IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v193y2025ics0960077925001481.html

The evolution of cooperation and punishment in spatial public goods games with bribery

Author

Listed:
  • Ding, Rui
  • Wang, Xianjia
  • Quan, Ji
  • Ma, Shuting
  • Zhao, Jinhua

Abstract

Altruistic punishment provides a compelling explanation for the prevalence of cooperation in social dilemmas. However, defectors often have an incentive to bribe punishers to avoid sanctions, while punishers may accept bribes and save punishment costs. To explore the impact of bribe on cooperation in structured populations, we introduce bribers and bribees into the spatial public goods game with punishment. By studying the phase transitions of this five-strategy system, we aim to uncover the underlying mechanisms behind strategy competition. Our findings reveal that bribers collude with bribees to invade other strategies, both directly at multi-strategy interface and indirectly through mediators such as defectors or bribers. Consequently, bribers protect bribe-recipients from invasion by defectors, leading to increased contribution levels when punishment intensity is relatively weak. However, when the fine is sufficiently high to empower punishers in confront defectors effectively, bribe can undermine the maintenance of punishment. Interestingly, the competitive advantage of bribers and bribe-recipients diminishes as bribery funds increase, leading to the emergence of a cyclic state involving three strategies: bribers, bribe-recipients and defectors.

Suggested Citation

  • Ding, Rui & Wang, Xianjia & Quan, Ji & Ma, Shuting & Zhao, Jinhua, 2025. "The evolution of cooperation and punishment in spatial public goods games with bribery," Chaos, Solitons & Fractals, Elsevier, vol. 193(C).
    • Handle: RePEc:eee:chsofr:v:193:y:2025:i:c:s0960077925001481
    DOI: 10.1016/j.chaos.2025.116135
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925001481
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2025.116135?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. Carpenter, Jeffrey P., 2007. "Punishing free-riders: How group size affects mutual monitoring and the provision of public goods," Games and Economic Behavior, Elsevier, vol. 60(1), pages 31-51, July.
    2. Wang, Xianjia & Ding, Rui & Zhao, Jinhua & Chen, Wenman & Gu, Cuiling, 2022. "Competition of punishment and reward among inequity-averse individuals in spatial public goods games," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    3. Nikos Nikiforakis & Hans-Theo Normann, 2008. "A comparative statics analysis of punishment in public-good experiments," Experimental Economics, Springer;Economic Science Association, vol. 11(4), pages 358-369, December.
    4. Dirk Helbing & Attila Szolnoki & Matjaž Perc & György Szabó, 2010. "Evolutionary Establishment of Moral and Double Moral Standards through Spatial Interactions," PLOS Computational Biology, Public Library of Science, vol. 6(4), pages 1-9, April.
    5. Ling, Ting & Li, Zhang & Feng, Minyu & Szolnoki, Attila, 2025. "Supervised cooperation on interdependent public goods games," Applied Mathematics and Computation, Elsevier, vol. 492(C).
    6. Ananish Chaudhuri, 2011. "Sustaining cooperation in laboratory public goods experiments: a selective survey of the literature," Experimental Economics, Springer;Economic Science Association, vol. 14(1), pages 47-83, March.
    7. Dirk Semmann & Hans-Jürgen Krambeck & Manfred Milinski, 2003. "Volunteering leads to rock–paper–scissors dynamics in a public goods game," Nature, Nature, vol. 425(6956), pages 390-393, September.
    8. Wang, Xianjia & Ding, Rui & Zhao, Jinhua & Chen, Wenman, 2021. "The rise and fall of donation behavior through reputation," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    9. Liu, Linjie & Chen, Xiaojie, 2022. "Effects of interconnections among corruption, institutional punishment, and economic factors on the evolution of cooperation," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    10. Ernst Fehr & Urs Fischbacher, 2003. "The nature of human altruism," Nature, Nature, vol. 425(6960), pages 785-791, October.
    11. Lee, Hsuan-Wei & Cleveland, Colin & Szolnoki, Attila, 2024. "Supporting punishment via taxation in a structured population," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    12. Martin A. Nowak & Karl Sigmund, 2005. "Evolution of indirect reciprocity," Nature, Nature, vol. 437(7063), pages 1291-1298, October.
    13. Lee, Hsuan-Wei & Cleveland, Colin & Szolnoki, Attila, 2022. "Mercenary punishment in structured populations," Applied Mathematics and Computation, Elsevier, vol. 417(C).
    14. Ding, Rui & Wang, Xianjia & Zhao, Jinhua & Gu, Cuiling, 2025. "Effects of punishment driven by inequity aversion on promoting cooperation in public goods games," Chaos, Solitons & Fractals, Elsevier, vol. 190(C).
    15. Joung-Hun Lee & Yoh Iwasa & Ulf Dieckmann & Karl Sigmund, 2019. "Social evolution leads to persistent corruption," Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, vol. 116(27), pages 13276-13281, July.
    16. Shi, Zhenyu & Wei, Wei & Zheng, Hongwei & Zheng, Zhiming, 2023. "Bidirectional supervision: An effective method to suppress corruption and defection under the third party punishment mechanism of donation games," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    17. Saptarshi Pal & Christian Hilbe, 2022. "Reputation effects drive the joint evolution of cooperation and social rewarding," Nature Communications, Nature, vol. 13(1), pages 1-11, December.
    18. Wang, Qiang & He, Nanrong & Chen, Xiaojie, 2018. "Replicator dynamics for public goods game with resource allocation in large populations," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 162-170.
    19. Prateek Verma & Supratim Sengupta, 2015. "Bribe and Punishment: An Evolutionary Game-Theoretic Analysis of Bribery," PLOS ONE, Public Library of Science, vol. 10(7), pages 1-22, July.
    20. Jörg Gross & Carsten K. W. Dreu, 2019. "The rise and fall of cooperation through reputation and group polarization," Nature Communications, Nature, vol. 10(1), pages 1-10, December.
    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. Zhang, Yongqiang & Zheng, Zehao & Zhang, Xiaoming & Ma, Jinlong, 2025. "A layered strategy updating mechanism for spatial public goods game with punishment," Chaos, Solitons & Fractals, Elsevier, vol. 201(P2).
    2. Liu, Xinyu & Jin, Wei & Chen, Guanrong & Tang, Changbing & Qian, Youhua & Jin, Weifeng, 2025. "Small groups nurturing collective wisdom: The punishment-prediction reinforcement learning mechanism for multi-group cooperation," Chaos, Solitons & Fractals, Elsevier, vol. 201(P1).

    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. Ding, Rui & Wang, Xianjia & Zhao, Jinhua & Gu, Cuiling, 2025. "Effects of punishment driven by inequity aversion on promoting cooperation in public goods games," Chaos, Solitons & Fractals, Elsevier, vol. 190(C).
    2. Ding, Rui & Wang, Xianjia & Quan, Ji & Ma, Shuting & Zhao, Jinhua, 2025. "Evolutionary dynamics of spatial public goods games with insurance mechanisms," Chaos, Solitons & Fractals, Elsevier, vol. 195(C).
    3. Wang, Xianjia & Ding, Rui & Zhao, Jinhua & Chen, Wenman & Gu, Cuiling, 2022. "Competition of punishment and reward among inequity-averse individuals in spatial public goods games," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    4. Ding, Rui & Wang, Xianjia & Zhao, Jinhua & Gu, Cuiling & Wang, Tao, 2023. "The evolution of cooperation in spatial public goods games under a risk-transfer mechanism," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    5. Ding, Rui & Wang, Xianjia & Liu, Yang & Zhao, Jinhua & Gu, Cuiling, 2023. "Evolutionary games with environmental feedbacks under an external incentive mechanism," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    6. Qin Li & Ting Ling & Minyu Feng & Attila Szolnoki, 2026. "Supervised tax compliance and evasion from a spatial evolutionary game perspective," Humanities and Social Sciences Communications, Palgrave Macmillan, vol. 13(1), pages 1-12, December.
    7. Takesue, Hirofumi, 2026. "Hinderance of cooperation by individual solutions: Evolutionary dynamics of three-strategy games combining the prisoner’s dilemma and stag hunt," Applied Mathematics and Computation, Elsevier, vol. 510(C).
    8. Simon Gaechter & Benedikt Herrmann, 2008. "Reciprocity, culture, and human cooperation: Previous insights and a new cross-cultural experiment," Discussion Papers 2008-14, The Centre for Decision Research and Experimental Economics, School of Economics, University of Nottingham.
    9. Takesue, Hirofumi, 2025. "Evolution of cooperation in a three-strategy game combining snowdrift and stag hunt games," Applied Mathematics and Computation, Elsevier, vol. 495(C).
    10. Gao, Meng & Li, Zhi & Wu, Te, 2023. "Evolutionary dynamics of friendship-driven reputation strategies," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    11. Han, Dun & He, Youxin, 2023. "The impact of labor subsidy, taxation and corruption on individual behavior," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    12. Kang, Hongwei & Li, Xin & Shen, Yong & Sun, Xingping & Chen, Qingyi, 2024. "Particle swarm optimization with historical return decay enhances cooperation in public goods games with investment risks," Chaos, Solitons & Fractals, Elsevier, vol. 189(P1).
    13. Simon Gaechter & Benedikt Herrmann, 2008. "Reciprocity, culture, and human cooperation: Previous insights and a new cross-cultural experiment," Discussion Papers 2008-14, The Centre for Decision Research and Experimental Economics, School of Economics, University of Nottingham.
    14. Kang, Hongwei & Li, Zhekang & Shen, Yong & Huang, Zhiyuan & Sun, Xingping & Chen, Qingyi, 2024. "From eligibility to suitability: Regulation and restriction of reputation-based access system on free-riding behavior in spatial public goods game," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    15. Shen, Yong & Deng, Zixuan & Kang, Hongwei & Sun, Xingping & Chen, Qingyi, 2025. "The impact of tax-based rewards and mercenary punishment on public goods games," Chaos, Solitons & Fractals, Elsevier, vol. 199(P1).
    16. Calabuig, Vicente & Fatas, Enrique & Olcina, Gonzalo & Rodriguez-Lara, Ismael, 2016. "Carry a big stick, or no stick at all," Journal of Economic Psychology, Elsevier, vol. 57(C), pages 153-171.
    17. Gächter, Simon & Herrmann, Benedikt, 2011. "The limits of self-governance when cooperators get punished: Experimental evidence from urban and rural Russia," European Economic Review, Elsevier, vol. 55(2), pages 193-210, February.
    18. Chaudhuri, Ananish & Cruickshank, Amy & Sbai, Erwann, 2015. "Gender differences in personnel management: Some experimental evidence," Journal of Behavioral and Experimental Economics (formerly The Journal of Socio-Economics), Elsevier, vol. 58(C), pages 20-32.
    19. Liu, Linjie & Chen, Xiaojie, 2022. "Effects of interconnections among corruption, institutional punishment, and economic factors on the evolution of cooperation," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    20. Lee, Hsuan-Wei & Cleveland, Colin & Szolnoki, Attila, 2024. "Supporting punishment via taxation in a structured population," Chaos, Solitons & Fractals, Elsevier, vol. 178(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:193:y:2025:i:c:s0960077925001481. 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.