Lin Longhao, Li Chengrui, Chen Xiaojie
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China.
Chaos. 2025 Feb 1;35(2). doi: 10.1063/5.0245574.
How to understand the evolution of cooperation remains a scientific challenge. Individual strategy update rule plays an important role in the evolution of cooperation in a population. Previous works mainly assume that individuals adopt one single update rule during the evolutionary process. Indeed, individuals may adopt a mixed update rule influenced by different preferences such as payoff-driven and conformity-driven factors. It is still unclear how such mixed update rules influence the evolutionary dynamics of cooperation from a theoretical analysis perspective. In this work, in combination with the pairwise comparison rule and the conformity rule, we consider a mixed updating procedure into the evolutionary prisoner's dilemma game. We assume that individuals adopt the conformity rule for strategy updating with a certain probability in a structured population. By means of the pair approximation and mean-field approaches, we obtain the dynamical equations for the fraction of cooperators in the population. We prove that under weak selection, there exists one unique interior equilibrium point, which is stable, in the system. Accordingly, cooperators can survive with defectors under the mixed update rule in the structured population. In addition, we find that the stationary fraction of cooperators increases as the conformity strength increases, but is independent of the benefit parameter. Furthermore, we perform numerical calculations and computer simulations to confirm our theoretical predictions.
如何理解合作的演变仍然是一项科学挑战。个体策略更新规则在群体合作的演变中起着重要作用。先前的研究主要假设个体在进化过程中采用单一的更新规则。实际上，个体可能会采用受不同偏好（如收益驱动和从众驱动因素）影响的混合更新规则。从理论分析的角度来看，这种混合更新规则如何影响合作的进化动态仍不清楚。在这项工作中，结合成对比较规则和从众规则，我们在进化囚徒困境博弈中考虑了一种混合更新过程。我们假设个体在结构化群体中以一定概率采用从众规则进行策略更新。通过对近似法和平均场方法，我们得到了群体中合作者比例的动力学方程。我们证明，在弱选择条件下，系统中存在一个唯一的内部平衡点，且该平衡点是稳定的。因此，在结构化群体的混合更新规则下，合作者可以与背叛者共存。此外，我们发现合作者的固定比例随着从众强度的增加而增加，但与收益参数无关。此外，我们进行了数值计算和计算机模拟以证实我们的理论预测。
