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.

中文翻译：

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具有单向溢出的异构策略的动态Stackelberg-Cournot双头模型

许多作品都研究了古诺或斯塔克尔贝格游戏的复杂动力学，但很少有参考文献讨论了结合了古诺尔游戏阶段和斯塔克尔贝格游戏阶段的动态游戏模型。在R＆D溢出仅从R＆D领导者流向R＆D追随者的假设下，本文考虑了具有异类期望的双头垄断Stackelberg-Cournot博弈。具有不同研发能力的两家公司在Stackelberg研发阶段依次确定其研发投资，并在古诺生产阶段同时做出产出决策。模型中引入了研发溢出，研发投资和技术创新效率。我们发现：（i）动态Stackelberg-Cournot双寡头系统的边界均衡，如果系统的纳什均衡严格为正，则两个参与者分别采用有限理性预期和天真的预期是不稳定的。（ii）动态的Stackelberg-Cournot双寡头系统的纳什均衡（其中两个参与者分别采用有限理性期望和天真的期望）只有在模型参数满足某些条件的情况下才是局部渐近稳定的。特别是，结果表明，R＆D溢出值较小或输出调整速度较大时，可能会产生分歧甚至混乱。进行数值模拟以显示最大Lyapunov指数，分叉图，奇异吸引子以及对初始条件的敏感依赖性，以验证我们的发现。