In this paper, a repeated two-stage oligopoly game where two boundedly rational firms produce homogeneous product and apply gradient adjustment mechanism to decide their individual R&D investment is considered. Results concerning the equilibrium in the built model and the stability are discussed. The effects of system parameters on the complex dynamical behaviors of the built game are analyzed. We find that the system can lose stability through a flip bifurcation or a Neimark–Sacker bifurcation. In addition, the coexistence of multiattractors is also discussed using the so-called basin of attraction. At the end of this research, the social welfare of the given duopoly game is also studied.

中文翻译：

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研发溢出效应下的两阶段博弈的稳定性，全球动力和社会福利

本文考虑了一个重复的两阶段寡头博弈，其中两个有边界的理性公司生产同质产品并应用梯度调整机制来决定他们的个人R＆D投资。讨论了有关构建模型中的平衡和稳定性的结果。分析了系统参数对所构建游戏的复杂动力学行为的影响。我们发现该系统可能会由于翻转分叉或Neimark-Sacker分叉而失去稳定性。此外，还使用所谓的吸引盆讨论了多吸引子的共存。在本研究的最后，还研究了给定双寡头博弈的社会福利。