F 1 þ ½j 2 ½CðC 1 Þ : Vð½jÞ \ T 2 ¼ fig 1 i2T 2
¼ k1 ðjT 1 j þ jT 2 jÞ ðk2 k1 Þ
o X n k j 2 N0 n T 1 : fi;jg 2 F 1
i2T 1 n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ T 1 ¼ fig 1 ðk2 k1 Þ ! o n o X n k1 k1 j 2 N n T : fi; jg 2 F þ ½ j 2 ½Cð C Þ : Vð½ j Þ \ T ¼ fig 1 0 2 2 i2T 2
¼ k1 jT 1 [ T 2 j ðk2 k1 Þ
o X n k j 2 N0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1
i2T 1 [T 2 n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig 1 þ k1 jT 1 \ T 2 j ðk2 k1 Þ o n X n k k j 2 N 0 n ðT 1 \ T 2 Þ : fi; jg 2 F 1 þ ½j 2 ½CðC 1 Þ : Vð½jÞ \ ðT 1 \ T 2 Þ
i2T 1 \T 2 ¼ figgj 1 ¼ c ðT 1 [ T 2 Þ þ c ðT 1 \ T 2 Þ;
in which the first and last equalities follow from Lemma A.1 and the third equality is a direct consequence of Lemma A.3 (ii) and (iii). h
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Littlechild, S. C., & Owen, G. (1976). A further note on the nucleolus of the ‘‘airport game’’. International Journal of Game Theory, 5, 91–95. Macula, A. J. (1995). Lewis Carroll and the enumeration of minimal covers. Mathematics Magazine, 68(4), 269–274. Meertens, M., & Potters, J. (2006). The nucleolus of trees with revenues. Mathematical Methods of Operations Research, 64, 363–382. Moretti (2008). Cost allocation problems arising from connection situations in an interactive cooperative setting. PhD thesis, Tilburg University. ISBN: 978-905668-218-7. Norde, H., Moretti, S., & Tijs, S. (2004). Minimum cost spanning tree games and population monotonic allocation schemes. European Journal of Operational Research, 154, 84–97. Potters, J. A., Curiel, I. J., & Tijs, S. H. (1992). Traveling salesman games. Mathematical Programming, 53, 199–211. Schmeidler, D. (1972). Cores of exact games. Journal of Mathematical Analysis and Applications, 40, 214–225. Sharkey, W. (1982). Cooperative games with large cores. International Journal of Game Theory, 11, 175–182. Suijs, J., Borm, P., Hamers, H., Quant, M., & Koster, M. (2005). Communication and cooperation in public network situations. Annals of Operations Research, 137, 117–140. Tijs, S., Brânzei, R., Moretti, S., & Norde, H. (2006). Obligation rules for minimum cost spanning tree situations and their monotonicity properties. European Journal of Operational Research, 175, 121–134.
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
On the core of cost-revenue games: Minimum cost spanning tree games with revenues q Arantza Estévez-Fernández a,⇑, Hans Reijnierse b a b
Tinbergen Institute and Department of Econometrics & OR, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands CentER and Department of Econometrics & OR, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
a r t i c l e
i n f o
Article history: Received 21 November 2012 Accepted 24 January 2014 Available online 4 February 2014 Keywords: Game theory Cost-revenue allocation problem Cooperative game Core Minimum cost spanning tree problem
a b s t r a c t In this paper, we analyze cost sharing problems arising from a general service by explicitly taking into account the generated revenues. To this cost-revenue sharing problem, we associate a cooperative game with transferable utility, called cost-revenue game. By considering cooperation among the agents using the general service, the value of a coalition is defined as the maximum net revenues that the coalition may obtain by means of cooperation. As a result, a coalition may profit from not allowing all its members to get the service that generates the revenues. We focus on the study of the core of cost-revenue games. Under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members, it is shown that a cost-revenue game has a nonempty core for any vector of revenues if, and only if, the dual game of the cost game has a large core. Using this result, we investigate minimum cost spanning tree games with revenues. We show that if every connection cost can take only two values (low or high cost), then, the corresponding minimum cost spanning tree game with revenues has a nonempty core. Furthermore, we provide an example of a minimum cost spanning tree game with revenues with an empty core where every connection cost can take only one of three values (low, medium, or high cost). Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we study cost sharing problems arising from a general service by explicitly taking into account the revenues generated by this service. In addition, we apply our findings to minimum cost spanning tree (mcst) problems with revenues. The study of cost sharing problems combined with revenues has been considered in the literature before. Littlechild and Owen (1976) initiated this stream of literature within the framework of airport problems, where they provide an algorithm to compute the nucleolus of airport games. Brânzei, Iñarra, Tijs, and Zarzuelo (2006) is a recent follow-up to Littlechild and Owen (1976). Suijs, Borm, Hamers, Quant, and Koster (2005) analyze the allocation of costs and revenues within a public network communication structure within the context of cooperative games with transferable utility. Meertens and Potters (2006) provide an algorithm to
q The authors thank two referees for their valuable suggestions for improvement. Special thanks go to Peter Borm and Ruud Hendrickx for their helpful comments in a previous version of this paper. ⇑ Corresponding author. Tel.: +31 20 5982294; fax: +31 20 5986020. E-mail addresses: [email protected] (A. Estévez-Fernández), [email protected] (H. Reijnierse).
http://dx.doi.org/10.1016/j.ejor.2014.01.056 0377-2217/Ó 2014 Elsevier B.V. All rights reserved.
compute the nucleolus of fixed tree games with revenues. Estévez-Fernández, Borm, Meertens, and Reijnierse (2009) focus on the analysis of the core of routing games with revenues. Here, we consider the situation in which a group of agents may cooperate to reduce the (possibly nonlinear) costs associated with the performance of a service. Each agent has some fixed revenue from the performance of the service. The question is how to share the net revenues obtained from cooperation among the agents. To solve this allocation problem, we use cooperative game theoretic techniques. By considering cooperation among the agents, we define a cost-revenue game where the value of a coalition is defined as the maximum net revenues that the coalition can obtain by means of cooperation. Note that with this definition, a coalition may profit from not allowing all its members to get the service that generates the revenues. In order to provide an allocation of the cost-revenue sharing problem at hand, we focus on the core of the cost-revenue game. The core of a game is the set of allocations of the total net revenues obtained by the cooperation of all agents that are stable in the sense that no group of agents has an incentive to deviate. In this paper, we characterize the class of cost-revenue games with a nonempty core under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members. It turns out that
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a cost-revenue game has a nonempty core for any vector of revenues that guarantees full service to all members of the grand coalition if, and only if, the dual game of the corresponding cost game has a large core. We apply the general results found for cost-revenue games to the study of mcst allocation problems with revenues. In an mcst allocation problem, a group of customers needs to be connected to a common information source; the customer needs a physical connection to the source (for instance, the cable for TV transmission) and this connection may be done either directly, or indirectly through other costumers. The cheapest way to connect all customers to the source is through a spanning tree that can be constructed by means of cooperation among the customers. The mcst allocation problem is, then, how to share the connection costs among the agents. Claus and Kleitman (1973) introduced this problem, which has been extensively analyzed in the literature. Granot and Huberman (1981) define mcst games and show that mcst games have a nonempty core. Granot and Huberman (1984) define allocation rules for mcst allocation problems which lead to elements of the core of the corresponding mcst games and show that, for some instances, some of these rules lead to the nucleolus of the associated game. Granot and Maschler (1998) generalize mcst games by studying spanning network games. In a spanning network situation, costs are associated to both edges and vertices and the costs may be negative (reflecting rewards). They provide sufficient conditions for non-emptiness of the core of spanning network games and characterize the set of essential coalitions. Besides, they study the computational complexity of the core and the nucleolus of this class of games, and the decomposition of the core and the nucleolus for some special cases. Kuipers, Solymosi, and Aarts (2000) provide a method to compute the nucleolus of an mcst game in polynomial time. Norde, Moretti, and Tijs (2004) show that mcst games posses population monotonic allocation schemes and provide an algorithm to find such a scheme. Tijs, Brânzei, Moretti, and Norde (2006) analyze a class of rules for mcst problems that are cost monotonic and induce monotonic allocation schemes. For a more extensive overview on the literature of mcst games, we refer to Moretti (2008). When the revenues obtained by the customers by being connected to the source are taking into account, we face an mcst allocation problem with revenues. In this paper, we define mcst games with revenues and we provide an example of an mcst game with revenues where it is profitable that all members of the grand coalition connect to the source and whose core is empty. In this example, the connection costs between two distinct players can only take one of three values: high, medium, or low cost. We show that if the connection costs can only take one of two values (high or low cost), then, the associated mcst game with revenues has a nonempty core under the assumption that it is profitable for the grand coalition to have all its members connected to the source. The remainder of the paper is organized as follows. Section 2 provides the basic definitions and terminology of graph theory and cooperative games used in this paper. In Section 3, we analyze cost-revenue games. Section 4 analyzes mcst games with revenues. We conclude with some further remarks in Section 5.
fv l ; v lþ1 g 2 E. We denote by Vðpathðv ; wÞÞ the set of nodes of pathðv ; wÞ and by Eðpathðv ; wÞÞ the set of edges of pathðv ; wÞ. Forand Eðpathðv ; wÞÞ ¼ mally, Vðpathðv ; wÞÞ ¼ fv 0 ; v 1 ; . . . ; v p g ffv 0 ; v 1 g; . . . ; fv p1 ; v p gg. A path v 0 ; . . . ; v p is called simple if all nodes are distinct. A cycle, j, is a sequence of nodes v 0 ; v 1 ; . . . ; v p with v 0 ¼ v p ; p P 3; v 0 ; v 1 ; . . . ; v p1 distinct nodes and, for each l 2 f0; . . . ; p 1g; fv l ; v lþ1 g 2 E. We denote by VðjÞ the set of nodes of j and by EðjÞ the set of edges of j. Formally, VðjÞ ¼ fv 0 ; v 1 ; . . . ; v p1 g and EðjÞ ¼ ffv 0 ; v 1 g; . . . ; fv p1 ; v p gg. Given a graph C, we denote by CðCÞ the set of cycles of C. A subset V 0 of V is called connected if, for every v ; w 2 V 0 , there is a path from v to w using only nodes of V 0 ; we denote by CðCÞ the set of all connected subsets of V, formally, CðCÞ ¼ fV 0 # V : V 0 is connectedg. A subset U of V is called maximally connected or a component if U is connected and, for any w 2 V n U; U [ fwg is not connected. Given V 0 # V, we denote by V 0jC the set of components of V 0 with respect to C. A graph C ¼ ðV; EÞ is a forest if it has no cycles and it is a tree if it is a connected forest. A rooted tree with root v 0 2 V is a tree where node v 0 is singled out. 2.2. Cooperative games A cooperative (transferable utility) game in characteristic function form is an ordered pair ðN; v Þ where N is a finite set of players and v : 2N ! R is the characteristic function satisfying v ð;Þ ¼ 0. In general, v ðSÞ represents the joint payoff that this coalition can obtain when its members decide to cooperate and is called the value of coalition S. A cooperative game can reflect costs or rewards. A game reflecting costs is denoted by a mapping c, while a game reflecting rewards is denoted by a mapping v. The following properties and definitions apply to games reflecting rewards. The central question within a cooperative framework is how to share among the players the revenues v ðNÞ that are obtained by means of cooperation. One of the most studied solution concepts in cooperative game theory is the core of a game, first introduced in Gillies (1953). The core of a game ðN; v Þ; Coreðv Þ, is the set of efficient allocations of v ðNÞ (exactly v ðNÞ is shared among the players) that are coalitionally rational (to which no coalition can reasonably object). Formally,1
Coreðv Þ ¼ x 2 RN : xðNÞ ¼ v ðNÞ; xðSÞ P v ðSÞ for all S N : The upper core of a game ðN; v Þ; Uðv Þ, is the set of not necessarily efficient allocations that are coalitionally rational. Formally,
Uðv Þ ¼ x 2 RN : xðSÞ P v ðSÞ for all S # N : Largeness of the core was first introduced in Sharkey (1982). The core of ðN; v Þ is said to be large if, for every x 2 Uðv Þ, there is a y 2 Coreðv Þ with yi 6 xi for each i 2 N. The class of games with a large core has been characterized in Estévez-Fernández (2012) by means of minimal cover inequalities. Given a finite set N, a collection C 2N n f;; Ng is called a minimal cover of N if the following two conditions are satisfied: (i) [S2C S ¼ N; (ii) [S2CnfS0 g S – N for every S0 2 C.
2. Preliminaries 2.1. Graph theory A graph C is a tuple ðV; EÞ in which V is the finite set of nodes and E is the set of edges. A subgraph ðV 0 ; E0 Þ of C is a graph with V 0 # V and E0 ¼ ffv ; wg 2 E : v ; w 2 V 0 g; we denote CjV 0 ¼ ðV 0 ; E0 Þ. A path from node v to w; pathðv ; wÞ, is a sequence of nodes v 0 ; v 1 ; . . . ; v p with v 0 ¼ v ; v p ¼ w and, for each l 2 f0; . . . ; p 1g;
By condition (i), it follows that C covers N and by condition (ii), we have that no element of C is superfluous. We denote by MCðNÞ the set of minimal covers of N. Let PðNÞ denote the set of partitions of N. Note that PðNÞ n ffNgg # MCðNÞ. 1
Here and further, for any vector x 2 RN , we denote xðSÞ :¼
P
i2S
xi .
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Given a finite set N and C 2 MCðNÞ, for every i 2 N, we denote by nðC; iÞ the number of times that i is ‘‘extra covered’’ in C; formally, nðC; iÞ ¼ jfS 2 C : i 2 Sgj 1. We denote by EðCÞ the set of elements of N that are ‘‘extra covered’’ by C; formally, EðCÞ ¼ fi 2 N : nðC; iÞ > 0g. EðCÞ Given C 2 MCðNÞ with EðCÞ – ;, a vector l 2 R2þ nf;g is said to be a vector of minimal cover balanced weights (mc-balanced vector) for C if
X
lS e S ¼
S22EðCÞ nf;g
X
nðC; iÞefig
ð2:1Þ
i2EðCÞ
S2C
X
lS eS
ð2:2Þ
S22EðCÞ nf;g
for every l 2 MðCÞ. Note that if EðCÞ ¼ ;, then, C is a non-trivial parP tition of N and S2C eS ¼ eN . Given a game ðN; v Þ and a minimal cover C 2 MCðNÞ, the minimal cover inequality associated with C is
8 < X
X S2C
v ðSÞ 6 v ðNÞ þ lmax 2MðCÞ:
9 = lS v ðSÞ : ;
ð2:3Þ
S22EðCÞ nf;g
Theorem 2.1 Estévez-Fernández, 2012. A game ðN; v Þ has a large core if, and only if, all minimal cover inequalities are satisfied. Macula (1995) provides the number of minimal covers of a set N k Pn 1 Pak 2 k 1 m!Sðn; mÞ where with jNj ¼ n, which is k¼2 k! m¼k mk k
ak ¼ minfn; 2 1g and Sðn; mÞ is the Stirling number of the second Pm
m n kind, which is given by Sðn; mÞ ¼ l . However, l¼1 ð1Þ l in some situations, analyzing largeness of the core of a game may be easier. Let ðN; v Þ be a game, a coalition S # N is called essential if, for every non-trivial partition hS1 ; . . . ; St i of P S; v ðSÞ > tl¼1 v ðSl Þ. We denote by Essðv Þ the set of essential coalitions of ðN; v Þ. Notice that only essential coalitions are needed for the description of both the core and the upper core of a game. Therefore, following the comments in the Final Remarks of Estévez-Fernández (2012), we can rewrite Theorem 2.1 as follows: 1 m!
ml
A game ðN; v Þ has a large core if, and only if,
X S2C
LðcÞ ¼ x 2 RN : xðSÞ 6 cðSÞ for all S # N : Following the definition of largeness of the core for reward games, we say that the cost game ðN; cÞ has a large anti core if, for any x 2 LðcÞ, there is a y 2 ACoreðcÞ with xi 6 yi for each i 2 N. The dual game of ðN; cÞ; ðN; c Þ, is defined, for every S # N, by
c ðSÞ ¼ cðNÞ cðN n SÞ: It is easily seen that ACoreðcÞ ¼ Coreðc Þ.
with eR 2 RN defined by eRi ¼ 1 if i 2 R and eRi ¼ 0 if i R R for every R 2 2N n f;g. We denote by MðCÞ the set of mc-balanced vectors for C. Trivially, MðCÞ – ; if EðCÞ – ;. It follows that
X eS ¼ eN þ
The lower core of a cost game ðN; cÞ; Lðv Þ, is the set of not necessarily efficient allocations that are coalitionally rational. Formally,
8 < X
v ðSÞ 6 v ðNÞ þ lmax 2MðCÞ:
S22
EðCÞ
lS v ðSÞ
nf;g
9 = ;
for every C 2 MCðNÞ such that C # Essðv Þ. A game ðN; v Þ is said to be exact (Schmeidler, 1972) if, for every S N, there exists x 2 Coreðv Þ such that xðSÞ ¼ v ðSÞ. Given a forest C ¼ ðN; EÞ, the game ðN; v Þ is said Ptto be forest essential with respect to2 C if v ðNÞ ¼ maxhS1 ;...;St i2PðNÞ l¼1 v ðSl Þ and all essential coalitions are connected with respect to C, (Essðv Þ # CðCÞ). If C is a tree, we say that ðN; v Þ is tree-essential with respect to C. Let ðN; cÞ be a cost game. The anti core of ðN; cÞ; ACoreðcÞ, is the set of efficient allocations of cðNÞ to which no coalition can reasonably object. Formally,
ACoreðcÞ ¼ x 2 RN : xðNÞ ¼ cðNÞ; xðSÞ 6 cðSÞ for all S N :
3. Cost-revenue games In this section, we analyze situations in which a group of agents cooperate in order to obtain a higher joint revenue. Each time a coalition decides to cooperate, an additive reward is obtained and a (possibly nonadditive) cost is generated. The problem at hand is how to share the net revenue that is obtained by the grand coalition among the agents. We define cost-revenue games and concentrate on the study of their cores. It turns out that a costrevenue game has a nonempty core for any vector of revenues that provides maximal cooperation if, and only if, the dual game of the cost game has a large core. Let ðN; cÞ be a cost game. Consider the case in which, next to the costs that a coalition must confront, the coalition also obtains revenues when it decides to cooperate. Let b 2 RNþ be the vector of revenues, where bi is the revenue that player i generates if i gets the service under consideration. Then, the total revenue that a coalition S # N can obtain by cooperation is
pb ðSÞ ¼ bðSÞ cðSÞ: Notice that, due to the revenue structure of the game, it may be more profitable for coalition S not to form as a whole. Given a cost game ðN; cÞ and a vector of revenues b 2 RNþ , we define the costrevenue game, ðN; v b Þ, for every S # N, by
v b ðSÞ ¼ max fpb ðRÞg: R#S The following example illustrates the computation of costrevenues games and shows that full participation of the grand coalition (v b ðNÞ ¼ pb ðNÞ) is relevant in order to have non-emptiness of the core of the corresponding cost-revenue game. In order to exclude situations like this example we will introduce later on Assumption 3.1. Example 3.1. Consider the six players cost game ðN; cÞ with characteristic function defined by if jSj ¼ 1; cðSÞ ¼
if jSj ¼ 2;
if jSj ¼ 3;
if jSj ¼ 4; if jSj ¼ 5;
2 if S f1;2;3g;
1 8 > <4 cðSÞ ¼ 3 > : 2 8 > <6 cðSÞ ¼ 4 > : 3 5 cðSÞ ¼ 4 cðSÞ ¼ 5;
otherwise; if S f1;2;3g; if S 2 ff1;6g;f2;5g;f3;4gg; otherwise; if S ¼ f1;2;3g; if S 2 ff1;2;5g;f1;2;6g;f1;3;4g;f1;3;6g;f2;3;4g;f2;3;5gg; otherwise; if S 2 ff1;2;3;4g;f1;2;3;5g;f1;2;3;6gg; otherwise;
if S ¼ N; cðSÞ ¼ 6:
Let b ¼ ð2; 2; 2; 0; 0; 0Þ. The corresponding cost-revenue-game is given by 2 In fact, a forest-essential game with respect to C is a game with the total dependency property (see Kuipers et al. (2000)) and where the grand coalition is stable (see Derks & Kuipers (1997)).
v b ðSÞ ¼
1 if f1; 2; 4g # S; or f1; 3; 5g # S; or f2; 3; 6g # S; 0
otherwise:
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A. Estévez-Fernández, H. Reijnierse / European Journal of Operational Research 237 (2014) 606–616 Table 1 Coalitional values of the cost game, dual game, additive revenues, and cost-revenue game in Example 3.2. S
f1g
f2g
f3g
f4g
f5g
f1; 2g
f1; 3g
f1; 4g
f1; 5g
f2; 3g
f2; 4g
f2; 5g
f3; 4g
f3; 5g
f4; 5g
cðSÞ c ðSÞ bðSÞ v b ðSÞ
1 0 1 0
1 0 1 0
1 0 1 0
2 0 1 0
1 0 0 0
2 0 2 0
1 1 2 1
2 1 2 0
2 0 1 0
1 1 2 1
2 1 2 0
2 0 1 0
3 0 2 0
1 1 1 0
2 1 1 0
S
f1; 2; 3g
f1; 2; 4g
f1; 2; 5g
f1; 3; 4g
f1; 3; 5g
f1; 4; 5g
f2; 3; 4g
f2; 3; 5g
f2; 4; 5g
f3; 4; 5g
cðSÞ c ðSÞ bðSÞ v b ðSÞ
2 1 3 1
2 2 3 1
3 0 2 0
3 1 3 1
2 1 2 1
2 2 2 0
3 1 3 1
2 1 2 1
2 2 2 0
3 1 2 0
S
f1; 2; 3; 4g
f1; 2; 3; 5g
f1; 2; 4; 5g
f1; 3; 4; 5g
f2; 3; 4; 5g
N
cðSÞ c ðSÞ bðSÞ v b ðSÞ
3 2 4 1
3 1 3 1
3 2 3 1
3 2 3 1
3 2 3 1
3 3 4 1
Note that v b ðNÞ ¼ 1 > 0 ¼ bðNÞ cðNÞ. Below, we explain how to compute the value of coalition f1; 2; 4g.
v b ðf1;2;4gÞ ¼ maxfpb ðf;gÞ; pb ðf1gÞ; pb ðf2gÞ; pb ðf4gÞ; pb ðf1;2gÞ; pb ðf1;4gÞ; pb ðf2;4gÞ; pb ðf1;2;4gÞg ¼ maxf0;2 2;2 2;0 1;4 4;2 2;2 2;4 3g ¼ maxf0;0;0;1;0;0;0;1g ¼ 1: Note that ðN; cÞ has a nonempty anti core since ð1; 1; 1; 1; 1; 1Þ 2 ACoreðcÞ. However, ðN; v b Þ has an empty core. To show this, suppose that x 2 Coreðv b Þ, then, 1 6 x1 þ x2 þ x4 , 1 6 x1 þ x3 þ x5 , 1 6 x2 þ x3 þ x6 , 0 6 x4 þ x5 þ x6 . By adding all these equations, we obtain 3 6 2xðNÞ ¼ 2v b ðNÞ ¼ 2, establishing a contradiction. Here, the first equality follows from x 2 Coreðv b Þ. As we have seen in the example above, when not all members of the grand coalition get the service that generates the revenues, the core of the cost-revenue game may be empty. For that reason, we assume3 from here that it is profitable to form the grand coalition. Assumption 3.1.
v b ðNÞ ¼ pb ðNÞ ¼ bðNÞ cðNÞ:
ð3:1Þ
Example 3.2. Consider the five players exact cost game ðN; cÞ with5 ACoreðcÞ ¼ convfð0; 0; 1; 2; 0Þ, ð1; 1; 0; 0; 1Þg, and let b ¼ ð1; 1; 1; 1; 0Þ. In Table 1, we provide the coalitional values of ðN; cÞ; ðN; c Þ; ðN; bÞ and ðN; v b Þ. Note that bðSÞ P c ðSÞ for every S # N. However, ðN; v b Þ has an empty core. To show this, suppose that x 2 Coreðv b Þ, then, 1 6 x1 þ x2 þ x3 þ x4 , 1 6 x1 þ x2 þ x3 þ x5 , 1 6 x1 þ x2 þ x4 þ x5 , 1 6 x1 þ x3 þ x4 þ x5 , 1 6 x2 þ x3 þ x4 þ x5 . By adding all these equations, we obtain 5 6 4xðNÞ ¼ 4v b ðNÞ ¼ 4, establishing a contradiction. Here, the first equality follows from x 2 Coreðv b Þ. The first result of this section states that if the core of a costrevenue game is nonempty, then, its core can be expressed in terms of the vector of revenues and the anti core of the cost game. Lemma 3.1. Let ðN; cÞ be a cost game, let b 2 RN þ be a vector of revenues satisfying Assumption 3.1, and let ðN; v b Þ be the corresponding cost-revenue game. Then,
Coreðv b Þ ¼ fb y : y 2 ACoreðcÞ with y 6 bg: Proof. We first show ‘‘ # ’’. If Coreðv b Þ ¼ ;, we are done. Let x 2 Coreðv b Þ and let y ¼ b x, we have to show that (i) y 2 ACoreðcÞ and (ii) y 6 b.
Taking into account that pb ðNÞ ¼ maxR # N fbðRÞ cðRÞg, it follows that Assumption 3.1 is equivalent to
bðSÞ P c ðSÞ for every S # N:
(i) Let S N. We have
bðSÞ cðSÞ 6 maxfbðRÞ cðRÞg ¼ v b ðSÞ 6 xðSÞ;
ð3:2Þ
R#S
This is because v b ðNÞ ¼ bðNÞ cðNÞ is equivalent to bðNÞ cðNÞ P bðRÞ cðRÞ for every R N, which, taking R ¼ N n S, reduces to Eq. (3.2). Assumption 3.1 may seem somehow restrictive. It ensures that the grand coalition will effectively form without leaving any player behind. The following example shows that non-emptiness of the anti core of the cost game is not sufficient to have non-emptiness of the core of the corresponding cost-revenue game, even when the cost game is exact.4 Therefore, revenues have a real impact in the structure of the core of cost-revenue games. It further shows why the game v b of Example 3.2 has an empty core; there are no elements of AcoreðcÞ that are dominated by b.
where the last inequality follows because x 2 Coreðv b Þ. Therefore,
yðSÞ ¼ bðSÞ xðSÞ 6 cðSÞ: bðNÞ cðNÞ ¼ v b ðNÞ ¼ xðNÞ;
where the first equality follows by Assumption 3.1 and the second one because x 2 Coreðv b Þ. This gives
yðNÞ ¼ bðNÞ xðNÞ ¼ cðNÞ:
ð3:4Þ
By Eqs. (3.3) and (3.4), we have that y 2 ACoreðcÞ. (ii) Core elements are in particular imputations, so for every i 2 N,
xi P v b ðfigÞ P 0 3
This assumption is however not mandatory for having a nonempty core. Think of a situation with v b ðNÞ ¼ 0 . 4 Similarly as in (reward) games, a cost game is exact if, for every S N, there is an x 2 ACoreðcÞ such that xðSÞ ¼ cðSÞ.
ð3:3Þ
Furthermore,
and, therefore, yi ¼ bi xi 6 bi for every i 2 N. 5
By conv we refer to the convex hull.
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Next, we show ‘‘’’. If there does not exist a y 2 ACoreðcÞ satisfying y 6 b, then, we are done. Let x ¼ b y with y 2 ACoreðcÞ and y 6 b. We have
xðNÞ ¼ bðNÞ yðNÞ ¼ bðNÞ cðNÞ ¼ v b ðNÞ;
ð3:5Þ
where the second equality follows from y 2 ACoreðcÞ and the third one is a direct consequence of Assumption 3.1. Furthermore, for every S N, let RS # S be such that v b ðSÞ ¼ pb ðRS Þ ¼ bðRS Þ cðRS Þ. We have that
xðSÞ P xðRS Þ ¼ bðRS Þ yðRS Þ P bðRS Þ cðRS Þ ¼ v b ðSÞ;
ð3:6Þ
where the first inequality is a direct consequence of x ¼ b y P 0 and RS # S, and the second inequality follows by y 2 ACoreðcÞ. Eqs. (3.5) and (3.6) imply that x 2 Coreðv b Þ. h The following result characterizes the class of cost games in which, for any vector of revenues satisfying Assumption 3.1, it follows that the corresponding cost-revenue game has a nonempty core. Theorem 3.2. Let ðN; cÞ be a cost game. Then, ðN; v b Þ has a nonempty core for any b 2 RN þ satisfying Assumption 3.1 if, and only if, ðN; c Þ has a large core.
between two different villages can only take one of, at most, two possible values (low and high cost). We call such games 2-mcst games with revenues. Second, we answer this question by showing that, under Assumption 3.1, every 2-mcst game with revenues has a nonempty core. Let N ¼ f1; 2; . . . ; ng denote the set of villages that have to be connected to a source. Let N 0 ¼ N [ f0g, where 0 denotes the source. Let C ¼ ðcij Þ be an N 0 N 0 -matrix, where cij P 0, for every i; j 2 N 0 , represents the costs to go from village i to village j. Since the cost matrix is nonnegative, it turns out that the graph that connects all villages to the source with a minimum cost is a tree. Throughout this article we assume that:
ðiÞ cii ¼ 0 for all i 2 N0 ; ðiiÞ cij ¼ cji
for all i; j 2 N0
ðsymmetryÞ:
The network ðN 0 ; CÞ is usually represented by the complete graph on N 0 with costs cij on the edge between nodes i and j. Given a network ðN 0 ; CÞ and a subset S # N, we denote S0 :¼ S [ f0g. A tree ðV; EÞ is called a spanning tree of S if V ¼ S0 . Let TðSÞ denote the set of all spanning trees of S and, for C ¼ ðS0 ; EÞ 2 TðSÞ, let cðC; SÞ denote the cost associated with the tree, formally,
cðC; SÞ ¼
X
cij :
fi;jg2E
Proof. We first show the ‘‘if’’ part. Let ðN; c Þ have a large core. We have to show that ðN; v b Þ has a nonempty core for any b 2 RNþ satisfying Assumption 3.1. Such a b satisfies bðSÞ P c ðSÞ for every S # N and, therefore, b 2 Uðc Þ. By largeness of the core of ðN; c Þ, there is a yb 2 Coreðc Þ ¼ ACoreðcÞ with b P yb . By Lemma 3.1, it follows that b yb 2 Coreðv b Þ. In order to show the ‘‘only if’’ part, let ðN; cÞ be such that ðN; v b Þ has a nonempty core for every b 2 RN þ satisfying Assumption 3.1. We have to show that ðN; c Þ has a large core. Let b 2 Uðc Þ. Then, b satisfies Assumption 3.1 since bðSÞ P c ðSÞ for every S # N. Choose any x 2 Coreðv b Þ. First, we have that b P b x since x P 0 by non-negativity of ðN; v b Þ; second, b x 2 ACoreðcÞ ¼ Coreðc Þ by Lemma 3.1. Therefore, ðN; c Þ has a large core. h The following example illustrates that largeness of the core of ðN; c Þ and largeness of the anti core of ðN; cÞ can be unrelated.
By associating each village in N with a player, an mcst game, ðN; cÞ, is defined, for every S # N, by
cðSÞ :¼ min fcðC; SÞg: C2TðSÞ
We add revenues to the situation by assuming that if village i 2 N is connected to the source, a revenue bi P 0 is obtained. An mcst problem with revenues is denoted by a tuple ðN 0 ; C; bÞ and the corresponding mcst game with revenues by ðN; v b Þ. Note that, under Assumption 3.1, we have v b ðNÞ ¼ bðNÞ cðNÞ. Example 4.1. Consider the mcst problem with revenues represented in Fig. 1, in which the numbers at the edges represent the connection costs and the boldfaced numbers at the nodes represent the revenues. The value of coalition f1; 2g is computed as follows:
v b ðf1; 2gÞ ¼ max f0; b1 cðf1gÞ; b2 cðf2gÞ; b1 þ b2 cðf1; 2gÞg Example 3.3. Consider the cost game ðN; cÞ given in Table 2. It can be easily seen that Coreðc Þ ¼ ACoreðcÞ ¼ fð1; 1; 1; 1Þg. Since, for any x 2 Uðc Þ; xi P 1 for every i 2 N, the core of ðN; c Þ is large. On the other hand, it can be checked that x ¼ ð1; 1:1; 0:9; 0:9Þ 2 LðcÞ. Therefore, the anti core of ðN; cÞ is not large since x2 ¼ 1:1 > 1 ¼ y2 with y ¼ ð1; 1; 1; 1Þ and, hence, there is no y 2 ACoreðcÞ ¼ fð1; 1; 1; 1Þg with xi 6 yi for every i 2 N. 4. Minimum cost spanning tree games with revenues In this section, we study minimum cost spanning tree (mcst) games with revenues. First, we provide an example of an mcst game with revenues that has an empty core. In this example, the connection cost between two different villages can only take one out of three possible values (low, medium, and high cost). Therefore, the arising question is whether we can find an mcst game with revenues with an empty core in which the connection cost
¼ max f0; 6 5; 2 10; 6 þ 2 10g ¼ max f0; 1; 8; 2g ¼ 1: The associated mcst game with revenues is given in Table 3. It can be checked that Coreðv b Þ ¼ convfð3; 0; 4Þ; ð1; 2; 4Þ; ð1; 0; 6Þg. In general, mcst games with revenues satisfying Assumption 3.1 can have an empty core as the following example illustrates. Example 4.2. Consider the mcst network N ¼ f1; 2; 3; 4; 5; 6; 7g and cost matrix defined by
8 0 > > > < 3 cij ¼ > >4 > : 5
ðN 0 ; CÞ
with
if i ¼ j; if fi;jg 2 ff0;1g;f1;2g;f1;3g;f1;4g;f2;7g;f3;6g;f4;5gg; if fi;jg 2 ff0;2g;f0;3g;f0;4g;f2;5g;f3;7g;f4;6gg; otherwise:
Table 2 Coalitional values of the cost game and dual game in Example 3.3. S
f1g
f2g
f3g
f4g
f1; 2g
f1; 3g
f1; 4g
f2; 3g
f2; 4g
f3; 4g
f1; 2; 3g
f1; 2; 4g
f1; 3; 4g
f2; 3; 4g
N
cðSÞ c ðSÞ
2 1
2 1
1 1
1 1
4 2
2 2
2 2
2 2
2 2
2 0
3 3
3 3
3 2
3 2
4 4
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611
is a node connected to all other nodes (the graph is a star), then, there are 2n1 þ n 1; n ¼ jNj, connected coalitions. In this case, the number of minimal covers using only essential coalitions exceeds k Pn1 1 Pak 2 k1 m!Sðn1;mÞ, in which a ¼ minfn1;2k 1g k k¼2 k! m¼k mk Pm n1 ml m 1 and Sðn 1; mÞ ¼ m! l . l¼1 ð1Þ l In order to describe the collection of essential coalitions with respect to ðN; c Þ, we introduce some extra notation. Given a 2-mcst network ðN 0 ; CÞ, we define the k1 -graph Ck1 by N 0 ; Ek1 ,
Fig. 1. The mcst problem with revenues in Example 4.1.
where
Ek1 ¼ fi; jg N0 : cij ¼ k1 :
Table 3 Coalitional values of the mcst game with revenues in Example 4.1.
Let F k1 # Ek1 be the set of edges that belong to any mcst for N, i.e.,
S
f1g
f2g
f3g
f1; 2g
f1; 3g
f2; 3g
N
v b ðSÞ
1
0
4
1
5
4
7
Note that ðN; c Þ does not have a large core by considering the minimal cover C ¼ ff1; 2; 4; 5g; f1; 2; 3; 7g; f1; 3; 4; 6gg. Since c ðf1; 2; 4; 5gÞ ¼ c ðf1; 2; 3; 7gÞ ¼ c ðf1; 3; 4; 6gÞ ¼ 10; c ðNÞ ¼ 21, the inequality
c ðf1; 2; 4; 5gÞ þ c ðf1; 2; 3; 7gÞ þ c ðf1; 3; 4; 6gÞ ¼ 30 > 21 þ 8 9 8 = < X ¼ c ðNÞ þ max lS c ðSÞ ; ; l2MðCÞ: EðCÞ S22
nf;g
shows the non-largeness of the core of c . As an optimal solution to ^ can be chothe involved maximization problem above, the vector l ^ S ¼ 1 if S 2 ff1; 2; 3g; f1; 4gg and l ^ S ¼ 0 othersen to be defined by l wise. It can be checked that b ¼ ð1; 2; 2; 2; 5; 5; 5Þ satisfies Assumption 3.1 and ðN; v b Þ has an empty core since v b ðf4gÞ ¼ v b ðf1; 2gÞ ¼ v b ðf1; 3gÞ ¼ 0; v b ðf2; 5; 7gÞ ¼ v b ðf3; 6; 7gÞ ¼ v b ðf4; 5; 6gÞ ¼ v b ðNÞ ¼ 1 and, therefore, if x 2 Coreðv b Þ, then, 0 6 x4 ; 0 6 x1 þ x2 ; 0 6 x1 þ x3 ; 1 6 x2 þ x5 þ x7 ; 1 6 x3 þ x6 þ x7 ; 1 6 P x4 þ x5 þ x6 and 7i¼1 xi ¼ 1. By adding the inequalities, we obtain P7 3 6 2 i¼1 xi ¼ 2, establishing a contradiction. Every connection cost in the mcst network of Example 4.2 can take one of three given values: 3; 4, or 5. We now will analyze the case in which connection costs between two different villages in an mcst network can take, at most, two values: low and high cost. It will turn out that, under Assumption 3.1, the corresponding mcst game with revenues always has a nonempty core. We refer to this type of mcst network as a 2-mcst network. Formally, ðN 0 ; CÞ is a 2-mcst network if, for every i; j 2 N 0 ; i – j; cij 2 fk1 ; k2 g with 0 6 k1 6 k2 . We will show non-emptiness of the core of 2-mcst games with revenues under Assumption 3.1. Their duals will play a crucial role. Therefore, we first provide some results on the dual game of a 2mcst game. The first result states that the dual game of an mcst b for the game is tree-essential with respect to any optimal tree C grand coalition. It is based on results from Kuipers (1994) (see Theorem 4.4 in Kuipers (1994) and comments below) and the proof is, therefore, omitted. Theorem 4.1 Kuipers, 1994. Let ðN 0 ; CÞ be an mcst network and let b be an optimal mcst for N. ðN; cÞ be the associated mcst game. Let C Then, the associated dual game ðN; c Þ is forest-essential with respect b jN . to C This result served as the base of a polynomial time algorithm by Kuipers (1994) for computing the nucleolus of mcst games. Unfortunately, it does not limit the number of essential coalitions to polyb jN is such that there nomially bounded proportions. For instance, if C
F k1 ¼
8 < E k1
if Ck1 is a forest; : E n [j2CðCk1 Þ EðjÞ if Ck1 is not a forest: k1
With minor abuse of language, we write essential coalition when we refer to an essential coalition with respect to ðN; c Þ. By Theorem 4.1, all essential coalitions are connected in every mcst for N. Since F k1 is contained in any mcst for N, every connected coalition in F k1 is also connected in any mcst for N. We will now analyze those coalitions that are connected in every mcst for N and that have a nonempty intersection with [j2CðCk1 Þ VðjÞ. Lemma 4.2. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let S # N with S 2 CðCk1 Þ. If there exists j 2 CðCk1 Þ with 1 < jS \ VðjÞj < b jN Þ for at least one mcst C b for N. jVðjÞj, then, S R Cð C Proof. Let j 2 CðCk1 Þ with 1 < jS \ VðjÞj < jVðjÞj. Let i; j 2 S \ VðjÞ and let l 2 VðjÞ n S. Then, there are two disjoint paths in j connecting i and j, one of them visiting l. Fix an edge e on the path that does b not visit l. Using Kruskal’s algorithm, we can find an mcst C containing all edges of j with the exception of e. The only path b Þ visits village l. connecting i and j that uses only edges in Eð C b. h Therefore, S is not connected in C As an immediate consequence of Theorem 4.1 and Lemma 4.2, we have the following corollary: if an essential coalition has a nonempty intersection with the set of nodes of a cycle of the k1 -graph, then, the intersection is either a singleton, or the set of nodes of the cycle. Corollary 4.3. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, let ðN; cÞ be the associated 2-mcst game, and let S 2 Essðc Þ. If there exists
j 2 CðCk1 Þ with S \ VðjÞ – ;, then, either S \ VðjÞ ¼ VðjÞ, or S \ VðjÞ ¼ fig for some i 2 S. Given a 2-mcst network ðN 0 ; CÞ with k1 -graph Ck1 and 2-mcst game ðN; cÞ, as a result of Corollary 4.3, we have that if two cycles
j1 ; j2 2 CðCk1 Þ share two nodes, then, an essential coalition, S, of the dual game satisfies that either jS \ ðVðj1 Þ [ Vðj2 ÞÞj 6 1, or Vðj1 Þ [ Vðj2 Þ # S. Following this, we define an equivalence relation in order to describe the set of essential coalitions of the dual game of a 2-mcst game. Let C ¼ ðN 0 ; EÞ be a graph and let j1 ; j2 2 CðCÞ. We say that j1 and j2 are related, j1 Rj2 , if jVðj1 Þ \ Vðj2 Þj P 2. The binary relation R is reflexive and symmetric, but does not need to be transitive. Therefore, we consider the transitive closure of R; Rþ , which is an equivalence relation. We denote by ½j the equivalence class of j 2 CðCÞ with respect to Rþ and by ½CðCÞ the set of equivalence
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Fig. 2. Example of the numbering of C in the proof of Theorem 4.6.
classes in C, so ½CðCÞ ¼ f½j : j 2 CðCÞg. Given j 2 CðCÞ, we denote ~ Þ and Eð½jÞ ¼ [j~ 2½j Eðj ~ Þ. Vð½jÞ ¼ [j~ 2½j Vðj Let ðN 0 ; CÞ be a 2-mcst network. We define EðN 0 ; CÞ by
l1 l2 lr lr X X X X 1 c ðT l Þ ¼ c ðT l Þ þ c ðT l Þ þ þ c ðT l Þ ¼ c [ll¼1 Tl l¼1
l¼1
n
EðN0 ; CÞ ¼ S 2 CðCk1 Þ : jS \ Vð½jÞj 2 f0; 1; jVð½jÞjg o for every ½j 2 ½CðCk1 Þ :
þ
l¼l1 þ1
l¼lr1 þ1
r c [ll¼1 Tl \ T lþ1 þ þ c [ll¼l T l þ1 r1
l1 1 X l¼1
þ
The following result is a direct consequence of Theorem 4.1 and Corollary 4.3 and the proof is, therefore, omitted.
lr 1 X
c
r X s [ll¼lr1 þ1 Tl \ T lþ1 ¼ c [ll¼l Tl s1 þ1 s¼1
l¼lr1 þ1
þ Corollary 4.4. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let ðN; cÞ be the associated 2-mcst game. Then, Essðc Þ # EðN 0 ; CÞ. Another crucial result states that if two essential coalitions have a nonempty intersection, then, the sum of their dual values equals the sum of the dual values of their union and intersection. Due to the technicality of the proof, it is postponed to Appendix A.
r X
ls 1 X
s¼1 l¼ls1 þ1
þ
ls 1 r X X s¼1 l¼ls1 þ1
þ max
8 <
l2MðCÞ:
c [ll¼ls1 þ1 Tl \ T lþ1 6 c ðNÞ c
[ll¼ls1 þ1 Tl \ T lþ1 6 c ðNÞ
X
9 = lT c ðTÞ ; ;
T22EðCÞ nf;g
Lemma 4.5. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let ðN; cÞ be the associated 2-mcst game. Then,
in which the second equality is a consequence of successively applying Lemma 4.5 and the first inequality is valid because n o 1 r [ll¼1 T l ; . . . ; [ll¼l T l is a partition of N and ðN; c Þ is forest-essenr1 þ1
c ðT 1 Þ þ c ðT 2 Þ ¼ c ðT 1 \ T 2 Þ þ c ðT 1 [ T 2 Þ
tial with respect to any mcst C for N by Theorem 4.1. Finally, the last inequality follows because l defined by lT ¼ 1 for every T ¼ [ll¼l þ1 Tl \ T lþ1 with s 2 f1; . . . ; rg and l 2 fls1 þ 1; . . . ;
for every T 1 ; T 2 2 EðN 0 ; CÞ with T 1 \ T 2 – ;. We can finally provide the main result of this Section.
s1
ls 1g and Theorem 4.6. Every 2-mcst game Assumption 3.1 has a nonempty core.
with
revenues
Proof. Let ðN 0 ; CÞ be a 2-mcst network and let ðN; cÞ be the associated 2-mcst game. Let b 2 RNþ be a vector of revenues satisfying Assumption 3.1. By Theorem 3.2, it suffices to show that ðN; c Þ has a large core. By Theorem 2.1, we have to show that, for all C 2 MCðNÞ,
8 < X c ðTÞ 6 c ðNÞ þ max l2MðCÞ: T2C
X
lT ¼ 0 otherwise, is an mc-balanced vector for C.
h
satisfying
9 = lT c ðTÞ : ;
T22EðCÞ nf;g
As already mentioned in Section 2, we can restrict our analysis to minimal covers whose elements are essential coalitions of ðN; c Þ. Therefore, fix C 2 MCðNÞ such that T 2 Essðc Þ for every T 2 C. By Corollary 4.4, T 2 EðN 0 ; CÞ for every T 2 C. Let C ¼ fT 1 ; . . . ; T l1 ; . . . ; T lr1 ; . . . ; T lr g be ordered as follows: 1 1 1 r T 1 \ T 2 – ;; . . . ; [ll¼1 T l \ T l1 – ;; [ll¼1 T l \ [ll¼l T l ¼ ; and for 1 þ1 s 1 T l \ T ls – ; every s 2 f2; . . . ; rg, T ls1 þ1 \ T ls1 þ2 – ;; . . . ; [ll¼l s1 þ1 s r and [ll¼l T l \ [ll¼l T l ¼ ; (see Fig. 2). Note that s þ1 s1 þ1 n o 1 r is a partition of N. We have T l ; . . . ; [ll¼l T [ll¼1 l þ1 r1
It remains an open and very interesting question whether it is possible to verify Assumption 3.1 for an mcst game with revenues in polynomial time. To conclude this Section, note that the cost game in Example 3.1 is a 2-mcst game with k1 ¼ 1; k2 ¼ 2, and C defined by cij ¼ 1 if fi; jg 2 ff0; 4g; f0; 5g; f0; 6g; f1; 4g; f1; 5g; f2; 4g; f2; 6g; f3; 5g; f3; 6gg and cij ¼ 2 otherwise. Therefore, Assumption 3.1 is, indeed, necessary in Theorem 4.6. 5. Further remarks Littlechild and Owen (1976) were the first to analyze cost problems arising from a general service facility by taking into account the revenues that the service generates. They restricted their study to the framework of airport problems, with a more recent follow up by Brânzei et al. (2006). Meertens and Potters (2006) consider fixed tree games with revenues. In both settings, the underlying cost game is concave6 and, therefore, its dual game is convex7 and has a large core (see Sharkey, 1982). 6 A cost game ðN; cÞ is concave if, for every i 2 N and every S T # N n fig; cðS [ figÞ cðSÞ P cðT [ figÞ cðTÞ. 7 A game ðN; v Þ is convex if, for every i 2 N and every S T # N n fig; v ðS [ figÞ v ðSÞ 6 v ðT [ figÞ v ðTÞ.
613
A. Estévez-Fernández, H. Reijnierse / European Journal of Operational Research 237 (2014) 606–616 Table 4 Coalitional values of the routing game, associated dual game, additive revenues, and routing game with revenues in Example 5.1. S
f1g
f2g
f3g
f4g
f5g
f1; 2g
f1; 3g
f1; 4g
f1; 5g
f2; 3g
f2; 4g
f2; 5g
f3; 4g
f3; 5g
f4; 5g
cp^ ðSÞ cp^ ðSÞ bðSÞ v p^ b ðSÞ
4 3 10 6
2 0 10 8
20 10 10.5 0
4 16 10 6
20 10 10 0
5 4 20 15
24 13 20.5 6
5 13 20 15
24 13 20 6
21 13 20.5 8
5 16 20 15
22 10 20 8
14 3 20.5 6.5
40 20 20.5 0
14 4 20 6
S
f1; 2; 3g
f1; 2; 4g
f1; 2; 5g
f1; 3; 4g
f1; 3; 5g
f1; 4; 5g
f2; 3; 4g
f2; 3; 5g
f2; 4; 5g
f3; 4; 5g
cp^ ðSÞ cp^ ðSÞ bðSÞ v p^ b ðSÞ
24 14 30.5 15
8 12 30 22
25 14 30 15
18 6 30.5 15
44 23 30.5 6
15 7 30 15
15 4 30.5 15.5
41 23 30.5 8
15 4 30 15
24 23 30.5 6.5
S
f1; 2; 3; 4g
f1; 2; 3; 5g
f1; 2; 4; 5g
f1; 3; 4; 5g
f2; 3; 4; 5g
N
cp^ ðSÞ cp^ ðSÞ bðSÞ v p^ b ðSÞ
18 8 40.5 22.5
44 24 40.5 15
18 8 40 22
28 26 40.5 15
25 24 40.5 15.5
28 28 50.5 22.5
Granot and Maschler (1998) consider spanning network situations. These situations generalize mcst situations in the sense that they allow for ‘Steiner-points’ (vertices at which no players are resided), costs can be negative and can be located at vertices, and coalitions are allowed to use arcs and vertices not adjacent to their residences (making the associated game monotone). Unlike in our setting, a coalition must connect each of its members to the root. They provide conditions under which the core of the associated game is nonempty (Theorem 4.13 in their paper). Their condition that the grand coalition connects all vertices to the root is similar to our Assumption 3.1, however their assumption that each edge (including its end point) used by the grand coalition has nonnegative costs quite differs from our situation. The nature of our model implies that each branch of an optimal tree has nonnegative benefits. This makes both approaches difficult to compare. Estévez-Fernández et al. (2009) consider routing games with revenues. Routing games are well-known and broadly studied in the literature. A routing game is similar to a traveling salesman game (see Potters, Curiel, & Tijs, 1992), except that a proper coalition cannot choose the order of the visits of the salesman, but must use the order induced by an optimal order to visit all cities. EstévezFernández et al. (2009) show that routing games with revenues have a nonempty core under the assumption that cooperation among the members of the grand coalition grants that the optimal tour will go through all cities. Example 5.1 illustrates that the dual game of a routing game does not always have a large core. This
might seem in contradiction with Theorem 3.2 in this article. In Example 5.1, we provide an explanation for this phenomenon. Example 5.1. Consider the routing problem with revenues ðN 0 ; C; bÞ with N ¼ f1; 2; 3; 4; 5g,
;
^ : 0 1 2 3 4 b ¼ ð10; 10; 10:5; 10; 10Þ, and optimal tour p 5 0. In Table 4, we provide the coalitional values of the routing game ðN; cp^ Þ, the corresponding dual game ðN; cp^ Þ; ðN; bÞ and ðN; v p^ b Þ. Note that bðSÞ P cp^ ðSÞ for every S # N. Still, ðN; v p^ b Þ has an empty core. To show this, suppose that x 2 Coreðv p^ b Þ; then, 15 6 x1 þ x4 ; 8 6 x2 , and 0 6 x3 þ x5 . By adding all these equations P we obtain 23 6 5i¼1 xi ¼ v p^ b ðNÞ ¼ 22:5 establishing a contradiction. Therefore, Coreðv p^ b Þ ¼ ;, and, consequently, ðN; cp^ Þ does not have a large core.
Table 5 Coalitional values of the traveling salesman game, associated dual game, additive revenues, and net revenues in Example 5.1. S
f1g
f2g
f3g
f4g
f5g
f1; 2g
f1; 3g
f1; 4g
f1; 5g
f2; 3g
f2; 4g
f2; 5g
f3; 4g
f3; 5g
f4; 5g
cts ðSÞ cts ðSÞ bðSÞ bðSÞ cts ðSÞ
4 3 10 6
2 0 10 8
20 11 10.5 9.5
4 16 10 6
20 12 10 10
5 4 20 15
24 13 20.5 3.5
5 13 20 15
24 13 20 4
21 13 20.5 0.5
5 16 20 15
22 13 20 2
14 3 20.5 6.5
40 20 20.5 19.5
14 4 20 6
S
f1; 2; 3g
f1; 2; 4g
f1; 2; 5g
f1; 3; 4g
f1; 3; 5g
f1; 4; 5g
f2; 3; 4g
f2; 3; 5g
f2; 4; 5g
f3; 4; 5g
cts ðSÞ cts ðSÞ bðSÞ bðSÞ cts ðSÞ
24 14 30.5 6.5
8 12 30 22
25 14 30 5
15 6 30.5 15.5
44 23 30.5 13.5
15 7 30 15
15 4 30.5 15.5
41 23 30.5 10.5
15 4 30 15
24 23 30.5 6.5
S
f1; 2; 3; 4g
f1; 2; 3; 5g
f1; 2; 4; 5g
f1; 3; 4; 5g
f2; 3; 4; 5g
N
cts ðSÞ cts ðSÞ bðSÞ bðSÞ cts ðSÞ
16 8 40.5 24.5
44 24 40.5 3.5
17 8 40 23
28 26 40.5 12.5
25 24 40.5 15.5
28 28 50.5 22.5
614
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Hence, Assumption 3.1 is insufficient to ensure non-emptiness of the core for routing games with revenues. Estévez-Fernández et al. (2009) show that non-emptiness of the core is guaranteed if in the corresponding traveling salesman problem with revenues, it is optimal for N to visit all cities and to visit them in increasing order. This boils down to bðSÞ P cts ðSÞ for every S # N, where ðN; cts Þ denotes the traveling salesman game corresponding to distance matrix C. In Table 5, we provide the coalitional values of salesman game ðN; cts Þ, the corresponding dual game the traveling
N; cts , and ðN; bÞ. Note that without a prescribed order it is more profitable to only visit cities f1; 2; 3; 4g (in order 0–2–1–4–3–0) than to visit all cities in N. Besides, note that b is not an element of the upper core of ðN; cts Þ since b3 ¼ 10:5 < 11 ¼ cts ðf3gÞ and b5 ¼ 10 < 12 ¼ cts ðf5gÞ. Note that Example 5.1 illustrates that Assumption 3.1 is not a mandatory condition for non-emptiness of the core of cost games with revenues. Appendix A In this appendix, we show the proof of Lemma 4.5 in Section 4. Before giving the proof, we provide three auxiliary lemmas. Lemma A.1. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let ðN; cÞ be the corresponding 2-mcst game. Then,
o X n k j 2 N0 n T : fi; jg 2 F 1 i2T n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ T ¼ fig 1
c ðTÞ ¼ jTjk1 ðk2 k1 Þ
for every T 2 EðN 0 ; CÞ. Proof. By the definition of EðN 0 ; CÞ; T 2 CðCk1 Þ. Let U 2 N 0 k be the jC 1 component of Ck1 with T # U. We have c ðTÞ ¼ cðNÞ cðN n TÞ ¼ k1 jNj k1 jN 0jCk1 j 1 þ k2 jN 0jCk1 j 1 k1 jN n T j k1 ðjðN 0 n TÞjCk1 j 1Þ þ k2 ðjðN 0 n TÞjCk1 j 1Þ ¼ k1 jTj ðk2 k1 Þ jðN 0 n TÞjCk1 j jN 0jCk1 j ¼ k1 jTj ðk2 k1 Þ jðU n TÞjCk1 j 1 ¼ k1 jTj ðk2 k1 Þ ! o n o X n k1 k1 j 2 U n T : fi; jg 2 F þ ½j 2 ½CðC Þ : Vð½jÞ \ T ¼ fig 1 i2T
o X n k ¼ k1 jTj ðk2 k1 Þ j 2 N 0 n T : fi; jg 2 F 1 i2T n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ T ¼ fig 1 ;
in which the fourth equality is a direct consequence of T # U with U 2 N0jCk1 ; the fifth equality follows because the elements of ðU n TÞjCk1 are those components that we obtain when the edges between T and U n T are broken; the final equality is a direct consequence of T # U and fi; jg # U for every fi; jg 2 Ek1 with i 2 T. h Lemma A.2. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j. For any disjoint pair T 1 ; T 2 2 EðN 0 ; CÞ of essential coalitions we have (i) If jVð½jÞ \ T 1 j ¼ jVð½jÞ \ T 2 j ¼ 1 for some ½j 2 ½CðCk1 Þ, then, Vð½jÞ \ T 1 ¼ Vð½jÞ \ T 2 ; (ii) T 1 [ T 2 ; T 1 \ T 2 2 EðN 0 ; CÞ.
Proof. (i) On the contrary, suppose that Vð½jÞ \ T 1 ¼ fi1 g and Vð½jÞ \ T 2 ¼ fi2 g with i1 – i2 . Since jVð½jÞ \ T 1 j ¼ 1 and jVð½jÞ \ T 2 j ¼ 1, we have i1 2 T 1 n T 2 and i2 2 T 2 n T 1 . Since Vð½jÞ 2 CðCk1 Þ, there exists a path connecting i1 and i2 ; pathði1 ; i2 Þ, using only edges in Eð½jÞ. Let j 2 T 1 \ T 2 ; by connectivity of T 1 , there exists a path, pathði1 ; jÞ, connecting i1 and j in Ck1 using nodes of T 1 . Because jVð½jÞ \ T 1 j ¼ 1, no edge of pathði1 ; jÞ belongs to Eð½jÞ. Analogously, there exists a path, pathðj; i2 Þ, connecting j and i2 in Ck1 using nodes of T 2 and such that no edge in the path belongs to Eð½jÞ. Combining pathði1 ; jÞ and pathðj; i2 Þ, we can construct a path connecting i1 and i2 ; pathði1 ; i2 Þ , without using any edge in Eð½jÞ. Combining the paths 2 CðCk1 Þ pathði1 ; i2 Þ and pathði1 ; i2 Þ , there exists a cycle j Þ and ½j – ½j . This establishes a contradicwith i1 ; i2 2 Vð½j . tion since i1 ; i2 2 Vð½jÞ and, therefore, ½j ¼ ½j (ii) First, note that T 1 [ T 2 2 CðCk1 Þ since T 1 ; T 2 2 CðCk1 Þ with T 1 \ T 2 – ;. Moreover, for every ½j 2 ½CðCk1 Þ, we have that a. if Vð½jÞ # T 1 (analogously, Vð½jÞ # T 2 ), then, Vð½jÞ # T 1 [ T 2 ; b. if Vð½jÞ \ T 1 ¼ fig for some i 2 T 1 and jVð½jÞ \ T 2 j ¼ 1 (analogously, Vð½jÞ \ T 2 ¼ fig for some i 2 T 2 and jVð½jÞ \ T 1 j ¼ 1), then, Vð½jÞ \ T 2 ¼ fig by (i) of this lemma and Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig; c. if Vð½jÞ \ T 1 ¼ fig for some i 2 T 1 and Vð½jÞ \ T 2 ¼ ; (analogously, Vð½jÞ \ T 2 ¼ fig for some i 2 T 2 and Vð½jÞ \ T 1 ¼ ;), then, Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig; d. if Vð½jÞ \ T 1 ¼ ; and Vð½jÞ \ T 2 ¼ ;, then, Vð½jÞ\ ðT 1 [ T 2 Þ ¼ ;. Therefore, T 1 [ T 2 2 EðN 0 ; CÞ. Second, note that the intersection of two connected sets is connected and, therefore, T 1 \ T 2 2 CðCk1 Þ. Moreover, for every ½j 2 ½CðCk1 Þ, we have e. if Vð½jÞ \ T 1 ¼ ; (analogously, Vð½jÞ \ T 2 ¼ ;), then, Vð½jÞ \ ðT 1 \ T 2 Þ ¼ ;; f. if Vð½jÞ \ T 1 ¼ fig for some i 2 T 1 and jVð½jÞ \ T 2 j ¼ 1 (analogously, Vð½jÞ \ T 2 ¼ fig for some i 2 T 2 and jVð½jÞ \ T 1 j ¼ 1), then, Vð½jÞ \ T 2 ¼ fig by (i) of this lemma and Vð½jÞ \ ðT 1 \ T 2 Þ ¼ fig; g. if Vð½jÞ \ T 1 ¼ fig for some i 2 T 1 and Vð½jÞ # T 2 (analogously, Vð½jÞ \ T 2 ¼ fig for some i 2 T 2 and Vð½jÞ # T 1 ), then, Vð½jÞ \ ðT 1 \ T 2 Þ ¼ fig; h. if Vð½jÞ # T 1 and Vð½jÞ # T 2 , then, Vð½jÞ # T 1 \ T 2 . Therefore, T 1 \ T 2 2 EðN 0 ; CÞ.
h
Lemma A.3. Let ðN 0 ; CÞ be a 2-mcst network with cij 2 fk1 ; k2 g; 0 6 k1 6 k2 , for every i; j 2 N 0 with i – j, and let T 1 ; T 2 2 EðN 0 ; CÞ with T 1 \ T 2 – ;. Then, (i)
P
i2T 1 nT 2 jfj k1
2 T 2 n T 1 : fi; jg 2 F k1 gj þ
P
i2T 2 nT 1 jfj
2 T1 n T2 :
fi; jg 2 F gj ¼ 0; P P k1 (ii) i2T 1 jfj 2 N 0 n T 1 : fi; jg 2 F gj þ i2T 2 jfj 2 N 0 n T 2 : fi; jg P P k1 2 F gj ¼ i2T 1 [T 2 jfj 2 N 0 n ðT 1 [ T 2 Þ : fi; jg 2 F k1 gj þ i2T 1 \ (iii)
T 2 jfj 2 N n ðT 1 \ T 2 Þ : fi; jg 2 F k1 gj; o P P n 0 k1 i2T 1 ½j 2 ½CðC Þ : Vð½jÞ \ T 1 ¼ fig þ i2T 2 jf½j 2 n P k1 ½CðC Þ : Vð½jÞ \ T 2 ¼ figgj ¼ i2T 1 [T 2 ½j 2 ½CðCk1 Þ : n P Vð½jÞ \ ðT 1 [ T 2 Þ ¼ figgj þ i2T 1 \T 2 ½j 2 ½CðCk1 Þ : Vð½jÞ\ ðT 1 \ T 2 Þ ¼ figgj.
A. Estévez-Fernández, H. Reijnierse / European Journal of Operational Research 237 (2014) 606–616
Proof. First, note that T 1 [ T 2 ; T 1 \ T 2 2 EðN 0 ; CÞ by Lemma A.2 (ii). (i) On the contrary, suppose that X X jfj 2 T 2 n T 1 : fi; jg 2 F k1 gj þ jfj 2 T 1 n T 2 : fi;jg 2 F k1 gj – 0: i2T 1 nT 2
i2T 2 nT 1
Then, there exist i1 2 T 1 n T 2 and i2 2 T 2 n T 1 with fi1 ; i2 g 2 F k1 . By definition of F k1 ; fi1 ; i2 g R [j2CðCk1 Þ EðjÞ. Let j 2 T 1 \ T 2 ; by connectivity of T 1 , there exists a path, pathði1 ; jÞ, connecting i1 and j in Ck1 using nodes of T 1 . Analogously, there exists a path, pathðj; i2 Þ, connecting j and i2 in Ck1 using nodes of T 2 . By concatenating pathði1 ; jÞ and pathðj; i2 Þ, we construct a path, pathði1 ; i2 Þ, connecting i1 and i2 without using edge fi1 ; i2 g. As a result, there exists a cycle j 2 CðCk1 Þ with fi1 ; i2 g 2 EðjÞ, establishing a contradiction. o X n o X n k k (ii) j 2 N0 n T 1 : fi;jg 2 F 1 þ j 2 N0 n T 2 : fi;jg 2 F 1 i2T 1
i2T 2
o o X n X n k k j 2 N0 n T 1 : fi;jg 2 F 1 þ j 2 N0 n T 1 : fi;jg 2 F 1
¼
i2T 1 nT 2
þ
i2T 1 \T 2
o o X n X n k k j 2 N0 n T 2 : fi;jg 2 F 1 þ j 2 N0 n T 2 : fi;jg 2 F 1
i2T 2 nT 1
i2T 1 \T 2
o n o X n k k ¼ j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 T 2 n T 1 : fi; jg 2 F 1 i2T 1 nT 2
þ
o n o X n k k j 2 N0 n ðT 1 [ T 2 Þ : fi; jg 2 F 1 þ j 2 T 2 n T 1 : fi; jg 2 F 1
i2T 1 \T 2
b. ½j 2 ½CðCk1 ÞðT 1 Þ and ½j R ½CðCk1 ÞðT 2 Þ. In this case, we have to distinguish between two new cases: Vð½jÞ \ T 2 ¼ ; and Vð½jÞ \ T 2 ¼ Vð½jÞ. b.1. Vð½jÞ \ T 2 ¼ ;. In this case, ½j 2 ½CðCk1 ÞðT 1 [ T 2 Þ and ½j R ½CðCk1 ÞðT 1 \ T 2 Þ. b.2. Vð½jÞ \ T 2 ¼ Vð½jÞ. In this case, ½j R ½CðCk1 Þ ðT 1 [ T 2 Þ and ½j 2 ½CðCk1 ÞðT 1 \ T 2 Þ. c. ½j R ½CðCk1 ÞðT 1 Þ and ½j 2 ½CðCk1 ÞðT 2 Þ. Again we distinguish between two new cases: Vð½jÞ \ T 1 ¼ ; and Vð½jÞ \ T 1 ¼ Vð½jÞ. c.1. Vð½jÞ \ T 1 ¼ ;. In this case, ½j 2 ½CðCk1 ÞðT 1 [ T 2 Þ and ½j R ½CðCk1 ÞðT 1 \ T 2 Þ. c.2. Vð½jÞ \ T 1 ¼ Vð½jÞ. In this case, ½j R ½CðCk1 Þ ðT 1 [ T 2 Þ and ½j 2 ½CðCk1 ÞðT 1 \ T 2 Þ. d. ½j R ½CðCk1 ÞðT 1 Þ and ½j R ½CðCk1 ÞðT 2 Þ. In this case, ½j R ½CðCk1 ÞðT 1 [ T 2 Þ and ½j R ½CðCk1 ÞðT 1 \ T 2 Þ. Following these cases, we have j½CðCk1 ÞðT 1 Þjþ j½CðCk1 ÞðT 2 Þj ¼ j½CðCk1 ÞðT 1 [ T 2 Þj þ j½CðCk1 ÞðT 1 \ T 2 Þj and, therefore,
o X n k ½j 2 ½CðC 1 Þ : Vð½jÞ \ T 1 ¼ fig i2T 1
o n o X n k k þ j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 T 1 n T 2 : fi;jg 2 F 1
þ
n o ¼ ½j 2 ½CðCk1 Þ : jVð½jÞ \ T 1 j ¼ 1 n o þ ½j 2 ½CðCk1 Þ : jVð½jÞ \ T 2 j ¼ 1 n o ¼ ½j 2 ½CðCk1 Þ : jVð½jÞ \ ðT 1 [ T 2 Þj ¼ 1 n o þ ½j 2 ½CðCk1 Þ : jVð½jÞ \ ðT 1 \ T 2 Þj ¼ 1 o X n k ¼ ½j 2 ½CðC 1 Þ : Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig
o n o X n k k j 2 N0 n ðT 1 [ T 2 Þ : fi; jg 2 F 1 þ j 2 T 1 n T 2 : fi; jg 2 F 1
i2T 1 \T 2
o o X n X n k k j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 T 2 n T 1 : fi;jg 2 F 1
¼
i2T 1 [T 2
þ
i2T 1 nT 2
o o X n X n k k j 2 T 2 n T 1 : fi;jg 2 F 1 þ j 2 T 1 n T 2 : fi;jg 2 F 1
i2T 1 \T 2
þ
i2T 2 nT 1
o n o X n k k j 2 N0 n ðT 1 [ T 2 Þ : fi; jg 2 F 1 þ j 2 T 1 n T 2 : fi; jg 2 F 1
i2T 1 \T 2
¼
o o X n X n k k j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 T 2 n T 1 : fi; jg 2 F 1
i2T 1 [T 2
i2T 1 [T 2
i2T 1 \T 2
o n o X n k k þ j 2 N0 n ðT 1 [ T 2 Þ : fi; jg 2 F 1 þ j 2 T 1 n T 2 : fi; jg 2 F 1
þ
i2T 1 \T 2
¼
o X n k ½j 2 ½CðC 1 Þ : Vð½jÞ \ ðT 1 \ T 2 Þ ¼ fig :
i2T 1 \T 2
o o X n X n k k j 2 N 0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1 þ j 2 N0 n ðT 1 \ T 2 Þ : fi; jg 2 F 1 ;
i2T 1 [T 2
o X n k ½j 2 ½CðC 1 Þ : Vð½jÞ \ T 2 ¼ fig i2T 2
i2T 2 nT 1
þ
615
i2T 1 \T 2
in which the first equality follows from T 1 ¼ ðT 1 n T 2 Þ [ ðT 1 \ T 2 Þ and T 2 ¼ ðT 2 n T 1 Þ [ ðT 1 \ T 2 Þ. The second equality is a direct consequence of N 0 n T 1 ¼ ðN0 n ðT 1 [ T 2 ÞÞ [ ðT 2 n T 1 Þ and N 0 n T 2 ¼ ðN0 n ðT 1 [ T 2 ÞÞ [ ðT 1 n T 2 Þ. The third equality follows since T 1 [ T 2 ¼ ðT 1 n T 2 Þ [ ðT 2 n T 1 Þ [ ðT 1 \ T 2 Þ. The fourth equality is a consequence of (i) of this Lemma. The fifth equality follows because N 0 n ðT 1 \ T 2 Þ ¼ ðT 2 n T 1 Þ [ ðN 0 n ðT 1 [ T 2 ÞÞ [ ðT 1 n T 2 Þ. (iii) By definition of EðN 0 ; CÞ, if R 2 EðN 0 ; CÞ and ½j 2 ½CðCk1 Þ with Vð½jÞ \ R – ;, then, either Vð½jÞ \ R ¼ Vð½jÞ, or Vð½jÞ \ R ¼ fig for some i 2 R. As a result,
o Xn j ½j 2 ½CðCk1 Þ : Vð½jÞ \ R ¼ fig j i2R
n o ¼ j ½j 2 ½CðCk1 Þ : jVð½jÞ \ Rj ¼ 1 j:
n o For R # N, we denote ½CðCk1 ÞðRÞ ¼ ½j 2 ½CðCk1 Þ : jVð½jÞ \ Rj ¼ 1 . k1 k1 Next, we compare the sets ½CðC ÞðT 1 Þ and ½CðC ÞðT 2 Þ and the sets ½CðCk1 ÞðT 1 [ T 2 Þ and ½CðCk1 ÞðT 1 \ T 2 Þ. For this, we distinguish several cases: a. ½j 2 ½CðCk1 ÞðT 1 Þ and ½j 2 ½CðCk1 ÞðT 2 Þ. By Lemma A.2 (i), Vð½jÞ \ T 1 ¼ Vð½jÞ \ T 2 ¼ fig for some i 2 T 1 \ T 2 implying Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig and Vð½jÞ \ ðT 1 \ T 2 Þ ¼ fig. Therefore, ½j 2 ½CðCk1 ÞðT 1 [ T 2 Þ and ½j 2 ½CðCk1 Þ ðT 1 \ T 2 Þ.
Proof of Lemma 4.5. By Lemma A.2 (ii), we have that T 1 [ T 2 ; T 1 \ T 2 2 EðN 0 ; CÞ. This gives c ðT 1 Þ þ c ðT 2 Þ ¼ k1 jT 1 j ðk2 k1 Þ
o X n k j 2 N0 n T 1 : fi;jg 2 F 1
i2T 1 n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ T 1 ¼ fig 1 þ k1 jT 2 j ðk2 k1 Þ ! o n o X n k k j 2 N 0 n T 2 : fi; jg 2 F 1 þ ½j 2 ½CðC 1 Þ : Vð½jÞ \ T 2 ¼ fig 1 i2T 2
¼ k1 ðjT 1 j þ jT 2 jÞ ðk2 k1 Þ
o X n k j 2 N0 n T 1 : fi;jg 2 F 1
i2T 1 n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ T 1 ¼ fig 1 ðk2 k1 Þ ! o n o X n k1 k1 j 2 N n T : fi; jg 2 F þ ½ j 2 ½Cð C Þ : Vð½ j Þ \ T ¼ fig 1 0 2 2 i2T 2
¼ k1 jT 1 [ T 2 j ðk2 k1 Þ
o X n k j 2 N0 n ðT 1 [ T 2 Þ : fi;jg 2 F 1
i2T 1 [T 2 n o þ ½j 2 ½CðCk1 Þ : Vð½jÞ \ ðT 1 [ T 2 Þ ¼ fig 1 þ k1 jT 1 \ T 2 j ðk2 k1 Þ o n X n k k j 2 N 0 n ðT 1 \ T 2 Þ : fi; jg 2 F 1 þ ½j 2 ½CðC 1 Þ : Vð½jÞ \ ðT 1 \ T 2 Þ
i2T 1 \T 2 ¼ figgj 1 ¼ c ðT 1 [ T 2 Þ þ c ðT 1 \ T 2 Þ;
in which the first and last equalities follow from Lemma A.1 and the third equality is a direct consequence of Lemma A.3 (ii) and (iii). h
616
A. Estévez-Fernández, H. Reijnierse / European Journal of Operational Research 237 (2014) 606–616
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