环境反馈与混合更新规则驱动的合作演化
The Evolution of Cooperation Driven by Environmental Feedback and Hybrid Update Rules
DOI: 10.12677/orf.2026.162017, PDF,    科研立项经费支持
作者: 刘 婷, 蒋 君*, 王艺清, 陶 俊：武汉科技大学数学与系统科学学院，湖北 武汉
关键词: 合作演化；环境反馈；混合更新规则；囚徒困境博弈；Cooperative Evolution； Environmental Feedback； Hybrid Update Rules； Prisoner’s Dilemma
摘要: 在现实社会系统中，个体行为与环境状态相互作用并协同演化。本文在结构化种群的囚徒困境博弈框架下，引入一种同时作用于个体适应度与决策期望的环境反馈机制，研究环境反馈与期望–模仿混合更新规则耦合条件下合作行为的演化特征。数值模拟结果表明，引入环境反馈能够显著提高系统中的合作水平，并在较强背叛诱惑条件下拓展合作可持续存在的参数范围。进一步发现，期望敏感度参数β的促进作用具有明显的区间依赖性，过大的β反而削弱合作的稳定性。本文的研究为理解环境变化、个体预期与合作行为之间的相互作用提供了新的视角。
Abstract: In real social systems, individual behavior and environmental states interact and coevolve. Within the framework of the prisoner’s dilemma game on structured populations, this study introduces an environmental feedback mechanism that simultaneously influences individual fitness and decision expectations, and investigates the evolutionary dynamics of cooperation under the coupling of environmental feedback and an expectation imitation hybrid update rule. Numerical simulations show that environmental feedback can significantly enhance the overall level of cooperation and extend the parameter region in which cooperation can be sustained, even under strong temptation to defect. Furthermore, we find that the promotive effect of the expectation sensitivity parameter β exhibits a pronounced interval dependence, with excessively large values of β instead undermining the stability of cooperation. These results provide new insights into the interplay between environmental dynamics, individual expectations, and the evolution of cooperative behavior.
文章引用：刘婷, 蒋君, 王艺清, 陶俊. 环境反馈与混合更新规则驱动的合作演化[J]. 运筹与模糊学, 2026, 16(2): 31-43. https://doi.org/10.12677/orf.2026.162017

参考文献

[1] Von Neumann, J. and Morgenstern, O. (1947) Theory of Games and Economic Behavior. 2nd Revised Edition, Princeton University Press.
[2] Dugatkin, L.A. (1997) Cooperation among Animals: An Evolutionary Perspective. Oxford University Press.
[3] Nash, J.F. (1950) Equilibrium Points in n-Person Games. Proceedings of the National Academy of Sciences, 36, 48-49. [Google Scholar] [CrossRef] [PubMed]
[4] Smith, J.M. (1982) Evolution and the Theory of Games. In: Smith, J.M., Ed., Did Darwin Get It Right? Essays on Games, Sex and Evolution, Springer, 202-215. [Google Scholar] [CrossRef] 
[5] Nowak, M.A. (2006) Evolutionary Dynamics: Exploring the Equations of Life. Harvard University Press. [Google Scholar] [CrossRef] 
[6] Lieberman, E., Hauert, C. and Nowak, M.A. (2005) Evolutionary Dynamics on Graphs. Nature, 433, 312-316. [Google Scholar] [CrossRef] [PubMed]
[7] Maciejewski, W., Fu, F. and Hauert, C. (2014) Evolutionary Game Dynamics in Populations with Heterogenous Structures. PLOS Computational Biology, 10, e1003567. [Google Scholar] [CrossRef] [PubMed]
[8] Fehr, E. (2004) Don’t Lose Your Reputation. Nature, 432, 449-450. [Google Scholar] [CrossRef] [PubMed]
[9] Hu, Q., Jin, T., Jiang, Y. and Liu, X. (2024) Reputation Incentives with Public Supervision Promote Cooperation in Evolutionary Games. Applied Mathematics and Computation, 466, Article ID: 128445. [Google Scholar] [CrossRef] 
[10] Szolnoki, A. and Perc, M. (2012) Evolutionary Advantages of Adaptive Rewarding. New Journal of Physics, 14, Article ID: 093016. [Google Scholar] [CrossRef] 
[11] Miyaji, K. and Tanimoto, J. (2021) A Co-Evolutionary Model Combined Mixed-Strategy and Network Adaptation by Severing Disassortative Neighbors Promotes Cooperation in Prisoner’s Dilemma Games. Chaos, Solitons & Fractals, 143, Article ID: 110603. [Google Scholar] [CrossRef] 
[12] Wu, Z.X., Xu, X.J., Huang, Z.G., Wang, S.J. and Wang, Y.H. (2006) Evolutionary Prisoner’s Dilemma Game with Dynamic Preferential Selection. Physical Review E, 74, Article ID: 021107. [Google Scholar] [CrossRef] [PubMed]
[13] Lu, J., Wang, L., Wang, Y. and Zhang, X. (2017) Logit Selection Promotes Cooperation in Voluntary Public Goods Game. Applied Mathematics and Computation, 310, 134-138. [Google Scholar] [CrossRef] 
[14] Yan, F., Chen, X., Qiu, Z. and Szolnoki, A. (2021) Cooperator Driven Oscillation in a Time-Delayed Feedback-Evolving Game. New Journal of Physics, 23, Article ID: 053017. [Google Scholar] [CrossRef] 
[15] Yan, F., Hou, X., Tian, T. and Chen, X. (2023) Nonlinear Model Reference Adaptive Control Approach for Governance of the Commons in a Feedback-Evolving Game. Chaos, Solitons & Fractals, 174, Article ID: 113861. [Google Scholar] [CrossRef] 
[16] Perc, M., Jordan, J.J., Rand, D.G., Wang, Z., Boccaletti, S. and Szolnoki, A. (2017) Statistical Physics of Human Cooperation. Physics Reports, 687, 1-51. [Google Scholar] [CrossRef] 
[17] Ma, X., Quan, J. and Wang, X. (2023) Effect of Reciprocity Mechanisms on Evolutionary Dynamics in Feedback-Evolving Games. Nonlinear Dynamics, 112, 709-729. [Google Scholar] [CrossRef] 
[18] Weitz, J.S., Eksin, C., Paarporn, K., Brown, S.P. and Ratcliff, W.C. (2016) An Oscillating Tragedy of the Commons in Replicator Dynamics with Game-Environment Feedback. Proceedings of the National Academy of Sciences, 113, E7518-E7525. [Google Scholar] [CrossRef] [PubMed]
[19] Cao, L. and Wu, B. (2021) Eco-Evolutionary Dynamics with Payoff-Dependent Environmental Feedback. Chaos, Solitons & Fractals, 150, Article ID: 111088. [Google Scholar] [CrossRef] 
[20] Wang, X. and Fu, F. (2020) Eco-Evolutionary Dynamics with Environmental Feedback: Cooperation in a Changing World. Europhysics Letters, 132, Article No. 10001. [Google Scholar] [CrossRef] 
[21] Gong, L., Yao, W., Gao, J. and Cao, M. (2022) Limit Cycles Analysis and Control of Evolutionary Game Dynamics with Environmental Feedback. Automatica, 145, Article ID: 110536. [Google Scholar] [CrossRef] 
[22] Das Bairagya, J., Mondal, S.S., Chowdhury, D. and Chakraborty, S. (2021) Game-Environment Feedback Dynamics in Growing Population: Effect of Finite Carrying Capacity. Physical Review E, 104, Article ID: 044407. [Google Scholar] [CrossRef] [PubMed]
[23] Szolnoki, A. and Chen, X. (2017) Environmental Feedback Drives Cooperation in Spatial Social Dilemmas. EPL (Europhysics Letters), 120, Article No. 58001. [Google Scholar] [CrossRef] 
[24] 张艳玲, 莫廷钰, 李松涛, 等. 演化博弈与资源配置综述[J]. 工程科学学报, 2022, 44(3): 402-410.
[25] Chen, X. and Wang, L. (2008) Promotion of Cooperation Induced by Appropriate Payoff Aspirations in a Small-World Networked Game. Physical Review E, 77, Article ID: 017103. [Google Scholar] [CrossRef] [PubMed]
[26] Perc, M. and Wang, Z. (2010) Heterogeneous Aspirations Promote Cooperation in the Prisoner’s Dilemma Game. PLOS ONE, 5, e15117. [Google Scholar] [CrossRef] [PubMed]
[27] Du, J., Wu, B., Altrock, P.M. and Wang, L. (2014) Aspiration Dynamics of Multi-Player Games in Finite Populations. Journal of the Royal Society Interface, 11, Article ID: 20140077. [Google Scholar] [CrossRef] [PubMed]
[28] Zhang, J., Hu, B., Huang, Y.J., Deng, Z.H. and Wu, T. (2020) The Evolution of Cooperation Affected by Aspiration-Driven Updating Rule in Multi-Games with Voluntary Participation. Chaos, Solitons & Fractals, 139, Article ID: 110067. [Google Scholar] [CrossRef] 
[29] Xu, K., Li, K., Cong, R. and Wang, L. (2017) Cooperation Guided by the Coexistence of Imitation Dynamics and Aspiration Dynamics in Structured Populations. EPL (Europhysics Letters), 117, Article No. 48002. [Google Scholar] [CrossRef] 
[30] Wang, X., Gu, C., Zhao, J. and Quan, J. (2019) Evolutionary Game Dynamics of Combining the Imitation and Aspiration-Driven Update Rules. Physical Review E, 100, Article ID: 022411. [Google Scholar] [CrossRef] [PubMed]
[31] Arefin, M.R. and Tanimoto, J. (2020) Evolution of Cooperation in Social Dilemmas under the Coexistence of Aspiration and Imitation Mechanisms. Physical Review E, 102, Article ID: 032120. [Google Scholar] [CrossRef] [PubMed]
[32] Wang, X., Li, C. and Fan, S. (2021) Evolution of Cooperation in Prisoner’s Dilemma Game under the Coupling of Aspiration and Imitation Rules. Europhysics Letters, 136, Article No. 58004. [Google Scholar] [CrossRef] 
[33] Zhao, J., Zhang, R., An, T., Zhang, H., Tong, D., Luo, X., et al. (2025) The Effect of Heterogeneous Perceptions with Environmental Feedback on Spatial Social Dilemmas. Chaos, Solitons & Fractals, 192, Article ID: 115982. [Google Scholar] [CrossRef] 
[34] Gao, Y., Li, M., Hu, Y., Wang, R. and Wang, C. (2024) Evolutionary Dynamics in Voluntary Prisoner’s Dilemma Game with Environmental Feedbacks. Chaos, Solitons & Fractals, 181, Article ID: 114667. [Google Scholar] [CrossRef] 
[35] Guo, H., Shen, C., Dai, D., Zhang, M., Chu, C. and Shi, L. (2017) Environment Promotes the Evolution of Cooperation in Spatial Voluntary Prisoner’s Dilemma Game. Applied Mathematics and Computation, 315, 47-53. [Google Scholar] [CrossRef] 
[36] Yang, L. and Zhang, L. (2021) Environmental Feedback in Spatial Public Goods Game. Chaos, Solitons & Fractals, 142, Article ID: 110485. [Google Scholar] [CrossRef] 

为你推荐