The evaluation and updating of individual reputation plays a significant role in the evolution of game cooperation under the framework of reputation-based indirect reciprocity. However, the reputation-updating timescale is not necessarily consistent with the strategy-selection timescale. In this letter, we explore the effect of reputation-updating timescale on the cooperation behavior of second-order reputation-based public goods game, and it is assumed that the reputation-updating timescale is slower than the strategy-selection timescale. We observe that the low-frequency of reputation updating will result in some delay, and further hamper the emergency of cooperation behavior. Thus, we propose a novel adaptive reputation-update rule by combining the number of cooperative strategy in the historical memory. Through extensive numerical simulations, we find that under the appropriate reputation-updating timescale (not too fast or too slow), the emergence of cooperation behavior will be improved greatly, even the cooperation level is better than that under the conventional second-order reputation-based PGG. Taking together, the current results will be conducive to understanding the emergency of cooperation in theory and design a reasonable reputation-updating rule in practice.

在基于声誉的间接互惠框架下，个人声誉的评估和更新在游戏合作的发展中起着重要作用。但是，信誉更新时标不一定与策略选择时标一致。在这封信中，我们探讨了信誉更新时间尺度对基于二阶信誉的公共物品博弈合作行为的影响，并假设信誉更新时间尺度比策略选择时间尺度慢。我们发现信誉更新的频率较低会导致一些延迟，并进一步阻碍了合作行为的紧急性。因此，我们结合历史记忆中合作策略的数量，提出了一种新颖的自适应信誉更新规则。通过广泛的数值模拟，我们发现，在适当的信誉更新时间尺度下（不要太快或太慢），合作行为的出现将大大改善，即使合作水平也要优于传统的基于二阶信誉的PGG。综合起来，当前的结果将有助于从理论上理解合作的紧急性，并在实践中设计合理的声誉更新规则。