Nonlinear Dynamics of a Quantum Cournot Duopoly with Bounded Rationality and Relative Profit Maximization

Based on boundary rationality and relative proft maximization, quantum game theory is applied to develop a dynamic model of the quantum Cournot duopoly game. We investigate the efect of quantum properties on game stability and nonlinear dynamics. Te results show that quantum entanglement can improve the model’s stability and control the generation of bifurcation and chaos when comparing the classical game and the game without considering relative proft maximization. It facilitates the player’s choice of quantum entanglement to control the chaos that emerges in the production output. Numerical simulations verify the chaotic properties of the game by means of bifurcations, maximum Lyapunov exponents, and phase diagrams. Te results show that quantum entanglement has diferent efects on diferent games.


Introduction
Nonlinear economic models contain many complex phenomena. If there is confusion in an economic model, especially in a duopoly game, it can cause chaos and unpredictability in the market, such as [1][2][3]. Delaying or avoiding bifurcation and chaos is essential. Many scholars have attempted to describe games in the quantum domain, in particular the quantum scheme proposed by Li et al. [4,5]. Some new and exciting dynamical results have been discovered in this research on quantum games, which are diferent from the classical game models [6][7][8][9][10].
In recent years, classical dynamic Cournot games have continued to be of interest internationally. Elsadany [11] considered players seeking to maximise their relative profts rather than their own absolute profts and proposed the classical Cournot duopoly game based on bounded rationality and relative proft maximization. Subsequently, Pecora and Sodini [12] studied the Cournot duopoly game in a continuous time frame and discovered the dynamic behaviour of competitors when deciding on output decisions and compared how diferent time delays afect the stability of the economy. Cerboni Baiardi and Naimzada [13] examined competition between quantity-setting players in a nonlinear deterministic duopoly environment distinguished by an isoelastic demand curve. Tey observed a double instability of the Cournot-Nash equilibrium as a result of both the number of players and the percentage of imitators. Among n participants, Andaluz et al. [14] presented a nonlinear Cournot duopoly game showing isoelastic demand in general. Lian and Zheng [15] discussed the dynamic interactions of players in Cournot markets. Te market outcome of the stage game is presented, showing transfer probabilities and fnding the steady state of the system. However, in the classical dynamic Gounod duopoly game model described above, there is the dilemma that Nash equilibrium is inferior to Pareto optimal. Quantum game theory may provide an efective solution to this dilemma [4,5]. Moreover, even if both players behave "selfshly" in a quantum game, as they do in a classical game, they actually cooperate due to the quantum entanglement between them. It is not entirely rational for players to play a quantum game. Tey can determine the stability and dynamic behavior of the game model based on quantum properties (e.g., quantum entanglement). Several new questions will arise as a result. It is not entirely rational for players to play a quantum game. Tey can determine the stability and dynamic behavior of the game model based on quantum properties (e.g., quantum entanglement). Several new questions will arise as a result. Tis has been studied by many scholars and diferent conclusions have been obtained. Based on heterogeneous players, Shi et al. [16,17] proposed a dynamic quantum Cournot duopoly game. It is shown that the stability region increases with quantum entanglement. Zhang et al. [18] found that the stability region is infuenced by the diference in cost coefcients between the quadratic and linear cost models and quantum entanglement. Lo and Yeung [19,20] proposed that positive quantum entanglement is more proftable for the leader and destroys the second shot advantage, while Garcia-Perez et al. [21] demonstrated that the stability region is larger when the adjustment speed is in the negative zone than when it is in the positive zone. Due to the system's quicker stabilization, a signifcant economic advantage arises. Tus, there is still more uncertainty on how quantum entanglement would afect the dynamical benefts of Cournot duopoly games in various scenarios. Using quantum game theory, Zhang et al. [22] studied the chaotic dynamics of a quantum Cournot duopoly game with bounded rational participants. Te results reveal that quantum entanglement reduces the stability region. Chaotic behavior results from the speed at which bounded rational participants adjust, while quantum entanglement accelerates bifurcation and chaos. We will consider the quantum Cournot duopoly game based on bounded rationality and relative proft maximization compared with [11,22], to fnd the advantages and disadvantages of quantum entanglement for the players.
Tree main contributions are presented in this paper.
(1) Te classical Cournot duopoly game is a particular case of the dynamic quantum Cournot duopoly. (2) Compared to [11], the adjustment rate α 1 expands the stability region as c increases after the introduction of quantum entanglement c. It helps the game between the two players reach equilibrium, reducing adjustment costs and increasing profts. (3) Compared to [22], considering proft maximization makes the interval of equilibrium widen and bifurcation and chaos are delayed rather than advanced. Terefore, slight changes in the game can have an opposite efect on the outcome. Tis paper provides a comparable solution for both players.
Te paper follows the following structure. In Section 2, a quantum Cournot duopoly game model based on relative proft maximization and bounded rationality is proposed. In Section 3, quantum entanglement is discussed theoretically with equilibrium stability and nonlinear dynamics. In Section 4, the numerical simulations fully demonstrate the model's nonlinear dynamics. Section 5 contains the conclusion.

The Model
Elsadany [11] proposes a game in which two players produce (cross-sectional) diferentiated products of varieties 1 and 2 and concludes that the absolute proft of each player is, respectively, where b i , c i , α i are used to measure the product level difference, marginal cost, and adjustment rate of player i(i � 1, 2), respectively. Te relative proft of a player is defned by Elsadany as the diference between the absolute proft of the player and the average of the absolute profts of the other players. Players 1 and 2 each have a relative proft of Φ 1 , and Φ 2 denotes their relative profts. Te following equation gives the relative profts of the two players: Based on the Li-Du-Massar quantum scheme [4], the Cournot duopoly game described above can be recast as a quantum version with bound rationality and relative proft maximization. Let be "momentu" operator of j's electromagnetic feld. Set the fnal measurement be corresponding to the observables Y j � (b † is the creation (annihilation) operator of player j 's electromagnetic feld. Te following steps are available.
(1) Te game starts from |00〉. Te entangling operator F(c) is given by Using the squeeze parameter c⩾0, we can measure the degree of entanglement. Tus, the initial state is 2 Mathematical Problems in Engineering (2) Trough unitary operations, the two players execute their strategic moves and (3) Te two players' states are measured after a disentanglement operation Q(c) † . Te fnal state is carried Terefore, We obtain the fnal state Te fnal measurement of quantum entanglement, i.e., c ≠ 0, derives the yield of each of the two players: If there is no quantum entanglement, then c � 0, the quantum game returns to its classical form: q 1 � y 1 , q 2 � y 2 , where y 1 andy 2 represent the quantum output of the two players in the game.
Te scheme of quantum entanglement is introduced into equation (2) and the following quantum marginal profts are derived: From the above analysis, we obtain a novel discrete dynamical system with quantum entanglement after considering quantum properties (for convenience, let y 1 � x n , y 2 � y n ).
Mathematical Problems in Engineering 3

Quantum Equilibrium Points and Local Dynamics
Tis section presents a theoretical study of the complex dynamic behavior of model (12). It is easy to show that (12) has the following quantum equilibrium points: Equilibrium points E 0 , E 1 , E 2 are called quantum boundary equilibrium points, and the equilibrium point E * is called the quantum Cournot-Nash equilibrium point. Since the output of the yield must be positive, the following discussion is based on the condition that the quantum Cournot-Nash equilibrium point is positive.

Proposition 1. Based on the case that the quantum Cournot-Nash equilibrium point E * is positive, the condition for local asymptotic stability is
or Te following discussion produces the necessary conditions for generating fip bifurcation and Neimark-Sacker bifurcation. By [23], the Jacobian matrix A with eigenvalues λ 1 � 1, |λ 2 | ≠ 1 is a necessary condition for generating Flip bifurcation. From the previous analysis, we obtained (Replacing α 1 with α f 1 is to avoid confusion) and For the roots of G(λ) � 0 to be imaginary roots of mode 1, we need to satisfy Defnition 1. [23] Let F α be a one parameter family of map of R 2 satisfying is not an mth root of unity for m � 1, 2, 3, 4.

Mathematical Problems in Engineering
Proposition . When condition (22) is satisfed, α f 1 is the fip bifurcation critical point for the quantum Cournot-Nash equilibrium point E * at (2).

Numerical Simulations
We will verify the theory's validity through numerical simulations in this section to ensure its correctness. With diferent parameter values, the impact of quantum entanglement on the dynamic properties of the quantum Cournot duopoly game is visualized. For example, stability region, bifurcation and phase diagrams, maximum Lyapunov exponents (MLE), and singular attractors. Example 1. We choose the speed of adjustment α 1 as the bifurcation parameter and the other parameters as condition (x, y) � (3, 2.651), at which λ 1 � −1, λ 2 � −0.0660073, the model (12) undergoes fip bifurcation. Figure 1 shows the relationship between the quantum entanglement c and the stability region, where the stability region becomes larger as c increases, i.e., the bifurcation point of fip gradually increases, and the fip bifurcation is delayed. Figure 2(a) shows the bifurcation It is clearly demonstrated from numerical validation that the introduction of the quantum game into a Cournot's doupoly game based on relative proft maximization yields exactly the opposite result to that in [22], i.e., the increase in quantum entanglement prints the creation of the fip bifurcation, rather than accelerating it. Example 2. We choose the speed of adjustment α 1 as the bifurcation parameter, the other parameters are  Figure 4 shows the quantum entanglement c with respect to the stability region. Intuitively, it shows that as c increases, the bifurcation point gradually increases and the stability region expands. However, the bifurcation point is clearly more sensitive to c than in Example 1. c � 0 (in blue). Compared to Example 1, its stable region expands while the chaos-generating region shrinks as c increases, but does not delay at α 1 � 0.55. Figure 6(a) illustrates the invariant curve of (12) becoming smaller at c � 0.02. Figures 6(b)-6(f ) show the orbits of periods-7, 14 and 28 until the chaotic attractor appears. Tus when c is increased, the Neimark-Sacker bifurcation also occurs with a delay, but is more diferent from Flip. Tis is related to the initial conditions we have chosen.     Figure 6: Continued.

. Conclusion
With relative proft maximization and bounded rationality, this paper establishes the quantum Cournot duopoly game. Quantum entanglement is examined for its efect on the system's stability and dynamic behavior. Based on the results, this classical model is a particular case of the quantum form. Quantum entanglement delays the onset of bifurcation behavior and expands the stability region with increasing entanglement in Flip bifurcation and Neimark-Sacker bifurcation. Furthermore, complex dynamical processes such as stability regions, bifurcation diagrams, maximum Lyapunov exponents and phase diagrams (including periodic orbits and chaotic attractors) are described using numerical simulation methods. Due to the introduction of quantum entanglement, the two players actually "cooperate." Te classical model is a subset of the quantum model. It provides a more fexible way for players to control the production output, for example, by simply setting c to nonzero or zero and choosing to regulate the steady, chaotic state of the outcome. Tis is closely related to proft maximization.

Data Availability
Te data that support the fndings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.