Due to limited cognitive skills for perceptual error or other emotional reasons, players may keep their current strategies even if there is a more promising choice. Such behavior inertia has already been studied, but its consequences remained unexplored in the weak selection limit. To fill this gap, we consider a spatial public goods game model where inertia is considered during the imitation process. By using the identity-by-descent method, we present analytical forms of the critical synergy factor $r^\star$, which determines when cooperation is favored. We find that inertia hinders cooperation, which can be explained by the decelerated coarsening process under weak selection. Interestingly, the critical synergy conditions for different updating protocols, including death-birth and birth-death rules, can be formally linked by the extreme limits of the inertia factor. To explore the robustness of our observations, calculations are made for different lattices and group sizes. Monte Carlo simulations also confirm the results.

翻译：由于认知错误导致的有限认知能力或其他情绪原因，即使存在更有前景的选择，玩家也可能维持当前策略。此类行为惯性虽已有研究，但其在弱选择极限下的影响仍待探索。为弥补这一空白，我们考虑了一个在模仿过程中考虑惯性的空间公共品博弈模型。通过使用同源遗传方法，我们给出了决定合作何时被偏好的临界协同因子$r^\star$的解析形式。研究发现惯性会阻碍合作，这可通过弱选择下减缓的粗化过程来解释。有趣的是，不同更新协议（包括死亡-出生规则和出生-死亡规则）的临界协同条件可通过惯性因子的极端极限建立形式化关联。为验证观测结果的稳健性，我们针对不同晶格和群体规模进行了计算。蒙特卡洛模拟也证实了上述结果。