Feedback-evolving games characterize the interplay between the evolution of strategies and environments. Rich dynamics have been derived for such games under the premise of the replicator equation, which unveils persistent oscillations between cooperation and defection. Besides replicator dynamics, here we have employed aspiration dynamics, in which individuals, instead of comparing payoffs with opposite strategies, assess their payoffs by self-evaluation to update strategies. We start with a brief review of feedback-evolving games with replicator dynamics and then comprehensively discuss such games with aspiration dynamics. Interestingly, the tenacious cycles, as perceived in replicator dynamics, cannot be observed in aspiration dynamics. Our analysis reveals that a parameter θ—which depicts the strength of cooperation in enhancing the environment—plays a pivotal role in comprehending the dynamics. In particular, with the symmetric aspiration level, if replete and depleted states, respectively, experience Prisoner's Dilemma and Trivial games, the rich environment is achievable only when θ > 1. The case θ < 1 never allows us to reach the replete state, even with a higher cooperation level. Furthermore, if cooperators aspire less than defectors, then the enhanced state can be achieved with a relatively lower θ value compared with the opposite scenario because too much expectation from cooperation can be less beneficial.

反馈演化博弈表征了策略和环境演化之间的相互作用。在复制方程的前提下，此类博弈得到了丰富的动态，揭示了合作与背叛之间的持续振荡。除了复制器动力学之外，这里我们还采用了愿望动力学，其中个人不是将收益与相反的策略进行比较，而是通过自我评估来评估他们的收益以更新策略。我们首先简要回顾具有复制动力学的反馈进化游戏，然后全面讨论具有愿望动力学的此类游戏。有趣的是，在复制动力学中观察到的顽强循环在吸入动力学中无法观察到。我们的分析表明，参数θ——描绘了合作在改善环境方面的力量——在理解动态方面起着举足轻重的作用。特别是对于对称的吸入水平，如果充满和耗尽状态分别经历囚徒困境和琐碎博弈，则只有当θ  > 1时才能实现丰富的环境。θ  < 1的情况永远不允许我们达到充满状态，即使具有更高的合作水平。此外，如果合作者的渴望少于叛逃者，那么与相反的情况相比，可以以相对较低的θ值实现增强状态，因为对合作的太多期望可能不太有利。