Long homogeneous payoff records with the latest strategy promotes the cooperation.

In this study, we studied the fraction of cooperators in the public goods game, taking into account the memory effect that affects the strategy updating. Unlike previous studies where an agent learned the opponent's last strategy based on their last payoffs, agents with memory in this study choose to cooperate according to opponents' effective strategies by comparing their effective payoffs based on payoffs and strategies in their memories. The effective payoff of an agent is the weighted average of previous strategies in the agent's memory. The weight is the decay measuring the significance of previous strategies and former payoffs are less significant than latter ones upon the future strategy. And it is the same with the effective strategy. By this means, when the effective payoff and the effective strategy share a same memory length and a same set of decay, the numerical simulation shows increasing the memory length or a homogeneous decay promotes cooperation among agents. However, it is a surprise that the effective payoff and the effective strategy have opposite effects. Homogeneous payoff weights lead to a higher fraction of cooperators, while heterogeneous strategy weights favors the cooperation, especially when agents only consider the latest strategy. Comparing the effect of memorizing payoffs and strategies, the effect of memorizing payoffs plays a dominant role. Furthermore, when the total memory length is limited, agents should memorize as many historical payoffs as possible. In addition the qualitative result above is independent of the rational noise. • Homogeneous weights on historic payoffs promote the cooperation level most. • Agents only need to remember components' latest strategy. • Long memory with rational agents promotes the cooperation. [ABSTRACT FROM AUTHOR]