- Email: [email protected]
Noncooperative facility location games
Operations Research Letters 35 (2007) 151 – 154
Operations Research Letters www.elsevier.com/locate/orl
Noncooperative facility location games Lina Mallozzi∗ Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II” Via Claudio, 21-80125 Napoli, Italy Received 21 May 2005; accepted 12 March 2006 Available online 23 May 2006
Abstract A noncooperative game theoretical approach is considered for the multifacility location problem. It turns out that the facility location game is a potential game in the sense of Monderer and Shapley and some properties of the game are studied. © 2006 Elsevier B.V. All rights reserved. Keywords: Facility location problems; Noncooperative games; Minisum problems; Potential games
1. Introduction Location problems are concerned with finding the best location for one or more facilities optimizing a given objective; if more than one facility has to be located we deal with a multifacility location problem. In the classical Fermat–Weber problem or minisum problem, for example, the sum of weighted distances is minimized, in the minimax model the maximal distance between facilities and demand points is minimized (see, for example, [3,4]). There exists also some particular type of location problems: facility layout models, i.e. absence of demand points (see, for example, [4]), competitive models, i.e. facilities compete for customers and the objective is to maximize their market share (see, for example, [1]). The first competitive location model ∗ Fax: +39 081 2396733.
E-mail address: [email protected]. 0167-6377/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2006.03.003
is in [9] where the location of two duopolists, whose decision variables are locations and prices, is chosen. Another type of facility location models, the so-called uncapacitated facility location problems, deals with the location of facilities in order to provide service for a set of costumers (see, for example, [10]). Many of the above mentioned facility location problems have been studied from a game-theoretic point of view. In the competitive facility location problem, noncooperative game-theoretic tools have been used (see, for example, [1]). In the uncapacitated facility location problem, a fair cost allocation of the total cost to the costumers has been found by using cooperative game-theoretic concepts (see, for example, [5]). A noncooperative game-theoretic approach to a multifacility location problem has been proposed in [14], in the context of supermodular games. Each competitor controls one or more facilities trying to optimize the transportation cost, that is supposed to be a linear function of the distance between facilities. In the facility location game the players are
152
L. Mallozzi / Operations Research Letters 35 (2007) 151 – 154
competitors, the strategies are locations and the payoffs are cost functions. The solution of the location problem is a Nash equilibrium of the game. In [14] supermodularity properties of the facility location game have been given together with an existence theorem. In this paper we present the facility location game revisited by considering a more general payoff for each firm. In fact, as observed in [6], transportation costs are rarely linear with respect to the distance in the real world. In line with previous results [2], we prove that the revisited facility location game is a potential game in the sense of Monderer and Shapley [11] and a more general existence result for pure Nash equilibria is given. More precisely, in Section 2 the noncooperative game-theoretic approach is considered in a general setting and the facility location game turns out to be a potential game. Existence of pure Nash equilibria and existence of symmetric Nash equilibria are investigated. In Section 3 the relation between the proposed definition of facility location game and some multifacility location models is discussed. 2. Facility location game. The minisum case Multifacility location is concerned with the problem of locating n ∈ N new facilities (n > 1) with respect to existing facilities (demand points), in order to minimize a total cost function. We assume that there are n firms competing in order to locate the new facilities and the location problem can be stated as a Nash equilibrium problem. Definition 1. Let N = {1, . . . , n}; the facility location game is the n-player strategic form game N; S; {fi , i ∈ N}, where S and fi are defined by the following assumptions: (i) Each firm i has to set up a new facility in a point x i ∈ Si ⊂ R2 , where Si is the compact set of the feasible location of firm i. Let be S = ni=1 Si . Firms 1, . . . , n are the players and Si is the strategy set of player ith. (ii) m demand points has to be connected to any new facility; the kth point is denoted by t k ∈ R2 , k = 1, . . . , m. (iii) The function d(y, z) is a measure of the distance between any two points y and z in R2 .
(iv) The transportation cost between two new facilities x i and x j is given by a lower semicontinuous function of the distance, i.e. Cij (d(x i , x j )); the transportation cost between the new facility x i and the demand point t k is given by a lower semicontinuous function of the distance, i.e. Bik (d(x i , t k )). (v) The new facilities will be located in (x¯ 1 , . . . , x¯ n ) ∈ S such that each firm i wants to minimize the total transportation cost fi (x 1 , . . . , x n ) = Cij (d(x i , x j )) 1 j n,j =i
+
Bik (d(x i , t k )). (1)
1k m
The function fi defined on S is the cost function of player ith. Recall that f : X → R, X being a subset of a Euclidean space, is a (sequentially) lower semicontinuous function in X if for any x ∈ X and for any sequence (xn )n converging to x in X, we have lim inf n→+∞ f (xn ) f (x). For simplicity, in the following, we assume that Cij , for i > j is equal to Cj i and that Cii = 0 for all i ∈ N . We denote x = (x 1 , . . . , x n ) ∈ S and x −i = (x 1 , . . . , x i−1 , x i+1 , . . . , x n ) ∈ S−i where S−i = 1 j n,j =i Sj . A solution to the facility location game is a Nash equilibrium point of the game N ; S; {fi , i ∈ N }, i.e. a vector x¯ = (x¯ 1 , . . . , x¯ n ) ∈ S ⊂ R2n such that fi (x) ¯ fi (x i , x¯ −i )
∀x i ∈ Si
for any i ∈ N . Let us define for any x −i ∈ S−i the set of player i’s best responses by Ri (x −i ) = {x i ∈ Si : fi (x i , x −i ) fi (x˜ i , x −i )
∀x˜ i ∈ Si }.
A Nash equilibrium is a vector x¯ =(x¯ 1 , . . . , x¯ n ) ∈ S for which x¯ i ∈ Ri (x¯ −i ) for all i ∈ N . A very well known existence result of Nash equilibria is the theorem proved by Nash in 1950. One of the assumptions of this theorem is the quasi-convexity of the ith objective function with respect to the ith variable. Let X be a convex subset of a Euclidean space
L. Mallozzi / Operations Research Letters 35 (2007) 151 – 154
and let f : X → R. We say that f is convex if for any x, y ∈ X and for any ∈ [0, 1] we have that f (x+(1−)y)f (x)+(1−)f (y). We say that f is quasi-convex if for any x, y ∈ X and for any ∈ [0, 1] we have that f (x + (1 − )y) max[f (x), f (y)]. In a facility location game, even if we suppose that the distance d(y, z) is expressed by a lp norm in R2 , the function fi is not in general quasi-convex in x i . A lp norm is defined by v p = [|v1 |p + |v2 |p ]1/p , v = (v1 , v2 ) ∈ R2 and 1p + ∞; for p = 2 we have the Euclidean norm. Nevertheless, an existence theorem is given below taking advantage of the potential structure of the facility location game. Recall that the game N ; S; {fi , i ∈ N} is called a potential game [11] if there is a (potential) function P : S → R such that for all i ∈ N and for each x −i ∈ S−i fi (y, x −i ) − fi (z, x −i ) = P (y, x −i ) − P (z, x −i ) ∀y, z ∈ Si . Clearly, elements of argmin(P ) are Nash equilibria of the game. Theorem 1. The facility location game N; S; {fi , i ∈ N} is a potential game with potential function defined on S
P (x , . . . , x ) = 1
n
h
Chj (d(x , x ))
1 h
+
Bj k (d(x j , t k )). (2)
Proof. For any x ∈ S and for any i ∈ N the quantity 1 h
+
Chj (d(x h , x j ))
Bj k (d(x j , t k ))
1 j n,j =i 1 k m
does not depend on x i .
Usually each firm has to locate the new facility within a compact convex region Q of the plane (a rectangle or a square). In this case Si = Q ⊂ R2 for all i ∈ N and firms have the same feasible location set. If the transportation cost functions are all equal, firms have a symmetric role and all “prefer” the same position. More precisely, we prove in the following an existence result of a symmetric Nash equilibrium, i.e. a vector x¯ = (x¯ 1 , . . . , x¯ n ) having equal components x¯ 1 = x¯ 2 = · · · = x¯ n ∈ Q. Theorem 3. Assume that Q is a compact convex region of the plane, Cij (·)=c(·) and Bik (·)=b(·) for all i, j ∈ N (i = j ), k = 1, . . . , m, with c and b continuous, increasing and convex functions. Then the facility location game admits a symmetric Nash equilibrium. Proof. Under the assumptions on the transportation costs, by substituting the ith variable x i in (1) with the jth variable x j and also x j with x i , we obtain the cost function of player jth, i.e. fj and the game is called symmetrical game. The cost function of player ith c(d(x i , x j )) fi (x 1 , . . . , x n ) = 1 j n,j =i
b(d(x i , t k ))
1k m
1j n 1k m
P (x) − fi (x) =
Theorem 2. The facility location game has at least a Nash equilibrium.
+
j
153
is a continuous function, and turns out to be convex in its variable x i since c and b are increasing and convex (see, for example, [8] Chapter B). All the assumptions of the Nash’s theorem for symmetrical games (see [12, p. 115]) are satisfied and at least a symmetric Nash equilibrium exists. In this case 1 1 1 R1 (x(n−1) ) = R2 (x(n−1) ) = · · · = Rn (x(n−1) ) for any 1 1 1 1 x ∈ Q, where x(n−1) = (x , . . . , x ) ∈ Qn−1 . Given 1 ), the a fixed point x¯ 1 of the correspondence R1 (x(n−1) vector (x¯ 1 , . . . , x¯ 1 ) ∈ Qn is a symmetrical Nash equilibrium.
We assume that the distance between two facilities is expressed by a lp -norm, for each 1p + ∞. By mean of the lower semicontinuity of the potential function (2), the following existence result holds.
3. Discussion and relations with multifacility location problems The facility location game considered in [14] (paragraph 4.4.7) has linear transportation costs with
154
L. Mallozzi / Operations Research Letters 35 (2007) 151 – 154
respect to the distance between facilities, i.e. Cij (d(x i , x j )) = cij d(x i , x j ),
University of Technology. The visit was partially supported by the Short Mobility Program of the University of Naples “Federico II”.
Bik (d(x i , t k )) = bik d(x i , t k ) ∀i, j ∈ N ; k = 1, . . . , m
References
for some nonnegative coefficients cij , bik . The game turns out to be supermodular [14] and an existence result is given for particular choices of the distance d (not for the Euclidean distance). By using Theorem 1 the facility location game with transportation costs linearly depending on the distance is a potential game and by using Theorem 2 we have an existence result for any lp -norm. In this case the potential function is P (x 1 , . . . , x n ) = chj d(x h , x j ) 1 h
+
bj k d(x j , t k ). (3)
1j n 1k m
The potential approach proposed in this paper is directly related to the well known multifacility location problem ([3,4,10]...) . The multifacility location problem is to find optimal locations for the new facilities; in the minisum case the objective is the sum of functions of the distances from each new facility to each existing facility plus the sum of functions of those between each pair of new facilities. The objective function used for example by Hansen et al. in problem (M1) of Ref. [7], is precisely the potential function P defined in (2). In the Euclidean multifacility location problem, i.e. in the case of linear transportation costs in distance, the objective function used for example by Rosen and Xue in Ref. [13], is the potential function P defined in (3). Acknowledgments This research was initiated while the author was visiting the Systems Analysis Laboratory of Helsinki
[1] R.J. Aumann, S. Hart (Eds.), Handbook of Game Theory with Economic Applications, Handbooks in Economics, vol. 11, North-Holland, Amsterdam, 1992. [2] R. Branzei, L. Mallozzi, S. Tijs, Supermodular games and potential games, J. Math. Econom. 39 (2003) 39–49. [3] Z. Drezner (Ed.), Facility Location: a Survey of Applications and Methods, Springer, New York, 1995. [4] R.L. Francis, L.F. McGinnis Jr., J.A. White, Facility Layout and Location: an Analitycal Approach, second ed., PrenticeHall, Englewood Cliffs, NJ, 1992. [5] M.X. Goemans, M. Skutella, Cooperative facility location games, J. Algorithms 50 (2004) 194–214. [6] P. Hansen, D. Peeters, D. Richard, J.-F. Thisse, The minisum and minimax location problems revisited, Oper. Res. 33 (1985) 1251–1265. [7] P. Hansen, J. Perreur, J.-F. Thisse, Location theory, dominance, and convexity: some further results, Oper. Res. 28 (1980) 1241–1250. [8] J.B. Hiriart-Urruty, C. Lemarechal, Foundamental of Convex Analysis, Springer, Berlin, 2004. [9] H. Hotelling, Stability in Competition, Econom. J. 39 (1929) 41–57. [10] M. Labbé, G. Laporte, K. Tanczos, P. Toint (Eds.), Operations Research and Decision Aid Methodologies in Traffic and Transportation Management, Series F: Computer and Systems Sciences, NATO ASI Series, Springer, Berlin, 1997. [11] D. Monderer, L.S. Shapley, Potential games, Games Econom. Behav. 14 (1996) 124–143. [12] H. Moulin, Game Theory for the Social Sciences, second and revised ed., New York University Press, New York, 1986. [13] J.B. Rosen, G.L. Xue, On the convergence of a hyperboloid approximation procedure for the perturbed euclidean multifacility location problem, Oper. Res. 41 (1993) 1164–1171. [14] D. Topkis, Supermodularity and Complementarity, Princeton University Press, Princeton, NJ, 1998.
Operations Research Letters www.elsevier.com/locate/orl
Noncooperative facility location games Lina Mallozzi∗ Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II” Via Claudio, 21-80125 Napoli, Italy Received 21 May 2005; accepted 12 March 2006 Available online 23 May 2006
Abstract A noncooperative game theoretical approach is considered for the multifacility location problem. It turns out that the facility location game is a potential game in the sense of Monderer and Shapley and some properties of the game are studied. © 2006 Elsevier B.V. All rights reserved. Keywords: Facility location problems; Noncooperative games; Minisum problems; Potential games
1. Introduction Location problems are concerned with finding the best location for one or more facilities optimizing a given objective; if more than one facility has to be located we deal with a multifacility location problem. In the classical Fermat–Weber problem or minisum problem, for example, the sum of weighted distances is minimized, in the minimax model the maximal distance between facilities and demand points is minimized (see, for example, [3,4]). There exists also some particular type of location problems: facility layout models, i.e. absence of demand points (see, for example, [4]), competitive models, i.e. facilities compete for customers and the objective is to maximize their market share (see, for example, [1]). The first competitive location model ∗ Fax: +39 081 2396733.
E-mail address: [email protected]. 0167-6377/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2006.03.003
is in [9] where the location of two duopolists, whose decision variables are locations and prices, is chosen. Another type of facility location models, the so-called uncapacitated facility location problems, deals with the location of facilities in order to provide service for a set of costumers (see, for example, [10]). Many of the above mentioned facility location problems have been studied from a game-theoretic point of view. In the competitive facility location problem, noncooperative game-theoretic tools have been used (see, for example, [1]). In the uncapacitated facility location problem, a fair cost allocation of the total cost to the costumers has been found by using cooperative game-theoretic concepts (see, for example, [5]). A noncooperative game-theoretic approach to a multifacility location problem has been proposed in [14], in the context of supermodular games. Each competitor controls one or more facilities trying to optimize the transportation cost, that is supposed to be a linear function of the distance between facilities. In the facility location game the players are
152
L. Mallozzi / Operations Research Letters 35 (2007) 151 – 154
competitors, the strategies are locations and the payoffs are cost functions. The solution of the location problem is a Nash equilibrium of the game. In [14] supermodularity properties of the facility location game have been given together with an existence theorem. In this paper we present the facility location game revisited by considering a more general payoff for each firm. In fact, as observed in [6], transportation costs are rarely linear with respect to the distance in the real world. In line with previous results [2], we prove that the revisited facility location game is a potential game in the sense of Monderer and Shapley [11] and a more general existence result for pure Nash equilibria is given. More precisely, in Section 2 the noncooperative game-theoretic approach is considered in a general setting and the facility location game turns out to be a potential game. Existence of pure Nash equilibria and existence of symmetric Nash equilibria are investigated. In Section 3 the relation between the proposed definition of facility location game and some multifacility location models is discussed. 2. Facility location game. The minisum case Multifacility location is concerned with the problem of locating n ∈ N new facilities (n > 1) with respect to existing facilities (demand points), in order to minimize a total cost function. We assume that there are n firms competing in order to locate the new facilities and the location problem can be stated as a Nash equilibrium problem. Definition 1. Let N = {1, . . . , n}; the facility location game is the n-player strategic form game N; S; {fi , i ∈ N}, where S and fi are defined by the following assumptions: (i) Each firm i has to set up a new facility in a point x i ∈ Si ⊂ R2 , where Si is the compact set of the feasible location of firm i. Let be S = ni=1 Si . Firms 1, . . . , n are the players and Si is the strategy set of player ith. (ii) m demand points has to be connected to any new facility; the kth point is denoted by t k ∈ R2 , k = 1, . . . , m. (iii) The function d(y, z) is a measure of the distance between any two points y and z in R2 .
(iv) The transportation cost between two new facilities x i and x j is given by a lower semicontinuous function of the distance, i.e. Cij (d(x i , x j )); the transportation cost between the new facility x i and the demand point t k is given by a lower semicontinuous function of the distance, i.e. Bik (d(x i , t k )). (v) The new facilities will be located in (x¯ 1 , . . . , x¯ n ) ∈ S such that each firm i wants to minimize the total transportation cost fi (x 1 , . . . , x n ) = Cij (d(x i , x j )) 1 j n,j =i
+
Bik (d(x i , t k )). (1)
1k m
The function fi defined on S is the cost function of player ith. Recall that f : X → R, X being a subset of a Euclidean space, is a (sequentially) lower semicontinuous function in X if for any x ∈ X and for any sequence (xn )n converging to x in X, we have lim inf n→+∞ f (xn ) f (x). For simplicity, in the following, we assume that Cij , for i > j is equal to Cj i and that Cii = 0 for all i ∈ N . We denote x = (x 1 , . . . , x n ) ∈ S and x −i = (x 1 , . . . , x i−1 , x i+1 , . . . , x n ) ∈ S−i where S−i = 1 j n,j =i Sj . A solution to the facility location game is a Nash equilibrium point of the game N ; S; {fi , i ∈ N }, i.e. a vector x¯ = (x¯ 1 , . . . , x¯ n ) ∈ S ⊂ R2n such that fi (x) ¯ fi (x i , x¯ −i )
∀x i ∈ Si
for any i ∈ N . Let us define for any x −i ∈ S−i the set of player i’s best responses by Ri (x −i ) = {x i ∈ Si : fi (x i , x −i ) fi (x˜ i , x −i )
∀x˜ i ∈ Si }.
A Nash equilibrium is a vector x¯ =(x¯ 1 , . . . , x¯ n ) ∈ S for which x¯ i ∈ Ri (x¯ −i ) for all i ∈ N . A very well known existence result of Nash equilibria is the theorem proved by Nash in 1950. One of the assumptions of this theorem is the quasi-convexity of the ith objective function with respect to the ith variable. Let X be a convex subset of a Euclidean space
L. Mallozzi / Operations Research Letters 35 (2007) 151 – 154
and let f : X → R. We say that f is convex if for any x, y ∈ X and for any ∈ [0, 1] we have that f (x+(1−)y)f (x)+(1−)f (y). We say that f is quasi-convex if for any x, y ∈ X and for any ∈ [0, 1] we have that f (x + (1 − )y) max[f (x), f (y)]. In a facility location game, even if we suppose that the distance d(y, z) is expressed by a lp norm in R2 , the function fi is not in general quasi-convex in x i . A lp norm is defined by v p = [|v1 |p + |v2 |p ]1/p , v = (v1 , v2 ) ∈ R2 and 1p + ∞; for p = 2 we have the Euclidean norm. Nevertheless, an existence theorem is given below taking advantage of the potential structure of the facility location game. Recall that the game N ; S; {fi , i ∈ N} is called a potential game [11] if there is a (potential) function P : S → R such that for all i ∈ N and for each x −i ∈ S−i fi (y, x −i ) − fi (z, x −i ) = P (y, x −i ) − P (z, x −i ) ∀y, z ∈ Si . Clearly, elements of argmin(P ) are Nash equilibria of the game. Theorem 1. The facility location game N; S; {fi , i ∈ N} is a potential game with potential function defined on S
P (x , . . . , x ) = 1
n
h
Chj (d(x , x ))
1 h
+
Bj k (d(x j , t k )). (2)
Proof. For any x ∈ S and for any i ∈ N the quantity 1 h
+
Chj (d(x h , x j ))
Bj k (d(x j , t k ))
1 j n,j =i 1 k m
does not depend on x i .
Usually each firm has to locate the new facility within a compact convex region Q of the plane (a rectangle or a square). In this case Si = Q ⊂ R2 for all i ∈ N and firms have the same feasible location set. If the transportation cost functions are all equal, firms have a symmetric role and all “prefer” the same position. More precisely, we prove in the following an existence result of a symmetric Nash equilibrium, i.e. a vector x¯ = (x¯ 1 , . . . , x¯ n ) having equal components x¯ 1 = x¯ 2 = · · · = x¯ n ∈ Q. Theorem 3. Assume that Q is a compact convex region of the plane, Cij (·)=c(·) and Bik (·)=b(·) for all i, j ∈ N (i = j ), k = 1, . . . , m, with c and b continuous, increasing and convex functions. Then the facility location game admits a symmetric Nash equilibrium. Proof. Under the assumptions on the transportation costs, by substituting the ith variable x i in (1) with the jth variable x j and also x j with x i , we obtain the cost function of player jth, i.e. fj and the game is called symmetrical game. The cost function of player ith c(d(x i , x j )) fi (x 1 , . . . , x n ) = 1 j n,j =i
b(d(x i , t k ))
1k m
1j n 1k m
P (x) − fi (x) =
Theorem 2. The facility location game has at least a Nash equilibrium.
+
j
153
is a continuous function, and turns out to be convex in its variable x i since c and b are increasing and convex (see, for example, [8] Chapter B). All the assumptions of the Nash’s theorem for symmetrical games (see [12, p. 115]) are satisfied and at least a symmetric Nash equilibrium exists. In this case 1 1 1 R1 (x(n−1) ) = R2 (x(n−1) ) = · · · = Rn (x(n−1) ) for any 1 1 1 1 x ∈ Q, where x(n−1) = (x , . . . , x ) ∈ Qn−1 . Given 1 ), the a fixed point x¯ 1 of the correspondence R1 (x(n−1) vector (x¯ 1 , . . . , x¯ 1 ) ∈ Qn is a symmetrical Nash equilibrium.
We assume that the distance between two facilities is expressed by a lp -norm, for each 1p + ∞. By mean of the lower semicontinuity of the potential function (2), the following existence result holds.
3. Discussion and relations with multifacility location problems The facility location game considered in [14] (paragraph 4.4.7) has linear transportation costs with
154
L. Mallozzi / Operations Research Letters 35 (2007) 151 – 154
respect to the distance between facilities, i.e. Cij (d(x i , x j )) = cij d(x i , x j ),
University of Technology. The visit was partially supported by the Short Mobility Program of the University of Naples “Federico II”.
Bik (d(x i , t k )) = bik d(x i , t k ) ∀i, j ∈ N ; k = 1, . . . , m
References
for some nonnegative coefficients cij , bik . The game turns out to be supermodular [14] and an existence result is given for particular choices of the distance d (not for the Euclidean distance). By using Theorem 1 the facility location game with transportation costs linearly depending on the distance is a potential game and by using Theorem 2 we have an existence result for any lp -norm. In this case the potential function is P (x 1 , . . . , x n ) = chj d(x h , x j ) 1 h
+
bj k d(x j , t k ). (3)
1j n 1k m
The potential approach proposed in this paper is directly related to the well known multifacility location problem ([3,4,10]...) . The multifacility location problem is to find optimal locations for the new facilities; in the minisum case the objective is the sum of functions of the distances from each new facility to each existing facility plus the sum of functions of those between each pair of new facilities. The objective function used for example by Hansen et al. in problem (M1) of Ref. [7], is precisely the potential function P defined in (2). In the Euclidean multifacility location problem, i.e. in the case of linear transportation costs in distance, the objective function used for example by Rosen and Xue in Ref. [13], is the potential function P defined in (3). Acknowledgments This research was initiated while the author was visiting the Systems Analysis Laboratory of Helsinki
[1] R.J. Aumann, S. Hart (Eds.), Handbook of Game Theory with Economic Applications, Handbooks in Economics, vol. 11, North-Holland, Amsterdam, 1992. [2] R. Branzei, L. Mallozzi, S. Tijs, Supermodular games and potential games, J. Math. Econom. 39 (2003) 39–49. [3] Z. Drezner (Ed.), Facility Location: a Survey of Applications and Methods, Springer, New York, 1995. [4] R.L. Francis, L.F. McGinnis Jr., J.A. White, Facility Layout and Location: an Analitycal Approach, second ed., PrenticeHall, Englewood Cliffs, NJ, 1992. [5] M.X. Goemans, M. Skutella, Cooperative facility location games, J. Algorithms 50 (2004) 194–214. [6] P. Hansen, D. Peeters, D. Richard, J.-F. Thisse, The minisum and minimax location problems revisited, Oper. Res. 33 (1985) 1251–1265. [7] P. Hansen, J. Perreur, J.-F. Thisse, Location theory, dominance, and convexity: some further results, Oper. Res. 28 (1980) 1241–1250. [8] J.B. Hiriart-Urruty, C. Lemarechal, Foundamental of Convex Analysis, Springer, Berlin, 2004. [9] H. Hotelling, Stability in Competition, Econom. J. 39 (1929) 41–57. [10] M. Labbé, G. Laporte, K. Tanczos, P. Toint (Eds.), Operations Research and Decision Aid Methodologies in Traffic and Transportation Management, Series F: Computer and Systems Sciences, NATO ASI Series, Springer, Berlin, 1997. [11] D. Monderer, L.S. Shapley, Potential games, Games Econom. Behav. 14 (1996) 124–143. [12] H. Moulin, Game Theory for the Social Sciences, second and revised ed., New York University Press, New York, 1986. [13] J.B. Rosen, G.L. Xue, On the convergence of a hyperboloid approximation procedure for the perturbed euclidean multifacility location problem, Oper. Res. 41 (1993) 1164–1171. [14] D. Topkis, Supermodularity and Complementarity, Princeton University Press, Princeton, NJ, 1998.