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1. Introduction
In 1928, John von Neumann proved the basic principle of game theory [1]. Nowadays, game theory is not only a new field of modern mathematics but also an important subject of operational research. The game theory mainly studies the interaction between the mathematical theory and the incentive structure for studying the competitive phenomena [2]. It is one of the standard analysis tools for economics and is widely applied in finance, securities, international relations, computer science, political science, and many other fields [3–8].
As an important research object in the field of industrial organization and supply chain management, the location problem attracts attention more and more. In 1929, the game theory was applied to the positional problem by Hotelling and the classic Hotelling model was constructed [9]. In this model, it is assumed that consumers are uniformly distributed in a linear street, and there are two companies of the same size which determine their locations to maximize the profits. In the subsequent decades, various position problems developed from the classical model were considered, and many results were obtained. The result of d’Aspremont et al. [10] shows that the price equilibrium solution is ubiquitous for the modified Hotelling model and that the seller tends toward the difference of maximization. The Cournot competition with uneven distribution of consumers in a linear city model was studied in [11], and a necessary condition of agglomeration equilibrium was obtained. The author in [12] claimed that if there is no pure strategy equilibrium, the Hotelling model exhibits a mixed-strategy equilibrium. The Hotelling spatial competition model was extended by the author in [13] from three aspects: shape of the demand curve, the number of firms, and type of space. In [14], the Hotelling model for duopolistic competition with a class of utility functions was examined. In the meantime, when the curvature of the utility functions is high enough, the existence of an equilibrium was proven. The relationship between the equilibrium location of the Hotelling model and the consumer density was analyzed by the authors in [15], and it was pointed out that the higher the consumer density, the closer the equilibrium position. In [16], the author investigated the existence of equilibrium states in the Hotelling model in the case of
In fact, the market usually includes a variety of complex traffic networks. In order to accurately reflect the actual market, complex places such as spokes and circles are considered by many researchers. Based on the quadratic transportation cost function, the author in [24] considered the location space as a circular road and proved the existence and uniqueness of a unique price equilibrium in multiplayer location game. Furthermore, as for the circle market, the authors in [25] considered the problems of nonexistence and existence of an equilibrium for a location-price game. In [26], the authors explored a linear and circular model with spatial Cournot competition and examined the dependence between demand density and location equilibrium. For multiple participants in a circular market, the authors in [27] claimed that the unique equilibrium position is equidistantly distributed. By using a spoke model, the nonlocalised spatial competition was considered by the authors in [28], and the influence of the number of enterprises on the equilibrium price was also analyzed. In addition, an explicit partial game complete set of equilibrium positions was induced by the author in [29] by assuming that crossing finite roads and transport costs proportional to the distance square root. In the spoke model, the location choices and spatial price discrimination were considered by the author in [30].
In this paper, strongly motivated by the above discussion, we developed a location game in the spoke market, where two players make price competition in the market. The main problem is how to choose the optimal point on the spokes for each player as its location such that its profit is maximized.
2. Descriptions of the Spoke Model
In terms of geometry, the market is made up of
[figure omitted; refer to PDF]
A customer on a spoke
Suppose that each player prices the products. For any customer, the products from one of the players are sold for the same price. Also, the transportation costs are paid by the customer. Let
The net utility of a customer at
3. Main Results
It is natural that the customer located at
Lemma 1.
Suppose that firms 1 and 2 are both on spoke 1 and
(a)
(b)
(c)
(d)
Lemma 2.
Suppose that firm 1 is on spoke 1, firm 2 is on spoke 2, and
(a)
(b)
(c)
(d)
The structure of the game played by the two firms is as follows:
(1) Location stage: each firm determines its location
(2) Price stage: each firm chooses the price strategy
In the following, the backward induction method will be employed to solve the game. In the second stage, for given positions on the spokes, firms simultaneously determine their prices to ensure maximum profits in current location. In the following, we only discuss the case of the two firms in the same spoke. For the other case, where they are in different spokes, the discussion is very similar. Obviously, their profits
[figure omitted; refer to PDF]
According to Lemma 1 and by solving equation (6), we have the price equilibrium:
In the first stage, firms simultaneously determine their location based on optimal price strategy (7). Noting that
Summarizing the above discussion, we obtain the main result.
Proposition 1.
In a spoke market with transportation cost function (1) and net utility (2), if the two players develop the location game with price competition, the equilibrium location is that
Remark 1.
In the proposition, the equilibrium location reflects the principle of maximum differentiation, which makes the two players avoid vicious price war in the same place.
To illustrate the dynamic behaviors of two players in the market and verify the validity of the results, we design the computer simulation algorithm in the following. At the beginning, each player randomly chooses its position
Therefore, the algorithm for simulating dynamic behaviors in the spoke market can be given as follows:
(i) Step 1: initialize the move step size
(ii) Step 2: compute
(iii) Step 3: compute
(iv) Step 4: if
(v) Step 5: compute
(vi) Step 6: if
4. Conclusion
The location game with price competition in a spoke market is established for two players, where the transportation cost is linear. Employing a two-stage approach, the location equilibrium of the location game is proved for the considered market. The obtained result shows that one player should be at the center point of the market while another one should be at the extreme point when the location game is in equilibrium. This paper considers the geometry of the market as spokes. In fact, the geometry of the market could be very complex in the real world. Therefore, we will mainly consider the location games on complex grids in future.
Acknowledgments
This study was supported in part by the Science and Technology Research Program of Chongqing Municipal Education Commission (no. KJQN201900701), in part by the Basic and Frontier Research Projects of Chongqing (no. cstc2018jcyjAX0606), in part by the Team Building Project for Graduate Tutors in Chongqing (no. JDDSTD201802), and in part by the Group Building Scientific Innovation Project for Universities in Chongqing (no. CXQT21021).
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Abstract
This paper investigates the location game of two players in a spoke market with linear transportation cost. A spoke market model has been proposed, which is inspired by the Hotelling model and develops two-player games in price competition. Using two-stage (position and price) patterns and the backward guidance method, the existence of price and location equilibrium results for the position games is proved.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer