Due to the heterogeneity of strategy updating rules progressively destroying the cluster of cooperators, cooperation would be heavily suppressed when players adopt mixed strategy updating rules. Thus, how to improve the emergence of cooperation with the scenario of heterogeneous strategy updating rules becomes an important open issue. In this paper, we introduce the memory factor into the game model, and then study the joint effect of memory and heterogeneous strategy updating rules on the emergence of cooperation. Detailly, in our game model, memory-based imitation and innovation are chosen as two different strategy updating rules. Afterwards, the annealing and quenching rules are specifically used as the methods to mix the proposed two memory-based strategy updating rules. At last, Monte Carlo simulations are conducted to demonstrate the significance of our model. The simulation results show that for memory-based imitators, memory is not always effective in promoting cooperation. It’s more like a catalyst. When the proportion of the memory-based imitation is large, it promotes the emergence of cooperation. When the proportion of innovation is large, it accelerates the extinction of cooperation; for overall proportion of cooperation, memory effectively promotes the emergence of cooperation and improves the ability of cooperators to resist high temptation, but memory is a partial optimization, it cannot offset the deterioration of cooperation caused by heterogeneous strategies updating rules, memory only reduces the degree of deterioration. Finally, compared with the annealing rule, the change of quenching rule is more smooth, because it is easier for cooperators to form clusters under the quenching rule. Our investigation sheds some light to the role of memory in prisoner’s dilemma game under heterogeneous strategies updating rules.

中文翻译：

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多策略更新机制下囚徒困境博弈的记忆效应

由于策略更新规则的异质性逐渐破坏合作者集群，当玩家采用混合策略更新规则时，合作将受到严重抑制。因此，如何在异构策略更新规则的场景下改善协作的出现成为一个重要的开放性问题。在本文中，我们将记忆因素引入博弈模型，然后研究记忆和异构策略更新规则对合作出现的共同影响。具体来说，在我们的博弈模型中，基于记忆的模仿和创新被选为两种不同的策略更新规则。之后，具体使用退火和淬火规则作为混合所提出的两种基于内存的策略更新规则的方法。最后，进行蒙特卡罗模拟以证明我们模型的重要性。模拟结果表明，对于基于记忆的模仿者来说，记忆并不总是有效地促进合作。它更像是一种催化剂。当基于记忆的模仿比例大时，促进了合作的出现。创新占比大时，加速合作消亡；对于整体合作比例，记忆有效促进了合作的出现，提高了合作者抵抗高诱惑的能力，但记忆是局部优化，无法抵消异构策略更新规则造成的合作恶化，记忆只会降低程度的恶化。最后，与退火规则相比，淬火规则的变化更加平滑，因为在淬火规则下，合作者更容易形成集群。我们的调查揭示了异质策略更新规则下记忆在囚徒困境博弈中的作用。