Location and Price Competition on a Uniform Path with Different Pricing Policies

Abstract

This paper models duopolistic competition between an online retailer and a physical store retailer with the online retailer modelled as a firm with uniform delivered pricing policy and the physical store as a firm with a mill pricing policy. Both firms seek to maximize their respective profits through an appropriate choice of location and price. The market is assumed to be given by a “uniform path” - a tree whose node weights and arc lengths are equal. Modelling reality, we consider two alternate types of transportation costs faced by the online retailer: either dependent on the relative location of the firm and a customer or independent of it. Beginning with the framework of a Stackelberg, i.e., sequential game, optimal location and price strategies are analytically derived for both sequences of market entry by the two competitors. Cases in which the leader faces the first entry paradox or can become a monopolist by strategically deterring entry by the follower are delineated. Thereafter, Nash Equilibrium solutions to the simultaneous game are identified. Salient insights from the results include: (a) the competitive pressure faced by the physical store in the presence of online competition (b) the inability of an online retailer to compete in “large” markets under the first type of transportation cost and (c) the advantage to the physical retailer of being a market leader.

1 Introduction

Business-to-consumer (B2C) shopping has been a human endeavor since antiquity. In ancient Greece, the agora was the marketplace, in ancient Rome it was the forum. The first stores (other than the market stalls) appeared in the 13^th century, and the first shopping malls, the Royal Exchange and the New Exchange existed in 1609 London (The Royal Exchange and the New Exchange 2002). The first department store appears to have been Harding, Howell & Co, which opened in 1796 in London, and the first modern shopping mall in the United States dates back to 1922, when the Country Club Plaza opened its doors in Kansas City, Missouri. After WW II, the development of new shopping malls really took off with the Mall of America in Bloomington, Minnesota with its 5.6 million square feet of space at one time being the largest mall in the world. Today, it is the only mall located in either the North America or Europe that is among the 20 largest malls in the world (The 10 Biggest Shopping Malls In The World 2023). Just in comparison, the largest mall is presently the Iran Mall with 21million sq ft. The importance of retail location, be it that of a mega mall or that of a simple store, is well known; see, e.g., Berge (2023), Charalambous (2019), or many similar references. As far as the future of malls is concerned, they range from predictions of a clear demise (Levin 2022) to survival and reinvention (Lange 2022).

Another development started in the late 19^th century. In order to enable the (then significant) rural population to purchase its goods, the Montgomery Ward store started providing a catalog of some of its goods in 1872. These pioneers in catalog sales were joined in 1894 by the Sears Roebuck catalog that was published for close to 100 years until 1993. For a history of Sears & Roebuck, see, e.g., Emmett and Jeuck (1950) and Sears Archives (2023), and for the last catalog see Retail Archaeology (2018). The 1990s were a decade that, slowly at first but at an ever-increasing rate, the world turned online. In the mid- 1990s, Amazon.com first became an online bookstore and is now the largest e-commerce retailer worldwide. Chevalier (2022) shows that the percentage of internet sales as a proportion of total retail sales in the United States have increased from a paltry 5.8% in 2013 to 17.9% in 2023, while the increase in the United Kingdom in the same time frame was from 10.1% to 24.9% (Lewis 2023). Another source of evidence attesting to the growth of online retail in United States are the Monthly Trade Reports from US Census Bureau (2022).

The competition between offline, i.e., brick-and-mortar stores (separate, department store, or mall) and online retailers entails novel components, some of which did not exist before. For instance, as long as the two type of stores co-exist, behavioral patterns such as showrooming (customers obtain their information from a salesclerk at the store and then purchase online) and webrooming (customers obtain the required information online and then make the purchase in the store). While some store retailers retaliate by charging a “just looking” fee, this is not one-sided: especially when it comes to technical details, many customers engage in webrooming, thus relieving the brick-and-mortar retailers from having highly trained sales staff. A recent discussion of showrooming effects can be found in Wang and Wang (2022). Some of the problems between the offline and online retailers are discussed by Ratchford et al. (2022).

Another issue that crops up in the competition between online and offline retail facilities concerns the pricing policies. Some thoughts regarding pricing policies of firms that offer online and offline facilities, see, e.g., Grewal et al. (2010). Unless they deliver, most offline retail facilities will sell the goods at the store, i.e., they practice mill pricing. This means that the full price customers end up paying is the mill price charged at the facility plus the transport costs they incur. Thus, any customer traveling to an offline retailer, i.e., a brick-and-mortar store to purchase a product incurs the accompanying transportation cost, thus increasing the final price, also referred to as the full price, paid by the customer. On the other hand, online retailers will have the goods delivered to the door at a price, also referred to as delivered price to the customer, that may or may not depend on the location of the customer, at least not inside a country. If it does not, we refer to uniform delivered pricing (or its close relative, zone pricing, which means uniform delivered pricing in each of a—typically small—number of zones, e.g., postage for domestic and international mail). On the other hand, the delivery of goods from the firm to the customer may very well be dependent on the distance between firm and customer. As long as differences between the prices charged to the customers at different locations can be explained by the transportation costs, this does not present a problem. Once other considerations play a role, the Robinson-Patman Act of 1936 may prohibit such pricing. Location and pricing competition between online and offline firms is the topic of this paper.

In an effort to model this competition between the two types of retailers using the framework of competitive location theory, we consider a duopoly model in which both of the competing firms locate a single facility each, but one of the two firms follows a uniform delivered pricing policy while the other adopts a mill pricing policy in competing for a common market for a product. Thus, it is apparent that in our model, the firm that adopts a uniform delivered pricing is the assumed proxy for an online retailer and the other with a mill pricing policy is the same for a competing brick-and-mortar store retailer.

The remainder of this paper is organized as follows. The next section is devoted to giving a background of our model and then develop the basic model while demonstrating its inherent analytical complexity given the asymmetric pricing policies of the duopolists. Section 3 develops closed form solutions for optimal location and price strategies for both retailers when the online retailer is the leader that chooses its location and price first, followed by the bricks-and-mortar retailer doing the same, while Section 4 reproduces this analysis when the sequence of market entry by the two competing retailers is the opposite. Finally, Section 5 summarizes the model and discusses the key insights gleaned and how they relate to observed real world phenomena, ending with avenues for future research.

2 Background and Model Development

The investigation of competition is at the essence of economics. Research into location and price competition started with the seminal paper by Hotelling (1929), which should be read in conjunction with its correction and extension by d’Aspremont et al. (1979). The literature about the topic is vast: in May 2024, Google Scholar revealed in excess of 5,000 hits for the search argument “competitive location.” While the early contributions were in search of spatial competitive situations regarding equilibria, their existence and uniqueness, later work also included models, in which firms locate their facilities sequentially. Already in the 1940s, the economist von Stackelberg (1943) introduced the concept of leader and follower firms that locate sequentially and irrevocably. The first to put this concept to use in the locational context appears to have been Teitz (1968), who, after demonstrating the instability of a location model, in which competitors locate facilities on a line segment (a so-called “linear market”), which are allowed to relocate so as to maximize their profit, assuming that their competitor will not react. This relocation game is repeated, and the author shows that a stable solution may not exist. He considers this behavior unlikely and instead suggests “long-term or conservative maximization,” in which facilities are located sequentially and irrevocably. Rothschild (1976) further discusses the problem with existing facilities at the end of the line already present but not participating in the optimization. In a follow-up paper, Rothschild (1979) allows the demand, which was assumed to be fixed in his previous contribution, to be a decreasing function of the distance. A breakthrough was the paper by Prescott and Visscher (1977), who base their model on the fact that firms will react to their competitor’s courses of actions, since, as they so succinctly put it, “no firm mistakenly considers itself a profit-maximizer in a world of fools.” In other words, the leader firm will anticipate the reaction of the follower (which assumes that the leader firm is assumed to know that there will be a follower and what the follower’s objective is), while the follower observes what the leader has done and reacts accordingly. From a practical point of view, the leader needs to determine for each of his possible courses of action what the reaction of the follower will be, thus establishing what is known as the reaction function. As far as optimization is concerned, while the follower’s problem is a standard conditional optimization problem (locate your own facilities while considering that the leader’s facilities have already been placed), the leader’s problem optimizes his own objective, while assuming that the follower will react optimally based on his own, possibly different, objective, thus making the leader’s problem a (much more difficult) bi-level problem - see Serra and Revelle (1994).

Drezner (1982) was the first to discuss strategies of the leader and the follower in the two-dimensional plane. Only a year later, it was Hakimi (1983), who first put order and names to leader-follower location problems and described some of their basic properties. From that point on, the follower’s location choice was to be known as a medianoid (presumably due to the similarity of its objective function to that of the standard median problems), while the leader’s location was to be known as a centroid (probably as its objective that minimized the maximum market share the follower could obtain, resembled the objective of the standard center problems). In a follow-up paper, Hakimi (1990) shows some complexity results for (r, X_p) medianoids (the follower locates r facilities in the presence the leader’s facilities at the set X_p), and (r, p) centroids (the leader locates p facilities, knowing that the follower will locate r facilities later), namely, that both the problems are NP-Hard when r, p ≥ 2. He also discusses a number of different customer choice rules.

The contribution by Drezner (1994) is among the first to include a measure of attractiveness in its model. The origin of models with uni-dimensional quality measure dates back to Reilly (1931) and later Huff (1964); multidimensional measures of attractiveness were introduced by Nakanishi and Cooper (1974). Plastria and Carrizosa (2004) further investigate a number of attraction functions, examine their properties, and describe polynomial algorithms for some cases. Drezner (2011) studies a follower situation, in which facilities of two chains already exist on the market. Each of these facilities has its own attractiveness. One of the two chains now plans to locate a new facility with a given attractiveness. The research then determines the efficient frontier of the bi-objective problem that optimizes total market share of the locating firm and cannibalization of demand associated with its own facilities. Drezner et al. (2015) take the research a step further by allocating a budget to the follower that allows it to increase its market share by either locating additional facilities or increasing the quality or attractiveness of existing facilities. The paper by Küçükaydın et al. (2012) allows the follower to open new and/or close existing facilities and choose or modify their attractiveness in order to maximize its profit.

The work by Santos-Peñate et al. (2007) extends the basic leader follower location model in a different direction. The authors deviate from the “essential good” assumption made by most authors and allow elastic demand. This issue is followed up upon by Lüer-Villagra et al. (2022), who put the elastic demand assumption into the context of multipurpose shopping and demonstrate that if a leader locates with foresight, not only is does his own market share benefit, but also that of the follower.

Other lines of research were followed by Marianov and Eiselt (2016), who, in some sense, go back to the original Hotelling claim of (central) agglomeration and study the forces that result in agglomeration and dispersion in competitive leader-follower location problems. Yu (2019) introduces a different behavioral rule that is followed by customers. His “partially proportional rule” first determines the store a customer is most attracted to. The chain this store belongs to is then chosen and the customer’s demand is then split among all stores of that chain. Note that this rule differs from Hakimi’s “partially binary preferences,” in which customers choose a chain first, and then allocate their entire demand to the closest facility of that chain. Recently, Rahmani and Hosseini (2021) included inventory management in their stochastic competitive location model.

Finally, we would like to mention some reviews that have surveyed the field. Younies and Eiselt (2011) review the basics of leader-follower location models, Drezner and Drezner (2017) report on different features of leader-follower location problems under special consideration of problems in the plane, reporting on different solution approaches and computation issues. Drezner (2023) reviews the usual models, but under special consideration of their stochastic components. Drezner and Eiselt (2023) review leader-follower location models under special considerations of the contribution made by Tammy Drezner, while the contribution by Drezner and Eiselt (2023) demonstrate the computational results of a fast heuristic for the leader’s problem.

With the above background, we now turn to the development of our model and the notations used, beginning by designating the online firm that adopts a uniform delivered pricing by U and the other (physical store) with the mill pricing policy as M. To render the analysis tractable, we assume that the competition between M and U can be modeled by the sequential Stackelberg Game where one is a leader that locates and chooses its price first, followed by the other that does the same. Both M and U will be assumed to be profit maximizers. We will begin with the assumption that U is the leader that locates and chooses its price first, followed by M who does the same and the bulk of the analytical results in this paper will be produced with this assumption. Thereafter, in Section 4, it will be shown that these results are readily modified when the opposite is the case, i.e., when M is the leader and U the follower. In either case, the leader firm thus has to choose a location and price to maximize its own profit with the knowledge that having done so, the follower firm will do the same. Note that this is different from the location leadership and price leadership advocated by authors such as Anderson (1987).

As we will subsequently see, in our model there are instances in which even at optimality, the leader cannot make a positive profit implying that the leader would not enter the market, i.e., the opposite of first mover advantage mentioned above. If the leader does not enter the market, thereby preventing either firm from entering the market, any further progress of the game depends on additional assumptions concerning the competitors’ behaviors. For the purpose of our paper, we will assume that a new game will be played with the sequence of entry by the two firms reversed.

The case in which the leader is not able to make a positive profit is a special (and significantly stronger) case of Ghosh and Buchanan’s (1988) “first entry paradox,” which refers to situations, in which the leader does not have an advantage. Subsequent research has shown that the first mover advantage is more often the exception than the rule. This phenomenon has also been studied in Eiselt and Laporte (1991) who showed that it is related to non-uniqueness of Nash-Equilibria of locations by the competitors. For the purpose of this paper, we use the term strong first entry paradox for those cases, in which the leader is not able to make a positive profit. We will assume that in all cases of first entry paradox, non-entry by the leader implies the same by the follower (as without an actual leader, the follower will have to act as the leader whenever it enters the market), leading to an indeterminate case resulting in both competitors make zero profit and as such, we will delineate all conditions under which the strong first entry paradox will occur. By contrast, should the leader be able to ensure that through a strategic choice of location and price, the follower’s optimal profit will be non-positive, we will assume that then the leader, who clearly enjoys first mover advantage in this case, enters the market as a monopolist. In game theory and economics, this is referred to as entry deterrence by the leader (see Wilson 1992), that has also been observed in some competitive location models (Bhadury et al. 1994). In reality, for such cases in which the leader enjoys a first mover advantage that allows it to strategically deter entry by a follower thereby rendering it a monopoly, the leader could charge monopolistic prices and earn monopolistic profit. However, for analytical tractability, we will assume in this paper that a leader with such a first mover advantage that is able to deter entry by a follower will continue to charge the same optimal price as a monopolist as they would have if the follower had also entered the market.

As the next point of note in our model, when studying the competition between online and physical stores, one important aspect of consumer behavior is their preference when all other factors, especially delivered price, are equivalent between the two types of retailers. In that regard, a well trusted source of information comes from the marketing consultant firm Radyant.com (2022) and its annual “Report on State of Consumer Behavior”. As stated in the 2022 report, “brick-and-mortar is the preferred option of nearly half of all shoppers”. With this as a backdrop, we will assume throughout the paper that when delivered prices are equal, a customer prefers to shop with the brick-and-mortar store, i.e., M. In technical terms, this implies that ties are broken in favor of M. While the central results of the paper do not change if the opposite were the case, i.e., ties were broken in favor of U, we will proceed with our assumption given the above-referenced report on consumer behavior.

For notations used in our analysis, we assume that in general, the common market that the two retailers compete for is given by a graph G = (V, E), where V = {1, 2, …, N} is the set of nodes, E = {(i, j)} with i, j∈ V represents the set of arcs and |E|, represents the cardinality of E. Thus, G consists of N nodes and |E|, arcs. The “weight” (i.e., customer demand) at node i is assumed to be w_i > 0 for all nodes i, with w(G) representing the total weight of the tree, i.e. w(G) = Σ_i w_i. The arc between any pair of adjacent nodes i and j is represented as (i, j). Thereafter, for any point x ∈ G, let F(x) = Σ_i w_id(x, i), where d(x, y) represents the distance between the two points x and y in G, i.e., the length of a shortest path between these two points, with dia(G) representing the maximum of shortest paths between any two pairs of nodes in G. Thus, F(x) represents the weighted distance of x from all nodes in G. In location theory literature, x is referred to as the 1-median of G iff F(x) ≤ F(y) ∀ y ∈ G. From Hakimi (1964) we also know that without loss of generality (wlog), for any graph G, with w_i ≥ 0 for all nodes i and all arcs have positive lengths, at least one 1-median is located at a node of G.

As for the costs of transporting sold goods from a retailer to a customer, we assume wlog as is frequently done in competitive location models that the unit transportation cost is unity; thus if distance is measured in miles and the cost of transportation is $X/mile, then we assume $X to represent one unit of transportation cost. While we assume that the production costs/unit and operational costs/unit (besides transportation costs) for both U and M are normalized to zero, that is not a limiting assumption and has been made for easier readability of the analysis; since our paper produces closed form solutions for optimal sales volume and profits for both U and M in all cases, and hence, any retailer can then deduct applicable production and non-transportation operational costs from the optimal profit to determine whether or not the net profit meets its own desired benchmarks. Finally, we also assume that both M and U are uncapacitated, i.e. have no exogenously imposed limits on their production and operations capacities.

At this point, it merits to discuss the type of transportation costs incurred our model since, while the brick-and-mortar retailer M does not incur any transportation expenses, the online retailer U, which follows a uniform delivered pricing policy, does. In that regard, given the real-world practices of online retailers, we will assume two different types of transportation costs that could be incurred by U. The first, that we refer to as “Location-Dependent-Transportation-Cost” assumes a situation where U delivers a product to each customer in the market for the same delivered price while incurring an associated transportation cost that is linearly proportional to the distance of that customer from its own location. One example of an online retailer that incurs Location-Dependent-Transportation Cost is the prepared foods delivery service DoorDash (https://www.doordash.com/). The second type of transportation cost that could be incurred by U that is frequently observed for online retailers, will be referred to as the “Location-Independent-Transportation-Cost” and assumes instead that while each customer in the market will receive the product for the same delivered price from U, the transportation cost incurred by U will remain a constant for each customer regardless of the distance of the customer from its own location. Thus, in such cases, the total transportation expense incurred by U in serving its customers will vary linearly with the total volume of demand that it serves. In other words, the transportation cost incurred by U in serving a customer is defined not by the distance of that customer from U but by the number of units that will be consumed by that customer. This latter type of transportation cost is frequently observed for online retailers such as Amazon.com who have the products sold to every customer in the market for the same delivered price by having their products shipped by a delivery service such as UPS or a governmental postal service, to whom they pay a fixed shipping cost/unit that is a component of the delivered price to the customer. We will explore optimal pricing and location strategies for U with both types of transportation costs.

Now, with the understanding that U is the leader and M the follower and all ties are broken in favor of M, we develop some additional notation for our model. Assume that U charges a price of p_U and has x_U ∈ G. We will represent this combination of location and price by the leader by the vector U = {x_U, p_U}. Next, we assume that given any U, the follower M locates at x_M ∈ G and charges a price of p_M, which, we will similarly denote by the vector M = {x_M, p_M}. Then, for any location and price combination of U = {x_U,p_U} chosen by U, followed by M = {x_M, p_M}) chosen by M, let π_U(U, M) and π_M(U, M) represent the profit of U and M respectively, with Δ = (p_U-p_M) representing the price differential between U and M. Since U follows uniform delivered pricing policy, it is clear that π_U(U, M) = R_U(U, M) – T_U(U, M), where R_U(U, M) represents the total revenue earned by U given any M and T_U(U, M) represents the transportation cost incurred by U in serving its customers (i.e., those not served by M). By contrast, since the follower M follows mill pricing policy, π_M(U, M) is always equal to the revenue earned by M. Since both firms are assumed to be profit maximizers, given any location and price by the leader U that is denoted by U, the optimal, i.e., profit maximizing, location and price of the follower M is denoted by ${\pi }_{\mathrm{M}}^{*}$(U, M* = $\left\{{x}_{\mathrm{M}}^{*},\,{p}_{\mathrm{M}}^{*}\right\}$), or simply, ${\pi }_{\mathrm{M}}^{*}$(U, M*). Thereafter, the optimal, i.e., profit maximizing, location and price strategy for the leader U, taking into consideration the subsequent profit maximizing response by the follower M, is denoted as ${\pi }_{\mathrm{U}}^{*}$(U* = $\left\{{x}_{\mathrm{U}}^{*},\,{p}_{\mathrm{U}}^{*}\right\}$, M*) or simply as ${\pi }_{\mathrm{U}}^{*}$(U*, M*). Additionally, note that in Section 4, where we assume that the sequence of market entry is reversed and M is the leader and U the follower, we will similarly reverse the ordering of U and M in the profit expressions above to denote the same. Finally, with regards to the locations by U and M, i.e., x_M and x_U, while we make no restrictive assumptions about these only being on the nodes of G, as we will see, given our subsequent assumption of G being a “uniform path”, ${x}_{\mathrm{M}}^{*}$ and ${x}_{\mathrm{U}}^{*}$ can be assumed to occur at nodes of G, thus exhibiting a similarity to the node-optimality result of Hakimi (1964).

With the above notation, it is clear that, for any locations x_U and x_M by U and M respectively, the following statements can be assumed to be true.

• Neither U nor M will ever consider prices that are zero since that gives them a non-positive profit.

• If p_U < p_M, i.e., Δ = (p_U-p_M) < 0, then U captures all the demand in G since the final price charged by U to every customer in every mode is < p_M.

• If p_U ≥ (p_M + dia(G)), i.e., Δ ≥ dia(G), then M captures all the demand in G since the final price charged by M to every customer in every mode is ≤ p_U.

• Therefore, in order to avoid trivial instances of the problem when U is the leader and M the follower, we will assume that

  $$0 < {p}_{\mathrm{M}} \le {p}_{\mathrm{U}} < ({p}_{\mathrm{M}} + dia(G)),\,\mathrm{i.e. }0 \le \Delta < dia(G)$$

  (1)

In Section 3, where we assume that U is the leader and M the follower, prices p_U and p_M under consideration will thus always be assumed to be in compliance with (1) above; we will relax the assumption of p_M ≤ p_U only in Section 4 which assumes that M is the leader and U the follower.

With the above in place, we are ready to demonstrate the two primary sources of the inherent analytical complexity in our model. The first and most fundamental of these emanates from the competitive nature of the locations by U and M as they compete for the discrete nodes in G. As has already been mentioned, both the leader’s decision-making (i.e., the centroid) problem and the follower’s (i.e., the medianoid) problem are NP-Hard when G is a general graph (Hakimi 1990). As is apparent from that same study, even when the duopolists charge fixed prices that are determined a priori, the objective functions of the two competitors are a sum of quasi-convex functions, making it difficult to either obtain a closed form solutions or develop polynomial algorithms to determine them.

The second source of complexity arises from the asymmetric pricing policies of the competitors in our model - an illustration of this will be provided in the numerical example below that is based on Fig. 1, where we assume that the market comprises only two nodes A and B of weights 12 and 10 respectively, connected by a single arc of length 2. Further, we assume that U is the leader and M the follower and seek to find the optimal location and price for M, given the same for the leader U.

Fig. 1

A 2-node Graph G

Suppose that the leader U locates at node A and set its price at p_U = 4 and that in response, M charges a price of p_M while located at a distance d from node A. Then, Then M’s profit function π_M(d, p_M) is as follows

$$\begin{aligned}&{\mathrm{For}}\;d\leq 1,\qquad{\pi }_{\mathrm{M}}\left(d,\,{p}_{\mathrm{M}}\right)=\left\{\begin{array}{cl}22{p}_{\mathrm{M}}& 2+d\ge {p}_{\mathrm{M}}\le 4-d\\ 12{p}_{\mathrm{M}}& 2+d<{p}_{\mathrm{M}}\le 4-d\\ 0& 4-d<{p}_{\mathrm{M}}\end{array}\right.\\&\mathrm{and}\\&{\mathrm{For}}\;1<\,d\leq2,\qquad{\pi }_{\mathrm{M}}\left(d,\,{p}_{\mathrm{M}}\right)=\left\{\begin{array}{cl}22{p}_{\mathrm{M}}& {p}_{\mathrm{M}}\le 4-d\\ 10{p}_{\mathrm{M}}& 4-d<{p}_{\mathrm{M}}\le 2+d\\ 0& 2+d<{p}_{\mathrm{M}}\end{array}\right.\end{aligned}$$

Examining π_M(d, p_M)above, it is evident that the highest possible profit for M is 66 which occurs in case M locates as the follower at the center of the edge and set its price p_M = 3. Then, it would capture the entire market regardless of where U locates, thereby causing the strong first entry paradox for U. Thus, U will not enter the market, rendering M unable to enter the market as a follower and realize its profit, leaving both competitors with zero profit. Therefore, M’s optimal strategy would be to locate at the center of the edge and charge a price of p_M = 3 + ε = 3[, where ε is an arbitrarily small positive number. Doing so would leave M with capturing node A and a revenue of 36[ and U with node B, with a revenue of 10. Note that the first entry paradox, albeit not the strong first entry paradox applies.

With the above analytical complexity in mind, for the remaining paper we will assume that G is described by a uniform path, which we define as follows:

A uniform path is defined as a tree where all node weights and arc lengths are 1

This assumption that both the node weights and arc lengths are represented by 1 is made wlog and with the understanding that node weights and arc lengths are measured in different units. While this might appear at first glance to be a simplistic approximation of the real world, given the inherent analytical complexity of the problem as discussed above, making this assumption allows us to develop closed form solutions that, in turn, yield several key insights (presented as needed in the paper and summarized in the concluding Section 5) which closely approximate real-world instances of competition between online and brick-and-mortar retailers. For example, our model provides a theoretical basis for the adverse impact of online retailing on brick-and-mortar stores in USA over the past two decades, as substantiated in the empirical study by Chava et al. (2024).

The assumption that G is a uniform path enables us to present some additional definitions and results that will be needed for subsequent analysis. To begin with, since the market G is given by a uniform path, we can assume that the nodes are numbered 1 through N, from left to right and hence, dia(G) is defined by the entire length of the path, i.e., dia(G) = N-1. We will refer to any point x ∈ G such that 1 ≤ x ≤ N as node i iff x = i. Finally, for any given U and M, where Δ = (p_U-p_M) ≥ 0, we define a node of G as a Δ-interior node iff this node is at a distance equal to or greater than Δ from either of the two extremities of G, i.e., nodes 1 and N.

As for the additional results, the first of these is a corollary of Goldman (1971).

Goldman (1971). When G is a tree, the 1-median of G is given by any node such that the weight of the heaviest subtree spanned by that node is no more than w(G)/2.

In applying Goldman (1971) to G, where it is assumed to be a uniform path, we can readily see that the 1 median of G is given by:

$$\begin{aligned}&{\mathrm{Node}} \left(N/2\right)\mathrm{ or node} \left(N/2\right)+ 1 ={\mathrm{node}}(N+2)/2, {\mathrm{if}}\;N \mathrm{ is even},\; \mathrm{with}\\& \mathrm{a total weighted distance of the}\;1-\mathrm{median to all customers} = {N}^{2}/4\;\\&\mathrm{or}\end{aligned}$$

(2)

$$\begin{aligned}&{\mathrm{Node}}\;(N+1)/2, {\mathrm{if}}\;N \mathrm{ is even},\mathrm{ with a total weighted}\\& \mathrm{distance of the}\;1-\mathrm{median to all customers} =\\&({N}^{2}-1)/4.\end{aligned}$$

Finally, the unique structure of G as a uniform path also allows for the following second result that we will use in our analysis.

$$\begin{aligned}&\mathrm{The node with the property that it maximizes the distance}\\&\mathrm{from either of the two leaf}-\mathrm{nodes of}\;G, \mathrm{ i.e., nodes}\;1 {\mathrm{ and}}\;N,\\& \mathrm{is a}\mathrm{lways a}\;1-\mathrm{median of}\;G.\end{aligned}$$

(3)

Now, to illustrate the market shares captured by U and M, consider the example shown in Fig. 2 where G is a uniform path with N = 7 with all seven nodes equally weighted at 1 and |E|= 6. with all six arcs having a length of 1. Assume further that for any given U = {p_U, x_U} and p_M, M chooses to locate at a node i. Then, regardless of the specific node i, the maximum market share it can capture from U is the linear segment that extends up to a maximum of Δ on each side of node i. Since all arc lengths are 1, this implies that this market share of M is the linear segment covering all nodes that lie between max(1, i-Δ) and min(N, i + Δ). As such, this market share will contain exactly 2$\left\lfloor \Delta \right\rfloor$ + 1 nodes if i is a Δ-interior node and at most $\left\lfloor {2\Delta } \right\rfloor$ + 1 when i is a non-Δ-interior node (since at least one of the two points i-Δ or i-Δ would be outside of G). For instance, if p_U = 3 and p_M = 1.7, resulting in Δ = 1.3, the 1.3-interior nodes of G are 3 through 5 since the Δ-interior nodes are those at a distance greater than 1.3 from nodes 1 and 7. Thus, if M locates at the Δ-interior node 4, its market share will be represented by the 2$\left\lfloor \Delta \right\rfloor$ + 1 = 3 nodes in the line segment [i-Δ, i + Δ] = [2.7,5.3], i.e., nodes 3 through 5 and its revenue would be 5.1. By contrast, had M located at the non-Δ-interior node 7, its market share would be represented by nodes in the line segment [(i-Δ), min(N, i + Δ] = [5.7, 7] containing only the two nodes 6 and 7, resulting in an attendant revenue of 3.4.

Fig. 2

A Uniform Graph G

The above illustration can be generalized to state that for a configuration of U and M in G,

$$\pi_{{\mathrm{M}}} \left( {{\mathbf{U}},{\mathbf{M}}} \right) = p_{{\mathrm{M}}} \left( {\sum\nolimits_{i} {w_{i} } } \right) \forall\;i \in \left[ {\min \left( {N,\left\lfloor {i + \Delta } \right\rfloor } \right)},\;\max(\lceil i-\Delta\rceil \right]$$

(4)

As a final note in this section, a continuous location version of our model was first studied in Eiselt (1991), which investigated Nash Equilibria for a duopoly and thereafter, compared the same to social optima with minisum and minimax objectives. Another paper that studied the competition between online and physical stores on a uniform and continuous demand framework, but solely from the standpoint of price and service quality is that by Dan et al. (2014), whose primary import was to show that brick-and-mortar retailers can preserve profits by exercising power over their suppliers to reduce purchasing costs as well as by increasing their own service quality.

3 U is the Leader, M is the Follower

This Section 3 will focus exclusively on the case where, in a continuation of the assumption from Section 2, U is the leader that decides its optimal, i.e. profit-maximizing location and price with the knowledge that having done so, the follower M would do the same. We begin with a simple result in this case that will be necessary for further analysis. For that, note that as a follower M has the opportunity to observe U’s price p_U before setting its own price p_M. Then, assuming that U actually enters the market, i.e., there exists no strong first entry paradox for U, should M choose a price of p_M = p_U, while located at any node i, due to our tie-breaking rule, M’s market share will be given by node i and hence, M’s revenue and profit will be equal to w_i, which is positive. This leads to

Lemma 1

When U is the leader and M the follower and G is a uniform path, should U be able to make a positive profit and thus enter the market, M can always make a positive profit.

Lemma 1 underscores why it is never optimal for M as the follower to cause U not to enter the market, since—given that we assume that the leader not entering the market means that the game will not be played—that would leave M with a zero profit; by contrast, M can always make a positive profit as the follower. This will be used frequently in deriving the final results of Section 3.

3.1 The Follower’s Problem (Determining M* = {${x}_{\mathrm{M}}^{*},\,{p}_{\mathrm{M}}^{*}$} for a Given U)

As with all sequential games, we will begin with characterizing the follower’s optimal location and price strategy, for any given location and price set by the leader, i.e. ${\mathbf{M}}^{*}=\left\{{x}_{\mathrm{M}}^{*},\,{p}_{\mathrm{M}}^{*}\right\}$, for any given U = {x_U, p_U} determined by the leader; and that is the focus of this Section 3.1. As will become evident during the analyses presented in this Section, the final result given in Lemma 5 characterizes the optimal response by M, will necessitate the assumption that p_U ≥ 3.5, which, in turn, requires the additional assumption that N ≥ 4. In order to get there, we begin with the following observation.

Lemma 2

For any U and M, with Δ = p_U – p_M ≥ 0, ${x}_{\mathrm{M}}^{*}$ can be assumed to be a Δ-interior node of G.

Proof

First, note that since p_U and p_M are assumed to satisfy (1), at least one Δ-interior node will always exist. Next, we begin with the assumption that M only locates at nodes of G and in so doing, chooses to locate at node i. Then, it has already been established in (4) and the discussion leading up to it that the market share captured by M from U will be the linear segment [max(1, i_-Δ) and min(N, i₊Δ)]. If node i is a Δ-interior node of G, this market share of M will contain exactly 2 $\left\lfloor \Delta \right\rfloor$ + 1 nodes and if not, it will be at most 2 $\left\lfloor \Delta \right\rfloor$+ 1 nodes. Since all nodes are also equally weighted at 1, this leads to the conclusion that the total demand served by M is guaranteed to be 2 $\left\lfloor \Delta \right\rfloor$ + 1 if it locates at a Δ-interior node of G but could be less than that if it does not. Thus, it follows that for any given U and Δ determined by p_U and p_M, should the optimal location of M, i.e., ${x}_{\mathrm{M}}^{*}$, not be a Δ-interior node of G, then there is at least one Δ-interior node of G that provides M with equal or more revenue. On this basis, it can be assumed that in response to any U, should M choose to locate at a node, it will be at a Δ-interior node.

Next, we show that it can be assumed that M will always choose to locate at a node of G and never on any other interior point between two adjacent nodes, say x, that is between two adjacent nodes, and to prove the same, assume not. Then, since x is not a node, M’s market share is 2$\left\lfloor \Delta \right\rfloor$. However, as has been shown above, M can enjoy a higher market share of 2$\left\lfloor \Delta \right\rfloor$ + 1 by locating at a Δ-interior node. ■

Lemma 2 also reveals that for any given p_U and p_M, where p_M ≤ p_U, leading to Δ = (p_U-p_M) ≥ 0, the market share captured by M is 2$\left\lfloor \Delta \right\rfloor$ + 1 for every Δ-interior node of G. This leads straightforwardly to the next observation:

Lemma 3

It can be assumed that given any U and p_M, the optimal location of M, i.e. ${x}_{\mathrm{M}}^{*}$, is at any Δ-interior node of G, earning M a total revenue = profit = ${\pi }_{\mathrm{M}}^{*}(\mathbf{U},\;{\mathbf{M}=\{x}_{\mathrm{M}}^{*},\,{p}_{M}\})={p}_{\mathrm{M}}\times \left(2+1\right)$, with the nodes captured by M being the Δ-interior node that it is located on and the adjacent $\left\lfloor \Delta \right\rfloor$ nodes to the left and right of its location.

Having characterized the optimal location ${x}_{\mathrm{M}}^{*}$ of the follower M, we now turn to the optimal price ${p}_{\mathrm{M}}^{*}$ charged by M in response to a given U = {x_U,p_U}, knowing that p_M ≤ p_U resulting in Δ = p_U-p_M ≥ 0. For that, we already know from Lemma 3 above that the revenue and profit of M as function of p_M is as follows.

$${\uppi }_{{\mathrm{M}}} \left( {{\mathbf{U}},{ }{\mathbf{M}} = \left\{ {x_{{\mathrm{M}}}^{*} ,\,p_{{\mathrm{M}}} } \right\}} \right) = p_{{\mathrm{M}}} \times \left( {2\left\lfloor \Delta \right\rfloor + 1} \right) = p_{{\mathrm{M}}} \times \left( {2\left\lfloor {p_{{\mathrm{U}}} - p_{{\mathrm{M}}} } \right\rfloor + 1} \right)$$

(5)

First, note that the presence of the floor function in (5) above makes π_M(p_M) a step function, thereby eliminating the possibility of using first order derivative to calculate this optimal value of p_M. Instead, assume that p_M charges the highest possible price of p_U. Then, M’s market share is given only by the Δ-interior node it is located on, giving M a total revenue of p_M = p_U. As a next step, assuming that p_U ≥ 1, say M considers charging a lower price than p_U as it would lead to greater market share. Should p_U-1 < p_M < p_U, M’s market share would remain unchanged from p_M = p_U as only the Δ-interior node it is located on, thus generating a lower revenue than when p_M = p_U. Therefore, should it be optimal for M to charge a price that is less than p_U, the next candidate for consideration would be p_M = (p_U-1), which allows M to capture 3 nodes – the one that it is located on and the two adjacent nodes to the left and right, leading to a revenue of 3(p_U—1). Generalizing this argument, we can conclude that at p_M = p_U – k ≥ 0, where k is an integer ≥ 1, M’s market share by locating at a Δ-interior node is given by (2 k + 1) nodes and

$$\pi_{{\mathrm{M}}} \left( {{\mathbf{U}}, {\mathbf{M}} = \left\{ {x_{{\mathrm{M}}} ,\,p_{{\mathrm{M}}} = p_{{\mathrm{U}}} - k} \right\}} \right) = \left( {p_{{\mathrm{U}}} - k} \right)\left( {2k + 1} \right) = 2kp_{{\mathrm{U}}}+p_{{\mathrm{U}}}-2k^{2} - k$$

(6)

By reapplying (5), we obtain that when p_M = p_U –(k + 1) = p_U -k-1, where p_U -k-1 ≥ 0, with M still located at a Δ-interior node, it captures (2 k + 3) nodes and

$$\pi_{{\mathrm{M}}} \left( {{\mathbf{U}}, {\mathbf{M}} = \left\{ {x_{{\mathrm{M}}} ,\,p_{{\mathrm{M}}} = p_{{\mathrm{U}}} - k - 1} \right\}} \right) =2kp_{{\mathrm{U}}}+ 3p_{{\mathrm{U}}}-2k^{2} - 5k - 3$$

(7)

In other words, for any value of p_M that is between two consecutive integers k and (k + 1), i.e., when p_U-k-1 < p_M < p_U-k for the integer k, it is clear that since M’s market share remains unchanged at (2 k + 1) nodes and as its price decreases from p_U-k towards p_U-k-1, M’s profit function π_M(p_M) decreases linearly in p_M from (6). This observation leads us to conclude that:

Lemma 4

In the determination of the optimal price of M, i.e., ${p}_{\mathrm{M}}^{*}$, it suffices to consider prices given by p_M = p_U – k, where k is an integer ≥ 1, as long as k remains small enough to ensure that p_M = p_U – k ≥ 0.

With Lemma 4 in mind, we next try to determine the integer k*, which denotes the value of k that maximizes π_M(U, M). In order to do so, consider values of k for which ${\pi }_{\mathrm{M}}({\mathbf{U},\mathbf{M}=\{{x}_{\mathrm{M}},\,p}_{\mathrm{M}}={p}_{\mathrm{U}}-k\})\leq{\pi }_{\mathrm{M}}({\mathbf{U},\mathbf{M}=\{{x}_{\mathrm{M}},\,p}_{\mathrm{M}}={p}_{\mathrm{U}}-k-1\})$, i.e., in decreasing its price from p_U–k to p_U–k–1, M’s profit does not increase. This is obtained by requiring that the expression in (6) cannot exceed that in expression (7), which can be seen to be true as long as

$$k \le \left( {2p_{{\mathrm{U}}} {-} 3} \right)/4$$

(8)

Since k is an integer, (8) results in

$$k^{*} = \left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor$$

(9)

Finally, it can be verified that the two requirements of k* ≥ 1 and p_M = p_U – k* ≥ 0 hold only if p_U ≥ 3.5, which, by (1), necessitates the additional assumption that N ≥ 4.

Thus, we can state that assuming p_U ≥ 3.5 and N ≥ 4

$$p_{{\mathrm{M}}}^{*} = p_{{\mathrm{U}}} {-}\left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor = \left( {p_{{\mathrm{U}}} {-}\left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor } \right)$$

(10)

and, given a fixed x_M and U with p_U ≥ 3.5, at optimality M captures

$$\left( {2k^{*} + 1} \right) = 2\left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor + 1\;{\mathrm{nodes in}}\;G\;{\mathrm{and since all node weights}} = 1,$$

(11)

makes a revenue = profit of

$$\pi_{{\mathrm{M}}}^{*}({\mathbf{U}}, {\mathbf{M}} = \{ x_{{\mathrm{M}}}^{*} , p_{{\mathrm{M}}}^{*} \} ) = \left( {2p_{{\mathrm{U}}} - 1} \right)\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor - 2\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor^{2} + p_\mathrm{U}$$

(12)

We can summarize the above as:

Lemma 5

Given any U = {x_U,p_U} with p_U ≥ 3.5 and N ≥ 4, ${\mathbf{M}}^{*}=\left\{{x}_{\mathrm{M}}^{*},\;{p}_{\mathrm{M}}^{*}\right\}$  can be assumed to be given as follows,

$$\begin{aligned}&{x}_{\mathrm{M}}^{*}=any\;\Delta-interior\;node\;of\;G\\&p_{{\mathrm{M}}}^{*}=\left(p_\mathrm{U}\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor\right)\\&\mathrm{and}\\&\pi_{{\mathrm{M}}}^{*}({\mathbf{U}},\,{\mathbf{M}} = \left( {2p_{{\mathrm{U}}} - 1} \right)\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor - 2\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor^{2} + p_{U}\end{aligned}$$

In the literature, Lemma 5 is characterized as the reaction function of a follower in a Stackelberg Game. As a final observation, it is interesting to note that when M is the follower, its optimal pricing and location are entirely decided by the price charged by the leader U and is independent of U’s location. Note that this is not always the case for trees and is a direct consequence of the follower adopting a mill pricing policy; it is well known that if both firms adopted a uniform delivered pricing policy, the optimal location of the follower depends on where the leader has located, even on trees (see Eiselt and Laporte 1991).

3.2 The Leader’s Problem (Determining U* = {${x}_{\mathrm{U}}^{*},\;{p}_{\mathrm{U}}^{*}$})

Having characterized the optimal solution to the follower’s problem in Lemma 5, we now turn to the problem of the leader U, where it has to determine its own optimal location and price knowing that the response by the follower M will be characterized by the results of that lemma; i.e., the problem of determining ${\mathbf{U}}^{*}=\left\{{x}_{\mathrm{U}}^{*},\;{p}_{\mathrm{U}}^{*}\right\},$ given M* as characterized in Lemma 5.

To begin with, recall that for any U, the net profit earned by U is given by

$${\uppi }_{{\mathrm{U}}} \left( {{\mathbf{U}}, {\mathbf{M}}^{*} } \right) = {\mathrm{R}}_{{\mathrm{U}}} \left( {{\mathbf{U}}, {\mathbf{M}}^{*} } \right) {-} {\mathrm{T}}_{{\mathrm{U}}} \left( {{\mathbf{U}}, {\mathbf{M}}^{*} } \right)$$

(13)

where R_U(U, M*) represents the total revenue earned by U with M’s price and location given by M* and T_U(U, M) representing the transportation cost incurred by U in serving its customers for the same case. While at first glance it might appear that U needs to simultaneously consider both its price and location in the determination of its revenue and transportation cost components, we will show that it will suffice for U to optimize these two components independently and further, when U’s transportation expenses are determined as per a location-dependent transportation cost, the minimization of the transportation cost component need only consider U’s location x_U.

First, recall from Lemma 4 that at M*, M is located at a Δ-interior node of G capturing 2$\left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor$ + 1 nodes of G, namely those that are contained in the segment [${x}_{\mathrm{M}}^{*}$ – Δ, ${x}_{\mathrm{M}}^{*}$ + Δ] = [${x}_{\mathrm{M}}^{*}$ – p_U + ${p}_{\mathrm{M}}^{*}$, ${x}_{\mathrm{M}}^{*}$ + p_U – ${p}_{\mathrm{M}}^{*}$]. This leaves U to capture the remaining

$$N {-} 2\left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor {-} 1\;{\mathrm{non}} - {\Delta } - {\mathrm{interior nodes of }} G$$

(14)

Relation (14) then allows us to conclude that for any given U and resulting M*,

$$\begin{aligned}{\mathrm{R}}_{{\mathrm{U}}} \left( {{\mathbf{U}}, {\mathbf{M}}^{*} } \right) &= p_{{\mathrm{U}}} \left( {N{ }{-}2\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor - 1} \right) \\&= p_{{\mathrm{U}}} \left( {N - 1} \right) - 2p_{{\mathrm{U}}} \left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor\end{aligned}$$

(15)

Relation (15) reveals that R_U(U, M*) is independent of U’s location x_U. When U incurs transportation expenses in accordance with location-independent transportation cost, its total transportation expenses depend solely on the market share captured by U and therefore, independent of its location. By contrast, under the assumption of location-dependent transportation cost for U, T_U(U, M*) is also independent of p_U since, it represents the weighted sum of the distance of x_U from the N—2$\left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor$ – 1 non-Δ-interior nodes of G that will be served by U. This leads to:

Lemma 6

When U incurs transportation expenses in accordance with location-independent transportation cost, then its profit π_U(U, M*) can be maximized by maximizing its revenue R_U(U, M*) based on p_U. In cases where U’s transportation expenses are determined as per location-dependent transportation cost, π_U(U, M*) can be maximized by independently maximizing R_U(U, M*) based on p_U and minimizing T_U(U, M*) based on U’s location x_U.

With Lemma 6 allowing the separability of the two terms R_U(U, M*) and T_U(U, M*), we will represent R_U(U, M*) simply as R_U(p_U, M*) and T_U(U, M*) as T_U(x_U, M*), while noting that when U operates as per location-independent transportation cost, T_U(x_U, M*) is the same for all locations x_U by U.

3.2.1 The Optimal Price ${p}_{\mathrm{U}}^{*}$ When p _U ≥ 3.5

Having established Lemma 6 above, we now turn our attention to finding the optimal price for U, i.e. ${p}_{\mathrm{U}}^{*}$. We will continue with the assumption made in Section 3.1 that p_U ≥ 3.5 and N ≥ 4 and show that the results of this Section 3.2.1 will necessitate the stronger assumption that p_U ≥ 3.5 and N ≥ 9. In doing so, first note that by (15), R_U(p_U, M*) =$p_{{\mathrm{U}}} \left( {N{-}2\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor -1} \right) = p_{{\mathrm{U}}} \left( {N - 1} \right) - 2p_{{\mathrm{U}}} \left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor$ when p_U ≥ 3.5 and N ≥ 4. Once again, as was the case in (5) for M, due to the floor function $\left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor$ present in (15), R_U(p_U, M*) is also step function of p_U, eliminating the possibility of using first order derivative directly on R_U(p_U, M*) to find ${p}_{\mathrm{U}}^{*}$. Therefore, assume that $\left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor$ is denoted by a non-negative integer k.

First, we note that if p_U is such that (2p_U – 3)/4 assumes a value between any integer k ≥ 0 but less than the consecutive integer k + 1, i.e., when (4 k-3)/2 ≤ p_U < (4 k + 7)/2 for any integer k ≥ 0, then an examination of (15) reveals the following:

$$\begin{aligned} {\mathrm{R}}_{{\mathrm{U}}} (p_{\mathrm{U}}, \mathbf{M}^{*}) &= p_{{\mathrm{U}}} \left( {N - 1} \right) {-} 2p_{{\mathrm{U}}}\\ k &= p_{{\mathrm{U}}} \left( {N - 1} \right) {-} 2p_{{\mathrm{U}}} \left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor\\&\;\;\forall\;p_{{\mathrm{U}}} {\mathrm{ such that}}\;(4\left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor - 3)/2\\& \le p_{{\mathrm{U}}} < (4 \left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor + 7)/2 \end{aligned}$$

(16)

Since (16) is a continuous function of p_U in the range prescribed therein, it is possible to use the first derivative to examine the slope of R_U(p_U, M*) in this range. To do so, we note that when (2p_U – 3)/4 is a non-integer because p_U is in the range prescribed in (16),

$$\begin{aligned}&{\mathrm{R}}_{{\mathrm{U}}} \left( {p_{{\mathrm{U}}} , {\mathbf{M}}^{*} } \right) = p_{{\mathrm{U}}} \left( {N - 1} \right) {-} 2p_{{\mathrm{U}}} \\&k = p_{{\mathrm{U}}} \left( {N - 1} \right) {-} 2p_{{\mathrm{U}}} \left\lfloor {\left( {2p_{{\mathrm{U}}} - 3} \right)/4} \right\rfloor\end{aligned}$$

(17)

and hence,

$$\begin{aligned}&\frac{{\partial \left( {{\mathrm{R}}_{{\mathrm{U}}} \left( {p_{{\mathrm{U}}} , {\mathbf{M}}^{*} } \right)} \right)}}{{\partial p_{{\mathrm{U}}} }} = \left( {N - 1} \right) - 2\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor,\\&{\mathrm{ where}}\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor {\mathrm{is a constant}}\end{aligned}$$

(18)

Examining relation (18) above we conclude that as long as the value of p_U stays in the range given in (16), the slope of R_U(p_U, M*) is a constant, since N is fixed. To examine values of p_U when this slope is non-negative, thus making R_U(p_U, M*) a non-decreasing function of p_U, we set (18) ≥ 0 and find that this only occurs for values of p_U such that

$$\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor \le \frac{N - 1}{2}$$

(19)

and for all larger values of p_U, the slope of R_U(p_U, M*) is negative. Thus, R_U(p_U, M*) is a quasi-convex step function of p_U and as U increases p_U from the minimum permissible price of 3.5, its revenue R_U(p_U, M*) will continue to increase until p_U reaches a maximum that still allows:

$$\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor = \frac{N - 1}{2}$$

(20)

Recall that the derivation of ${p}_{\mathrm{U}}^{*}$ as characterized in (20) had initially assumed that p_U ≥ 3.5, which necessitated the additional assumption that N ≥ 4. Therefore, to investigate conditions under which ${p}_{\mathrm{U}}^{*}$ is guaranteed to be 3.5 or higher, we set

$$\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor =\left\lfloor \frac{N - 1}{2}\right\rfloor \ge 3.5$$

which can be shown to hold as long as N ≥ 9.

While (20) is a characterization of U’s optimal price ${p}_{\mathrm{U}}^{*}$, it does not prescribe the precise value of ${p}_{\mathrm{U}}^{*}$. In order to do so, we will have to optimize within the interval defined by (20). For this, we note that

$$\left\lfloor {\frac{N - 1}{2}} \right\rfloor = \left\{ {\begin{array}{ll}{\frac{N - 1}{2}} & \mathrm{when}\ N{\mathrm{ is odd}} \\ {\frac{N - 2}{2}} & \mathrm{when}\ N{\mathrm{ is even}}\end{array}}\right.$$

(21)

Using (21) along with the relationship between p_U and N given in (20), we obtain that when N is odd, ${p}_{\mathrm{U}}^{*}$ is a value of p_U that satisfies the following inequality

$$\begin{aligned}&\left\lfloor {\frac{{2p_{{\mathrm{U}}} - 3}}{4}} \right\rfloor = \frac{N - 1}{2}\;\mathrm{which is equivalent }\\&\mathrm{to stating}\;\frac{N - 1}{2} \le \frac{{2p_{{\mathrm{U}}} - 3}}{4} <\frac{N - 1}{2} + 1 = \frac{N + 1}{2},\end{aligned}$$

which leads to

$$N-1\le {p}_{\mathrm{U}}<\frac{2N+5}{2}$$

(22)

Since compliance with (1) already assures that N–1 < p_U and because R_U(p_U, M*) increases continuously with p_U (because the number of nodes captured by U remains unchanged) within the range assumed in (22), we can conclude that when N is odd

$${p}_{\mathrm{U}}^{*}=](2N+5)/2$$

(23)

where, in order to simplify the notation used in the rest of the paper, we use “]” to denote a value that is arbitrarily smaller than (2N + 5)/2, i.e., ](2N + 5)/2 is assumed to represent ((2N + 5)/2) – ε, where ε is an arbitrarily small positive number. Then, by using similar reasoning as used for (23), we obtain that when N is even,

$${p}_{\mathrm{U}}^{*}=](2N+3)/2$$

(24)

Finally, with ${p}_{\mathrm{U}}^{*}$ as defined in (23) and (24), we can calculate U’s optimal revenue, i.e., ${\mathrm{R}}_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},\,{\mathbf{M}}^{*}\right)$. By substituting (23) and (24) as the value of p_U into (15), we obtain the following result.

$${R}_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},{\mathbf{M}}^{*}\right)=\left\{\begin{array}{cc}0& \mathrm{when }N\ge 9 \mathrm{ and is odd} \\ ](2N+3)/2 & \mathrm{when }N\ge 9 \mathrm{ and is even}\end{array}\right.$$

(25)

Further, since the profit of U is revenue minus its transportation cost, we can now combine Lemma 6 and (25) to conclude that

Lemma 7

When p_U ≥ 3.5 and N ≥ 9 and even, = ${p}_{\mathrm{U}}^{*}$ ](2N + 5)/2. Additionally, ${R}_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},\,{\mathbf{M}}^{\mathbf{*}}\right)$ = ](2N + 3)/2. In cases where p_U ≥ 3.5 and N ≥ 9 but odd, ${R}_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},\,{\mathbf{M}}^{\mathbf{*}}\right)=0$.

3.2.2 M* and ${p}_{\mathrm{U}}^{*}$ When p _U < 3.5

All derivations of M’s optimal location and price in Section 3.1 pertaining to the follower’s problem as resolved in Lemma 4 needed the assumption that p_U ≥ 3.5, which, in turn, necessitated the additional assumption that N ≥ 4 to retain compliance with (1). Thereafter in Section 3.2.1, deriving the value of ${p}_{\mathrm{U}}^{*}$ per the relationship in (20) further required the stronger assumption that p_U ≥ 3.5 and N ≥ 9, making p_U ≥ 3.5 and N ≥ 9 the binding assumptions of Lemma 7 and thus, all our results up to now. This section will explore the remaining case, in which p_U < 3.5 and as we will see, that does not require any assumptions about the minimum value of N, allowing the delineation in the subsequent Section 3.3 of the optimal location and price for both U and M when U is the leader and M the follower and U is subject to location-dependent transportation cost.

Since we assume that p_U < 3.5, we will see that in combination with some of our prior results, this upper bound makes it possible to solve this case of p_U < 3.5 using simple total enumeration. As before, the minimum number of nodes that M can capture is 1 (the node that it is located at) which occurs when it charges a price of p_U. However, with p_U < 3.5, regardless of the market size N, the maximum number of nodes M can capture by lowering its price is limited to 7, which occurs when M can charge a non-negative price that is (p_U-3) or lower, provided that p_U > 3. In determining M’s optimal price given any U, we begin by noting that even when p_U < 3.5, the results of Lemma 4 still hold as long as p_M ≤ p_U. Thus, the only four prices that M would consider in maximizing its profit ${\pi }_{\mathrm{M}}^{*}\left({\mathbf{U},\mathbf{M}}^{*}\right)$ are as below, where, just as the case before for p_U ≥ 3.5, M faces the tradeoff between price and market captured – increasing one lowers the other.

$$\begin{aligned}&\left\{ {p_{{\mathrm{U}}} ,\,p_{{\mathrm{U}}} - 1,\;p_{{\mathrm{U}}} - 2 {\mathrm{ and }} p_{{\mathrm{U}}} - 3} \right\},\\&{\mathrm{ while ensuring that }}p_\mathrm{M}^{*} >0\end{aligned}$$

(26)

Relation (26) above coupled with the assumption of p_U < 3.5 is what renders simple total enumeration feasible in solving this case.

Now, as done in Section 3.1 when p_U ≥ 3.5, we begin with M’s optimal response for any given U, i.e., M* = {${x}_{\mathrm{M}}^{*}$, ${p}_{\mathrm{M}}^{*}\}$. Of M’s price and location, we can discard the determination of M’s optimal location since Lemma 3 holds even when p_U < 3.5 as long as p_M ≤ p_U, and hence, ${x}_{\mathrm{M}}^{*}$ will continue to be assumed to be at a Δ-interior node of G, thus leaving us to focus solely on ${p}_{\mathrm{M}}^{*}$. Further, for any price p_U charged by U, the revenue earned by U depends solely on the number of nodes that would be captured by M. By exploiting this fact along with (26), total enumeration allows us to verify that the optimal, i.e., profit maximizing response of M for any given value of p_U, varies as follows.

$$\begin{aligned}&{\mathrm{Case}}\;\mathrm{I}: {p}_{\mathrm{U}}<1: \, {\mathrm{M}\mathrm{'s optimal price is set to }}{p}_\mathrm{M}^{*}={p}_{\mathrm{U}}.\\&\;{\mathrm{By doing so, M captures 1 node }}{(}{\mathrm{the}} \, {\mathrm{one}} \, {\mathrm{that}}\;{\mathrm{it}} \\& {\mathrm{is}} \, {\mathrm{located}} \, {\mathrm{on}}{)} \, {\mathrm{and}} \, {\mathrm{earns}} \, {\mathrm{the}} \, {\mathrm{optimal}} \, {\mathrm{revenue}} \, = {\pi }_{\mathrm{M}}^{*}\left(\mathbf{U}, \mathbf{M}^{*}\right)\\&={p}_{\mathrm{U}}.\;{\mathrm{By}} \, {\mathrm{Lemma}} \, \, 3,\;{x}_{\mathrm{M}}^{*}\;{\mathrm{is}} \, {\mathrm{assumed}} {\mathrm{to}} \, {\mathrm{be}} \, {\mathrm{any}}\Delta\\&-{\mathrm{interior}} \, {\mathrm{node}} \, {\mathrm{of }} \, G.\end{aligned}$$

$$\begin{aligned}&{\mathrm{Case II}}: 1 \le {p}_{\mathrm{U}}<3.5: \, {\mathrm{M}\mathrm{'s}} \, {\mathrm{optimal}} \, {\mathrm{price}} \, {\mathrm{is}} \, {\mathrm{set}} \, {\mathrm{to}} \, {p}_{\mathrm{M}}^{*}={p}_{\mathrm{U}}-1. \\&{\mathrm{By}} \, {\mathrm{doing}} \, {\mathrm{so}}, \, {\mathrm{M}} \, {\mathrm{captures}}\;3\;{\mathrm{nodes}}\;({\mathrm{the}} \, {\mathrm{one}} \, {\mathrm{that}} \, {\mathrm{it}} \, {\mathrm{is}} \, {\mathrm{located}} \, {\mathrm{on}} \, {\mathrm{and}} \\& {\mathrm{the}} \, {\mathrm{two}} \, {\mathrm{neighboring}} \, {\mathrm{nodes}}) \, {\mathrm{and}} \, {\mathrm{earns}} \, {\mathrm{the}} \, {\mathrm{optimal}} \, {\mathrm{revenue}} \, =\\&{\pi }_{\mathrm{M}}^{*}\left(\mathbf{U}, \mathbf{M}^{*}\right)=3\left({p}_{\mathrm{U}}-1\right).\;{\mathrm{By}} \, {\mathrm{Lemma}}\;3,\;{x}_{\mathrm{M}}^{*}\;{\mathrm{is}} \, {\mathrm{assumed}} \, {\mathrm{to}} \, {\mathrm{be}} \\& {\mathrm{any}}\;\Delta -{\mathrm{interior}} \, {\mathrm{node}} \, {\mathrm{of}} \, G.\end{aligned}$$

(27)

As for the two other prices that M would consider, namely p_U-2 and p_U-3, total enumeration can be used to verify that they do not yield any higher revenue for M than as given in (27). Next, note that by Lemma 6, which is true even when p_U < 3.5, for any given M*, U would set its profit-maximizing price p_U so as to maximize its own revenue R_U(p_U, M*). The two cases in (27) above now enable us to determine the optimal (revenue maximizing) price for U ${p}_{\mathrm{U}}^{*}$ as follows:

$$\begin{aligned} &{\mathrm{Case I}}: p_{{\mathrm{U}}}^{*} = ]1,{\mathrm{ which allows U to capture }}N{-}1{\mathrm{ nodes,}}{\mathrm{ resulting in optimal revenue of }}\\&R_{{\mathrm{U}}}^{*} \left( {{\mathbf{U}}^{*} ,{\mathbf{M}}^{*} } \right) = ]\left( {N {-} 1} \right). \hfill \\ \end{aligned}$$

$$\begin{aligned} &{\mathrm{Case II}}: p_{{\mathrm{U}}}^{*} = ]3.5,{\mathrm{ which allows U to capture }}N{-}3{\mathrm{ nodes,}}{\mathrm{ resulting in optimal revenue of }}\\&R_{{\mathrm{U}}}^{*} \left( {{\mathbf{U}}^{*} ,{\mathbf{M}}^{*} } \right) = \left] {3.5\left( {N {-} 3} \right) = } \right] \left( {3.5N {-} 10.5} \right). \hfill \\ \end{aligned}$$

(28)

Examining (28) by comparing the two different equations for U’s revenue at optimality, it becomes evident that when N ≤ 3, it is optimal for U to set its price using Case I, i.e., ${p}_{\mathrm{U}}^{*}$, M* = ]1 whereas when N ≥ 4, U would set its optimal price using Case II, i.e., ${p}_{\mathrm{U}}^{*}$ = ]3.5. Further, by examining the values of ${R}_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},{\mathbf{M}}^{\mathbf{*}}\right)$, i.e., U’s optimal revenue in either case, it is also verified that whenever N ≥ 2, U is always guaranteed a positive revenue, albeit not positive profit, at optimality.

3.2.3 M* and ${p}_{\mathrm{U}}^{*}$ Without any Assumption about p _U and N.

Lemma 6 allowed us to focus exclusively on U’s price p_U to maximize its revenue to ${R}_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},{\mathbf{M}}^{\mathbf{*}}\right)$, which is what was done in Sections 3.2.1 and 3.2.2. It is apparent that U will always choose its optimal price as less than 3.5 (as analyzed in Section 3.2.2) or at least 3.5 (as analyzed in Section 3.2.1) depending on whichever gives it higher revenue which, in turn, is determined by M’s optimal response (as discussed in Section 3.1). Hence, in identifying firm U’s optimal pricing strategy, we can compare the values of ${R}_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},{\mathbf{M}}^{\mathbf{*}}\right)$ as given in (25), which holds only when p_U ≥ 3.5 and N ≥ 9 with what it is in (28), which holds for p_U < 3.5 and all values of N. Doing so, it can be verified that the optimal revenue of U, as determined in (28), is always larger than the one produced by (25) regardless of the value of N. This leads to:

Lemma 8

Given profit-maximizing duopolists U and M, where U is the leader and adopts a uniform delivered pricing policy and M is the follower with a mill pricing policy, and the graph G is a uniform path with N ≥ 2, it can be assumed that revenue-maximization for U and M is achieved as follows:

$$\begin{aligned}&Case\;{\mathrm{I}}:If\;N\le 3,\mathrm{ then }{p}_{\mathrm{U}}^{*}=\left]1,\;{\mathbf{M}}^{\mathbf{*}}=\left\{{p}_{\mathrm{M}}^{\boldsymbol{*}}={p}_{\mathrm{U}}=\right.\right]1, {\mathrm{ and }}{ x}_{\mathrm{M}}^{*}\\& is\;any\;\Delta -interior\;node\;of\;\left.G\right\}.\;Further,{\pi }_{\mathrm{M}}^{*}\left(\mathbf{U},\;{\mathbf{M}}^{*}\right)\left.=\right] \\&1 {\mathrm{ and }} {R}_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},\;{\mathbf{M}}^{\mathbf{*}}\right)\left.=\right]\left(N-1\right).\end{aligned}$$

$$\begin{aligned}&Case\;{\mathrm{II}}:If\;N\ge 4,\mathrm{ then }{p}_{\mathrm{U}}^{*}=\left]3.5,{\mathbf{ M}}^{\mathbf{*}}=\left\{{p}_{\mathrm{M}}^{\boldsymbol{*}}=\right.\right]2.5, and\;{ x}_{\mathrm{M}}^{*} \, is \\&any \; \Delta -interior\,node\,of \left.G\right\}.\;Further,{\pi }_{\mathrm{M}}^{*}\left(\mathbf{U},\,{\mathbf{M}}^{*}\right)\left.=\right]\\& 7.5\,{\mathrm{ and }}\, {R}_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},\,{\mathbf{M}}^{\mathbf{*}}\right)\left.=\right]3.5\left(N-3\right).\end{aligned}$$

We can also show that both M and U, are guaranteed a positive revenue for all values of N.

Lemma 8 demonstrates that in our model, if N ≥ 4, despite the fact that the model does not impose any upper bound on the price charged by U, the optimal pricing strategy of U is to set its own price to ]3.5 even when the total demand in the market, represented by N, is indefinitely large. This is to prevent the follower M, which follows a mill pricing policy, from capturing any more than 3 nodes, i.e., a small share of the market. Having adopted this optimal pricing strategy, U’s revenue can then be assured to grow with the size of the market, i.e., N.

3.2.4 Optimal Location of U (${x}_{\mathbf{U}}^{*}$) with Location-Dependent Transportation Cost

Having established the optimal location and price of M, i.e., ${x}_{\mathrm{M}}^{*}$ and ${p}_{\mathrm{M}}^{*}$, in Lemma 5 and determined U’s optimal price ${p}_{\mathrm{U}}^{*}$ in Lemma 8, we now turn our attention to finding U’s optimal location ${x}_{\mathrm{U}}^{*}$ under the assumption that U incurs transportation expenses in accordance with location-dependent transportation cost. That, in turn, will allow us to obtain in the next Section 3.3, the complete optimal pricing and location strategy for U for this type of transportation cost.

First, note that minimizing T_U(x_U, M*) with location-dependent transportation cost is equivalent to finding a 1-median of a modification of G, which we denote by G(M*), where all nodes have a weight of 1 except those captured by M, which are assumed to have a weight of 0 in G(M*). Further, we also know from Hakimi (1964) that a 1-median of G(M*), which is also the optimal location of U ${x}_{\mathrm{U}}^{*}$, is always at a node of G.

Next, we turn our attention to the simpler cases of N = 2 and N = 3, i.e., Case I in Lemma 8 above. In these two cases, we know from Lemma 8 that ${p}_{\mathrm{M}}^{*}$ = ${p}_{\mathrm{U}}^{*}$ = ]1, leading to Δ = 0 and thus, all nodes of G are Δ-interior nodes and M could locate on any node of G. Then, when N = 2, it is easy to see that while both nodes are the 1-medians of G, the 1-median of G(M*) is a node other than the one where M locates. Further, if the optimal locations of M and U are coincident, then U makes a negative profit of ]1–1 < 0 but when these optimal locations are distinct, U’s profit is ]1. With that and recalling that as a follower, M has an opportunity to observe U’s location before deciding its own, it will always locate to avoid a strong first entry paradox for U since Lemma 1 guarantees M a positive optimal profit. Thus, we can conclude that when U is the leader and M the follower, their optimal locations will be on distinct nodes of G with each earning a profit of ]1.

For N = 3, node 2 is a 1-median of G but the 1-median of G(M*) is characterized as follows. If M locates at node 2, then G(M*) is a 3-node uniform path where nodes 1 and 3 have a weight of 1 and node 2 has a weight of 0. Thus the 1-median of G(M*) is also node 2 and that is where U will locate at optimality. Using the same logic, in cases where M locates on node 1 (respectively, node 3), the 1-median nodes of G(M*) are node 2 and node 3 (respectively, node 1 and node 2). The optimal profit of U for these configurations are as follows: when the optimal locations of M and U coincide as what happens if M locates at node 2, then the optimal profit of U is ]2 – 2 < 0, i.e., negative; but if they are located on two distinct nodes, then it is ]2–1 > 0, i.e. positive. Thereafter, using the same reasoning as N = 2 above, we infer that when U is the leader and M the follower, the optimal locational configurations can be assumed to be at two different nodes of G and U’s optimal profit is ]2 – 1 = ]1 which is the same as the optimal profit of M.

Now, for the more complex case of N ≥ 4, where, by Lemma 8, ${p}_{\mathrm{U}}^{*}$ = ]3.5, ${p}_{\mathrm{M}}^{*}$ = ]2.5, leading to Δ = 1 resulting in M capturing the three adjacent nodes (${x}_{\mathrm{M}}^{*}$–1), ${x}_{\mathrm{M}}^{*}$ and (${x}_{\mathrm{M}}^{*}$+1), while located at any of 1-interior nodes or nodes 2 through N–1. As we also know, U’s optimal location ${x}_{\mathrm{U}}^{*}$ must be a 1-median of G(M*) and hence, we will attempt to characterize the 1-median of G(M*) in terms of where M chooses to locate at optimality, i.e.,${x}_{\mathrm{M}}^{*}$. In order to do, so, we need the result of Goldman (1971) and repeatedly apply it to G(M*), which in this case is entirely determined by M’s location, i.e., ${x}_{\mathrm{M}}^{*}$. Since exactly 3 nodes of G have been captured by M at optimality, the total weight of G(M*) is N–3. Therefore, by Goldman (1971), a node of G is a 1-median of G(M*) iff the weight of both the subtrees spanned by it (one to its left and the one to its right) in G(M*) is at most (N–3)/2. Now, let us denote by node i a 1-median of G, with i = (N/2) being the lower-numbered 1-median node when N is even and (N + 1)/2 when N is odd. Further, for any two points x and y ϵ G, let w[x, y] represent the sum of the weights of all nodes in the line segment [x, y]. Then, by Goldman (1971), we know that in G, w[1, i–1] and w[i + 1, N] are both at most N/2. By the same token, if a node j ϵ G is a 1-median of G(M*), then w[1, j–1] and w[j + 1,N] in G(M*) must both be at most (N-3)/2, where G(M*) differs from G in only that the three adjacent nodes (${x}_{\mathrm{M}}^{*}$–1), ${x}_{\mathrm{M}}^{*}$ and (${x}_{\mathrm{M}}^{*}$+1) of G have 0 weights each in G(M*). Thereafter, by iteratively applying the necessary and sufficient conditions of Goldman (1971), we can verify that a 1-median of G(M*), which is also the optimal location of U, i.e., ${x}_{\mathrm{U}}^{*}$, may be characterized by M’s optimal location ${x}_{\mathrm{M}}^{*}$ as follows:

Lemma 9

Given M* = {${x}_{\mathrm{M}}^{*},{p}_{\mathrm{M}}^{*}$} and ${p}_{\mathrm{U}}^{*}$ as defined in Lemma 8, should U incur transportation expenses in accordance with location-dependent transportation cost, we can assume that the optimal location of U, i.e. ${x}_{\mathrm{U}}^{*}$, is as follows when N ≥ 4:

$$\begin{aligned}&\begin{aligned}If\;N\;is\;od&d,\;with\;i=(N+1)/2,\;which\;is\;the\;1-median\;of\;G:\\&For\;2\;\le{x}_\mathrm{M}^*\le{i,}\;{x}_\mathrm{U}^*\;is\;a\;1-median\;of\;G(\mathbf{M}^*),\;\mathrm{i.e.,}\;node\;(i+1)\\&For\;{x}_\mathrm{M}^*=i,\;and\;{x}_\mathrm{U}^*\;is\;a\;1-median\;of\;G(\mathbf{M}^*),\;\mathrm{i.e.,}\;node\;i\\&For\;i+1\le{x}_{M}^*\le{N-1,}\;{x}_\mathrm{U}^*\;is\;a\;1-median\;of\;G(\mathbf{M}^*),\mathrm{i.e.,}\,node\;(i-1)\end{aligned}\\&\begin{aligned}If\;N\;is\;ev&en,\;with\;i=N/2,\;which\;is\;the\;lower\;numbered\;1-median\;of\;G:\\&For\;2\;\le{x}_\mathrm{M}^*\le{i,}\;{x}_\mathrm{U}^*\;is\;a\;1-median\;of\;G(\mathbf{M}^*),\;\mathrm{i.e.,}\;node\;(i+2)\\&For\;i+1\le{x}_\mathrm{M}^*\le{N-1,}\;{x}_\mathrm{U}^*\;is\;a\;1-median\;of\;G(\mathbf{M}^*),\mathrm{i.e.,}\,node\;(i-1)\end{aligned}\end{aligned}$$

The net import of Lemma 9 above is that when N ≥ 4 and U operates with location-dependent transportation cost, if at optimality M locates to the left (respectively, right) of the 1-median of G (respectively, lower numbered 1-median when N is even), then the optimal location of U that minimizes U’s total transportation cost can be assumed to be the immediate right (respectively, left) neighboring node of the 1-median (respectively, higher numbered 1-median when N is even). An interesting case occurs when N is odd and hence the 1-median of G is unique – in that case, should M locate at this unique 1-median at optimality, then that same unique 1-median is also the optimal location of U.

Before ending this section, we make a special note of two particular cases of the configuration of optimal locations of M and U when N ≥ 4 as described in Lemma 9 above. Lemma 5 asserts that at optimality, M can choose to locate on any of the Δ-interior nodes 2 through N–1. By virtue of Lemma 9, should M choose node 2 or node N–1 as its optimal location, i.e., ${x}_{\mathrm{M}}^{*}$ is at node 1 or node N–1, then by Lemma 9, U’s optimal location ${x}_{\mathrm{U}}^{*}$ is at the farthest distance from M among all solutions in Lemma 9. We will refer to this as a case of maximal locational differentiation between the optimal locations of M and U. An examination of Lemma 9 reveals that in this case of maximal locational differentiation, when M locates on node 2 (respectively, node N–1), it causes U to be optimally located at node (N + 3)/2 when N is odd and node (N + 4)/2 when N is even (respectively, at node (N–1)/2 when N is odd and node (N-2)/2 when N is even). By contrast, the opposite occurs when M chooses to locate at a 1-median of G, the optimal locations of M and U are closest among all solutions in Lemma 9. We will refer to this as minimal locational differentiation between the optimal locations of M and U. In such cases of minimal locational differentiation, when N is odd, both M and U are located at node (N + 1)/2 at optimality and when N is even, M locates at node N/2 (respectively, node (N + 2)/2) and U locates at node (N + 4)/2 (respectively, node (N-2)/2). These configurations of maximal and minimal locational differentiation will be useful in the analysis below on the total transportation expenses incurred by U with location-dependent transportation cost.

3.3 U* and M*: U is the Leader and M is the Follower (In the Case of Location-Dependent Transportation Cost)

In Section 3.2.4 above we characterized the optimal locations and profits of U and M when N ≤ 3 and thereafter, the optimal location of U, i.e., ${x}_{\mathrm{U}}^{*}$, based on the optimal location of M, i.e.,${x}_{\mathrm{M}}^{*}$, for all N ≥ 4 in Lemma 9. Hence we will begin the present Section 3.3 by focusing on ${\mathrm{T}}_{\mathrm{U}}^{*}$ (${x}_{\mathrm{U}}^{*}$, M*), i.e., U’s optimal transportation cost with location-dependent transportation cost when N ≥ 4. It is apparent that ${\mathrm{T}}_{\mathrm{U}}^{*}$ (${x}_{\mathrm{U}}^{*}$, M*) depends on the precise Δ-interior node of G where M locates at optimality (i.e., $\mathrm{at}{x}_{\mathrm{M}}^{*}$), since the latter determines the nodes that would then be subsequently served by U. As an illustration of this situation, consider the case where G is given by Fig. 2 where N = 7, leading to i = node 4 by (2). Then, by Lemma 8, U’s optimal price is ]3.5, which allows U to earn optimal revenue of ]14. Further, M charges an optimal price of ]2.5, leading to Δ = 1, allowing M to be optimally located at any of the 1-interior nodes 2 through 6. If M chooses to locate at node 4, then, by Lemma 9 above, that is also U’s optimal location, with M capturing nodes 3, 4 and 5, leaving U to serve nodes 1, 2, 6 and 7, resulting in U’s total transportation cost being 10 and thus, an optimal profit of ]4 for U. By contrast, if M chooses to locate at node 2 at optimality, then it captures nodes 1, 2 and 3, leaving nodes 4 through 7 to be served by U. In this case, U’s optimal location by Lemma 9 is node 5 which renders U a total transportation cost of 7, resulting in U’s optimal profit being ]7.

Thus when N ≥ 4 and U is optimally located as per Lemma 9, the realized value of its minimum transportation cost, i.e., ${\mathrm{T}}_{\mathrm{U}}^{*}$ (${x}_{\mathrm{U}}^{*}$, M*), will depend on ${x}_{\mathrm{M}}^{*}$, i.e., which of the 1-interior nodes 2 through N–1 that M locates on at optimality, since that defines the nodes the 3 consecutive nodes that will not be served by U. It can then be verified that with U’s optimal location determined by Lemma 9, the precise value of U’s total transportation costs in serving its (N–3) nodes is always a quadratic and increasing function of N, regardless of where M locates, since it is always the sum of an arithmetic series with each term representing the distance of U from a node that it serves and the total number of such terms equal to the total number of nodes served by U, i.e., N–3. By contrast, as we have seen in Lemma 8, U’s optimal revenue grows only linearly with N. As such, we can expect that for a sufficiently high value of N, U’s minimum transportation costs achieved by locating optimally at ${x}_{\mathrm{U}}^{*}$ will exceed U’s maximum revenue, thereby causing U to incur a loss even at optimality and precipitating the strong first entry paradox. Therefore, we will now turn our attention to determining lower and upper bounds on the market size N that guarantee the existence of the strong first-entry paradox.

In order to do so, we begin by recalling that when N ≥ 4, the optimal location of M causes U to optimally locate at ${x}_{\mathrm{U}}^{*}$ as per Lemma 9 to serve only (N–3) non-zero-weight nodes in G(M*). Further, these (N–3) non-zero-weight nodes in G(M*) comprise of all N nodes in G except the following 3, viz., the 1-interior node that M is located on and its two immediate neighboring nodes to the left and right. Therefore, the farther M locates from U at optimality, the lower will be the realized value of U’s minimum transportation costs since the nodes not served by U will then also be those that have the highest transportation costs. In other words, ${\mathrm{T}}_{\mathrm{U}}^{*}$ (${x}_{\mathrm{U}}^{*}$, M*) is minimized when ${x}_{\mathrm{M}}^{*}$ is at node 2 or node N–1 and the optimal locations of M and U, i.e., ${x}_{\mathrm{M}}^{*}$ and${x}_{\mathrm{U}}^{*}$, exhibit maximal locational differentiation. In all these cases of maximal locational differentiation between U and M, by summing the respective arithmetic series representing U’s total transportation cost in servicing its N–3 nodes can be seen to be equal to (N²–6N + 13)/4 when N is odd and (N²–6N + 8)/4 when N is even, representing a lower bound on U’s minimum total transportation cost ${\mathrm{T}}_{\mathrm{U}}^{*}$ (${x}_{\mathrm{U}}^{*}$, M*) and hence, an upper bound on U’s optimal profit ${\pi }_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},{\mathbf{M}}^{*}\right)$. Since, by Lemma 8, U’s optimal profit is known to be ]3.5N – 10.5 when N ≥ 4, we can verify that with maximal locational differentiation U’s optimal profit is maximized to:

$${\pi }_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},\,{\mathbf{M}}^{*}\right)=\left\{\begin{array}{cc}\frac{20N-{N}^{2}-55}{4}& \mathrm{when } N \mathrm{ is odd} \\ \frac{20N-{N}^{2}-50}{4}& \mathrm{when } N \mathrm{ is even}\end{array}\right.$$

(29)

Thereafter, examining the upper bound on U’s optimal profit as given in (29) reveals that when N ≥ 4, U’s optimal profit is guaranteed to be non-positive (respectively, positive) for N ≥ 17 (respectively, N ≤ 16). Thus, we establish that when U incurs transportation expenses in accordance with location-dependent transportation cost, it will be unable to compete in large (i.e., N ≥ 17) markets, and should U also happen to be the leader, it will face the strong first-entry paradox. However, as a follower, should M choose its own optimal location to be maximally differentiated from that of U by locating at node 2 or node N–1, then U’s optimal profit is guaranteed to be positive for 4 ≤ N ≤ 16.

While we have investigated the lower bound of U’s minimum transportation cost ${\mathrm{T}}_{\mathrm{U}}^{*}$ (${x}_{\mathrm{U}}^{*}$, M*) for maximal locational differentiation, now we will now do the same for the upper bound, i.e., that optimal location of M which causes the realized value of U’s minimum transportation cost to be highest and therefore its optimal profit to be lowest. While this is not necessary for this section, that will be useful in the analysis done in the next Section 4 where we will assume M is the leader and U the follower. The discussion above on how U’s minimum transportation cost varies with the optimal location of M leads us to conclude that such maximization of U’s optimal transportation cost and corresponding minimization of U’s optimal profit occurs when the optimal locations of M and U exhibit minimal locational differentiation, i.e., when M chooses to locate at a 1-median of G and by Lemma 9, U’s optimal location coincides with that of M when N is odd or is 2 nodes to the left or right of M when N is even. By redoing the same analysis as done above for (29), it can be seen that with minimal locational differentiation, this maximum of U’s transportation expenses due to M choosing to locate on a 1-median node of G is (N²–9)/4 when N is odd and (N²–16)/4 when N is even. This, in turn, causes U’s optimal profit to be minimized to the following lower bound.

$${\pi }_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},\,{\mathbf{M}}^{*}\right)=\left\{\begin{array}{cc}\frac{14N-{N}^{2}-33}{4}& \mathrm{when }N \mathrm{ is odd}\\ \frac{14N-{N}^{2}-26}{4}& \mathrm{when }N \mathrm{ is even}\end{array}\right.$$

(30)

Examining (30) above, it can be verified that if M’s optimal location is a 1-median of G thus causing M and U’s optimal locations to be minimally differentiated, U’s optimal profit is guaranteed to be non-positive (respectively, positive) for all odd values of N ≥ 11 (respectively, N ≤ 9) and non-positive (respectively, positive) for all even values of N ≥ 12 (respectively, N ≤ 10). With this, our results pertaining to instances of first-entry paradox for U can now be summarized as follows:

Lemma 10

If U incurs transportation expenses in accordance with location-dependent transportation cost, it will fail to compete in the market because of non-positive optimal profit for all N ≥ 17, thereby resulting in the strong first entry paradox when U is the leader. For 2 ≤ N ≤ 16, whether or not U faces the strong first-entry paradox as a leader is determined by the optimal location chosen by M, which, in turn, determines the optimal location and profit of U. Further,

When the optimal locations of M and U exhibit maximal locational differentiation, U’s optimal profit is guaranteed to be positive for all 2 ≤ N ≤ 16 and hence, if U is the leader, it is guaranteed not to face the strong first entry paradox.

When the optimal locations of M and U exhibit minimal locational differentiation, U’s profit is non-positive for 2 ≤ N ≤ 3 and thereafter, for all N ≥ 11 and should U be the leader, it will face the strong first-entry paradox.

Thus, the only cases where U is guaranteed a positive profit regardless of where M locates at optimality are for 4 ≤ N ≤10 and therefore, when U is the leader, this range of N also defines the cases where U is guaranteed to not face the strong first-entry paradox.

Based on this observation along with the fact that as a follower M can observe U’s location in selecting its own and would do so to avoid causing the strong first-entry paradox for U if possible (i.e. when N ≤ 16), we can therefore assume that when U is the leader and M the follower, the optimal locations of M and U will exhibit maximal locational differentiation. With this, we can now combine the results of Lemmas 8, 9 and 10 to state.

Theorem 1 (Optimal Solutions When U is Leader, M is follower and U incurs Location-Dependent Transportation-Cost).

Given profit-maximizing duopolists U and M, competing for a common market G with N ≥ 2, where U is the leader and adopts a uniform delivered pricing policy while incurring location-dependent transportation cost and M is the follower with a mill pricing policy, it can be assumed that a solution to the Stackelberg game is as follows:

• For N = 2, U and M locate at distinct nodes, so that and both charge the optimal price of ]1 and make a profit of ]1.

• For N = 3, either M locates on node 1 and U locates on node 2 or node 3 or M locates at node 3 and U locates at node 1 or node 2. Both charge an optimal price of ]1 and earn the same optimal profit of ]1.

• For 4 ≤ N ≤ 16, M’s optimal price is ]2.5 and optimal profit is ]7.5. M and U optimally locate to be maximally differentiated as follows: either M locates at node 2 and U at node (N + 3)/2 when N is odd and node (N + 4)/2 when N is even OR M locates at node N-1 with U locating at node (N-1)/2 when N is odd and node (N-2)/2 when N is even. In such case of maximal locational differentiation, U’s optimal profit is given by (29) as

  $${\pi }_{\mathrm{U}}^{*}\left({\mathbf{U}}^{*},\,{\mathbf{M}}^{*}\right)=\left\{\begin{array}{cc}\frac{20N-{N}^{2}-55}{4}& when\;N\;is\;odd \\ \frac{20N-{N}^{2}-50}{4}& when\;N\;is\;even\end{array}\right.$$

• For N ≥ 17, U’s optimal profit is guaranteed to be non-positive and hence U will face first-entry paradox. Thus neither U nor M can enter the market.

Thus, Theorem 1 above completes the central observation from Lemma 8 to show that although U benefits from the revenue maximization standpoint by adopting the unform delivered pricing policy, should its transportation expenses be determined based on location-dependent transportation cost, it will face the strong first-entry paradox for all “large” (N ≥ 17) markets.

3.4 U* and M*: U is Leader and M is Follower (Location-Independent Transportation Cost)

The results obtained above in Section 3.3 assumed a location-dependent transportation cost incurred by U. As posited in the introduction however, many online retailers incur transportation costs that is determined solely by the number of units sold and not the location of the sale. Typically, this is achieved by using a delivery service such as a governmental Postal Service, which delivers a product to the customer for the same price throughout a given region; as an example, when Netflix started, it’s DVDs were delivered throughout USA using first class postage offered by US Postal Service.

Should U’s transportation expenses be determined as per location-independent transportation cost, its total transportation expenses would increase only and linearly with its market share. Instantiated for our model, that would imply that U’s total transportation cost would be a multiple of the nodes it captures at optimality. Thus, if the transportation cost per shipped-unit, also referred to as shipping cost per unit, that is incurred by U is a constant denoted by c, and U intends to charge a delivered price of p_U per unit to every customer, the profit that it makes per unit sold equal to (p_U–c), which is positive when c < p_U, which is what we will assume.

With this backdrop, it is readily seen that all the results of Sections 3.1 through 3.3 hold true, with the exception of the results pertaining to the optimal location of U, such as Lemma 9. In other words, making the assumption that U’s transportation expenses are determined as per location-independent transportation cost instead of location-dependent transportation cost does not alter the results pertaining to the optimal revenue earned by U and M (as given in Lemma 8) and the optimal location of M (as characterized in Lemma 3 and reiterated in Lemma 8). Further, since U’s total revenue is determined exclusively by p_U (which, in turn, also determines the number of nodes captured by U), the optimal location of U with location-independent transportation cost could be any node of G since every customer served by U imposes the same unit transportation cost of c on U regardless of the distance between U and the customer. Thus, we readily obtain.

Theorem 2 (Optimal solutions when U is leader, M is follower and U incurs location-independent transportation cost).

Given profit-maximizing duopolists U and M, competing for a common market G with N ≥ 2, where U is the leader and adopts a uniform delivered pricing policy while incurring location-independent transportation cost with a shipping cost of c per unit-sold where c < ]1 when N ≤ 3 and < ]3.5 when N ≥ 4, and M is the follower with a mill pricing policy, the solution to the Stackelberg game is as follows:

For N ≤ 3, U’s optimal location is at any node of G while it’s optimal price ${p}_{\mathrm{U}}^{*}$ = ]1. M’s optimal price = ${p}_{\mathrm{M}}^{*}$ = ]1 and its optimal location is also any node of G. Further, the optimal profit of M is ]1 and that of U is (]1–c)(N–1).

For N ≥ 4, U’s optimal location is at any node of G while its optimal price ${p}_{\mathrm{U}}^{*}$ = ]3.5. M’s optimal price = ${p}_{\mathrm{M}}^{*}$ ]2.5 and its optimal location is on any 1-interior node of G, i.e. on any node 2 through N–1. Further, the optimal profit of M is ]7.5 and that of U is (]3.5-c)(N–3).

Theorem 2 above shows that with location-independent transportation cost, as long as U’s unit transportation cost of c per shipped-unit remains below its optimal price, which is either]1 or]3.5 depending on whether N ≤ 3 or N ≥ 4, U is guaranteed to make a positive profit at optimality. This stands in contrast with the result of Lemma 10 and Theorem 1 where U incurred location-dependent transportation cost where it encounters the strong first-entry paradox for all N ≥ 17. Further, with location-independent transportation cost, U’s optimal profit increases with the size of the market given by N, whereas by contrast, M’s profit is always upper bounded and does not increase indefinitely with an increase in market size.

As a common observation between the two central results, namely Theorem 1 in Section 3.3 and the Theorem 2 in this Section 3.4, it is evident that in our model, the unit transportation cost faced by U plays a significant role in determining its net profitability at optimality regardless of what kind of transportation cost framework, location-dependent or location-independent, is assumed. It is interesting to note that this criticality of transportation costs in the determination of net profitability has been well documented in the practitioner literature for online retailers. See for example, the recent coverage in Reuters for Amazon.com (Baertlein, 2023). Further, this importance of unit transportation cost remains equally valid even when a delivery service provider can guarantee the online retailer the same unit transportation cost regardless of the location of the customer. For example, in the aforementioned case of Netflix when it used to ship DVDs using the US Postal Service, Netflix’s profitability suffered significantly when the US Postal Service raised its first-class mail service charges (Epstein 2010). This importance of shipping costs for Netflix was further reinforced when its delivery arrangement eventually triggered an audit by the U.S. Postal Service's Inspector General (Manjoo 2007) which recommended that “the Post Office force Netflix to redesign its mailers or be assessed a 17-cent per DVD surcharge,” triggering a downgrading of Netflix’s valuation by some Wall Street analysts.

4 M is the Leader and U is the Follower

This section is devoted to finding the optimal solution to our Stackelberg game when, in contrast to Section 3, M following the mill pricing policy is the leader that selects its price and location first, followed by the follower U which does the same while observing a uniform delivered pricing policy. As in Section 3, both firms are assumed to be profit maximizers and we will obtain results for both cases of transportation cost (location-dependent or location-independent) for U.

It helps to begin by noting one of the results from our prior discussion holds true even in this case when the sequence of market entry by the two firms is reversed from what it was assumed before in Section 3. The results of Lemma 6 are still true since they made no assumptions about the sequence of market entry by U and M, thereby establishing the irrelevance of U’s location to its optimal revenue even when U is the follower. Thus, even when M is the leader and U the follower, we can determine U’s optimal price ${p}_{\mathrm{U}}^{*}$ independently of its optimal location ${x}_{\mathrm{U}}^{*},$ which is the approach we will take.

As done in Section 3, but this time with U as follower, we will first determine U’s optimal price ${p}_{\mathrm{U}}^{*}$, given a location and price configuration by M of M = {x_M, p_M}. For this, we delineate U’s optimal response by considering the following two mutually exclusive cases: Case I: p_U < p_M, causing M to lose the entire market to the follower U and Case II: p_U ≥ p_M. Then, it is clear that if U sets its price as per Case I, M faces the strong first-entry paradox as a leader, thereby preventing market entry by U also, leading to zero profit for U. Thus, by our assumption made in Section 2 about the consequence of a strong first-entry paradox on the follower and thereafter applying it just as we did in Section 3.2.4 to M in selecting its own optimal location for N = 2 and N = 3, we can conclude that U will not consider Case I as an option when it follows M into the market as a follower.

This brings us to Case II, in which, even as a follower, U chooses a price p_U ≥ p_M. Recall that this was a basic assumption made in (1) for all of Section 3 when U was leader and M the follower, to avoid the trivial cases of the strong first-entry paradox for U. Next, note all results in Sections 3.2.1, 3.2.2 and 3.2.3, while derived when U was the leader and M the follower, involved U optimally responding to a given M, which in these sections happened to be M*. The sole additional assumption made was at the start of Section 3.2 that the optimal location of M, i.e., ${x}_{\mathrm{M}}^{*}$ was a Δ-interior node of G. However, since Case II assumes that p_U ≥ p_M and hence, Δ ≥ 0, we can assert that the results of Lemma 2 and, its corollary Lemma 3, still hold even though M is now the leader and U the follower. Therefore, we can state that while as a leader, M would not know what the optimal price of U would be, it will still seek to locate at a node that is always guaranteed to be a Δ-interior node of G regardless of U’s optimal price, knowing that it will be at least as high as M’s price (i.e., p_U ≥ p_M). Based on (3), we can claim that the node which satisfies this requirement is a 1-median of G. On this basis, we can assume that when M is the leader, its optimal location should always be at a 1-median of G, since it is always guaranteed to be a Δ-interior node for any value of p_U, provided that p_U ≥ p_M. Thus, with this assumption that M is located at a 1-median of G, we can assert that even in Case II with M as the leader and U the follower, the results of Lemma 2, Lemma 3, Lemma 4 and Lemma 7 still hold. However, Lemma 8 needs to be restated to accommodate M’s optimal location at a 1-median and thus is as follows.

Lemma 13

Given profit-maximizing duopolists U and M, where M is the leader who adopts a mill pricing policy and U is the follower with a uniform delivered pricing policy, if the graph G is a uniform path with N ≥ 2, it can be assumed that revenue-maximization for U and M is achieved as follows:

$$\begin{aligned}&Case \mathrm{ I}: When\;N \le 3,\;{\mathbf{M}}^{\mathbf{*}}=\left\{{p}_{\mathrm{M}}^{*}=\right]1\;and\;{x}_{\mathrm{M}}^{*} {\mathrm{ is }}\;a\;1-median\;of\;G\}\;and\;{p}_\mathrm{U}^{*}=]1.\;Further,\;{\pi }_{\mathrm{M}}^{*}\left({\mathbf{M}}^{\mathbf{*}},\,{\mathbf{U}}^{\mathbf{*}}\right)\\&=]1 {\mathrm{ and }} {R}_{\mathrm{U}}^{*}{(\mathbf{M}}^{\mathbf{*}},{\mathbf{U}}^{\mathbf{*}})=](N- 1).\end{aligned}$$

$$\begin{aligned}&Case \mathrm{ II}: When\;N \ge 4,\;{\mathbf{M}}^{\mathbf{*}}=\left\{{p}_\mathrm{M}^{*}=\right]2.5\;and\;{x}_{\mathrm{M}}^{*}\;is\;a\;1-median\;of\;G\}\;and\;{p}_{\mathrm{U}}^{*}=]3.5.\;Further,\\&{\pi }_{M}^{*}\left({\mathbf{M}}^{\mathbf{*}},\,{\mathbf{U}}^{\mathbf{*}}\right)=]7.5\mathrm{ and }{R}_{\mathrm{U}}^{*}\left({\mathbf{M}}^{\mathbf{*}},\,{\mathbf{U}}^{\mathbf{*}}\right)=]3.5(N- 3)\end{aligned}$$

Thus, both M and U are guaranteed a positive revenue for all values of N.

Lemma 13 having established M and U’s optimal prices and M’s optimal location when M is the leader and U the follower, we now turn to the determination of the optimal location of U, i.e., ${x}_{\mathrm{U}}^{*}$, for this assumed sequence of market entry, beginning with the more complex case where U incurs transportation expenses in accordance with location-dependent transportation cost. Since as a leader M is assumed to always locate at a 1-median of G at optimality, U’s optimal location will continue to be determined as per Lemma 9, which still holds because it did not assume any given sequence of market entry. Thereafter, using the observations we made in Section 3.2.4 for N = 2 and N = 3 but this time with U as the follower who has an opportunity to observe M’s optimal location prior to selecting its own, and using Lemma 9, we can state that.

Lemma 14

Given profit-maximizing duopolists U and M, where M is the leader and adopts a mill pricing policy and U is the follower with a uniform delivered pricing policy that incurs transportation expenses in accordance with location-dependent transportation cost, for N ≥ 2, we can assume that the optimal locations of M and U are as follows:

For N ≤ 3,

if N = 2, ${x}_{\mathrm{M}}^{*}$ = node 1 (respectively, node 2) and ${x}_{\mathrm{U}}^{*}$ is at node 2 (respectively, node 1)

if N = 3, ${x}_{\mathrm{M}}^{*}$ is at node 2 and U does not enter the market because its optimal profit is non-positive

For N ≥ 4,

if N is odd, ${x}_{\mathrm{M}}^{*}$ = ${x}_{\mathrm{U}}^{*}$ is at the unique 1-median of G = node (N + 1)/2.

if N is even, and both nodes N/2 and (N + 2)/2 are 1-median of G, ${x}_{\mathrm{M}}^{*}$ is at node N/2 (respectively, node (N + 2)/2) and ${x}_{\mathrm{U}}^{*}$ is at node (N + 4)/2 (respectively, node (N-2)/2)

As Lemma 14 makes evident, when N ≥ 4 with M as a leader and U the follower, and U operating as per location-dependent transportation cost, their respective optimal locations exhibit minimal locational differentiation and hence, U’s optimal profit earned as a follower is determined by (30). This also implies that the results in Lemma 10 for minimal locational differentiation will now apply in the determination of whether U’s optimal profit is positive.

With all the above results in place, we can now synthesize them to present the solution to our model when M is the leader and U the follower. Since we have assumed two different frameworks for assessing U’s transportation expenses (location-dependent vs location independent), we present the respective analogs of Theorems 1 and 2 as Theorems 3 and 4 below.

Theorem 3 (Optimal solutions when M is leader, U is follower and U incurs location-dependent transportation cost).

Given profit-maximizing duopolists U and M competing for a market in the shape of a uniform path G with N ≥ 2, where M is the leader and adopts a mill pricing policy and U is the follower with a uniform delivered pricing policy and incurs transportation expenses in accordance with the location-dependent transportation cost, the solution to the Stackelberg game is as follows:

• For N = 2, U and M optimally locate at distinct nodes, and both charge the optimal price of ]1, with each making an optimal profit of ]1.

• For N = 3, M locates at the node 2, the 1-median of G, charging a price of ]1, thereby ensuring that U’s optimal profit is non-positive and deterring entry by U. Firm M’s profit as a monopolist is ]3.

• For 4 ≤ N ≤ 10, at optimality, M locates at the 1-median of G and charges a price = ]2.5 while U charges a price of ]3.5. U’s optimal location is node (N + 1)/2 when N is odd. When N is even, it is node (N + 4)/2 in case M is located at N/2 or at node (N-2)/2 if M is located at node (N + 2)/2. M’s optimal revenue is ]7.5 and U’s optimal profit is as per (30) and as follows

  $${\pi }_{\mathrm{U}}^{*}\left({\mathbf{M}}^{*},\,{\mathbf{U}}^{*}\right)=\left\{\begin{array}{cc}\frac{14N-{N}^{2}-33}{4}& when\;N\;is\;odd\\ \frac{14N-{N}^{2}-26}{4}& when\;N\;is\;even\end{array}\right.$$

• For N ≥ 11, M deters entry by U by locating at a 1-median of G and charging a price of ]2.5. U’s optimal profit is non-positive, and U does not enter the market, rendering M as the monopolist and earning monopolistic profit = ]2.5N.

Theorem 4 (Optimal solutions when M is leader, U is follower and U incurs location-independent transportation cost).

Given profit-maximizing duopolists U and M, competing for a common market G with N ≥ 2, where M is the leader and adopts a mill pricing policy and U the follower with uniform delivered pricing policy while incurring location-independent transportation cost with a cost of c per unit-sold where c < ]1 when N ≤ 3 and < ]3.5 when N ≥ 4, the solution to the Stackelberg game is as follows:

• When N ≤ 3, M’s optimal location is at a 1-median of G while it’s optimal price = ]1. U’s optimal price = ]1 and its optimal location is at any node of G. Further, the optimal profit of M is ]1 and that of U is (]1–c)(N–1).

• When N ≥ 4, M’s optimal location is at a 1-median of G while it’s optimal price = ]2.5. U’s optimal location is at any node of G and its optimal price = ]3.5 Further, the optimal profit of M is ]7.5 and that of U is (]3.5-c)(N–3).

The central results of Sections 3 and 4, as stated in Theorems 1, 2, 3 and 4 are summarized as follows in Table 1.

Table 1 Solutions to duopoly Stackelberg game with different pricing policies when the market is a uniform path

5 Nash Equilibrium Location and Price for U and M

Thus far we have assumed that the two competitors M and U enter the market sequentially and the leader chooses its location and price optimally to maximize its own profit with the knowledge that having done so, the follower would then enter the market to do the same. This therefore led our competitive location and price problem to be modeled as a Stackelberg game. By contrast, in a simultaneous game, it would be assumed that both competitors M and U simultaneously select their respective locations and prices. For such simultaneous games, a well-known solution is the Nash Equilibrium which prescribes a choice of location and price for each of the two competitors such that neither have an incentive to unilaterally change either their own location or price (or both). Thereafter, we note that if a certain configuration of location and price choices for both competitors is the optimal solution regardless of the sequence of market entry by the two competitors, then it must represent Nash equilibria for our problem. In doing so, an examination of the summary of results provided in Table 1 allows us to delineate several instances of Nash Equilibrium for our problem. Note that in producing the results below, we avoid the degenerate cases where either competitor makes a non-positive profit at optimality; and as seen in Table 1, this only occurs for U when its transportation expenses are determined as per location-dependent transportation cost.

We begin with the case where U operates with location-dependent transportation cost. From here onwards, we will use the subscript “NE” to indicate that the parameter/variable in question belongs to a Nash equilibrium. Let us examine the following solution when 4 ≤ N ≤ 10:

If N is odd,

$${\mathbf{M}}_{\mathbf{N}\mathbf{E}}=\{{x}_{\mathrm{M}} \mathrm{ is at node } (N+1)/2,\,{p}_{\mathrm{M}}=]2.5\} {\mathrm{ and }} {\mathbf{U}}_{\mathbf{N}\mathbf{E}} =\{{x}_{\mathrm{U}} \mathrm{ is at node } (N+1)/2 {\mathrm{ and }}{p}_{\mathrm{U}}=]3.5\}.$$

If N is even,

$$\begin{aligned}&{\mathrm{Either }} {\mathbf{M}}_{\mathbf{N}\mathbf{E}} = \{{x}_{\mathrm{M}} \mathrm{ is at node } N/2,\,{p}_{\mathrm{M}}=]2.5\} {\mathrm{ and }} {\mathbf{U}}_{\mathbf{N}\mathbf{E}} = \{{x}_{\mathrm{U}} \mathrm{ is at node } (N+4)/2 {\mathrm{ and }} {p}_{\mathrm{U}}=]3.5\},\\&{\mathrm{or }} {\mathbf{M}}_{\mathbf{N}\mathbf{E}}=\{{x}_{\mathrm{M}} \mathrm{ is at node } (N+2)/2,\,{p}_{\mathrm{M}}=]2.5\} {\mathrm{ and }} {\mathbf{U}}_{\mathbf{N}\mathbf{E}}=\{{x}_{\mathrm{U}} \mathrm{ is at node } (N-2)/2 {\mathrm{ and }} {p}_{\mathrm{U}}=]3.5\}\end{aligned}$$

(31)

Then, it is readily observable by examining Table 1 that the above configuration of M_NE and U_NE in (31) represents a Nash Equilibrium should both M and U enter the market simultaneously and U incurs transportation expenses in accordance with location-dependent transportation cost.

As for the case when U operates with location-independent transportation cost, with a unit shipping cost of c that is less than < ]1 when N ≤ 3 and less than ]3.5 when N ≥ 4, consider the following solution:

$$\begin{aligned}\mathrm{When }N&\le3\\&{\mathbf{M}}_{\mathbf{N}\mathbf{E}}=\{{x}_{\mathrm{M}} \mathrm{ is at any node of }G,\,{p}_{\mathrm{M}}=]1\} {\mathrm{ and }} {{\mathbf{U}}}_{\mathbf{N}\mathbf{E}}=\{{x}_{\mathrm{U}} =\mathrm{any node of } G {\mathrm{ and }} {p}_{\mathrm{U}}=]1\}\end{aligned}$$

When N ≥ 4

$$\begin{aligned}&\mathrm{When}\;N\;\ge4\\&\qquad\qquad{\mathbf{M}}_{\mathbf{N}\mathbf{E}}=\{{x}_{\mathrm{M}} \mathrm{ is at any node 2 through } N-1\mathrm{ of }G,\,{p}_{\mathrm{M}}= ]2.5\} {\mathrm{ and }}{\mathbf{U}}_{\mathbf{N}\mathbf{E}}=\{{x}_{\mathrm{U}} \mathrm{ is at any node of }\\&G {\mathrm{ and }} {p}_{\mathrm{U}}=]3.5\}\end{aligned}$$

(32)

Then, it is readily observable by examining Table 1 that the above configuration of M_NE and U_NE in (32) represents a Nash Equilibrium should both M and U enter the market simultaneously and U incurs transportation expenses in accordance with location-independent transportation cost. This allows us to conclude:

Theorem 5 (Nash equilibrium solutions when M and U enter the market simultaneously).

Given profit-maximizing duopolists U and M, competing for a common market G with N ≥ 2, where both enter the market (i.e., decide their respective locations and prices) simultaneously,

• When U incurs transportation expenses in accordance with location-dependent transportation cost and 4 ≤ N ≤ 10, (31) represents a Nash Equilibrium.

• When U incurs transportation expenses in accordance with location-independent transportation costs, (32) represents a Nash Equilibrium.

6 Implications and Future Research

In this paper, we modelled the duopolistic competition between an online retailer and a brick-and-mortar (i.e., physical store) retailer that compete for common market by modeling the online retailer as a firm with uniform delivered pricing policy and the physical store as a firm with a mill pricing policy. Both firms were assumed to be profit maximizing through an appropriate choice of location and price. The common market that these duopolists compete for is assumed to be given by a uniform path, i.e., a path whose node weights and arc lengths are assumed to be unity. In observance of real-worlds practices, we assumed two different types of transportation costs faced by the online retailer: either dependent on the relative location of the firm in the customers or independent of it (and thus decided exclusively based on market share). Beginning with the framework of a Stackelberg, i.e., sequential game, we consider the two separate cases when one of the firms is the leader that chooses its location and price first, followed by the follower firm which does the same. Optimal location and price strategies are analytically derived for all cases and cases where the leader faces first entry paradox or can become a monopolist by strategically deterring entry by the follower are delineated. Thereafter, the solutions to the sequential game allow us to identify some Nash Equilibrium solutions to the simultaneous game.

The final results of our paper, as summarized in Table 1 for the sequential game and Lemma 17 for the simultaneous game, allow for a number of interesting conclusions, some of which have implications for real-world competition between an online retailer and a physical store.

• 1. While our model imposes no upper limits on the prices than can be charged by either retailer, it turns out that at optimality, should both competitors be profitable, neither charge arbitrarily high prices and in fact, this price does not increase with the size of the market – this is true for the sequential as well as the simultaneous game. This upper bounding of the optimal prices charged by the two competing retailers is a direct result of the competition between them as well as the asymmetry between their pricing policies. As comparison, it is interesting to note that this invariance of the optimal prices to market size is not observed for the Nash Equilibrium in the continuous location analog of our model studied in Eiselt (1991). Further, for the sequential version of modelling this competition, we notice that when the online retailer’s transportation costs are decided based on the distance of each of its customers from its location, the optimal locations by the two competitors exhibit maximal locational differentiation when the online retailer is the leader but exhibit minimal locational differentiation when the physical store is the leader.

• 2. In cases where the online retailer can profitably compete in the market, as is the case when, similar to Amazon.com, the online retailer’s transportation expenses are independent of the relative locations of this retailer and its customers and therefore determined solely by its sales volume, upper-bounding on the optimal profit of the brick-and-mortar retailer can be a source of significant competitive pressure. In practice this would imply that the brick-and-mortar retailer would have to ensure that its optimal profit, which is invariant to the size of the total market, always stays sufficiently larger than its own production and/or operational costs to ensure that its net profit meets necessary benchmarks to stay in business. An interesting instance of the adverse impact of this competitive pressure has been documented in the recent empirical study conducted in Chava et al. (2024) who have demonstrated that over the past two years, the presence of online retailing has had a significantly adverse impact on the brick-and-mortar stores in USA.

• 3. The online retailer’s profitability is saliently determined by the type of transportation costs it incurs. When this is based on the distance of each of its customers from its location, the online retailer is often unable to compete in the market regardless of whether it is the leader who enters the market first (faces the strong first entry paradox) or a follower that responds to the existing location and pricing decisions of a physical store (gets shut out of the market due to the leader adopting an entry-deterrent location strategy). This occurrence of non-positive profit for the online retailer is guaranteed to happen for large markets (i.e., N ≥ 11 or 17, depending on the location of the physical store) regardless of whether it is a leader or follower in the case where its transportation costs are determined based on its distance from customers. By contrast, should the transportation cost incurred by the online retailer be independent of the distance of the customers from its location and depend solely on sales volume, the online retailer is guaranteed a positive profit at optimality regardless of whether it’is a leader or a follower; further, this profit grows linearly with the size of the market. Thus, if the choice of the type of transportation cost is also a decision variable under the control of the online retailer, it is better for it to adopt the second transportation cost model. We posit that this is why so many online retailers such as Amazon.com frequently partner with delivery services such as UPS, US Postal Service, etc., to ensure that all customers in a specified region they serve receive their products for a standard shipping cost charged to the customer or the online retailer. Overall, this underscores the importance of transportation costs for online retailers as an important determinant of optimal profit when competing with physical stores, as also reinforced in the practitioner literature (Epstein 2010, Baertlein 2023).

• 4. As for the brick-and-mortar retailer, i.e., a physical store that is in competition with an online retailer, Table 1 indicates the following additional result. If the brick-and-mortar retailer has a choice in selecting the sequence of market entry, then it should choose to be a leader and enter the market first while adopting an entry-deterrent location strategy of locating at the “center” of the market. Doing so keeps its optimal profit at least as high as when it would be a follower and further, makes it a monopolist in large markets when the online retailer incurs transportation costs based on the distance of each of its customers from its location. Note that for most retail sectors, brick-and-mortar stores existed before the emergence of online retailing, thus making them the equivalent of the market leader in our model. In that regard, this result of location the center of the market nears a resemblance to the classical spatial agglomeration results of Hotelling (1929). Also note that most brick-and-mortar stores are natural leaders, as many of them have existed for a long time, while online stores are a more recent phenomenon.

  In contrast, should the brick-and-mortar retailer only have the option to enter the market as a follower with the knowledge that the online retailer incurs transportation costs based on the distance of each of its customers from its location, then the optimal location of the brick-and-mortar store is to be maximally differentiated from that of the online retailer towards one end of the market. An instance of this in practice has been observed in the stores located on some of the remote islands of Scotland in Schiffling et al. (2015), who observed that such stores often offer additional services such as housing the local post office, to their customers.

Future research on the general topic of duopolistic competition with different pricing policies could take one of several directions. An immediate avenue of extension of our model is to consider one where the two competing retail firms are allowed to locate multiple facilities within a market; motivation behind this emanates from the fact that when the online retailer incurs location-dependent transportation cost, its optimal profit is positive for “small” markets but not for “large” ones. The second avenue for future research could incorporate demand uncertainty into the model and seek optimal strategies under the same. Additionally, another enhancement of our model could look at duopolistic competition between an online and a physical store retailer but with multiple products and/or customer demand being determined not solely by cost to the customer but other factors such as attractiveness, convenience, etc. Finally, it would also be interesting to investigate our basic model but within the context of other commonly used pricing policies by retailers such as: dynamic pricing, value-based pricing, penetration pricing, etc.