A Dynamic Stackelberg–Cournot Duopoly Model with Heterogeneous Strategies through One-Way Spillovers

Abstract

Many works studied on complex dynamics of Cournot or Stackelberg games, but few references discussed a dynamic game model combined with the Cournot game phase and Stackelberg game phase. Under the assumption that R&D spillovers only flow from the R&D leader to the R&D follower, a duopoly Stackelberg–Cournot game with heterogeneous expectations is considered in this paper. Two firms with different R&D capabilities determine their R&D investments sequentially in the Stackelberg R&D phase and make output decisions simultaneously in the Cournot production phase. R&D spillovers, R&D investments, and technological innovation efficiency are introduced in our model. We find that: (i) the boundary equilibrium of the dynamic Stackelberg–Cournot duopoly system, where two players adopt boundedly rational expectation and naïve expectation, respectively, is unstable if the Nash equilibrium of the system is strictly positive. (ii) The Nash equilibrium of the dynamic Stackelberg–Cournot duopoly system, where two players adopt boundedly rational expectation and naïve expectation, respectively, is locally asymptotically stable only if the model parameters meet certain conditions. Especially, results indicate that small value of R&D spillovers or big value of output adjustment speed may yield bifurcations or even chaos. Numerical simulations are performed to exhibit maximum Lyapunov exponents, bifurcation diagrams, strange attractors, and sensitive dependence on initial conditions to verify our findings. It is also shown that the chaotic behaviors can be controlled with the state variables feedback and parameter variation method.

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

“…which indicates that the eigenvalues are real. According to Jury conditions [35], the necessary and sufficient conditions for the local stability of E 4 are as follows:…”

Stability of equilibrium price-production of dynamic duopoly Bertrand–Cournot game in a transnational supply chain with asymmetric information

“…Complete information requires participants to master all information, such as competitors' costs and market environment, but in reality, there are limitations in cognitive ability and computing ability, and therefore an incomplete information economy is more common. In addition, due to the gap in research and development ability, professional information acquisition ability and other aspects, information asymmetry exists widely among oligopolists, and some of its manifestations are the asymmetric information acquisition degree of market demand between a manufacturer and a retailer [16] and the information asymmetry of production cost between a leading firm and a following firm [17]. Therefore, it would be worthy to further study the hybrid game with upstream and downstream oligopolists in a transnational supply chain from the perspective of bounded rationality and asymmetric information.…”

“…As we know, asymmetric information is an important research direction of supply chain, and from the perspective of the content of asymmetric information, the mainstream research focuses on the asymmetry of cost information [21,22] and demand information [23], while there are also some papers in the literature studying information asymmetry from other aspects, such as innovation efficiency [24], productivity [25], etc. However, these articles basically assumed supply chain enterprises to be completely rational; and although some papers [17,26] studied asymmetric information in dynamic games with bounded rationality in recent years, they have not applied it to the domain of supply chains. Therefore, unlike the aforementioned references, on the one hand, our paper introduces both bounded rationality and asymmetric information into a Bertrand-Cournot game model, and on the other hand, we extend the dynamic hybrid game to a transnational supply chain; therefore, the main objective of our study is to discuss the stability condition of the discrete supply chain system and analyse the effect of asymmetric information on dynamic complexity.…”

Evolution Dynamics Model of Private Enterprises under Simultaneous and Sequential Innovation Decisions

“…However, in economic reality, the game among enterprises is continuous, and reaching the Nash equilibrium state immediately is impossible. Additionally, enterprises do not have sufficient information, and corporate decisions are made by individuals who are limited by cognitive abilities and language constraints, resulting in limited rationality (Agiza et al [18]; Agiza et al [23]; Long et al [24]; Williamson [25]). Long et al [24] examined a dynamic Stackelberg-Cournot duopoly game with one-way spillovers and found that complex dynamic behaviors, such as cycles and chaos, can occur as model parameters vary.…”