# Asymmetric Information on Price Can Affect Bertrand Duopoly Players with the Gradient-Based Mechanism

Authors: S. S. Askar
Journal: Mathematical Problems in Engineering
 (2020)

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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 typeZ0−Z2 or Z1−Z3, while the classical model is of type Z0−Z2 only. This explains the quite complicated basins of attraction given for the proposed map.


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## Body

## 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 Section2. 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 Z0−Z2 or Z1−Z3 depending on the nature of preimages coexisted. Furthermore, in this section, we compare the influences of asymmetric price information with the classical model without such information. Finally, the obtained results are concluded in Section 4.


## 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)Uq1,q2=aq1+q2−12q12+q22+2bq1q2,q1,q2>0,where a is a constant and b is restricted in the interval −1,1. Both q1 and q2 are the quantities produced by each firm. The following characterize this utility:(1)
U is concave which means ∂2U/∂qi2=−1<0;i=1,2. Economically, the marginal utility of each quantity is decreasing.(2)
∂2U/∂q1q2=−b≠0. This means that the marginal utility of quantity depends on the other.(3)
Uθq1,θq2≠θUq1,q2. This means it is not homogeneous.(4)
Using a budget constraint∑i=12piqi≤m, we get(2)MaxUq1,q2,s.t.∑i=12piqi≤m.Problem (2) is a maximization problem with a consumer’s budget constraint. pi denotes the price of quantity qi, and m is a constant. Solving (2) gives the following demand functions:(3)q1=a1+b−p11−b2+bp21−b2,q2=a1+b−+bp11−b2−p21−b2.We should mention here thatb denotes the horizontal differentiation degree. If b=0, we have two monopolistic firms, and b=1 represents two identical firms. Negative values for b refer to complementarity between the firms. Assuming linear costs, Ciqi=c1qi, where ∂Ci/∂qi=ci is a constant marginal cost. Now, the profit of each firm becomes(4)π1=p1−c1q1=p1−c1a1+b−p11−b2+bp21−b2,π2=p2−c2q1=p1−c1a1+b−+bp11−b2−p21−b2.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:(5)p1,t+1=p1,t+kp1,t∂π1p1,t,p2,t∂p1,p2,t+1=p2,t+kp2,t∂π2p1,t+1,p2,t∂p2.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:(6)Tp1,p2:p1,t+1=p1,t+kp1,t1−b2a1−b+c1−2p1,t+bp2,t,p2,t+1=p2,t+kp2,t1−b2a1−b+c2+bp1,t−2p2,t+k2bp1,tp2,t1−b22a1−b+c1−bc2−2p1,t+2bp2,t.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:(7)T1p1,p2:p1,t+1=p1,t+kp1,t1−b2a1−b+c1−2p1,t+bp2,t,p2,t+1=p2,t+kp2,t1−b2a1−b+c2+bp1,t−2p2,t.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:(8)E0=0,0,E1=0,a1−b+c22,E2=a1−b+c12,0,E3=p^1,p^2,where(9)p^1=14kb2A−B1k+B12k2+Dk+A2,p^2=14kb2A+B2k+B12k2+Dk+A2,A=1−b24−b2,B1=b2a−c2b−a−c1,B2=b2a+c2b−a−c1,D=2b21−b2ab3−c2b3+3ab2−c1b2−4a−4c1.The pointsE0,E1, and E2 are called corner fixed points, while E3 is an interior fixed point. It is clear that E0,E1, and E2 are positive since −1<b<1. Due to the complicated form of the interior point, we will discuss its positivity and stability by assuming values for the parameters a,b,c1,c2, and k. The Jacobian of map (6) becomes(10)Jm=J11J12J21J22,where(11)J11=1+ka1−b+c1−4p1+bp21−b2,J12=bkp11−b2,J21=kbp21−b2+bk2p2a1−b+c1−bc2+2bp2−4bp11−b22,J22=1+ka1−b+c2+bp1−4p21−b2+bp1a1−b+c2+bp1−4p2k21−b22.The stability of the above fixed points depends on the eigenvaluesλ1 and λ2 of the Jacobian (10). These eigenvalues have the following properties:(1)
A fixed point is a locally stable attracting node if its eigenvalues satisfyλ1,2<1(2)
A fixed point is an unstable repelling node if its eigenvalues satisfyλ1,2>1(3)
A fixed point is a saddle point if its eigenvalues satisfyλ1<1 and λ2>1 (or λ1>1 and λ2<1)(4)
A fixed point is a nonhyperbolic point if its eigenvalues satisfyλ1=1 and λ2≠1 (or λ1≠1 and λ2=1)Proposition 1.
The fixed pointE0 is an unstable repelling point.Proof.
The Jacobian at this point becomes(12)JE0=1+ka1−b+c11−b2,0,0,1+ka1−b+c21−b2,whose eigenvalues are λi=1+ka1−b+ci/1−b2,i=1,2. Since b<1 and ci>0, we have λi>1. This completes the proof.Proposition 2.
The pointE1 is a stable point provided that 0<ka1−b+c2/1−b2<2 and 0<ka1−b2+b+2c1+bc2/1−b2<4.Proof.
The Jacobian atE1(13)JE1=1+ka1−b2+b+2c1+bc221−b2,0,bka1−b+c21−b2+ka+c1−ab2k21−b22,1−ka1−b+c21−b2,has two real eigenvalues(14)λ1=1−ka1−b+c21−b2,λ2=1−ka1−b2+b+2c1+bc221−b2.
The conditionsλ1<1 and λ2<1 complete the proof. The same proof is for E2.Proposition 3.
Ata=2,b=0.2,c1=1,c2=0.3, and k=0.5, we have E3=1.41028,1.10282. It is locally asymptotically stable.Proof.
The Jacobian at those parameters andE3 is(15)JE3=−0.469040.14690−0.044276−0.11502,and the eigenvalues are λ1≈−0.4496 and λ2≈−0.13446 with λ1,2<1, and then E3 is stable. It is easy to see that the determinant of JE3 equals δ=0.060454<1 which means that map (6) is dissipative. This numerical proof of the stability of E3 gives rise to more studies in order to investigate and get more insights on the regions on where this fixed point can be destabilized. This will be discussed in the simulation given in the next section.


### 3.2. Dynamic Analysis via Numerical Simulation

Here, we investigate the influences of parameterk on the dynamics of map (6). This makes us to assume different sets of parameter values as follows.


#### 3.2.1. Set 1

In this set, we assume the following values:a=2,b=0.2,c1=1,c2=0.3, and k=0.5. This set indicates that we have two firms with different marginal costs (c1>c2). In this set, we have E3=1.41028,1.10282. This requires to investigate the impact of one parameter of those parameters on the stability of this point. Let us assume that parameter k will be taken as the bifurcation parameter. The first numerical experiment displayed in Figure 1(a) shows the effect of bifurcation parameter k on the stability of E3. One can see that it is locally stable for all the values of k except for k≈0.704 on which the first period-2 cycle is born. Periodic cycles may possess complicated basins of attraction, and this makes us to analyze these basins for cycles. At k=0.88, a period 2 with its basins is given in Figure 1(b) with the fixed point E3. From now on, we will fix the colors as follows. The grey color denotes the divergent and nonconvergent points, while the other two colors refer to the basins of the plotted cycle. Increasing k further to 0.915 of a period-4 cycle rises with quite complicated basins of attraction than those of period-2 cycle. Figure 1(d) shows complicated basins for period-8 cycle at k=0.924. Further increase in the parameter gives rise to more complicated attractive basins of higher periodic cycles, for instance, at k=0.95, the dynamic changes into two disconnected chaotic areas which gather together to become one chaotic attractor at k=0.995. Figures 1(e) and 1(f) present those chaotic behaviors of map (6).Figure 1
(a) Flip bifurcation diagram on varyingk. (b) Basins of attraction of the period-2 cycle at k=0.88. (c) Basins of attraction of the period-4 cycle at k=0.915. (d) Basins of attraction of the period-8 cycle at k=0.924. (e) The phase plane of the two chaotic attractors at k=0.95. (f) The phase plane of the chaotic attractor at k=0.995. Other parameter values are a=2,b=0.2,c1=1,andc2=0.3.
(a)(b)(c)(d)(e)(f)Using the parameter values in set 1, it is observed through numerical experiments that when the marginal costs are equal (c1=c2), another dynamic behavior is emerged. At c1=c2=1, the fixed point E3 can be destabilized via Neimark–Sacker bifurcation. The Jacobian (10) at these parameter values becomes(16)JE3=−0.806590.18066−0.13619−0.75027,with complex eigenvalues λ1,2=−0.77843±0.15431i and λ1,2=0.79358<1, and then E3 is locally stable. It is easy to see that the determinant of JE3 equals δ=0.62976<1 which means that map (6) is also dissipative. Now, we analyze the influence of k on the stability of this point. When we increase k further, dynamic behaviors arise such as spiral and closed curves around the fixed point. Figure 2(b) gives those behaviors for different values of k. When k reaches 0.774, a period-17 cycle appears with quite complicated basins of attraction as given in Figure 2(c). As k increases further, high-periodic cycles coexist with complicated basins until the dynamic turns into a closed chaotic attractor. This chaotic attractor is born at k=0.79, and its phase plane is displayed in Figure 2(d). This chaotic attractor becomes very complicated as k increases to 0.818, and its phase plane is shown in Figure 3(a). This unexpected dynamic gives no predictions about the evolution of map (6) and makes the price game with asymmetric information more complicated. The asymmetry and nonlinearity of this map give more complicated behaviors that we should report here. At k=0.827, a six-piece chaotic attractor is emerged. The phase plane of this dynamic behavior around the fixed point is given in Figure 3(b). In this figure, we enlarge one piece of this chaotic behavior. These six pieces are gathered together as k increases further to form one chaotic and complicated attractor in Figure 3(c). At k=0.877, a period-7 cycle coexists. As given in Figure 3(d), it has very complicated attractive basins. Increasing k to 0.88, another period cycle is raised. This period is of cycle 14, and its basins are plotted in Figure 3(e). Finally, for any other increase for k, the dynamic behavior of map (6) turns to one chaotic attractor. Figure 3(f) displays the phase plane of this chaotic attractor.Figure 2
(a) Neimark–Sacker bifurcation diagram on varyingk. (b) The phase plane of spiral and closed curves around the fixed point for different values of k. (c) Basins of attraction of the period-17 cycle at k=0.774. (d) The phase plane of the chaotic attractor at k=0.79. (e) Basins of attraction of the period-19 cycle at k=0.795. (f) The phase plane of the chaotic attractor at k=0.8. Other parameter values are a=2,b=0.2,c1=1,andc2=1.
(a)(b)(c)(d)(e)(f)Figure 3
(a) The phase plane of the chaotic attractor atk=0.818. (b) The phase plane of six-piece chaotic attractors at k=0.827. (c) The phase plane of the chaotic attractor at k=0.835. (d) Basins of attraction of the period-7 cycle at k=0.877. (e) Basins of attraction of the period-14 cycle at k=0.88. (f) The phase plane of the chaotic attractor at k=0.89. Other parameter values are a=2,b=0.2,c1=1,andc2=1.
(a)(b)(c)(d)(e)(f)


#### 3.2.2. Set 2

In this set, we assume the following values:a=b=c1=c2=0.5 and k=1.6. The fixed point at set 2 becomes E3=0.5,0.5, and its Jacobian is(17)JE3=−1.13330.53333−0.60444−0.56444,with complex eigenvalues λ1,2=−0.84889±0.49139i and λ1,2=0.98086<1, and then E3 is locally stable. It is easy to see that the determinant of JE3 equals δ=0.96207<1 which means that map (6) is again dissipative. As previously stated, we study the effects of k on the stability of the fixed point. It is not a repeated analysis for the above simulation; it gives other interesting and hidden dynamics of map (6). Increasing k makes the fixed point unstable due to Neimark–Sacker bifurcation. For instance, at different values of k, we give in Figure 4(a) different closed curves around the fixed point. Those closed invariant curves are changed to a period-5 cycle as parameter k increases to 1.83. It is plotted in Figure 4(b) with its attractive basins. Another interesting observation is the coexistence of period-5 and period-10 cycles together at k=1.9. They have quite complicated basins of attraction which are given in Figure 4(c). Increasing k further makes those cycles convert into five unconnected and complicated chaotic areas. The phase plane of these chaotic areas is given in Figure 4(d) with the fixed point. These chaotic areas become larger and more complicated as k increases to 1.95. They are displayed in Figure 4(e). After that, the dynamic behavior of map (6) becomes chaotic as k increases to 1.97, as given in Figure 4(f).Figure 4
(a) The phase plane of closed invariant curves at different values ofk=0.818. (b) Basins of attraction of the period-5 cycle at k=1.83. (c) Basins of attraction of period-5 and period-10 cycles at k=1.9. (d) The phase plane of five-piece chaotic attractors at k=1.92. (e) The phase plane of five-piece chaotic attractors at k=1.95. (f) The phase plane of the chaotic attractor at k=1.97. Other parameter values are a=0.5,b=0.5,c1=0.5,andc2=0.5.
(a)(b)(c)(d)(e)(f)We should highlight here that the role of parameterk is affected by parameter b. Figure 5(a) shows that the fixed point can be destabilized due to flip bifurcation on varying parameter k and for the same set of parameter values except for b=−0.55. We give some numerical simulations of the dynamic behavior of k at negative values for b. At k=1.51, Figure 5(b) shows the coexistence of unconnected period-3 cycles. It has an attractive basins given in Figure 5(b) which are more complicated than those obtained previously. At b=−0.5, the effect of k on the fixed point makes it unstable due to flip bifurcation. Figure 5(c) shows a stable fixed point for all k until the coexistence of the unconnected period-6 cycle. This cycle has basins of attraction which are given in Figure 5(d).Figure 5
(a) The flip bifurcation on varyingk at b=−0.55. (b) The basins of attraction of the period-3 cycle at k=1.51andb=−0.55. (c) The flip bifurcation on varying k at b=−0.5. (d) The basins of attraction of the period-6 cycle at k=1.66andb=−0.5. The other parameter values are a=0.5,c1=0.5,andc2=0.5.
(a)(b)(c)(d)


### 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 thata=2,b=0.2,c1=1,andc2=0.3, Figure 6(a) shows the impact of k on map (7). Comparing Figure 1(a) with Figure 6(a), we see that the asymmetric information possessed by the second firm broadens the stability region for both players simultaneously. Numerical experiments show that the basins of attraction detected for map (6) are quite complicated than those of map (7). Furthermore, we show by experiments that, at sets 1 and 2, the stability region of the fixed point for map (5) is bigger than that detected for the fixed point of map (7). For more details about the properties of classic model (7), refer [31]. In Figure 6(b), we present the attractive basins of the period-4 cycle for map (7). Comparing these basins with others given in Figure 1(c), we get that the dynamics of map (6) are more complex than those of map (7). Therefore, the future prediction for map (6) becomes more complicated.Figure 6
(a) Bifurcation diagram for map (7) on varying k. (b) The basins of attraction for the period-4 cycle of map (7) at k=0.70. Other parameters are a=2,b=0.2,c1=1,andc2=0.3.
(a)(b)The previous analysis shows that the structure of basins is quite complicated, and this requires to study extra characteristics of these maps. Settingp1,t+1=p´1 and p2,t+1=p´2 in (6), where ′ refers to the time evolution, we get(18)Tp1,p2:p´1=p1+kp11−b2a1−b+c1−2p1+bp2,p´2=p2+kp21−b2a1−b+c2+bp1−2p2+k2bp1p21−b22a1−b+c1−bc2−2p1+2bp2.Complicated nonlinear map (18) makes us to study its noninvertibility characteristic by substituting a=2,b=0.2,c1=1,c2=0.3, and k=0.88 in (18) as follows:(19)Tp1,p2:p´1=p12.2833−1.8333p1+0.18333p2,p´2=p21.6417+0.40853p1+0.067222p1p2−1.8333p2−0.33611p12.This means that a pointp1,p2 can be characterized by many rank-1 preimages. Those preimages are mapped in p1,p2 in one iteration. Still, map (19) is in a complicated form to solve it algebraically with respect to the variables p1 and p2. Here, we follow the same procedure carried out in [28] by calculating the number of preimages of points belonging to the vertical axis (p1=0). Therefore, substituting the point 0,p´2 in (13), we get(20)p1=0orp1=1.245459+0.1p2.This means if we substitutep1=0 in the second part of (13), we get two real preimages for the point 0,p´2 provided that p´2∈−∞,0.36753; otherwise, we have no preimages. This concludes that T is a noninvertible map of type Z0−Z2, and hence, the phase plane is divided into two regions that are Z0 and Z2. On the contrary, substituting p1=1.245459+0.1p2 in the second part gives that the point 0,p´2 has three real preimages provided that p´2∈0.37050,∞; otherwise, we have only one real preimage. This means the preimages of the point in this case divide the phase plane into two regions Z1 and Z3, and hence, the map T is of type Z1−Z3. The same procedure can be used for the points belonging to the horizontal axis (p2=0) and take the form p´1,0. It is worth to mention here that both critical lines LC and LC−1 are obtained at those parameter values, but they have long forms to be written here or even to be plotted. We have performed the same procedure for map T1 for the points 0,p´2 and p´1,0. The results show that map T1 is of type Z0−Z2 only.


## 3.1. Fixed Points and Stability

Map (6) has four fixed points:(8)E0=0,0,E1=0,a1−b+c22,E2=a1−b+c12,0,E3=p^1,p^2,where(9)p^1=14kb2A−B1k+B12k2+Dk+A2,p^2=14kb2A+B2k+B12k2+Dk+A2,A=1−b24−b2,B1=b2a−c2b−a−c1,B2=b2a+c2b−a−c1,D=2b21−b2ab3−c2b3+3ab2−c1b2−4a−4c1.The pointsE0,E1, and E2 are called corner fixed points, while E3 is an interior fixed point. It is clear that E0,E1, and E2 are positive since −1<b<1. Due to the complicated form of the interior point, we will discuss its positivity and stability by assuming values for the parameters a,b,c1,c2, and k. The Jacobian of map (6) becomes(10)Jm=J11J12J21J22,where(11)J11=1+ka1−b+c1−4p1+bp21−b2,J12=bkp11−b2,J21=kbp21−b2+bk2p2a1−b+c1−bc2+2bp2−4bp11−b22,J22=1+ka1−b+c2+bp1−4p21−b2+bp1a1−b+c2+bp1−4p2k21−b22.The stability of the above fixed points depends on the eigenvaluesλ1 and λ2 of the Jacobian (10). These eigenvalues have the following properties:(1)
A fixed point is a locally stable attracting node if its eigenvalues satisfyλ1,2<1(2)
A fixed point is an unstable repelling node if its eigenvalues satisfyλ1,2>1(3)
A fixed point is a saddle point if its eigenvalues satisfyλ1<1 and λ2>1 (or λ1>1 and λ2<1)(4)
A fixed point is a nonhyperbolic point if its eigenvalues satisfyλ1=1 and λ2≠1 (or λ1≠1 and λ2=1)Proposition 1.
The fixed pointE0 is an unstable repelling point.Proof.
The Jacobian at this point becomes(12)JE0=1+ka1−b+c11−b2,0,0,1+ka1−b+c21−b2,whose eigenvalues are λi=1+ka1−b+ci/1−b2,i=1,2. Since b<1 and ci>0, we have λi>1. This completes the proof.Proposition 2.
The pointE1 is a stable point provided that 0<ka1−b+c2/1−b2<2 and 0<ka1−b2+b+2c1+bc2/1−b2<4.Proof.
The Jacobian atE1(13)JE1=1+ka1−b2+b+2c1+bc221−b2,0,bka1−b+c21−b2+ka+c1−ab2k21−b22,1−ka1−b+c21−b2,has two real eigenvalues(14)λ1=1−ka1−b+c21−b2,λ2=1−ka1−b2+b+2c1+bc221−b2.
The conditionsλ1<1 and λ2<1 complete the proof. The same proof is for E2.Proposition 3.
Ata=2,b=0.2,c1=1,c2=0.3, and k=0.5, we have E3=1.41028,1.10282. It is locally asymptotically stable.Proof.
The Jacobian at those parameters andE3 is(15)JE3=−0.469040.14690−0.044276−0.11502,and the eigenvalues are λ1≈−0.4496 and λ2≈−0.13446 with λ1,2<1, and then E3 is stable. It is easy to see that the determinant of JE3 equals δ=0.060454<1 which means that map (6) is dissipative. This numerical proof of the stability of E3 gives rise to more studies in order to investigate and get more insights on the regions on where this fixed point can be destabilized. This will be discussed in the simulation given in the next section.


## 3.2. Dynamic Analysis via Numerical Simulation

Here, we investigate the influences of parameterk on the dynamics of map (6). This makes us to assume different sets of parameter values as follows.


### 3.2.1. Set 1

In this set, we assume the following values:a=2,b=0.2,c1=1,c2=0.3, and k=0.5. This set indicates that we have two firms with different marginal costs (c1>c2). In this set, we have E3=1.41028,1.10282. This requires to investigate the impact of one parameter of those parameters on the stability of this point. Let us assume that parameter k will be taken as the bifurcation parameter. The first numerical experiment displayed in Figure 1(a) shows the effect of bifurcation parameter k on the stability of E3. One can see that it is locally stable for all the values of k except for k≈0.704 on which the first period-2 cycle is born. Periodic cycles may possess complicated basins of attraction, and this makes us to analyze these basins for cycles. At k=0.88, a period 2 with its basins is given in Figure 1(b) with the fixed point E3. From now on, we will fix the colors as follows. The grey color denotes the divergent and nonconvergent points, while the other two colors refer to the basins of the plotted cycle. Increasing k further to 0.915 of a period-4 cycle rises with quite complicated basins of attraction than those of period-2 cycle. Figure 1(d) shows complicated basins for period-8 cycle at k=0.924. Further increase in the parameter gives rise to more complicated attractive basins of higher periodic cycles, for instance, at k=0.95, the dynamic changes into two disconnected chaotic areas which gather together to become one chaotic attractor at k=0.995. Figures 1(e) and 1(f) present those chaotic behaviors of map (6).Figure 1
(a) Flip bifurcation diagram on varyingk. (b) Basins of attraction of the period-2 cycle at k=0.88. (c) Basins of attraction of the period-4 cycle at k=0.915. (d) Basins of attraction of the period-8 cycle at k=0.924. (e) The phase plane of the two chaotic attractors at k=0.95. (f) The phase plane of the chaotic attractor at k=0.995. Other parameter values are a=2,b=0.2,c1=1,andc2=0.3.
(a)(b)(c)(d)(e)(f)Using the parameter values in set 1, it is observed through numerical experiments that when the marginal costs are equal (c1=c2), another dynamic behavior is emerged. At c1=c2=1, the fixed point E3 can be destabilized via Neimark–Sacker bifurcation. The Jacobian (10) at these parameter values becomes(16)JE3=−0.806590.18066−0.13619−0.75027,with complex eigenvalues λ1,2=−0.77843±0.15431i and λ1,2=0.79358<1, and then E3 is locally stable. It is easy to see that the determinant of JE3 equals δ=0.62976<1 which means that map (6) is also dissipative. Now, we analyze the influence of k on the stability of this point. When we increase k further, dynamic behaviors arise such as spiral and closed curves around the fixed point. Figure 2(b) gives those behaviors for different values of k. When k reaches 0.774, a period-17 cycle appears with quite complicated basins of attraction as given in Figure 2(c). As k increases further, high-periodic cycles coexist with complicated basins until the dynamic turns into a closed chaotic attractor. This chaotic attractor is born at k=0.79, and its phase plane is displayed in Figure 2(d). This chaotic attractor becomes very complicated as k increases to 0.818, and its phase plane is shown in Figure 3(a). This unexpected dynamic gives no predictions about the evolution of map (6) and makes the price game with asymmetric information more complicated. The asymmetry and nonlinearity of this map give more complicated behaviors that we should report here. At k=0.827, a six-piece chaotic attractor is emerged. The phase plane of this dynamic behavior around the fixed point is given in Figure 3(b). In this figure, we enlarge one piece of this chaotic behavior. These six pieces are gathered together as k increases further to form one chaotic and complicated attractor in Figure 3(c). At k=0.877, a period-7 cycle coexists. As given in Figure 3(d), it has very complicated attractive basins. Increasing k to 0.88, another period cycle is raised. This period is of cycle 14, and its basins are plotted in Figure 3(e). Finally, for any other increase for k, the dynamic behavior of map (6) turns to one chaotic attractor. Figure 3(f) displays the phase plane of this chaotic attractor.Figure 2
(a) Neimark–Sacker bifurcation diagram on varyingk. (b) The phase plane of spiral and closed curves around the fixed point for different values of k. (c) Basins of attraction of the period-17 cycle at k=0.774. (d) The phase plane of the chaotic attractor at k=0.79. (e) Basins of attraction of the period-19 cycle at k=0.795. (f) The phase plane of the chaotic attractor at k=0.8. Other parameter values are a=2,b=0.2,c1=1,andc2=1.
(a)(b)(c)(d)(e)(f)Figure 3
(a) The phase plane of the chaotic attractor atk=0.818. (b) The phase plane of six-piece chaotic attractors at k=0.827. (c) The phase plane of the chaotic attractor at k=0.835. (d) Basins of attraction of the period-7 cycle at k=0.877. (e) Basins of attraction of the period-14 cycle at k=0.88. (f) The phase plane of the chaotic attractor at k=0.89. Other parameter values are a=2,b=0.2,c1=1,andc2=1.
(a)(b)(c)(d)(e)(f)


### 3.2.2. Set 2

In this set, we assume the following values:a=b=c1=c2=0.5 and k=1.6. The fixed point at set 2 becomes E3=0.5,0.5, and its Jacobian is(17)JE3=−1.13330.53333−0.60444−0.56444,with complex eigenvalues λ1,2=−0.84889±0.49139i and λ1,2=0.98086<1, and then E3 is locally stable. It is easy to see that the determinant of JE3 equals δ=0.96207<1 which means that map (6) is again dissipative. As previously stated, we study the effects of k on the stability of the fixed point. It is not a repeated analysis for the above simulation; it gives other interesting and hidden dynamics of map (6). Increasing k makes the fixed point unstable due to Neimark–Sacker bifurcation. For instance, at different values of k, we give in Figure 4(a) different closed curves around the fixed point. Those closed invariant curves are changed to a period-5 cycle as parameter k increases to 1.83. It is plotted in Figure 4(b) with its attractive basins. Another interesting observation is the coexistence of period-5 and period-10 cycles together at k=1.9. They have quite complicated basins of attraction which are given in Figure 4(c). Increasing k further makes those cycles convert into five unconnected and complicated chaotic areas. The phase plane of these chaotic areas is given in Figure 4(d) with the fixed point. These chaotic areas become larger and more complicated as k increases to 1.95. They are displayed in Figure 4(e). After that, the dynamic behavior of map (6) becomes chaotic as k increases to 1.97, as given in Figure 4(f).Figure 4
(a) The phase plane of closed invariant curves at different values ofk=0.818. (b) Basins of attraction of the period-5 cycle at k=1.83. (c) Basins of attraction of period-5 and period-10 cycles at k=1.9. (d) The phase plane of five-piece chaotic attractors at k=1.92. (e) The phase plane of five-piece chaotic attractors at k=1.95. (f) The phase plane of the chaotic attractor at k=1.97. Other parameter values are a=0.5,b=0.5,c1=0.5,andc2=0.5.
(a)(b)(c)(d)(e)(f)We should highlight here that the role of parameterk is affected by parameter b. Figure 5(a) shows that the fixed point can be destabilized due to flip bifurcation on varying parameter k and for the same set of parameter values except for b=−0.55. We give some numerical simulations of the dynamic behavior of k at negative values for b. At k=1.51, Figure 5(b) shows the coexistence of unconnected period-3 cycles. It has an attractive basins given in Figure 5(b) which are more complicated than those obtained previously. At b=−0.5, the effect of k on the fixed point makes it unstable due to flip bifurcation. Figure 5(c) shows a stable fixed point for all k until the coexistence of the unconnected period-6 cycle. This cycle has basins of attraction which are given in Figure 5(d).Figure 5
(a) The flip bifurcation on varyingk at b=−0.55. (b) The basins of attraction of the period-3 cycle at k=1.51andb=−0.55. (c) The flip bifurcation on varying k at b=−0.5. (d) The basins of attraction of the period-6 cycle at k=1.66andb=−0.5. The other parameter values are a=0.5,c1=0.5,andc2=0.5.
(a)(b)(c)(d)


## 3.2.1. Set 1

In this set, we assume the following values:a=2,b=0.2,c1=1,c2=0.3, and k=0.5. This set indicates that we have two firms with different marginal costs (c1>c2). In this set, we have E3=1.41028,1.10282. This requires to investigate the impact of one parameter of those parameters on the stability of this point. Let us assume that parameter k will be taken as the bifurcation parameter. The first numerical experiment displayed in Figure 1(a) shows the effect of bifurcation parameter k on the stability of E3. One can see that it is locally stable for all the values of k except for k≈0.704 on which the first period-2 cycle is born. Periodic cycles may possess complicated basins of attraction, and this makes us to analyze these basins for cycles. At k=0.88, a period 2 with its basins is given in Figure 1(b) with the fixed point E3. From now on, we will fix the colors as follows. The grey color denotes the divergent and nonconvergent points, while the other two colors refer to the basins of the plotted cycle. Increasing k further to 0.915 of a period-4 cycle rises with quite complicated basins of attraction than those of period-2 cycle. Figure 1(d) shows complicated basins for period-8 cycle at k=0.924. Further increase in the parameter gives rise to more complicated attractive basins of higher periodic cycles, for instance, at k=0.95, the dynamic changes into two disconnected chaotic areas which gather together to become one chaotic attractor at k=0.995. Figures 1(e) and 1(f) present those chaotic behaviors of map (6).Figure 1
(a) Flip bifurcation diagram on varyingk. (b) Basins of attraction of the period-2 cycle at k=0.88. (c) Basins of attraction of the period-4 cycle at k=0.915. (d) Basins of attraction of the period-8 cycle at k=0.924. (e) The phase plane of the two chaotic attractors at k=0.95. (f) The phase plane of the chaotic attractor at k=0.995. Other parameter values are a=2,b=0.2,c1=1,andc2=0.3.
(a)(b)(c)(d)(e)(f)Using the parameter values in set 1, it is observed through numerical experiments that when the marginal costs are equal (c1=c2), another dynamic behavior is emerged. At c1=c2=1, the fixed point E3 can be destabilized via Neimark–Sacker bifurcation. The Jacobian (10) at these parameter values becomes(16)JE3=−0.806590.18066−0.13619−0.75027,with complex eigenvalues λ1,2=−0.77843±0.15431i and λ1,2=0.79358<1, and then E3 is locally stable. It is easy to see that the determinant of JE3 equals δ=0.62976<1 which means that map (6) is also dissipative. Now, we analyze the influence of k on the stability of this point. When we increase k further, dynamic behaviors arise such as spiral and closed curves around the fixed point. Figure 2(b) gives those behaviors for different values of k. When k reaches 0.774, a period-17 cycle appears with quite complicated basins of attraction as given in Figure 2(c). As k increases further, high-periodic cycles coexist with complicated basins until the dynamic turns into a closed chaotic attractor. This chaotic attractor is born at k=0.79, and its phase plane is displayed in Figure 2(d). This chaotic attractor becomes very complicated as k increases to 0.818, and its phase plane is shown in Figure 3(a). This unexpected dynamic gives no predictions about the evolution of map (6) and makes the price game with asymmetric information more complicated. The asymmetry and nonlinearity of this map give more complicated behaviors that we should report here. At k=0.827, a six-piece chaotic attractor is emerged. The phase plane of this dynamic behavior around the fixed point is given in Figure 3(b). In this figure, we enlarge one piece of this chaotic behavior. These six pieces are gathered together as k increases further to form one chaotic and complicated attractor in Figure 3(c). At k=0.877, a period-7 cycle coexists. As given in Figure 3(d), it has very complicated attractive basins. Increasing k to 0.88, another period cycle is raised. This period is of cycle 14, and its basins are plotted in Figure 3(e). Finally, for any other increase for k, the dynamic behavior of map (6) turns to one chaotic attractor. Figure 3(f) displays the phase plane of this chaotic attractor.Figure 2
(a) Neimark–Sacker bifurcation diagram on varyingk. (b) The phase plane of spiral and closed curves around the fixed point for different values of k. (c) Basins of attraction of the period-17 cycle at k=0.774. (d) The phase plane of the chaotic attractor at k=0.79. (e) Basins of attraction of the period-19 cycle at k=0.795. (f) The phase plane of the chaotic attractor at k=0.8. Other parameter values are a=2,b=0.2,c1=1,andc2=1.
(a)(b)(c)(d)(e)(f)Figure 3
(a) The phase plane of the chaotic attractor atk=0.818. (b) The phase plane of six-piece chaotic attractors at k=0.827. (c) The phase plane of the chaotic attractor at k=0.835. (d) Basins of attraction of the period-7 cycle at k=0.877. (e) Basins of attraction of the period-14 cycle at k=0.88. (f) The phase plane of the chaotic attractor at k=0.89. Other parameter values are a=2,b=0.2,c1=1,andc2=1.
(a)(b)(c)(d)(e)(f)


## 3.2.2. Set 2

In this set, we assume the following values:a=b=c1=c2=0.5 and k=1.6. The fixed point at set 2 becomes E3=0.5,0.5, and its Jacobian is(17)JE3=−1.13330.53333−0.60444−0.56444,with complex eigenvalues λ1,2=−0.84889±0.49139i and λ1,2=0.98086<1, and then E3 is locally stable. It is easy to see that the determinant of JE3 equals δ=0.96207<1 which means that map (6) is again dissipative. As previously stated, we study the effects of k on the stability of the fixed point. It is not a repeated analysis for the above simulation; it gives other interesting and hidden dynamics of map (6). Increasing k makes the fixed point unstable due to Neimark–Sacker bifurcation. For instance, at different values of k, we give in Figure 4(a) different closed curves around the fixed point. Those closed invariant curves are changed to a period-5 cycle as parameter k increases to 1.83. It is plotted in Figure 4(b) with its attractive basins. Another interesting observation is the coexistence of period-5 and period-10 cycles together at k=1.9. They have quite complicated basins of attraction which are given in Figure 4(c). Increasing k further makes those cycles convert into five unconnected and complicated chaotic areas. The phase plane of these chaotic areas is given in Figure 4(d) with the fixed point. These chaotic areas become larger and more complicated as k increases to 1.95. They are displayed in Figure 4(e). After that, the dynamic behavior of map (6) becomes chaotic as k increases to 1.97, as given in Figure 4(f).Figure 4
(a) The phase plane of closed invariant curves at different values ofk=0.818. (b) Basins of attraction of the period-5 cycle at k=1.83. (c) Basins of attraction of period-5 and period-10 cycles at k=1.9. (d) The phase plane of five-piece chaotic attractors at k=1.92. (e) The phase plane of five-piece chaotic attractors at k=1.95. (f) The phase plane of the chaotic attractor at k=1.97. Other parameter values are a=0.5,b=0.5,c1=0.5,andc2=0.5.
(a)(b)(c)(d)(e)(f)We should highlight here that the role of parameterk is affected by parameter b. Figure 5(a) shows that the fixed point can be destabilized due to flip bifurcation on varying parameter k and for the same set of parameter values except for b=−0.55. We give some numerical simulations of the dynamic behavior of k at negative values for b. At k=1.51, Figure 5(b) shows the coexistence of unconnected period-3 cycles. It has an attractive basins given in Figure 5(b) which are more complicated than those obtained previously. At b=−0.5, the effect of k on the fixed point makes it unstable due to flip bifurcation. Figure 5(c) shows a stable fixed point for all k until the coexistence of the unconnected period-6 cycle. This cycle has basins of attraction which are given in Figure 5(d).Figure 5
(a) The flip bifurcation on varyingk at b=−0.55. (b) The basins of attraction of the period-3 cycle at k=1.51andb=−0.55. (c) The flip bifurcation on varying k at b=−0.5. (d) The basins of attraction of the period-6 cycle at k=1.66andb=−0.5. The other parameter values are a=0.5,c1=0.5,andc2=0.5.
(a)(b)(c)(d)


## 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 thata=2,b=0.2,c1=1,andc2=0.3, Figure 6(a) shows the impact of k on map (7). Comparing Figure 1(a) with Figure 6(a), we see that the asymmetric information possessed by the second firm broadens the stability region for both players simultaneously. Numerical experiments show that the basins of attraction detected for map (6) are quite complicated than those of map (7). Furthermore, we show by experiments that, at sets 1 and 2, the stability region of the fixed point for map (5) is bigger than that detected for the fixed point of map (7). For more details about the properties of classic model (7), refer [31]. In Figure 6(b), we present the attractive basins of the period-4 cycle for map (7). Comparing these basins with others given in Figure 1(c), we get that the dynamics of map (6) are more complex than those of map (7). Therefore, the future prediction for map (6) becomes more complicated.Figure 6
(a) Bifurcation diagram for map (7) on varying k. (b) The basins of attraction for the period-4 cycle of map (7) at k=0.70. Other parameters are a=2,b=0.2,c1=1,andc2=0.3.
(a)(b)The previous analysis shows that the structure of basins is quite complicated, and this requires to study extra characteristics of these maps. Settingp1,t+1=p´1 and p2,t+1=p´2 in (6), where ′ refers to the time evolution, we get(18)Tp1,p2:p´1=p1+kp11−b2a1−b+c1−2p1+bp2,p´2=p2+kp21−b2a1−b+c2+bp1−2p2+k2bp1p21−b22a1−b+c1−bc2−2p1+2bp2.Complicated nonlinear map (18) makes us to study its noninvertibility characteristic by substituting a=2,b=0.2,c1=1,c2=0.3, and k=0.88 in (18) as follows:(19)Tp1,p2:p´1=p12.2833−1.8333p1+0.18333p2,p´2=p21.6417+0.40853p1+0.067222p1p2−1.8333p2−0.33611p12.This means that a pointp1,p2 can be characterized by many rank-1 preimages. Those preimages are mapped in p1,p2 in one iteration. Still, map (19) is in a complicated form to solve it algebraically with respect to the variables p1 and p2. Here, we follow the same procedure carried out in [28] by calculating the number of preimages of points belonging to the vertical axis (p1=0). Therefore, substituting the point 0,p´2 in (13), we get(20)p1=0orp1=1.245459+0.1p2.This means if we substitutep1=0 in the second part of (13), we get two real preimages for the point 0,p´2 provided that p´2∈−∞,0.36753; otherwise, we have no preimages. This concludes that T is a noninvertible map of type Z0−Z2, and hence, the phase plane is divided into two regions that are Z0 and Z2. On the contrary, substituting p1=1.245459+0.1p2 in the second part gives that the point 0,p´2 has three real preimages provided that p´2∈0.37050,∞; otherwise, we have only one real preimage. This means the preimages of the point in this case divide the phase plane into two regions Z1 and Z3, and hence, the map T is of type Z1−Z3. The same procedure can be used for the points belonging to the horizontal axis (p2=0) and take the form p´1,0. It is worth to mention here that both critical lines LC and LC−1 are obtained at those parameter values, but they have long forms to be written here or even to be plotted. We have performed the same procedure for map T1 for the points 0,p´2 and p´1,0. The results show that map T1 is of type Z0−Z2 only.


## 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 ofZ0−Z2 or Z1−Z3 depending on the parameter value sets, while the model of classical Bertrand belongs to the type Z0−Z2 only.



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*Source: 6620570-2020-12-10.xml*