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Equilibrium stability of dynamic duopoly Cournot game under heterogeneous strategies, asymmetric information, and one-way R&D spillovers

  • Jianjun Long and Fenglian Wang EMAIL logo
Published/Copyright: August 28, 2023
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

Bounded rationality, asymmetric information, and R&D spillovers are widely existed in monopoly markets, and they have been researched separately by a large number of literatures; however, there are few works that discussed both R&D spillovers and asymmetric information in oligopolistic games with bounded rational firms. Considering that R&D spillovers only flow from the R&D leader to the R&D follower, a duopoly Cournot game with heterogeneous expectations and asymmetric information is presented. In our model, a firm with private information of his marginal cost is designed, and the coefficient of R&D spillovers is introduced. Interesting findings show the following: (i) In a static duopoly Cournot game with perfect rationality, the equilibrium output of firm 1 with private information is negatively related to R&D spillovers and the probability of high marginal cost, while firm 2’s equilibrium output is positively correlated with them. (ii) In a dynamic duopoly Cournot game with asymmetric information and heterogeneous expectations, if firms adopt adaptive expectation and naïve expectation respectively, the Nash equilibrium is always globally asymptotically stable; if they use adaptive expectation and gradient dynamical expectation respectively, the Nash equilibrium tends to be locally asymptotically stable under certain conditions. Furthermore, the bigger the probability of high marginal cost or R&D spillovers are, the more volatile the monopoly market is, while higher technology innovation efficiency (TIE) of firm 1 is conducive to the stability of the product market. Our study would have theoretical and practical significance to the technological innovation activities of homogeneous products in oligopoly markets.

Keywords: Cournot; Bayesian Nash equilibrium; asymmetric information; bifurcation; Chaos control

1 Introduction

Oligopoly means that a tiny number of firms produce the same or similar products and supply the whole market, and it attracts attentions of mathematicians, economists, and management scientists for its complexity, diversity, and social authenticity. One of the classic oligopoly games is the Cournot model, which was pioneered by Augustin Cournot in 1838 [1]. In this typical model, two sellers compete simultaneously with no collusion in a product market and they determine the optimal outputs based on each other’s actions.

The classical Cournot game requires participants to have all the information and to be completely rational, but this strong assumption often contradicts the reality. Due to the complexity and uncertainty of economic environment, as well as the limitation of computing power and cognition ability, incomplete information is ubiquitous in market economy. Asymmetric information is a special case of incomplete information, it mainly comes from the asymmetry of information held by market participants due to the imperfection of the market itself, the difference of information acquisition ability between buyers and sellers, and the difference of industry information caused by the division of labor. Participants with access to more valid information tend to be in a good position to trade in the market. The widespread existence of information asymmetry in economic society has attracted the attention of a large number of scholars, and relevant papers mainly focus on cost information asymmetry [2,3] and demand information asymmetry [4,5]; there are also some studies on information asymmetry from the perspectives of innovation efficiency asymmetry [6] and productivity asymmetry [7].

Traditional economics has always been premised on complete rationality, that is, the actor has access to all information, and so he can choose the one that maximizes the utility among various schemes. However, decision-makers are bounded rational in reality, and they are influenced by many factors such as physiology, motivation, and ability. First, it is almost impossible to make the optimal decision completely rational in a short time in the face of many changing factors. Second, decision-makers also have limited access to information, knowledge, and abilities, and also their abilities to process information are different. Third, decision-makers will also be affected by emotion, value bias, and other irrational factors, and cannot be completely objective and rational when making decisions. Therefore, complete rationality has been criticized for its overstepping reality [8]; bounded rationality, which is closer to reality, has attracted more and more attention; and expectation rules with bounded rationality is more and more valued and applied in modeling monopoly games. As far as we know, naïve [9–11] expectation, adaptive expectation [3,12,13], gradient dynamical expectation [14,15], local monopolistic approximation [16–18], and expectation are the most commonly used decision criteria in extant references.

Research and development (R&D) refers to the systematic activities with clear objectives carried out continuously by research institutions and enterprises. R&D is mainly aimed at acquiring new scientific and technological knowledge and be used to improve technologies, products, and services. It generally refers to product R&D and technological R&D. As an innovation activity, R&D is a crucial source of competitive advantage and an important guarantee of sustainable development of enterprises. As mentioned in many works [19–24], spillovers inevitably occurs in R&D activities. Because of its positive externality, R&D spillovers can reduce the product costs of R&D enterprises, but it can also reduce the enthusiasm of enterprises engaged in R&D activities for malicious free-riding behavior. Besides, due to the differences in R&D capabilities, one-way R&D spillovers is common in reality [19,20], that is, R&D spillovers only flow from R&D leaders to R&D followers. In addition to one-way R&D spillovers, our article also considers another important economic phenomenon: the enterprise cluster, which refers to the geographical aggregation of some closely related enterprises and their supporting institutions in an industry with few leading enterprises as the core [25]. Due to the geographical proximity, organizational proximity, and cognitive proximity, enterprises in the cluster show stronger competitiveness than general enterprises [26] for R&D spillovers. So this article mainly studies the complexity of a duopoly Cournot game with asymmetric information, bounded rationality, and one-way R&D spillovers.

Incomplete information, bounded rationality and one-way R&D spillovers are widespread in the real economic society, and have been studied a lot in oligopoly games. But as far as we know, there are few works integrating asymmetry information and R&D spillovers into dynamic Cournot models with perfect or imperfect rationality. This gives us the impetus to solve the following problems: (i) Whether Bayesian Nash equilibrium outputs exist in a static duopoly Cournot game with asymmetric information and perfect rationality? If it exists, what is the condition? Furthermore, what are the effects of asymmetric information and R&D spillovers on the equilibrium outputs? (ii) In a dynamic duopoly Cournot game with bounded rationality, what are the effects of asymmetric information and R&D spillovers on the equilibrium outputs? (iii) When duopoly firms adopt different bounded expectations, does Bayesian Nash equilibrium exist in the dynamic system? If so, what are the conditions? To address the aforementioned issues, this article constructs a duopoly Cournot model with asymmetric information, where firm 1’s marginal cost is private and firm 2’s marginal cost is well known. We mainly intend to study influences of R&D spillovers and asymmetric information on Nash equilibrium under perfect and bounded rationality, and provide the region where equilibrium outputs exist.

Our research expands the existing works on complex dynamics of oligopoly Cournot games from the perspective of bounded rationality and asymmetric information. Ever since the Cournot model was proposed in 1838, studies revolving around it continued to spring up, and earlier works focused on perfect rationality and complete information [27,28]. As bounded rationality gets more and more attention, many scholars have analyzed Cournot models with bounded rationality and complete information. Reference [29] analyzed complex dynamics in a triopoly Cournot game with different strategies. Reference [30] discussed an investment process in a Cournot game with heterogenous players. Reference [31] established a duopoly Cournot model with relative profits maximization and investigated the effect of the degree of product differentiation on Nash equilibrium. Reference [24] studied the complexities of product differentiation and R&D spillover in a two-stage duopoly Cournot game. Our article differs from the aforementioned references in two ways. First, as we can see the extant works rarely researched asymmetric information in dynamic Cournot games, by contrast, this article introduces asymmetric information of marginal cost and discusses its effect on the stability of equilibrium outputs. Second, our research extends the application of Cournot competition to the enterprise cluster. Since Rand introduced the chaos theory into oligopoly model as early as 1978 [32], scholars are increasingly concerned about the application of bounded rationality in the economic market and social environment, such as Chinese air-conditioning market [33] and enterprise network [21]. Despite the widespread monopoly phenomenon in enterprises clusters, few literature applied bounded rationality to enterprises clusters, References [3,34] discussed bounded rationality and asymmetric information in an enterprise cluster with duopoly firms, but these two articles, respectively, analyzed Bertrand game and Cournot-Bertrand game, which are fundamentally different from the Cournot game studied in this article.

Our research also extends the application of R&D spillovers to nonlinear discrete systems. R&D spillovers refers to the positive externality brought by the knowledge and technology generated by enterprises’ R&D activities. Other cluster enterprises with close distance and frequent staff turnover receive part or all of the knowledge and technology for free to gain profits, and R&D spillovers inevitably occurs [9]. Many works have studied R&D spillovers under perfect rationality [19,20]. The limitations of information acquisition, cognition, computation and other abilities have attracted more and more scholars to pay attention to R&D spillover under bounded rationality [9,21–24]. Reference [3] analyzed the influence of cluster spillovers on price equilibrium under bounded rationality and asymmetric information and found that when duopoly firms adopt different strategies, high cluster spillovers are conducive to market stability. Reference [34] studied the complexity of cluster spillover effect in Cournot–Bertrand mixed game, and found that the influence of cluster spillover effect on the stability of nonlinear system depends on the substitutability between products. Reference [9] investigated the effects of spillovers on the existence of the Nash equilibrium, and figured out that the cost-reduction effect caused by R&D spillovers affected the stability of output deeply. References [21] and [35] discussed R&D externalities in a two-stage monopoly game along networks. Reference [24] analyzed R&D spillover effect in a Cournot duopoly game with product differentiation. Reference [3] analyzed the influence of cluster spillovers on price equilibrium under bounded rationality and asymmetric information, and found that when duopoly firms adopt different strategies, high cluster spillovers is conducive to market stability. Reference [34] studied the complexity of cluster spillover effect in Cournot-Bertrand mixed game, and found that the influence of cluster spillover effect on the stability of nonlinear system depends on the substitutability between products. Our article differs in the following respects: first, we consider one-way R&D spillovers in the process of R&D activities in Cournot games with bounded rational firms, which is different from the assumption of bilateral spillovers in most previous works [3,34]. Second, most existing references discussed R&D spillovers in oligopoly games with bounded rationality or asymmetric information separately, and this article investigates R&D spillovers in a Cournot game combined with bounded rationality and asymmetric information, different from Bertrand game in ref. [3] and hybrid game in ref. [34]. Third, we extend the application of chaos theory to the firms’ R&D process in an enterprise cluster, which is an important economic phenomenon. Except the analysis of R&D spillovers, like most previous articles, we also study the influence of R&D investments, TIE, and asymmetric information on the dynamic equilibrium outputs with the chaos theory.

Several key findings are obtained. First, the equilibrium output and its stable regions are proposed in the dynamic duopoly Cournot game with perfect rationality and asymmetric information, which implies that, the equilibrium output of firm 1 with private information is negatively correlated with R&D spillovers and the probability of firm 1’s high marginal cost, while the equilibrium output of firm 2 is positively correlated with them. The extant references rarely discussed this issue. Second, the Nash equilibrium output is always globally asymptotically stable in a dynamic duopoly Cournot game with asymmetric information and heterogeneous expectations, where two firms adopt adaptive expectation and gradient dynamical expectation, respectively. This finding is different from our common sense that Bayesian Nash equilibrium exists only if players use same expectations, which is verified and simulated in the main reference [36]. Third, in a dynamic duopoly Cournot game with asymmetric information and heterogeneous expectations, where adaptive expectation and gradient dynamical expectation are used, respectively, the Nash equilibrium output is locally asymptotically stable only when the parameters meet certain conditions. Specially, we find that large probability of high marginal cost and big value of R&D spillovers can yield bifurcation, or even chaos, while high TIE is conducive to the stability of the production market.

The rest of our article is arranged as follows. We introduce asymmetric information and one-way R&D spillovers into a static duopoly Cournot model with perfect rationality in Section 2, and the Bayesian Nash equilibrium output is calculated. Section 3 proposes a dynamic Cournot duopoly model with heterogeneous expectations and asymmetric information through one-way R&D spillovers, and the existence of Bayesian Nash equilibrium and its stability region are discussed. In Section 4, various numerical simulation tools, including 1D and 2D bifurcation diagrams, largest Lyapunov exponents, attraction basin, and sensitive dependence on initial conditions, are presented to verify our theoretical analysis, and the management significance is also given. In Section 5, the state variables feedback and parameter variation method is used to control chaos. Section 6 gives a summary of our research finally.

2 Equilibrium outputs of duopoly firms under perfect rationality and asymmetric information

Bayesian Nash equilibrium is used to describe the static game equilibrium with incomplete information, and its definition [37] is as follows: in a static Bayesian game G = { A 1 , … , A n ; T 1 , … , T n ; p 1 , … , p n ; u 1 , … , u n } , which specifies n -players’ action spaces A 1 , … , A n , type spaces T 1 , … , T n , beliefs p 1 , … , p n , and their payoff functions u 1 , … , u n . The strategy s ∗ = ( s 1 ∗ , … , s n ∗ ) is a (pure-strategy) Bayesian Nash equilibrium, where s i ∗ ( t i ) solves max a i ∈ A i ∑ t − i ∈ T − i u i ( s 1 ∗ ( t 1 ) , … , s i − 1 ∗ ( t i − 1 ) , a i , s i + 1 ∗ ( t i + 1 ) , … , s n ∗ ( t n ) ; t i ) p i ( t − i ∣ t i ) , s i ( t i ) represents a strategy for player i for each type t i in T i , player i ’s belief p i ( t − i ∣ t i ) describes i ’s uncertainty about the n − 1 other players’ possible types t − i ( t − i = ( t 1 , … , t i − 1 , t i + 1 , … , t n ) ) , given i ’s own type t i . Player i ’s type, t i , is privately known to himself, and it is well known to other n − 1 players as a random variable with a known probability distribution. This article studies the dynamic Cournot game based on asymmetric information, and so Bayesian Nash equilibrium will be used to analyze the influencing factors of yield stability.

It is assumed that two firms, denoted by i = 1 , 2 , carry out a Cournot game. They produce the same or homogenous commodities with production quantity q i in an enterprise cluster, and firm 1 is the core enterprise. Both firms have considerable differences in innovation ability, and firm 1 has a stronger innovation ability and is called as the R&D leader, while firm 2 is named as the R&D follower for his weaker innovation ability. To get higher unshared returns, two firms carry out independent innovation, and then R&D spillovers will inevitably exist in the cluster due to the geographical proximity, organization proximity and cognitive proximity, which differentiate cluster enterprises and non-cluster enterprises [26]. Moreover, we assume that R&D spillovers only flow from firm 1 to firm 2 by one-way [19], based on the gaps in innovation ability between firms.

We assume the inverse demand function is linear and has the form p ( Q ) = a − Q , where a > 0 is the maximum capacity of the market, and Q = q 1 + q 2 represents the total supply. We also assume that the cost function of firm 2 is C 2 ( q 2 ) = c 2 q 2 , and the marginal cost c 2 = c is known to both firms, while the cost function of firm 1 C 1 ( q 1 ) = c 1 q 1 is private. Firm 1’s marginal c 1 is assumed to have two values c h = c + ε and c l = c − ε , ( c 1 > 0 , c > 0 , ε > 0 ) , and the respective probability P ( c 1 = c h ) = θ , P ( c 1 = c l ) = 1 − θ , ( θ ∈ ( 0 , 1 ) ) is known to each other.

In the process of R&D, the cost of an enterprise is not only related to its own R&D investments but also closely related to the R&D spillovers. With the assumption that firms have no fixed costs, we consider that the unit cost of an enterprise has a marginal decreasing relationship with its own investment, which means the cost reduction is y i = β i x i (the same hypothesis as ref. [20]), where x i is firm i ’s investments and β i is the TIE of firm i . After successful R&D, the marginal cost of firm 1 is reduced to c 1 ′ = c 1 − β 1 x 1 , while firm 2’s marginal cost reduction is not only related to its own input and TIE, but also related to R&D spillover spilling out from firm 1. Therefore, the marginal cost of firm 2 is given by c ′ = c − β 2 ( x 2 + γ x 1 ) , where γ ∈ [ 0 , 1 ] represents one-way R&D spillovers flowing from firm 1 to fimr 2. γ = 0 indicates that no R&D spillovers flows out, and γ = 1 implies that all R&D spillovers flows out. To ensure firms’ marginal costs after innovation positive, we assume x 1 ∈ 0 , c 1 2 β 1 2 , x 2 ∈ 0 , c β 2 − γ x 1 2 .

Under all aforementioned assumptions, it is easy to know:

E c 1 ′ = E c 1 − β 1 x 1 = θ ( c + ε ) + ( 1 − θ ) ( c − ε ) − β 1 x 1 = c + ( 2 θ − 1 ) ε − β 1 x 1 .

To ensure the validity of subsequent propositions, we propose the following condition, which can be worked out from Eqs. (4) and (6) with backward induction:

(1) a − c ≥ max { β 2 ( x 2 + γ x 1 ) + 2 ε − 2 β 1 x 1 , β 1 x 1 + ε − 2 β 2 ( x 2 + γ x 1 ) } .

To acquire the Bayesian Nash equilibrium, we need to obtain the optimal response functions of the two firms. Firm 1’s best reply function q 1 ( c 1 ) can be computed by:

max q 1 E π 1 = q 1 [ a − q 1 − q 2 − ( c 1 − β 1 x 1 ) ] − x 1 .

Then the marginal profit of firm 1 is:

(2) ∂ E π 1 ∂ q 1 = a − 2 q 1 − q 2 − ( c 1 − β 1 x 1 ) .

We set ∂ E π 1 ∂ q 1 = 0 , and then the best reply function of firm 1 is:

(3) q 1 ( q 2 ; c 1 ) = a − q 2 − c 1 + β 1 x 1 2 q 2 ≤ a − c 1 + β 1 x 1 0 q 2 > a − c 1 + β 1 x 1 c 1 = c h , c l

or

q 1 ( q 2 ; c h ) = a − q 2 − c h + β 1 x 1 2 q 2 ≤ a − c h + β 1 x 1 0 q 2 > a − c h + β 1 x 1

q 1 ( q 2 ; c l ) = a − q 2 − c l + β 1 x 1 2 q 2 ≤ a − c l + β 1 x 1 0 q 2 > a − c l + β 1 x 1 .

The mathematical expectation of the best reply function q 1 with respect to q 2 and c 1 is

(4) E q 1 = θ q 1 ( q 2 ; c h ) + ( 1 − θ ) q 1 ( q 2 ; c l ) = a − c + ( 1 − 2 θ ) ε − q 2 + β 1 x 1 2 q 2 ≤ a − c h + β 1 x 1 ( 1 − θ ) ( a − c + ε − q 2 + β 1 x 1 ) 2 a − c h + β 1 x 1 < q 2 ≤ a − c l + β 1 x 1 0 q 2 > a − c l + β 1 x 1 .

After obtaining the optimal response for firm 1’s output, we would calculate the best reply function for firm 2. As firm 2 only knows the probability distribution of firm 1’s marginal cost c 1 , then his maximum expected profit is solved through q 2 , i.e.,

max q 2 E π 2 = a − E q 1 ( c 1 ) − q 2 − ( c − β 2 x 2 − γ β 2 x 1 ) .

The marginal profit of firm 2 is:

(5) ∂ E π 2 ∂ q 2 = a − E q 1 ( c 1 ) − 2 q 2 − ( c − β 2 x 2 − γ β 2 x 1 ) .

We set the first order ∂ E π 2 ∂ q 2 = 0 , and then the best response is:

(6) q 2 ( E q 1 ) = a − ( c − β 2 x 2 − γ β 2 x 1 ) − E q 1 ( c 1 ) 2 E q 1 ( c 1 ) ≤ a − ( c − β 2 x 2 − γ β 2 x 1 ) 0 E q 1 ( c 1 ) > a − ( c − β 2 x 2 − γ β 2 x 1 ) ,

where E q 1 ( c 1 ) is given by Eq. (4).

Bayesian Nash equilibrium can be obtained in two steps, assuming that the duopoly has complete information. Step 1: after calculating the expected output of firm 1 in Eq. (4), firm 2 can determine his optimal quantity through Eq. (6). Step 2: firm 1 would determine his optimal output in Eq. (3) after firm 2 ensures its output in step 1. Then the Bayesian Nash equilibrium can be acquired by combining Eqs. (3), (4), and (6).

Propositions 1 and 2, proved in Appendix A, can be obtained through the aforementioned calculations.

Proposition 1

When parameters meet the condition Eq. (1), the expectation of firm 1’s best reply function is E q 1 = a − c + ( 1 − 2 θ ) ε − q 2 + β 1 x 1 2 .

Proposition 2

When parameters meet the condition Eq. (1), Bayesian Nash equilibrium ( q 1 ∗ , q 2 ∗ ) of static Cournot duopoly model with asymmetric information is expressed as follows:

(7) q 1 ∗ ( c 1 ) = a − c − ( 1 + θ ) ε + 2 β 1 x 1 − β 2 ( x 2 + γ x 1 ) 3 c 1 = c h a − c + ( 2 − θ ) ε + 2 β 1 x 1 − β 2 ( x 2 + γ x 1 ) 3 c 1 = c l

(8) q 2 ∗ = a − c − ( 1 − 2 θ ) ε − β 1 x 1 + 2 β 2 ( x 2 + γ x 1 ) 3 .

In this section, Bayesian Nash equilibrium of a static Cournot model with asymmetric information and complete rationality is proposed and given by Eqs. (7) and (8). Eq. (8) implies that, the bigger a , β 2 , x 2 , γ or θ is, the bigger firm 2’s equilibrium output is, and the bigger the marginal cost c is, the smaller firm 2’s equilibrium output is. The correlation between x 1 and q 2 ∗ is related to the algebraic expression 2 γ β 2 − β 1 : if 2 γ β 2 − β 1 > 0 , the equilibrium output q 2 ∗ is positively correlated with x 1 ; if 2 γ β 2 − β 1 < 0 , the equilibrium output q 2 ∗ is negatively correlated with x 1 ; if 2 γ β 2 − β 1 = 0 , and there is no correlation between x 1 and q 2 ∗ . The equilibrium output of firm 1 depends on his private type of marginal cost c 1 . The greater a or β 1 is, the greater firm 1’s equilibrium output is. The greater the marginal cost c , θ , γ , x 2 or β 2 is, the smaller firm 1’s equilibrium output is. The correlation between x 1 and q 1 ∗ is related to the expression 2 β 1 − γ β 2 : if 2 β 1 − γ β 2 > 0 , the equilibrium output q 1 ∗ is positively correlated with x 1 ; if 2 β 1 − γ β 2 < 0 , the equilibrium output q 1 ∗ is negatively correlated with x 1 ; if 2 β 1 − γ β 2 = 0 , and there is no correlation between x 1 and q 1 ∗ . Particularly, q 1 ( c l ) ∗ = q 1 ( c h ) ∗ + ε .

3 Stability of dynamic duopoly Cournot model with heterogeneous expectations and asymmetric information

3.1 Adaptive expectation and naïve expectation adopted under asymmetric information

It is assumed that firm 1 has a high marginal cost with the probability θ and a low marginal cost with the probability 1 − θ in the market at every period t ( t = 0 , 1 , 2 , 3 , … ) . He adopts an adaptive expectation, which means he adjusts the quantity at period t + 1 according to his output and best reply at period t . Firm 2 is a naïve player, which implies that he determines his quantity at period t + 1 just on the best response at period t . Therefore, the dynamic three-dimensional system has the following form:

(9) q 1 h ( t + 1 ) = α q 1 h ( t ) + ( 1 − α ) a − c h + β 1 x 1 − q 2 ( t ) 2 q 1 l ( t + 1 ) = α q 1 l ( t ) + ( 1 − α ) a − c l + β 1 x 1 − q 2 ( t ) 2 q 2 ( t + 1 ) = a − c + β 2 ( x 2 + γ x 1 ) − θ q 1 h ( t ) − ( 1 − θ ) q 1 l ( t ) 2 ,

where α ∈ ( 0 , 1 ) denotes firm 1’s output adjustment speed, q i ( t + 1 ) is firm i ’s ( i = 1 h , 1 l , 2 ) output at period t + 1 , and q i ( t ) represents firm i ’s output at period t . q 1 h ( t ) and q 1 l ( t ) represent the output when the marginal cost of firm 1 is c h and c l respectively. Therefore, the Bayesian Nash equilibrium ( q 1 h ∗ , q 1 l ∗ , q 2 ∗ ) of system (9) is easily obtained, described in Eqs. (7) and (8), and further we obtain the following proposition.

Proposition 3

On the hypothesis that firm 1 uses an adaptive expectation and firm 2 adopts a naive expectation, the Bayesian Nash equilibrium of system (9) is always globally asymptotically stable.

We can see the proof of Proposition 3 in Appendix A. Proposition 3 shows that the Bayesian Nash equilibrium may exist stably even if the bounded rational firms adopt different output adjustment strategies. This is very different from findings in most existing works, which usually point out that equilibrium points is asymptotically stable only in two cases, one is that parameters meet certain conditions and the other is that players use homogeneous expectations.

3.2 Adaptive expectation and gradient dynamical expectation adopted under asymmetric information

In this subsection, we consider that firm 1 still adopts adaptive expectation, firm 2 acts as a gradient dynamical expectation player, so firm 2 decides his output on the basis of marginal profit ∂ π 2 ∂ q 2 , i.e., he adjusts output of next period according to his output of current period and estimation of marginal profit. Then the discrete system has the following format:

(10) q 1 h ( t + 1 ) = α q 1 h ( t ) + ( 1 − α ) a − c h + β 1 x 1 − q 2 ( t ) 2 q 1 l ( t + 1 ) = α q 1 l ( t ) + ( 1 − α ) a − c l + β 1 x 1 − q 2 ( t ) 2 q 2 ( t + 1 ) = q 2 ( t ) + v q 2 ( t ) [ a − c + β 2 ( x 2 + γ x 1 ) − θ q 1 h ( t ) − ( 1 − θ ) q 1 l ( t ) − 2 q 2 ( t ) ] ,

where v > 0 is firm 2’s output adjustment speed.

To study the stability of equilibrium points of system (10), we set q i ( t + 1 ) = q i ( t ) ( i = 1 h , 1 l , 2 ) , namely:

(11) q 1 h − a − c h + β 1 x 1 − q 2 2 = 0 q 1 l − a − c l + β 1 x 1 − q 2 2 = 0 q 2 [ a − c + β 2 ( x 2 + γ x 1 ) − θ q 1 h − ( 1 − θ ) q 1 l − 2 q 2 ] = 0 .

Then system (10) has two equilibrium points:

E 1 a − c h + β 1 x 1 2 , a − c l + β 1 x 1 2 , 0 , E 2 ( q 1 h ∗ , q 1 l ∗ , q 2 ∗ ) .

Next we need to study the local stability of equilibrium points, and to this end, it is necessarily to calculate the Jacobian matrix of system (10) at any point ( q 1 h , q 1 l , q 2 ) , namely:

J = α 0 − 1 − α 2 0 α − 1 − α 2 − v θ q 2 − v ( 1 − θ ) q 2 1 + v [ a − c + β 2 ( x 2 + γ x 1 ) − θ q 1 h − ( 1 − θ ) q 1 l − 4 q 2 ] .

Plug in the specific value of equilibrium points E 1 and E 2 , Propositions 4 and 5 can be obtained according to the stability theory.

Proposition 4

If firm 1 chooses an adaptive expectation and firm 2 uses a gradient dynamical expectation, the boundary equilibrium point E 1 a − c h + β 1 x 1 2 , a − c l + β 1 x 1 2 , 0 is a saddle point.

Proof

The Jacobian matrix at E 1 is

J = α 0 − 1 − α 2 0 α − 1 − α 2 0 0 1 + v [ a − c + 2 β 2 ( x 2 + γ x 1 ) − β 1 x 1 + ( 2 θ − 1 ) ε ] 2 .

Obviously three eigenvalues are λ 1 = λ 2 = α , λ 3 = 1 + v [ a − c + 2 β 2 ( x 2 + γ x 1 ) − β 1 x 1 + ( 2 θ − 1 ) ε ] 2 . Given the assumption of condition (1), it is easy to know that ∣ λ i ∣ < 1 ( i = 1 , 2 ) for α ∈ ( 0 , 1 ) and λ 3 > 1 , and hence, Proposition 4 is proved.

Proposition 5

If firm 1 chooses an adaptive expectation and firm 2 uses a gradient dynamical expectation, the Nash equilibrium point E 2 ( q 1 h ∗ , q 1 l ∗ , q 2 ∗ ) is locally asymptotically stable if v < V ∗ = 4 ( 1 + α ) ( 5 + 3 α ) q 2 ∗ .

The proof of Proposition 5 is in Appendix A, and it gives a stable region S :

S = { α ∈ ( 0 , 1 ) , 0 < v < V ∗ } ,

where V ∗ = 4 ( 1 + α ) ( 5 + 3 α ) q 2 ∗ . Therefore, when the parameter value falls in the region S , E 2 will be asymptotically locally stable, and otherwise, the system will fall into bifurcation or even chaos.

Substitute q 2 ∗ = a − c − ( 1 − 2 θ ) ε − β 1 x 1 + 2 β 2 ( x 2 + γ x 1 ) 3 into v ∗ , and we can obtain the following:

(12) V ∗ = 12 ( 1 + α ) ( 5 + 3 α ) [ a − c − ( 1 − 2 θ ) ε − β 1 x 1 + 2 β 2 ( x 2 + γ x 1 ) ] .

By combining the stable region S and Eq. (12), we can obtain Proposition 6.

Proposition 6

Consider two firms with heterogenous strategies where firm 1 chooses an adaptive expectation and firm 2 adopts a gradient dynamical expectation, and the stability of Bayesian Nash equilibrium increases as α , c , or β 1 increases, and decreases as a , θ , v , β 2 , or x 2 goes up. As to x 1 , and the stability of Bayesian Nash equilibrium raises as x 1 increases if 2 γ β 2 − β 1 < 0 , and decreases as x 1 increases if 2 γ β 2 − β 1 > 0 .

4 Numerical simulation and management significance

In this section, we carry on some numerical experiments to verify typical features of nonlinear dynamical systems (9) and (10), and put forward the management significance in practice. We use various tools, including 1D and 2D bifurcation diagrams, strange attractors, 1D and 2D largest Lyapunov exponents, basins of attraction, and sensitive dependence on initial conditions, to exhibit the existence of bifurcation and chaos.

Figures 1 and 2 show Bayesian Nash equilibrium of system (9) with respect to α and θ . Just as Proposition 3 verifies, the Nash equilibrium outputs in both figures approach to stable fixed points, while the Nash equilibrium output in Figure 1 stables at ( q 1 h ∗ , q 1 l ∗ , q 2 ∗ ) = ( 3.013 , 3.513 , 2.673 ) , and it changes with the probability θ in Figure 2. As we can see, in the case of perfect rationality and asymmetric information, the equilibrium output of monopoly firms is independent of the output adjustment speed, but depends on the probability of firm 1’s high marginal cost. When the probability becomes bigger, firm 1 will reduce its output for the reduced competitiveness due to increased costs. Meanwhile, firm 2 would take the opportunity to increase product share.

Figure 1 Bayesian Nash equilibrium of system (9) with respect to α \alpha . Other parameters are ( a , c , ε , β 1 , β 2 , γ , θ , x 1 , x 2 ) \left(a,c,\varepsilon ,{\beta }_{1},{\beta }_{2},\gamma ,\theta ,{x}_{1},{x}_{2}) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , (10,2,0.5,0.6,0.4, 0.2 , 0.2 , 4 , 2.25 ) 0.2,0.2,4,2.25) .
Figure 1

Bayesian Nash equilibrium of system (9) with respect to α . Other parameters are ( a , c , ε , β 1 , β 2 , γ , θ , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.2 , 4 , 2.25 ) .

Figure 2 Bayesian Nash equilibrium of system (9) with respect to θ \theta . Other parameters are ( a , c , ε , β 1 , β 2 , γ , α , x 1 , x 2 ) \left(a,c,\varepsilon ,{\beta }_{1},{\beta }_{2},\gamma ,\alpha ,{x}_{1},{x}_{2}) = ( 10 , 2 , 0.5 , 0.6 , (10,2,0.5,0.6, 0.4 , 0.2 , 0.2 , 4 , 2.25 ) 0.4,0.2,0.2,4,2.25) .
Figure 2

Bayesian Nash equilibrium of system (9) with respect to θ . Other parameters are ( a , c , ε , β 1 , β 2 , γ , α , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.2 , 4 , 2.25 ) .

Next, we will simulate the trajectory of system (10) under different values of parameters α , v , θ , γ , β 1 , β 2 , x 1 , x 2 .

Figures 3 and 4 present bifurcation diagrams of system (10) with respect to α and v . It can be seen form Figure 3 that the system experiences a two-period bifurcation as α increases and becomes stable when α > 0.569 . Figure 4 shows that system (10) stays with an equilibrium state when v < v ∗ = 0.326 , undergoes a flip bifurcation at ( q 1 h ∗ , q 1 l ∗ , q 2 ∗ ) = ( 2.683 , 3.183 , 2.933 ) when v = v ∗ = 0.326 , and then it fluctuates until falling into chaos when v > 0.326 . We also figure out the largest Lyapunov exponents to analyze the properties of system (10) quantitatively, as shown in Figure 4, and positive values indicates that the system falls in chaos, while zero means that the system is bifurcating.

Figure 3 The bifurcaiton diagram of system (10) with respect to α \alpha . Other parameters are ( a , c , ε , β 1 , β 2 , γ , θ , v , x 1 , x 2 ) \left(a,c,\varepsilon ,{\beta }_{1},{\beta }_{2},\gamma ,\theta ,v,{x}_{1},{x}_{2}) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , (10,2,0.5,0.6,0.4,0.2, 0.2 , 0.35 , 4 , 2.25 ) 0.2,0.35,4,2.25) .
Figure 3

The bifurcaiton diagram of system (10) with respect to α . Other parameters are ( a , c , ε , β 1 , β 2 , γ , θ , v , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.2 , 0.35 , 4 , 2.25 ) .

Figure 4 The bifurcaiton diagram of system (10) with respect to v v . Other parameters are ( a , c , ε , β 1 , β 2 , γ , θ , α , x 1 , x 2 ) \left(a,c,\varepsilon ,{\beta }_{1},{\beta }_{2},\gamma ,\theta ,\alpha ,{x}_{1},{x}_{2}) = ( 10 , 2 , 0.5 , 0.4 , 0.5 , 0.2 , (10,2,0.5,0.4,0.5,0.2, 0.2 , 0.7 , 4 , 2.25 ) 0.2,0.7,4,2.25) .
Figure 4

The bifurcaiton diagram of system (10) with respect to v . Other parameters are ( a , c , ε , β 1 , β 2 , γ , θ , α , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.4 , 0.5 , 0.2 , 0.2 , 0.7 , 4 , 2.25 ) .

Figures 3 and 4 show the opposite effects of the output adjustment speed on the stability of outputs, and this is mainly because the two firms are in a competitive relationship. Larger α indicates that the output of firm 1 with private marginal cost information is closer to that in previous period, while larger v implies the more uncertainty brought to firm 2, which can lead to market disruption. Therefore, in practice, we can maintain market stability by increasing the output adjustment speed of firms with opaque marginal cost and reducing the output adjustment speed of firms with transparent marginal cost.

Figure 5(a) shows 2D bifurcation diagram in the ( α , v ) plane. The color bar in the figure shows the colors for different periods. Brown means the system is stable; green, orange, yellow, dark green, red, blue, and purple regions, respectively, indicate that the system is in a 2 to 8 periodic bifurcation state; and black region means that the system is in a chaotic state. From Figure 5(a), we can see that if the output adjustment speed of firm 2 is relatively small, the system is more likely to tend to a stable state no matter what value firm 1’s output adjustment speed takes, and in other words, when there exists information asymmetry of marginal cost, low output adjustment speed of enterprises with transparent information is conducive to market stability. We can also find that when v gradually becomes larger, system (10) will move to an unstable state, experiencing 2-period bifurcation, 4-period bifurcation, 8-period bifurcation, and so on, until it finally falls into chaos, just as described in Figure 4. Figure 5(b) shows the 2D maximum Lyapunov exponents, the black region indicates that the system is in a stable or periodic bifurcation state where the maximum Lyapunov exponent is less than 1, the gray region shows that the system is in a chaotic state and the corresponding maximum Lyapunov exponent is between 0 and 2, while the white areas refer to the escape region corresponding to the maximum Lyapunov exponents greater than 2.

Figure 5 (a) 2D bifurcation diagram in the ( α , v ) \left(\alpha ,v) plane in system (10), other parameters are set as: ( a , c , ε a,c,\varepsilon , β 1 , β 2 {\beta }_{1},{\beta }_{2} , θ , γ , x 1 , x 2 \theta ,\gamma ,{x}_{1},{x}_{2} ) = ( 10 , 2 , 0.5 , 0.4 , 0.5 , (10,2,0.5,0.4,0.5, 0.2 , 0.2 , 4 , 2.25 ) 0.2,0.2,4,2.25) . (b) The corresponding 2D largest Lyapunov exponents.
Figure 5

(a) 2D bifurcation diagram in the ( α , v ) plane in system (10), other parameters are set as: ( a , c , ε , β 1 , β 2 , θ , γ , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.4 , 0.5 , 0.2 , 0.2 , 4 , 2.25 ) . (b) The corresponding 2D largest Lyapunov exponents.

Figures 6 and 7 shows bifurcation diagrams of system (10) with respect to θ and γ . The system experiences from equilibrium to bifurcation as θ or γ increases. Figure 6 indicates that high probability of firm 1 with high marginal cost may yield bifurcation when θ > 0.325 . Figure 7 implies that the one-way R&D spillover may drive the stable market to unstable when γ is large enough, namely γ > 0.290 . Hence, the higher the probability of high marginal cost or one-way R&D spillover is, the more likely it is to cause market disorder, and this is because the increase of probability with high marginal cost or R&D spillover will reduce the cost advantage of leading firms, leading to the dilemma of innovation failure, and as a result, market output would become unstable. So firms can stable outputs by reducing the uncertainty of high marginal costs and lowing R&D spillover.

Figure 6 The bifurcaiton diagram of system (10) with respect to θ \theta . Other parameters are ( a , c , ε a,c,\varepsilon , β 1 , β 2 , γ {\beta }_{1},{\beta }_{2},\gamma , α , v , x 1 , x 2 \alpha ,v,{x}_{1},{x}_{2} ) = ( 10 , 2 , 0.5 , 0.6 , (10,2,0.5,0.6, 0.4 , 0.2 , 0.85 , 0.315 , 4 , 2.25 ) 0.4,0.2,0.85,0.315,4,2.25) .
Figure 6

The bifurcaiton diagram of system (10) with respect to θ . Other parameters are ( a , c , ε , β 1 , β 2 , γ , α , v , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.85 , 0.315 , 4 , 2.25 ) .

Figure 7 The bifurcaiton diagram of system (10) with respect to γ \gamma . Other parameters are ( a , c , ε , β 1 a,c,\varepsilon ,{\beta }_{1} , β 2 , θ , α , v , x 1 , x 2 {\beta }_{2},\theta ,\alpha ,v,{x}_{1},{x}_{2} ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.2 , (10,2,0.5,0.6,0.4,0.2,0.2, 0.315 , 4 , 2.25 ) 0.315,4,2.25) .
Figure 7

The bifurcaiton diagram of system (10) with respect to γ . Other parameters are ( a , c , ε , β 1 , β 2 , θ , α , v , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.2 , 0.315 , 4 , 2.25 ) .

Figures 8 and 9 present bifurcation diagrams of system (10) with respect to β 1 and β 2 . Figure 8 shows that the equilibrium outputs become stable from bifurcation phenomenon if β 1 > 0.324 . Figure 9 shows that the equilibrium output is locally stable if the TIE β 2 < β 2 ∗ = 0.094 , then the system becomes bifurcated or even chaotic as β 2 increases. Simulation results are consistent with Propositions 5 and 6. As β 1 increases, the cost advantage of firm 1 is more significant, and then firm 1 has a stronger passion for continued innovation. But the improvement of β 2 would bring the cost advantage to firm 2, and firm 2’s free-riding in the innovation process aggravates the dilemma of that whether firm 1 should continue to innovate.

Figure 8 Bayesian Nash equilibrium of system (10) with respect to β 1 {\beta }_{1} . Other parameters are ( a , c , ε a,c,\varepsilon , β 2 , θ {\beta }_{2},\theta , γ , α , v , x 1 , x 2 \gamma ,\alpha ,v,{x}_{1},{x}_{2} ) = ( 10 , 2 , 0.5 , 0.4 , (10,2,0.5,0.4, 0.2 , 0.2 , 0.2 , 0.315 , 4 , 2.25 ) 0.2,0.2,0.2,0.315,4,2.25) .
Figure 8

Bayesian Nash equilibrium of system (10) with respect to β 1 . Other parameters are ( a , c , ε , β 2 , θ , γ , α , v , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.4 , 0.2 , 0.2 , 0.2 , 0.315 , 4 , 2.25 ) .

Figure 9 Bayesian Nash equilibrium of system (10) with respect to β 2 {\beta }_{2} . Other parameters are ( a , c , ε a,c,\varepsilon , β 1 , θ {\beta }_{1},\theta , γ , α , v , x 1 , x 2 \gamma ,\alpha ,v,{x}_{1},{x}_{2} ) = ( 10 , 2 , 0.5 , 0.6 , (10,2,0.5,0.6, 0.2 , 0.2 , 0.2 , 0.375 , 4 , 2.25 ) 0.2,0.2,0.2,0.375,4,2.25) .
Figure 9

Bayesian Nash equilibrium of system (10) with respect to β 2 . Other parameters are ( a , c , ε , β 1 , θ , γ , α , v , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.2 , 0.2 , 0.2 , 0.375 , 4 , 2.25 ) .

Figures 10 and 11 show the bifurcation diagrams with respect to x 1 and x 2 . Figure 10(a) exhibits that the equilibrium output experiences a flip bifurcation at x 1 ∗ = 6.339 and keeps stable when x 1 > x 1 ∗ , where 2 γ β 2 − β 1 < 0 . It indicates that, if the technology spillover flowing from firm 1 to firm 2 is relatively small, the bigger the value of firm 1’s innovation investments is, the more stable the market is. Conversely, Figure 10(b) shows that the equilibrium output moves from stable to 2-period bifurcation as x 1 increases, where 2 γ β 2 − β 1 > 0 . That is, if the technology spillover is relatively large, the bigger the value of firm 1’s innovation investments is, the more unstable the system is. Figure 11 implies that, the bigger firm 2’s innovation investments is, the more unstable the outputs are.

Figure 10 The bifurcation diagrams with respect to x 1 {x}_{1} in system (10). (a) The bifurcation diagram when 2 γ β 2 − β 1 < 0 2\gamma {\beta }_{2}-{\beta }_{1}\lt 0 , and other parameters are set as ( a , c , ε , β 1 a,c,\varepsilon ,{\beta }_{1} , β 2 , θ {\beta }_{2},\theta , γ , α , v , x 2 \gamma ,\alpha ,v,{x}_{2} ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.2 , 0.2 , 0.33 , 2.25 ) \left(10,2,0.5,0.6,0.4,0.2,0.2,0.2,0.33,2.25) . (b) The bifurcation diagram when 2 γ β 2 − β 1 > 0 2\gamma {\beta }_{2}-{\beta }_{1}\gt 0 , and other parameters are set as ( a , c , ε , β 1 a,c,\varepsilon ,{\beta }_{1} , β 2 , θ , γ {\beta }_{2},\theta ,\gamma , α , v , x 2 \alpha ,v,{x}_{2} ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.8 , 0.2 , 0.287 , 2.25 ) \left(10,2,0.5,0.6,0.4,0.2,0.8,0.2,0.287,2.25) .
Figure 10

The bifurcation diagrams with respect to x 1 in system (10). (a) The bifurcation diagram when 2 γ β 2 − β 1 < 0 , and other parameters are set as ( a , c , ε , β 1 , β 2 , θ , γ , α , v , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.2 , 0.2 , 0.33 , 2.25 ) . (b) The bifurcation diagram when 2 γ β 2 − β 1 > 0 , and other parameters are set as ( a , c , ε , β 1 , β 2 , θ , γ , α , v , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.8 , 0.2 , 0.287 , 2.25 ) .

Figure 11 The bifurcaiton diagram of system (10) with respect to x 2 {x}_{2} .
Figure 11

The bifurcaiton diagram of system (10) with respect to x 2 .

Figures 10 and 11 further validate our theoretical analysis, and the impact of firm 1’s innovation investment on the stability of equilibrium output depends on TIE and R&D spillover between firms. When the firm 1’s TIE is relatively large, the increase of x 1 will be beneficial to raise his output and stable product market, otherwise it may lead to bifurcation or even chaos. Although high investment of firm 2 would bring him cost advantage, firm 1’s enthusiasm for innovation will wane, and equilibrium outputs will become unstable.

Two important features of chaos are the strange attractor and the sensitivity to initial conditions. Figure 12 shows a chaos attractor at ( a , c , ε , β 1 , β 2 , θ , γ , θ , α , v , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.2 , 0.85 , 0.51 , 4 , 2.25 ), Figure 13 shows the attractor trajectories from different perspectives, and Figure 13(c) further verifies that the gap between firm 1’s output at high marginal cost and low marginal cost is fixed. The sensitivity behavior is simulated in Figure 14, and we can see that the difference becomes obvious after a series of iterations with tiny change 0.0001 of the initial values at the beginning.

Figure 12 The strange attractor of the system (10) for v = 0.51 v=0.51 .
Figure 12

The strange attractor of the system (10) for v = 0.51 .

Figure 13 The strange attractor of system (10) in different planes.
Figure 13

The strange attractor of system (10) in different planes.

Figure 14 Sensitive dependence of system (10) on initial conditions. The system orbits in the time periods [ 0 , 100 ] \left[0,100] are plotted with other parameters values ( a , c , β 1 , β 2 , θ , v ) \left(a,c,{\beta }_{1},{\beta }_{2},\theta ,v) = ( 10 , 2 , 0.6 , 0.3 , 0.2 , 0.42 ) \left(10,2,0.6,0.3,0.2,0.42) and ( q 1 h ( 0 ) , q 1 l ( 0 ) , q 2 ( 0 ) ) ({q}_{1h}\left(0),{q}_{1l}\left(0),{q}_{2}\left(0)) = ( 3.2 , 3.5 , 2.6 ) \left(3.2,3.5,2.6) . (a) q 1 h {q}_{1h} -coordinate with initial points ( 3.2 , 3.5 , 2.6 ) \left(3.2,3.5,2.6) and ( 3.201 , 3.5 , 2.6 ) \left(3.201,3.5,2.6) . (b) q 1 l {q}_{1l} -coordinate with initial points ( 3.2 , 3.5 , 2.6 ) \left(3.2,3.5,2.6) and ( 3.2 , 3.501 , 2.6 ) \left(3.2,3.501,2.6) . (c) q 2 {q}_{2} -coordinate with initial points ( 3.2 , 3.5 , 2.6 ) \left(3.2,3.5,2.6) and ( 3.2 , 3.5 , 2.601 ) \left(3.2,3.5,2.601) .
Figure 14

Sensitive dependence of system (10) on initial conditions. The system orbits in the time periods [ 0 , 100 ] are plotted with other parameters values ( a , c , β 1 , β 2 , θ , v ) = ( 10 , 2 , 0.6 , 0.3 , 0.2 , 0.42 ) and ( q 1 h ( 0 ) , q 1 l ( 0 ) , q 2 ( 0 ) ) = ( 3.2 , 3.5 , 2.6 ) . (a) q 1 h -coordinate with initial points ( 3.2 , 3.5 , 2.6 ) and ( 3.201 , 3.5 , 2.6 ) . (b) q 1 l -coordinate with initial points ( 3.2 , 3.5 , 2.6 ) and ( 3.2 , 3.501 , 2.6 ) . (c) q 2 -coordinate with initial points ( 3.2 , 3.5 , 2.6 ) and ( 3.2 , 3.5 , 2.601 ) .

Figures 15, 16, 17, 18, 19, and 20 simulate the attraction basin of system (10) with different values of θ , γ , v , β 2 , and x 1 , where yellow, blue, and red represent the attraction domain, the escape domain, and the attractor. Other parameter settings are the same as in Figures 4, 6, 7, 9, and 10, respectively. Simulations of the attraction basin once again verify Proposition 6, and the system is more likely to be stable in the case of low probability of firm 1’s high marginal cost, moderate R&D spillover, low TIE, or small output adjustment speed, of firm 2 with transparent cost information. Conversely, excessive R&D spillover, big probability of high marginal cost, high output adjustment speed or TIE of firm 2 may trigger market volatility and even lead to chaos. As can be seen from Figures 15(a), 16(a), 17(a), 18(a), 19(b), and 20(a), when the initial decision outputs are in the domain of attraction, the system will eventually converge to the Nash equilibrium point after some games. However, in the periodic bifurcation and chaotic state, the firms’ output will fluctuate periodically and even become unpredictable. In addition, from these figures, we can see that the attraction domain of Nash equilibrium point is larger than that of the attractors in unstable state. Therefore, maintaining market stability is more conducive to reducing the interference of firms’ decision-making errors.

Figure 15 The attraction basin of system (10) with different values of θ. (a) θ = 0.2; (b) θ = 0.6.
Figure 15

The attraction basin of system (10) with different values of θ. (a) θ = 0.2; (b) θ = 0.6.

Figure 16 The attraction basin of system (10) with different values of γ. (a) γ = 0.1; (b) γ = 0.6.
Figure 16

The attraction basin of system (10) with different values of γ. (a) γ = 0.1; (b) γ = 0.6.

Figure 17 The attraction basin of system (10) with different values of v. (a) v = 0.2; (b) v = 0.49; (c) v = 0.505.
Figure 17

The attraction basin of system (10) with different values of v. (a) v = 0.2; (b) v = 0.49; (c) v = 0.505.

Figure 18 The attraction basin of system (10) with different values of β 2. (a) β 2 = 0.05; (b) β 2 = 0.4; (c) β 2 = 0.9; (d) β 2 = 0.99.
Figure 18

The attraction basin of system (10) with different values of β 2. (a) β 2 = 0.05; (b) β 2 = 0.4; (c) β 2 = 0.9; (d) β 2 = 0.99.

Figure 19 The attraction basin of system (10) with different values of x 1 {x}_{1} when 2 γ β 2 ‒ β 1 < 0 2\gamma {\beta }_{2}&#x2012;{\beta }_{1}\lt 0 . (a) x 1 {x}_{1} = 4; (b) x 1 {x}_{1} = 8.
Figure 19

The attraction basin of system (10) with different values of x 1 when 2 γ β 2 ‒ β 1 < 0 . (a) x 1 = 4; (b) x 1 = 8.

Figure 20 The attraction basin of system (10) with different values of x 1 {x}_{1} when 2 γ β 2 ‒ β 1 > 0 2\gamma {\beta }_{2}&#x2012;{\beta }_{1}\gt 0 . (a) x 1 {x}_{1} = 1; (b) x 1 {x}_{1} = 6.
Figure 20

The attraction basin of system (10) with different values of x 1 when 2 γ β 2 ‒ β 1 > 0 . (a) x 1 = 1; (b) x 1 = 6.

5 Chaos control

In a chaotic market, the product outputs can not be predicted, which is not conducive to the long-term development of cluster enterprises. Therefore, it is necessary to adopt corresponding methods and measures to rid the market of chaos. Prior literature shows that parameter variation and feedback are two effective methods to control chaos [10,12, 13,19,20,38–42]. A state variables feedback and parameter variation method was proposed [43] and had been used in many works [44–46], and it will applied in our article. Hence, the three-dimensional discrete dynamic system (10) are changed into the following format:

(13) q 1 h ( t + 1 ) = ( 1 − k ) α q 1 h ( t ) + ( 1 − α ) a − c h + β 1 x 1 − q 2 ( t ) 2 + k q 1 h ( t ) q 1 l ( t + 1 ) = ( 1 − k ) α q 1 l ( t ) + ( 1 − α ) a − c l + β 1 x 1 − q 2 ( t ) 2 + k q 1 l ( t ) q 2 ( t + 1 ) = ( 1 − k ) { q 2 ( t ) + v q 2 ( t ) [ a − c + β 2 ( x 2 + γ x 1 ) − θ q 1 h ( t ) − ( 1 − θ ) q 1 l ( t ) − 2 q 2 ( t ) ] } + k q 2 ( t ) ,

where k ∈ [ 0 , 1 ] is the controlling factor.

From Figure 4, system (10) falls into chaos when v = 0.51 . However, after adding the controlling factor k to the chaotic state, the complex situation could be forced to steady. Figure 21 shows that chaos in system (13) can be controlled at the four period-doubling bifurcation, two period-doubling bifurcation, until at fixed points when k ≥ 0.248 . Figure 22 exhibits that chaos are controlled successfully at k = 0.4 with other parameters set as ( a , c , ε , β 1 , β 2 , γ , θ , α = 0.85 , v , x 1 , x 2 ) = ( 10 , 2 , 0.5 , 0.6 , 0.4 , 0.2 , 0.2 , 0.85 , 0.51 , 4 , 2.25 ).

Figure 21 The bifurcation diagram of the system (13) with respect to the controlling factor k k .
Figure 21

The bifurcation diagram of the system (13) with respect to the controlling factor k .

Figure 22 The time series of system (13) when control parameter k = 0.4 k=0.4 .
Figure 22

The time series of system (13) when control parameter k = 0.4 .

In the real word, as well as the theoretical analysis of Section 3, we can vary parameters to maintain the stability of outputs, for example, increasing the adjustment speed of firms with uncertain cost, slowing the adjustment speed of firms with transparent cost, reducing R&D spillovers, increasing TIE of firms with uncertain cost, and so on. As described in system (13) and verified in Figures 21 and 22, another way to obtain the market out of mess is that, the outputs adjustment in the next period should takes more account of the outputs in current period.

6 Conclusion

It is an attractive topic to study dynamic games with imperfect rationality and incomplete information, which scholars paid less attention. In this article, a dynamic duopoly Cournot model with asymmetric information and bounded rationality is constructed, where two firms adopt different output adjustment strategies. We introduce two important realistic assumptions into our model, the first is that firms have different R&D capabilities, R&D spillovers only flow from firm 1 with stronger R&D capabilities to firm 2 with weaker R&D capabilities by one-way. The second is the information asymmetry between these two firms, that is, firm 2’s marginal cost is well known, while firm 1’s marginal cost is a private information.

We discuss the existence and the stability of Bayesian Nash equilibrium in discrete systems with heterogenous expectations in two cases, and interestingly, we obtain two opposite findings. In the first case, where firms adopt adaptive expectation and naïve expectation, it is found that firms’ outputs are always stable no matter what the values of other model parameters are. In the second case, where firms use adaptive expectation and gradient dynamical expectation, equilibrium points are locally asymptotically stable only when the model parameters met certain conditions, the dynamic system could go from equilibrium to bifurcation, or from unstable state to steady with model parameters varying. In particular, when the probability of high cost or technology spillovers of firm 1 is larger, the system is more likely to be away from equilibrium. When firm 1’s TIE is larger than firm 2’s, it would enlarge the stability of dynamic output system as R&D investment increases; on the contrary, it would decrease.

The numerical simulation verifies our theoretical analysis, and we describes the dynamic system via calculating the largest Lyapunov exponents, 2D bifurcation diagram, attractor basin, and sensitive dependence on initial conditions. Moreover, we stabilize the chaotic behavior of the system to a stable fixed point by introducing an appropriate controlling parameter with the state variables feedback and parameter variation method.

Acknowledgments

The authors would like to thank the reviewers and editor for their careful reading and helpful comments on the revision of paper.

  1. Funding information: Science and Technology Research Project of Chongqing Municipal Education Commission (Grant No. KJQN202000832), Humanities and Social Sciences Project of the Ministry of Education (Grant No. 21YJC630130), Humanities and Social Sciences Research Major Project of Anhui Province University (Grant No. 2022AH040133), High-level Talents Program of Chongqing Technology and Business University (Grant No. 1955046), and On-campus Scientific Research Project of Chongqing Technology and Business University (Grant No. 2151018).

  2. Author contributions: Jianjun Long was responsible for formulating the method, performing numerical simulation, and drafting the manuscript; and Fenglian Wang for supervising the research and revising and finalising the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The data used to support the findings of this study are available from the corresponding author upon request. No conflict of interest exits in the this article. This article has not been copy-righted, or submitted for publication elsewhere.

Appendix A Proofs of Propositions

Proof of Proposition 1

Proposition 1 is proved if neither E q 1 = 0 nor E q 1 = ( 1 − θ ) ( a − c + ε − q 2 + β 1 x 1 ) 2 could establish under the premise of the condition (1).

Case 1: if E q 1 = 0 , we know q 2 = a − ( c − β 2 x 2 − γ β 2 x 1 ) 2 > a − c l + β 1 x 1 from Eqs. (4) and (6), where c l = c − ε . Then we can obtain a − c < β 2 ( x 2 + γ x 1 ) − 2 ε − 2 β 1 x 1 , which is inconsistent with Eq. (1), so, E q 1 = 0 does not work.

Case 2: if E q 1 = ( 1 − θ ) ( a − c + ε − q 2 + β 1 x 1 ) 2 , from Eqs. (1) and (4), we can deduce E q 1 ≤ a − ( c − β 2 x 2 − γ β 2 x 1 ) . Then q 2 = ( 1 + θ ) ( a − c ) − ( 1 − θ ) ε + 2 β 2 ( x 2 + γ x 1 ) − ( 1 − θ ) β 1 x 1 3 + θ is calculated from Eq. (6), and it is also easy to prove that q 2 > a − c − ε + β 1 x 1 could not establish under the assumption of condition (1). Therefore, E q 1 = ( 1 − θ ) ( a − c + ε − q 2 + β 1 x 1 ) 2 is not possible either.

Finally, E q 1 = a − c + ( 1 − 2 θ ) ε − q 2 + β 1 x 1 2 .

Proof of Proposition 2

Firstly, if E q 1 ≤ a − c + β 2 ( x 2 + γ x 1 ) and q 2 ≤ a − c h + β 1 x 1 , from Eqs. (4) and (6), we have q 2 ∗ = a − c − ( 1 − 2 θ ) ε − β 1 x 1 + 2 β 2 ( x 2 + γ x 1 ) 3 ; thus, Eq. (8) is solved.

Second, from Eq. (3), when q 2 ∗ ≤ a − c h + β 1 x 1 ,

q 1 ∗ ( c 1 ) = a − c − ( 1 + θ ) ε + 2 β 1 x 1 − β 2 ( x 2 + γ x 1 ) 3 c 1 = c h a − c + ( 2 − θ ) ε + 2 β 1 x 1 − β 2 ( x 2 + γ x 1 ) 3 c 1 = c l ,

so Eq. (7) establishes. That means we only need to verify that E q 1 ∗ ≤ a − c + β 2 ( x 2 + γ x 1 ) and q 2 ∗ ≤ a − c h + β 1 x 1 establish to make Proposition 2 true, and this could be easily derived from the assumption of Eq. (1).

Thus, Proposition 2 is proved on the hypothesis of Eq. (1).

Proof of Proposition 3

To illustrate the stability of ( q 1 h ∗ , q 1 l ∗ , q 2 ∗ ) , one way is to prove that the characteristic roots of the Jacobian matrix in system (9) ∣ λ i ∣ < 1 . The Jacobian matrix for the system (9) at any point has the following form:

J = α 0 − 1 − α 2 0 α − 1 − α 2 − θ 2 − 1 − θ 2 0 .

The characteristic polynomial of J is

f ( λ ) = ∣ λ I − J ∣ = λ − α 0 1 − α 2 0 λ − α 1 − α 2 θ 2 1 − θ 2 λ = λ − α 4 [ 4 λ 2 − ( 4 α − θ α + θ ) λ + ( 1 − α ) ( θ α + θ − 1 ) ] .

Therefore, we can easily obtain three characteristic roots of f ( λ ) , which are λ 1 = α , λ 2 = 4 α − θ α + θ + ( 4 α − θ α + θ ) 2 − 16 ( 1 − α ) ( θ α + θ − 1 ) 8 , and λ 3 = 4 α − θ α + θ − ( 4 α − θ α + θ ) 2 − 16 ( 1 − α ) ( θ α + θ − 1 ) 8 , respectively. It is obvious that ∣ λ 1 ∣ < 1 , and then Proposition 3 is true only if λ 2 < 1 and λ 3 > − 1 .

λ 2 < 1 is equivalent to the following inequality θ α 2 < 3 ( 1 − α ) + θ α , while λ 3 > − 1 is equivalent to the following inequality θ α 2 < 3 ( 1 − α ) + θ α . Obviously, both formulas always hold for α ∈ ( 0 , 1 ) and θ ∈ [ 0 , 1 ] .

Thus, the Bayesian Nash equilibrium ( q 1 h ∗ , q 1 l ∗ , q 2 ∗ ) is always asymptotically stable in system (9).

Proof of Proposition 5

The Jacobian matrix at boundary equilibrium E 2 is

J = α 0 − 1 − α 2 0 α − 1 − α 2 − v θ q 2 ∗ − v ( 1 − θ ) q 2 ∗ 1 − 2 v q 2 ∗ .

The characteristic polynomial of J is

f ( λ ) = ∣ λ I − J ∣ = λ − α 0 1 − α 2 0 λ − α 1 − α 2 v θ q 2 ∗ v ( 1 − θ ) q 2 ∗ λ + 2 v q 2 ∗ − 1 = λ − α 2 [ 2 λ 2 + 2 ( 2 v q 2 ∗ − α − 1 ) λ + 2 α − v q 2 ∗ − 3 α v q 2 ∗ ] .

Apparently λ 1 = α ∈ ( 0 , 1 ) is one characteristic root, the other two characteristic roots are λ 2 = − 2 v q 2 ∗ + α + 1 + ( 2 v q 2 ∗ − α − 1 ) 2 − 2 ( 2 α − v q 2 ∗ − 3 α v q 2 ∗ ) 2 and λ 3 = − 2 v q 2 ∗ + α + 1 − ( 2 v q 2 ∗ − α − 1 ) 2 − 2 ( 2 α − v q 2 ∗ − 3 α v q 2 ∗ ) 2 . Proposition 5 is true only if λ 2 < 1 and λ 3 > − 1 .

λ 2 < 1 is equivalent to 3 α v q 2 ∗ < 3 v q 2 ∗ , and this always holds for α ∈ ( 0 , 1 ) . λ 3 > − 1 is equivalent to ( 3 α + 5 ) v q 2 ∗ < 4 ( 1 + α ) . Therefore, E 2 is asymptotically stable if v < v ∗ = 4 ( 1 + α ) ( 5 + 3 α ) q 2 ∗ .

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Received: 2023-03-27
Revised: 2023-07-03
Accepted: 2023-08-03
Published Online: 2023-08-28

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2022-0313
Keywords for this article
Cournot; Bayesian Nash equilibrium; asymmetric information; bifurcation; Chaos control
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