We show that the replicator dynamics for zero-sum games arises as a result of a non-canonical bracket that is a hybrid between a Poisson Bracket and a Nambu Bracket. The resulting non-canonical bracket is parameterized both the by the skew-symmetric payoff matrix and a mediating function. The mediating function is only sometimes a conserved quantity, but plays a critical role in the determination of the dynamics. As a by-product, we show that for the replicator dynamics this function arises in the definition of a natural metric on which phase flow-volume is preserved. Additionally, we show that the non-canonical bracket satisfies all the same identities as the Poisson bracket except for the Jacobi identity (JI), which is satisfied for special cases of the mediating function. In particular, the mediating function that gives rise to the replicator dynamics yields a bracket that satisfies JI. This neatly explains why the mediating function allows us to derive a metric on which phase flow is conserved and suggests a natural geometry for zero-sum games that extends the Symplectic geometry of the Poisson bracket and potentially an alternate approach to quantizing evolutionary games.
Eco-evolutionary games for harvesting self-renewing common resource: Effect of growing harvester population
公地悲剧（英语：Tragedy of the Commons，TOC）是一个普遍存在的社会困境，在一群生物与共享资源之间的互动中被见证：群体中的个体倾向于自私地过度开发公共资源，因为这可以说是理性的选择，或者在非人类的情况下，这可能是进化上不可侵犯的行为。如何避免TOC是一个关系到节约资源的重大问题。不难设想这样的情况：资源可以自我更新，人口的规模可能取决于资源的状况，因为人口的不同部分采用不同的开采率。如果资源的自我更新率位于最大和最小开采率之间，那么在什么条件下可以避免TOC就不是先验的。在本文中，我们解决这个问题的分析和数值模拟使用的进化博弈理论复制子方程模型的达尔文主义的自然选择的宗旨设置。通过复制子方程，我们在研究复制子群体如何利用共享资源的同时，也不能忽略后者对前者的动态反馈。我们还提出了一个透明的自下而上的游戏资源反馈模型的推导，以方便未来的研究对本文提出的研究结果的随机效应。
