IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v168y2023ics0960077923001169.html

Coevolution of consensus and cooperation in evolutionary Hegselmann–Krause dilemma with the cooperation cost

Author

Listed:
  • Huang, Changwei
  • Hou, Yongzhao
  • Han, Wenchen

Abstract

In this work, we focus on the opinion consensus involved with evolutionary games based on Hegselmann–Krause model. Agents can choose to be defective, unwilling to change their opinions, or cooperative, willing to change opinions following the ordinary Hegselmann–Krause rule. The payoff of an agent is assumed as the local benefit, depicting the local consensus around this agent, paying the opinion variation cost, the effort to change the agent’s opinion. Increasing the cost hinders the preference of cooperation and suppresses forging consensus. On the contrary, increasing the bounded confidence, as well as the average degree, helps agents gain a higher benefit and reach consensus. The consensus is the tradeoff between supportive factors, a high confidence bound or a large average degree, and the drawback, a high variation cost. When these two competitive factors share a same strength, it costs the longest time for the population reaching consensus. It is verified by numerical simulations on not only scale-free networks but also Erdös-Rényi random networks.

Suggested Citation

  • Huang, Changwei & Hou, Yongzhao & Han, Wenchen, 2023. "Coevolution of consensus and cooperation in evolutionary Hegselmann–Krause dilemma with the cooperation cost," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    • Handle: RePEc:eee:chsofr:v:168:y:2023:i:c:s0960077923001169
    DOI: 10.1016/j.chaos.2023.113215
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077923001169
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2023.113215?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Ren, Yizhi & Chen, Xiangyu & Wang, Zhen & Shi, Benyun & Cui, Guanghai & Wu, Ting & Choo, Kim-Kwang Raymond, 2018. "Neighbor-considered migration facilitates cooperation in prisoner’s dilemma games," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 95-105.
    2. Rainer Hegselmann & Ulrich Krause, 2002. "Opinion Dynamics and Bounded Confidence Models, Analysis and Simulation," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 5(3), pages 1-2.
    3. Shun Gao & Changwei Huang & Wenchen Han & Junzhong Yang, 2020. "General consensus with circular opinion under attractive and repulsive mechanisms," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 31(05), pages 1-11, March.
    4. Xi Chen & Xiao Zhang & Yong Xie & Wei Li, 2017. "Opinion Dynamics of Social-Similarity-Based Hegselmann–Krause Model," Complexity, Hindawi, vol. 2017, pages 1-12, December.
    5. Huang, Changwei & Dai, Qionglin & Han, Wenchen & Feng, Yuee & Cheng, Hongyan & Li, Haihong, 2018. "Effects of heterogeneous convergence rate on consensus in opinion dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 499(C), pages 428-435.
    6. Guillaume Deffuant & David Neau & Frederic Amblard & Gérard Weisbuch, 2000. "Mixing beliefs among interacting agents," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 3(01n04), pages 87-98.
    7. Xinhao Yang & Sheng Xu & Ze Li, 2017. "Consensus Congestion Control in Multirouter Networks Based on Multiagent System," Complexity, Hindawi, vol. 2017, pages 1-10, June.
    8. Han, Wenchen & Gao, Shun & Huang, Changwei & Yang, Junzhong, 2022. "Non-consensus states in circular opinion model with repulsive interaction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 585(C).
    9. Alessandro Di Mare & Vito Latora, 2007. "Opinion Formation Models Based On Game Theory," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 18(09), pages 1377-1395.
    10. Baumann, Fabian & Sokolov, Igor M. & Tyloo, Melvyn, 2020. "A Laplacian approach to stubborn agents and their role in opinion formation on influence networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    11. Pineda, M. & Buendía, G.M., 2015. "Mass media and heterogeneous bounds of confidence in continuous opinion dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 420(C), pages 73-84.
    12. Fei Ding & Yun Liu & Yong Li, 2009. "Co-Evolution Of Opinion And Strategy In Persuasion Dynamics: An Evolutionary Game Theoretical Approach," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 479-490.
    13. Fu, Guiyuan & Zhang, Weidong & Li, Zhijun, 2015. "Opinion dynamics of modified Hegselmann–Krause model in a group-based population with heterogeneous bounded confidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 558-565.
    14. Han, Wenchen & Huang, Changwei & Yang, Junzhong, 2019. "Opinion clusters in a modified Hegselmann–Krause model with heterogeneous bounded confidences and stubbornness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 531(C).
    15. Quanbo Zha & Gang Kou & Hengjie Zhang & Haiming Liang & Xia Chen & Cong-Cong Li & Yucheng Dong, 2020. "Opinion dynamics in finance and business: a literature review and research opportunities," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 6(1), pages 1-22, December.
    16. Xi Chen & Zhan Wu & Hongwei Wang & Wei Li, 2017. "Impact of Heterogeneity on Opinion Dynamics: Heterogeneous Interaction Model," Complexity, Hindawi, vol. 2017, pages 1-10, April.
    17. Han, Wenchen & Feng, Yuee & Qian, Xiaolan & Yang, Qihui & Huang, Changwei, 2020. "Clusters and the entropy in opinion dynamics on complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 559(C).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Xu, Zhixiong & Xu, Zhehang & Zhang, Wei & Han, Xiao-Pu & Meng, Fanyuan, 2024. "Memory-based spatial evolutionary prisoner’s dilemma," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    2. Zhang, Wei & Brandes, Ulrik, 2023. "Conformity versus credibility: A coupled rumor-belief model," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    3. Cui, Peng-Bi, 2023. "Exploring the foundation of social diversity and coherence with a novel attraction–repulsion model framework," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 618(C).
    4. Zhu, Enqiang & Chen, Jia & Liu, Shiqi & Zhang, Xiaokai & Liu, Chanjuan, 2025. "The influence of heterogeneous investment on the evolution of trust in complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 677(C).
    5. Peng, Yuan & Zhao, Yiyi & Dong, Jianglin & Hu, Jiangping, 2025. "On demonstrating liberating effect in complex social networks: Modeling multiple pressure-coping strategies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 674(C).
    6. Huang, Changwei & Luo, Yijun & Han, Wenchen, 2023. "Cooperation and synchronization in evolutionary opinion changing rate games," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    7. Luo, Yijun & Huang, Changwei & Han, Wenchen, 2024. "The evolution of cooperation and global synchronization in the evolutionary Kuramoto dilemma combined with the prisoner's dilemma," Applied Mathematics and Computation, Elsevier, vol. 482(C).
    8. Huang, Changwei & Bian, Huanyu & Han, Wenchen, 2024. "Breaking the symmetry neutralizes the extremization under the repulsion and higher order interactions," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    9. Li, Jiayuan & Xu, Haojie & Huang, Fuying & Han, Wenchen & Huang, Changwei, 2026. "Depolarization in the coevolutionary dynamics of opinion and cooperation," Applied Mathematics and Computation, Elsevier, vol. 520(C).
    10. Ding, Haixin & Xie, Li, 2024. "The applicability of positive information in negative opinion management: An attitude-laden communication perspective," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 645(C).
    11. Yang, Yizhou & Li, Haihong & Gao, Shun & Dai, Qionglin & Yang, Junzhong, 2025. "Interdependent evolutionary dynamics of opinion and strategy on two-layer networks," Chaos, Solitons & Fractals, Elsevier, vol. 191(C).
    12. Zhang, Huizhen & An, Tianbo & Yan, Pingping & Hu, Kaipeng & An, Jinjin & Shi, Lijuan & Zhao, Jian & Wang, Jingrui, 2024. "Exploring cooperative evolution with tunable payoff’s loners using reinforcement learning," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Huang, Changwei & Bian, Huanyu & Han, Wenchen, 2024. "Breaking the symmetry neutralizes the extremization under the repulsion and higher order interactions," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    2. Han, Wenchen & Gao, Shun & Huang, Changwei & Yang, Junzhong, 2022. "Non-consensus states in circular opinion model with repulsive interaction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 585(C).
    3. Takesue, Hirofumi, 2023. "Relative opinion similarity leads to the emergence of large clusters in opinion formation models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).
    4. Catherine A. Glass & David H. Glass, 2021. "Social Influence of Competing Groups and Leaders in Opinion Dynamics," Computational Economics, Springer;Society for Computational Economics, vol. 58(3), pages 799-823, October.
    5. Han, Wenchen & Feng, Yuee & Qian, Xiaolan & Yang, Qihui & Huang, Changwei, 2020. "Clusters and the entropy in opinion dynamics on complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 559(C).
    6. Cahill, Patrick & Gottwald, Georg A., 2025. "A modified Hegselmann–Krause model for interacting voters and political parties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 665(C).
    7. Ding, Haixin & Xie, Li, 2024. "The applicability of positive information in negative opinion management: An attitude-laden communication perspective," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 645(C).
    8. Wang, Chaoqian, 2021. "Opinion dynamics with bilateral propaganda and unilateral information blockade," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).
    9. Zhang, Lan & Gao, Shun, 2025. "The impact of environment-based heterogeneous bounded confidence on opinion dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 201(P2).
    10. Xi Chen & Shen Zhao & Wei Li, 2019. "Opinion Dynamics Model Based on Cognitive Styles: Field-Dependence and Field-Independence," Complexity, Hindawi, vol. 2019, pages 1-12, February.
    11. Glass, Catherine A. & Glass, David H., 2021. "Opinion dynamics of social learning with a conflicting source," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 563(C).
    12. Antonio Parravano & Ascensión Andina-Díaz & Miguel A Meléndez-Jiménez, 2016. "Bounded Confidence under Preferential Flip: A Coupled Dynamics of Structural Balance and Opinions," PLOS ONE, Public Library of Science, vol. 11(10), pages 1-23, October.
    13. Xu, Yuan & Liu, Shifeng & Cheng, T.C.E. & Feng, Xue & Wang, Jun & Shang, Xiaopu, 2025. "Opinion convergence and management: Opinion dynamics in interactive group decision-making," European Journal of Operational Research, Elsevier, vol. 323(3), pages 938-951.
    14. Huang, Changwei & Luo, Yijun & Han, Wenchen, 2023. "Cooperation and synchronization in evolutionary opinion changing rate games," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    15. Hou, Jian & Li, Wenshan & Jiang, Mingyue, 2021. "Opinion dynamics in modified expressed and private model with bounded confidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 574(C).
    16. Tan, Eugene & Stemler, Thomas & Small, Michael, 2024. "Cognitive dissonance and introversion effects on opinion dynamics and echo chamber formation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 654(C).
    17. Yang, Yizhou & Li, Haihong & Gao, Shun & Dai, Qionglin & Yang, Junzhong, 2025. "Interdependent evolutionary dynamics of opinion and strategy on two-layer networks," Chaos, Solitons & Fractals, Elsevier, vol. 191(C).
    18. Bashari, Masoud & Akbarzadeh-T, Mohammad-R., 2020. "Controlling opinions in Deffuant model by reconfiguring the network topology," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 544(C).
    19. Dong Jiang & Qionglin Dai & Haihong Li & Junzhong Yang, 2024. "Opinion dynamics based on social learning theory," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 97(12), pages 1-9, December.
    20. Peng, Yuan & Zhao, Yiyi & Dong, Jianglin & Hu, Jiangping, 2025. "On demonstrating liberating effect in complex social networks: Modeling multiple pressure-coping strategies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 674(C).

    More about this item

    Keywords

    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:168:y:2023:i:c:s0960077923001169. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.