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1. Introduction
The Bertrand competition was introduced as a model describing an economic game by the famous scientist Joseph Louis François Bertrand [1]. Such game was used in the literature to simulate interactions among firms (players) who set prices as their strategies. Bertrand claimed but not formalized that firms set prices than quantities as their outputs in the competition would take place with prices and marginal costs that are equal. Comparing Cournot games (on where players set quantities as their strategies) with Bertrand, we report few studies on Bertrand games in the literature. For instance, the minimal differentiation principle was tested in a Bertrand competition in [2], and it was found that it was applied to spatial competition. The capacity precommitment as an entry-deterring device was reexamined in [3] on the price competition of Bertrand–Edgeworth model. In [4], a repeated game of price was studied to detect that when the number of competing firms increased above two competitive players, the outcomes become less likely. For the case of spatial discrimination, a comparison between Bertrand and Cournot in the context of spatial duopoly was analyzed in [5]. In [6], a Bertrand game with an uncertain number of active firms was analyzed. In the case of differentiated quantities, a price setting game has been analyzed in a continuous time scale with random demands [7].
All the above reported studies handled the price setting game statically. There are few studies of Bertrand games about the complex dynamic characteristics of their equilibrium points. We report some of them from the literature as follows. In [8], a Bertrand game whose players adopted the bounded rationality mechanism to update their prices investigated that bifurcation and chaos may occur when the speed of adjustment of players also increases. Ma and Wu [9] introduced the game of Bertrand triopoly with bounded rational players and investigated that the time delay may not improve the stability region of the game. Ma and Sun studied the multiteam Bertrand game in [10]. In [11], a Bertrand game with delayed bounded rational players was considered. In that study, it was pointed out that lagged structure may affect the stable region of the stable state. For substitutable products in a supply chain [12], the game of price setting analyzed the influences of different competitive strategies on optimum decisions of prices. In [13], an advertising cost-dependent demand for a two-echelon supply chain was investigated, and the obtained results showed that it was beneficial for the firms to use different wholesale pricing strategies. With differentiated products, the complex dynamic characteristics of a Bertrand game were discussed by Fanti et al. in [14]. They have deduced that the interior fixed point can be destabilized when differentiation between firms increased, and therefore, chaotic attractors with complicated structure may arise. Other studies on Bertrand and Cournot–Bertrand games and more information on their complex dynamic behaviors have been reported in the literature [15, 16]. We should also highlight some recent studies on the Bertrand game. For instance, in [17], the bounded rationality and naive expectation mechanisms were used to study a dynamic model of the quantum Bertrand game with differentiated products. In [18], a Bertrand game in the downstream market was investigated. A triopoly Bertrand game based on differentiated products was investigated in [19]. A Cournot–Bertrand game was introduced and studied in [20]. In [21], necessary and sufficient conditions for a unique Nash equilibrium in a standard Bertrand duopoly game based on homogeneous products were analyzed.
There are different types of adjustment mechanisms that have been adopted to model the maps describing such games. For instance, there are the gradient-based mechanisms such as bounded rationality. Bounded rationality has been intensively used for this purpose in the literature [22–27]. This mechanism depends on the estimation of players’ profits for updating their outputs. If the profits are increased (or decreased), this will affect the prices whether they will be increased (or decreased). There are also other mechanisms that have been used to formulate the Bertrand and Cournot games such as the local monopolistic approximation (LMA) mechanism [28] and tit-for-tat approach [29]. In the paper at hand, we recall the bounded rationality mechanism to formulate our Bertrand game.
The current paper belongs to the research direction of the Bertrand game with players adopting the bounded rationality, and one of the players possesses information about the price its opponent adopts. We model the game by introducing a discrete-time dynamical map whose variables are prices, and it is nonlinear. The main results in this paper focus on many things. We concern with the destabilization of the interior equilibrium and the routes responsible for that. We investigate the flip and Neimark–Sacker bifurcations under different types of parameter sets causing chaotic behavior of the game’s map. Depending on a rich numerical analysis, we show different types of quite complicated attractive basins of periodic cycles. Furthermore, the noninvertibility aspect is numerically discussed for the map.
The paper presents the map describing the game in Section 2. In Section 3, we discuss the main results which include investigation of the stability of the equilibrium points of the map. It also gives local and global analyses about the interior equilibrium point and the routes by which it can be destabilized. This section includes numerical simulation about the basins of attraction and their complicated structure. We also prove in the same section that the game’s map is a noninvertible map of type
2. The Model
The model in this paper consists of two firms (players) whose products are differentiated, and their prices are derived from the utility function introduced by Singh and Vives [30] as follows:
(1)
(2)
(3)
(4) Using a budget constraint
Problem (2) is a maximization problem with a consumer’s budget constraint.
We should mention here that
Now, we consider a Bertrand game whose players adopt a gradient-based mechanism in order to update their prices according to the following discrete dynamical map:
It should be noted that the second equation in (5) differs from the first because the second player in the game has some advantages. He knows by some way the price of the first player next time step. This kind of asymmetric information gives with (4) and (5) the following discrete time map:
Map (6) is nonlinear, and due to the asymmetric information possessed by the second firm, it becomes more complex. Without this asymmetric information, we get the following map:
In this paper, we will compare between maps (6) and (7) in order to see the influences of asymmetric information on the complex dynamic characteristics of the game’s equilibrium points.
3. Main Results of Map (6)
3.1. Fixed Points and Stability
Map (6) has four fixed points:
The points
The stability of the above fixed points depends on the eigenvalues
(1) A fixed point is a locally stable attracting node if its eigenvalues satisfy
(2) A fixed point is an unstable repelling node if its eigenvalues satisfy
(3) A fixed point is a saddle point if its eigenvalues satisfy
(4) A fixed point is a nonhyperbolic point if its eigenvalues satisfy
Proposition 1.
The fixed point
Proof.
The Jacobian at this point becomes
Proposition 2.
The point
Proof.
The Jacobian at
The conditions
Proposition 3.
At
Proof.
The Jacobian at those parameters and
3.2. Dynamic Analysis via Numerical Simulation
Here, we investigate the influences of parameter
3.2.1. Set 1
In this set, we assume the following values:
[figures omitted; refer to PDF]
Using the parameter values in set 1, it is observed through numerical experiments that when the marginal costs are equal (
[figures omitted; refer to PDF]
[figures omitted; refer to PDF]
3.2.2. Set 2
In this set, we assume the following values:
[figures omitted; refer to PDF]
We should highlight here that the role of parameter
[figures omitted; refer to PDF]
3.3. Map (6) versus Map (7)
Here, we compare the two models in order to investigate the influences of asymmetric information in their dynamics. Assuming that
[figures omitted; refer to PDF]
The previous analysis shows that the structure of basins is quite complicated, and this requires to study extra characteristics of these maps. Setting
Complicated nonlinear map (18) makes us to study its noninvertibility characteristic by substituting
This means that a point
This means if we substitute
4. Conclusion
The current paper has introduced and studied a Bertrand duopoly game with asymmetric price information. A gradient-based mechanism that is the bounded rationality has been adopted by game’s players to update their prices in the next time step. The interior equilibrium point of the proposed game has been calculated, and its stability conditions have been discussed. We have compared our model with the classical Bertrand model without asymmetric information as both players in the two models use the same gradient-based mechanism. Our contributions have shown that the asymmetric price information has broadened the stability region of the interior point. Furthermore, the structure of basins of attraction for periodic cycles due to this information has shown quite complicated structure for those basins in comparison with those obtained for classical Bertrand. Moreover, the local and global analysis performed have shown that the proposed game’s map is noninvertible and belongs to the type of
Acknowledgments
This work was supported by the Research Supporting Project Number (RSP-2020/167), King Saud University, Riyadh, Saudi Arabia.
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Abstract
We study a Bertrand duopoly game in which firms adopt a gradient-based mechanism to update their prices. In this competition, one of the firms knows somehow the price adopted by the other firm next time step. Such asymmetric information of the market price possessed by one firm gives interesting results about its stability in the market. Under such information, we use the bounded rationality mechanism to build the model describing the game at hand. We calculate the equilibrium points of the game and study their stabilities. Using different sets of parameter values, we show that the interior equilibrium point can be destabilized through flip and Neimark–Sacker bifurcations. We compare the region of stability of the proposed model with a classical Bertrand model without asymmetric information. The results show that the proposed game’s map is noninvertible with type
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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