LOCALIZATION AND PRICES IN THE QUADRATIC HOTELLING MODEL WITH UNCERTAINTY

For the quadratic Hotelling model, we study the optimal localization and price strategies under incomplete information on the production costs of the firms. We compute explicitly the pure Bayesian-Nash price duopoly equilibrium and we prove that it does not depend upon the distributions of the production costs of the firms, except on their first moments. We find when the maximal differentiation is a local optimum for the localization strategy of both firms.


(Communicated by Frank Page)
Abstract. For the quadratic Hotelling model, we study the optimal localization and price strategies under incomplete information on the production costs of the firms. We compute explicitly the pure Bayesian-Nash price duopoly equilibrium and we prove that it does not depend upon the distributions of the production costs of the firms, except on their first moments. We find when the maximal differentiation is a local optimum for the localization strategy of both firms.
1. Introduction. Since the seminal work of Hotelling [14], the model of spatial competition has been seen by many researchers as an attractive framework for analyzing oligopoly markets (see [13,16,20,19,21,22]). In his model, Hotelling presents a city represented by a line segment where a uniformly distributed continuum of consumers have to buy a homogeneous good. Consumers have to support linear transportation costs when buying the good 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. [1] show that the "Principle of Minimum Differentiation" is invalid, since there was no price equilibrium solution for all possible locations of the firms, in particular when they are not far enough from each other. Moreover, in the same article, D'Aspremont et al. [1] introduce a modification in the Hotelling model, considering quadratic transportation costs instead of linear. The introduction of this feature removed the discontinuities verified in the profit and demand functions, which was a problem in Hotelling model and they show that, under quadratic transportation costs, a price equilibrium exists for all locations and a location equilibrium exists and involves maximum product differentiation, i.e. the firms opt to locate at the extremes of the line.
Hotelling and D'Aspremont et al. consider that the production costs of both firms are equal to zero. Ziss [23] introduced a modification in the model of D'Aspremont 2 ALBERTO A. PINTO AND TELMO PARREIRA et al. [1] allowing for different production costs between the two firms and examines the effect of heterogeneous production technologies on the location problem. Ziss showed that a price equilibrium exists for all locations and concluded that when the difference between the production costs is small, a price and location equilibrium exists in which the firms prefer to be located at the opposite extremes of the line. However, if the difference between the production costs is sufficiently large, a location equilibrium does not exist.
Boyer et al. [4] and Biscaia and Sarmento [2] extended the work of Ziss considering that the uncertainty on the productions costs only during the first subgame in location strategies. Then the production costs are revealed to the firms before the firms have to choose their optimal price strategies and so the second subgame has complete information.
In this paper, we study the quadratic Hotelling model with incomplete information in the production costs of both firms. The incomplete information consists in each firm to know its production cost but to be uncertain about the competitor cost as usual in oligopoly theory (see [5,6,7,8,9,10,11,12,17,18]). However, in contrast with Boyer et al. [4], the production costs are not revealed to the firms before the firms have to choose their price strategy. Furthermore, our results are universal, in the incomplete information scenario, in the sense that they apply to all probability distributions in the production costs.
We say that the Bayesian-Nash price strategy has the duopoly property if both firms have non-empty market for every pair of production costs. We introduce the bounded uncertain costs and location BU CL1 condition that defines a bound for the production costs in terms only of the exogenous variables that are the transportation cost; the road length of the segment line; and the localization of both firms (see section 6). We prove that there is a local optimum price strategy with the duopoly property if and only if the bounded uncertain costs and location BU CL1 condition holds. We compute explicitly the formula for the local optimum price strategy that is simple and leaves clear the influence of the relevance economic exogenous quantities in the price formation. In particular, we observe that the local optimum price strategy do not depend on the distributions of the production costs of the firms, except on their first moments.
We introduce two mild additional bounded uncertain costs and location BU CL2 and BU CL3 conditions. Under the BU CL1 and BU CL2 conditions, we prove that the local optimum price strategy is a Bayesian-Nash price strategy. Assuming that the firms choose the Bayesian-Nash price strategy, under the BU CL3 condition, we prove that the maximal differentiation is a local optimum for the localization strategy of both firms.
2. Local optimum price strategy under complete information. The buyers of a commodity will be supposed uniformly distributed along a line with length l, where two firms A and B located at respective distances a and b from the endpoints of the line sell the same commodity with unitary production costs c A and c B . We assume without loss of generality that a ≥ 0, b ≥ 0 and l − a − b ≥ 0. No customer has any preference for either seller except on the ground of price plus transportation cost t.
Denote A's price by p A and B's price by p B . The point of division x = x(p A , p B ) ∈ ]0, l[ between the regions served by the two entrepreneurs is determined by the condition that at this place it is a matter of indifference whether one buys from A or  Figure 1). The point x is the location of the indifferent consumer to buy from firm A or firm B, if Solving for x, we obtain Both firms have a non-empty market share if, and only if, x ∈]0, l[ . Hence, the prices will have to satisfy Assuming inequality (1), both firms A and B have a non-empty demand (x and l − x) and the profits of the two firms are defined respectively by and Two of the fundamental economic quantities in oligopoly theory are the consumer surplus CS and the welfare W . The consumer surplus is the gain of the consumers community for given price strategies of both firms. The welfare is the gain of the state that includes the gains of the consumers community and the gains of the firms for given price strategies of both firms.
Let us denote by v T the total amount that consumers are willing to pay for the commodity. The total amount v(y) that a consumer located at y pays for the commodity is given by The consumer surplus CS is the difference between the total amount that a consumer is willing to pay v T and the total amount that the consumer pays v(y) The welfare W is given by adding the profits of firms A and B with the consumer surplus

ALBERTO A. PINTO AND TELMO PARREIRA
Definition 2.1. A price strategy (p A , p B ) for both firms is a local optimum price strategy if (i) for every small deviation of the price p A the profit π A of firm A decreases, and for every small deviation of the price p B the profit π B of firm B decreases (local optimum property); and (ii) the indifferent consumer exists, i.e. 0 < x < l (duopoly property).
Let us compute the local optimum price strategy (p A , p B ). Differentiating π A with respect to p A and π B with respect to p B and equalizing to zero, we obtain the first order conditions (FOC). The FOC imply that and We note that the first order conditions refer to jointly optimizing the profit function (2) with respect to the price p A and the profit function (3) with respect to the price p B . Since the profit functions (2) and (3) are concave, the second-order conditions for this maximization problem are satisfied and so the prices (6) and (7) are indeed maxima for the functions (2) and (3), respectively. The corresponding equilibrium profits are given by and Furthermore, the indifferent consumer location corresponding to the maximizers p A and p B of the profit functions π A and π B is Finally, for the pair of prices (p A , p B ) to be a local optimum price strategy, we need assumption (1) to be satisfied with respect to these pair of prices. We observe that assumption (1) is satisfied with respect to the pair of prices (p A , p B ) if and only if the following condition with respect to the production costs is satisfied.
We note that under the BCL condition the prices are higher than the production costs p A > c A and p B > c B . Hence, there is a local optimum price strategy if and only if the BCL condition holds. Furthermore, under the BCL condition, the pair of prices (p A , p B ) is the local optimum price strategy.
A strong restriction that the BCL condition imposes is that ∆ C converges to 0 when m tends to 0, i.e. when the differentiation in the localization tends to vanish.
Throughout this paper, consider and X 2 = m t 36 (45 l 2 + 6 l ∆ l + 5 ∆ 2 l ). By equation (4), the consumer surplus CS with respect to the local optimum price strategy (p A , p B ) is given by Hence, Adding (8), (9) and (10), we obtain the welfare 3. Nash price strategy under complete information. We note that, if a Nash price equilibrium satisfies the duopoly property then it is a local optimum price strategy. However, a local optimum price strategy is only a local strategic maximum. Hence, the local optimum price strategy to be a Nash equilibrium must also be global strategic maximum. In this section, we are going to show that this is the case. Following D'Aspremont et al. [1], we note that the profits of the two firms, valued at local optimum price strategy are globally optimal if they are at least as great as the payoffs that firms would earn by undercutting the rivals' price and supplying the whole market.
Let (p A , p B ) be the local optimum price strategy. Firm A may gain the whole market, undercutting its rival by setting In this case the profit amounts to A similar argument is valid for store B. Undercutting this rival, setting The conditions for such undercutting not to be profitable are π A ≥ π M A and π B ≥ π M B . Hence, proving that is sufficient to prove that π B ≥ π M B . However, conditions (12) and (13) are satisfied because they are equivalent to is a local optimum price strategy then (p A , p B ) is a Nash price equilibrium.

4.
Optimum localization equilibrium under complete information. We are going to find when the maximal differentiation is a local optimum strategy assuming that the firms in second subgame choose the Nash price equilibrium strategy. For a complete discussion see Ziss [23].
We note that from (6) and (8), we can write the profit of firm A as

5.
Incomplete information on the production costs. The incomplete information consists in each firm to know its production cost but to be uncertain about the competitor's cost. In this section, we introduce a simple notation that is fundamental for the elegance and understanding of the results presented in this paper.
Let the triples (I A , Ω A , q A ) and (I B , Ω B , q B ) represent (finite, countable or uncountable) sets of types I A and I B with σ-algebras Ω A and Ω B and probability measures q A and q B , over I A and I B , respectively.
We define the expected values E A (f ), E B (f ) and E(f ) with respect to the probability measures q A and q B as follows: and Let c A : I A → R + 0 and c B : I B → R + 0 be measurable functions where c z A = c A (z) denotes the production cost of firm A when the type of firm A is z ∈ I A and c w B = c B (w) denotes the production cost of firm B when the type of firm B is w ∈ I B . Furthermore, we assume that the expected values of c A and c B are finite We assume that dq A (z) denotes the probability of the belief of the firm B on the production costs of the firm A to be c z A . Similarly, we assume that dq B (w) denotes the probability of the belief of the firm A on the production costs of the firm B to be c w B . The simplicity of the following cost deviation formulas is crucial to express the main results of this paper in a clear and understandable way. The cost deviations of firm A and firm B The cost deviation between the firms Since the meaning is clear, we will use through the paper the following simplified notation: The expected cost deviation ∆ E between the firms is given by Let V A and V B be the variances of the production costs c A and c B , respectively. We observe that Furthermore, 6. Local optimal price strategy under incomplete information. In this section, we introduce incomplete information in the classical Hotelling game and we find the local optimal price strategy. We introduce the bounded uncertain costs condition that allows us to find the local optimum price strategy. A price strategy (p A , p B ) is given by a pair of functions p A : I A → R + 0 and p B :

ALBERTO A. PINTO AND TELMO PARREIRA
The ex-post profit of the firms is the effective profit of the firms given a realization of the production costs for both firm. Hence, it is the main economic information for both firms. However, the incomplete information prevents the firms to have access to their ex-post profits except after the firms have already decided their price strategies. The ex-post profits π EP A : . The ex-ante profit of the firms is the expected profit of the firm that knows its production cost but are uncertain about the production cost of the competitor firm. The ex-ante profits π EA A : We note that, the expected profit . The incomplete information forces the firms to have to choose their price strategies using their knowledge of their ex-ante profits, to which they have access, instead of the ex-post profits, to which they do not have access except after the price strategies are decided. Definition 6.1. A price strategy (p A , p B ) for both firms is a local optimum price strategy if (i) for every z ∈ I A and for every small deviation of the price p z A the ex-ante profit π EA A (z) of firm A decreases, and for every w ∈ I B and for every small deviation of the price p w B the ex-ante profit π EA B (w) of firm B decreases (local optimum property); and (ii) for every z ∈ I A and w ∈ I B the indifferent consumer exists, i.e. 0 < x z,w < l (duopoly property).
We introduce the BU CL1 condition that has the crucial economical information that can be extracted from the exogenous variables. The BU CL1 condition allow us to know if there is, or not, a local optimum price strategy in the presence of uncertainty for the production costs of both firms.
for all z ∈ I A and for all w ∈ I B .
A strong restriction that the BU CL1 condition imposes is that ∆ c converges to 0 when m tends to 0, i.e. when the differentiation in the localization tends to vanish.
For i ∈ {A, B}, we define Thus, the bounded uncertain costs and location BU CL1 is implied by the following stronger SBU CL1 condition.
The following theorem is a key economical result in oligopoly theory. First, it tell us about the existence, or not, of a local optimum price strategy only by accessing a simple inequality in the exogenous variables and so available to both firms. Secondly, give us explicit and simple formulas that allow the firms to know the relevance of the exogenous variables in their price strategies and corresponding profits.
Furthermore, the local optimum price strategy (p A , p B ) is unique and it is given by We observe that the difference between the expected prices of both firms has a very useful and clear economical interpretation in terms of the localization and expected cost deviations.
Furthermore, for different production costs, the differences between the optimal prices of a firm are proportional to the differences of the production costs for all z 1 , z 2 ∈ I A and w 1 , w 2 ∈ I B . Hence, half of the production costs value is incorporated in the price. The ex-post profit of the firms is the effective profit of the firms given a realization of the production costs for both firms. Hence it is the main economic information for both firms. By equation (23), the ex-post profit of firm A is 72 t m and the ex-post profit of firm B is

ALBERTO A. PINTO AND TELMO PARREIRA
The ex-ante profit of a firm is the expected profit of the firm that knows its production cost but is uncertain about the production costs of the competitor firm. Since π EP A (z, w) is given by Similarly, the ex-ante profit of firm B is Let α A and α B be given by The following corollary gives us the information of the market size of both firms by giving the explicit localization of the indifferent consumer with respect to the local optimum price strategy. Corollary 1. Under the BU CL1 condition, the indifferent consumer x z,w is given by The pair of prices (p A , p B ) satisfies Proof of Theorem 6.4 and Corollary 1. Under incomplete information, each firm seeks to maximize its ex-ante profit. From (19), the ex-ante profit for firm A is given by From the first order condition FOC applied to the ex-ante profit of firm A we obtain Similarly, and, by the FOC, we obtain Then, from (29) and (30), Solving the system of two equations, we obtain that Hence, equalities (21) and (22) are satisfied. Replacing (22) in (29) and replacing (21) in (30) we obtain that Hence, equation (23) is satisfied. Replacing in equation (18) the values of p A and p B given by the equation (23) we obtain that the indifferent consumer x z,w is given by Hence, equation (26) is satisfied. Therefore, (p A , p B ) satisfies property (ii) if and only if the BU CL1 condition holds.
Since the ex-ante profit functions (28) and (6) are concave, the second-order conditions for this maximization problem are satisfied and so the prices p z A and p w B are indeed maxima for the functions (28) and (6), respectively. Therefore, the pair (p z A , p w B ) satisfies property (i) and so (p z A , p w B ) is a local optimum price strategy. Let us prove that p z A and p w B satisfy inequalities (27). By equation (23), By the BU CL1 condition, for every w ∈ I B , we obtain Similarly, by the BU CL1 condition, for every z ∈ I A , we obtain Hence, inequalities (27) are satisfied.
7. Bayesian-Nash equilibrium. We note that, if a Bayesian-Nash price equilibrium satisfies the duopoly property then it is a local optimum price strategy. However, a local optimum price strategy is only a local strategic maximum. Hence, the local optimum price strategy to be a Bayesian-Nash equilibrium must also be global strategic maximum. In this section, we are going to show that this is the case. Following D'Aspremont et al. [1], we note that the profits of the two firms, valued at local optimum price strategy are globally optimal if they are at least as great as the payoffs that firms would earn by undercutting the rivals's price and supplying the whole market for all admissible subsets of types I A and I B .
Definition 7.1. A price strategy (p A , p B ) for both firms is a Bayesian-Nash, if for every z ∈ I A and for every deviation of the price p z A the ex-ante profit π EA A (z) of firm A decreases, and for every w ∈ I B and for every deviation of the price p w B the ex-ante profit π EA B (w) of firm B decreases. Let (p A , p B ) be the local optimum price strategy. Given the type w 0 of firm B, firm A may gain the whole market, undercutting its rival by setting We observe that if firm A chooses the price p M A (w 0 ) then by equalities (18) and (23) the whole market belongs to Firm A for all types w of firm B with c w ≥ c w0 . Let Thus, the expected profit with respect to the price p M A (w 0 ) for firm A is Given the type z 0 of firm A, firm B may gain the whole market, undercutting its rival by setting We observe that if firm B chooses the price p M B (z 0 ) then by equalities (18) and (23) the whole market belongs to Firm B for all types z of firm A with c z ≥ c z0 . Let Thus, the expected profit with respect to the price p M B (z 0 ) of firm B is and Thus, the bounded uncertain costs condition BU CL2 is implied by the following stronger SBU CL2 condition. Proof. By equalities (24) and (25), we obtain that π EA A (z M ) ≤ π EA A (z) and π EA B (w M ) ≤ π EA B (w) for all z ∈ I A and for all w ∈ I B . Hence, putting conditions (31), (32) and (33) together, we obtain the following sufficient condition for the local optimal strategic prices (p A , p B ) to be a Bayesian Nash equilibrium: By equalities (24) and (25) we obtain that Also, from (23), we know that Hence, condition (36) holds if Finally, we note that inequality (37) is equivalent to inequality (34) and that inequality (38) is equivalent to inequality (35).

Optimum localization equilibrium under incomplete information.
We note that from (23) and (24), we can write the profit of firm A as .
. Similarly, we obtain that Therefore, the maximal differentiation (a, b) = (0, 0) is a local optimum strategy if and only if ∂π EA  9. Comparative profit analysis. From now on, we assume that the BU CL1 condition holds and that the price strategy (p A , p B ) is the local optimum price strategy determined in Theorem 6.4.
Let ∆ 1 = ∆ A + ∆ B and ∆ 2 = ∆ A − ∆ B . We observe that the difference between the ex-post profits of both firms, π EP A (z, w) − π EP B (z, w), has a very useful and clear economical interpretation in terms of the expected cost deviations and is given by Furthermore, for different production costs, the differences between the ex-post profit of firm A, π EP A (z 1 , w) − π EP A (z 2 , w), is given by ) 24 t m for all z, z 1 , z 2 ∈ I A and w, w 1 , w 2 ∈ I B .
We observe that the difference between the ex-ante profits of both firms has a very useful and clear economical interpretation in terms of the expected cost deviations.
Furthermore, for different production costs, the differences between the ex-ante profit of firm A, π EA A (z 1 ) − π EA A (z 2 ), is given by ) 24 t m and, similarly, π EA B (w 1 ) − π EA B (w 2 ) is given by ) 24 t m for all z, z 1 , z 2 ∈ I A and w, w 1 , w 2 ∈ I B .
The difference between the ex-post and the ex-ante profit for a firm is the real deviation from the realized gain of the firm and the expected gain of the firm knowing its own production cost but being uncertain about the production cost of the other firm. It is the best measure of the risk involved for the firm given the uncertainty in the production costs of the other firm. The difference between the ex-post profit and the ex-ante profit for firm A is The difference between the ex-post profit and the ex-ante profit for firm B is The following corollary tells us that the sign of the risk of a firm has the opposite sign of the deviation of the competitor firm realized production cost from its average. Hence, under incomplete information the sign of the risk of a firm is not accessible to the firm. However, the probability of the sign of the risk of a firm to be positive or negative is accessible to the firm.
The proof of the above corollary follows from a simple manipulation of the previous formulas for the ex-post and ex-ante profits.
The expected profit of the firm is the expected gain of the firm. We observe that the ex-ante and the ex-posts profits of both firms are strictly positive with respect to the local optimum price strategy. Hence, the expected profits of both firms are also strictly positive. Since the ex-ante profit π EA A (z) of firm A is equal to from (14), we obtain that the expected profit of firm A is given by Similarly, the expected profit of firm B is given by The difference between the ex-ante and the expected profit of a firm is the deviation from the expected realized gain of the firm given the realization of its own production cost and the expected gain in average for different realizations of its own production cost, but being in both cases uncertain about the production costs of the competitor firm. It is the best measure of the quality of its realized production cost in terms of the expected profit over its own production costs.
Corollary 4. The difference between the ex-ante profit and the expected profit for firm A is The difference between the ex-ante profit and the expected profit for firm B is and so equality (41) holds. The proof of equality (42) follows similarly. 10. Comparative consumer surplus and welfare analysis. Consider throughout this section that X = t m (3 l + ∆ l ). The ex-post consumer surplus is the realized gain of the consumers community for given outcomes of the production costs of both firms. Under incomplete information, by equation (4), the ex-post consumer surplus is The expected value of the consumer surplus is the expected gain of the consumers community for all possible outcomes of the production costs of both firms. The expected value of the consumer surplus E(CS EP ) is given by We note that, from equalities (14) and (17), the expected value of 144 t m is given by The difference between the ex-post consumer surplus and the expected value of the consumer surplus measures the difference between the gain of the consumers for the realized outcomes of the production costs of both firms and the expected gain of the consumers for all possible outcomes of the production costs of both firms. Hence, it measures the risk taken by the consumers for different outcomes of the production costs of both firms.
Corollary 5. The difference between the ex-post consumer surplus and the expected value of the consumer surplus, Proof.
The ex-post welfare is the realized gain of the state that includes the gains of the consumers community and the gains of the firms for a given outcomes of the production costs of both firms. By equation (5), the ex-post welfare is The expected value of the welfare is the expected gain of the state for all possible outcomes of the production costs of both firms. The expected value of the welfare E(W EP ) is given by We note that, from equalities (14) and (17), the expected value of X 3 is given by The difference between the ex-post welfare and the expected value of the welfare measures the difference in the gains of the state between the realized outcomes of the production costs of both firms and the expected gain of the state for all possible outcomes of the production costs of both firms. Hence, it measures the risk taken by the state for different outcomes of the production costs of both firms. The difference between the ex-post welfare and the expected value of welfare is 11. Example: Symmetric Hotelling. A Hotelling game is symmetric, if (I A , Ω A , q A ) = (I B , Ω B , q B ) and c = c A = c B . Hence, we observe that all the formulas of this section hold with the following simplifications The bounded uncertain costs in the symmetric case can be written in the following simple way.
for all z ∈ I A and for all w ∈ I B .
The Hotelling model with incomplete symmetric information satisfies the bounded uncertain costs (BU CL2) condition, if Under the BU C1 condition, the expected prices of the local optimum price strategy have the simple expression By Proposition 6.4, for the Hotelling game with incomplete symmetric information, the local optimum price strategy (p A , p B ) has the form The ex-post profit of firm A and firm B are, respectively The difference between the ex-post profits, π EP A (z, w) − π EP B (z, w), of both firms is given by 24 t m .
Furthermore, for different production costs, the difference between the ex-post profit of firm A, π EP A (z 1 , w) − π EP A (z 2 , w), is given by (c z2 A − c z1 A ) (4 t m (3 l + ∆ l ) + 3 (c w B + E(c A ) − c z1 A − c z2 A )) 24 t m and, for different production costs, the difference between the ex-post profit of firm B, π EP B (z, w 1 ) − π EP B (z, w 2 ), is given by (c w2 B − c w1 B ) (4 t m (3 l − ∆ l ) + 3 (c z A + E(c B ) − c w1 B − c w2 B )) 24 t m for all z, z 1 , z 2 ∈ I A and w, w 1 , w 2 ∈ I B . The ex-ante profit of firm A and of firm B are, respectively π EA A (z) =  The difference between the ex-ante profits of both firms is given by π EA A (z) − π EA B (w) = (4 t m l − ∆ A − ∆ B ) (4 t m ∆ l − 3 ∆ C ) 24 t m Furthermore, for different production costs, the differences between the ex-ante profits of a firm are given by ) 24 t m for all z, z 1 , z 2 ∈ I A and w, w 1 , w 2 ∈ I B . The difference between the ex-post profit and the ex-ante profit for firm A is The difference between the ex-post profit and the ex-ante profit for firm B is We observe that that the A − BU CL and B − BU CL conditions are implied by the BU CL1 condition. Hence, Corollary 3 can be rewritten without any restriction, i.e. The difference between the ex-ante profit and the expected profit for firm A is E(π EP A ) − π EA A (z) = ∆ A (4 t m (3 l + ∆ l ) − 3 ∆ A ) + 3 V 24 t m .
The difference between the ex-ante profit and the expected profit for firm B is The ex-post consumer surplus is The expected value of the consumer surplus is E(CS EP ) = X 1 − E(c) l + 4 t 2 m 2 (3 l + ∆ l ) 2 + 18 V 144 t m .
The difference between the ex-post consumer surplus and the expected value of the consumer surplus is The ex-post welfare is The expected value of the welfare E(W EP ) is given by The difference between the ex-post welfare and the expected value of welfare is 12. Conclusion. We proved that there is a local optimum price strategy with the duopoly property if and only if the bounded uncertain costs and location BU CL1 condition holds. The explicit formulas of the local optimum price strategy determine prices for both firms that are affine with respect to the expected costs of both firms and to its own costs. Under the BU CL1 and BU CL2 conditions, we proved that the local optimum price strategy is a Bayesian-Nash price strategy. Assuming that the firms choose the Bayesian-Nash price strategy, under the BU CL3 condition, we proved that the maximal differentiation is a local optimum for the localization strategy of both firms.