On Duopolistic Competition with the Gradient Adjustment Mechanism Under Constant and Decreasing Returns to Scale

Abstract

This paper investigates duopoly competition under both constant and decreasing returns to scale in a market characterized by an isoelastic demand function, where firms adjust their strategies using a gradient adjustment mechanism. To establish the stability conditions of the model, we adopt different analytical approaches depending on the type of returns to scale. Under constant returns to scale, we employ a traditional approach by deriving the closed-form solution of the Nash equilibrium and analyzing the Jacobian matrix to verify whether the moduli of all eigenvalues are less than one. In contrast, under decreasing returns to scale, we analyze the local stability of the Nash equilibrium using symbolic computation methods without deriving a closed-form solution. The results show that when firms have heterogeneous costs, the model can exhibit both period-doubling and Neimark–Sacker bifurcations under both types of returns to scale. However, when costs are homogeneous, only period-doubling bifurcations occur. Numerical simulations support these analytical results and further demonstrate the emergence of complex dynamics, including chaotic behavior.

1. Introduction

It is widely acknowledged that Cournot [1] proposed the first formal theoretical framework for oligopolistic competition. This seminal work on oligopoly games has been widely applied in industrial organization economics. By introducing the assumption of static expectations regarding rivals’ outputs, Cournot developed a duopoly model with a linear map and a unique Nash equilibrium. It was shown that the equilibrium is globally stable in the presence of two competitors but may become unstable when three firms are involved. This conclusion can also be found in [2,3]. If the assumption of a linear map is relaxed, multiple equilibria may arise. Palander [4] and Wald [5] were among the first to employ nonlinear (piecewise linear) demand functions in oligopoly games, which may generate multiple Nash equilibria. Their work experienced a revival in the 1990s due to the development and application of analytical and numerical tools for global analysis.

Different from piecewise linear demand functions, many scholars have introduced nonlinearities into oligopoly games by employing isoelastic demand functions. For example, Puu [6] considered an inverse demand function of the form $p\left(Q\right)=1/Q$, where Q denotes the total supply in the market. In Puu’s dynamic duopoly game, firms are assumed to be boundedly rational and adjust their outputs using the best-response function based on naive expectations. The results indicate that the Nash equilibrium may become unstable, and complex dynamics, such as chaos, may arise if the difference in marginal costs between the two firms is sufficiently large. In a subsequent study, Ahmed and Agiza [7] extended Puu’s model to an oligopoly game with n firms and obtained similar conclusions. Moreover, Agliari et al. [8] adopted the same demand function as in Puu [6] and focused on the case with three competitors. Using the critical line method, they investigated various bifurcations of the attractors and their corresponding basins.

In addition to the above-mentioned best response mechanism based on naive expectations, other types of boundedly rational adjustment mechanisms have also been investigated in the existing literature; see [9,10,11]. For instance, Bischi et al. [9] and Naimzada and Tramontana [10] studied the output adjustment mechanism based on the local monopolistic approximation (LMA). Their results show that this mechanism requires less information while leading to stronger stability properties. It is worth noting that Bischi et al. [9] considered the same cost function as in our study, which also allows for the presence of diseconomies of scale. Furthermore, to capture bounded rationality, time delays have also been incorporated into models of oligopolistic competition; see [12,13].

The gradient adjustment mechanism described in this paper has been extensively analyzed in the literature; see [14,15,16,17,18,19,20]. Tramontana [19] and Tramontana and Elsadany [20] introduced a heterogeneous oligopoly model in which one firm follows a gradient adjustment rule. Fanti et al. [16] and Andaluz et al. [15] investigated the dynamic behavior of the gradient adjustment mechanism in a homogeneous-agent framework, adopting a general isoelastic demand function. In addition, Agiza et al. [14] and Li et al. [17] also focused on homogeneous games with gradient adjustment but employed a nonlinear inverse demand function of the form $p\left(Q\right)=a-b\sqrt{Q}$. However, these studies consider only linear cost functions, corresponding to the case of constant returns to scale. In contrast, this paper examines both constant and decreasing returns to scale in order to assess whether the impact of the gradient adjustment mechanism on market stability is robust.

This paper examines a duopoly market characterized by isoelastic demand under the assumptions of constant and decreasing returns to scale. It is assumed that the firms are boundedly rational and adjust their outputs according to the gradient adjustment mechanism. We establish the stability conditions for the Nash equilibrium of the considered game and show that both the cost parameters and the adjustment speed parameters act as destabilizing factors, meaning that their increases may lead to the destabilization of the Nash equilibrium. Furthermore, we demonstrate that when firms have heterogeneous costs, the model may exhibit both period-doubling and Neimark–Sacker bifurcations under both constant and decreasing returns to scale. In contrast, when costs are homogeneous, only period-doubling bifurcations may occur, while Neimark–Sacker bifurcations cannot arise. Finally, numerical simulations are conducted to explore more complex dynamics, and the results confirm the correctness of the analytical findings obtained above.

The contributions of this paper are twofold. From an economic perspective, we examine the case of decreasing returns to scale, which has received relatively little attention in the literature. This case often differs from that of constant returns to scale and may lead to distinct economic implications (see [21]). However, our analysis shows that, under the present setting, the results for decreasing returns to scale are qualitatively similar to those obtained under constant returns to scale. From a methodological perspective, we employ different approaches to analyze the local stability and bifurcations of the Nash equilibrium under different types of returns to scale. In particular, under decreasing returns to scale, the local stability and bifurcation properties of the Nash equilibrium are investigated using symbolic computation methods, including triangular decomposition and partial cylindrical algebraic decomposition, without the need to derive a closed-form solution.

The structure of this paper is as follows. Section 2 introduces a duopoly game in which firms face either constant or decreasing returns to scale and adjust their outputs according to a gradient adjustment mechanism. In Section 3, we analyze the local stability and bifurcations of the model using different methodological approaches. Section 4 presents numerical simulations to explore more complex dynamics, including chaotic behavior. Finally, Section 5 concludes the paper with some final remarks.

2. Model Setup

We consider a duopoly market in which two firms produce homogeneous products. Let the output of Firm i be denoted by $q_i$. The market demand is assumed to be isoelastic, with the isoelastic demand function given by

$$p\left(Q\right)={\displaystyle \frac{1}{Q}},$$

where $Q=q_1+q_2$ represents the total market supply. This specification is consistent with the assumption that consumers have a Cobb–Douglas utility function. For more details on the isoelastic demand function, readers are referred to [6].

In the economics literature, the analysis typically focuses only on the case of constant returns to scale. In this paper, we consider both constant and decreasing returns to scale in order to examine whether firms’ output adjustment strategies have similar effects on market stability. To analyze the constant returns to scale case, we assume that the cost function of Firm i is linear, i.e., $C_i\left(q_i\right)=c_i{q}_i$ with $c_i>0$. In contrast, to capture decreasing returns to scale, we consider a quadratic cost function, $C_i\left(q_i\right)=c_i{q}_i^2$ with $c_i>0$. Moreover, let period $t+1$ denote the current period and period t the previous period. The profit of Firm i in period t is then given by

$${\Pi }_i\left(t\right)=p\left(Q\left(t\right)\right)·q_i\left(t\right)-C_i\left(q_i\left(t\right)\right).$$

Furthermore, in the economics literature, firms are often assumed to follow a best-response mechanism to adjust output [1,6], adjusting instantaneously to the profit-maximizing optimal level. In reality, however, output adjustments are constrained by costs and time, so firms tend to adopt a gradual adjustment strategy. In our game, the two players employ the gradient adjustment mechanism to determine their outputs, specifically increasing or decreasing them based on the marginal profit observed in the previous period. The marginal profit of Firm i at period t is

$${\displaystyle \frac{\partial {\Pi }_i\left(t\right)}{\partial q_i\left(t\right)}}={\displaystyle \frac{q_{-i}\left(t\right)}{{(q_i\left(t\right)+q_{-i}\left(t\right))}^2}}-c_i\mathrm{if}{C}_i\left(q_i\right)=c_i{q}_i,$$

and

$${\displaystyle \frac{\partial {\Pi }_i\left(t\right)}{\partial q_i\left(t\right)}}={\displaystyle \frac{q_{-i}\left(t\right)}{{(q_i\left(t\right)+q_{-i}\left(t\right))}^2}}-2c_i{q}_i\left(t\right)\mathrm{if}{C}_i\left(q_i\right)=c_i{q}_i^2,$$

where $q_{-i}\left(t\right)$ is the output of the rival at Period t.

Based on the gradient adjustment mechanism, Firm i adjusts its output at Period $t+1$ according to the following rule:

$$q_i(t+1)=q_i\left(t\right)+K_i{q}_i\left(t\right){\displaystyle \frac{\partial {\Pi }_i\left(t\right)}{\partial q_i\left(t\right)}},$$

where the parameter $K_i>0$ controls the adjustment speed. It is important to note that the adjustment speed depends not only on the parameter $K_i$, but also on $q_i\left(t\right)$, which represents the size of Firm i at period t. This specification reflects the idea that larger firms tend to make larger changes in output per adjustment.

In short, the duopoly game can be formulated as the following discrete dynamical system:

$$\left\{\begin{array}{c}{q}_1(t+1)=q_1\left(t\right)+K_1{q}_1\left(t\right){\displaystyle \frac{\partial {\Pi }_1\left(t\right)}{\partial q_1\left(t\right)}},\hfill \\ q_2(t+1)=q_2\left(t\right)+K_2{q}_2\left(t\right){\displaystyle \frac{\partial {\Pi }_2\left(t\right)}{\partial q_2\left(t\right)}},\hfill \end{array}\right.$$

(1)

where $K_1>0$ and $K_2>0$.

3. Local Stability and Bifurcations

In this section, we first examine the case of constant returns to scale ($C_i\left(q_i\right)=c_i{q}_i$) and then consider the case of decreasing returns to scale ($C_i\left(q_i\right)=c_i{q}_i^2$). For these two cases, we use different approaches to analyze the local stability. When $C_i\left(q_i\right)=c_i{q}_i$, the closed-form solution for the Nash equilibrium can be readily determined, thus facilitating the establishment of conditions for its local stability and bifurcations. However, when $C_i\left(q_i\right)=c_i{q}_i^2$, the closed-form solution for the Nash equilibrium is highly intricate. Consequently, we employ symbolic computation algorithms to analyze the stability and bifurcations without attempting to find the closed-form equilibrium. This illustrates the advantage of symbolic computation approaches in cases where the closed form of the equilibrium is intricate or challenging to compute.

3.1. Constant Returns to Scale

Let $C_1\left(q_1\right)=c_1{q}_1$ and $C_2\left(q_2\right)=c_2{q}_2$. According to Map (1), the equilibrium equations are

$$\left\{\begin{array}{c}{q}_1^{*}=q_1^{*}+K_1{q}_1^{*}\left({\displaystyle \frac{q_2^{*}}{{(q_1^{*}+q_2^{*})}^2}}-c_1\right),\hfill \\ q_2^{*}=q_2^{*}+K_2{q}_2^{*}\left({\displaystyle \frac{q_1^{*}}{{(q_1^{*}+q_2^{\ast })}^2}}-c_2\right).\hfill \end{array}\right.$$

Since $K_1,K_2>0$, we have

$$\left\{\begin{array}{c}{q}_1^{\ast }\left({\displaystyle \frac{q_2^{\ast }}{{(q_1^{\ast }+q_2^{\ast })}^2}}-c_1\right)=0,\hfill \\ q_2^{\ast }\left({\displaystyle \frac{q_1^{\ast }}{{(q_1^{\ast }+q_2^{\ast })}^2}}-c_2\right)=0,\hfill \end{array}\right.$$

from which we derive the closed form of the Nash equilibrium as follows:

$$(q_1^{\ast },q_2^{\ast })=\left({\displaystyle \frac{c_2}{{(c_1+c_2)}^2}},{\displaystyle \frac{c_1}{{(c_1+c_2)}^2}}\right).$$

Therefore, the following theorem can be derived.

Theorem 1.

Let $C_1\left(q_1\right)=c_1{q}_1$ and $C_2\left(q_2\right)=c_2{q}_2$. The Nash equilibrium is locally stable if

$$K_1{K}_2(c_1+c_2)-2(K_1+K_2)<0,$$

and

$$(K_1{K}_2(c_1+c_2)-4(K_1+K_2))c_1{c}_2+4(c_1+c_2)>0.$$

Furthermore, a Neimark–Sacker bifurcation and a period-doubling bifurcation may occur when

$$K_1{K}_2(c_1+c_2)-2(K_1+K_2)=0,$$

and

$$(K_1{K}_2(c_1+c_2)-4(K_1+K_2))c_1{c}_2+4(c_1+c_2)=0,$$

respectively. There are no fold bifurcations.

Proof. 

The local stability of the Nash equilibrium can be determined by the eigenvalues of the Jacobian matrix of Map (1):

$$J=\left\{\begin{array}{cc}{J}_{11}& J_{12}\\ J_{21}& J_{22}\end{array}\right\}=\left[\begin{array}{cc}{\displaystyle \frac{\partial q_1(t+1)}{\partial q_1\left(t\right)}}{|}_{(q_1^{\ast },q_2^{\ast })}& {\displaystyle \frac{\partial q_1(t+1)}{\partial q_2\left(t\right)}}{|}_{(q_1^{\ast },q_2^{\ast })}\\ {\displaystyle \frac{\partial q_2(t+1)}{\partial q_1\left(t\right)}}{|}_{(q_1^{\ast },q_2^{\ast })}& {\displaystyle \frac{\partial q_2(t+1)}{\partial q_2\left(t\right)}}{|}_{(q_1^{\ast },q_2^{\ast })}\end{array}\right],$$

where

$$\begin{array}{c}{\displaystyle \frac{\partial q_1(t+1)}{\partial q_1\left(t\right)}}{|}_{(q_1^{\ast },q_2^{\ast })}=1-2K_1{\displaystyle \frac{c_1{c}_2}{c_1+c_2}},\hfill \\ {\displaystyle \frac{\partial q_1(t+1)}{\partial q_2\left(t\right)}}{|}_{(q_1^{\ast },q_2^{\ast })}=K_1{\displaystyle \frac{(c_2-c_1)c_2}{c_1+c_2}},\hfill \\ {\displaystyle \frac{\partial q_2(t+1)}{\partial q_1\left(t\right)}}{|}_{(q_1^{\ast },q_2^{\ast })}=K_2{\displaystyle \frac{(c_1-c_2)c_1}{c_1+c_2}},\hfill \\ {\displaystyle \frac{\partial q_2(t+1)}{\partial q_2\left(t\right)}}{|}_{(q_1^{\ast },q_2^{\ast })}=1-2K_2{\displaystyle \frac{c_1{c}_2}{c_1+c_2}}.\hfill \end{array}$$

The eigenvalues of J can be written as

$${\lambda }_{1,2}=1-{\displaystyle \frac{c_1{c}_2(K_1+K_2)\pm \sqrt{c_1{c}_2(c_1{K}_1-c_2{K}_2)(c_2{K}_1-c_1{K}_2)}}{c_1+c_2}}.$$

It is well known that the Nash equilibrium is locally stable if $|{\lambda }_{1,2}|<1$.

If $(c_1{K}_1-c_2{K}_2)(c_2{K}_1-c_1{K}_2)\ge 0$, then ${\lambda }_{1,2}$ are real numbers. Thus, the stability condition becomes $-1<{\lambda }_1\le {\lambda }_2<1$. One can see that

$$c_1{c}_2(K_1+K_2)>\sqrt{c_1{c}_2(c_1{K}_1-c_2{K}_2)(c_2{K}_1-c_1{K}_2)},$$

which implies that ${\lambda }_2<1$ always holds. In addition,

$${\lambda }_1>-1\iff (K_1{K}_2(c_1+c_2)-4(K_1+K_2))c_1{c}_2+4(c_1+c_2)>0.$$

If $(c_1{K}_1-c_2{K}_2)(c_2{K}_1-c_1{K}_2)<0$, then ${\lambda }_{1,2}$ are a pair of conjugate complex numbers. Accordingly, we have

$$|{\lambda }_{1,2}|=1+{\displaystyle \frac{(K_1{K}_2(c_1+c_2)-2(K_1+K_2))c_1{c}_2}{c_1+c_2}}.$$

Furthermore,

$$|{\lambda }_{1,2}|<1\iff K_1{K}_2(c_1+c_2)-2(K_1+K_2)<0.$$

Therefore, the Nash equilibrium is locally stable if

$$K_1{K}_2(c_1+c_2)-2(K_1+K_2)<0,$$

and

$$(K_1{K}_2(c_1+c_2)-4(K_1+K_2))c_1{c}_2+4(c_1+c_2)>0.$$

According to the classical bifurcation theory, a Neimark–Sacker bifurcation and a period-doubling bifurcation may occur when

$$K_1{K}_2(c_1+c_2)-2(K_1+K_2)=0,$$

and

$$(K_1{K}_2(c_1+c_2)-4(K_1+K_2))c_1{c}_2+4(c_1+c_2)=0,$$

respectively. The proof is completed.    □

Figure 1 depicts the two-dimensional cross-sections of the four-dimensional stability region, wherein the algebraic varieties (bifurcation curves) $(K_1{K}_2(c_1+c_2)-4(K_1+K_2))c_1{c}_2+4(c_1+c_2)=0$ and $K_1{K}_2(c_1+c_2)-2(K_1+K_2)=0$ are colored in blue and red, respectively. As shown in Figure 1a, it can be observed that when the cost parameters and adjustment speed parameters are identical for both players, i.e., $c_1=c_2$ and $K_1=K_2$, raising these variables may result in the destabilization of the Nash equilibrium. It is noteworthy that the blue and red curves are in alignment, indicating the necessity for further analysis to ascertain the type of bifurcation. As illustrated in Figure 1b, when $K_1=1$ and $K_2=1/2$, two potential routes also exist for the destabilization of the Nash equilibrium, i.e., through a Neimark–Sacker bifurcation or through a period-doubling bifurcation. To be precise, if the values of $c_1$ and $c_2$ are in close proximity to one another, then only a period-doubling bifurcation may occur. Conversely, if the values of $c_1$ and $c_2$ are sufficiently distinct, then a Neimark–Sacker bifurcation may ensue. Similarly, Figure 1c illustrates that when $c_1=1$ and $c_2=1/2$, two potential routes exist for the destabilization of the Nash equilibrium. When the discrepancy between $K_1$ and $K_2$ is sufficiently pronounced, the model may undergo a period-doubling bifurcation. In contrast, when the discrepancy between $K_1$ and $K_2$ is relatively minor, the model may undergo a Neimark–Sacker bifurcation. Figure 1d clearly shows that when the costs of the two firms are identical (but $K_1\ne K_2$), the Nash equilibrium is only likely to lose its local stability through a period-doubling bifurcation, which is formally proven in Corollary 1.

Corollary 1.

Let $C_i=c{q}_i$ for $i=1,2$. The Nash equilibrium is locally stable if $c<2/K_1^2$ and $c<2/K_2^2$. Furthermore, provided that $K_1\ne K_2$, a period-doubling bifurcation may occur when $c=2/K_1^2$ or $c=2/K_2^2$, and there are no other bifurcations.

Proof. 

Recall Theorem 1. Plugging $c_1=c_2=c$ into $K_1{K}_2(c_1+c_2)-2(K_1+K_2)<0$ yields

$$2c{K}_1{K}_2-2(K_1+K_2)<0,$$

which is equivalent to

$$c<{\displaystyle \frac{K_1+K_2}{K_1{K}_2}}.$$

(2)

Moreover, plugging $c_1=c_2=c$ into $(K_1{K}_2(c_1+c_2)-4(K_1+K_2))c_1{c}_2+4(c_1+c_2)>0$ yields

$$2c(c{K}_2-2)(c{K}_1-2)>0,$$

which can be transformed into

$$c<{\displaystyle \frac{2}{K_1}}\mathrm{and}c<{\displaystyle \frac{2}{K_2}},$$

or

$$c>{\displaystyle \frac{2}{K_1}}\mathrm{and}c>{\displaystyle \frac{2}{K_2}}.$$

However, if $c>2/K_1$ and $c>2/K_2$, then we have

$$c>{\displaystyle \frac{1}{K_1}}+{\displaystyle \frac{1}{K_2}}={\displaystyle \frac{K_1+K_2}{K_1{K}_2}},$$

which means that $c>2/K_1$ and $c>2/K_2$ are in conflict with (2). On the other hand, it can be easily verified that $c<2/K_1$ and $c<2/K_2$ implies (2). Accordingly, we can conclude that the Nash equilibrium is locally stable if $c<2/K_1$ and $c<2/K_2$. Furthermore, provided that $K_1\ne K_2$, a period-doubling bifurcation may occur when $c=2/K_1$ or $c=2/K_2$, and there are no other bifurcations.    □

3.2. Decreasing Returns to Scale

Let $C_1\left(q_1\right)=c_1{q}_1^2$ and $C_2\left(q_2\right)=c_2{q}_2^2$. According to Map (1), the equilibrium equations are

$$\left\{\begin{array}{c}{q}_1^{\ast }=q_1^{\ast }+K_1{q}_1^{\ast }\left({\displaystyle \frac{q_2^{\ast }}{{(q_1^{\ast }+q_2^{\ast })}^2}}-2c_1{q}_1^{\ast }\right),\hfill \\ q_2^{\ast }=q_2^{\ast }+K_2{q}_2^{\ast }\left({\displaystyle \frac{q_1^{\ast }}{{(q_1^{\ast }+q_2^{\ast })}^2}}-2c_2{q}_2^{\ast }\right).\hfill \end{array}\right.$$

Since $K_1,K_2>0$, we obtain the equations as follows:

$$\left\{\begin{array}{c}{q}_1^{\ast }\left({\displaystyle \frac{q_2^{\ast }}{{(q_1^{\ast }+q_2^{\ast })}^2}}-2c_1{q}_1^{\ast }\right)=0,\hfill \\ q_2^{\ast }\left({\displaystyle \frac{q_1^{\ast }}{{(q_1^{\ast }+q_2^{\ast })}^2}}-2c_2{q}_2^{\ast }\right)=0.\hfill \end{array}\right.$$

The closed-form solutions of the above equations are intricate and challenging to analyze further (for example, see [9,21]). Accordingly, the triangular decomposition method is employed below to describe the Nash equilibrium. For further details regarding the triangular decomposition, the reader is referred to [22,23,24,25]. In essence, the triangular decomposition method can decompose a system of polynomial equations into some systems of polynomial equations in the triangular form (i.e., triangular sets) without altering the solutions.

We use $(·)$ to denote the numerator. Then the solutions of the equilibrium equations are the same as those of

$$\mathcal{P}=\left[Num\left(q_1^{\ast }\left({\displaystyle \frac{q_2^{\ast }}{{(q_1^{\ast }+q_2^{\ast })}^2}}-2c_1{q}_1^{\ast }\right)\right),Num\left(q_2^{\ast }\left({\displaystyle \frac{q_1^{\ast }}{{(q_1^{\ast }+q_2^{\ast })}^2}}-2c_2{q}_2^{\ast }\right)\right)\right].$$

The triangular decomposition method allows us to decompose $\mathcal{P}$ into two triangular sets, $[q_2^{\ast },q_1^{\ast }]$ and $[T_2,T_1]$, where

$$\begin{array}{c}{T}_2=(c_1{c}_2+c_2^2)q_2^{\ast }-(2c_1^3-4c_1^2{c}_2+2c_1{c}_2^2)q_1^{\ast 3}-2c_1{c}_2{q}_1^{\ast },\hfill \\ T_1=(4c_1^3-8c_1^2{c}_2+4c_1{c}_2^2)q_1^{\ast 4}+8c_1{c}_2{q}_1^{\ast 2}-c_2.\hfill \end{array}$$

It is evident that the zero of $[q_2^{\ast },q_1^{\ast }]$, that is, $q_1^{\ast }=q_2^{\ast }=0$, is not defined in the equilibrium equations. Consequently, our attention is limited to the triangular set $[T_2,T_1]$, which corresponds to the Nash equilibrium.

It can be observed that the polynomial $T_2$ has two variables, namely $q_1^{\ast }$ and $q_2^{\ast }$, yet is linear with respect to $q_2^{\ast }$. Therefore, we can conclude that

$$q_2^{\ast }={\displaystyle \frac{(2c_1^3-4c_1^2{c}_2+2c_1{c}_2^2)q_1^{\ast 3}+2c_1{c}_2{q}_1^{\ast }}{c_1{c}_2+c_2^2}}.$$

The solution for $q_1^{\ast }$ can be derived from $T_1$, but they are complex and therefore difficult to analyze in terms of local stability. In a previous study [26], the first author of this paper and his coauthor proposed a symbolic computation approach (for a comparable approach, the reader is directed to [27]) to analyze the equilibrium of static models without solving its closed form. In what follows, we will borrow the main idea of this approach to derive the conditions for the local stability and bifurcations of the Nash equilibrium.

The Sylvester resultant (or, for brevity, called the resultant) plays an important role in our approach. Let us consider two univariate polynomials in the variable x, designated by the symbols $A\left(x\right)$ and $B\left(x\right)$. In this context, the notation res$(A,B,x)$ is used to denote the resultant of polynomials $A\left(x\right)$ and $B\left(x\right)$ with respect to x. As demonstrated in [28], there exist two polynomials, $F\left(x\right)$ and $G\left(x\right)$, such that res$(A,B,x)$ can be expressed as $FA+GB$. This implies that res$(A,B,x)=0$ if and only if $A\left(x\right)$ and $B\left(x\right)$ have a common zero in the field of complex numbers.

Remark 1.

Let us consider a triangular set $\mathcal{S}=[S_1\left(x\right),S_2(x,y)]$ and a polynomial $P(x,y)$. Denote

$$res(P,\mathcal{S})\equiv res(res(P,S_2,y),S_1,x).$$

If we have $S_1=0$ and $S_2=0$ (or simply denoted as $\mathcal{S}=0$), then it can be shown that $P=0$ implies that $res(P,\mathcal{S})=0$. That is to say, $res(P,\mathcal{S})=0$ is a necessary condition for $P=0$.

In order to investigate the local stability of the Nash equilibrium, we calculate the Jacobian matrix associated with (1) as follows:

$$H=\left\{\begin{array}{cc}{H}_{11}& H_{12}\\ H_{21}& H_{22}\end{array}\right\},$$

where

$$\begin{array}{c}{H}_{11}=K_1\left({\displaystyle \frac{1}{q_1^{\ast }+q_2^{\ast }}}-{\displaystyle \frac{q_1^{\ast }}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^2}}-2c_1{q}_1^{\ast }\right)+K_1\left({\displaystyle \frac{2q_1^{\ast }}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^3}}-{\displaystyle \frac{2}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^2}}-2c_1\right)q_1^{\ast }+1,\hfill \\ H_{12}=K_1\left({\displaystyle \frac{2q_1^{\ast }}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^3}}-{\displaystyle \frac{1}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^2}}\right)q_1^{\ast },\hfill \\ H_{21}=K_2\left({\displaystyle \frac{2q_2^{\ast }}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^3}}-{\displaystyle \frac{1}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^2}}\right)q_2^{\ast },\hfill \\ H_{22}=K_2\left({\displaystyle \frac{1}{q_1^{\ast }+q_2^{\ast }}}-{\displaystyle \frac{q_2^{\ast }}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^2}}-2c_2{q}_2^{\ast }\right)+K_2\left({\displaystyle \frac{2q_2^{\ast }}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^3}}-{\displaystyle \frac{2}{{\left(q_1^{\ast }+q_2^{\ast }\right)}^2}}-2c_2\right)q_2^{\ast }+1.\hfill \end{array}$$

The characteristic polynomial of H is denoted as

$$CP\left(\lambda \right)={\lambda }^2-Tr\left(H\right)\lambda +Det\left(H\right),$$

where $Tr\left(H\right)=H_{11}+H_{22}$ and $Det\left(H\right)=H_{11}{H}_{22}-H_{12}{H}_{21}$ are the trace and the determinant of H, respectively. In accordance with the Jury criterion [29], the conditions for the local stability of the Nash equilibrium are as follows:

$$\begin{array}{cc}\hfill & C{D}_1^H\equiv CP\left(1\right)=1-Tr\left(H\right)+Det\left(H\right)>0,\hfill \\ \hfill & C{D}_2^H\equiv CP(-1)=1+Tr\left(H\right)+Det\left(H\right)>0,\hfill \\ \hfill & C{D}_3^H\equiv 1-Det\left(H\right)>0.\hfill \end{array}$$

Remark 2.

In accordance with the classical bifurcation theory, it is established that a discrete dynamic system may exhibit a fold, period-doubling, or Neimark–Sacker bifurcation when the equilibrium point loses its stability when $C{D}_1^H=0$, $C{D}_2^H=0$, or $C{D}_3^H=0$, respectively.

Following Remark 2, we conduct a bifurcation analysis first. It is necessary to ascertain the parameter conditions for $C{D}_1^H=0$, $C{D}_2^H=0$ and $C{D}_3^H=0$ at the Nash equilibrium. Note that $C{D}_1^H$, $C{D}_2^H$ and $C{D}_3^H$ are rational functions. Thus, the problem of identifying bifurcations is transformed into finding the conditions for $Num\left(C{D}_1^H\right)=0$, $Num\left(C{D}_2^H\right)=0$ and $Num\left(C{D}_3^H\right)=0$ when $T_1\left(q_1^{\ast }\right)=0$ and $T_2(q_1^{\ast },q_2^{\ast })=0$ (or simply $\mathcal{T}=0$ if we denote $\mathcal{T}=[T_2,T_1]$).

By computing the resultants, we acquire that

$$\begin{array}{cc}\hfill & res(Num\left(C{D}_1^H\right),\mathcal{T})=-4294967296c_1^{14}{c}_2^{17}{\left(c_1+c_2\right)}^{16}{\left(c_1-c_2\right)}^{22}{K}_1^4{K}_2^4,\hfill \\ \hfill & res(Num\left(C{D}_2^H\right),\mathcal{T})=-67108864c_1^{10}{c}_2^{13}{(c_1+c_2)}^{16}{(c_1-c_2)}^{22}·R_1,\hfill \\ \hfill & res(Num\left(C{D}_3^H\right),\mathcal{T})=-4194304c_1^{11}{c}_2^{14}{(c_1+c_2)}^{16}{(c_1-c_2)}^{22}·R_2,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill R_1=& -64c_1^4{c}_2^4{K}_1^4{K}_2^4+96c_1^4{c}_2^3{K}_1^4{K}_2^2-32c_1^4{c}_2^3{K}_1^3{K}_2^3-32c_1^3{c}_2^4{K}_1^3{K}_2^3+96c_1^3{c}_2^4{K}_1^2{K}_2^4+c_1^5{c}_2{K}_1^4\hfill \\ \hfill & -18c_1^4{c}_2^2{K}_1^4+32c_1^4{c}_2^2{K}_1^3{K}_2-18c_1^4{c}_2^2{K}_1^2{K}_2^2+81c_1^3{c}_2^3{K}_1^4+96c_1^3{c}_2^3{K}_1^3{K}_2-92c_1^3{c}_2^3{K}_1^2{K}_2^2\hfill \\ \hfill & +96c_1^3{c}_2^3{K}_1{K}_2^3+81c_1^3{c}_2^3{K}_2^4-18c_1^2{c}_2^4{K}_1^2{K}_2^2+32c_1^2{c}_2^4{K}_1{K}_2^3-18c_1^2{c}_2^4{K}_2^4+c_1{c}_2^5{K}_2^4+8c_1^4{c}_2{K}_1^2\hfill \\ \hfill & -8c_1^4{c}_2{K}_1{K}_2-16c_1^3{c}_2^2{K}_1^2-120c_1^3{c}_2^2{K}_1{K}_2-120c_1^3{c}_2^2{K}_2^2-120c_1^2{c}_2^3{K}_1^2-120c_1^2{c}_2^3{K}_1{K}_2\hfill \\ \hfill & -16c_1^2{c}_2^3{K}_2^2-8c_1{c}_2^4{K}_1{K}_2+8c_1{c}_2^4{K}_2^2-4c_1^4+16c_1^3{c}_2-24c_1^2{c}_2^2+16c_1{c}_2^3-4c_2^4,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill R_2=& -1024c_1^3{c}_2^3{K}_1^4{K}_2^4+384c_1^3{c}_2^2{K}_1^4{K}_2^2+384c_1^3{c}_2^2{K}_1^3{K}_2^3+384c_1^2{c}_2^3{K}_1^3{K}_2^3+384c_1^2{c}_2^3{K}_1^2{K}_2^4+c_1^4{K}_1^4\hfill \\ \hfill & -18c_1^3{c}_2{K}_1^4-32c_1^3{c}_2{K}_1^3{K}_2-18c_1^3{c}_2{K}_1^2{K}_2^2+81c_1^2{c}_2^2{K}_1^4+288c_1^2{c}_2^2{K}_1^3{K}_2+420c_1^2{c}_2^2{K}_1^2{K}_2^2\hfill \\ \hfill & +288c_1^2{c}_2^2{K}_1{K}_2^3+81c_1^2{c}_2^2{K}_2^4-18c_1{c}_2^3{K}_1^2{K}_2^2-32c_1{c}_2^3{K}_1{K}_2^3-18c_1{c}_2^3{K}_2^4+c_2^4{K}_2^4.\hfill \end{array}$$

It is a known fact that $c_1,c_2>0$ and $K_1,K_2>0$. In accordance with Remarks 1 and 2, it can be deduced that $c_1-c_2=0$ is a necessary condition for the existence of fold bifurcations provided that $\mathcal{T}=0$. It can also be seen that if period-doubling bifurcations occur, then either $c_1-c_2=0$ or $R_1=0$, while if Neimark–Sacker bifurcations take place, then either $c_1-c_2=0$ or $R_2=0$. However, further analysis is required to determine whether these bifurcations are indeed taking place, and this will involve an examination of the local stability.

To analyze the local stability, it is necessary to determine the signs of $C{D}_1^H$, $C{D}_2^H$ and $C{D}_3^H$ at the equilibrium. We use $Den(·)$ to denote the denominator. If $Den\left(C{D}_i^H\right)\ne 0$, then $C{D}_i^H$ and $Num\left(C{D}_i^H\right)·Den\left(C{D}_i^H\right)$ have the same signs. It can be computed that

$$\begin{array}{cc}\hfill & res(Num\left(C{D}_1^H\right)·Den\left(C{D}_1^H\right),\mathcal{T})=17592186044416c_1^{20}{c}_2^{26}{\left(c_1+c_2\right)}^{28}{\left(c_1-c_2\right)}^{40}{K}_1^4{K}_2^4,\hfill \\ \hfill & res(Num\left(C{D}_2^H\right)·Den\left(C{D}_2^H\right),\mathcal{T})=274877906944c_1^{16}{c}_2^{22}{(c_1+c_2)}^{28}{(c_1-c_2)}^{40}·R_1,\hfill \\ \hfill & res(Num\left(C{D}_3^H\right)·Den\left(C{D}_3^H\right),\mathcal{T})=17179869184c_1^{17}{c}_2^{23}{(c_1+c_2)}^{28}{(c_1-c_2)}^{40}·R_2.\hfill \end{array}$$

The feasible parameter set consists of all tuples of the form $(c_1,c_2,K_1,K_2)$ where $c_1,c_2>0$ and $K_1,K_2>0$. The algebraic varieties $res(Num\left(C{D}_1^H\right)·Den\left(C{D}_1^H\right),\mathcal{T})=0$, $i=1,2,3$, divide the feasible parameter set into several regions. In each region, the signs of $Num\left(C{D}_i^H\right)·Den\left(C{D}_i^H\right)$ (equivalently, the signs of $C{D}_i^H$), $i=1,2,3$, remain constant. Therefore, it is sufficient to select at least one sample point from each region and identify the signs of $C{D}_i^H$, $i=1,2,3$, at the selected sample point. The selection of sample points may be a challenging process in general and can be automated using, for example, the partial cylindrical algebraic decomposition (PCAD) method [30].

As demonstrated by our implementation, the PCAD method generates a total of 2869 sample points. Due to space limitations, it is not feasible to report all of the sample points here. (These sample points are available from the corresponding author upon request. Also, the reader can compute the sample points by directly using the SamplePoints function in the RegularChains package of Maple 2022.) However, a selection of these points is presented in

Table 1. Some sample points in $\left\{(c_1,c_2,K_1,K_2)\right|c_1,c_2>0\mathrm{and}{K}_1,K_2>0\}$.

$({\mathit{c}}_1,{\mathit{c}}_2,{\mathit{K}}_1,{\mathit{K}}_2)$	Stable?	${\mathit{R}}_1$	${\mathit{R}}_2$	$({\mathit{c}}_1,{\mathit{c}}_2,{\mathit{K}}_1,{\mathit{K}}_2)$	Stable?	${\mathit{R}}_1$	${\mathit{R}}_2$
$(1,2,5/4,1)$	Yes	−	+	$(1,2,5/4,19/16)$	No	+	+
$(1,2,21/16,1)$	Yes	−	+	$(1,2,21/16,2)$	No	+	+
$(1,21/4,5/4,1/2)$	Yes	−	+	$(1,21/4,5/4,1)$	No	+	+
$(1,6,27/32,1/2)$	Yes	−	+	$(1,6,27/32,1)$	No	−	−
$(1,51/2,45/32,1/4)$	Yes	−	+	$(1,51/2,45/32,1)$	No	+	+
$(1,26,3/4,1/2)$	Yes	−	+	$(1,26,3/4,1)$	No	−	−
$(1,34,31/32,1/2)$	Yes	−	+	$(1,34,31/32,45/64)$	No	+	+
$(1,82,13/8,1/4)$	Yes	−	+	$(1,82,13/8,1)$	No	+	+

, which also provides information regarding the signs of $R_1,R_2$ and the local stability (this is equivalent to the simultaneous fulfillment of the conditions $C{D}_1^H>0$, $C{D}_2^H>0$, and $C{D}_3^H>0$) of the Nash equilibrium.

Upon verification of all 2869 sample points, it can be observed that $C{D}_1^H>0$, $C{D}_2^H>0$, and $C{D}_3^H>0$ if and only if $R_1<0$ and $R_2>0$. This leads to the conclusion of the following theorem.

Theorem 2.

Let $C_1\left(q_1\right)=c_1{q}_1^2$ and $C_2\left(q_2\right)=c_2{q}_2^2$. The Nash equilibrium is locally stable if $R_1<0$ and $R_2>0$. Furthermore, a period-doubling bifurcation may occur when $R_1=0$, while a Neimark–Sacker bifurcation may occur when $R_2=0$. There are no fold bifurcations.

The computational procedure is summarized as follows:

Step 1. Compute the triangular sets ${\mathcal{T}}_k$, $k=1,\dots ,s$, of the equilibrium equations using the triangular decomposition method.

Step 2. Evaluate the Jacobian matrix H at the Nash equilibrium point. The corresponding stability conditions are denoted $C{D}_i^H>0$ for $i=1,2,3$.

Step 3. For $k=1,\dots ,s$ and $i=1,2,3$, compute $res(Num\left(C{D}_i^H\right),{\mathcal{T}}_k)$. This yields the parameter conditions under which the Nash equilibrium may undergo bifurcations.

Step 4. For $k=1,\dots ,s$ and $i=1,2,3$, compute $res(Num\left(C{D}_i^H\right)·Den\left(C{D}_i^H\right),{\mathcal{T}}_k)$. Using the PCAD method, sample points can be selected within the parameter regions divided by the algebraic varieties $res(Num\left(C{D}_i^H\right)·Den\left(C{D}_i^H\right),{\mathcal{T}}_k)=0$. This provides the parameter conditions under which the Nash equilibrium is locally stable.

Figure 2 depicts the two-dimensional cross-sections of the four-dimensional stability region, wherein the algebraic varieties $R_1=0$ and $R_2=0$ are represented by blue and red, respectively. As illustrated in Figure 2a, it can be observed that when the cost parameters and adjustment speed parameters are identical for both players, an increase in these variables may result in the destabilization of the Nash equilibrium. It is worth noting that the blue and red curves coincide with each other, meaning that the type of bifurcation needs to be identified by further analysis. As illustrated in Figure 2b, when $K_1=1$ and $K_2=1/2$, if the value of $c_2$ is sufficiently large, an increase in $c_1$ will result in the loss of stability through a Neimark–Sacker bifurcation. Nevertheless, if the value of $c_2$ is sufficiently small, an increase in $c_1$ will result in the model losing its stability through a period-doubling bifurcation. We should mention that the effects of $c_1$ and $c_2$ revealed by Figure 2b are starkly divergent from those observed in Figure 1b. Figure 2c demonstrates that when $c_1=1$ and $c_2=1/2$, two potential routes exist for the destabilization of the Nash equilibrium, i.e., through a Neimark–Sacker bifurcation or through a period-doubling bifurcation. In particular, when the discrepancy between $K_1$ and $K_2$ is sufficiently pronounced, the model may exhibit a period-doubling bifurcation. In contrast, when the discrepancy between $K_1$ and $K_2$ is relatively minor, the model may undergo a Neimark–Sacker bifurcation. Figure 2d indicates that when the costs of the two firms are identical, the Nash equilibrium is only likely to lose stability through a period-doubling bifurcation. This observation is formally proven in Corollary 2.

Corollary 2.

Let $C_i=c{q}_i^2$ for $i=1,2$. The Nash equilibrium is locally stable if $c<2/K_1^2$ and $c<2/K_2^2$. Furthermore, provided that $K_1\ne K_2$, a period-doubling bifurcation may occur when $c=2/K_1^2$ or $c=2/K_2^2$, and there are no other bifurcations.

Proof. 

Recall Theorem 2. Plugging $c_1=c_2=c$ into $R_1<0$ yields

$$64c^4\left(8c{K}_1^2{K}_2^2+{(K_1+K_2)}^2\right)\left(2c{K}_1^2{K}_2^2-{(K_1+K_2)}^2\right)<0,$$

which is equivalent to

$$c<{\displaystyle \frac{{(K_1+K_2)}^2}{2K_1^2{K}_2^2}}.$$

(3)

Moreover, plugging $c_1=c_2=c$ into $R_2>0$ yields

$$32c^5\left(2c{K}_1^2{K}_2^2+{(K_1+K_2)}^2\right)(c{K}_1^2-2)(c{K}_2^2-2)>0,$$

which can be transformed into

$$c<{\displaystyle \frac{2}{K_1^2}}\mathrm{and}c<{\displaystyle \frac{2}{K_2^2}},$$

or

$$c>{\displaystyle \frac{2}{K_1^2}}\mathrm{and}c>{\displaystyle \frac{2}{K_2^2}}.$$

However, if $c>2/K_1^2$ and $c>2/K_2^2$, then we have

$$c>{\displaystyle \frac{1}{K_1^2}}+{\displaystyle \frac{1}{K_2^2}}>{\displaystyle \frac{{(K_1+K_2)}^2}{2K_1^2{K}_2^2}},$$

which means that $c>2/K_1^2$ and $c>2/K_2^2$ are in conflict with (3). On the other hand, it can be verified that $c<2/K_1^2$ and $c<2/K_2^2$ implies (3).

First, assume that $K_1\ge K_2$. We have

$${\displaystyle \frac{{(K_1+K_2)}^2}{2K_1^2{K}_2^2}}-{\displaystyle \frac{2}{K_1^2}}={\displaystyle \frac{{(K_1+K_2)}^2-4K_2^2}{2K_1^2{K}_2^2}}={\displaystyle \frac{K_1^2+2K_1{K}_2+K_2^2-4K_2^2}{2K_1^2{K}_2^2}}\ge 0.$$

Thus, $c<2/K_2^2$ implies

$$c<{\displaystyle \frac{{(K_1+K_2)}^2}{2K_1^2{K}_2^2}}.$$

Furthermore, in the case where $K_1<K_2$, we can similarly prove that $c<2/K_1^2$ implies

$$c<{\displaystyle \frac{{(K_1+K_2)}^2}{2K_1^2{K}_2^2}}.$$

Accordingly, we can conclude that the Nash equilibrium is locally stable if $c<2/K_1^2$ and $c<2/K_2^2$. Furthermore, provided that $K_1\ne K_2$, a period-doubling bifurcation may occur when $c=2/K_1^2$ or $c=2/K_2^2$, and there are no other bifurcations. □

4. Numerical Simulations

In this section, numerical simulations are conducted under both constant and decreasing returns to scale. The results confirm the conclusions obtained in the previous section and provide further insight into the complex dynamics of the model considered in this paper.

4.1. Constant Returns to Scale

Figure 3 depicts the one-dimensional bifurcation diagrams under constant returns to scale. In these diagrams, the initial conditions are chosen in the neighborhood of the unique equilibrium to illustrate the process by which the equilibrium loses stability. (Under constant returns to scale, the basin of attraction is smaller, so trajectories tend to diverge even with slight deviations from the equilibrium. This explains why the initial conditions selected under constant returns to scale are not the same.)

In Figure 3a,b, $K_2$ is treated as the bifurcation parameter, and the initial condition is set to $(q_1\left(0\right),q_2\left(0\right))=(0.2,0.4)$. Panel (a) shows that, when $K_1=1.35$, as $K_2$ increases, the equilibrium of the model undergoes a cascade of period-doubling bifurcations leading to chaotic dynamics. In contrast, panel (b) illustrates that, when $K_1=2.7$, the model loses stability via a Neimark–Sacker bifurcation as $K_2$ increases. Moreover, it can be observed that periodic orbits of different orders may emerge in the parameter intervals between the appearance of invariant closed curves as $K_2$ varies. In short, variations in $K_2$ may cause the model to undergo different types of bifurcations, depending on the values of the other parameters.

In Figure 3c,d, $c_1$ is taken as the bifurcation parameter. It should be noted that the initial conditions differ across the two panels: $(q_1\left(0\right),q_2\left(0\right))=(0.08,0.16)$ in panel (c) and $(q_1\left(0\right),q_2\left(0\right))=(0.056,0.111)$ in panel (d). Panel (c) shows that, when $c_2=3$, increasing $c_1$ destabilizes the model through a sequence of period-doubling bifurcations. In contrast, panel (d) illustrates that, when $c_2=1.9$, an increase in $c_1$ causes the Nash equilibrium to lose stability via a Neimark–Sacker bifurcation. Similarly, the above observations indicate that variations in the parameter $c_1$ may lead to the occurrence of different types of bifurcations.

Phase portraits providing further details of the dynamics under constant returns to scale are presented in Figure 4 and Figure 5. The first 2000 trajectory points are shown in gray, while the last 1000 points are highlighted in black.

Figure 4 depicts several representative phase portraits illustrating the process by which the model loses stability through a cascade of period-doubling bifurcations when $K_1=1.35$, $c_1=1$, and $c_2=0.5$. As shown in panels (a)–(c), 1-cycle, 2-cycle, and 4-cycle orbits can be observed. Specifically, when $K_2=3.200000$, the trajectory converges rapidly to the unique Nash equilibrium. Panel (b) shows a stable 2-cycle orbit at $K_2=3.398947$, while panel (c) displays a stable 4-cycle orbit at $K_2=3.624040$. Moreover, when $K_2$ is sufficiently large, chaotic attractors may emerge. For instance, as illustrated in panels (d) and (e), chaotic attractors with four and two pieces arise at $K_2=3.653333$ and $K_2=3.661414$, respectively.

On the other hand, Figure 5 presents typical phase portraits illustrating the process by which the model undergoes a Neimark–Sacker bifurcation as $K_2$ varies, with $K_1=2.7$, $c_1=1$, and $c_2=0.5$ held fixed. As shown in panel (a), when $K_2=2.620000$, the trajectory converges to the equilibrium in a rotational manner. In panel (b), the trajectory approaches an invariant closed curve when $K_2=2.636583$. High-order periodic orbits may also arise; for instance, panel (c) shows such a case at $K_2=2.638090$. From panel (d), it can be seen that the trajectory is attracted to an invariant closed curve with a different shape, while panel (e) illustrates the emergence of a broken invariant closed curve. Finally, when $K_2$ is sufficiently large, chaotic dynamics may occur, as shown in panel (f). A comparison of Figure 4 and Figure 5 reveals that the routes through which the equilibrium loses stability are entirely different in the two cases.

For additional insights into the dynamics of the considered game under constant returns to scale, see the two-dimensional bifurcation diagrams in Figure 6. In these diagrams, different colors are used to indicate convergence to periodic orbits of different orders. Specifically, dark blue points correspond to trajectories converging to a 1-cycle (i.e., an equilibrium), while light blue points represent convergence to a 2-cycle. Yellow points indicate trajectories with periods greater than or equal to 24, and gray points denote trajectories that diverge (i.e., tend to ∞).

Figure 6a–d correspond to Figure 1a–d, respectively. Panel (a) considers the case of homogeneous costs, i.e., $c_1=c_2$, and homogeneous adjustment speeds, i.e., $K_1=K_2$. It can be observed that the Nash equilibrium may lose stability through a Neimark–Sacker bifurcation when either $c_1=c_2$ or $K_1=K_2$ is sufficiently large. Panel (b) illustrates that, when $K_1=1$ and $K_2=1/2$, both period-doubling and Neimark–Sacker bifurcations may arise when $c_1$ and $c_2$ are treated as free parameters. We also note that Arnold tongues can be observed in panel (b).

In contrast, Figure 6c,d explore the dynamics of the model when $K_1$ and $K_2$ are treated as free parameters. The cases of heterogeneous costs ($c_1=1$ and $c_2=1/2$) and homogeneous costs ($c_1=c_2=1$) are illustrated in panels (c) and (d), respectively. Panel (c) shows that the Nash equilibrium can lose stability via two possible routes, namely, through a period-doubling bifurcation or a Neimark–Sacker bifurcation. In contrast, panel (d) indicates that the equilibrium undergoes only period-doubling bifurcations, which confirms the result of Corollary 1. In conclusion, the numerical simulations presented in Figure 6 are fully consistent with the analytical results shown in Figure 1.

4.2. Decreasing Returns to Scale

Figure 7 presents the one-dimensional bifurcation diagrams under decreasing returns to scale, with the initial state chosen as $(q_1\left(0\right),q_2\left(0\right))=(0.05,0.05)$. In Figure 7a,b (respectively, Figure 7c,d), $K_2$ (respectively, $c_1$) is treated as the bifurcation parameter.

Panel (a) shows that, when $K_1=1$, as $K_2$ increases, the Nash equilibrium of the model transitions to complex dynamics through a cascade of period-doubling bifurcations. Panel (b) illustrates that, when $K_1=1.6$, the Nash equilibrium loses stability via a Neimark–Sacker bifurcation as $K_2$ increases. In summary, variations in $K_2$ may cause the model to undergo different types of bifurcations, depending on the values of the other parameters.

From Figure 7c, it can be observed that, when $c_2=4$, increasing $c_1$ destabilizes the model through a sequence of period-doubling bifurcations. Panel (d) shows that, when $c_2=12$, an increase in $c_1$ leads to a loss of stability via a Neimark–Sacker bifurcation. In short, variations in $c_1$ may cause the model to undergo different types of bifurcations, depending on the values of the other parameters.

Further details regarding the period-doubling bifurcation are provided in Figure 8, which presents several representative phase portraits for $c_2=4$, $K_1=1$, and $K_2=0.5$.

It can be observed that, when $c_1=1.775758$, the trajectory converges to the unique Nash equilibrium. Panel (b) of Figure 8 depicts a stable 2-cycle orbit at $c_1=1.788889$, while panel (c) shows a stable 4-cycle orbit at $c_1=2.734343$. Moreover, when $c_1$ is sufficiently large, chaotic attractors may emerge. For instance, panel (e) illustrates a chaotic attractor with two pieces at $c_1=3.131658$. Further increasing $c_1$ to the same value (e.g., $c_1=3.131658$) results in a strange attractor with a single piece, as shown in panel (f).

On the other hand, Figure 9 presents several phase portraits for $c_2=12$, $K_1=1$, and $K_2=0.5$, illustrating the occurrence of a Neimark–Sacker bifurcation.

It can be observed that, when $c_1=1.800000$, the trajectory rotationally converges to the Nash equilibrium. Panel (b) of Figure 9 shows that the trajectory approaches an invariant closed curve in a rotational manner when $c_1=1.906061$. High-order periodic orbits may also occur; for instance, a 9-cycle orbit is observed at $c_1=2.153535$, as illustrated in panel (c). As $c_1$ increases further, broken invariant closed curves may emerge, as shown in panels (d) and (e). When $c_1$ becomes sufficiently large, chaotic dynamics may appear, as illustrated in panel (f).

Similarly, in Figure 10, we present the two-dimensional bifurcation diagrams under decreasing returns to scale, providing additional insights into the dynamics of the considered game.

It should be noted that Figure 10a–d correspond to Figure 2a–d, respectively. Panel (a) shows that, in the homogeneous case where $c_1=c_2$ and $K_1=K_2$, a period-doubling bifurcation occurs if $c_1=c_2$ or $K_1=K_2$ is sufficiently large, which is markedly different from Figure 6a. Moreover, Arnold tongues can be observed in panel (b). In summary, the numerical simulations presented in Figure 10 are consistent with the analytical results shown in Figure 2.

5. Concluding Remarks

In this paper, we examine a duopoly market under constant and decreasing returns to scale, assuming that the firms’ cost functions are linear and quadratic, respectively. The market is characterized by an isoelastic demand function. Moreover, we assume that the firms are boundedly rational and adjust their output according to a gradient adjustment mechanism.

From a methodological point of view, we employ different approaches to analyze the local stability and bifurcations of the Nash equilibrium under constant and decreasing returns to scale. Under constant returns to scale, we use the traditional analytical approach, i.e., solving the closed-form solution for the Nash equilibrium and then substituting it into the Jacobian matrix to determine whether the moduli of all eigenvalues are less than one. Under decreasing returns to scale, we employ symbolic computation methods, including triangular decomposition and partial cylindrical algebraic decomposition (PCAD). The basic idea is: first, the equilibrium equations are transformed into triangular sets, then the Sylvester resultant of the stability conditions with respect to the triangular set is computed; the algebraic variety of the resultant partitions the parameter space into regions; the stability of the Nash equilibrium can be determined in each region by selecting a sample point via the PCAD method.

We establish analytical conditions for the local stability and bifurcations of the Nash equilibrium (see Theorems 1 and 2). The results indicate that the stability region of the Nash equilibrium exhibits a similar structure under both constant and decreasing returns to scale. In particular, we find that the cost parameters $c_1,c_2$ and the adjustment speed parameters $K_1,K_2$ act as destabilizing factors, implying that an increase in any of these parameters may lead to the destabilization of the Nash equilibrium (see Figure 1 and Figure 2).

From a perspective of managerial implications, the above results suggest that under both constant and decreasing returns to scale, market regulators should design policies that encourage firms to adopt relatively moderate adjustment speeds when changing output, thereby promoting more stable market dynamics. In addition, regulators need to implement rules or use market-based mechanisms to exclude rather inefficient firms from the market, which also serves as an effective means to maintain market stability.

Moreover, we analyzed the local bifurcations of the model and found that period-doubling and Neimark–Sacker bifurcations may occur for general parameter values. In the case where $c_1=c_2$ and $K_1\ne K_2$, it was shown that the Nash equilibrium can undergo only period-doubling bifurcations, and not Neimark–Sacker bifurcations, under both constant and decreasing returns to scale (see Corollaries 1 and 2).

To gain further insight into the complex dynamics of the game, we performed numerical simulations that confirm the analytical results described above. Specifically, the simulations show that the Nash equilibrium may lose stability and transition to chaos through two possible routes: period-doubling or Neimark–Sacker bifurcations. When chaotic dynamics occur, the underlying pattern of firms’ output and profits becomes extremely difficult to predict, even for fully rational players.

Given the feasibility of symbolic computation, we adopt a highly specific inverse demand function, $p\left(Q\right)=1/Q$, along with very particular cost functions (linear and quadratic polynomials). These settings constitute the limitations of this study. Extending the results to more general inverse demand and cost functions (e.g., under only mild assumptions on monotonicity and convexity/concavity) represents a valuable direction for future research, which we leave for further investigation.