Local market structure in a Hotelling town

We develop a theoretical framework to study the location-price competition in a Hotelling-type network game, extending the Hotelling model, with linear transportation costs, from a line (city) to a network (town). 
 We show the existence of a pure Nash equilibrium price if, and only if, some explicit conditions on the production costs and on the network structure hold. Furthermore, we prove that the local optimal localization of the firms are at the cross-roads of the town.

Hotelling [13] presented a city represented by a line segment where a uniformly distributed continuum of consumers buy a single commodity. The consumers have to support linear transportation costs when buying the commodity in one of the two firms of the city. The firms compete in a two-staged location-price game, where simultaneously choose their location and afterwards set their prices in order to maximize their profits. Hotelling concluded that firms would agglomerate at the center of the line, an observation referred as the "Principle of Minimum Differentiation". In 1979, D'Aspremont et al. [2] showed that the "Principle of Minimum Differentiation" is invalid, since there was no (pure) price equilibrium solution for all possible locations of the firms, in particular when they are not far enough from each other.
Other models have been developed where the line in the Hotelling model is replaced by other topologies as for example in the Salop Model [25], where the line is replaced by the circle, or in the Spokes model [6] (see, also, [15,14]).
In this work, we study the Hotelling town model (see [19,26]). The Hotelling town model extends the Hotelling model from a line (city) to a network (town). Trying to mimic a real town, the roads of the town are the edges of the network, the crossroad are the vertices with degree higher than two and the ends of no-exit roads are the vertices with degree one. The firms are spread over the town and the consumers are uniformly distributed along the roads (similar size houses). The roads can have different lengths (market sizes) and the firms can have different unitary productions costs (firm's heterogeneity).
There is also a vast literature in network games (see, for instance [4,9,8,10]). Usually, following the modeling methodology common in social network analysis, these studies locate firms and consumers at nodes and the edges are used to establish relevant information among the agents. The Hotelling town model, presented here, is different because the consumers are assumed uniformly distributed along the edges of the network and not at the nodes.
A price strategy consists in associating to each firm a selling price of the commodity. As in the original Hotelling model, the expenditure of a consumer that chooses to buy in a shop consists in the sum of the price practiced by that shop plus the transportation cost that is proportional to the minimal distance between his house (position at the network) and the shop.
The firms compete in a two-stage location-price game, where simultaneously choose their location and afterwards set their prices in order to maximize their profits; and each consumer will buy in the shop that will minimize its expenditure.
In the price subgame, the main goal is to compute the price strategy that has the following two essential economic properties: (i) any small deviation of a price of a firm provokes a decrease in its own profit (local strategic optimum); and (ii) all firms have non-empty market (local market structure). Property (i) stabilizes the prices because the firms do not have an incentive to do small changes in their prices. Property (ii) stabilizes the set of competing firms because every firm has a non-empty market (positive profit) and so, does not go to bankruptcy, and a sufficiently high entry cost can avoid new firms to appear in the town. We call a price strategy satisfying these two properties a local market optimum price strategy.
We prove that the price subgame has a local market optimum price strategy if, and only if, some explicit conditions on the production costs and road lengths hold. We show that if there is a local market optimum price strategy then it is unique. Furthermore, we introduce the weak bounded costs condition that gives a simple bound on the maximum difference between the production costs and on the maximum difference between the road lengths in terms of the transportation cost and the minimal road length. The weak bounded costs condition is a simple sufficient condition for the existence of the local market optimum price strategy.
We give an explicit closed formula and an explicit series expansion formula for the local market optimum price strategy. The series expansion formula shows explicitly how the local market optimum price strategy of a firm depends on the production costs, road market sizes and firms locations. Furthermore, the influence of a firm in the local market optimum price strategy of other firm decreases exponentially fast with the distance between the firms.
Assuming that the firms might not know the entire network, we introduce the idea of space bounded information that determines how far from its location each firm knows the network structure in terms of the production costs, node degrees and road sizes. We show that each firm is able to compute an approximation of its LOCAL MARKET STRUCTURE IN A HOTELLING TOWN 3 own local market optimum price strategy that improves exponentially fast with the space bounded information knowledge of the firm.
We say that a price strategy has the profit degree growth property if the profits of the firms increase with the degree of the nodes in the neighborhoods in which they are located. We give an example where the local market optimum price strategy does not have the profit degree growth property. Hence, we introduce the degreebound condition, that is similar to the weak bounded condition, and we prove, under this condition, that the local market optimum price strategy has the profit degree growth property.
For the price-subgame, we note that a Nash equilibrium price with the local market structure property is a local market optimum price strategy, but a local market optimum price strategy might not be a Nash equilibrium price. However, we prove that the local market optimum price strategy is a Nash equilibrium price if, and only if, some explicit conditions on the production costs and road lengths hold. Furthermore, we introduce the strong bounded condition that in comparison with the weak bounded condition has the additional feature of depending also on the maximum node degree of the network. The strong bounded costs condition is a simple sufficient condition for the existence of the Nash equilibrium price. We note that this condition avoids that firms can get too close, and so D'Aspremont et al. [2] objection to the existence of a Nash equilibrium price does not occur.
For the location-subgame, we assume that, for every location allowed by the weak bounded condition, the firms choose the corresponding local market optimum price strategy for the price-subgame. A localization strategy for the firms in the network is for every firm to choose his position in the Hotelling town. A local optimal localization strategy is achieved when small perturbations in location do not result in improved profits. Similarly to the original Hotteling model in the line, we prove that the firms do not prefer to be located at the ends of no-exit roads. However, in contrast with the original Hotteling model in the line, we prove that the firms prefer maximum differentiation, in the sense that they prefer to be located at the crossroads of the network. This result is observed in real towns because the owners of the shops usually prefer to have them located at crossroads than along the roads.
2. Hotelling town. The Hotelling town model (see [19]) consists of a network of consumers and firms. The consumers (buyers) are located along the edges (roads) of the network. Every road has two endpoints (vertices). For simplicity of the model, we assume that in a neighborhood of every vertex is located a single firm (shop). The degree k of the vertex is given by the number of incident edges. If the degree k is greater that 2 then the vertex is a crossroad of k roads; if the degree k is equal to 2 then the vertex is a junction between two roads; and if k is equal to 1 the vertex is in the end of a road with no exit. Every consumer will buy one unit of the commodity from only one firm in the network and each firm will charge its customers the same price for the commodity.
2.1. Prices and profits. A Hotelling town price strategy P consists of a vector whose coordinates are the prices p i of each firm F i . Every firm F i is located at a position y i in a neighborhood of a vertex i ∈ V . A consumer located at a point x of the network who decides to buy at firm F i spends E(x; i, P) = p i + t d(x, y i ) the price p i charged by the firm F i plus the transportation cost that is proportional t to the minimal distance measured in the network between the position y i of the firm F i and the position x of the consumer. Given a price strategy P, the consumer will choose to buy in the firm F v(x,P) that minimizes his expenditure v(x, P) = argmin i∈V E(x; i, P), where V is the set of all vertices of the Hotelling town. Hence, for every firm F i , the market M (i, P) = {x : v(x, P) = i} consists of all consumers who minimize their expenditures by opting to buy in firm F i . The road market size l i,j of a road R i,j is the Lebesgue measure (or length) of the road R i,j , because the buyers are uniformly distributed along the roads. The market size S(i, P) of the firm F i is the Lebesgue measure of M (i, P). The Hotelling town production cost C is the vector whose coordinates are the production costs c i of the firms F i . The Hotelling town profit Π(P, C) is the vector whose coordinates are the profits of the firms F i .

2.2.
Local market network structure. The local firms of a consumer located at a point x in a road R i,j with vertices i and j are the firms F i and F j . For every vertex i let N i be the set of all neighboring vertices j for which there is a road R i,j connecting the vertices. A price strategy P determines a local market structure if every consumer buys from one of his local firms, i.e.
If a price strategy P determines a local market structure then for every road R i,j there is one consumer located at a point x i,j ∈ R i,j who is indifferent to the local firm from which he going to buy his commodity, i.e. E(x; i, P) = E(x; j, P).

2.3.
Inner network structure. The Hotelling town admissible market size L is the vector whose coordinates are the admissible local firm market sizes The Hotelling town neighboring market structure K is the matrix whose coordinates are (i) k i,j = k −1 i , if there is a road R i,j between the firms F i and F j ; and (ii) k i,j = 0, if there is not a road R i,j between the firms F i and F j .

2.4.
Nash equilibrium network price. The candidate Nash equilibrium with a local market structure is where 1 is the identity matrix. We note that K is a stochastic matrix and so has spectrum radium one. Equation (1) gives an explicit closed formula and an explicit series expansion formula for the the candidate Nash equilibrium with a local market structure. The series expansion formula shows explicitly how the candidate Nash equilibrium with a local market structure of a firm depends on the production costs, road market sizes and firms locations. Furthermore, the influence of a firm in the the candidate Let k M (resp. k m ) be the maximum (resp. minimum) node degree of the Hotelling town 3. Local best response price strategy. In the following sections, for simplicity of exposition, we will assume that every firm is located at the corresponding node, i.e. y i = i. In the section 10, we extend all the results to the general case where the firms are not necessarily located at the nodes, but can choose their locations in the neighbourhoods of the nodes. Furthermore, in the section 10, we give the proofs of all the results presented through the paper.
We observe that, for every road R i,j , there is an indifferent buyer located at a distance of firm F i if, and only if, |p i − p j | < t l i,j . Thus, a price strategy P determines a local market structure if, and only if, |p i − p j | < t l i,j for every road R i,j . The firms F i and F j (or vertices i and j) are neighbors if there is a road R i,j with end nodes i and j. Let N i the set of all vertices that are neighbors of the vertex i and, so, k i is the cardinality of the set N i that is equal to the degree of the vertex i. If the price strategy determines a local market structure then S(i, P) = j∈Ni x i,j and Given a pair of price strategies P and P * and a firm F i , we define the price vectorP(i, P, P * ) whose coordinates arep i = p * i andp j = p j , for every j ∈ V \ {i}. Let P and P * be price strategies that determine local market structures. The price strategy P * is a local best response to the price strategy P, if for every i ∈ V the price strategyP(i, P, P * ) determines a local market structure and ∂π i (P(i, P, P * ), C) ∂p i = 0 and ∂ 2 π i (P(i, P, P * ), C) Recall from subsection 2.3 the definitions of the admissible market size L and of the neighboring market structure K. 6 ALBERTO A. PINTO, JOÃO P. ALMEIDA AND TELMO PARREIRA Lemma 3.1. Let P and P * be price strategies that determine local market structures. The price strategy P * is the local best response to price strategy P if, and only if, and the price strategiesP(i, P, P * ) determine local market structures for all i ∈ V .
4. Local market optimum price strategy. We prove that the price subgame has a local market optimum price strategy if, and only if, some explicit conditions on the production costs and road lengths hold. We show that if there is a local market optimum price strategy then it is the candidate Nash equilibrium with a local market structure P L introduced in subsection 2.4. We introduce the weak bounded W B costs condition that gives a simple bound on the ∆(c) and ∆(l) in terms of the transportation cost t and the minimal road length l m of the network. We prove that if the W B holds then P L is the price subgame local market optimum price strategy. A price strategy P * is a local market optimum price strategy if (i) P * is the local best response to P * ; and (ii) P * determines a local market structure.
Recall from subsection 2.4 the definition of the candidate Nash equilibrium P L with a local market structure.
Theorem 4.1. If there is a local market optimum price strategy then it is the the candidate price strategy P L . Furthermore, there is a local market optimum price strategy if, and only if, |p L i − p L j | < t l i,j for all firms F i and F j that are neighbors.
Recall from subsection 2.5 the definition of heterogeneous network measures.

Definition 4.2. A Hotelling town satisfies the weak bounded length and costs (WB)
Theorem 4.3. If the Hotelling town satisfies the WB condition, then the candidate price strategy is the unique local market optimum price strategy. Furthermore, the local market optimum prices p L i are uniformly bounded and the local market optimum profits The marginal rates of the local market optimum prices p L i are positive with respect to the production costs c a , admissible local firm market sizes L a , transportation costs t and road lengths l b,c . The marginal rates of the local market optimum profits π L i are negative with respect to the production costs c i and positive with respect to the production costs c d , admissible local firm market sizes L a , transportation costs t and road lengths l b,c . 5. Nash equilibrium price strategy. For the price-subgame, we note that a Nash equilibrium price with the local market structure property is the local market optimum price strategy P L , but the local market optimum price strategy might not be a Nash equilibrium price. We introduce the strong bounded SB costs condition that gives a simple bound on the ∆(c) and ∆(l) in terms of the transportation cost t, the minimal road length l m of the network and also on the maximum node degree k M of the network. We prove that if the SB holds then P L is the Nash equilibrium price.
The price strategy P * is a best response to the price strategy P, if , for all i ∈ V and for all price strategies P ′ i whose coordinates satisfy p ′ i ≥ c i and p ′ j = p j for all j ∈ V \ {i}. A price strategy P * is a Hotelling town Nash equilibrium if P * is the best response to P * .
Lemma 5.1. In a Hotelling town satisfying the WB condition, if there is a Nash price P * then P * is unique and P * = P L .
Hence, the local market optimum price strategy P L is the only candidate to be a Nash equilibrium price strategy. However, P L might not be a Nash equilibrium price strategy because there can be a firm F i that by decreasing his price is able to absorb markets of other firms in such a way that increases its own profit. Therefore, the best response price strategy P L, * to the local market optimum price strategy P L might be different from P L . Let Lemma 5.2. In a Hotelling town satisfying the WB condition, Hence, a consumer x ∈ R j,k might not buy in its local firms F j and F k . However, the consumer x ∈ R j,k still has to buy in a firm F i that is a neighboring firm of its local firms F j and F k , i.e. i ∈ N j ∪ N k .
For every firm F i and every 0 The SB condition implies the WB condition, and so under the SB condition the only candidate to be a Nash equilibrium price strategy is the local market optimum price strategy P L . On the other hand, the condition implies the SB condition.
Theorem 5.5. If a Hotelling town satisfies the SB condition then there is a unique Hotelling town Nash equilibrium price strategy P * = P L .
Hence, the Nash equilibrium price strategy for the Hotelling town satisfying the SB condition determines a local market structure, i.e. every consumer located at x ∈ R i,j spends less by shopping at his local firms F i or F j than in any other firm in the town and so the consumer at x will buy either at his local firm F i or at his local firm F j .
6. Firm position stability. For the location-subgame, we assume that, for every location allowed by the weak bounded condition, the firms choose the corresponding local market optimum price strategy for the price-subgame. A localization strategy for the firms in the network is for every firm F i to choose his position in the neighborhood of its vertex i. A local optimal localization strategy is achieved when small perturbations in location do not result in improved profits. We prove that a Hotelling town network satisfying the W B condition and with the minimum node degree k m ≥ 3 has a local optimal localization strategy, whereby every firm F i is located at the corresponding node i.
Consider that a firm F i located a node i changes its location to a point y i in a road R i,j at distance x for the node i. In the last section, we prove that there is an ǫ > 0 such that, for every x < ǫ, the price vector P(x; i, j) is the Nash equilibrium price strategy, given by the local market optimum price strategy, taking in account the new localization of the firm F i . Let π i (x; i, j) denote the profit of firm F i with respect to the price strategy P(x; i, j). Hence, a Hotelling town network satisfying the WB condition and with k m ≥ 3 has a local market optimum price strategy, whereby every firm F i is located at the corresponding node i.
In the subsection 10.4, we observe that firms F i , with node degree k i = 1, are node local unstable. Firms F i , with k i = 2 , are node local unstable, except for networks satisfying special symmetric properties. Firms F i , with k i = 3, whose neighboring firms have nodes degree greater or equal to 3 are node local stable. Furthermore, firms F i , with k i ≥ 4, whose neighboring firms have nodes degree greater or equal to 2 are node local stable. 7. Space bounded information. Assuming that the firms might not know the entire network, we introduce the idea of n-space bounded information. We say that a firm has n-space bounded information, if the firm knows the production costs of the other firms and the road lengths of the network up to n consecutive nodes of distance. Given a Hotelling town network satisfying the W B condition, every firm with n-space bounded information can compute a price p i (n) that estimates its own local market optimum price p L i , with exponential precision depending upon n. In addition, the firm can also estimate its profit with exponential precision depending upon n. Given m + 1 vertices x 0 , . . . , x m with the property that there are roads R x0,x1 , . . . , R xm−1,xm the (ordered) m path R is R = (R x0,x1 , . . . , R xm−1,xm ).
Let R(i, j; m) be the set of all m (ordered) paths R = (R x0,x1 , . . . , R xm−1,xm ) starting at i = x 0 and ending at j = x m . Given a m order path R = (R x0,x1 , . . . , R xm−1,xm ), the corresponding weight is The matrix K 0 is the identity matrix and, for n ≥ 1, the coordinates of the matrix The n local market optimum price vector is We observe that in a n-I Hotelling town, the firms might not be able to compute K, C or L. However, every firm F i is able to compute his n local market optimum price p i (n) Let N V denote the number of nodes in the network.
Theorem 7.2. A Hotelling town satisfying the WB condition has a local market optimum price strategy P L that is well approximated by the n local market optimum price P(n) with the following 2 −n bound . Furthermore, P(n + 1) is the best response to P(n) for n sufficiently high.
Hence, by theorem 7.2, the profit π i (P L ) is well approximated by π i (P(n)) with the following bound 8. Profit degree growth. We say that a price strategy has the profit degree growth property if the profits of the firms increase with the degree of the nodes in the neighborhoods in which they are located. We give an example where the local market optimum price strategy does not have the profit degree growth property. Hence, we introduce the degree-bound condition, that is similar to the weak bounded condition, and we prove, under this condition, that the local market optimum price strategy has the profit degree growth property. Hence, we introduce the degreebound DB condition that gives a new bound for ∆(c) and ∆(l). We prove that for a Hotelling town network satisfying the DB condition the local market optimum price strategy P L has the profit degree growth property.
We say that a price strategy P has the profit degree growth property if for every i, j ∈ V .
Lemma 8.1. Let F i be a firm located in a node of degree k i and F j a firm located in a node of degree k j . Then, π L i > π L j if, and only if, Recall from subsection 2.5 the definition of heterogeneous network measures. 9. Examples: Homogeneous town models. We show that a homogeneous Hotelling town satisfies the SB and DB conditions. We present an example of a network with a local market optimum price strategy that is not a Nash equilibrium price. Therefore, the condition (2 k M + 1)∆(c) ≤ t l.
implies the SB condition. The DB condition is reduced to A Hotelling town is homogeneous if c m = c M and l m = l M . For a homogeneous Hotelling town, the WB, SB and DB conditions are satisfied independently of the degree of the nodes. Hence, there is a Nash equilibrium price. Furthermore, the Nash equilibrium price satisfies the profit degree growth property.
We are going to present an example satisfying the WB condition but not the SB condition. Furthermore, we will show that in this example the local market optimum price strategy do not form a Nash equilibrium price. We will show that the local market optimum price strategy is not a Nash equilibrium. The profits of the firms are given by π L i = p i S(i, P L ), and the local market sizes S(i, P L ) are Hence, the local market sizes are Suppose that firm F 2 decides to lower its price in order to capture the market of firm F 1 . The firm F 2 captures the market of F 1 , excluding F 1 from the game, if the firm F 2 charges a price p 2 such that p 2 + 4 t < p L 1 or, equivalently p 2 < 4/3 t. Let us consider p 2 = 4/3 t − ǫ, where ǫ is sufficiently small. Hence, for this new price, firm F 2 keeps the market M (2, P L ) and, since the price of F 2 at location of F 1 is less that p L 1 , firm F 2 gains at least the market of firm F 1 . Thus, the new market M (2, P) of firm F 2 is such that S(2, P) > S(1, P L ) + S(2, P L ). Therefore, S(2, P) > 31/3 and so Thus π 2 > 98 t/9 = π L 2 , and so firm F 2 prefers to alter its price p L 2 . Therefore, P L is not a Nash equilibrium price.
We are going to present an example satisfying the W B condition but not the DB condition. Furthermore, we will show that this example does not have the profit degree growth property. Consider the Hotelling town network presented in Figure  2. We observe that k 1 > k 3 and π L 3 > π L 1 . Hence, the profit degree growth property is not satisfied and so the DB condition does not hold. 10. Firms at nodes neighborhoods. In this section, we extend all the results to the general case where the firms are not necessarily located at the nodes, but can choose their locations in the neighbourhoods of the nodes. Furthermore, we give the proofs of all the results presented through the paper.
For every v ∈ V , let ǫ v = d(v, y v ) and j(v) be the node with the property that y v is at the road R v,j(v) . The shift location matrix S(v) associated to node v is defined by The distancel i,j = d(y i , y j ) between the location of firms F i and F j is given bỹ Let ǫ = max v∈V ǫ v . Hence, for every i, j ∈ V we have 10.1. Local best response price strategy. Under a local market structure, for every road R i,j there is an indifferent buyer located at a distance of firm F i . Thus, a price strategy P determines a local market structure if, and only if, |p i − p j | < tl i,j for every road R i,j . Hence, if then condition (10) is satisfied. Therefore, if condition (11) holds then the price strategy P determines a local market structure. If the price strategy determines a local market structure then The Hotelling town firm deviation is the vector Y whose coordinates are Lemma 10.1. Let P and P * be price strategies that determine local market structures. The price strategy P * is the local best response to price strategy P if, and only if, 13) and the price strategiesP(i, P, P * ) determine local market structures for all i ∈ V .
Proof of Lemma 10.1. By (12), the profit function π i (P, C) of firm F i , in a local market structure, is given by LetP(i, P, P * ) be the price vector whose coordinates arep i = p * i andp j = p j , for every j ∈ V \{i}. Since P and P * are local price strategies, the local best response of firm F i to the price strategy P, is given by computing ∂π i (P(i, P, P * ), C)/∂p i = 0. Hence, By (9), we obtain We note that j∈Ni v∈{i,j} Hence, Therefore, since ∂ 2 π i (P(i, P, P * ), C)/∂p 2 i = −k i /t < 0, the local best response strategy prices P * is given by Hence, the WB condition implies ǫ < l m /6. Let Theorem 10.3. If there is a local market optimum price strategy then it is the the candidate price strategy P L . Furthermore, there is a local market optimum price strategy if, and only if, |p L i − p L j | < tl i,j for all firms F i and F j that are neighbors. If the Hotelling town satisfies the WB condition, then the candidate price strategy is the unique local market optimum price strategy. Furthermore, the local market optimum prices p L i are uniformly bounded The local market optimum profit π L i (P, C) of firm F i is given by and it is uniformly bounded Proof of Theorem 10.3. The matrix K is a stochastic matrix (i.e., j∈V k i,j = 1, for every i ∈ V ) we have K = 1. Hence, the matrix Q is well-defined by and Q is also a non-negative and stochastic matrix. By Lemma 10.1, a local market optimum price strategy satisfies equality (13). Therefore, and so P L satisfies (16). By construction, Let us prove that the price strategy P L is local, i.e., the indifferent consumer x i,j satisfies 0 < x i,j <l i,j for every R i,j ∈ E. We note that We note that and if k v ≥ 3 then Hence, Therefore, from (21), (22) and (23), we have Since Q is a nonnegative and stochastic matrix, we obtain Hence, putting (19), (20) and (24) together we obtain that Since the last relation is satisfied for every firm, we obtain Hence, by the WB condition, we conclude that |p L i − p L j | < t l m − 2 t ǫ. Thus, by equation (11), we obtain that the indifferent consumer is located at 0 < x i,j <l i,j for every road R i,j ∈ E. Hence, the price strategy P L is local and is the unique local market optimum price strategy.
From (19), we have that From (20) and (24), we obtain Similarly, we obtain and so the local local market optimum prices p L i are uniformly bounded and satisfy (17).
We can write the the profit function (12) of firm F i for the price strategy P L as   (25) Since P L satisfies the best response function (14), we have and replacing this sum in the profit function (25), we obtain Hence, using the price bounds (17), we conclude 10.3. Nash equilibrium price strategy.
Lemma 10.4. In a Hotelling town satisfying the WB condition, if there is a Nash price P * then P * is unique and P * = P L .
Proof of Lemma 10.4. Suppose that P * is a Nash price strategy and that P * = P L . Hence, P * does not determine a local market structure, i.e., there exists i ∈ V such that M (i, P * ) ⊂ ∪ j∈Ni R i,j .
Hence, there exists j ∈ N i such that M (j, P * ) = 0 and, therefore, π * j = 0. Moreover, in this case, we have that p * j > p * i + tl i,j . Consider, now, that F j changes his price to p j = c j + t ∆(l) + 4 t ǫ. Since p * i > c i and c j − c i ≤ ∆(c) we have that Since the Hotelling town satisfies the WB condition, ∆(c)+t ∆(l)+4 t ǫ < t l m −2 t ǫ, we have Hence, M (j,P(j, P * , P)) > 0 and π j = (c j + t ∆(l) + 4 t ǫ) S(j,P(j, P * , P)) > 0. Therefore, F j will change its price and so P * is not a Nash equilibrium price strategy. Hence, if there is a Nash price P * then P * = P L . Proof of Lemma 10.5. By contradiction, let us consider a consumer z ∈ M (i, P(i, P L , P L, * )) and z / ∈ N (i, 2). The price that consumer z pays to buy in firm F i is given by where p i = p L, * i is the coordinate of the vectorP(i, P L , P L, * ) and for the 2-path (R i1,i2 , R i2,i3 ) with i 1 = i. If the consumer z buys at firm F i3 , then the price that has to pay isẽ = p L i3 + t d (y i3 , z). Since, by hypothesis, z ∈ M (i,P(i, P L , P L, * )), we have e <ẽ. Therefore By (17), Furthermore, By the WB condition, p i − c i < 0. Hence, π L, * i < 0 which contradicts the fact that p i is the best response to P L (since π L i > 0). Therefore, z ∈ N (i, 2) and M (i,P(i, P L , P L, * )) ⊂ N (i, 2).
For every firm F i and every 0 < p < p L i , letN i (p) ⊂ N i be the set of all j ∈ N i such that |p − p j | < tl i,j . LetŜ Theorem 10.6. The local market optimum price strategy P L is a Nash equilibrium price if, and only if,Ŝ i (p) p ≤ π L i for every firm F i and every 0 < p < p L i . Theorem 10.6 implies Theorem 5.3.
Proof. For every firm F i and every 0 < p < p L i , let By Lemma 10.5 and by construction ofŜ i (p), we obtain Hence, Definition 10.7. A Hotelling town satisfies the strong bounded length and costs (SB) condition, if Theorem 10.8. If a Hotelling town satisfies the SB condition then there is a unique Hotelling town Nash equilibrium price strategy P * = P L .
Proof of Theorem 10.8. By Theorem 10.3 and Lemma 10.4, if there is a Nash equilibrium price strategy P * then P * is unique and P * = P L . We note that if M (i,P(i, P L , P L, * )) ⊂ N (i, 1) for every i ∈ V thenP(i, P L , P L, * ) = p L i and so P L is a Nash equilibrium. By Lemma 10.5, we have that M (i,P(i, P L , P L, * )) ⊂ N (i, 2) for every i ∈ V . Now, we will prove that condition (26) implies that firm F i earns more competing only in the 1-neighborhood than competing in a 2-neighborhood.
By Lemma 10.5, where p i = p L, * i is the coordinate of the vectorP(i, P L , P L, * ). Hence, By contradiction, let us consider a consumer z ∈ M (i,P(i, P L , P L, * )) and z / ∈ N (i, 1). Let i 2 ∈ N i be the vertex such that z ∈ N (i 2 , i). The price that consumer z pays to buy in firm F i is given by If the consumer y buys at firm F i2 , then the price that has to pay is e = p L i2 + t d (y i2 , z). Since, by hypothesis, z ∈ M (i,P(i, P L , P L, * )), we have e <ẽ. Therefore By (17), p L i2 ≤ t l M + 2 t ǫ + 1 2 (c M + c i2 ). Since l i,i2 ≥ l m , we have Thus, Hence, from (27) we obtain By the SB condition, By Theorem 10.3 and (28), which contradicts the fact that p i is the best response to P L . Therefore, z ∈ N (i, 1) and M (i,P(i, P L , P L, * )) ⊂ N (i, 1). Hence,P(i, P L , P L, * ) = p L i and so P L is a Nash equilibrium.
10.4. Firm position stability. For every firm F i , with k i = 2, let v(i) ∈ N i \{j(i)} be the neighboring vertex of i that is different from j(i). Let us denote Theorem 10.9. The marginal rate of the price of a firm F i with respect to the deviation of the localization of the firm from the node is given by The marginal rate of the profit of a firm F i with respect to the deviation of the localization of the firm from the node is given by and v the other adjacent node of i.
Proof of Theorem 10.9. From Theorem 10.8, we have and Hence, Hence, to study the influence of ǫ i in the profit π L i , we only have to study the signal of ∂p L i /∂ǫ i . By (29), we have

Recall that
and for v / ∈ N i , we have ∂Y i /∂ǫ i = 0. Therefore, Furthermore, if k i ≥ 4 and k v ≥ 2, for every v ∈ N i , then Definition 10.10. A Hotelling town has n-space bounded information (n-I) if for every 1 ≤ m ≤ n, for every firm F i and for every non-empty set R(i, j; m): (i) firm F i knows the cost c j and the average length road L j and the firm deviation Y j of firm F j ; (ii) for every m path R ∈ R(i, j; m), firm F i knows the corresponding weight k(R).
The n-local market optimum price vector is We observe that in a n-I Hotelling town, the firms might not be able to compute K, C, L or Y. However, every firm F i is able to compute his n local market optimum price p i (n) Theorem 10.11. A Hotelling town satisfying the WB condition has a local market optimum price strategy P L that is well approximated by the n local market optimum price P(n) with the following 2 −n bound ). Furthermore, P(n + 1) is the best response to P(n) for n sufficiently high. Proof of Theorem 10.11. By Theorem 10.3, if a Hotelling town satisfies the WB condition then there is local market optimum price strategy P L given by Considering Q = ∞ m=0 2 −(m+1) K m , we can write the equilibrium prices as For the space bounded information Hotelling town, the n local market optimum price P(n) is given by and The difference R i (n) between p L i and p i (n) is positive and is given by 23 We note that
10.6. Profit degree growth. Let F i be a firm located in a node of degree k i and F j a firm located in a node of degree k j . Letp i = p L i − c i andp j = p L j − c j represent the unit profit of firm F i and F j , respectively.
Lemma 10.12. Given the local market optimum price strategy P L , π L i > π L j if and only if k i − k j k j >p Proof of Lemma 10.12. If F j is a firm located in a node of degree k j , then π L j = (2t) −1 k j (p L j − c j ) 2 = (2t) −1 k jp 2 j . Similarly, if F i is a firm located in a node of degree k i , then Hence, 2 t π L i = k jp 2 j + k j θ(p) (2p j + θ(p)) + θ(k) (p j + θ(p)) 2 = 2 t π L j + k j θ(p) (p j +p i ) + θ(k)p 2 i , and so 2 t (π L i − π L j ) = k j (p i −p j ) (p j +p i ) + θ(k)p 2 i = k j (p 2 i −p 2 j ) + (k i − k j )p 2 i .
Theorem 10.14. A Hotelling town network satisfying the WB and DB conditions has the profit degree growth property.
Proof of Theorem 10.14. Let F i and F j be firms in the Hotelling town network such that k i > k j . We need to prove that π L i > π L j . From Lemma 10.12 we say that π L i > π L j , if and only, if k j θ(p) (p j +p i ) + θ(k)p 2 i > 0.
We note that f (θ(k), k j ) > 0 and θ(p) − = −f (θ(k), k j )p i . If θ(p) − < θ(p) < 0 then condition (33) is satisfied. By hypothesis θ(p) < 0 and, so, if then (33) is satisfied. Since θ(p) =p i −p j , from (17) then (36) is satisfied. Hence, if condition (37) is satisfied, then (32) is satisfied, π L j > π L i for every firms F i and F j such that k j > k j , and, so, the network has the profit degree growth property. 11. Conclusion. Under the weak bounded costs condition, we proved that the price subgame has a unique local market optimum price strategy. We gave an explicit closed formula and an explicit series expansion formula for the local market optimum price strategy. We showed that the influence of a firm in the local market optimum price strategy of other firm decreases exponentially fast with the distance between the firms. We showed that each firm is able to compute an approximation of its own local market optimum price strategy that improves exponentially fast with the space bounded information knowledge of the firm. Under the strong bounded condition, we proved that the local market optimum price strategy is a Nash equilibrium price strategy. We proved that the firms prefer to be located at the crossroads of the network but they do not prefer to be located at the ends of no-exit roads.