Hotelling’s Duopoly in a Two-Sided Platform Market on the Plane

: Equilibrium in a two-sided market represented by network platforms on the plane and heterogeneous agents is investigated. The advocated approach is based on the duopoly model which implies a continuum of agents of limited size on each side of the market and examines the agents’ heterogeneous utility with the Hotelling speciﬁcation. The exact values were found for the equilibrium in the case of duopoly in a two-sided market with two platforms on the plane. The dependence of the platforms’ beneﬁts on network externalities was investigated. The problem of the optimal location of platforms in the market was considered.


Introduction
Digital economy has formed a paradigm of accelerated economic development. A central position in it is occupied by network technologies, which have led to the establishment of network markets. Network markets are set up on network platforms, which are a novel business element in the modern economy, providing benefits for both the platform's operator and the community.
The two-sided market involving a platform is a network type of market, where users belonging to two different groups interact. Consumption by any one of the groups generates external effects on the other group. Assuming a network effect exists between the two sides of the market (network externalities), the problem of optimizing the platform's profit can be formulated. Reference [1] suggested a model of a monopoly platform in a two-sided market with the total number of agents on each side equaling one. The agents have different transport costs to access the platform via Hotelling specification. As an illustration, imagine that agents in the model are sellers and customers in an online store, job-seekers and employers in a job centre, men and women in a dance floor, and so forth. Following the Hotelling specification [2], agents compare their costs before taking the decision about visiting the platform. Our study assumes agents to be heterogeneous and demand to be endogenous, and investigates the platform's behavior and revenues under these assumptions.
A duopoly in a two-sided in-plane platform market is considered, equilibrium in such a market and the dependence of platform revenues on network externalities are determined. For the case of identical agents and identical platforms, the conditions were detected enabling services in such agents, while Section 4 finds the solution for the model with identical groups of agents and platforms occupying different positions in the market; Section 5 investigates the optimal location of platforms; Section 6 finds the equilibrium in the general case of the pricing problem; the article is completed with the conclusions and visions for further research in the area.

Description of the Model
Suppose there are two non-intersecting groups of agents in the market: group 1 (also known as group of buyers), and group 2 (also known as group of sellers). Agents in both groups, whose size equals a unit, are distributed uniformly over a plane in the square S = [−1, 1] × [−1, 1]. The location of the members of groups 1 and 2 in the square S is defined by the points (x 1 , y 1 ) and (x 2 , y 2 ) respectively, where −1 ≤ x i ≤ 1 and −1 ≤ y i ≤ 1, i = 1, 2. Agents from both groups meet in platforms I and I I, located at the boundary of the square S in diametrically opposite points, for example, I(−1, 0) and I I(1, 0).
The following notations are introduced: n (j) i is the size of the group i in the platform j (i = 1, 2; j = I, I I); p (j) i is the price of visiting the platform j by an agent of the group i (i = 1, 2; j = I, I I); α, β are the degrees of the effect that the number of second (first) group agents in the platform has on the payoff of the first (second) group; t i is the effect of the transport costs for visiting the ith platform where t i > 0 (i = 1, 2).
In a two-sided market with two platforms, agents in both groups choose between them based on the utility they can derive from visiting the respective platform. For group 1 agents, the utility of visiting the platform I, situated in the point (−1, 0) has the form and the utility of visiting the platform I I, in the point (1, 0) is In the above expressions, the first term captures the utility from network externality of the other side of the market, the second term is the cost of price the agent has to pay to access the platform, and the third term is essentially the agent's heterogeneous transport cost of accessing the platform, via Hotelling specification.
The boundary between regions of the market for group 1 is found from the equation u (I) 2t 1 , given that n Figure 1). Parameter s 1 is the coordinate value at which the hyperbola crosses the abscissa axis. In the symmetric case, when the market is divided equally, s 1 = 0. If s 1 > 0 (s 1 < 0), then platform I (II) is more attractive for agents. The market being divided into two regions, the respective numbers of group 1 agents visiting platforms I and I I are The utilities of group 2 agents from visiting platforms I and I I are determined similarly, and are equal, respectively − (x 2 − 1) 2 + y 2 2 · t 2 , and the boundary between market regions for group 2 is 2t 2 , given that n n (I I) 2 We thus arrive at the pricing game for the two platforms I and I I, which respectively serve

Identical Platforms and Identical Groups of Agents
Consider the case of identical platforms and identical agents, where the effect of the other group's size on the payoff and the transport costs of agents of the two groups are equal, that is, α = β and t 1 = t 2 = t, and the service costs of the platforms are equal, g 1 = g 2 = g. In this case, it suffices to find the optimal solution for one of the platforms, for example, for the platform I. The price equilibrium is found from the first-order optimality condition ∂H (I) ∂p (I) 1 = 0, which has the following form: From Equations (1) and (2) we get that , from where we find that It follows from the symmetry of the problems that in the equilibrium the prices set by both platforms for agents of groups 1 and 2 have to be equal, that is, p Furthermore, the size of both groups in the platforms should be equal, that is, n Thus, Equations (6) and (7) in the form , are substituted into (5) to find Derive from here the price in the equilibrium, which is equal to Observe that competition exists only if p ≥ g, or if Test the sufficient conditions for the maximum of the function H I p I 1 , p I I 2 . To this end, find the . Considering the symmetry of the problem we have It follows from the symmetry of the problem that ∂n and ∂n , and also that has a maximum at (p, p), where p is found from Formula (8). Hence, the following theorem is valid for the case of identical platforms and identical groups of agents. Theorem 1. In the Hotelling model for a two-sided platform market on a plane with identical groups of agents, competitive service will take place given that , and the price in the equilibrium in this case will have the form (8).

Identical Platforms and Different Groups of Agents
Suppose agents in the two groups differ in their parameters. Remark that owing to their symmetric location in the market the platforms are identical. In this case, the price equilibrium is derived from the first-order optimality condition for each group of agents in one of the platforms, for example, platform I, which has the form The equation From the Equations (1)-(4), we find that It follows from the symmetry of the problem that in the equilibrium the prices set by the platforms for each group of customers should be equal, although they may differ for different groups, that is, It follows that the expression (9) has the form Similar reasoning for equation = 0, which has the form leads to the equation From (10) and (11), we find the prices for the groups in the equilibrium Hence, the following theorem is valid for the case with identical platforms and different groups of agents. Theorem 2. In the Hotelling model for a two-sided platform market on the plane with identical platforms, competitive service will take place given that and the service prices in the equilibrium in this case will have the form (12).

Different Platforms and Identical Groups of Agents
Consider the model of a two-sided market with different platforms located in points (a, 0) and (b, 0) of the square S, and with identical groups of agents, which have the same parameters of the power of the group size effect and the transport costs. Assume for definiteness that a ≤ b. In this case, the utility for group 1 agents from visiting the platform I in the point (a, 0) has the form u (I) while from visiting the platform I I in the point (b, 0) it is The boundary between regions of the market for group 1 is determined from the equation 1 , which is solved to get that 2 ) + p The utilities of group 2 agents from visiting platforms I and I I are determined similarly, and equal, respectively − (x 2 − b) 2 + y 2 2 · t, and the boundary between market regions for group 2 is found from the equation of the form Since the groups of agents in this model are identical, the price equilibrium can be found from the first-order optimality condition for any one of the groups of agents (e.g., the first one) in both platforms: It follows from the symmetry of the problem that in the equilibrium the prices in the platforms should be equal for each group of customers, although they may differ for different groups, that is, , and the size of the groups in each platform is equal, that is, and ∂n (I) 1 Denote the expression (17) as the function D(s). Then, it follows from (6)- (7) and (14)- (15) that The resultant expressions are substituted into system (13): where From system (18) we find that and subsequently the optimal prices are found using the formulas Theorem 3. In the Hotelling model for a two-sided platform market on the plane with identical agents and different platforms, service prices in the equilibrium satisfy Equations (16), (19) and (20).
From Equations (16) and (20), we find that wherefore Together with (19), this leads to the equation for finding the parameter s in the equilibrium: Thus, optimal prices in the pricing game can be found by finding the parameter s from Equation (22), then finding the size of the group of agents in the first platform using Formula (21), and substituting the resultant parameters into Formula (20). The results of the numerical calculations are shown in Table 1. The table shows that when the second platform is shifted to the center, s shifts to the left, which reduces the market for platform I. This means that customers in the center are getting closer to platform I I.

Optimal Location of Platforms
Observe the results of numerical calculations in Table 1. When the location of the first platform I is fixed in the position a = −1 and the position of the second platform b changes from 1 to 0.6 the second player's payoff H (I I) first grows and then declines. The maximum is reached in the position b = 0.8. This means that if the position of the first platform is fixed, the second player should put his/her platform in the position b = 0.8.
If, however, the second player puts his/her platform in that position, the first player may also relocate. Table 2 shows the calculations simulating this situation. Let the players switch roles. The first player is now in the fixed position a = −0.8, while the second player's position is variable. It follows from the numerical calculations shown in Table 2 that his/her optimal position in this situation is b = 0.9. Continuing this sequence of the best responses, we arrive at the equilibrium location of platforms in the market a = −0.85, b = 0.85.

Different Platforms and Different Groups of Agents
Consider the case where different platforms serve different groups of agents, which are described by the parameters α = β and t 1 = t 2 . In this case, the payoffs of both platforms have the form Therefore, where the respective derivatives satisfy the following equations: The system of Equation (30) yields formulas for finding the optimal prices in the equilibrium: Thus, the following theorem is valid for the case of different platforms and different agents.
from where the number of agents in the first platform is determined using the formulas n (I) The Equations (36) and (37) together with (33) and (34) lead to the system of equations for finding the parameters s 1 and s 2 in the equilibrium: Thus, the values of the parameters s 1 and s 2 are found from the system of Equation (38), then the sizes of the first and second groups on the first platform are found using the Formulas (36) and (37), and the optimal prices the platforms charge different groups are determined by the Formula (35). The results of numerical calculations are shown in Table 3 for different values of the parameters α and β. Table 3. Numerical results for α = 1 and β = 0.7 when t 1 = t 2 = 1 and g 1 = g 2 = 0.

Conclusions
This paper investigated the structure of prices in the equilibrium in a two-sided market on the plane. Service is provided in two platforms for heterogeneous agents and endogenous demand. Agents are uniformly distributed in a square. The target functions are the platform's maximum profit. The exact expressions were found for the prices which depend on the cost structure and network externalities. Analytical expressions were derived for equilibrium prices. The numerical experiments for finding the equilibrium prices in the market for different parameters of the problem and different locations of platforms in the market were described. The study was concerned with the case where agents of both groups were uniformly distributed over the market. We plan to study this problem in application to different distributions of agents in space.
The paper demonstrated the validity of the problem of the optimal platform location in the market. In the future, the optimal platform location problem will be considered in more detail for various distributions of agents over the market and for a greater number of competing platforms.