Soft Computing

, Volume 22, Issue 4, pp 1287–1294 | Cite as

Heterogeneous investment in spatial public goods game with mixed strategy

  • Hong Ding
  • Yao Zhang
  • Yizhi RenEmail author
  • Benyun ShiEmail author
  • Kim-Kwang Raymond Choo
Methodologies and Application


Understanding and maximizing the effects of heterogeneous investment, particularly in a socially diverse society, on the evolution of cooperation have been the focus of recent research. In the most existing studies, individuals are limited to make binary decisions (i.e., either cooperate or defect). This is unrealistic in many real-world situations. In this paper, we investigate the effect of a heterogeneous investment on the evolution of cooperation in mixed strategy public goods games, wherein individuals have different probability of cooperation. Specifically, players are able to distribute heterogeneous investments into different groups, and they tend to allocate their investment into the group which achieves a higher return on investment (e.g., payoffs). Simulation results show that the formation of cooperative clusters allows cooperative players to resist the exploitation of defective players; subsequently, the cooperation level of the whole population significantly increases. Moreover, the results also show that cooperative clusters become more robust when the investment redistribution decision relies on more recent information. Our study may offer new insights into how strategy diversity promotes the evolutionary of cooperation in realistic situations.


Spatial public goods game Mixed strategy Heterogeneous investment 



Authors like to appreciate the anonymous referees for their valuable comments and suggestions. This work is supported by the National Science Foundation of China (Grant Nos. 61100194, 61402141, 61100039, 61272173 and 61403059) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20131277).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Bo X, Wang J, Ruipu D (2014) Relational diversity promotes cooperation in prisoner’s dilemma games. PLoS ONE 9:e114464CrossRefGoogle Scholar
  2. Brandt H, Hauert C, Sigmund K (2003) Punishment and reputation in spatial public goods games. Biol Sci 270:1099–1104CrossRefGoogle Scholar
  3. Cao XB, Du WB, Rong ZH (2010) The evolutionary public goods game on scale-free networks with heterogeneous investment. Phys A Stat Mech Appl 389:1273–1280CrossRefGoogle Scholar
  4. Chen X, Liu Y, Zhou Y (2012) Adaptive and bounded investment returns promote cooperation in spatial public goods games. PLoS ONE 7:e36895CrossRefGoogle Scholar
  5. Christoph H, Arne T, Hannelore B (2007) Via freedom to coercion: the emergence of costly punishment. Science 316:1905–1907MathSciNetCrossRefzbMATHGoogle Scholar
  6. Fu Z, Ren K, Shu J, Sun X, Huang F (2015) Enabling personalized search over encrypted outsourced data with efficiency improvement. IEEE Trans Parallel Distrib Syst 27(9):2546–2559CrossRefGoogle Scholar
  7. Fu Z, Sun X, Liu Q, Zhou L, Shu J (2015) Achieving efficient cloud search services: multi-keyword ranked search over encrypted cloud data supporting parallel computing. IEICE Trans Commun 98(1):190–200CrossRefGoogle Scholar
  8. Gao J, Li Z, Wu T (2007) Diversity of contribution promotes cooperation in public goods games. Phys A Stat Mech Appl 389:3166–3171CrossRefGoogle Scholar
  9. Gross T (2010) Towards a new human-centred computing methodology for cooperative ambient intelligence. J Ambient Intell Humaniz Comput 1(1):31–42MathSciNetCrossRefGoogle Scholar
  10. Hauert C, Monte S, Hofbauer J (2002) Volunteering as red queen mechanism for cooperation in public goods games. Science 296:1129–1132CrossRefGoogle Scholar
  11. Huang K, Wang T, Cheng Y (2015) Effect of heterogeneous investments on the evolution of cooperation in spatial public goods game. PLoS ONE 10:e0120317CrossRefGoogle Scholar
  12. Jiang W, Wang G, Zakirul Alam Bhuiyan Md, Wu J (2016) Understanding graph-based trust evaluation in online social networks: methodologies and challenges. ACM Comput Surv 49(1):10CrossRefGoogle Scholar
  13. Killingback T, Doebeli M, Hauert C (2010) Diversity of cooperation in the tragedy of the commons. Biol Theory 1:243–253Google Scholar
  14. Kohei M, Tanimoto J, Aya H (2013) Influence of stochastic perturbation of both action updating and strategy updating in mixed-strategy 2\(\times \)2 games on evolution of cooperation. Phys Rev E 88:062149CrossRefGoogle Scholar
  15. Kokubo S, Wang Z, Tanimoto J (2015) Spatial reciprocity for discrete, continuous and mixed strategy setups. Appl Math Comput 259:552–568MathSciNetGoogle Scholar
  16. Lei C, Wu T, Jia JY (2010) Heterogeneity of allocation promotes cooperation in public goods games. Phys A Stat Mech Appl 389:4708–4714CrossRefGoogle Scholar
  17. Liang X, Xiao Y (2014) A survey of biological collaboration models. J Ambient Intell Humaniz Comput 5(4):551C563CrossRefGoogle Scholar
  18. Ma T, Zhou J, Tang M, Tian Y, Al-Dhelaan A, Al-Rodhaan M, Lee S (2015) Social network and tag sources based augmenting collaborative recommender system. IEICE Trans Inform Syst E98–D(4):902–910CrossRefGoogle Scholar
  19. Michael K, Friedrich B (2016) Risk sensitivity and assortment in social dilemmas. Soft Comput. doi: 10.1007/s00500-016-2090-5 Google Scholar
  20. Nax H, Perc M, Szolnoki A (2015) Stability of cooperation under image scoring in group interactions. Sci Rep 5:12145Google Scholar
  21. Nowak M (2006) Five rules for the evolution of cooperation. Science 314:1560–1563CrossRefGoogle Scholar
  22. Nowak M, Sigmund K (1998) Evolution of indirect reciprocity by image scoring. Nature 393:573–577CrossRefGoogle Scholar
  23. Peng D, Yang H, Wang W (2010) Promotion of cooperation induced by nonuniform payoff allocation in spatial public goods game. Eur Phys J B 73:455–459CrossRefzbMATHGoogle Scholar
  24. Perc M (2011) Success-driven distribution of public goods promotes cooperation but preserves defection. Phys Rev E 84:1915–1924CrossRefGoogle Scholar
  25. Rand D, Anna B, Tore E (2009) Positive interactions promote public cooperation. Science 325:1272–1275MathSciNetCrossRefzbMATHGoogle Scholar
  26. Ren Y, Li M, Sakurai K (2011) Finetrust: a fine-grained trust model for peer-to-peer networks. Secur Commun Netw 4(1):61–69CrossRefGoogle Scholar
  27. Ren Y, Li M, Xiang Y, Cui Y, Sakurai K (2013) Evolution of cooperation in reputation system by group-based scheme. J Supercomput 63(1):171–190CrossRefGoogle Scholar
  28. Ren Y, Shen J, Wang J, Han J, Lee S (2015) Mutual verifiable provable data auditing in public cloud storage. J Intern Technol 16(2):317–323Google Scholar
  29. Santos F, Santos M, Pacheco J (2008) Social diversity promotes the emergence of cooperation in public goods games. Nature 454:213–216CrossRefGoogle Scholar
  30. Schoenmakers S, Hilbe C, Blasius B (2014) Sanctions as honest signals—the evolution of pool punishment by public sanctioning institutions. J Theor Biol 356:1272–1275MathSciNetCrossRefGoogle Scholar
  31. Shen J, Tan H, Wang J, Wang J, Lee S (2015) A novel routing protocol providing good transmission reliability in underwater sensor networks. J Intern Technol 16(1):171–177Google Scholar
  32. Smith J (1982) Evolution and the theory of games. Cambridge university press, CambridgeCrossRefzbMATHGoogle Scholar
  33. Szolonki A, Perc M (2012) Evolutionary advantages of adaptive rewarding. New J Phys 14:93016–93029CrossRefGoogle Scholar
  34. Szolonki A, Perc M (2012) Conditional strategies and the evolution of cooperation in spatial public goods games. Phys Rev E 85:287–300Google Scholar
  35. Szolonki A, Perc M, Szab G (2009) Topology independent impact of noise on cooperation in spatial public goods games. Phys Rev E 80:2142–2152Google Scholar
  36. Szolonki A, Szab G, Perc M (2011) Phase diagrams for the spatial public goods game with pool punishment. Phys Rev E 83:856–875Google Scholar
  37. Szolonki A, Wang Z, Perc M (2012) Wisdom of groups promotes cooperation in evolutionary social dilemmas. Sci Rep 2:576CrossRefGoogle Scholar
  38. Tanimoto J (2015) Correlated asynchronous behavior updating with a mixed strategy system in spatial prisoner’s dilemma games enhances cooperation. Chaos Soliton Fract 80:39–46MathSciNetCrossRefzbMATHGoogle Scholar
  39. Teng X, Yan S, Tang S (2014) Individual behavior and social wealth in the spatial public goods game. Phys A Stat Mech Appl 402:141–149MathSciNetCrossRefGoogle Scholar
  40. Tian L, Li M, Lu K (2013) The influence of age-driven investment on cooperation in spatial public goods games. Chaos Solitons Fract 54:65–70MathSciNetCrossRefzbMATHGoogle Scholar
  41. Vukov J, Santos F, Pacheco JM (2011) Escaping the tragedy of the commons via directed investments. J Theor Biol 287:37–41MathSciNetCrossRefGoogle Scholar
  42. Wang Z, Szolonki A, Perc M (2012) Percolation threshold determines the optimal population density for public cooperation. Phys Rev E 85:922–938Google Scholar
  43. Xu B, Li M, Deng R (2015) The evolution of cooperation in spatial prisoner’s dilemma games with heterogeneous relationships. Phys A Stat Mech Appl 424:168–175CrossRefGoogle Scholar
  44. Yuan W, Xia C (2014) Role of investment heterogeneity in the cooperation on spatial public goods game. PLoS ONE 9:e91012CrossRefGoogle Scholar
  45. Zeng W, Li M, Chen F, Nan G (2016) Risk consideration and cooperation in the iterated prisoner’s dilemma. Soft Comput 20(2):567–587CrossRefGoogle Scholar
  46. Zhang H, Yang H, Du W (2010) Evolutionary public goods games on scale-free networks with unequal payoff allocation mechanism. Phys A Stat Mech Appl 389:1099–1104CrossRefGoogle Scholar
  47. Zhang H, Liu R, Wang Z (2012) Aspiration-induced reconnection in spatial public goods game. Europhys Lett 94:73–79Google Scholar
  48. Zhang J, Zhang C, Cao M (2015) How insurance affects altruistic provision in threshold public goods games. Sci Rep 5:9098CrossRefGoogle Scholar
  49. Zhong W, Kokubo S, Tanimoto J (2012) How is the equilibrium of continuous strategy game different from that of discrete strategy game. Biosystems 107:88–94CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of CyberspaceHangzhou Dianzi UniversityHangzhouChina
  2. 2.School of Computer Science and TechnologyHangzhou Dianzi UniversityHangzhouChina
  3. 3.Key Laboratory of Complex Systems Modeling and Simulation, Ministry of EducationHangzhouChina
  4. 4.Department of Information Systems and Cyber SecurityUniversity of Texas at San AntonioSan AntonioUSA
  5. 5.School of Information Technology and Mathematical SciencesUniversity of South AustraliaAdelaideAustralia

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