Sequential multi-store location in a duopoly

Regional Science and Urban Economics 43 (2013) 491–506

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Sequential multi-store location in a duopoly Masaya Takaki, Nobuo Matsubayashi ⁎ Department of Administration Engineering, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan

a r t i c l e

i n f o

Article history: Received 13 February 2012 Received in revised form 18 February 2013 Accepted 19 February 2013 Available online 26 February 2013 JEL classification: L13 D43 R12 Keywords: Multi-store competition Hotelling model Maximum and minimum differentiation Sequential location

a b s t r a c t This paper focuses on multi-store sequential locations between two firms within a confined geographical area over the short term. Based on the model of Teitz (1968), we incorporate a fixed cost for opening stores, as well as every possible asymmetry regarding an upper limit on the number of store openings. These two factors have an impact on firms' location strategies as constraints, which yield only two opposing types of equilibrium strategies for the leader. One is the segmentation strategy, where the leader monopolizes a market segment by partially deterring the follower's entry. The other is the equidistant location strategy, where stores are opened at equidistant locations throughout the market. Both maximum and minimum differentiation can result in equilibrium at the firm level. This seems to reflect real-world location patterns well, particularly those observed in some retail industries such as cafes and fast fashion retailers. We also obtain welfare implications of multi-store competition by analyzing the case where the social planner can optimize the upper limit on the number of store openings. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Many modern retailers sell their goods through chain stores, particularly those in retail industries such as cafes, fast fashion retailing, convenience stores, supermarkets, and fast food restaurants. As Janssen et al. (2005) noted, this is because retailers gain considerable benefits from chains, such as increased consumer recognition, increased bargaining power in purchasing, more effective advertisement, and lower distribution costs. Determination of store location in competitive environments is an important consideration for maximizing the benefits of establishing chains. We note two characteristics of multi-store competition. One is that multi-store firms generally adopt a uniform pricing policy within geographical areas where product pricing is not differentiated between stores. Dobson and Waterson (2005) theoretically explain this observation by showing that uniform prices can relax price competition among stores. On the other hand, in many industries resale price maintenance precludes price competition. The other characteristic is that all chain stores owned by the same firm are generally homogeneous, employing similar designs and product offerings. These characteristics can be observed in the chain stores of major international brands such as Starbucks, McDonald's, and Subway. A theoretical explanation for this phenomenon is examined in Loertscher and Schneider

⁎ Corresponding author. Tel.: +81 45 563 1151; fax: +81 45 566 1617. E-mail addresses: [email protected] (M. Takaki), [email protected] (N. Matsubayashi). 0166-0462/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.regsciurbeco.2013.02.004

(2011), where a chain with standardized products and homogeneous stores has superior profitability as consumer mobility increases. We therefore note that short-term strategic focus is on the number of stores to open and where to open them, given current financial constraints.1 Observing such real-world examples, focusing in particular on those in geographically confined areas, we can find characteristic patterns in competitive multi-store placement. One frequently observed pattern is concentration of investment efforts in a particular geographical area in the manner of a local monopoly, resulting in market segmentation. Closely examining first-entrant locations indicates that opening many stores too closely results in cannibalization of sales. Fig. 1(a) shows the locations of Starbucks in Chicago and Shinjuku (Tokyo) as an example. As seen in the figure, Starbucks monopolizes the region by opening many stores in a confined area, as compared with its competitors (Caribou Coffee and Tully's Coffee are the strongest Starbucks competitor in these respective areas). As a result, the firms asymmetrically segment the market. Such location patterns, where both firms open fewer stores, are also observed in relatively small cities (Fig. 1(b)). This type of segmentation is also observed for major convenience stores 1 In fact, multi-store firms face financial constraints that limit the number of store openings, as indicated by Starbucks' announcement pertaining plans for new store openings (see Starbucks' fiscal report at http://news.starbucks.com/article_display.cfm?article_id=651, 26 April 2012). Store openings may also be constrained by other factors, such as limited available real estate. However, some factors may constrain not only the number of stores but also their locations. To maintain consistency with a model having no locational constraints, we assume that the cap on the number of stores is due to financial constraints, and maintain this assumption throughout the paper.

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(a) Locations in Chicago and Shinjuku (Tokyo).

(b) Locations in Shin-Yokohama and Tsukuba.

Fig. 1. Segmentation between Starbucks and competitors.

like Seven-Eleven Japan, as well as for fast-fashion retailers such as H&M, Mango, and Zara. On the other hand, a different location pattern is seen in other areas. As shown in Fig. 2, both Starbucks and its competitor opened fewer stores, with each store in close proximity to a rival. Similarly, for H&M, Mango, and Zara, in competitive locations such as London's Oxford St. and Regent St., we can also observe a quite different location pattern from segmentation, in which several stores are in close proximity to rivals. These are head-to-head locations, which exhibit a special type of interlacing. These two location patterns are opposites in terms of differentiation at the chain or firm level. In the former case, the market segmentation is a variant of maximum differentiation, because the chains do not intersect. In contrast, the latter location pattern is clearly a minimum differentiation between chains or firms. 2 Our interest is in how

2 The areas in Fig. 2 are not necessarily small relative to those in Fig. 1, which would suggest that the location pattern with fewer stores in Fig. 2 is not due to market size. In fact, according to investigations in Japan by the East Japan Railways Company (http:// www.jreast.co.jp/e/index.html) and the Metropolitan Intercity Railway Company (http://www.mir.co.jp/en/), the number of passengers per day using Gotanda station (Fig. 2a) and Higashi-Totsuka station (Fig. 2b) are 127,966 and 57,520, respectively, while those using Shin-Yokohama station and Tsukuba station (Fig. 1b) are 56,666 and 15,638, respectively.

these opposing patterns can both result in equilibrium. In this paper, we attempt to explain these outcomes by analyzing multi-store Stackelberg competition between two firms. Hotelling (1929) provided a seminal analysis of location competition, formulating a duopoly game on a line segment onto which consumers are uniformly distributed. Teitz (1968) incorporated multiple stores into the Hotelling model. Unlike the original Hotelling model, Teitz (1968) assumes that price competition between firms is not involved, and thus they determine only locations. However, that analysis unfortunately leads to the negative result that no pure-strategy equilibrium exists in the model where firms simultaneously determine the locations of multiple stores. Nevertheless, we should note that Teitz (1968) attempts to solve this problem by introducing the Stackelberg framework and, in fact, finds an equilibrium with a kind of minimum differentiation when the leader firm opens multiple stores at equidistant locations. Indeed, this outcome partially coincides with our results. However, our other location pattern does not appear in Teitz's results, implying that such assumptions are insufficient to explain real-world multi-store locations such as those mentioned above. In this paper, we incorporate a fixed cost for opening stores into the original framework of Teitz (1968). Specifically, given a fixed opening cost and an upper limit on the number of store openings, two firms determine a number of store openings and store locations

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(a) Locations in Chapel Hill and Gotanda (Tokyo).

(b) Locations in Higashi-Totsuka.

Fig. 2. Minimum differentiation between Starbucks and competitors.

in a Stackelberg fashion. The fixed costs, as well as the reservation prices of consumers, are identical for the two firms.3 Firms may, however, have different upper limits on the number of store openings. We explore every possible scenario under this asymmetry, including those not considered in Teitz (1968). 4 These two factors, fixed cost and limited store openings, determine the firm's capability for multi-store openings, which plays an important role in its location strategy. Our analysis indicates that while multiple equilibrium locations exist due to the multiplicity of the follower's best response, the location strategy of the leader is uniquely determined and can be classified into one of two opposing strategies: a segmentation strategy, where the leader partly monopolizes the market by opening stores at short and equal distances sufficient to deter rival entry, or an equidistant location strategy, where every store is opened at equidistant locations throughout the market. The boundary separating those two strategies

3 While these assumptions of symmetry and uniformly distributed consumers are restrictive, they are appropriate for analyzing store openings within a sufficiently constrained area such as a small town, downtown area, or fashion district, as in the examples above. 4 Only symmetric cases are explored by Iida and Matsubayashi (2011) in the operations research literature. They also discuss the possibility of second-mover advantage.

is determined by the demand potential obtained by the leader firm under its equidistant location strategy, which heavily depends on the asymmetry of the constraints on the number of stores between firms. If the leader implements a segmentation strategy, market segmentation is necessarily achieved in the equilibrium. On the other hand, if the leader implements an equidistant location strategy, minimum differentiation is completely achieved in the equilibrium when the constraints are symmetric, and partially achieved for asymmetric cases. As mentioned above, therefore, both maximum and minimum differentiations at the chain level can be sustained in the equilibrium. Our central contribution to the literature is the presentation of a model illustrating this interesting outcome, which is often observed in the real world. Another contribution is the obtainment of some welfare implications of multi-store competition. We specifically consider the situation where social planners can optimize the upper limit of store openings. In other words, since only two leader strategies can achieve equilibrium, we investigate which strategy should be implemented for social efficiency. Our analysis shows that in highly competitive environments, that is, in the case of symmetric upper limits, the social planner fundamentally should set the upper limit such that an equidistant location strategy is implemented. As an exception, however, a segmentation strategy should be implemented in the case where

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the fixed cost is high and the reservation price is low. We also find that, in many cases, the upper limit on the number of store openings should be set at the minimum level that would allow each firm to solely cover the whole of the market without competition. Although our study is strongly motivated by real-world geographic competitions such as those mentioned above, we believe that it can be applied to the analysis of some other competitions. For instance, for competitive positioning of free-to-air radio or TV stations in the space of music formats, we can consider mergers and repositioning among some firms (see, e.g., Berry and Waldfogel (2001) and Sweeting (2010)). In most media markets, consumers are not directly charged and thus, firms do not engage in price competition, at least as an issue of first-order importance; instead, they simply compete in terms of only the locations of their stations in product space. Therefore, if there are two merged groups including multiple stations, our study might be able to give some insights into their sequential positioning. In addition to such competition in media industries, which is often introduced as an example of twosided markets (see, e.g., Gabszewicz et al. (2001) and Anderson and Coate (2005)), an election is also an important example of a competition where price competition is absent (see, e.g., Palfrey (1984) and Weber (1992)). If we could consider cooperation among political parties, as an extension of existing models with single locations, our model of multiple locations might be applicable to analyzing their competition. We also note that our assumption of an exogenously given upper limit of the number of stores would be suitable for such situations. We briefly review the related literature. Martinez-Giralt and Neven (1988) show a negative result from multi-store competition based on the model of d'Aspremont et al. (1979), where they reach equilibrium when each firm opens only a single store, reducing the model to single-store competition. Given this troublesome history, multiple-store competition has long received little attention. Recently, however, some authors have attempted to develop models that describe more varied situations. For example, Pal and Sarkar (2002) investigated quantity–location competition and Peng and Tabuchi (2007) analyzed two-stage location-then-variety competition. Janssen et al. (2005) incorporated consumer heterogeneity into their model to analyze competition between horizontally differentiated firms that first determine store locations and then compete by price. Karamychev and van Reeven (2009) utilized a random utility model to explore competition between multi-store firms selling horizontally differentiated products. None of these papers, however, derived the outcomes of our interest, where both maximum differentiation and minimum differentiation are sustained in the equilibrium between chains. We note that our study is also closely related to several papers in some other areas. In particular, Prescott and Visscher (1977) and Loertscher and Muehlheusser (2011) are very closely related, and thus carefully explained in a later section. On the other hand, Neven (1987) and Götz (2005) approached the issue of sequential locations by single-store firms in the presence of ex-post price competition. Also, Brenner (2005) analyzed location-then-price competition by multiple firms, where locations are simultaneously chosen in the first stage. These settings with price competition lead to different strategic interactions between firms than do those under our settings, as well as differing outcomes. Also, our model concerns the context of product line problems in the sense that multi-product positioning would be determined in Hotelling's framework. In this context, Judd (1985) and Brander and Eaton (1984) obtained interesting findings, where market segmentation might be sustained in the equilibrium, although they employ different frameworks from ours, assuming at most two consumer segments and two possible products. The rest of this paper is organized as follows: Section 2 formulates a model of multi-store competition. Section 3 analyzes the equilibrium locations. Section 4 discusses the welfare implications of multi-store competition. Section 5 summarizes our findings. Proofs of our propositions are included as appendices.

2. Model Two firms r = 1,2 have given upper limits on the number of store openings Nr(≥1). Each store sells a single product. Each firm r opens nr(0 ≤ nr ≤ Nr) stores I r = {(r, 1), (r, 2), …, (r, nr)} at locations
 xr1 ; xr2 ; …; xrnr 0≤ xrk b xrkþ1 ≤ 1; k ¼ 1; …; nr −1 in a linear market [0,1]. In addition, for notational convenience, we put two dummy stores L and R at locations xL = 0 and xR = 1. Let Ω ≡ I 1 ∪ I 2 ∪ {L, R}. We suppose that the two firms can be differentiated in terms of both the upper limit on the number of stores and their locations, but are otherwise identical. Specifically, both firms incur a fixed opening cost F(>0) to establish each store. Also, the reservation price for each store is defined as V + 1(V > 0). Consumers are uniformly distributed on a unit interval [0,1] with density 1. As mentioned above, we do not consider price competition between the two firms, and thus we normalize the product price as 1. Therefore, the surplus of a consumer located at point x ∈ [0,1] who buys a product from store i ∈ I1 ∪ I2 at location xi is given as follows 5: i

U ðxÞ ¼ V−jx−xi j: Every consumer purchases a product from one store offering the maximum surplus from among all the stores that provide positive U i. When two stores i and k fit this requirement, the consumer chooses either of the stores with equal probability. For welfare analyi sis, the surplus of a consumer for each store is evaluated as U 2ðxÞ and U k ðxÞ 2 , respectively. Furthermore, if the surplus is less than zero for any store, the consumer chooses not to make a purchase. 6 When locations of three neighboring stores h, i, j ∈ Ω are given as xh b xi b xj, we denote the midpoint of stores h and i, and that of i and xi þxj i ^ j by x^hi ¼ xh þx 2 and x ij ¼ 2 , respectively. However, as an exceptional case, when store i is the closest store to the left corner in the market, implying h = L, let x^hi ¼ 0. Likewise, when store i is closest to the right corner, implying j = R, let x^ij ¼ 1. We denote the demand quantity of store i where it is the only store at xi by d i ¼ minfV; xi −x^hi gþ   min V; x^ij −xi . However, if the rival firm opens a store k at the same point xi ; d i is equally split between stores i and k. Therefore, the demand of store i is given by 8 < di ðwhen there exists a rival store k such that xi ¼ xk Þ di ¼ 2 : d i ðotherwiseÞ:

ð1Þ

Thus, each store incurs the opening cost F and then obtains the demand di. We define the profit of store i when it is opened as follows: πi ¼ di −F: Therefore, when we denote by Dr the sum of demand of firm r's stores, the profit of firm r is given by Πi ¼ ∑ πi i∈I r

¼ ∑ di −nr F i∈Ir

¼ Dr −nr F: We suppose that the two firms sequentially determine the number of store openings and their locations. Thus, we formulate a Stackelberg 5 When we do not need to describe explicitly which firm chooses this store location, we use this simple notation. 6 We assume linear transportation costs with coefficient 1 for expositional simplicity. We are sure that our main results concerning the equilibrium do not change under a symmetric cost function, which increases with distance traveled. On the other hand, our welfare analysis does have a sensitive dependence on the form of the cost function.

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Fig. 3. An example of entry deterrence by firm 1.

game where firm 1 first determines n1 ; x11 ; …; x1n1 to maximize its profit Π1 as the leader, and then firm 2 determines n2 ; x21 ; …; x2n2 to maximize its profit Π2 as the follower. For expositional convenience, we assume a tie-breaking rule by which firm 2 opens only stores that can earn a strictly positive profit. Therefore, we must have πi > 0 for all i ∈ I2. We seek a subgame perfect Nash equilibrium (hereafter called simply an equilibrium) of this game. We next define social welfare as pertaining to our model. When the number of stores and their locations are determined for both firms, producer surplus is defined by the sum of both firms' profits as follows: Π1 þ Π2 ¼ ∑ πi þ∑ πj i∈I1

j∈I2

¼ D1 þ D2 −ðn1 þ n2 ÞF: Consumer surplus is defined in our model as the sum of the surplus obtained from purchasing over all consumers in [0,1]. Specifically, when we denote by d(i) the set of points where a consumer purchases from store at i ∈ I1 ∪ I2, consumer surplus is formulated as follows:

of equilibrium (except for the mirror case). 7 In addition, we assume F≤ 2N1 1 , which excludes the trivial case where the leader firm can completely deter the rival's entry. Without loss of generality, we assume that x11 ≤ 12. We now define two key strategies which exactly constitute the equilibrium. First, we define the most efficient segmentation, that is, partial entry deterrence strategy for the leader. Specifically, we call the strategy in which the leader firm opens its n1 stores at xk1 = (2k − 1)F, k = 1, 2, …, n1 the segmentation strategy; this strategy is denoted by SSeg n1 . We explain in the next subsection why this strategy can achieve the most efficient segmentation. On the other hand, we call the strategy in which the leader firm opens its stores at equal intervals, formally x1k ¼ 2n11 ð2k−1Þ; k ¼ 1; 2; …; n1 , the equidistant location strategy, and denote it by SEqd n1 . Under this strategy, firm 1 opens its stores at the center of each subdivided market, where the whole market is evenly divided into n1 parts.   Now, for any firm 1's location strategy with n1 stores Sn1 , let D1 Sn1    and Π1 Sn1 be the demand and profit obtained by firm 1 respectively, provided that firm 2 takes the best response to Sn1 .

i

U ¼ ∑ ∫x∈dðiÞ U ðxÞdx

3.2. Best response of the follower

i∈I1 ∪I2

¼ ∑ ∫x∈dðiÞ ðV−jx−xi jÞdx: i∈I 1 ∪I 2

Thus, if two firms r = 1,2 open nr(0 ≤ nr ≤ Nr) stores I r = {(r,1), …,(r,nr)} and store i ∈ I1 ∪ I 2 is located at xi(0 ≤ xi ≤ 1), then the social welfare is given by SW ¼ Π1 þ Π2 þ U ¼ D1 þ D2 −ðn1 þ n2 ÞF þ ∑ ∫x∈dðiÞ ðV−jx−xi jÞdx i∈I 1 ∪I 2

¼ D1 þ D2 −ðn1 þ n2 ÞF þ V ðD1 þ D2 Þ− ∑ ∫x∈dðiÞ jx−xi jdx

ð2Þ

i∈I 1 ∪I 2

¼ ð1 þ V ÞðD1 þ D2 Þ−ðn1 þ n2 ÞF− ∑ ∫x∈dðiÞ jx−xi jdx: i∈I∪I 2

3. Equilibrium analysis In this section, we analyze the equilibrium outcomes in our game. To begin with, the following subsection presents some preliminaries for the analysis of this section. 3.1. Preliminaries We hereafter assume V≥ 2 minf1N1 ;N2 g, which implies that the reservation price is sufficiently high such that every firm could solely cover the whole of the market without competition. This assumption excludes cases of multiple equilibria and thus ensures the uniqueness

In advance of deriving our equilibrium results, we consider the best response of the follower to the leader's chosen locations. Unfortunately, for a given leader location, the best response may not be unique. Nevertheless, we can show that the leader's resulting profit is uniquely determined for any leader location. To see this, we first consider the case of (partial) entry deterrence. If firm 1 chooses locations for its n1 stores such that x11 ≤ F, x1n1 b1−F and xk1 − xk1 − 1 ≤ 2F(k = 2, 3, …, n1), firm 2 cannot profitably open h i any store in 0; x1n1 .8 In other words, firm 1 succeeds in partially deterring firm 2's entry. This is illustrated in Fig. 3: if the follower's stores (2,1) and (2,2) are established in the market, using Eq. (1) of Section 2 we can calculate the demand quantity of each store. However, since it can be easily seen that dð2;1Þ ¼ F− 2 and d(2,2) = F hold, both π(2,1) and π(2,2) are non-positive. Also, if the follower establishes two stores between any two of the leader's stores, the profit earned by these stores cannot be positive. We thus have that firm 2 does not enter the market. Clearly, among such entry deterring strategies, the segmentation strategy is the most efficient one. If the leader uses the segmentation strategy, the follower's best response is to open one store at a distance V from the

7 This includes somewhat troublesome cases other than the trivial ones where every store is a spatial monopoly. For example, when N1 = N2 = 2, V = 0.18 and F = 0, we obtain an equilibrium with full market coverage where firms open at x11 = 0.18, x21 = 0.54, x12 = 0.36, x22 = 0.82, while x11 = x12 = 0.18, x21 = 0.54, x22 = 0.82 is also equilibrium. Both indeed have different consumer surplus values, which affects welfare implications. 8 This is consistent with the assumption of x11 ≤ 12, because the assumption of 2N1F ≤ 1 requires F≤ 12.

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Fig. 4. The leader's segmentation strategy and the best response by the follower.

Fig. 5. An example of a partial segmentation strategy and the best response by the follower.

opposite end of the market, and other stores at intervals of 2V. With this rule, we note that depending on the distance between firm 1's N1-th store and firm 2's second store, the location of firm 2's first store and thus the resulting demand of firm 1 can follow either of the two patterns shown in Fig. 4. That is, if the best response is to open firm 2's first store in close proximity to firm 1's N1-th store, as in Fig. 4(a), then we have

 D1 SSeg ¼ ð2N1 −1ÞF þ . Otherwise, as in Fig. 4(b), D1 SSeg is given N1 N1 as the midpoint between x1N1 and x12. We next consider the case where the leader partially adopts a segmentation strategy. Specifically, firm 1 opens two consecutive stores at xk1 and xk1 + 1 such that xk1 + 1 − xk1 > 2F, and two other consecutive stores at xl1 and xl1+ 1 such that xl1+ 1 − xl1 = 2F. Then, if firm 2 can open sufficiently many stores to capture the interval [xk1,xk1 + 1], firm 2 can profitably do so, resulting in the demand for firm 1 in [xk1,xk1 + 1] being at most F. This is because if the demand for firm 1 is F + , where  is some positive number, then appropriate relocation and addition of firm 2's stores can necessarily increase profit (Fig. 5). As seen in Fig. 5, although the best response may not be unique, the resulting profit of the leader is uniquely determined

(in Fig. 5, it can be easily seen that all best-response locations for the follower can achieve the demand d2 þ F þ 2 ). We finally consider the best response to the leader's strategy of completely allowing the rival's entry (i.e., firm 1 opens two consecutive stores at xk1 and xk1 + 1 such that xk1 + 1 − xk1 > 2F). If the leader uses the equidistant location strategy with n1 stores, then the best response is that when N2 ≤ n1, the follower chooses to locate its N2 stores at the same locations as the leader's, and when N2 > n1, the follower sandwiches the leader's N2 − n1 stores by opening two stores in sufficiently close proximity to each store and locates the other 2n1 − N2(=N2 − 2(N2 − n1)) stores at the same points as the leader's, see Fig. 6.9 On the other hand, if the leader uses any other strategy which completely allows the rival's entry, then clearly the follower can obtain more profit by opening its stores at larger intervals. In all cases, the best

9 For example, suppose that the leader's equidistant locations are x11 = x0, x21 = 3x0, x31 = 5x0. Then, it can be directly verified that the follower's demand achieved by locating at x12 = x11 − ( > 0), x22 = x11 +  and x32 = x21 −  is3x0 − 32 , which is strictly less than the demand 3x0 − 34  achieved by locating at x 21 ¼ x11 −; x22 ¼ x11 þ  and x32 = x21.

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The stronger leader case (N1 > N2)

The symmetric case (N1 = N2)

The stronger follower case (N1 < N2) Fig. 6. The leader's equidistant location strategy and the best response by the follower.

response is not necessarily unique, but the leader's resulting profit is uniquely determined. 3.3. Equilibrium locations We are now ready to state our main results. Indeed, the equilibri
 um can be derived as follows, classified by the value of D1 SSeg N1 , that is, the demand obtained by firm 1 under its segmentation strategy while opening as many stores as possible. Proposition 1. The case of stronger leader (N1 ≥ N2) 1. When V≥ 2N1 2 , there exists an equilibrium, in which the market is fully covered. In this equilibrium, firm 1 necessarily uses either the segmentation strategy with N1 stores, SSeg N 1 , or the equidistant location ⁎ strategy with n1⁎ stores, SEqd n1 , where n1 is the integer satisfying qffiffiffiffiffiffiffiffiffiffiffiffi

qffiffiffiffiffiffiffiffiffiffiffiffi

max N2 ; − 12 þ 14 þ N2F2 ≤n1 ≤ min N 1 ; 12 þ 14 þ N2F2 .
 Seg  N2 ≥ 1 − 2n 2. If firm 1 uses SSeg  þ N1 in the equilibrium, then D1 SN1 1   Eqd  N1 −n1 F holds, and if firm 1 uses Sn in the equilibrium, then this 1
   N2 ≤ 1 − 2n implies that D1 SSeg N 1 −n1 F holds. The Appendix  þ N1 1
 gives explicit expressions for D1 SSeg and the parameter region in N1 (N1,N2) space for each strategy to be sustained in the equilibrium. Proposition 2. The case of stronger follower (N1 b N2) 1. When V≥ 2N1 2 , there exists an equilibrium, in which the market is fully covered. In this equilibrium, firm 1 necessarily uses either the

segmentation strategy with N1 stores, SSeg N1 , or the equidistant location strategy with N1 stores, SEqd N1 .


Seg  N2 > 1− 2N holds, 2. If firm 1 uses SSeg N 1 in the equilibrium, then D1 SN 1 1

and if firm 1 uses SEqd N 1 in the equilibrium, then that condition fails.
 and the parameter region in (N1,N2) Explicit expressions for D1 SSeg N1

space for each strategy to be sustained in the equilibrium are given in the Appendix.

The first parts of Propositions 1 and 2 show that the leader's equilibrium strategies are completely classified into only two opposing strategies, regardless of the relation between N1 and N2. As seen in the previous subsection, if the segmentation strategy is adopted by the leader, the best response of the follower is to open its stores on the opposite side of the market. As a result, the market segmentation between the two firms is achieved in the equilibrium. On the other hand, the best response to the leader's equidistant location strategy with n1 stores produces an equilibrium where minimum differentiation is completely attained in the symmetric case (n1 = N2), and is partially attained in some of the subdivided markets in asymmetric cases (n1 ≠ N2). Figs. 4 and 6 illustrate equilibrium outcomes with leader segmentation and equidistant location strategies, respectively. Note that, in the case of a stronger leader N1 ≥ N2, the leader does not necessarily open as many stores as possible when using its equidistant location strategy. Instead, firm 1 chooses the number that achieves a trade-off between total demand and total fixed costs. In contrast, in the case of a stronger follower N1 b N2, the leader necessarily opens as many stores as possible. However, the propositions ensure that in any case, the market is fully covered in the equilibrium under the assumption of V≥ 2 minf1N1 ;N2 g.

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a)

b) Fig. 7. The equilibrium leader strategy in (N1,N2) space.

The second parts of the propositions show that the boundary equations separating the equilibrium leader strategies are given as a relation between the demands possibly obtained by firm 1 under its segmentation and equidistant strategies. Therefore, taking the demand possibly obtained under the equidistant strategy as the threshold, given in the right-hand side of the equation, firm 1 adopts the segmentation strategy if it can obtain higher demand with this strategy, and adopts the equidistant strategy otherwise. Note that this threshold is decreasing in N2. More specifically, it is 12 in the symmetric case (N1 = N2), is greater than 12 in the case of a strictly stronger leader (N1 > N2), and vice versa. This implies that for a given N1, as N2 increases the leader is better off adopting the segmentation strategy.
 The explicit expression for D1 SSeg is complicated because there are N1 two possible best response patterns for firm 2, as illustrated in Fig. 4. Also, it is rather complicated to describe the boundary equation only in terms of parameters. Explicit expressions are given in the Appendix, but we can roughly illustrate them in (N1,N2) space as shown in Fig. 7, where we assume that the best response of firm 2 is given as the pat
 tern in Fig. 4(a), that is, D1 SSeg ˜ ð2N1 −1ÞF, and n1⁎ attains the value N1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi

min N1 ; 12 þ 12 þ N2F2 . An intuitive explanation of why only two strategies are sustained in the equilibrium is as follows. First, note that the leader firm benefits most by either deterring the follower's entry to as large a part of the market as possible, or by locating its stores as efficiently as possible while accommodating entry. As explained in the previous subsection, if firm 1 opens two consecutive stores at xk1 and xk1 + 1 such that xk1 + 1 − xk1 > 2F, and two other consecutive stores at xl1 and xl1+ 1 such that xl1+ 1 − xl1 = 2F, then the demand for firm 1 in [xk1, xk1 + 1] is at most F. In contrast, if firm 1 chooses xk1 and xk1 + 1 such that xk1 + 1 − xk1 = 2F, it can deter the rival's entry, resulting in obtaining its demand of 2F. By this logic, locating N1 stores each at a distance of 2F (i.e., SSeg N 1 ) is superior to any partial segmentation strategy. On the other hand, as explained in the previous subsection, the equidistant location strategy SEqd n is clearly the most efficient of the accommodation 1

strategies (completely allowing the rival's entry). Thus, we have that Eqd only SSeg N1 or Sn can be the optimal strategy for the leader.

which are negatively correlated, provided that it is equal to its bound
 ⁎ ary value. In fact, as D1 SSeg N1 ≒ ð2N 1 −1ÞF, let N1 be the solution of qffiffiffiffiffiffiffiffiffiffiffiffi

  N2 N2 1 1   ð2N1 −1ÞF ¼ 1− 2n  þ N 1 −n1 F, where n1 ¼ min N 1 ; 2 þ 4 þ 2F , 1

∂N 

satisfying N1 ≥ N2. Then, we can easily show that ∂F1 b 0 always holds. That is, the leader is better off adopting the segmentation strategy as the fixed cost increases.10 The real-world example of cafe chains, where market segmentation is frequently observed in metropolitan areas (Fig. 1), reflects this implication well. We should note from Fig. 7(b), however, that when both F and N2 are too low, the leader is always better off taking the equidistant location strategy, even if it can open sufficient stores to flood the market. To see this, consider the monopoly case where N2 = 0. Then the equidistant location is clearly optimal. The equidistant location is an accommodation strategy once N2 is positive. However, under very small F and N2, it remains relatively efficient compared to the segmentation strategy, as explained in detail later. In contrast, in the case of a stronger follower (N2 > N1), the leader should adopt the segmentation strategy as N1 is lower relative to N2. As explained in Subsection 3.2 (Fig. 6), in such a case the leader's equidistant location strategy results in most stores being unfortunately sandwiched by follower stores. Therefore, deterring the rival's entry with the segmentation strategy is optimal, even if it can be adopted only for a very small part of the market. The intuition behind these results is as follows. As explained above, the leader's optimal strategy is determined by the relation between the Eqd entry deterrence effect of SSeg N1 and the efficiency of Sn1 . As in Fig. 7(a), provided that the fixed cost is not so low that the segmentation strategy faces severe cannibalization, it follows that if the leader can afford to invest significant money in establishing its stores, it is better off adopting a segmentation strategy to deter entry to a larger part of the market. However, if the leader faces severe financial constraints, allowing monopolization of only a small part of the market, then greater benefit arises from allowing the rival's entry and instead seeking to efficiently cover a larger part of the market. Therefore, if the leader cannot capture relatively more demand through a segmentation strategy than through equidistant location strategy, the latter is the better choice. However, under too low a fixed cost, segmentation will suffer from inefficiency

1

3.4. The effect of caps on the number of stores on the equilibrium locations We now interpret our equilibrium results in terms of caps on the number of stores. It can be seen from Fig. 7(a) that, for the case of a stronger leader, the leader is better off adopting the segmentation strategy when N1 is higher, unless F is too low. In addition, it is
 is an increasing function of parameters N1 and F, clear that D1 SSeg N1

10 Provided that the fixed cost is symmetric between two firms, the product N1 F can be interpreted as the financial constraint for firm 1 to open its stores in our linear market. Therefore, we can interpret the result as being that firm 1 should adopt the segmentation strategy, if it can afford to invest significant amounts of money into establishing its stores. However, we should note that the superiority of the segmentation strategy is essentially determined by the fixed cost incurred by the rival. In fact, if firms incur asymmetric fixed costs, denoted by F1 and F2 respectively, the superiority of firm 1's segmentation strategy is determined by the value of N1 F2, which is not the financial constraint of firm 1. This is because firm 1 can deter the entry of firm 2 by opening its stores at the distance of at most F2.

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due to severe cannibalization, even if it can open sufficient stores to flood the market. The equidistant location becomes more efficient as the follower becomes weaker, as in Fig. 7(b). Furthermore, the result for the stronger follower case is also straightforward, as explained above.

3.5. Comparison with previous literature We end this section with a comparison of our results with those in the previous literature. First, as mentioned in Section 1, the outcome of market segmentation can be interpreted as a variation of maximum differentiation, because the store chains do not intersect. This possibility of maximum differentiation is in sharp contrast to that in most of the existing literature, where it can appear in the presence of price competition (e.g., d'Aspremont et al., 1979; Martinez-Giralt and Neven, 1988). Next, in our model only two types of equilibrium can be formed in terms of the leader's strategy. In this sense, our results are different from those of the previous literature, including Judd (1985), Peng and Tabuchi (2007), and Tabuchi (2012), where various types of interlacing can be in equilibrium. Moreover, we point out that leader segmentation where each store is established at a distance of 2F is similar to the result in Prescott and Visscher (1977), where multiple firms open single stores. However, theirs is slightly different from ours, in that in their equilibrium the second entrant opens its store at a distance of F from the end opposite the store of the first entrant. In addition, more importantly, no entry accommodation, including minimum differentiation, appears in their equilibrium. This is robust in comparison with the setting of Loertscher and Muehlheusser (2011), who extended the model of Prescott and Visscher (1977) to the situation where consumers are non-uniformly distributed on the line segment. Interestingly, Loertscher and Muehlheusser (2011) also explore sequential locations of multiple stores.11 However, in their model where each firm can open as many stores as it wants, in the equilibrium the first entrant always opens sufficiently many stores to deter subsequent entry, which indicates that our results strongly depend on the cap on the number of stores assumed in our model. Finally, in the context of competition between two merged groups of free-to-air radio stations, which was introduced in Section 1, our result suggests that the equilibrium format can be drastically altered by changes in the number of members included in each group.

the situation, and find the most socially efficient upper limit of store openings. We restrict the following analysis to the symmetric case where both firms can open the same number of stores, that is, N ≡ N1(=N2). As Teitz (1968) noted, in the absence of price competition a monopolistic situation is clearly the most efficient. That is, if the social planner could set both N1 and N2 independently, one should be set to an appropriate value, and the other should be set to zero. Therefore, we focus on the second-best case, where both firms are forced into competition. 1 As in the previous section, the following discussion assumes 2V ≤ 1 N≤ 2F , which implies that F ≤ V is also assumed. As shown in Propositions 1 and 2, we ensure that under this assumption, the line market is fully covered in the equilibrium. Therefore, we note that our welfare analysis is not affected by our restrictive assumption that the price is exogenously given as 1. We hereafter assume that N is a continuous variable for analytical tractability. 12 Furthermore, in this  sec tion, we restrict our analysis to the case of V b 1 and Fb 12. If 1≤V F≥ 12 , then the follower (leader) firm necessarily opens exactly one store regardless of the value of V (F), provided that the leader firm takes the segmentation strategy. However, under our assumption of a noninteger value, the number of follower (leader) stores is monotonically smaller as V (F) increases. This increases the approximation error so that it can change the qualitative implications of our results. With these assumptions, we calculate the social welfare under the two possible equilibrium strategies for the leader. Let SWSeg(N) and SWEqd(N) be the social welfare under the segmentation strategy and that under the equidistant location strategy, respectively. The results are stated in the following two lemmas. Lemma 1. When firm 1 takes the segmentation strategy in the equilibrium, ÞFþV the number of firm 2's stores is given as n2 ¼ 1−ð2N−1 . Then, the social 2V welfare is given by

SW Seg ðNÞ ¼ 1 þ

 V F 1 F −F− − N− ð1−V ÞðV−F Þ; 2 2V 2 V

which is a linearly decreasing function of the upper limit N. Lemma 2. When firm 1 takes the equidistant location strategy in the equilibrium, the social welfare is given by SW Eqd ðNÞ ¼ 1−2NF þ V−

4. Welfare implications In this section, based on the analysis of the equilibrium, we explore the social efficiency of multi-store competition to examine welfare implications. Specifically, we investigate optimization concerning the upper limit of store openings. As seen in the previous section, for a given upper limit, the leader firm takes either a segmentation or equidistant location strategy in the equilibrium. Our question now is which strategy should be implemented in terms of social efficiency? As it turns out, each strategy has advantages and disadvantages. Namely, the segmentation strategy increases consumer surplus because the leader firm opens multiple stores at short and even distances. This results in cannibalization of sales between stores, however, decreasing firm surplus. In contrast, while cannibalization does not occur in the equidistant location strategy, it leads to a configuration where two different but homogeneous stores are established at the same point, which is socially inefficient. We therefore perform a formal analysis of 11 In the operations research literature, Dasci and Laporte (2005) also studied a multi-store Stackelberg game on Hotelling's line where the number of stores and their locations are unconstrained, provided firms incur fixed costs for opening each store, allowing asymmetry. When consumers are uniformly distributed on the line, the unique equilibrium outcome is also that the leader firm completely deters rival entry by opening too many stores.

499

1 : 4N

Moreover, the optimal number of stores that maximizes SWEqd(N) is ^ ¼ p1ffiffiffiffi . given as N 2 2F Lemmas 1 and 2 show that for each of two possible equilibrium strategies, the value of social welfare is derived as a function of N. However, we note that the equilibrium strategy is uniquely given as one of these strategies, as shown in Proposition 1. Therefore, we have the following lemma, which precisely describes the social welfare in the equilibrium. Lemma 3. The social welfare in the equilibrium is specifically given as

SW Eq ðN Þ ¼

8 > < SW Eqd ðNÞ > : SW

Seg ðN Þ

1 1 1 ≤ N≤ þ 2V 4F 2 1 1 1 þ ≤ N≤ : if 4F 2 2F if

 1 1  1 1 In addition, we have SW Eqd 4F þ 2 b SW Seg 4F þ2 . From Proposition 1 (Fig. 7), it follows that for the symmetric case, equilibrium is attained with the leader's equidistant location strategy

12 This analytical technique refers to that used in Karamychev and van Reeven (2009), as well as much of the literature on free entry, e.g., Mankiw and Whinston (1986).

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at lower values of N, and with the leader's segmentation strategy at higher values, regardless of the value of F. Therefore, the form of SWEq(N) shown in Lemma 3 is straightforward. However, Lemma 3 also shows that at the boundary separating the two strategies, social welfare is strictly higher under the segmentation strategy than under the equidistant location strategy, although leader firm profits are identical. Utilizing the result of Lemma 3, we can easily find the socially optimal upper limit of stores, as well as the equilibrium strategy. In fact, 1 1 it suffices to find the N* that maximizes SWEq(N) under 2V ≤N≤ 2F . These results are summarized by the following proposition. Proposition 3. Suppose that the social planner sets N* to maximize ð1−V Þ 1 1 SWEq(N) under 2V ≤N≤ 2F . If F > V3−5V and Vb 35, then N* is chosen such that equilibrium is attained when the leader takes the segmentation strategy. Otherwise, it is chosen such that equilibrium is attained when the leader takes the equidistant location strategy. Proposition 3 shows that the social planner should determine which equilibrium strategy the leader should implement according to the relation between the value of the reservation price and the fixed cost. We illustrate the result of the proposition in Fig. 8. As seen in the figure, the segmentation strategy should be implemented only if the fixed cost is high and the reservation is low. We note that in the symmetric case, the leader's equidistant location strategy uniquely results in minimum differentiation in the equilibrium. Therefore, the proposition shows that the minimum differentiation is fundamentally superior to the segmentation from a normative viewpoint. Roughly speaking, when the fixed cost F is high and the reservation price V is low under F b V, the overall location pattern under segmentation approaches that of the monopoly's optimal location. In this case, the distance between each leader store is relatively long, so cannibalization diminishes. In contrast, the follower opens its stores at relatively short distances, which increases its consumers' surplus. Therefore, these efficiencies dominate those of minimum differentiation. In other cases, however, the relation is reversed and thus the minimum differentiation is more efficient, despite duplicate locations being allowed. We next obtain the following Proposition 4, which describes the socially optimal upper limit of store openings. Proposition 4. Suppose that the social planner sets N* to maximize 1 1 1 V 3 SWEq(N) under 2V ≤ N≤ 2F . Then, N* is 2V for 2ð1−V Þ ≤ F; V b 5 and V2 2

ð1−V Þ 1 ≤ F≤ V3−5V , or 35 ≤V and V2 ≤ F. Otherwise, it is strictly higher than 2V . 2

The results of Proposition 4 are illustrated in Fig. 9. We note that the four parts in the figure are obtained by further dividing each part in Fig. 8 into two parts. Overall, in about half of the areas, the

Fig. 9. Optimal upper limit of store openings.

social planner should set the upper limit of store openings as low as possible. However, the intuition behind the results is somewhat different between each of the four parts. In what follows, we discuss this in detail. V Part (i): 2ð1−V Þ ≤ F: From Proposition 3, the leader should implement the segmentation strategy. To achieve this, it follows from Propositions 1 and 2 that the social planner needs to set a higher product value NF. In this parameter region the value of F is sufficiently high, however, so the planner should set the minimum level of N. ð1−V Þ V b F b 2ð1−V Part (ii): Vb 35 and V3−5V Þ: From Proposition 3, the leader should implement the segmentation strategy. However, because F is relatively low, the social planner 1 þ 12, which is the boundary value between should set a higher N ¼ 4F segmentation and minimum differentiation given in Lemma 3. 2 2 ð1−V Þ ; or 35 ≤V and V2 ≤ F: Part (iii): Vb 35 and V2 ≤ F ≤ V3−5V From Proposition 3, the leader should implement the equidistant location strategy. Therefore, by Propositions 1 and 2, the social planner needs to set a lower value of NF. Nevertheless, F is relatively high in this range, so the planner should set the minimum level of N. 2 Part (iv): Fb V2 : The leader should implement the equidistant location strategy. F is lower, however, so the social planner should set N at the optimal level under minimum differentiation given in Lemma 2, which is slightly higher than the minimum level.

5. Conclusion

Fig. 8. Parameter ranges where segmentation and equidistant location should be implemented.

We analyzed multi-store competition between two firms in a Stackelberg fashion. Based on the model of Teitz (1968), we incorporated a fixed cost for opening stores, as well as every possible asymmetry concerning the upper limit on the number of store openings. As a result of our analysis, we found that only two opposite leader strategies can be sustained in the equilibrium. The leader should adopt the segmentation strategy, where it partly deters the rival's entry by opening stores at short distances, or should give up deterring the rival's entry and instead take an equidistant location strategy where every store is located over the market at equal distances. In the former case, market segmentation between the two firms is achieved in the equilibrium. In contrast to the minimum differentiation obtained in the latter case, this outcome can be interpreted as a variation of maximum differentiation in the sense that both store chains do not intersect. This possibility of maximum differentiation does not appear in Teitz (1968) or in other existing literature on multi-store competition.

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We then explored the situation where a social planner can optimize the upper limit of store openings. As a result, we showed that in the most competitive environment (the case of symmetric upper limits), the social planner normally should set the upper limit such that the equidistant location strategy is implemented. The exceptional case is where the fixed cost is high and the reservation price is low, where the overall location pattern in the equilibrium approaches that of optimal monopoly location. We furthermore found that, in many cases, the social planner should set the upper limit of store openings to the minimum level at which either (symmetric) firm could cover the entire market in the absence of competition. Acknowledgments We thank the Editor, Yves Zenou, and two anonymous referees for helpful comments. The second author is supported by the Grants-in-Aid for Scientific Research (C) 24510201 of the Ministry of Education, Culture, Sports, Science and Technology of Japan. Appendix This appendix is divided into three parts. Appendix A gives explicit expressions for Propositions 1 and 2. Appendix B gives proofs for the results on the equilibria. Appendix C gives proofs for the results on social efficiency. Appendix A. Explicit expressions for Propositions 1 and 2
 We first give an explicit expression of D1 SSeg N 1 , described in Propositions 1 and 2. Then, we illustrate the results of Propositions 1 and 2 by showing the parameter region in (N1,N2) space in which the strategies described in the propositions are sustained in the equilibrium.
 Seg Explicit
expression of D1 SN1  is assigned one of two values, depending on the best D1 SSeg N1 response pattern for firm 2. If the best response is to open firm 2's first store in close proximity to firm 1's N1-th store, as in Fig. 4(a),
 ¼ ð2N1 −1ÞF þ . Otherwise, as in Fig. 4(b), then we have D1 SSeg N1
 Seg D1 SN1 is given as the midpoint between x1N1 and x12. Because firm 2's best response is to open its stores with the interval of 2V, we must have x12 = 1 − (2γ − 1)V, where γ ¼
 ¼ ð2N1 −1ÞFþð21−ð2γ−1ÞV Þ. fore, we have D1 SSeg N1

t

ð1þV Þ−ð2N 1 −1ÞF 2V

b

. There-

extreme solution. For other cases, we can obtain the results in a qffiffiffiffiffiffiffiffiffiffiffiffi similar manner. Since N1 ≤ 12 þ 14 þ N2F2 is equivalent to N2 ≥ 2FN12 − 2FN1, this implies that n1⁎ = N1 if and only if N2 ≥ 2FN12 − 2FN1. Also
 ¼ ð2N1 −1ÞF apfor expositional purposes, we assume that D1 SSeg N1 proximately holds, which implies that the equilibrium location is as

  shown in Fig. 4(a). Therefore, by solving D1 SSeg ¼ ð2N1 −1ÞF ¼ N1   2 N2  þ N −n , we can obtain N = −4FN 1− 2n F for N  2 2 1 + 2(1 + F)N1 1 1 1

when N2 ≥ 2FN12 − 2FN1, and N2 ¼ 2F N21 − ð1 þ F ÞN1 þ N2 ≤ 2FN12 − 2FN1.

Let

2ð1 þ F ÞN1 ; and hðN1 Þ≡

x1N1 ≤1, we must consider only the region of 1 ≤ N1 ≤ 1þF 2F . Also, if N2 ≥ 2N1, then the follower can completely sandwich the leader's stores by opening N2 stores. Therefore, we must consider only N2 b 2N1. In this   region, we first have that f(N1) is increasing, f(1) = 0, and f 1þF 2F ≥ pffiffiffi ; g ð 1 Þ ¼ 2−2F≥0 1⇔F≤ 2−1. Next, g(N1) is maximized at N1 ¼ 1þF 4F   (because 2N1F ≤ 1 requires F≤ 12), and g 1þF 2F ¼ 0. Finally, h(N1) is depffiffi   2 3−3 creasing, and h 1þF 2F ≥1⇔F≤ 3 . Together with these facts, we can restate Propositions 1 and 2 in terms of N1 and N2 as follows: pffiffi (1) When F≥ 2 33−3, there exists N 1 such that for N 1 ≤N1 , regardless of N2, firm 1 takes the segmentation strategy in the equilibrium. If N 1 ≤N 1 , then there exists N 2 ðN1 Þ such that in the equilibrium with N2 ≥N 2 ðN1 Þ, firm 1 takes the segmentation strategy, while for N2 ≤ N 2 ðN1 Þ firm 1 takes the equidistant location strategy. pffiffi ˜ 2 ðN1 Þ such that for N2 ≥ N ˜ 2 ðN1 Þ (2) When F≤ 2 33−3, there exists N in the equilibrium, firm 1 takes the segmentation strategy, ˜ 2 ðN1 Þ firm 1 takes the equidistant location while for N2 ≤ N strategy. Appendix B. Proofs for results about the equilibria Before proving Proposition 1, we give the following lemma. Lemma A1. Suppose that firm 1's upper limit on the number of store openings is higher than firm 2's (N1 ≥ N2). If firm 1 takes the equidistant location strategy with n1(N2 ≤ n1 ≤ N1) stores, then the number of stores that maximizes firm 1's profit, n1⁎, is the integer satisfying ( ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 1 1 N2 1 1 N2  þ þ max N2 ; − þ ≤ n1 ≤ min N1 ; þ : 2 4 2F 2 4 2F

Proof of Lemma A1. When firm 1 takes the equidistant location strategy 1 1 SEqd n1 ðN 2 ≤n1 ≤N 1 Þ and opens its stores at xk ¼ 2n1 ð2k−1Þðk ¼ 1; …; n1 Þ, firm 2 appropriately chooses to locate its N2 stores at the same locations as firm 1's. As a result, because V≥ 2N1 2 > 2n11 , firm 1's demand and profit are given as, D1 ðn1 Þ ¼ 1−

N2 2n1

Π1 ðn1 Þ ¼ 1−

Explicit expression of the results of Propositions 1 and 2 in (N1,N2) space From Propositions 1 and 2, the boundary separating the optimal
 N2 leader strategies is obtained as the equation D1 SSeg ¼ 1− 2n  þ N1 1
   Seg  N2  N1 −n1 F if N1 ≥ N2, and D1 SN1 ¼ 1− 2N1 otherwise. For exposiqffiffiffiffiffiffiffiffiffiffiffiffi tional purposes, we suppose that n1⁎ = N1 or n1 ¼ 12 þ 14 þ N2F2 as an

2Fþ1 2F

when

f ðN1 Þ ≡ 2FN 21 − 2FN1 ; g ðN 1 Þ ≡ −4FN 21 þ F 2 2Fþ1 2 N 1 −ð1 þ F ÞN 1 þ 2F , respectively. Because

501

N2 −n1 F: 2n1

We next derive the number of store openings that maximizes Π1(n1). The second-order partial derivative of Π1(n1) with respect to n1 is given by ∂2 Π1 ðn1 Þ N ¼ − 32 b 0: ∂n21 n1 Thus, Π1(n1) is a concave function of n1, implying that the optimal number of stores n1⁎(N2 ≤ n1⁎ ≤ N1) is the integer satisfying Π1(n1⁎ − 1) ≤ Π1(n1⁎) and Π(n1⁎) ≥ Π1(n1⁎ + 1). Thus,    N N  Π1 ðn1 −1Þ ≤ Π1 ðn1 Þ⇔1−   2  − n1 −1 F ≤ 1− 2 −n1 F 2n1 2 n1 −1 N2 ≤0 2F rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 N2 1 1 N2  þ þ : ⇔ − ≤ n1 ≤ þ 2 4 2F 2 4 2F

 ⇔n2 1 −n1 −

ð3Þ

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each has a distance of more than 2F. 14 Furthermore, we should note that firm 2 succeeds in obtaining a larger or equal demand than firm 1 in the intervals not monopolized by both firms. Therefore, the difference between the demand of firm 1 and that of firm 2is at least in proportion to that of the number of the monopolized intervals, p − (N2 − n1 + p) = n1 − N2. That is, we have that        D1 Sn1 bD2 T Sn1 þ 2F ðn1 −N2 Þ≤1−D1 Sn1 þ 2F ðn1 −N 2 Þ. More  n1 −N2 N2 over, for F≤ 2n11 , we  obtainN2 that 2D1 Sn 1 b 1 þ n1 ¼ 2− n1 .  Hence, we have D1 Sn1 b1− 2n1 . Case 1-2: pbn1 − N22 . We next suppose that firm 1 chooses Sn1 satisfying pbn1 − N22 . In this case, because N2 b 2(n1 − p), firm 2 opening only N2 stores implies that it necessarily has at least one interval where it has only one store. We define the number of intervals where firm 2 profitably opens two stores as t. If firm 2 opens at least one store in every interval, then t = N2 − (n1 − p) holds. If t ≥ 0, the difference between the two firms in the number of intervals where each has a length of more than 2F must be p − {N2 − (n1 − p)} = n1 − N2. Therefore, in a manner similar to that in Case 1-1, we  N2 have D1 Sn 1 b1− 2n . 1 On the other hand, t b 0 implies that firm 2 cannot open stores in every interval. In that case, firm 2 benefits from opening its stores in wider intervals (we regard
theedges of the market as a single interval with length x11 þ 1−x1n1 ). Under this action by firm 2, we must have the following outcome: first, firm 1 monopolizes p intervals where each has a length of 2F. Furthermore, firm 1 divides the remaining length of 1–2Fp into n1 − p parts by locating its stores, where it monopolizes at least n1 − p − N2 intervals. As a result, the demand of firm 2 is at most half of the total lengths of the larger N2 parts. Therefore, we can evaluate the difference   between 1 and that of firm 2 as D1 Sn1 −   the  demand of firm n1 −p−N2 D2 T Sn1 ≤2Fp þ ð1−2FpÞ n1 −p , implying that

Also, 



Π1 ðn1 Þ ≥ Π1 ðn1 þ 1Þ⇔1−

   N2 N  −n1 F ≥ 1−   2  − n1 þ 1 F 2n1 2 n1 þ 1

N2 ≥0 2F rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 N2 1 1 N2   þ ;− þ þ ⇔n1 ≤ − − ≤ n1 : 2 4 2F 2 4 2F  ⇔n2 1 þ n1 −

ð4Þ

From Eqs. (3) and (4) and N2 ≤ n1⁎ ≤ N1, firm 1's optimal number of stores n1⁎ under the equidistant location strategy SEqd n1 is the integer satisfying ( ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) 1 1 N2 1 1 N2  þ þ ≤ n1 ≤ min N 1 ; þ : max N 2 ; − þ 2 4 2F 2 4 2F □ Proof of Proposition 1. In advance of proving the results by backward induction, we should note that for V > F, any location of firm 1 which does not satisfy x11 ≥ F and xk1 − xk1 − 1 ≥ 2F(k = 2, 3, …, N1) is dominated. In fact, to deter the rival's entry in some [xk1 − 1,xk1], it suffices that firm 1 chooses xk1 − 1 and xk1 such that xk1 − xk1 − 1 = 2F, which strictly dominates any inefficient location with xk1 − xk1 − 1 b 2F. Therefore, given S as any strategy of firm 1 other than one using these dominated locations, we consider the best response of firm 2 and the resulting profit of firm 1. As shown below, although the best response may not be unique, the resulting profit is uniquely determined for every S and thus the equilibrium always exists. We denote the best response by T(S) and the demand obtained by firm 2 under T(S) as D2(T(S)). In addition, let D1⁎(S) and Π1⁎(S) be, respectively, the demand and profit of firm 1 under S, provided firm 2 uses T(S). In the following, we first consider the case where firm 1 opens n1(N2 ≤ n1 ≤ N1) stores, and then the case where firm 1 opens n1(b N2) stores. Firm 1 opens n1(N2 ≤ n1 ≤ N1) stores. Let Sn1 be any strategy where firm 1 opens n1 stores, except for SSeg n1   and SEqd n1 . Then, with respect to T Sn1 , there are two possible situations: case 1, where firm 2 opens all N2 stores, and case 2, where firm 2 opens n2(b N2) stores. In the following, by considering the best response of firm 2, we show that in case 1 the resulting demand for firm 1 must N2 always be less than 1− 2n , implying that Sn1 is inferior to SEqd n1 under 1 V≥ 2N1 2 . We also show that in case 2 Sn1 mustindeed be SSeg n1 .  Case 1: firm 2 opens N2 stores under T Sn1 . We first divide case 1 into two subcases in terms of Sn1 , as follows. Here, we denote by P the set of k such that xk1 − xk1 − 1 = 2F(k = 2, 3, …, n1) and by p the number of elements of P. Therefore, the follower firm is able to enter only n1 − p intervals. Specifically, Case 1-1 is the case of p≥n1 − N22 , and Case 1-2 is the case of pbn1 − N22 .13 Case 1-1: p≥n1 − N22 . First, we assume that firm 1 chooses Sn1 satisfying p≥n1 − N22 . Then, firm 2 opening only N2 stores ensures that the following locations of N2 stores are feasible for firm 2: Since p≥n1 − N22 ; N 2 ≥2ðn1 −pÞ holds. Thus, firm 2 can open two or more stores in any of n1 − p − 1 intervals, so it can monopolize n1 − p − 1 segments where each has a length of more than 2F. In addition, firm 2 can appropriately open one store at the larger edge of the market. Furthermore, utilizing the remaining N2 − 2n1 + 2p + 1(≥1) stores, firm 2 can add (N2 − 2n1 + 2p + 1) intervals where each has a length of more than 2F. Therefore, firm 2 holds (n1 − p − 1) + (N2 − 2n1 + 2p + 1) = N2 − n1 + p intervals where






 n −p−N2 N2  þ 2Fp 1− 1 þ 1− D1 Sn1 ≤ D2 T Sn1 n1 −p n1 −p

 N2 ¼ D2 T Sn1 þ 1 þ ð2Fp−1Þ n1 −p

 p−n1 N2 þ1þ b D2 T Sn1 n1 n1 −p
 N  2 : ≤ 2−D1 Sn1 − n1

  N2 Hence, we have D1 Sn1 b1− 2n . 1   Case 2: Firm 2 opens n2bN2 stores under T Sn 1 . In this case, the resulting demand for firm 1 must be no more than 2F in every [xk1 − 1, xk1](k = 2, 3,.., n1), because otherwise firm 2 can profitably open stores in some interval, contradicting the fact that firm 2 opens only n2 stores under the best response. Indeed, this implies Sn1 ¼ SSeg n1 . Hence, when firm 1 opens n1(N2 ≤ n1 ≤ N1) stores, either SSeg n1 or
 Eqd Seg Eqd Sn1 is superior to all other strategies Sn1 ≠Sn1 ; Sn1 . Firm 1 opens n1 (b N2) stores We suppose that firm 1 opens n1(b N2) stores. As in the previous case, let Sn1 be any strategy of firm 1 with n1 stores except for   Eqd SSeg S . n1 ; Sn1 , and let T Sn1 be the best response of firm 2 to  n1 First, we assume that firm 2 opens N2 stores under T Sn1 . Then, in a manner similar to that in the previous case, we can prove that firm 2 can monopolize N2 − n1 more intervals with lengths of more than 2F than can firm 1, implying that firm 2 can obtain at least a 2F(N2 − n1) 14

When x11 = F and x1n1 ¼ 1−F, by appropriately opening N2 − 2n1 + 2p + 2 stores i in x11 ; x1n1 , firm 2 can monopolize at least N2 − n1 + p + 1 intervals where each has h

13 Only when x11 = F and x1n1 ¼ 1−F, we call the case of p≥n1 − N22þ1 as Case 1-1, otherwise as Case 1-2.

a distance of more than 2F.

M. Takaki, N. Matsubayashi / Regional Science and Urban Economics 43 (2013) 491–506

   larger demand than that of firm 1. Thus, we obtain that D2 T Sn1 >         D1 Sn1 þ 2F ðN2 −n1 Þ. Since D2 T Sn1 ≤1−D1 Sn1 , we have that     D1 Sn1 b 12 −ðN2 −n1 ÞF. Therefore, the profit of firm 1 is ∏1 Sn1 b 1 2 −ðN 2 −n1 ÞF−n1 F

¼ 12 −N 2 F. On the other hand, since V≥ 2N1 2 , the
 Eqd  ¼ 12 −N2 F. This implies that demand of firm 1 under SEqd N 2 is ∏1 SN2 Sn1 is inferior to SEqd N2 .

  We next suppose that firm 2 opens q(b N2) stores under T Sn1 .

In this case, Sn1 must be SSeg n1 for the same reason as that in case 2.

 Seg Seg  However, since it is obvious that ∏1 SSeg n1 b∏1 SN2 , SN 2 is superior to all strategies where firm 1 opens only n1 stores. Hence, it follows that all strategies where firm 1 opens n1(b N2) Eqd stores are necessarily inferior to either SSeg N 2 or SN2 .

locating its 2n1 − N2(= N2 − 2(N2 − n1)) stores at the same points as the rival's stores. As a result of this action by firm 2, because V≥ 2N1 1 from the assumption, firm 1's demand and profit are given as D1 ðn1 Þ ¼ 1−

N2 N þ ðN2 −n1 Þ; ∏1 ðn1 Þ ¼ 1− 2 þ ðN2 −n1 Þ−n1 F; 2n1 2n1

where  is sufficiently small, determined depending on N2 − n1. In the following, we consider the maximization of ∏ 1(n1) with respect to n1. By comparing ∏1 (n1) and ∏ 1(n1 − 1), ∏1 ðn1 Þ−∏1 ðn1 −1Þ ¼

Eqd ⁎ the other hand, by Lemma A1 SEqd n is superior to Sn1 ≠n , where n1 is 1

1

the integer given in Lemma A1. Hence, in the case where the leader has a higher upper limit of store openings, the leader's equilibrium  Eqd   strategy is given by either SSeg N 1 or Sn N 2 ≤n1 ≤N 1 . The profit of the 1

1−

− 1−

N2 þ ðN 2 −n1 Þ−n1 F 2n1



N2 þ ðN 2 −ðn1 −1ÞÞ−ðn1 −1ÞF 2ðn1 −1Þ N2 þ ðN 2 −n1 Þ−ðN 2 −ðn1 −1ÞÞ−F ¼ 2ðn1 −1Þn1  N −ðn1 −1Þ 1 : þ ðN 2−n1 Þ−ðN 2 −ðn1−1ÞÞ from F < > 2 2ðn1 −1Þn1 2n1

Eqd As a result, it follows that in all cases, either SSeg n1 or Sn1 ðN 2 ≤n1 ≤N 1 Þ
 Eqd is superior to all other strategies Sn1 ≠SSeg n1 ; Sn1 ; 1≤n1 ≤N 1 . Moreover, Seg in the segmentation strategy SSeg N 1 is obviously superior to Sn1 ≠N 1 . On

503

Since N2 ≥ N1 + 1 and are sufficiently small, ∏1 (n1) > ∏1 (n1 − 1) holds. Hence ∏1 (n1) is an increasing function, and it follows that the optimal number of stores under the equidistant location strategy is N1. □

leader under each strategy is given by, respectively,

Proof of Proposition 2. We can show, in the same way as in Propo-



  Seg  Seg ∏1 SN1 ¼ D1 SN1 −N 1 F

Eqd sition 1, that any firm 1 strategy Sn1 is inferior to either SSeg n1 or Sn1 when firm 1 opens n1(b N2) stores. To avoid repeating that lengthy discussion, we omit the details of the proof. In addition, also as in the case of Proposition 1, any other firm 1 segmentation strategy


 N  Eqd  ∏1 Sn ¼ 1− 2 −n1 F: 1 2n1

Seg SSeg n1 ð1≤n1 bN 1 Þ is inferior to SN 1 . Furthermore, with respect to the

Therefore, comparing these profits derives the statement of the proposition concerning the leader's equilibrium strategy. Finally, we prove the statement concerning full market coverage in the equilibrium. To show this, we consider the best response of firm 2 for the leader's equilibrium strategy. In the case of segmentation strategy, the best response of firm 2 is opening its stores at 1 − V, 1 − 3V,.... As a result, every consumer is supported by at least one firm. Suppose that this is not the case and the equilibrium number of stores each firm opens is given as n1 and n2, respectively. This immediately implies x21 −x1n1 ¼ 2V þ , where is some positive number. Note that we must have n2 b N2, from the assumption that 2N2V ≥ 1. Then, however, firm 2 can necessarily increase its profit by opening a new n2 + 1-st store at x2n2 þ1 ¼ x2n2 −2V, because the profit  −F, which is positive under V > F, a contraof this store is at leastV þ − 2

diction. On the other hand, in the case of the equidistant location strategy, as mentioned in the proof of Lemma A1 the assumption of V≥ 2N1 2 ensures full coverage in the equilibrium.

equidistant location strategy, it follows from Lemma A2 that Eqd SEqd n1 ð1≤n1 bN 1 Þ is inferior to SN 1 . Therefore, the equilibrium strategy Eqd of firm 1 is either SSeg N1 or SN 1 . The profit of firm 1 under the respective

strategies is given by

 Seg  Seg ∏1 SN1 ¼ D1 SN1 −N 1 F
 N Eqd ∏1 SN1 ¼ 1− 2 þ ðN2 −N1 Þ−N1 F: 2N 1 Since  (N2 − N1) is sufficiently small, we obtain the statement of Proposition 2 concerning the equilibrium strategy. The result of full market coverage can be proved in the same way as in the case of Proposition 1. □ Appendix C. Proofs for the results about social efficiency

□

We next prove Proposition 2. To do this, we first give the following lemma. Lemma A2. Suppose that firm 1's upper limit on the number of store openings is smaller than firm 2's (N1 b N2). If firm 1 takes the equidistant location strategy with n1 stores, then the number of stores that maximizes firm 1's profit is N1. Proof of Lemma A2. Clearly, it suffices to consider the case of N2 2 bn1 ≤N 1 , because otherwise firm 2 can completely sandwich all stores of firm 1 under V≥ 2N1 1 . When firm 1 takes the equidistant loca1 1 tion strategy SEqd n1 and opens its stores at xk ¼ 2n1 ð2k−1Þðk ¼ 1; …; n1 Þ, firm 2 can sandwich the rival's N2 − n1 stores by opening its two stores at sufficiently close proximity to each store at xk1(k = 1, 2,.., N2 − n1), allowing it to capture N2 − n1 intervals. In addition, it can capture half of the consumers from the remaining intervals by

In this section, we prove several lemmas and propositions related to social efficiency in terms of optimizing the upper limit on store openings. Here, we focus only on the symmetric case where the upper limits are identical between both firms (N1 = N2 = N). In addition, note that N is regarded as a continuous variable. Lemma 1. When firm 1 takes the segmentation strategy in the equilibrium, ÞFþV . Then, the social the number of firm 2's stores is given as n2 ¼ 1−ð2N−1 2V welfare is given by

SW Seg ðNÞ ¼ 1 þ

 V F 1 F −F− − N− ð1−V ÞðV−F Þ; 2 2V 2 V

which is a linearly decreasing function of the upper limit N. Proof of Lemma 1. When firm 1 takes a segmentation strategy with N stores, it is clear that the best response of firm 2 is opening its stores

504

M. Takaki, N. Matsubayashi / Regional Science and Urban Economics 43 (2013) 491–506

at 1 − V, 1 − 3V, …. Therefore, because we regard the number of firm 2's stores as a continuous one, we can express it as 

n2 ¼

1−ð2N−1ÞF þ V : 2V



By substituting this n2⁎ into Eq. (2), we can derive the social welfare

SW Seg ðNÞ ¼ 1 þ V−ðN þ n2 ÞF−∑ ∫x∈dðiÞ jx−xi jdx−∑ ∫x∈dðjÞ jx−xi jdx i∈I 1

¼1þ

1 1 1 F þ ¼ −F þ V− 4F 2 2 1 þ 2F  1 1 3 3V 5F F þ ¼ þ − − SW Seg 4F 2 4 4 4 4V   1 1 1 1 þ −SW Seg þ ΔSW 1 ≡ SW Eqd 4F 2 4F 2 SW Eqd

as

¼ 1 þ V−ðN þ

We next By Lemmas 1 and 2, the values  1 prove  the latter  1 statement.  of SW Eqd 4F þ 12 ; SW Seg 4F þ 12 and the difference between them, denoted by ΔSW 1 , are respectively given by

j∈I 2

1 F V F F − : ¼− þ þ þ 4 4 4 4V 1 þ 2F

   V2 − 2n2 −1 2 2

F n2 ÞF−ð2N−1Þ

2

 V F 1 F −F− − N− ð1−V ÞðV−F Þ: 2 2V 2 V

From the assumptions that V b 1 and F b V, we have that SWSeg(N) is linearly decreasing in N. □ Lemma 2. When firm 1 takes the equidistant location strategy in the equilibrium, the social welfare is given by

SW Eqd ðNÞ ¼ 1−2NF þ V−

1 : 4N

Moreover, the optimal number of stores that maximizes SWEqd(N) is ^ ¼ p1ffiffiffiffi. given as N 2 2F Proof of Lemma 2. In the symmetric case (N1 = N2), both firms open their N stores at the same locations under the equidistant location strategy, that is, xrk ¼ 2k−1 2N ; r ¼ 1; 2; k ¼ 1; …; N. Therefore, from Eq. (2), the social welfare SWEqd(N) is given by

We show that ΔSW 1 b0 for all V(F b V b 1) as follows. The secondorder derivatives of ΔSW 1 is given by ∂2 ΔSW 1 F ¼ > 0: ∂V 2 2V 3 Thus, ΔSW 1 is a convex function. Noting that Fb 12, we have 1 F F F F − ΔSW 1 ðV ¼ F Þ ¼ − þ þ þ 4 4 4 4F 1 þ 2F F F b0 ¼ − 2 1 þ 2F 1 F 1 F F ΔSW 1 ðV ¼ 1Þ ¼ − þ þ þ − 4 4 4 4 1 þ 2F F F b 0: ¼ − 2 1 þ 2F Therefore, for all V(F b V b 1), we have ΔSW 1 b0. □ The following lemma is trivial, but plays an important role in proving Propositions 3 and 4.

1 : SW Eqd ðNÞ ¼ 1−2N F þ V− 4N

Lemma A3. The second-order derivatives of SWEqd is given by 2

∂ SW Eqd ðNÞ ∂N 2

¼−

1 b 0: 2N3

Hence, SWEqd(N) is a concave function of N and thus, by solving ^ ¼ p1ffiffiffiffi. ¼ 0, we have N □ 2 2F

∂SW Eqd ðN Þ ∂N

Lemma 3. The social welfare in the equilibrium is specifically given as

SW Eq ðNÞ ¼

Seg ðN Þ

1 1 1 ≤ N≤ þ 2V 4F 2 1 1 1 þ ≤N≤ : if 4F 2 2F

In addition, we have SW Eqd

1 4F

  1 1 þ 12 b SW Seg 4F þ2 .

Proof of Lemma 3. From Proposition 1, under the symmetric case the equilibrium locations are changed at N where firm 1's demand is just 12 under its segmentation strategy. This implies that ð2N−1ÞF ¼

1 1 1 ⇔N ¼ þ : 2 4F 2

if 0 ≤ V b

3 V ð1−V Þ and F ≥ 5 3−5V

otherwise:

Proof  of Lemma  1 A3.  By Lemmas 1 and 2, the values of 1 SW Eqd 2V þ 12 and the difference between them, denoted ; SW Seg 4F by ΔSW 2 , are respectively given by 

8 > > < SW Eqd ðN Þ if > > : SW

  8 1 1 1 > > > < SW Eqd 2V ≤ SW Seg 4F þ 2   > 1 1 1 > > > SW Seg þ : SW Eqd 2V 4F 2

SW Eqd

1 2V

¼ 1−

F V þ V 2

 SW Seg

1 1 3 3V 5F F þ ¼ þ − − 4F 2 4 4 4 4V   1 1 1 −SW Seg þ ΔSW 2 ≡ SW Eqd 2V 4F 2

1 F 1−V− ð3−5V Þ : ¼ 4 V

Therefore,

ΔSW 2 ≤ 0 ⇔ 0 ≤ V b

3 V ð1−V Þ and F ≥ : 5 3−5V

□

M. Takaki, N. Matsubayashi / Regional Science and Urban Economics 43 (2013) 491–506

505

In what follows, we find the maximum value of SWEq(N) for each case.

Fig. 10. Parameter regions corresponding to Cases (a), (b1), (b2), and (c).

V Case (a): F≥ 2ð1−V Þ: Fig. 11(a) shows the SWEq(N) corresponding to this parameter region, with the relation between the four points to be compared. In fact, Lemma 3 ensures that in this case, SWEq(N) is equal to SWSeg(N) for every feasible N. Since, from Lemma 1, SWSeg(N) is decreasing in N, it follows that the maximum value of social wel1 . fare in this case is obtained as SW Seg 2V V ð 1−V Þ V Case (b1): Vb 35 and 3−5V bFb 2ð1−V : Þ 1 As seen in Fig. 11(b1), in this case it suffices to compare SW Eqd 2V  1 1   ð1−V Þ 1 , Lemma A3 ensures SW Eqd 2V and SW Seg 4F þ 2 . Since F > V3−5V b  1 1 þ 2 . Hence, the maximum value of social welfare is SW Seg 4F  1 1 þ2 . obtained as SW Seg 4F

ð1−V Þ Case (b2): Vb 35 and V2 ≤F≤ V3−5V ; or 35 ≤V and V2 ≤F: 1  1 1 þ2 and SW Seg 4F As in Case (b2), it suffices to compare SW Eqd 2V (see Fig. 11(b2)). However, in contrast to Case (b1), from Lemma 1  1 1 þ . Hence, the maximum > SW A3 we have that SW Eqd 2V  1  Seg 4F 2 value is obtained as SW Eqd 2V . 2 Case (c): Fb V2 :
 and As seen in Fig. 11(c), it suffices to compare SW Eqd 2p1ffiffiffiffi 2F 1 V ð1−V Þ V 2 V ð1−V Þ 3 SW Seg 2V . However, from 2 b 3−5V in Vb 5, we have Fb 3−5V , 1  1 1 þ 2 . Hence >SW Seg 4F implying from Lemma A3 that SW Eqd 2V
  1 1 1 > SW Seg 4F þ 2 , and thus the maximum we also have SW Eqd 2pffiffiffiffi 2F
 . value is obtained as SW Eqd 2p1ffiffiffiffi 2F 2

Proof of Proposition 3. To prove the proposition, it suffices to find the N⁎ that maximizes SWEq(N), as well as the corresponding equilibrium strategy. Nevertheless, from the form of SWEq(N) shown in Lemma 3, this can be reduced to comparing the social welfare at the following four points: the social optimum under the minimum
 differentiation N ¼ 2p1ffiffiffiffi , the boundary of the equilibrium strategy 2F   1 1 N ¼ 4F þ 2 , and the two extreme points of the feasible region   1 1 N ¼ 2V ; 2F . However, the relation between these values of social welfare sensitively depends on the parameters V and F. Therefore, according to the orders of these points, we first consider the following three cases (it can be easily verified from direct calculation that p1ffiffiffiffi ≤ 1 þ 1 holds for all F). 4F 2 2 2F
 1 1 V Case (a) 2p1ffiffiffiffi b 1 þ 12 b 2V b 2F i:e:; F > 2ð1−V Þ 2F 4F
 2 1 V b 1 b 1 þ 12 b 2F i:e:; V2 b F b 2ð1−V Case (b) 2p1ffiffiffiffi Þ 2F 2V 4F
 2 1 1 b 2p1ffiffiffiffi b 1 þ 12 b 2F i:e:; F b V2 Case (c) 2V 2F 4F

2

Summarizing cases (a) through (c), we conclude that the social optimum is attained with the segmentation strategy in cases (a) and (b1), and with the equidistant location strategy otherwise. □

In addition, considering the conditions shown in Lemma A3, Case (b) is divided into two subcases. Fig. 10 displays parameter regions corresponding to these four cases.

Proof of Proposition 4. Also from cases (a) through (c) in the proof of Proposition 3, it can be easily seen that the upper limit of store 1 openings that maximizes social welfare is 2V in cases (a) and (b2), in other words for

V 3 2ð1−V Þ ≤ F; Vb 5

and

V2 2

1 Otherwise, it is strictly higher than 2V .

Case (a)

Case (b1)

Case (b2)

Case (c) Fig. 11. SWEq in cases (a) through (c).

ð1−V Þ ≤ F≤ V3−5V , or 35 ≤V and

V2 2

≤ F. □

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