Location–Price Competition with Externality: an Application to the Tenerife Tram

INTERNATIONAL TRANSACTIONS IN OPERATIONAL Intl. Trans. in Op. Res. 15 (2008) 583–598 RESEARCH

Location–price competition with externality: an application to the Tenerife tram

P. Dorta-Gonza´lez, D. R. Santos-Pen˜ ate and R. Sua´rez-Vega Departamento de Me´todos Cuantitativos en Economı´a y Gestio´n, Universidad de Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain E-mail: [email protected] [Dorta-Gonza´lez]

Received 22 February 2007; received in revised form 14 March 2008; accepted 14 March 2008

Abstract This paper presents a location–price equilibrium problem on a tree. A suﬃcient condition for having a Nash equilibrium in a spatial competition model that incorporates price, transport, and externality costs is given. This condition implies both competitors are located at the same point, a vertex that is the unique median of the tree. However, this is not an equilibrium necessary condition. Some examples show that not all medians are equilibria. Finally, an application to the Tenerife tram is presented.

Keywords: duopoly; location; price; externality; equilibrium

1. Introduction

Models in spatial competition involve decisions on location and price (or production) made by firms in a spatial market. Equilibrium problems on a network have been studied by Lederer and Thisse (1990), Labbeánd Hakimi (1991), and Sarkar et al. (1997), among others. A situation is a Nash equilibrium if no firm unilaterally finds change profitable. Formally, in a two-facility location problem, a pair of locations, xA and xB, is a Nash equilibrium if and only if

pAðxA; xBÞ¼maxxpAðx; xBÞ; pBðxA; xBÞ¼maxxpBðxA; xÞ; where pi represents the profits for firm i, i 5 A, B. Two different approaches are usually considered when firms have to choose the location and the price. Some authors assume a simultaneous choice of price and location. However, most models in this context follow Hotelling’s (1929) formulation and use a refinement of the Nash equilibrium. More precisely, firms are supposed to choose location and price, one at a time, in a two-stage process, with the aim of maximizing their own profit. The division into two stages is motivated by the fact that the choice of location is usually prior to the decision on price. In r The Authors. Journal compilation r 2008 The Authors Published by Blackwell Publishing, 9600 Garsington Road, Oxford, OX4 2DQ, UK and 350 Main St, Malden, MA 02148, USA. 584 P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 the first stage, firms simultaneously choose their location. Given any outcome of the first stage, firms then simultaneously choose their price in the second stage. The corresponding two-stage solution concept is called a subgame perfect Nash equilibrium (Selten, 1975). It captures the idea that, when firms select their location, they all anticipate the consequences of their choice on price competition. Externalities have been little discussed in the literature on spatial competition. Equilibrium models incorporating location and externality cost have been investigated by Brandeau and Chiu (1994a, b), although in these works there is no competition in price. Dorta-Gonza´lez et al. (2002) propose a spatial competition model in networks incorporating price and externality cost. In this model, firms are already located and therefore the location equilibrium is not studied. A regulating agent assigns the demand, taking into account the price, transport, and externality costs, and minimizes the joint consumer cost in order to obtain a cooperative allocation that is a Pareto optimum. An allocation is a Pareto optimum if it is impossible to reduce the cost of one consumer without harming at least one other consumer. Assuming this cooperative allocation, each firm selects the price, for fixed locations, in order to maximize its profit. However, to our knowledge, no work considers decisions on location and price. The contribution of this paper with respect to previous work on spatial equilibrium on networks is based on trying, for the first time, location and price decisions in a market where there is externality. This paper studies the location–price equilibrium on trees based on the model by Dorta-Gonza´lez et al. (2002). Some of the assumptions considered in this work are often made in the economic literature on equilibria, as there is a single good and it is essential (the demand is perfectly inelastic). Considering location on trees is due to the complexity of the problem and the important properties that are satisfied by trees. On the other hand, it is common in the literature on location to start with a tree and then to extend the results to a general network. In this paper, cooperative allocation that minimizes the joint consumer cost is assumed, and each firm first selects the location of a facility and then selects the price in order to maximize its profit. A result describes a sufficient condition under which the Nash equilibria on locations and prices are guaranteed. This condition implies that both competitors are located at the same point, that is a vertex that is the unique median of the tree. However, this is not an equilibrium necessary condition. Some examples show that not all medians are equilibria. Finally, an application with real data is presented. The rest of the paper is organized as follows: the location–price equilibrium is discussed in Section 2. Section 3 contains some examples illustrating situations where a median of a tree is not a Nash equilibrium on location. An application to the Tenerife tram is presented in Section 4. Finally, some concluding remarks are included in Section 5.

2. Equilibrium analysis

Let T(V, E) be a tree with node set V and edge set E. Two firms, A and B, located on the tree at points xA and xB, respectively (any point on an edge), provide a product (or service) to customersP A n at nodes vk V, k 5 1,...,n. The demand at node vk is lk and the total demand is L ¼ k¼1 lk. The demand is perfectly inelastic and is totally satisfied; therefore, L 5 LA1LB, where Li is the r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 585 market share captured by firm i, i 5 A, B. The marginal cost for firm i is assumed to be independent of the quantity supplied and is denoted by Ci. Consider also the following notation: d(x, y) is the distance between x and y, tki 5 t(vk, xi) is the transport cost from the demand node k to the firm located at xi (a nondecreasing function of the distance), pi is the mill price determined by firm i, and ei40 is the unit externality cost of firm i and Ei(Li) 5 eiLi, that is, the externality cost functions are assumed to be linear. In this section, assuming cooperative allocation, the location–price equilibrium is studied as a two-stage game in a manner similar to other equilibrium problems analyzed in the literature (Labbeánd Hakimi, 1991; Sarkar et al., 1997). Firstly, the locations are chosen and then, given these, the prices are determined. To obtain the equilibrium an inverse process is applied; given the locations, the price equilibrium is inferred and then, using this information, the location equilibrium is obtained. Let xA and xB be a pair of fixed locations for the firms,

Dk ¼tkB þ pB tkA pA; k ¼ 1; 2; :::; n;

D0 ¼þ1; Dnþ1 ¼1:

The difference in delivered costs (price plus transport cost) between firm B and firm A, Dk,isa measure of optimality. Nodes can be renamed and, without loss of generality, assume that

D1 > D2 > ... > Dn1 > Dn:

If Dk 5 Dk11 then the demand nodes k and k11 are aggregated. We also deﬁne Xj fj ¼ lk; j ¼ 1; 2; :::; n; f0 ¼ 0; k¼1

Lj ¼ tjB tjA þ 2ðeA þ 2eBÞL 6ðeA þ eBÞfj; j ¼ 1; :::; n;

Tj1 ¼ tjB tjA þ 2ðeA þ 2eBÞL 6ðeA þ eBÞfj1; j ¼ 1; :::; n:

These values, Lj and Tj 1, allow us to make a partition of the real line. Notice that

Ln < Tn1 < Ln1 < ... < Tj < Lj < Tj1 < Lj1 < ... < T1 < L1 < T0:

Let j0A{1, . . ., n} such that ) ) Lj0 CA CB Tj01: Then, from Dorta-Gonza´lez et al. (2002), Propositions 7 and 8, there exists an equilibrium in ðpA; pBÞ, 1 p ¼ 2C þ C þ t t þ 2ðe þ 2e ÞL ; A 3 A B j0B j0A A B 1 p ¼ C þ 2C þ t t þ 2ð2e þ e ÞL : B 3 A B j0A j0B A B

r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies 586 P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 Now, using this price equilibrium, the location equilibrium is studied. A median of a network N is a point xmAN minimizing the total weighted transport cost from nodes V that is a solution to the problem: Xn min tðx; vkÞlk: ð1Þ x2N k¼1 In the particular case of a tree T, it is known that there is always a median in a node. Furthermore, if the demand is odd then this median is unique. When the demand is even there could be only one median (a node) or there could be an edge of medians. In order to provide a characterization of medians in a tree, the following notation is introduced. For v AV, let j VðvjÞ¼ v 2 V : Eðvj; vÞ2E ; Vðvj; uÞ¼ v 2 V : dðv; vjÞ < dðv; uÞ ; 8u 2 VðvjÞ; X lðVðvj; uÞÞ ¼ lv;

v2Vðvj ;uÞ Zj ¼ min lðVðvj; uÞÞ ; u2Vðvj Þ gj ¼ max tðvj; vÞ : v2V

Set V(vj) is determined by the adjacent nodes to vj and V(vj, u) contains those nodes on the tree that are closer to vj than to u. The minimum total demand of these sets V(vj, u)isZj. Finally, gj is the maximum transport cost between the nodes of the tree and vj. Note that Zj and gj can be defined for any point xAT considering an artificial node at x. Then, a characterization of medians in trees is as follows: L x is a median of T , Z * : ð2Þ m m 2 Moreover, L Z > ) x is the unique median of T: ð3Þ m 2 m The following assumption establishes a bound for the difference of marginal costs with respect to the minimum total demand Zj and the maximum transport cost gj. This assumption will be used later in Proposition 1 as a sufficient condition for the existence of a Nash equilibrium on location.

Assumption 1 There exists vmAV such that rm)CA CB)sm; where

rm ¼ gm þ 2ðeA þ 2eBÞL 6ðeA þ eBÞZm; ð4Þ

sm ¼gm þ 2ðeA þ 2eBÞL 6ðeA þ eBÞðL ZmÞ: ð5Þ

Note that rm4sm if and only if ) 2gm 6ðeA þ eBÞðL þ 2ZmÞ; r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 587 that is

* L gm Zm þ : ð6Þ 2 6ðeA þ eBÞ

As gm40 then L Z > ð7Þ m 2 and from (3) we conclude that a vertex vm satisfying Assumption 1 is the unique median of the tree. Nevertheless, not all medians satisfy Assumption 1, for two reasons. Firstly, because condition (7) does not imply (6), and secondly because it is possible that CA CB 2½= rm; sm: Then, Assumption 1 is more restrictive than the one that characterized the median of a tree. The relative competitiveness, which is expressed by the difference of firms’ marginal costs, is one determining factor. Thus, Assumption 1 establishes that the difference in marginal costs is not too large, that is, both firms have similar competitiveness. In order to introduce other determining factors and discuss when Assumption 1 is satisfied, consider the following reformulation of condition (6):

gm eA þ eB* : ð8Þ 3ð2Zm LÞ

From (7), 2Zm4L and therefore condition (8) is well deﬁned. Consider the following scenarios:

(i) gm large and 2Zm L small. In this case, condition (8) is satisﬁed for large values of eA1eB. Note that gm ¼ maxfgtðvm; vÞ is large when there are nodes far from vm (if t(vm, v) 5 d(vm, v)). v2V On the other hand, 2Zm L is small when the demand of each subtree containing vm resulting ^ from deleting edge E(v, vm)islðTvÞ’2. This means a similar demand distribution in subtrees Tv with E(v, vm)AE that is to say a similar demand distribution in subtrees Tv\{vm}. (ii) gm small and 2Zm L large. In this case, condition (8) is satisﬁed for small values of eA1eB. Note that gm ¼ maxfgtðvm; vÞ is small when the tree diameter is also small (if v2V L t(vm, v) 5 d(vm, v)), and 2Zm L is large when Zm 2 ; that is, when there is a demand concentration in the median. The following example illustrates a situation where Assumption 1 occurs. Figure 1 shows a tree with 3 nodes and the transport cost between nodes. The demands are l1 5 2, l2 5 7, l3 5 5. Note that vm 5 v2 is the median of the tree. Furthermore, g2 5 4.5, Z2 5 9, and condition (8) is g2 eA þ eB* ¼ 0:375: 3ð2Z2 LÞ

Considering that the unit externality costs are eA 5 0.2, eB 5 0.3, then

rm ¼0:1; sm ¼ 2:9;

3 4.5 v1 v2 v3

Fig. 1. Tree satisfying Assumption 1.

r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies 588 P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 and, therefore, for values CA and CB such that 0:1)CA CB)2:9; Assumption 1 holds, in particular when CA 5 CB. The following result establishes that under the previous assumption a Nash equilibrium on location exists.

Proposition 1 Let T(V, E) be a tree and the transport cost a linear function of the distance, i.e. t(x, y) 5 a(d(x, y)), a40. Under Assumption 1, ðxA; xBÞ¼ðvm; vmÞ is a Nash equilibrium on locations with vm the unique median of the tree.

Proof. It will be proved that each firm maximizes its profit at vm when the other firm is located at vm. (a) Consider xA ¼ vm; xB 2 T (if xB is not a node then an artificial node is created), and the shortest path containing xA and xB. The nodes outside this path can be aggregated to the nearest node in the shortest path, resulting in a linear tree. Denoting u1 5 vm, uh 5 xB,

L1 ¼ t1B þ 2ðeA þ 2eBÞL 6ðeA þ eBÞf1; ) g1 þ 2ðeA þ 2eBÞL 6ðeA þ eBÞf1; ) g1 þ 2ðeA þ 2eBÞL 6ðeA þ eBÞZ1;

)CA CB:

Moreover,

T0 ¼ t1B þ 2ðeA þ 2eBÞL; * g1 þ 2ðeA þ 2eBÞL 6ðeA þ eBÞðL Z1Þ;

*CA CB:

Because L14CA CB4T0, from Dorta-Gonza´lez et al. (2002), Proposition 8, the proﬁt function for ﬁrm B is

1 2 pBðxA; xBÞ¼ ðÞCA CB þ t1A t1B þ 2ð2eA þ eBÞL ; 18ðeA þ eBÞ

1 2 ¼ ðÞCA CB t1B þ 2ð2eA þ eBÞL : 18ðeA þ eBÞ

Note that

) t1B 2ð2eA þ eBÞL g1 þ 2ðeA þ 2eBÞL 6ðeA þ eBÞZ1;

)CA CB; r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 589 that is

CA CB t1B þ 2ð2eA þ eBÞL*0; and therefore pB is maximum when t1B is minimum, i.e. when xB ¼ u1 ¼ vm: (b) Consider xB ¼ vm; xA 2 T (if xA is not a node then an artiﬁcial node is created), and the shortest path containing xB and xA. The nodes outside this path can be aggregated to the nearest node in the shortest path, resulting in a linear tree. Denoting u1 5 xA, uh 5 vm,

Lh ¼thA þ 2ðeA þ 2eBÞL 6ðeA þ eBÞL; ) gh þ 2ðeA þ 2eBÞL 6ðeA þ eBÞL; ) gh þ 2ðeA þ 2eBÞL 6ðeA þ eBÞZh;

)CA CB:

Moreover,

Th1 ¼thA þ 2ðeA þ 2eBÞL 6ðeA þ eBÞðL lhÞ; * gh þ 2ðeA þ 2eBÞL 6ðeA þ eBÞðL lhÞ; * gh þ 2ðeA þ 2eBÞL 6ðeA þ eBÞðL ZhÞ;

*CA CB:

Since Lh4CA CB4Th 1, from Dorta-Gonza´lez et al. (2002), Proposition 8, the proﬁt function for ﬁrm A is

1 2 pAðxA; xBÞ¼ ðÞCB CA þ thB thA þ 2ðeA þ 2eBÞL ; 18ðeA þ eBÞ

1 2 ¼ ðÞCB CA thA þ 2ðeA þ 2eBÞL : 18ðeA þ eBÞ Note that * thA þ 2ðeA þ 2eBÞL gh þ 2ðeA þ 2eBÞL 6ðeA þ eBÞðL ZhÞ;

*CA CB; that is

CB CA thA þ 2ðeA þ 2eBÞL*0;

and therefore pA is maximum when thA is minimum, i.e. when xA ¼ uh ¼ vm. From (a) and (b), it follows that ðxA; xBÞ¼ðvm; vmÞ is a Nash equilibrium on locations. &

r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies 590 P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 The price equilibrium and the proﬁt functions can be obtained from Dorta-Gonza´lez et al. (2002), Propositions 7 and 8, considering that xA ¼ xB ¼ vm: That is, ðpA; pBÞ with 1 p ¼ ½2C þ C þ 2ðe þ 2e ÞL ; A 3 A B A B 1 p ¼ ½C þ 2C þ 2ð2e þ e ÞL B 3 A B A B is a Nash equilibrium in the second stage and the proﬁt functions are

1 2 pAðxA; xBÞ¼ ðÞCB CA þ 2ðeA þ 2eBÞL ; 18ðeA þ eBÞ

1 2 pBðxA; xBÞ¼ ðÞCA CB þ 2ð2eA þ eBÞL : 18ðeA þ eBÞ In Proposition 1, an equilibrium suﬃcient condition is presented for a spatial competition model that incorporates price, transport and externality costs. However, assuming ﬁrms locate only in points where there is price equilibrium, this is not a location equilibrium necessary condition as shown in the following example. Figure 2 shows a tree with two nodes and the transport cost between nodes. Suppose l1 5 3, l2 5 1, eA 5 0.2, eB 5 0.3. In this case, the median is vm 5 v1. Note that Zm 5 3, gm 5 4, and therefore

L gm Zm < þ ¼ 3:33: 2 6ðeA þ eBÞ That is, Assumption 1 is not satisﬁed. Consider xA 5 xB 5 vm. In this case all nodes can be aggregated to u1 5 vm and

L1 ¼5:6; T0 ¼ 6:4:

Suppose CA 5 4, CB 5 2. Then L14CA CB4T0 and, with price equilibrium,

pAðvm; vmÞ¼2:15;

pBðvm; vmÞ¼6:42:

Now consider xA 5 vm and ﬁrm B moves to xl, a distance l from xA,0ol44. Denoting u1 5 vm and u2 5 v2,

L1 ¼ l 2:6; T0 ¼ l þ 6:4; L1)CA CB)T0; 8l and as t1A t1B 5 lo0, in equilibrium, ﬁrm B does not improve its proﬁt.

4 v1 v2

Fig. 2. Tree showing Assumption 1 is not an equilibrium necessary condition. r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 591

Finally, consider xB 5 vm and ﬁrm A moves to xl, a distance l from xB,0ol44. Denoting u1 5 v2 and u2 5 vm,

L2 ¼l 5:6; T1 ¼l þ 3:4; L1 ¼ l þ 3:4:

If 0ol41.4 then L2oCA CB4T1, and as t2B t2A 5 lo0 in equilibrium, firm A does not improve its profit. On the other hand, if 1.4ol44 then T1oCA CBoL1, and there is not equilibrium in price. Therefore, (vm, vm) is a local Nash equilibrium and it is a Nash equilibrium assuming firms locate only in points where there is price equilibrium. Some examples showing medians that are not Nash equilibria are given in the following section.

3. Examples

These examples show that not all medians are equilibria. In the ﬁrst example there is a unique median and in the second there is an edge of medians. In both cases, condition (8) does not occur and none of the medians is a Nash equilibrium.

3.1. Example 1

Figure 3 shows a tree with ﬁve nodes and the transport cost between nodes. Suppose l3 5 2, lk 5 1, k6¼3, eA 5 0.3, eB 5 0.2. In this case, the median vm 5 v3 is unique. Note that Z3 5 4, g3 5 4, and therefore

L g3 Z3 < þ : 2 6ðeA þ eBÞ That is, Assumption 1 is not satisﬁed. Consider xA 5 xB 5 v3. In this case all nodes can be aggregated to u1 5 v3 and

L1 ¼9:6; T0 ¼ 8:4:

Suppose CA CB 5 6. Then L14CA CB4T0 and, in equilibrium,

pBðv3; v3Þ¼1:44:

Now consider xA 5 v3 and xB 5 v4. In this case, nodes can be aggregated to edge E(v3, v4). Denoting u1 5 v3 and u2 5 v4,

L2 ¼11:6; T1 ¼5:6; L2)CA CB)T1; and, in equilibrium,

pBðv3; v4Þ¼3:48:

2 2 2 2 v1 v2 v3 v4 v5

Fig. 3. Tree with an unique median v3.

r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies 592 P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 Therefore,

pBðvm; vmÞ < pBðvm; v4Þ and then (vm, vm) is not a Nash equilibrium.

3.2. Example 2

Now consider the tree in Fig. 4 and suppose lk 5 3, 8k; eA 5 0.2, eB 5 0.3. In this case, any point on the edge E(v2, v3) is a median of the tree. Consider vm a median (if it is not a node then an L artiﬁcial node is created). Note that Zm ¼ 2 ¼ 6, gm40, and therefore

L gm Zm < þ : 2 6ðeA þ eBÞ That is, Assumption 1 is not satisﬁed. Consider v24xA4xB4v3. In this case nodes can be aggregated to edge E(v2, v3). Denoting u1 5 v2 and u2 5 v3,

L1 ¼ 1:2 þ tðxA; xBÞ; T0 ¼ 19:2 þ tðxA; xBÞ:

Suppose CA CB 5 5. Then L14CA CB4T0 and, in equilibrium, 1 p ðx ; x Þ¼ ðÞ14:2 þ tðx ; x Þ 2; A A B 9 A B 1 p ðx ; x Þ¼ ðÞ21:8 tðx ; x Þ 2: B A B 9 A B

Then pA is maximum when t(xA, xB) is also maximum, i.e. xA 5 v2, xB 5 v3,andpB is maximum when t(xA, xB) is minimum, i.e. xA 5 xB. Therefore, there is no equilibrium in edge E(v2, v3) because firm A tends to move away from B (maximum differentiation), while firm B tends to move close to A (minimal differentiation). As a particular case, profits at medians v2 and v3 are shown in Table 1.

v 3 3 4.5 1 v2 v3 v4

Fig. 4. Tree with an edge of medians E(v1, v3).

4. Application to the Tenerife tram

In June 2007, the ﬁrst line of the Tenerife tram (line 1), in the metropolitan area ‘‘Santa Cruz-La Laguna’’, was inaugurated. The length of the line is 12.5 km, with 21 stops, and the tram has a capacity for 200 people. During peak hours, the tram runs every 5 min and the maximum waiting time is 30 min in the early hours at weekends. r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 593 Table 1 Proﬁts at medians v2 and v3 of Fig. 4

(pA, pB) xB 5 v2 xB 5 v3 xA 5 v2 (22.40, 52.80) (32.87, 39.27) xA 5 v3 (32.87, 39.27) (22.40, 52.80)

Table 2 Tram stops, travelling time between stops, and the number of ticket validations in October 2007 Node Tram stop Time (minutes) Cumulative time Validations

1 Intercambiador – 0 81,583 2 Fundacio´n 3 3 51,905 3 Teatro Guimera´ 1 4 90,033 4 Weyler 2 6 99,522 5 La Paz 2 8 64,013 6 Puente Zurita 2 10 46,132 7 Cruz del Sen˜ or 2 12 89,050 8 Conservatorio 2 14 14,717 9 Chimisay 1 15 26,040 10 Principes de Espan˜ a 2 17 51,155 11 Hospital La Calendaria 2 19 46,604 12 Taco 1 20 53,837 13 El Cardonal 3 23 17,169 14 Hospital Universitario 1 24 45,365 15 Las Mantecas 3 27 12,131 16 Campus Guajara 2 29 48,362 17 Gracia 1 30 27,941 18 Museo de la Ciencia 2 32 20,746 19 Cruz de Piedra 1 33 36,099 20 Padre Anchieta 3 36 41,369 21 La Trinidad 1 37 115,977 Total 37 1,079,751

The information about the tram stops, the travelling time between stops, and the number of ticket validations at the tram stops in October 2007 are shown in Table 2. This line, the median, and population density in the metropolitan area are shown in Fig. 5. In October 2007, the daily average number of validations was around 36,000. Most of them correspond to frequent users with monthly passes. The monthly pass offers unlimited travel on metropolitan transport for one month at a fixed price (40 euros). The tram operator ‘‘Metropolitano de Tenerife’’ (MTSA) is 85% publicly owned. It has two information offices at stops 1 and 21. These offices deal with customer enquiries and provide general information on the service (routes, timetables, bus connections) as well as information on standard tickets, multi-journey tickets and monthly passes, all of which can be purchased directly from the offices. They also handle complaints and suggestions from customers and act as lost property offices.

r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies 594 P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598

Fig. 5. Tenerife metropolitan area and tram stops.

At present, the tram operator MTSA is considering the possibility to expand and relocate the information oﬃces. Furthermore, it is examining the adequacy of outsourcing this service and introducing competition. In this context, in the following subsections two diﬀerent problems are considered.

4.1. The problem without competition

Tables 3 and 4 show the solution to the p-median and p-center problems, respectively. The p-median problem consists of determining p facility locations that minimize the total transport time (eﬃciency criterion). The p-center problem considers an equity criterion and consists of determining p facility locations that minimize the longest transport time between a customer and its nearest facility. Many applications of these problems arise both in the public and in the private sectors. As can be seen in Table 3, the 1-median solution is node 9, that is, the node should be chosen according to the minimization of the transport time of all users. Locating two facilities at nodes r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 595 Table 3 Solutions to the p-median problems (14p46) Total time Reduction p Median (minutes) from p 5 1 (%)

1 9 11,461,290 – 2 4,17 5,324,918 54 3 4,12,20 3,212,012 73 4 3,7,12,20 2,364,542 80 5 3,7,12,17,21 1,742,100 85 6 1,4,7,12,17,21 1,371,575 88 Total time Reduction Increment p Median (1,21 ﬁxed) (minutes) from p 5 2 (%) from no ﬁxed (%)

2 1,21 8,198,776 – 35 211 1,21;10 4,158,064 49 23 212 1,21;5,11 2,761,774 66 14 213 1,21;4,7,12 2,053,556 75 15 214 1,21;4,7,12,17 1,371,575 83 0

Table 4 Solutions to the p-center problems (14p46) Time Reduction p Center (maximum) from p 5 1 (%)

11119– 2 6,15 10 47 3 4,12,19 6 68 4 3,8,13,19 4 79 5 3,7,12,16,19 4 79 6 2,6,10,14,18,21 3 84 Time Reduction Increment p Center (1,21 Fixed) (maximum) from p 5 2 (%) from no ﬁxed (%)

2 1,21 18 – 80 211 1,21;11 9 50 50 212 1,21;7,14 6 67 50 213 1,21;5,10,15 5 72 25 214 1,21;4,7,12,16 4 78 33

4 and 17 would lead to a reduction in the total transport time of 54%. By introducing a third facility, the additional reduction would be 19%. New facilities produce additional reductions below 10%. In the present situation (two facilities at nodes 1 and 21), the transport time for all users is 35% greater than the 2-median solution. In case it is not possible to relocate facilities, a new facility in node 10 reduces this transport time by 49%. As can be seen in Table 4, the 1-center solution is node 11, obtaining in this case a maximum user transport time of 19 min. Two facilities at nodes 6 and 15 reduce this time to 10 min and a

r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies 596 P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 third service reduces this time to 6 min. In the present situation (facilities at nodes 1 and 21), the maximum transport time is 18 min, 80% greater than the 2-center solution. A new facility in node 11 reduces this time by half, but it is still 50% greater than the 3-center solution. Comparing Tables 3 and 4, two nodes in both the median and the center solutions are the same (4 and 12) while the others are close (19 and 20). Therefore, these data suggest the location of at least two facilities at nodes 4 and 17 (or three at nodes 4, 12, and 19 or 20).

4.2. The problem with competition

Consider that two firms, A and B, located on the tree at tram stops xA and xB, respectively, provide the information service to customers at tram stops vk,14k421. Suppose the cost of this service is covered by the tram operator, that is, the company MTSA pays a fixed amount (per customer served) to the firms. In this case, because the service is free for customers, in the model pi 5 0, i 5 A, B. The demand at each node is estimated as a function of the validations at the tram stops. Let Vk be the number of validations at tram stop k. Then the daily average number of validations at tram stop k is Vk/30 and the daily average number of users at the information office is 2 lk ¼ Vk=30 considering users’ demand the information service, on average, once a month. Assuming only half of the validations as a measure of the demand, understanding that most customers use the tram twice a day (destinationP and return journey), then the demand at node k is 1 21 1 1 2lk. Therefore, the daily demand is L ¼ k¼1 lk ¼ 1200 (2 L considering 2 lk). Customers incur a transport cost that is a function of the travelling time to the nearest facility. We consider that the transport cost from the demand node k to the firm located at xi is a linear function of the transport time (destination and return journey), that is t(vk, xi) 5 a(2time(vk, xi)), where 04time(vk, xi)437. In the parametric analysis a 5 0.001, 0.002, 0.005, is considered and therefore t(vk, xi)A[0, 0.37] euros. In addition, customers incur an externality cost that is a function of the waiting time to be served. It is assumed, without loss of generality, that eA4eB. This unitary externality cost is related to capacity and technology, among others. In the parametric analysis we consider eiA[0.002, 0.010]. Then, the externality cost corresponding to 300 customers, for example, is Ei(300) 5 300eiA[0.6, 3] euros. As has been indicated previously, v9 is the median of the tree. Furthermore, g9 5 44a (transport 1 cost from node 21) and Z9 5 603.10 (301.55 considering 2 lk). 1 Tables 5 and 6 show the interval for CA CB in Assumption 1 for daily demands lk and 2 lk.As can be seen in Table 5, the interval is empty in 18 of 54 situations considered. In cases where the interval is not empty, this range increases with the externality cost e and reduces with the unitary transport cost a. Furthermore, the interval length increases with demand. In Table 6, different externality costs are considered (1% and 5% of the difference, respectively). Notice that as the difference in externality cost increases, the intervals reduce and most of them become positive (CA4CB). As a conclusion, in those scenarios where the interval is not empty, in equilibrium both competitors locate at the same node, the median of the tree (tram stop 9). In this solution both firms could share the facility, employing their own staff. Therefore, in equilibrium customers would not perceive a difference with respect to who provides the service (all employees with the same uniform, r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 597 Table 5 Interval for CA CB in Assumption 1 considering eA 5 eB 5 e Demand e a 5 0.001 a 5 0.002 a 5 0.005 lk 0.002 [ 0.0338, 0.0338] – – 0.003 [ 0.0728, 0.0728] [ 0.0288, 0.0288] – 0.004 [ 0.1117, 0.1117] [ 0.0677, 0.0677] – 0.005 [ 0.1506, 0.1506] [ 0.1066, 0.1066] – 0.006 [ 0.1896, 0.1896] [ 0.1456, 0.1456] [ 0.0136, 0.0136] 0.007 [ 0.2285, 0.2285] [ 0.1845, 0.1845] [ 0.0525, 0.0525] 0.008 [ 0.2674, 0.2674] [ 0.2234, 0.2234] [ 0.0914, 0.0914] 0.009 [ 0.3063, 0.3063] [ 0.2623, 0.2623] [ 0.1303, 0.1303] 0.010 [ 0.3453, 0.3453] [ 0.3013, 0.3013] [ 0.1693, 0.1693] 1 2lk 0.002 – – – 0.003 [ 0.0144, 0.0144] – – 0.004 [ 0.0338, 0.0338] – – 0.005 [ 0.0533, 0.0533] [ 0.0093, 0.0093] – 0.006 [ 0.0728, 0.0728] [ 0.0288, 0.0288] – 0.007 [ 0.0922, 0.0922] [ 0.0482, 0.0482] – 0.008 [ 0.1117, 0.1117] [ 0.0677, 0.0677] – 0.009 [ 0.1312, 0.1312] [ 0.0872, 0.0872] – 0.010 [ 0.1506, 0.1506] [ 0.1066, 0.1066] –

Table 6 1 Interval for CA CB in Assumption 1 considering 2lk, eB 5 1.01eA, and eB 5 1.05eA eA, eB a 5 0.001 a 5 0.002 a 5 0.005

0.002, 0.00202 – – – 0.003, 0.00303 [0.0033, 0.0327] – – 0.004, 0.00404 [ 0.0102, 0.0582] – – 0.005, 0.00505 [ 0.0238, 0.0838] [0.0203, 0.0397] – 0.006, 0.00606 [ 0.0374, 0.1094] [0.0067, 0.0653] – 0.007, 0.00707 [ 0.0509, 0.1349] [ 0.0069, 0.0909] – 0.008, 0.00808 [ 0.0645, 0.1605] [ 0.0205, 0.1165] – 0.009, 0.00909 [ 0.0781, 0.1860] [ 0.0341, 0.1420] – 0.010, 0.0101 [ 0.0916, 0.2116] [ 0.0476, 0.1676] – 0.002, 0.00210 – – – 0.003, 0.00315 [0.0741, 0.1058] – – 0.004, 0.00420 [0.0842, 0.1558] – – 0.005, 0.00525 [0.0942, 0.2057] [0.1382, 0.1617] – 0.006, 0.00630 [0.1043, 0.2557] [0.1483, 0.2117] – 0.007, 0.00735 [0.1143, 0.3056] [0.1583, 0.2616] – 0.008, 0.00840 [0.1243, 0.3555] [0.1683, 0.3115] – 0.009, 0.00945 [0.1344, 0.4055] [0.1784, 0.3615] – 0.010, 0.0105 [0.1444, 0.4554] [0.1884, 0.4114] – for example). However, with competition the service leads to eﬃciency in minimizing transport and waiting times. In this solution, ﬁrms have the incentive of attending to the maximum number of customers (increasing the number of employees at peak hours, for example).

r 2008 The Authors. Journal compilation r 2008 International Federation of Operational Research Societies 598 P. Dorta-Gonza´lez et al. / Intl. Trans. in Op. Res. 15 (2008) 583–598 5. Concluding remarks

In this paper, assuming cooperative allocation, a location–price equilibrium problem with linear externality costs is studied. Under certain conditions, a Nash equilibrium on location exists. Furthermore, both competitors are located at the same point, a vertex that is the unique median of the tree. This sufficient condition is satisfied from small values of externality cost when the tree diameter is small and there is a demand concentration in the median. However, this is not an equilibrium necessary condition. Some examples show that there are medians that are not Nash equilibria. A parametric analysis with real data shows that the interval length for CA CB in Assumption 1 increases with the demand and externality cost and reduces with the transport cost. In the scenarios where the interval is not empty (44 of 108 cases analyzed), in equilibrium, both competitors locate at the same node, the median of the tree (tram stop 9). This result is known in spatial competition theory as the minimum differentiation principle. In this solution, both competitors could share the facility reducing costs and favoring user access. Furthermore, this solution minimizes user transport and waiting times.

Acknowledgments

This research was partially ﬁnanced by FEDER and Ministerio de Ciencia y Tecnologı´a, grant BFM2002-04525-C02-01, and Universidad de Las Palmas de Gran Canaria, grant UNI2004/12.

References

Brandeau, M., Chiu, S., 1994a. Facility location in a user-optimizing environment with market externalities: analysis of customer equilibria and optimal public facility locations. Location Science 2, 129–147. Brandeau, M., Chiu, S., 1994b. Location of competing private facilities in a user-optimizing environment with market externalities. Transportation Science 28, 125–140. Dorta-Gonza´lez, P., Santos-Pen˜ ate, D.R., Sua´rez-Vega, R., 2002. Pareto optimal allocation and price equilibrium for a duopoly with negative externality. Annals of Operation Research 116, 129–152. Hotelling, H., 1929. Stability in competition. Economic Journal 39, 41–57. Labbe´, M., Hakimi, L., 1991. Market and location equilibrium for two competitors. Operation Research 39, 749–756. Lederer, P., Thisse, J., 1990. Competitive location on networks under delivered pricing. Operational Research Letters 9, 147–153. Sarkar, J., Gupta, B., Pal, D., 1997. Location equilibrium for Cournot oligopoly in spatially separated markets. Journal of Regional Science 2, 195–212. Selten, R., 1975. Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory 4, 25–55.

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