Chinese Science Bulletin

, Volume 57, Issue 7, pp 724–728 | Cite as

Spatial prisoner’s dilemma games with increasing size of the interaction neighborhood on regular lattices

  • Juan Wang
  • ChengYi Xia
  • YiLing Wang
  • Shuai Ding
  • JunQing Sun
Open Access
Article Statistical Physics

Abstract

We studied the evolution of cooperation in the prisoner’s dilemma game on a square lattice where the size of the interaction neighborhood is considered. Firstly, the effects of noise and the cost-to-benefit ratio on the maintenance of cooperation were investigated. The results indicate that the cooperation frequency depends on the noise and cost-to-benefit ratio: cooperation reaches a climax as noise increases, but it monotonously decreases and even vanishes with the ratio increasing. Furthermore, we investigated how the size of the interaction neighborhood affects the emergence of cooperation in detail. Our study demonstrates that cooperation is remarkably enhanced by an increase in the size of the interaction neighborhood. However, cooperation died out when the size of the interaction neighborhood became too large since the system was similar to the mean-field system. On this basis, a cluster-forming mechanism acting among cooperators was also explored, and it showed that the moderate range of the neighborhood size is beneficial for forming larger cooperative clusters. Finally, large-scale Monte Carlo simulations were carried out to visualize and interpret these phenomena explicitly.

Keywords

prisoner’s dilemma game emergence of cooperation size of interaction neighborhood 

References

  1. 1.
    Hamilton W D. The evolution of social behavior. J Theor Biol, 1964, 7: 1–16CrossRefGoogle Scholar
  2. 2.
    Nowak M A, Sigmund K. Evolution of indirect reciprocity by image scoring. Nature, 1998, 393: 573–577CrossRefGoogle Scholar
  3. 3.
    Axelrod R, Hamilton W D. The evolution of cooperation. Science, 1981, 211: 1390–1396CrossRefGoogle Scholar
  4. 4.
    Wang Z, Du W B, Cao X B, et al. Integrating neighborhoods in the evaluation of fitness promotes cooperation in the spatial prisoner’s dilemma game. Physica A, 2011, 390: 1234–1240CrossRefGoogle Scholar
  5. 5.
    Fehr E, Gachter S. Altruistic punishment in humans. Nature, 2002, 415: 137–140CrossRefGoogle Scholar
  6. 6.
    Nowak M A, Sigmund K. Evolution of indirect reciprocity. Nature, 2005, 437: 1291–1298CrossRefGoogle Scholar
  7. 7.
    Rong Z H, Wu Z X. Effect of the degree correlation in public goods game on scale-free networks. Europhys Lett, 2009, 87: 30001CrossRefGoogle Scholar
  8. 8.
    Hauert C, De Monte S, Hofbauer J, et al. Volunteering as red queen mechanism for cooperation in public goods game. Science, 2002, 296: 1129–1132CrossRefGoogle Scholar
  9. 9.
    Semmann D, Krambeck H J, Milinski M. Volunteering leads to rock-paper-scissors dynamics in a public goods game. Nature, 2003, 425: 390–393CrossRefGoogle Scholar
  10. 10.
    Wang Z, Xu Z J, Zhang L Z. Maintenance of cooperation induced by punishment in public goods games. Chin Phys B, 2009, 19: 110201CrossRefGoogle Scholar
  11. 11.
    Szabó G, Fáth G. Evolutionary games on graphs. Phys Rep, 2007, 446: 97–216CrossRefGoogle Scholar
  12. 12.
    Cao X B, Du W B, Rong Z H. Evolutionary public goods game on scale-free networks with heterogeneous investment. Physica A, 2010, 389: 1273–1280CrossRefGoogle Scholar
  13. 13.
    Wang Z, Perc M. Aspiring to the fittest and promotion of cooperation in the prisoner’s dilemma game. Phys Rev E, 2010, 82: 021115CrossRefGoogle Scholar
  14. 14.
    Xia C Y, Zhao J, Wang J, et al. Influence of vertex weight on cooperative behavior in a spatial snowdrift game. Phys Scr, 2011, 84: 025802CrossRefGoogle Scholar
  15. 15.
    Du W B, Cao X B, Yang H X, et al. Evolutionary prisoner’s dilemma on Newman-Watts social networks with an asymmetric payoff distribution mechanism. Chin Phys B, 2010, 19: 010204CrossRefGoogle Scholar
  16. 16.
    Wang Z, Xu Z J, Zhang L Z. Punishment in optional public goods games. Chin Phys B, 2009, 19: 100204Google Scholar
  17. 17.
    Doebeli M, Hauert C. Models of cooperation based on the prisoner’s dilemma and the snowdrift game. Ecol Lett, 2005, 8: 748–766CrossRefGoogle Scholar
  18. 18.
    Wang Z, Perc M. Aspiring to the fittest and promotion of cooperation in the prisoner’s dilemma. Phys Rev E, 2010, 82: 021115CrossRefGoogle Scholar
  19. 19.
    Perc M. Coherence resonance in a spatial prisoner’s dilemma game. New J Phys, 2006, 8: 22CrossRefGoogle Scholar
  20. 20.
    Nowak M A, Bonhoeffer S, May R M. A simple rule for the evolution of cooperation on graphs. Nature, 2006, 441: 502–505CrossRefGoogle Scholar
  21. 21.
    Du W B, Cao X B, Hu M B. The effect of asymmetric payoff mechanism on evolutionary networked prisoner’s dilemma game. Physica A, 2009, 388: 5005–5012CrossRefGoogle Scholar
  22. 22.
    Hardin G. Extensions of “The tragedy of the commons”. Science, 1998, 280: 682–683CrossRefGoogle Scholar
  23. 23.
    Chen Y, Qin S M, Yu L C, et al. Emergence of synchronization induced by the interplay between two prisoner’s dilemma games with volunteering in small-world networks. Phys Rev E, 2008, 77: 032103CrossRefGoogle Scholar
  24. 24.
    Liu Z L, Peng C H, Xiang W H, et al. Application of artificial neural networks in global climate change and ecological research: An overview. Chin Sci Bull, 2010, 55: 3853–3863CrossRefGoogle Scholar
  25. 25.
    Liu H T, Li W W. Language clusters based on linguistic complex networks. Chin Sci Bull, 2010, 55: 3458–3465CrossRefGoogle Scholar
  26. 26.
    Chang S, Gong X Q, Jiao X, et al. Network analysis of protein-protein interaction. Chin Sci Bull, 2010, 55: 814–822CrossRefGoogle Scholar
  27. 27.
    Liu H T. Statistical properties of Chinese semantic networks. Chin Sci Bull, 2009, 54: 2781–2785CrossRefGoogle Scholar
  28. 28.
    Liu T, Li X, Liu X P. Integration of small world networks with multiagent systems for simulating epidemic spatiotemporal transmission. Chin Sci Bull, 2010, 55: 1285–1293CrossRefGoogle Scholar
  29. 29.
    Xia C Y, Sun S W, Liu Z X, et al. Epidemics of SIRS model with nonuniform transmission on scale-free networks. Int J Mod Phys B, 2009, 23: 2203–2213CrossRefGoogle Scholar
  30. 30.
    Santos F C, Pacheco J M. Scale-free networks provide a unifying framework for the emergence of cooperation. Phys Rev Lett, 2005, 95: 098104CrossRefGoogle Scholar
  31. 31.
    Wu Z X, Rong Z H, Holme P. Diversity of reproduction time scale promotes cooperation in spatial prisoner’s dilemma games. Phys Rev E, 2009, 80: 036106CrossRefGoogle Scholar
  32. 32.
    Szolnoki A, Szabó G. Cooperation enhanced by inhomogeneous activity of teaching for evolutionary prisoner’s dilemma games. Europhys Lett, 2007, 77: 30004CrossRefGoogle Scholar
  33. 33.
    Chen X J, Wang L. Cooperation enhanced by moderate tolerance ranges in myopically selective interactions. Phys Rev E, 2009, 80: 046109CrossRefGoogle Scholar
  34. 34.
    Perc M, Wang Z. Heterogeneous aspirations promote cooperation in the prisoner’s dilemma game. PLoS ONE, 2010, 5: e15117CrossRefGoogle Scholar
  35. 35.
    Szabó G, Vukov J, Szolnoki A. Phase diagrams for an evolutionary prisoner’s dilemma game on two-dimensional lattices. Phys Rev E, 2005, 72: 047107CrossRefGoogle Scholar
  36. 36.
    Zhang H F, Liu R R, Wang Z, et al. Aspiration-induced reconnection in spatial public goods game. Eur Phys Lett, 2011, 94: 18006CrossRefGoogle Scholar
  37. 37.
    Wang Z, Murks A, Du W B, et al. Coveting the neighbors fitness as a means to resolve social dilemmas. J Theor Biol, 2011, 277: 19–26CrossRefGoogle Scholar
  38. 38.
    Szabó G, Szolnoki A. Cooperation in spatial prisoner’s dilemma games with two types of players for increasing number of neighbors. Phys Rev E, 2009, 79: 016106CrossRefGoogle Scholar
  39. 39.
    Xu Z J, Wang Z, Zhang L Z. Bounded rationality in volunteering public goods games. J Theor Biol, 2010, 264: 19–23CrossRefGoogle Scholar
  40. 40.
    Szolnoki A, Perc M, Szabó G. Phase diagrams for three-strategy evolutionary prisoner’s dilemma games on regular graphs. Phys Rev E, 2009, 80: 056104CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Juan Wang
    • 1
    • 2
  • ChengYi Xia
    • 1
    • 2
  • YiLing Wang
    • 3
  • Shuai Ding
    • 4
  • JunQing Sun
    • 1
    • 2
  1. 1.Tianjin Key Laboratory for Control Theory and Complicated Industry SystemsTianjin University of TechnologyTianjinChina
  2. 2.Key Laboratory of Computer Vision and System (Ministry of Education)Tianjin University of TechnologyTianjinChina
  3. 3.School of Life ScienceShanxi Normal UniversityLinfenChina
  4. 4.Institute of Computer Network SystemsHefei University of TechnologyHefeiChina

Personalised recommendations