FrontiersNonlinear dynamics of a duopoly Stackelberg game with marginal costs
Introduction
The conflict of interests between the players in duopoly market; for instance, Chinese petroleum industry, has been widely concerned. Both firms play decisive role in the complex competition relationship and try their best to maximize self-interests. For exploring the theory of this phenomenon and provide optimal strategy to solve the problem, extensive researches have been proposed [1]. As one of the fundamental oligopoly games, monopoly theory can be dated back to 1838 when Antoine Augustin Cournot first put forward the mathematical model of duopoly competition [2]. As the earliest application of Nash equilibrium, Cournot model became a starting point for the analysis of oligopoly theory. Many duopoly markets base on neither perfect information nor completely monopoly, which mean each firm in the model plays a key role for the final price. The introduction of complex phenomena in Cournot models has been well documented in mathematical economics literature [3], [4], [5]. In order to maximize profits, participants of the game take not only opponents strategy but also consumers reaction into consideration, however, humans interaction networks are complicated [6], [7] and the evolutionary game model has been investigated explicitly [8], [9], [10]. The mechanisms of social cooperation are discussed in [11], [12], [13]. Quantum Cournot game was researched and properties of equilibrium are exhibits [14], [15]. Refs. [16], [17] pointed out that the Lyapunov exponents can fully characterize the invariance of regular and irregular linearization under the change of coordinates. Refs. [18], [19], [20], [21] explained the evolution of cooperation and found that interactions among living organisms are more complex than the non-living matter. In addition, Refs. [22], [23], [24] analyzed some extensions of the classical economic situation where two firms producing an identical good compete with full information for the market. Other than these applications of Cournot model, more and more literature have utilized Stackelberg game to discuss complex dynamics, because the Stackelberg game is more natural for human interaction [25], [26], [27], [28]. A nonlinear dynamical Stackelberg system which described the time evolution with bounded rationality was analyzed in Refs. [29], and with different strategies was researched in Refs. [30], [31], [32]. Furthermore, some discussions of the synchronization system with delay were presented [33], [34], [35].
According to previous works, there are few existing literature based on the Stackelberg models with marginal costs. However, more often than not, duopoly Stackelberg games which involve marginal cost are more in line with reality situation. The nonlinear duopoly games with heterogeneous players have been investigated in Ref. [36]. Ref. [37] constructed an example which studied the Puu’s model with isoelastic demand function and constant marginal costs. Ref. [38] researched the Cournot duopoly model with marginal costs and showed complicated behaviors of the dynamic system. Moreover, the complex conducts of Bayesian game involving marginal costs has been analyzed [39].
This paper pays attention to a nonlinear discrete-time duopoly Stackelberg game with marginal costs in which the players have bounded rationality expectations. Simulation of the complex dynamical behaviors and chaos control are presented. For the position between competing firms is not symmetrical, the leader firm and the follower firm take dynamic adjustment strategies to achieve best interests respectively.
The rest of this paper is organized as follows. In Section 2, the description of the nonlinear duopoly Stackelberg model with marginal cost is presented. Moreover, the Nash equilibrium points and local stability region are discussed. In Section 3, we simulate the major features of this game. The results show that effective chaos control method can put chaos to be stable. Finally, we summarize this work in Section 4.
Section snippets
Duopoly Stackelberg game
In the classical Stackelberg model, the yield competition applied in homogeneous market which consisted of two firms was considered. For the asymmetrical position between two players, differences in the order of action exist. In the planning phase, the leader firm takes the lead in announcing forward production according to the rule of thumb. Then the follower firm makes production decision in view of leaders strategy. Owing to this situation, the leader must consider how the follower will
Numerical simulations
In order to provide further studies to research system (9), we carry numerical simulations in this section. Simulations confirm the discussions in previous parts by taking some specific values into the model. Treat v2, the adjustment speed of firm 2, as the argument and fix Fig. 1 compares the Lyapunov exponent with v2 in different values. The results demonstrate that the market is stable for small values of v2, but as v2 increasing, chaotic behaviors occur
Conclusion
We investigate a nonlinear duopoly Stackelberg game with marginal cost in output decision process, which contains two bounded rational players. The announced plan products have been solved by backward induction, and as some parameters of the game are varied, the evolution results demonstrate that the stability of Nash equilibrium points will lost through period doubling bifurcation. The chaotic features are analyzed numerically via computing Lyapunov exponents, sensitive dependence on initial
Acknowledgments
We would like to thank the referees for valuable comments and suggestions that greatly help us to improve this paper.
References (41)
- et al.
Oligopoly and stability
Chaos Solitons Fractals
(2009) - et al.
Note on cournot competition under yield uncertainty
Manuf Serv Oper Manage
(2017) - et al.
Exploiting a cognitive bias promotes cooperation in social dilemma experiments
Nat Commun
(2018) - et al.
Evolution of cooperation in spatial iterated prisoners dilemma games under localized extremal dynamics
Physica A
(2016) - et al.
Evolutionary dynamics of group interactions on structured populations: a review
J R Soc Interface
(2013) - et al.
Complex nonlinear dynamics and controlling chaos in a cournot duopoly economic model
Nonlinear Anal Real World Appl
(2010) - et al.
Nonlinear dynamics in the cournot duopoly game with heterogeneous players
Physica A
(2003) - et al.
Backward integration strategy in a retailer Stackelberg supply chain
Omega
(2018) - et al.
A Stackelberg model of pricing of complementary goods under information asymmetry
Int J Prod Econ
(2011) - et al.
Chaotic dynamics in nonlinear duopoly Stackelberg game with heterogeneous players
Physica A
(2018)
Synchronization of weakly coupled oscillators: coupling, delay and topology
J Phys A
On synchronized regions of discrete-time complex dynamical networks
J Phys A
Isochronal synchronization of time delay and delay-coupled chaotic systems
J Phys A
Oligopoly games with local monopolistic approximation
J Econ Behav Organ
Complex dynamics analysis for a remanufacturing duopoly model with nonlinear cost
Physica A
Attractors, bifurcations and chaos
Nonlinear Phenom Econ
Researches into the principles of the theory of wealth
Dynamics of a cournot duopoly game with bounded rationality based on relative profit maximization
Appl Math Comput
Punishment diminishes the benefits of network reciprocity in social dilemma experiments
Proc Natl Acad Sci USA
Doubly effects of information sharing on interdependent network reciprocity
New J Phys
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