Zero-determinant strategies are a class of strategies in repeated games which unilaterally control payoffs. Zero-determinant strategies have attracted much attention in studies of social dilemma, particularly in the context of evolution of cooperation. So far, not only general properties of zero-determinant strategies have been investigated, but zero-determinant strategies have been applied to control in the fields of information and communications technology and analysis of imitation. Here, we provide another example of application of zero-determinant strategies: control of a particle on a lattice. We first prove that zero-determinant strategies, if exist, can be implemented by some one-dimensional transition probability. Next, we prove that, if a two-player game has a non-trivial potential function, a zero-determinant strategy exists in its repeated version. These two results enable us to apply the concept of zero-determinant strategies to control the expected potential energies of two coordinates of a particle on a two-dimensional lattice.

翻译：零行列式策略是重复博弈中一类能够单方面控制收益的策略。此类策略在社会困境研究（尤其是合作演化领域）中备受关注。迄今不仅探讨了零行列式策略的通用性质，还将其应用于信息通信技术领域与模仿分析中的控制问题。本文提供了零行列式策略的另一个应用实例：晶格上粒子的控制。我们首先证明，若零行列式策略存在，则可通过某种一维转移概率实现。接着证明，若双人博弈存在非平凡势函数，则其重复博弈版本存在零行列式策略。这两个结果使我们能够应用零行列式策略的概念，控制二维晶格上粒子两个坐标的期望势能。