The game interactions among individuals in nature are often uncertain and dynamically evolving, significantly influencing the persistence of cooperation. However, it remains a formidable challenge to effectively characterize these dynamic properties in structured populations, derive theoretical conditions for cooperation, and identify the optimal game distribution for promoting cooperation. To address these issues, we propose the variable game framework in a structured population, where the game interactions between different individuals change over time. By means of the Markov chain and the pair approximation method, we derive theoretical conditions under which cooperation is favored by natural selection and when it is favored over defection under weak selection. Furthermore, we respectively formulate and solve two optimization problems to determine the optimal game distribution that most effectively fosters the evolution of cooperation by maximizing the gradient of cooperation selection and minimizing the fitness difference between defectors and cooperators. The theoretical predictions regarding both the conditions for cooperation and optimal game distribution are further validated by numerical calculations and extensive Monte Carlo simulations. Our findings offer novel insights into the mechanisms driving cooperative behavior in complex systems and provide theoretical guidance for designing optimal game environments that facilitate the evolution of cooperation.

翻译：自然界的个体间博弈互动常具有不确定性与动态演化特征，这显著影响了合作的维持。然而，如何在结构群体中有效刻画这些动态特性、推导合作的理论条件并识别促进合作的最优博弈分布仍是严峻挑战。为解决上述问题，我们提出结构群体中的变博弈框架，其中不同个体间的博弈互动随时间变化。借助马尔可夫链和对近似方法，我们推导了弱选择条件下自然选择偏向合作及偏向背叛的理论条件。进一步地，我们分别建立并求解两个优化问题：通过最大化合作选择梯度与最小化背叛者与合作者间的适应度差，确定最有效促进合作演化的最优博弈分布。关于合作条件与最优博弈分布的理论预测，均通过数值计算与大规模蒙特卡洛模拟得到验证。本研究为复杂系统中合作行为的驱动机制提供了新见解，并为设计促进合作演化的最优博弈环境提供了理论指导。