A systematic framework for analyzing dynamical attributes of games has not been well-studied except for the special class of potential or near-potential games. In particular, the existing results have shortcomings in determining the asymptotic behavior of a given dynamic in a designated game. Although there is a large body literature on developing convergent dynamics to the Nash equilibrium (NE) of a game, in general, the asymptotic behavior of an underlying dynamic may not be even close to a NE. In this paper, we initiate a new direction towards game dynamics by studying the fundamental properties of the map of dynamics in games. To this aim, we first decompose the map of a given dynamic into contractive and non-contractive parts and then explore the asymptotic behavior of those dynamics using the proximity of such decomposition to contraction mappings. In particular, we analyze the non-contractive behavior for better/best response dynamics in discrete-action space sequential/repeated games and show that the non-contractive part of those dynamics is well-behaved in a certain sense. That allows us to estimate the asymptotic behavior of such dynamics using a neighborhood around the fixed point of their contractive part proxy. Finally, we demonstrate the practicality of our framework via an example from duopoly Cournot games.

翻译：针对博弈动力学属性进行系统性分析的框架，除势博弈或近势博弈这类特殊情形外尚未得到充分研究。现有方法在判定特定动态在指定博弈中的渐近行为时存在局限性。尽管已有大量文献致力于开发收敛至博弈纳什均衡的动态机制，但一般而言，底层动态的渐近行为可能甚至与纳什均衡相去甚远。本文通过研究博弈动力学映射的基本性质，开创了博弈动力学研究的新方向。为此，我们首先将给定动态的映射分解为收缩部分与非收缩部分，进而利用此类分解与收缩映射的近似程度来探究这些动态的渐近行为。具体而言，我们分析了离散动作空间序贯/重复博弈中优选/最优反应动态的非收缩行为，并证明这些动态的非收缩部分在特定意义下具有良好性质。这使得我们能够通过收缩部分代理映射不动点邻域来估计此类动态的渐近行为。最后，我们通过双寡头古诺博弈实例展示了该框架的实用性。