具有异质费用的动态古诺博弈

Dynamic Cournot Game with Heterogeneous Costs

摘要: 非对称微分博弈纳什均衡的求解及其性质的分析是博弈论中极具挑战的难题。本文提出具有异质(非对称)费用的动态古诺博弈模型，分别采用Pontryagin极大值原理和动态规划方法求得开环、闭环和反馈信息结构的纳什均衡，并重点分析了非对称纳什均衡的极限性质。当异质性通过费用函数的线性项表示时，纳什均衡的极限性质与对称情形是一致的。特别地，非对称的反馈纳什均衡仍然是渐近稳定的，这与Fershtman和Kamien的断言不同。当异质性通过费用函数的非线性项表示时，运用动态规划(值函数)方法不能得到反馈纳什均衡的解析解，此时非对称性对均衡极限性质的影响尚不明确。本文的理论突破了经典微分博弈纳什均衡求解和分析的界限，拓宽了应用。

Abstract: The computation of Nash equilibrium for asymmetric differential games and the analysis of its properties are challenging problems in game theory. In this paper, a dynamic Cournot game model with heterogeneous (asymmetric) costs is proposed. The open-loop, closed-loop and feedback Nash equilibria are obtained using Pontryagin’s maximum principle and dynamic programming method respectively, and the limit properties of asymmetric Nash equilibria are emphatically analyzed. When the heterogeneity is expressed by the linear term of the cost function, the limit property of the Nash equilibrium is consistent with the symmetric case. In particular, the asymmetric feedback Nash equilibrium is still asymptotically stable, which violates the assertion of Fershtman and Kamien. When the heterogeneity is expressed by the nonlinear term of the cost function, the analytic form of the feedback Nash equilibrium cannot be obtained by the dynamic programming (value function) method. In this case, the influence of asymmetry on the limit property of equilib-rium is not clear. The theory of this paper breaks through the boundary of computation and anal-ysis of Nash equilibrium in classical differential games and broadens its application.