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：由于认知能力有限导致的感知误差或其他情感因素，即使存在更优选择，参与者仍可能维持当前策略。此类行为惯性虽已得到研究，但其在弱选择极限下的影响尚未被探索。为填补这一空白，我们构建了考虑模仿过程中惯性效应的空间公共物品博弈模型。通过使用同源遗传方法，我们推导出决定合作演化方向的关键协同因子$r^\star$的解析形式。研究发现惯性会抑制合作，这可通过弱选择下缓慢的粗化过程来解释。有趣的是，不同更新协议（包括死亡-出生与出生-死亡规则）的关键协同条件可通过惯性因子的极端极限形式建立关联。为验证结论的稳健性，我们针对不同晶格结构与群体规模进行了计算分析，蒙特卡洛模拟结果也证实了上述发现。