Abstract
This paper considers the two-player location game in a closed-loop market with quantity competition. Based on the Cournot and Hotelling models, a circle model is established for a closed-loop market in which two players (firms) play a location game under quantity competition. Using a two-stage (location-then-quantity) pattern and backward induction method, the existence of subgame-perfect Nash equilibria is proved for the location game in the circle model with a minimum distance transportation cost function. In addition, sales strategies are proposed for the two players for every local market on the circle when the players are in the equilibrium positions. Finally, an algorithm for simulating the competitive dynamics of the closed-loop market is designed, and two numerical simulations are provided to substantiate the effectiveness of the obtained results.
Highlights
Game theory, the science of strategy, was pioneered by John von Neumann when he proved the basic principles in 1928 [1]
In [14], the authors analyzed the relationship between consumer density and the equilibrium locations of the Hotelling model and noted that the equilibrium locations are closer if the density is higher
To research the two-player location game in the closed-loop market, we consider the shape of the market as a circle. e location game with quantity competition between two players in a circular market is established
Summary
The science of strategy, was pioneered by John von Neumann when he proved the basic principles in 1928 [1]. In. Complexity [16], the authors discussed the Hotelling duopoly model with network effects and brand loyalty and showed that a pure strategy price equilibrium exists if the transportation costs are linear functions. It was proved that a pure strategy price-location Nash equilibrium exists in the Hotelling duopoly model under general conditions on the cost-of-location function in [18]. In [26], the authors investigated spatial Cournot competition in a circular city and showed that nonmaximum dispersion is the unique location equilibrium when duopoly firms deliver products in different transportation modes. Motivated by the above discussion, in this paper, we investigate the two-player location game in a closed-loop market. We give some notations used in this paper. x ∈ [0, 1) stands for the position of the local market in a closed-loop market. x1 and x2 represent the positions of firms A and B, respectively. p is the price of the product, while q1 and q2 are the quantities produced by the two players. e function ci represents the transportation cost per unit product from the position xi to x, where i 1, 2
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