# Dynamics and Stability Analysis of a Stackelberg Mixed Duopoly Game with Price Competition in Insurance Market

Authors: Longfei Wei; Haiwei Wang; Jing Wang; Jialong Hou
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This paper investigates the dynamical behaviors of a Stackelberg mixed duopoly game with price competition in the insurance market, involving one state-owned public insurance company and one private insurance company. We study and compare the stability conditions for the Nash equilibrium points of two sequential-move games, public leadership, and private leadership games. Numerical simulations present complicated dynamic behaviors. It is shown that the Nash equilibrium becomes unstable as the price adjustment speed increases, and the system eventually becomes chaotic via flip bifurcation. Moreover, the time-delayed feedback control is used to force the system back to stability.


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## Body

## 1. Introduction

The insurance market in most countries has shown typical characteristics of an oligopoly market. Most of the existing literature on oligopoly games in the insurance market concentrates on static games, while less on dynamic games. The competition among oligarchs in the insurance market is mainly reflected in price competition. One of the most famous price game models is the Bertrand model. For a long time, the complex dynamics in Bertrand oligopoly games have been researched widely. For instance, Zhang and Ma [1] investigated a nonlinear Bertrand game of insurance market wherein one of the oligarchs made the decision only with bounded rationality without delay, while the other part made the delayed decision with one period and two periods. Xu and Ma [2] established a price game model with delay based on the insurance market and discussed the existence and stability of equilibrium points. Ahmed et al. [3] analyzed the dynamic behaviors of a differentiated Bertrand duopoly game, in which boundedly rational players apply a gradient adjustment mechanism to update their price in each period. Ma and Si [4] introduced a continuous Bertrand duopoly game model with a two-stage delay and investigated the influence of delay parameters on the dynamic characteristics of the system. Zhao [5] studied the dynamic properties of a Bertrand game model with three oligarchs in which enterprises have heterogeneous expectations. Askar and Al-khedhairi [6] introduced two different Bertrand duopoly models where the first one is the competition of price in which each player wants to maximize its relative profit, and the second model is the classic Bertrand competition in which the players want to maximize their profits. There is still some literature which studies the Bertrand game theory and the complexity of the dynamical system; see [7–12]. The above price game models assume that firms play simultaneous-move games; however, the competition among oligarchs in the real market is mostly a dynamic game or sequential game. It is well known that the most classical sequential game is the Stackelberg game. Shi et al. [13] proposed a price-Stackelberg duopoly game model with boundedly rational players and studied the complex dynamical behaviors. Shah et al. [14] applied the Stackelberg game with stochastic demand for the vendor-retailer system. Wang [15] investigated a manufacturer-Stackelberg game in a price competition supply chain under a fuzzy decision environment. Xiao et al. [16] analyzed a nonlinear two-dimensional duopoly Stackelberg game, including two types of heterogeneous players which are bounded rational players and adaptable players. Other interesting works that can be used to extend the applications of such economic games have been reported in [17–20].Most countries have state-owned public firms that have a substantial influence on their market competitors. The competition in mixed oligopolies, in which a state-owned public firm competes against a private firm, is widespread in the real market. Such mixed oligopolies occur in various industries, such as automobile, postal services, hospitals, education, banking, and insurance [21]. Analysis of mixed oligopolies can be dated to 1966 when Merrill and Schneider first put forward the assumption that a public firm maximizes welfare (consumer surplus plus firm profits), while private firms maximize profits [22]. Some discussions of the mixed oligopolies were presented in [23–25]. In related works, there are few studies on Stackelberg mixed oligopolies. For instance, Wang and Mukherjee [26] showed welfare under different numbers of private firms under the assumption of a public firm as the Stackelberg leader and private firms as Stackelberg followers. Wang and Lee [27] examined the influence of the order of the firms’ moves on the social efficiency with foreign ownership and free entry in a mixed oligopoly market. Tao et al. [28] studied and compared total welfare in Stackelberg mixed duopolies when either the public firm or the private firm acts as the leader. Gelves and Heywood [29] compared the merger between the public leader and the private follower with unilateral privatization of the public leader. Hirose and Matsumura [21] compared welfare and profit under price and quantity competition in Stackelberg mixed duopolies, wherein a state-owned public firm competes against a private firm. These studies mainly focus on quantity competition, less on price competition, and all of them discuss directly the decisions of competitors in Nash equilibrium, and their dynamics have not been studied. However, the dynamic adjustment process converging to Nash equilibrium, and the stability of Nash equilibrium are important in the real market.In this paper, we pay attention to a Stackelberg mixed duopoly game of price competition between a state-owned public insurance company and a private insurance company in the insurance market and study how this duopoly game evolves to different Nash equilibriums in two sequential-move games, public leadership and private leadership games. Simulations of the complicated dynamic behaviors and chaos control are presented, and the welfare and profit in Nash equilibrium are also discussed.The rest of this paper is organized as follows: in Section2, the Stackelberg mixed duopoly game model with price competition is briefly described. In Section 3, the existence of equilibrium points, the instability of bounded equilibrium, and the local stability conditions of Nash equilibrium in two sequential-move cases are analyzed. In Section 4, numerical simulations are used to show the dynamic features of the game, including bifurcation diagram, maximum Lyapunov exponents, phase portrait, and sensitive dependence on initial conditions. The comparison of welfare and profit in Nash equilibrium under two sequential-move cases is also shown by the figures in this section. In Section 5, time-delayed feedback control is used to suppress the appearance of chaotic behavior for the proposed system. Finally, the paper is concluded in Section 6.


## 2. The Model

We consider the dynamical behaviors of a duopoly price game in the insurance market with two insurance companies (ICs), labeled byi=1,2. We assume IC1 is a state-owned public insurance company, while IC2 is a private insurance company, and the two ICs use different decisional mechanisms. Each ICi chooses a nonnegative real number pi, which is the price of its own product. The strategy profile p=p1,p2 results in a corresponding market quantity demanded q=q1,q2. We adopt a standard duopoly model with differentiated products and linear demand [30, 31]. The utility function of the representative consumer is given by(1)Uq1,q2=αq1+q2−β2q12+2δq1q2+q22.Parametersα and β are positive constants and δ∈0,1 measures the degree of horizontal differentiation, where a smaller δ indicates a larger degree of insurance product differentiation. Then, the inverse demand functions are(2)p1=α−βq1−βδq2,p2=α−βq2−βδq1.According to equation (2), the direct demand functions can be given by(3)q1=α−αδ−p1+δp2β1−δ2,q2=α−αδ−p2+δp1β1−δ2.We denote the marginal cost ofICi with a positive constant ci, assuming α>c1≥c2. In addition, we assume that α is sufficiently large and that c1−c2 is not too large to assure interior solutions in the following games. Since IC1 is a public firm, its payoff is the domestic social surplus (welfare) [21, 28] and is given by(4)π1=p1−c1q1+p2−c2q2+αq1+q2−βq12+2δq1q2+q222−p1q1−p2q2.For the convenience of expression, equation (4) can be simplified to the following form:(5)π1=α−c1q1+α−c2q2−βq12+2δq1q2+q222,and IC2 is a private firm and its payoff is its own profit:(6)π2=p2−c2q2.In this study, we consider two sequential-move games, public leadership and private leadership games. To construct and investigate the dynamic characteristics of the two games, we assume that both ICs are bounded rational, in which the players modify their price decisions dynamically according to the marginal payoff [32]. When ICi (ICj) is the leader (follower), i≠j, the dynamic system for two ICs has the following form:(7)pit+1=pit+kipit∂πipit,pjt∂pit,pjt+1=pjt+kjpjt∂πjpit+1,pjt∂pjt,where ki and kj are positive parameters known as the speed of price adjustment and t=0,1,2,…. At t+1, the leader ICi takes the lead in determining the price pit+1. Then, the follower ICj chooses pjt+1 to maximize its own payoff after observing pit+1, which can be seen in the second equation of system (7).


## 3. Equilibrium Points and Local Stability

### 3.1. Public Leadership

In this case, we analyze a Stackelberg model in whichIC1 (IC2) is the leader (follower). At t+1, the leader IC1 takes the lead in determining the price p1t+1 on the basis of marginal payoff. The marginal payoff of IC1 is(8)∂π1p1t,p2t∂p1t=δα1−δ−c2+2c12−δ2+3δ2−4p1t4β1−δ2.Then, we can get(9)p1t+1=p1t+k1p1t4β1−δ2δα1−δ−c2+2c12−δ2+3δ2−4p1t.The followerIC2 has an advantage over the leader. IC2 has known the current price of IC1 when it chooses its price p2t+1. Hence, the price of IC2 at period t+1 is determined by its own price of period t and IC1’s price of period t+1. Then, the marginal payoff of IC2 is given by(10)∂π2p1t+1,p2t∂p2t=1β1−δ2α1−δ+c2+δp1t−2p2t+δk1p1t4β21−δ22δα1−δ−c2+2c12−δ2+3δ2−4p1t.Then, we have(11)p2t+1=p2t+k2p2tβ1−δ2α1−δ+c2+δp1t−2p2t+δk1k2p1tp2t4β21−δ22δα1−δ−c2+2c12−δ2+3δ2−4p1t.Thus, the duopoly game can be described by a discrete-time dynamic map as follows:(12)p1t+1=p1t+k1p1t4β1−δ2δα1−δ−c2+2c12−δ2+3δ2−4p1t,p2t+1=p2t+k2p2tβ1−δ2α1−δ+c2+δp1t−2p2t+δk1k2p1tp2t4β21−δ22δα1−δ−c2+2c12−δ2+3δ2−4p1t.When the market structure is stable enough at timet, pit+1 are approximately equal to pit. Setting p1t+1=p1t and p2t+1=p2t in equation (12), we can get the following equilibria:(13)E1=0,0,E2=0,α1−δ+c22,E3=δα1−δ−c2+2c12−δ24−3δ2,0,E4=p1∗,p2∗,where(14)p1∗=δα1−δ−c2+2c12−δ24−3δ2,p2∗=2−δ2α1−δ+c1δ+2c21−δ24−3δ2.The pointsE1, E2, and E3 are known as boundary equilibrium points while E4 is the Nash equilibrium point. For the stake of economic significance, the equilibrium points should be nonnegative. It is easily concluded that E2, E3, and E4 are all positive according to the conditions that α,β,c1, and c2 are positive parameters, δ∈0,1, and α>c1≥c2. E1 represents that every IC has no price; E2 and E3 represent the monopolies IC1 and IC2, respectively; and E4 represents both ICs competing in a duopoly game with equilibrium prices of p1∗ and p2∗.To analyze the local stability of the equilibrium points, we consider the Jacobian matrix of system (12), which can be given by(15)Jp1,p2=1+k1μA+k13δ2−4μp104k2δμp21+k1μA+k13δ2−4μp11+4k2μB−8k2μp2+4k1k2δμ2Ap1,where(16)A=δα1−δ−c2+2c12−δ2+3δ2−4p1,B=α1−δ+c2+δp1−2p2,μ=14β1−δ2.Then, the conditions for a stable equilibrium point can be obtained based on the following lemma [33, 34].Lemma 1.
Suppose the Jacobian matrix (15) at any fixed point Ep˜1,p˜2 has two eigenvalues λ1 and λ2, then(i)
Ifλ1<1 and λ2<1, then Ep˜1,p˜2 is locally asymptotically stable and Ep˜1,p˜2 is called an attracting node(ii)
Ifλ1>1 and λ2>1, then Ep˜1,p˜2 is an unstable repelling node(iii)
Ifλ1<1 and λ2>1 (or λ1>1 and λ2<1), then Ep˜1,p˜2 is an unstable saddle point(iv)
Ifλ1=1 and λ2≠1 (or λ1≠1 and λ2=1), then Ep˜1,p˜2 is a nonhyperbolic pointTheorem 1.
The boundary equilibriumE1 is an unstable repelling node.Proof.
AtE1, the Jacobian matrix takes the form:(17)JE1=1+k1δα1−δ−c2+2c12−δ24β1−δ2001+k2α1−δ+c2β1−δ2.
The eigenvalues ofJE1 are given by(18)λ1=1+k1δα1−δ−c2+2c12−δ24β1−δ2,λ2=1+k2α1−δ+c2β1−δ2.
They are both greater than 1, so the pointE1 is an unstable repelling node.Theorem 2.
The boundary equilibriumE2 is unstable. More precisely, we have the following:(i)
If0<k2<β1−δ2/α1−δ+c2, then E2 is a saddle point(ii)
Ifk2=β1−δ2/α1−δ+c2, then E2 is a nonhyperbolic point(iii)
Ifk2>β1−δ2/α1−δ+c2, then E2 is a repelling nodeProof.
AtE2, the Jacobian matrix becomes(19)JE2=1+k1δα1−δ−c2+2c12−δ24β1−δ20C1−2k2α1−δ+c2β1−δ2,where(20)C=k2δα1−δ+c22β1−δ2+k1k2δα1−δ+c2δα1−δ−c2+2c12−δ28β21−δ22.
The eigenvalues ofJE2 are given by(21)λ1=1+k1δα1−δ−c2+2c12−δ24β1−δ2,λ2=1−2k2α1−δ+c2β1−δ2,λ1>1 is always satisfied, so E2 is unstable. Simple calculations show that λ2<1 if 0<k2<β1−δ2/α1−δ+c2, λ2=1 if k2=β1−δ2/α1−δ+c2, and λ2>1 if k2>β1−δ2/α1−δ+c2. This concludes the proof.Theorem 3.
The boundary equilibriumE3 is unstable. More precisely, we have the following:(i)
If0<k1<8β1−δ2/δα1−δ−c2+2c12−δ2, then E3 is a saddle point(ii)
Ifk1=8β1−δ2/δα1−δ−c2+2c12−δ2, then E3 is a nonhyperbolic point(iii)
Ifk1>8β1−δ2/δα1−δ−c2+2c12−δ2, then E3 is a repelling nodeProof.
AtE3, the Jacobian matrix is(22)JE3=1−k1δα1−δ−c2+2c12−δ24β1−δ2001+2k22−δ2α1−δ+c1+2c11−δ2β1−δ24−3δ2.
The eigenvalues ofJE3 are given by(23)λ1=1−k1δα1−δ−c2+2c12−δ24β1−δ2,λ2=1+2k22−δ2α1−δ+c1+2c11−δ2β1−δ24−3δ2,λ2>1 is always satisfied, so E3 is unstable. Simple calculations show that λ1<1 if 0<k1<8β1−δ2/δα1−δ−c2+2c12−δ2, λ1=1 if k1=8β1−δ2/δα1−δ−c2+2c12−δ2, and λ1>1 if k1>8β1−δ2/δα1−δ−c2+2c12−δ2. This concludes the proof.
The boundary equilibrium points correspond to the situation of one or both ICs going bankrupt. That is, the duopoly market becomes a monopoly, or both of the ICs are out of the insurance market at the same time. However, from an economic point of view, we should pay more attention to the situation of the duopoly market. Hence, we are more interested in investigating the local stability properties of the Nash equilibrium pointE4. The Jacobian matrix at E4 takes the form:(24)JE4=1−k14−3δ2p1∗4β1−δ20k2δp2∗β1−δ2−k1k2δ4−3δ2p1∗p2∗4β21−δ221−2k2p2∗β1−δ2.
The characteristic equation is(25)λ2−TrJE4λ+DetJE4=0,where Tr is the trace and Det is the determinant, which are given by(26)TrJE4=2−k14−3δ2p1∗4β1−δ2−2k2p2∗β1−δ2,DetJE4=1−k14−3δ2p1∗4β1−δ2−2k2p2∗β1−δ2+k1k24−3δ2p1∗p2∗2β21−δ22.
Then, we have(27)TrJE42−4DetJE4=k14−3δ2p1∗4β1−δ2−2k2p2∗β1−δ22,which indicates that the eigenvalues are real. According to Jury conditions [35], the necessary and sufficient conditions for the local stability of E4 are as follows:(28)1−DetJE4>0,1−TrJE4+DetJE4>0,1+TrJE4+DetJE4>0.
The above conditions are, respectively, equivalent to(29)ik14−3δ2p1∗4β1−δ2+2k2p2∗β1−δ2−k1k24−3δ2p1∗p2∗2β21−δ22>0,(30)iik1k24−3δ2p1∗p2∗2β21−δ22>0,(31)iii4−k14−3δ2p1∗2β1−δ2−4k2p2∗β1−δ2+k1k24−3δ2p1∗p2∗2β21−δ22>0.
Clearly, the condition (ii) is always satisfied. Then, the following result can be obtained from the derivation of conditions (i) and (iii).Theorem 4.
The Nash equilibrium pointE4 is asymptotically locally stable if(32)k1<8β1−δ24−3δ2p1∗,k2<β1−δ2p2∗.Proof.
Inequality (29) can be rewritten as(33)8β1−δ2k14−3δ2p1∗+β1−δ2k2p2∗>2.
Inequality (31) can be modified as(34)4−k14−3δ2p1∗2β1−δ2−4k2p2∗β1−δ2+k1k24−3δ2p1∗p2∗2β21−δ22=41−k14−3δ2p1∗8β1−δ21−k2p2∗β1−δ2>0.
Inequality (34) holds if and only if the following two conditions are satisfied:(35)k1<8β1−δ24−3δ2p1∗,k2<β1−δ2p2∗,or(36)k1>8β1−δ24−3δ2p1∗,k2>β1−δ2p2∗.
It is obvious that the first condition implies inequality (33). On the other hand, inequality (33) is impossible in the k1,k2-plane determined by the second condition. This concludes the proof.
Condition (32) defines a stability region in the plane of the price adjustment speed k1,k2 (see Figure 1(a)). The boundary intersects the axes k1 and k2 at points R1 and R2, respectively, whose coordinates are(37)R1=8β1−δ24−3δ2p1∗,0,R2=0,β1−δ2p2∗.
Simple calculations show that ifk1=8β1−δ2/4−3δ2p1∗ or k2=β1−δ2/p2∗, one of the absolute values of eigenvalues is equal to 1. Inequalities (29) and (31) define a bounded region of stability beyond which a flip bifurcation and a Neimark–Sacker bifurcation occur, respectively [33, 36]. According to Theorem 4, we can get that the Nash equilibrium point E4 loses its stability only via flip bifurcation when one or both values of k1 and k2 are greater than the boundary values of the stability region.
According to the expressions of the coordinate value of the boundary points, we can clearly find out the effects of the changing values of parametersα,β,c1, and c2 on the stability region, respectively, but it is difficult to directly observe how parameter δ affects the stability regions from the expression. By computer work on the stability conditions for four cases δ=0.05,0.3,0.5,0.7, the stability regions in the k1,k2-plane are numerically obtained and are plotted in Figure 1(a). We can see that increasing δ reduces the stability region, and the stability of system (12) is more sensitive to IC2. If k2 is relatively lower, even if k1 is relatively higher, system (12) is stable.Figure 1
(a) The stability regions in thek1,k2-plane of Nash equilibrium point E4 for system (12) with different levels of δ. (b) The stability regions in the k1,k2-plane of Nash equilibrium point E8 for system (38) with different levels of δ. The other values of the parameters are α=0.7, β=0.2, c1=0.13, and c2=0.1.
(a)(b)


### 3.2. Private Leadership

WhenIC2 (IC1) is the leader (follower), we can write system (7) as(38)p1t+1=p1t+k1p1tβ1−δ2c1−δc2−p1t+δp2t+δk1k2p1tp2tβ21−δ22α1−δ+δc1+1−2δ2c2−21−δ2p2t,p2t+1=p2t+k2p2tβ1−δ2α1−δ+δc1+1−2δ2c2−21−δ2p2t.Settingp1t+1=p1t and p2t+1=p2t in equation (38), we have the following equilibria:(39)E5=0,0,E6=0,α1−δ+δc1+1−2δ2c221−δ2,E7=c1−δc2,0,E8=p1∗∗,p2∗∗,where(40)p1∗∗=δα1−δ−c2+2−δ2c121−δ2,p2∗∗=α1−δ+δc1+1−2δ2c221−δ2.PointsE5, E6, and E7 are boundary equilibrium points and E8 is the unique Nash equilibrium point. It is clear that E6, E7, and E8 are all positive according to the conditions that α,β,c1, and c2 are positive parameters, δ∈0,1, and α>c1≥c2.The Jacobian matrix of system (38) can be given by(41)Jp1,p2=1+k1μ′B′−k1μ′p1+k1k2δμ′2A′p2k1δμ′p11+k2μ′A′−2k21−δ2μ′p201+k2μ′A′−2k21−δ2μ′p2,where(42)A′=α1−δ+δc1+1−2δ2c2−21−δ2p2,B′=c1−δc2−p1+δp2,μ′=1β1−δ2.Theorem 5.
The boundary equilibriumE5 is an unstable repelling node.Proof.
The Jacobian matrix (41) at the point E5 takes the form:(43)JE5=1+k1c1−δc2β1−δ2001+k2α1−δ+δc1+1−2δ2c2β1−δ2.
The eigenvalues ofJE5 are given by(44)λ1=1+k1c1−δc2β1−δ2,λ2=1+k2α1−δ+δc1+1−2δ2c2β1−δ2.
It is clear that they are both greater than 1, so the pointE5 is an unstable repelling node.Theorem 6.
The boundary equilibriumE6 is unstable. More precisely, we have the following:(i)
If0<k2<2β1−δ2/α1−δ+δc1+1−2δ2c2, then E6 is a saddle point(ii)
Ifk2=2β1−δ2/α1−δ+δc1+1−2δ2c2, then E6 is a nonhyperbolic point(iii)
Ifk2>2β1−δ2/α1−δ+δc1+1−2δ2c2, then E6 is a repelling nodeProof.
AtE6, the Jacobian matrix becomes(45)JE6=1+k12c1−δc2+αδ1−δ2β1−δ22001−k2α1−δ+δc1+1−2δ2c2β1−δ2.
The eigenvalues ofJE6 are given by(46)λ1=1+k12c1−δc2+αδ1−δ2β1−δ22,λ2=1−k2α1−δ+δc1+1−2δ2c2β1−δ2,λ1>1 is always satisfied, so E6 is unstable. Simple calculations show that λ2<1 if 0<k2<2β1−δ2/α1−δ+δc1+1−2δ2c2, λ2=1 if k2=2β1−δ2/α1−δ+δc1+1−2δ2c2, and λ2>1 if k2>2β1−δ2/α1−δ+δc1+1−2δ2c2. This concludes the proof.Theorem 7.
The boundary equilibriumE7 is unstable. More precisely, we have the following:(i)
If0<k1<2β1−δ2/c1−δc2, then E7 is a saddle point(ii)
Ifk1=2β1−δ2/c1−δc2, then E7 is a nonhyperbolic point(iii)
Ifk1>2β1−δ2/c1−δc2, then E7 is a repelling nodeProof.
AtE7, the Jacobian matrix is(47)JE7=1−k1c1−δc2β1−δ2k1δc1−δc2β1−δ2+k1k2δc1−δc2α1−δ+δc1+1−2δ2c2β21−δ2201+k2α1−δ+δc1+1−2δ2c2β1−δ2.
The eigenvalues ofJE7 are given by(48)λ1=1−k1c1−δc2β1−δ2,λ2=1+k2α1−δ+δc1+1−2δ2c2β1−δ2,λ2>1 is always satisfied, so E7 is unstable. Simple calculations show that λ1<1 if 0<k1<2β1−δ2/c1−δc2, λ1=1 if k1=2β1−δ2/c1−δc2, and λ1>1 if k1>2β1−δ2/c1−δc2. This concludes the proof.
Next, we investigate the local stability properties of the Nash equilibrium pointE8. The Jacobian matrix at the point E8 takes the form:(49)JE8=1−k1p1∗∗β1−δ2k1δp1∗∗β1−δ2−2k1k2δp1∗∗p2∗∗β21−δ201−2k2p2∗∗β.
The trace and determinant ofJE8 are(50)TrJE8=2−k1p1∗∗β1−δ2−2k2p2∗∗β,DetJE8=1−k1p1∗∗β1−δ2−2k2p2∗∗β+2k1k2p1∗∗p2∗∗β21−δ2.
Then, we have(51)TrJE82−4DetJE8=k1p1∗∗β1−δ2−2k2p2∗∗β2.
It indicates that the eigenvalues are real. According to Jury conditions, the necessary and sufficient conditions for the local stability ofE8 can be given by(52)ik1p1∗∗β1−δ2+2k2p2∗∗β−2k1k2p1∗∗p2∗∗β21−δ2>0,(53)ii2k1k2p1∗∗p2∗∗β21−δ2>0,(54)iii4−2k1p1∗∗β1−δ2−4k2p2∗∗β+2k1k2p1∗∗p2∗∗β21−δ2>0.
Clearly, condition (ii) is always satisfied. According to conditions (i) and (iii), we have the following result.Theorem 8.
The Nash equilibrium pointE8 is asymptotically locally stable if(55)k1<2β1−δ2p1∗∗,k2<βp2∗∗.Proof.
Inequality (52) can be rewritten as(56)2β1−δ2k1p1∗∗+βk2p2∗∗>2.
Inequality (54) can be modified as(57)4−2k1p1∗∗β1−δ2−4k2p2∗∗β+2k1k2p1∗∗p2∗∗β21−δ2=41−k1p1∗∗2β1−δ21−k2p2∗∗β>0.
We complete the proof by imitating the discussions in Theorem4.
The stability region for the Nash equilibrium pointE8 is defined by the inequalities in condition (55). The boundary curve intersects the axes k1 and k2 at points R3 and R4, respectively, whose coordinates are(58)R3=2β1−δ2p1∗∗,0,R4=0,βp2∗∗.
Simple calculations show that ifk1=2β1−δ2/p1∗∗ or k2=β/p2∗∗, one of the absolute values of eigenvalues is equal to 1. According to Theorem 8, we can get that the Nash equilibrium point E8 is stable inside the stability region, and loses its stability through flip bifurcation. By computer work on the stability conditions for four cases δ=0.05,0.3,0.5,0.7, the stability regions of Nash equilibrium point E8 in the k1,k2-plane are shown in Figure 1(b). Comparing Figures 1(a) and 1(b), we see that when the parameter δ is close to zero, the stability regions of Nash equilibrium points E4 and E8 are similar, but the difference between them becomes larger with the increase of the parameter δ.


## 3.1. Public Leadership

In this case, we analyze a Stackelberg model in whichIC1 (IC2) is the leader (follower). At t+1, the leader IC1 takes the lead in determining the price p1t+1 on the basis of marginal payoff. The marginal payoff of IC1 is(8)∂π1p1t,p2t∂p1t=δα1−δ−c2+2c12−δ2+3δ2−4p1t4β1−δ2.Then, we can get(9)p1t+1=p1t+k1p1t4β1−δ2δα1−δ−c2+2c12−δ2+3δ2−4p1t.The followerIC2 has an advantage over the leader. IC2 has known the current price of IC1 when it chooses its price p2t+1. Hence, the price of IC2 at period t+1 is determined by its own price of period t and IC1’s price of period t+1. Then, the marginal payoff of IC2 is given by(10)∂π2p1t+1,p2t∂p2t=1β1−δ2α1−δ+c2+δp1t−2p2t+δk1p1t4β21−δ22δα1−δ−c2+2c12−δ2+3δ2−4p1t.Then, we have(11)p2t+1=p2t+k2p2tβ1−δ2α1−δ+c2+δp1t−2p2t+δk1k2p1tp2t4β21−δ22δα1−δ−c2+2c12−δ2+3δ2−4p1t.Thus, the duopoly game can be described by a discrete-time dynamic map as follows:(12)p1t+1=p1t+k1p1t4β1−δ2δα1−δ−c2+2c12−δ2+3δ2−4p1t,p2t+1=p2t+k2p2tβ1−δ2α1−δ+c2+δp1t−2p2t+δk1k2p1tp2t4β21−δ22δα1−δ−c2+2c12−δ2+3δ2−4p1t.When the market structure is stable enough at timet, pit+1 are approximately equal to pit. Setting p1t+1=p1t and p2t+1=p2t in equation (12), we can get the following equilibria:(13)E1=0,0,E2=0,α1−δ+c22,E3=δα1−δ−c2+2c12−δ24−3δ2,0,E4=p1∗,p2∗,where(14)p1∗=δα1−δ−c2+2c12−δ24−3δ2,p2∗=2−δ2α1−δ+c1δ+2c21−δ24−3δ2.The pointsE1, E2, and E3 are known as boundary equilibrium points while E4 is the Nash equilibrium point. For the stake of economic significance, the equilibrium points should be nonnegative. It is easily concluded that E2, E3, and E4 are all positive according to the conditions that α,β,c1, and c2 are positive parameters, δ∈0,1, and α>c1≥c2. E1 represents that every IC has no price; E2 and E3 represent the monopolies IC1 and IC2, respectively; and E4 represents both ICs competing in a duopoly game with equilibrium prices of p1∗ and p2∗.To analyze the local stability of the equilibrium points, we consider the Jacobian matrix of system (12), which can be given by(15)Jp1,p2=1+k1μA+k13δ2−4μp104k2δμp21+k1μA+k13δ2−4μp11+4k2μB−8k2μp2+4k1k2δμ2Ap1,where(16)A=δα1−δ−c2+2c12−δ2+3δ2−4p1,B=α1−δ+c2+δp1−2p2,μ=14β1−δ2.Then, the conditions for a stable equilibrium point can be obtained based on the following lemma [33, 34].Lemma 1.
Suppose the Jacobian matrix (15) at any fixed point Ep˜1,p˜2 has two eigenvalues λ1 and λ2, then(i)
Ifλ1<1 and λ2<1, then Ep˜1,p˜2 is locally asymptotically stable and Ep˜1,p˜2 is called an attracting node(ii)
Ifλ1>1 and λ2>1, then Ep˜1,p˜2 is an unstable repelling node(iii)
Ifλ1<1 and λ2>1 (or λ1>1 and λ2<1), then Ep˜1,p˜2 is an unstable saddle point(iv)
Ifλ1=1 and λ2≠1 (or λ1≠1 and λ2=1), then Ep˜1,p˜2 is a nonhyperbolic pointTheorem 1.
The boundary equilibriumE1 is an unstable repelling node.Proof.
AtE1, the Jacobian matrix takes the form:(17)JE1=1+k1δα1−δ−c2+2c12−δ24β1−δ2001+k2α1−δ+c2β1−δ2.
The eigenvalues ofJE1 are given by(18)λ1=1+k1δα1−δ−c2+2c12−δ24β1−δ2,λ2=1+k2α1−δ+c2β1−δ2.
They are both greater than 1, so the pointE1 is an unstable repelling node.Theorem 2.
The boundary equilibriumE2 is unstable. More precisely, we have the following:(i)
If0<k2<β1−δ2/α1−δ+c2, then E2 is a saddle point(ii)
Ifk2=β1−δ2/α1−δ+c2, then E2 is a nonhyperbolic point(iii)
Ifk2>β1−δ2/α1−δ+c2, then E2 is a repelling nodeProof.
AtE2, the Jacobian matrix becomes(19)JE2=1+k1δα1−δ−c2+2c12−δ24β1−δ20C1−2k2α1−δ+c2β1−δ2,where(20)C=k2δα1−δ+c22β1−δ2+k1k2δα1−δ+c2δα1−δ−c2+2c12−δ28β21−δ22.
The eigenvalues ofJE2 are given by(21)λ1=1+k1δα1−δ−c2+2c12−δ24β1−δ2,λ2=1−2k2α1−δ+c2β1−δ2,λ1>1 is always satisfied, so E2 is unstable. Simple calculations show that λ2<1 if 0<k2<β1−δ2/α1−δ+c2, λ2=1 if k2=β1−δ2/α1−δ+c2, and λ2>1 if k2>β1−δ2/α1−δ+c2. This concludes the proof.Theorem 3.
The boundary equilibriumE3 is unstable. More precisely, we have the following:(i)
If0<k1<8β1−δ2/δα1−δ−c2+2c12−δ2, then E3 is a saddle point(ii)
Ifk1=8β1−δ2/δα1−δ−c2+2c12−δ2, then E3 is a nonhyperbolic point(iii)
Ifk1>8β1−δ2/δα1−δ−c2+2c12−δ2, then E3 is a repelling nodeProof.
AtE3, the Jacobian matrix is(22)JE3=1−k1δα1−δ−c2+2c12−δ24β1−δ2001+2k22−δ2α1−δ+c1+2c11−δ2β1−δ24−3δ2.
The eigenvalues ofJE3 are given by(23)λ1=1−k1δα1−δ−c2+2c12−δ24β1−δ2,λ2=1+2k22−δ2α1−δ+c1+2c11−δ2β1−δ24−3δ2,λ2>1 is always satisfied, so E3 is unstable. Simple calculations show that λ1<1 if 0<k1<8β1−δ2/δα1−δ−c2+2c12−δ2, λ1=1 if k1=8β1−δ2/δα1−δ−c2+2c12−δ2, and λ1>1 if k1>8β1−δ2/δα1−δ−c2+2c12−δ2. This concludes the proof.
The boundary equilibrium points correspond to the situation of one or both ICs going bankrupt. That is, the duopoly market becomes a monopoly, or both of the ICs are out of the insurance market at the same time. However, from an economic point of view, we should pay more attention to the situation of the duopoly market. Hence, we are more interested in investigating the local stability properties of the Nash equilibrium pointE4. The Jacobian matrix at E4 takes the form:(24)JE4=1−k14−3δ2p1∗4β1−δ20k2δp2∗β1−δ2−k1k2δ4−3δ2p1∗p2∗4β21−δ221−2k2p2∗β1−δ2.
The characteristic equation is(25)λ2−TrJE4λ+DetJE4=0,where Tr is the trace and Det is the determinant, which are given by(26)TrJE4=2−k14−3δ2p1∗4β1−δ2−2k2p2∗β1−δ2,DetJE4=1−k14−3δ2p1∗4β1−δ2−2k2p2∗β1−δ2+k1k24−3δ2p1∗p2∗2β21−δ22.
Then, we have(27)TrJE42−4DetJE4=k14−3δ2p1∗4β1−δ2−2k2p2∗β1−δ22,which indicates that the eigenvalues are real. According to Jury conditions [35], the necessary and sufficient conditions for the local stability of E4 are as follows:(28)1−DetJE4>0,1−TrJE4+DetJE4>0,1+TrJE4+DetJE4>0.
The above conditions are, respectively, equivalent to(29)ik14−3δ2p1∗4β1−δ2+2k2p2∗β1−δ2−k1k24−3δ2p1∗p2∗2β21−δ22>0,(30)iik1k24−3δ2p1∗p2∗2β21−δ22>0,(31)iii4−k14−3δ2p1∗2β1−δ2−4k2p2∗β1−δ2+k1k24−3δ2p1∗p2∗2β21−δ22>0.
Clearly, the condition (ii) is always satisfied. Then, the following result can be obtained from the derivation of conditions (i) and (iii).Theorem 4.
The Nash equilibrium pointE4 is asymptotically locally stable if(32)k1<8β1−δ24−3δ2p1∗,k2<β1−δ2p2∗.Proof.
Inequality (29) can be rewritten as(33)8β1−δ2k14−3δ2p1∗+β1−δ2k2p2∗>2.
Inequality (31) can be modified as(34)4−k14−3δ2p1∗2β1−δ2−4k2p2∗β1−δ2+k1k24−3δ2p1∗p2∗2β21−δ22=41−k14−3δ2p1∗8β1−δ21−k2p2∗β1−δ2>0.
Inequality (34) holds if and only if the following two conditions are satisfied:(35)k1<8β1−δ24−3δ2p1∗,k2<β1−δ2p2∗,or(36)k1>8β1−δ24−3δ2p1∗,k2>β1−δ2p2∗.
It is obvious that the first condition implies inequality (33). On the other hand, inequality (33) is impossible in the k1,k2-plane determined by the second condition. This concludes the proof.
Condition (32) defines a stability region in the plane of the price adjustment speed k1,k2 (see Figure 1(a)). The boundary intersects the axes k1 and k2 at points R1 and R2, respectively, whose coordinates are(37)R1=8β1−δ24−3δ2p1∗,0,R2=0,β1−δ2p2∗.
Simple calculations show that ifk1=8β1−δ2/4−3δ2p1∗ or k2=β1−δ2/p2∗, one of the absolute values of eigenvalues is equal to 1. Inequalities (29) and (31) define a bounded region of stability beyond which a flip bifurcation and a Neimark–Sacker bifurcation occur, respectively [33, 36]. According to Theorem 4, we can get that the Nash equilibrium point E4 loses its stability only via flip bifurcation when one or both values of k1 and k2 are greater than the boundary values of the stability region.
According to the expressions of the coordinate value of the boundary points, we can clearly find out the effects of the changing values of parametersα,β,c1, and c2 on the stability region, respectively, but it is difficult to directly observe how parameter δ affects the stability regions from the expression. By computer work on the stability conditions for four cases δ=0.05,0.3,0.5,0.7, the stability regions in the k1,k2-plane are numerically obtained and are plotted in Figure 1(a). We can see that increasing δ reduces the stability region, and the stability of system (12) is more sensitive to IC2. If k2 is relatively lower, even if k1 is relatively higher, system (12) is stable.Figure 1
(a) The stability regions in thek1,k2-plane of Nash equilibrium point E4 for system (12) with different levels of δ. (b) The stability regions in the k1,k2-plane of Nash equilibrium point E8 for system (38) with different levels of δ. The other values of the parameters are α=0.7, β=0.2, c1=0.13, and c2=0.1.
(a)(b)


## 3.2. Private Leadership

WhenIC2 (IC1) is the leader (follower), we can write system (7) as(38)p1t+1=p1t+k1p1tβ1−δ2c1−δc2−p1t+δp2t+δk1k2p1tp2tβ21−δ22α1−δ+δc1+1−2δ2c2−21−δ2p2t,p2t+1=p2t+k2p2tβ1−δ2α1−δ+δc1+1−2δ2c2−21−δ2p2t.Settingp1t+1=p1t and p2t+1=p2t in equation (38), we have the following equilibria:(39)E5=0,0,E6=0,α1−δ+δc1+1−2δ2c221−δ2,E7=c1−δc2,0,E8=p1∗∗,p2∗∗,where(40)p1∗∗=δα1−δ−c2+2−δ2c121−δ2,p2∗∗=α1−δ+δc1+1−2δ2c221−δ2.PointsE5, E6, and E7 are boundary equilibrium points and E8 is the unique Nash equilibrium point. It is clear that E6, E7, and E8 are all positive according to the conditions that α,β,c1, and c2 are positive parameters, δ∈0,1, and α>c1≥c2.The Jacobian matrix of system (38) can be given by(41)Jp1,p2=1+k1μ′B′−k1μ′p1+k1k2δμ′2A′p2k1δμ′p11+k2μ′A′−2k21−δ2μ′p201+k2μ′A′−2k21−δ2μ′p2,where(42)A′=α1−δ+δc1+1−2δ2c2−21−δ2p2,B′=c1−δc2−p1+δp2,μ′=1β1−δ2.Theorem 5.
The boundary equilibriumE5 is an unstable repelling node.Proof.
The Jacobian matrix (41) at the point E5 takes the form:(43)JE5=1+k1c1−δc2β1−δ2001+k2α1−δ+δc1+1−2δ2c2β1−δ2.
The eigenvalues ofJE5 are given by(44)λ1=1+k1c1−δc2β1−δ2,λ2=1+k2α1−δ+δc1+1−2δ2c2β1−δ2.
It is clear that they are both greater than 1, so the pointE5 is an unstable repelling node.Theorem 6.
The boundary equilibriumE6 is unstable. More precisely, we have the following:(i)
If0<k2<2β1−δ2/α1−δ+δc1+1−2δ2c2, then E6 is a saddle point(ii)
Ifk2=2β1−δ2/α1−δ+δc1+1−2δ2c2, then E6 is a nonhyperbolic point(iii)
Ifk2>2β1−δ2/α1−δ+δc1+1−2δ2c2, then E6 is a repelling nodeProof.
AtE6, the Jacobian matrix becomes(45)JE6=1+k12c1−δc2+αδ1−δ2β1−δ22001−k2α1−δ+δc1+1−2δ2c2β1−δ2.
The eigenvalues ofJE6 are given by(46)λ1=1+k12c1−δc2+αδ1−δ2β1−δ22,λ2=1−k2α1−δ+δc1+1−2δ2c2β1−δ2,λ1>1 is always satisfied, so E6 is unstable. Simple calculations show that λ2<1 if 0<k2<2β1−δ2/α1−δ+δc1+1−2δ2c2, λ2=1 if k2=2β1−δ2/α1−δ+δc1+1−2δ2c2, and λ2>1 if k2>2β1−δ2/α1−δ+δc1+1−2δ2c2. This concludes the proof.Theorem 7.
The boundary equilibriumE7 is unstable. More precisely, we have the following:(i)
If0<k1<2β1−δ2/c1−δc2, then E7 is a saddle point(ii)
Ifk1=2β1−δ2/c1−δc2, then E7 is a nonhyperbolic point(iii)
Ifk1>2β1−δ2/c1−δc2, then E7 is a repelling nodeProof.
AtE7, the Jacobian matrix is(47)JE7=1−k1c1−δc2β1−δ2k1δc1−δc2β1−δ2+k1k2δc1−δc2α1−δ+δc1+1−2δ2c2β21−δ2201+k2α1−δ+δc1+1−2δ2c2β1−δ2.
The eigenvalues ofJE7 are given by(48)λ1=1−k1c1−δc2β1−δ2,λ2=1+k2α1−δ+δc1+1−2δ2c2β1−δ2,λ2>1 is always satisfied, so E7 is unstable. Simple calculations show that λ1<1 if 0<k1<2β1−δ2/c1−δc2, λ1=1 if k1=2β1−δ2/c1−δc2, and λ1>1 if k1>2β1−δ2/c1−δc2. This concludes the proof.
Next, we investigate the local stability properties of the Nash equilibrium pointE8. The Jacobian matrix at the point E8 takes the form:(49)JE8=1−k1p1∗∗β1−δ2k1δp1∗∗β1−δ2−2k1k2δp1∗∗p2∗∗β21−δ201−2k2p2∗∗β.
The trace and determinant ofJE8 are(50)TrJE8=2−k1p1∗∗β1−δ2−2k2p2∗∗β,DetJE8=1−k1p1∗∗β1−δ2−2k2p2∗∗β+2k1k2p1∗∗p2∗∗β21−δ2.
Then, we have(51)TrJE82−4DetJE8=k1p1∗∗β1−δ2−2k2p2∗∗β2.
It indicates that the eigenvalues are real. According to Jury conditions, the necessary and sufficient conditions for the local stability ofE8 can be given by(52)ik1p1∗∗β1−δ2+2k2p2∗∗β−2k1k2p1∗∗p2∗∗β21−δ2>0,(53)ii2k1k2p1∗∗p2∗∗β21−δ2>0,(54)iii4−2k1p1∗∗β1−δ2−4k2p2∗∗β+2k1k2p1∗∗p2∗∗β21−δ2>0.
Clearly, condition (ii) is always satisfied. According to conditions (i) and (iii), we have the following result.Theorem 8.
The Nash equilibrium pointE8 is asymptotically locally stable if(55)k1<2β1−δ2p1∗∗,k2<βp2∗∗.Proof.
Inequality (52) can be rewritten as(56)2β1−δ2k1p1∗∗+βk2p2∗∗>2.
Inequality (54) can be modified as(57)4−2k1p1∗∗β1−δ2−4k2p2∗∗β+2k1k2p1∗∗p2∗∗β21−δ2=41−k1p1∗∗2β1−δ21−k2p2∗∗β>0.
We complete the proof by imitating the discussions in Theorem4.
The stability region for the Nash equilibrium pointE8 is defined by the inequalities in condition (55). The boundary curve intersects the axes k1 and k2 at points R3 and R4, respectively, whose coordinates are(58)R3=2β1−δ2p1∗∗,0,R4=0,βp2∗∗.
Simple calculations show that ifk1=2β1−δ2/p1∗∗ or k2=β/p2∗∗, one of the absolute values of eigenvalues is equal to 1. According to Theorem 8, we can get that the Nash equilibrium point E8 is stable inside the stability region, and loses its stability through flip bifurcation. By computer work on the stability conditions for four cases δ=0.05,0.3,0.5,0.7, the stability regions of Nash equilibrium point E8 in the k1,k2-plane are shown in Figure 1(b). Comparing Figures 1(a) and 1(b), we see that when the parameter δ is close to zero, the stability regions of Nash equilibrium points E4 and E8 are similar, but the difference between them becomes larger with the increase of the parameter δ.


## 4. Numerical Simulation

In this section, we perform some numerical simulations for the complex dynamical behaviors of systems (12) and (38) and show how the systems evolve under different levels of parameters. Such simulations include a bifurcation diagram, maximum Lyapunov exponents, phase portrait, and sensitive dependence on initial conditions to further study the unpredictable behavior of the game. In all numerical simulations, parameters α, β, δ, c1, and c2 are commonly set as α=0.7, β=0.2, δ=0.5, c1=0.13, and c2=0.1. We perform numerical simulations for the following two situations, respectively.


### 4.1. Public Leadership

In a real insurance market, the demand function and their marginal costs are relatively certain, so the price adjustment speed is regarded as an important strategy for ICs to pursue profit maximization. In this case, we show by numerical simulations how system (12) evolves under different levels of the parameter k1, the price adjustment speed of IC1. We fix the parameter k2=0.5, and the bifurcation diagram with respect to the fact that parameter k1 is plotted in Figure 2(a). It shows that the equilibrium point begins stable; increasing the value of k1 gives the appearance of a stable 2-cycle period through flip bifurcation, then increasing the value of k1 further shows a sequence of period-doubling bifurcations followed by cycles with high periodicity; and chaotic scenario occurs in the end. The corresponding maximum Lyapunov exponents are plotted in Figure 2(b) to show bifurcation and chaos, where positive values show the chaotic behaviors.Figure 2
Bifurcation diagram and MLEs for system (12) with respect to parameter k1 when k2=0.5.
(a)(b)The observations from Figure2(a) tell that system (12) becomes unstable through the period-doubling bifurcation when the parameter takes suitable values. About the case in Figure 2, five two-dimension phase portraits for different values of k1 are shown in Figure 3, which give a more detailed description of the orbits of system (12). The phase portraits show a flip bifurcation process, where chaos occurs when k1 takes its value big enough, and strange attractors can be seen in the fifth phase portrait in Figure 3. The strange attractor reflects the complexity of ICs’ dynamic price competition in chaos.Figure 3
Phase portraits for Figure2 with various values of k1.The sensitivity to initial conditions is also one of the important characteristics of chaos. Figure4 reflects the case that the initial state p1 ranges from 0.1785 to 0.1786 with other parameters keeping fixed. Figures 4(a) and 4(b) show the orbits of IC1’s price and IC2’s price, respectively, where the blue ones (labeled by superscript (1)) start from the initial point p110,p210=0.1785,0.2696 and the red ones (labeled by superscript (2)) start from the initial point p120,p220=0.1786,0.2696. After a series of iterations, great impacts will emerge in both ICs’ prices, even though the initial price of IC1 alters a little.Figure 4
Sensitive dependence on initial conditions for system (12) in the time periods [900, 1000] when k1=3 and k2=0.5.
(a)(b)


### 4.2. Private Leadership

In this case, the numerical simulations show the effect of the parameterk2 and the price adjustment speed of IC2, on the stability of system (38). Figures 5(a) and 5(b) show the bifurcation diagram with respect to k2 and the corresponding maximum Lyapunov exponents of system (38), respectively. Figure 6 shows five situations of phase portraits with different k2 of system (38). We can see system (38) loses its stability through flip bifurcation, and chaotic attractors occur after the accumulation of a period-doubling cascade. The results demonstrate that the insurance market is stable for relatively small values of k2, and a faster adjustment speed is disadvantageous for system (38) to keep the stability. Figure 7 shows the sensitive dependence on the initial conditions of system (38) when k1=0.7 and k2=0.95. Figures 7(a) and 7(b) show the orbits of IC1’s price and IC2’s price, respectively, where the blue ones (labeled by superscript (1)) start from the initial point p110,p210=0.2351,0.3101 and the red ones (labeled by superscript (2)) start from the initial point p120,p220=0.2351,0.3102. We can see that the difference between the orbits with slightly deviated initial values builds up rapidly after a series of iterations, although their states are indistinguishable at the beginning.Figure 5
Bifurcation diagram and MLEs for system (38) with respect to parameter k2 when k1=0.7.
(a)(b)Figure 6
Phase portraits for Figure5 with various values of k2.Figure 7
Sensitive dependence on initial conditions for system (38) in the time periods [900, 1000] when k1=0.7 and k2=0.95.
(a)(b)


### 4.3. The Comparison of Welfare and Profit for Two Games

We now compare the welfare and profit levels for the above two sequential-move games (public leadership and private leadership). Keeping fixedα=0.7, β=0.2, δ=0.5, c1=0.13, and c2=0.1, Figure 8(a) shows the values of the welfare of IC1, and Figure 8(b) shows the values of the profit of IC2. The values of the welfare and profit in the Nash equilibrium state of both games are marked, where the red points refer to the welfare and profit at the Nash equilibrium point E4 of the game (12), and the blue points refer to the welfare and profit at the Nash equilibrium point E8 of the game (38). We can see that the values of the welfare and profit at the two Nash equilibrium points are positive, where π1E4 is greater than π1E8 and π2E4 is less than π2E8. The results show that an IC, whether public or private, is more profitable in the Nash equilibrium state when it is the leader in the price competition game.Figure 8
The welfare ofIC1 and the profit of IC2.
(a)(b)


## 4.1. Public Leadership

In a real insurance market, the demand function and their marginal costs are relatively certain, so the price adjustment speed is regarded as an important strategy for ICs to pursue profit maximization. In this case, we show by numerical simulations how system (12) evolves under different levels of the parameter k1, the price adjustment speed of IC1. We fix the parameter k2=0.5, and the bifurcation diagram with respect to the fact that parameter k1 is plotted in Figure 2(a). It shows that the equilibrium point begins stable; increasing the value of k1 gives the appearance of a stable 2-cycle period through flip bifurcation, then increasing the value of k1 further shows a sequence of period-doubling bifurcations followed by cycles with high periodicity; and chaotic scenario occurs in the end. The corresponding maximum Lyapunov exponents are plotted in Figure 2(b) to show bifurcation and chaos, where positive values show the chaotic behaviors.Figure 2
Bifurcation diagram and MLEs for system (12) with respect to parameter k1 when k2=0.5.
(a)(b)The observations from Figure2(a) tell that system (12) becomes unstable through the period-doubling bifurcation when the parameter takes suitable values. About the case in Figure 2, five two-dimension phase portraits for different values of k1 are shown in Figure 3, which give a more detailed description of the orbits of system (12). The phase portraits show a flip bifurcation process, where chaos occurs when k1 takes its value big enough, and strange attractors can be seen in the fifth phase portrait in Figure 3. The strange attractor reflects the complexity of ICs’ dynamic price competition in chaos.Figure 3
Phase portraits for Figure2 with various values of k1.The sensitivity to initial conditions is also one of the important characteristics of chaos. Figure4 reflects the case that the initial state p1 ranges from 0.1785 to 0.1786 with other parameters keeping fixed. Figures 4(a) and 4(b) show the orbits of IC1’s price and IC2’s price, respectively, where the blue ones (labeled by superscript (1)) start from the initial point p110,p210=0.1785,0.2696 and the red ones (labeled by superscript (2)) start from the initial point p120,p220=0.1786,0.2696. After a series of iterations, great impacts will emerge in both ICs’ prices, even though the initial price of IC1 alters a little.Figure 4
Sensitive dependence on initial conditions for system (12) in the time periods [900, 1000] when k1=3 and k2=0.5.
(a)(b)


## 4.2. Private Leadership

In this case, the numerical simulations show the effect of the parameterk2 and the price adjustment speed of IC2, on the stability of system (38). Figures 5(a) and 5(b) show the bifurcation diagram with respect to k2 and the corresponding maximum Lyapunov exponents of system (38), respectively. Figure 6 shows five situations of phase portraits with different k2 of system (38). We can see system (38) loses its stability through flip bifurcation, and chaotic attractors occur after the accumulation of a period-doubling cascade. The results demonstrate that the insurance market is stable for relatively small values of k2, and a faster adjustment speed is disadvantageous for system (38) to keep the stability. Figure 7 shows the sensitive dependence on the initial conditions of system (38) when k1=0.7 and k2=0.95. Figures 7(a) and 7(b) show the orbits of IC1’s price and IC2’s price, respectively, where the blue ones (labeled by superscript (1)) start from the initial point p110,p210=0.2351,0.3101 and the red ones (labeled by superscript (2)) start from the initial point p120,p220=0.2351,0.3102. We can see that the difference between the orbits with slightly deviated initial values builds up rapidly after a series of iterations, although their states are indistinguishable at the beginning.Figure 5
Bifurcation diagram and MLEs for system (38) with respect to parameter k2 when k1=0.7.
(a)(b)Figure 6
Phase portraits for Figure5 with various values of k2.Figure 7
Sensitive dependence on initial conditions for system (38) in the time periods [900, 1000] when k1=0.7 and k2=0.95.
(a)(b)


## 4.3. The Comparison of Welfare and Profit for Two Games

We now compare the welfare and profit levels for the above two sequential-move games (public leadership and private leadership). Keeping fixedα=0.7, β=0.2, δ=0.5, c1=0.13, and c2=0.1, Figure 8(a) shows the values of the welfare of IC1, and Figure 8(b) shows the values of the profit of IC2. The values of the welfare and profit in the Nash equilibrium state of both games are marked, where the red points refer to the welfare and profit at the Nash equilibrium point E4 of the game (12), and the blue points refer to the welfare and profit at the Nash equilibrium point E8 of the game (38). We can see that the values of the welfare and profit at the two Nash equilibrium points are positive, where π1E4 is greater than π1E8 and π2E4 is less than π2E8. The results show that an IC, whether public or private, is more profitable in the Nash equilibrium state when it is the leader in the price competition game.Figure 8
The welfare ofIC1 and the profit of IC2.
(a)(b)


## 5. Chaos Control

As can be seen from the above numerical simulations that price adjustment speeds have a great influence on the stability of systems (12) and (38), the dynamical behaviors of both two systems will be chaotic when the parameters fail to locate in the stable region. In practical application, the appearance of chaos is not expected, we hope that the occurrence of chaos can be avoided or controlled, and the insurance market can be controlled to a balance when it runs irregularly. In this section, we use the time-delayed feedback control method to control the chaotic phenomenon [37–40]. Similar to the fourth section, we divide the following two cases for numerical simulations.


### 5.1. Public Leadership

We modify the equations of system (12) by inserting the control action ϕpit−pit+1, where ϕ>0 is the controlling coefficient. Then, the controlled system can be given by(59)p1t+1=p1t+k1p1t4β1−δ2δα1−δ−c2+2c12−δ2+3δ2−4p1t+ϕp1t−p1t+1,p2t+1=p2t+k2p2tβ1−δ2α1−δ+c2+δp1t−2p2t+δk1k2p1tp2t4β21−δ22δα1−δ−c2+2c12−δ2+3δ2−4p1t+ϕp2t−p2t+1,which can be rewritten as(60)p1t+1=p1t+k1p1t4β1−δ21+ϕδα1−δ−c2+2c12−δ2+3δ2−4p1t,p2t+1=p2t+k2p2tβ1−δ21+ϕα1−δ+c2+δp1t−2p2t+δk1k2p1tp2t4β21−δ221+ϕδα1−δ−c2+2c12−δ2+3δ2−4p1t.It is easy to see that the controlled system (60) has the same Nash equilibrium point E4 as the original system (12). At E4, the Jacobian matrix of system (60) takes the form(61)JE4=1−k14−3δ2p1∗4β1−δ21+ϕ0k2δp2∗β1−δ21+ϕ−k1k2δ4−3δ2p1∗p2∗4β21−δ21+ϕ1−2k2p2∗β1−δ21+ϕ.Figure2(a) shows that chaotic behavior of the original system (12) occurs when parameter values are fixed as α,β,δ,c1,c2,k1,k2=0.7,0.2,0.5,0.13,0.1,3,0.5. By a similar approach in Section 3 to get the stability conditions (i)-(iii) for the original system (12), we can get that all the eigenvalues of the matrix (61) will lie within the unit circle provided that ϕ>0.45, when the other parameters take above values. In other words, when ϕ>0.45, the controlled system (60) will be asymptotically locally stable. This result can be numerically shown by Figure 9.Figure 9
(a) Bifurcation diagram for system (60) with respect to the controlling factor ϕ. (b) Evolutions for system (60) with various values of ϕ.
(a)(b)Figure9(a) is the bifurcation diagram with respect to ϕ, where we see that, with the value of ϕ increasing, the system is gradually controlled from the chaotic state, 8, 4, 2-period bifurcation to a stable state. Figure 9(b) shows the stable behaviors of the orbits of the controlled system (60) beginning from the initial state p10,p20=0.19,0.27 for different levels of ϕ. We can see that the larger the feedback value is, the faster the system tends to be stable.


### 5.2. Private Leadership

Adding control actionψpit−pit+1 to system (38) and simplifying the system, we get the following form of the controlled system:(62)p1t+1=p1t+k1p1tβ1−δ21+ψc1−δc2−p1t+δp2t+δk1k2p1tp2tβ21−δ221+ψα1−δ+δc1+1−2δ2c2−21−δ2p2t,p2t+1=p2t+k2p2tβ1−δ21+ψα1−δ+δc1+1−2δ2c2−21−δ2p2t.It is obvious that the controlled system (62) has the same Nash equilibrium point E8 as the original system (38). At E8, the Jacobian matrix of system (62) becomes(63)JE8=1−k1p1∗∗β1−δ21+ψk1δp1∗∗β1−δ21+ψ−2k1k2δp1∗∗p2∗∗β21−δ21+ψ01−2k2p2∗∗β1+ψ.As has been shown in Figure5(a), the chaotic behavior of the original system (38) occurs when parameter values are fixed as α,β,δ,c1,c2,k1,k2=0.7,0.2,0.5,0.13,0.1,0.7,0.95. By computer work on the Jury stability conditions, we can infer that when the other parameters take the above values, the controlled system (62) will be asymptotically locally stable provided that ψ>0.4725. As shown in Figure 10(a), with the value of ψ increasing, the system gradually gets out of chaos and achieves stability when ψ>0.4725. When ψ=0.52, 0.55, and 0.58, the stable behaviors of the orbits of the controlled system (62) beginning from the initial state p10,p20=0.24,0.32 are plotted in Figure 10(b). From Figure 10, we can see that the chaotic system is controlled at the fixed point when the controlling coefficient ψ is properly large.Figure 10
(a) Bifurcation diagram for system (62) with respect to the controlling factor ψ. (b) Evolutions for system (62) with various values of ψ.
(a)(b)In a real insurance market, we can consider the control action as the regulation on the price adjustment speed, and we can also consider the control action as the learning ability or adaptability of the market [40]. The time-delayed feedback control method can be used to make the system from a chaotic state to stable state, which ensures that the insurance market develops in an orderly way.


## 5.1. Public Leadership

We modify the equations of system (12) by inserting the control action ϕpit−pit+1, where ϕ>0 is the controlling coefficient. Then, the controlled system can be given by(59)p1t+1=p1t+k1p1t4β1−δ2δα1−δ−c2+2c12−δ2+3δ2−4p1t+ϕp1t−p1t+1,p2t+1=p2t+k2p2tβ1−δ2α1−δ+c2+δp1t−2p2t+δk1k2p1tp2t4β21−δ22δα1−δ−c2+2c12−δ2+3δ2−4p1t+ϕp2t−p2t+1,which can be rewritten as(60)p1t+1=p1t+k1p1t4β1−δ21+ϕδα1−δ−c2+2c12−δ2+3δ2−4p1t,p2t+1=p2t+k2p2tβ1−δ21+ϕα1−δ+c2+δp1t−2p2t+δk1k2p1tp2t4β21−δ221+ϕδα1−δ−c2+2c12−δ2+3δ2−4p1t.It is easy to see that the controlled system (60) has the same Nash equilibrium point E4 as the original system (12). At E4, the Jacobian matrix of system (60) takes the form(61)JE4=1−k14−3δ2p1∗4β1−δ21+ϕ0k2δp2∗β1−δ21+ϕ−k1k2δ4−3δ2p1∗p2∗4β21−δ21+ϕ1−2k2p2∗β1−δ21+ϕ.Figure2(a) shows that chaotic behavior of the original system (12) occurs when parameter values are fixed as α,β,δ,c1,c2,k1,k2=0.7,0.2,0.5,0.13,0.1,3,0.5. By a similar approach in Section 3 to get the stability conditions (i)-(iii) for the original system (12), we can get that all the eigenvalues of the matrix (61) will lie within the unit circle provided that ϕ>0.45, when the other parameters take above values. In other words, when ϕ>0.45, the controlled system (60) will be asymptotically locally stable. This result can be numerically shown by Figure 9.Figure 9
(a) Bifurcation diagram for system (60) with respect to the controlling factor ϕ. (b) Evolutions for system (60) with various values of ϕ.
(a)(b)Figure9(a) is the bifurcation diagram with respect to ϕ, where we see that, with the value of ϕ increasing, the system is gradually controlled from the chaotic state, 8, 4, 2-period bifurcation to a stable state. Figure 9(b) shows the stable behaviors of the orbits of the controlled system (60) beginning from the initial state p10,p20=0.19,0.27 for different levels of ϕ. We can see that the larger the feedback value is, the faster the system tends to be stable.


## 5.2. Private Leadership

Adding control actionψpit−pit+1 to system (38) and simplifying the system, we get the following form of the controlled system:(62)p1t+1=p1t+k1p1tβ1−δ21+ψc1−δc2−p1t+δp2t+δk1k2p1tp2tβ21−δ221+ψα1−δ+δc1+1−2δ2c2−21−δ2p2t,p2t+1=p2t+k2p2tβ1−δ21+ψα1−δ+δc1+1−2δ2c2−21−δ2p2t.It is obvious that the controlled system (62) has the same Nash equilibrium point E8 as the original system (38). At E8, the Jacobian matrix of system (62) becomes(63)JE8=1−k1p1∗∗β1−δ21+ψk1δp1∗∗β1−δ21+ψ−2k1k2δp1∗∗p2∗∗β21−δ21+ψ01−2k2p2∗∗β1+ψ.As has been shown in Figure5(a), the chaotic behavior of the original system (38) occurs when parameter values are fixed as α,β,δ,c1,c2,k1,k2=0.7,0.2,0.5,0.13,0.1,0.7,0.95. By computer work on the Jury stability conditions, we can infer that when the other parameters take the above values, the controlled system (62) will be asymptotically locally stable provided that ψ>0.4725. As shown in Figure 10(a), with the value of ψ increasing, the system gradually gets out of chaos and achieves stability when ψ>0.4725. When ψ=0.52, 0.55, and 0.58, the stable behaviors of the orbits of the controlled system (62) beginning from the initial state p10,p20=0.24,0.32 are plotted in Figure 10(b). From Figure 10, we can see that the chaotic system is controlled at the fixed point when the controlling coefficient ψ is properly large.Figure 10
(a) Bifurcation diagram for system (62) with respect to the controlling factor ψ. (b) Evolutions for system (62) with various values of ψ.
(a)(b)In a real insurance market, we can consider the control action as the regulation on the price adjustment speed, and we can also consider the control action as the learning ability or adaptability of the market [40]. The time-delayed feedback control method can be used to make the system from a chaotic state to stable state, which ensures that the insurance market develops in an orderly way.


## 6. Conclusion

In this paper, we have studied the complex dynamic behaviors of a Stackelberg mixed duopoly game with price competition in an insurance market, wherein a public insurance company competes against a private insurance company. We investigated this problem in two different sequential-move scenarios, public leadership and private leadership games. The equilibrium points including the Nash equilibrium point have been obtained as functions of the system parameters in two cases, and the conditions for the stability of equilibria have been found. We have made some numerical simulations for the system evolution, including stability region, bifurcation diagram, maximal Lyapunov exponents, phase portrait, and sensitive dependence on initial conditions. They show that the price adjustment speeds have a great influence on the system stability, and while varying the values of the price adjustment speeds, complex dynamic behaviors would occur, such as period bifurcations and chaos. Meanwhile, we have compared the welfare and profit levels for two sequential-move games. When an insurance company, whether public or private, is the Stackelberg leader in the price competition game, it yields greater welfare (or profit) than as the follower. In addition, we have also shown that the time-delayed feedback control method can be used to force the system back to its stable state from chaos.



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*Source: 3985367-2021-06-21.xml*