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The spatial public goods game is one of the most popular models for studying the emergence and maintenance of cooperation among selfish individuals. A public goods game with costly punishment and self-questioning updating mechanism is studied in this paper. The theoretical analysis and Monte Carlo simulation are involved to analyze this model. This game model can be transformed into Ising model with an external field by theoretical analysis. When the costly punishment exists, the effective Hamiltonian includes the nearest-, the next-nearest-and the third-nearest-neighbor interactions and non-zero external field. The interactions are only determined by costly punishment. The sign of the interaction is always greater than zero, so it has the properties of ferromagnetic Ising. The external field is determined by the factor r of the public goods game, the fine F on each defector within the group, and the relevant punishment cost C. The Monte Carlo simulation results are consistent with the theoretical analysis results. In addition, the phase transitions and critical behaviors of the public goods game are also studied using the finite size scaling theory. The results show that the discontinuous phase transition has the same finite size effects as the two-dimensional Ising model, but the continuous phase transitions is inconsistent with Ising model.
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Keywords:
- ising model /
- finite size scaling theory /
- Monte Carlo simulations /
- self-questioning update rules
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[2] Szab G, Borsos I 2016 Phys. Rep. 624 1
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[6] Nowak M A (translated by Li Z Q, Wang S C) 2010 Evolutionary Dynamics:Exploring the Equations of Life (Beijing:Higher Education Press) pp108-125 (in Chinese)[Martin A. Nowak 著 (李振清, 王世畅 译) 2010 进化动力学-探索生命的方程(北京:高等教育出版社)第108125页]
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[10] Wang X W 2015 M. D. Dissertation (Anhui:University of Science and Technology of China) (in Chinese)[王旭文 2015 博士学位论文 (安徽:中国科学技术大学)]
[11] Wang B K 2013 M. S. Dissertation (Xi'an:Xidian University) (in Chinese)[王保魁 2013 硕士学位论文 (西安:西安电子科技大学)]
[12] Gao J 2012 Ph. D. Dissertation (Xi'an:Xidian University) (in Chinese)[高佳 2012 博士学位论文 (西安:西安电子科技大学)]
[13] Hauert C, Traulsen A, Brandt H, Nowak M A, Sigmund K 2007 Science 316 1905
[14] Dreber A, Rand D G, Fudenberg D, Nowak M A 2008 Nature 452 348
[15] Helbing D, Szolnoki A, Perc M, Szab G 2010 New J. Phys. 12 083005
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[17] Ohtsuki H, Iwasa Y, Nowak M A 2009 Nature 457 79
[18] Szolnoki A, Szab G, Perc M https://journals.aps.org/pre/abstract/10.1103/PhysRevE.83.036101 2011 Phys. Rev. E 83 036101
[19] Hauert C, Doebeli M 2004 Nature 428 643
[20] Santos F C, Santos M D, Pacheco J M 2008 Nature 454 213
[21] Szab G, Tőke C https://journals.aps.org/pre/abstract/10.1103/PhysRevE.58.69 1998 Phys. Rev. E 58 69
[22] Szab G, Hauert C 2002 Phys. Rev. Lett. 89 118101
[23] Szab G, Vukov J, Szolnoki A 2005 Phys. Rev. E 72 047107
[24] Wang W X, Ren J, Chen G, Wang B H 2006 Phys. Rev. E 74 056113
[25] Gao K, Wang W X, Wang B H 2007 Physica A 380 528
[26] Liu Y K, Li Z, Chen X J, Wang L 2009 Chin. Phys. Lett. 26 377
[27] Liu Y K 2010 Ph. D. Dissertation (Xi'an:Xidian University) (in Chinese)[刘永奎2010博士学位论文(西安:西安电子科技大学)]
[28] Yang B, Li X T, Chen W, Liu J, Chen X S 2016 Commun. Theor. Phys. 66 439
[29] McKelvey R D, Palfrey T R 1995 Games Econ. Behav. 10 6
[30] He J T 2006 Behavioral Economics The Economic Behavior of China (Beijing:China Financial Economic Publishing House) p79 (in Chinese)[贺京同 2006 行为经济学与中国经济行为 (北京:中国财政经济出版社) 第 79 页]
[31] Zhuang Q, Di Z R, Wu J S https://doi.org/10.1371/journal.pone.0105391 2014 PloS One 9 e105391
[32] Fisher M E 1967 Rep. Prog. Phys. 30 615
[33] Privman V 1990 Finite Size Scaling and Numerical Simulation of Statistical Systems (Singapore:World Scientific) p223
[34] Landau D P, Binder K 2014 A Guide to Monte Carlo Simulations in Statistical Physics (United Kingdom:Cambridge University Press) p82
[35] Binder K, Heermann D 2010 Monte Carlo Simulation in Statistical Physics An Introduction (Berlin Heidelberg:Springer-Verlag) p56
[36] Binder K, Landau D P https://journals.aps.org/prb/abstract/10.1103/PhysRevB.30.1477 1984 Phys. Rev. B 30 1477
[37] Lee J, Kosterlitz J M 1991 Phys. Rev. B 43 3265
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[1] Szab G, Fath G 2007 Phys. Rep. 446 97
[2] Szab G, Borsos I 2016 Phys. Rep. 624 1
[3] He D R, Liu Z H, Wang B H 2009 Complex Systems and Complex Networks (Beijing:Higher Education Press) p73 (in Chinese)[何大韧, 刘宗华, 汪秉宏 2009 复杂系统与复杂网络 (北京:高等教育出版社) 第 73 页]
[4] Nowak M A 2006 Science 314 1560
[5] Nowak M A, May R M 1992 Nature 359 826
[6] Nowak M A (translated by Li Z Q, Wang S C) 2010 Evolutionary Dynamics:Exploring the Equations of Life (Beijing:Higher Education Press) pp108-125 (in Chinese)[Martin A. Nowak 著 (李振清, 王世畅 译) 2010 进化动力学-探索生命的方程(北京:高等教育出版社)第108125页]
[7] Hardin G 1968 Science 162 1243
[8] Feeny D, Berkes F, McCay B J, Acheson J M https://link.springer.com/article/10.1007/BF00889070 1990 Hum. Ecol. 18 1
[9] Palfrey T R, Rosenthal H 1984 J. Public Econ. 24 171
[10] Wang X W 2015 M. D. Dissertation (Anhui:University of Science and Technology of China) (in Chinese)[王旭文 2015 博士学位论文 (安徽:中国科学技术大学)]
[11] Wang B K 2013 M. S. Dissertation (Xi'an:Xidian University) (in Chinese)[王保魁 2013 硕士学位论文 (西安:西安电子科技大学)]
[12] Gao J 2012 Ph. D. Dissertation (Xi'an:Xidian University) (in Chinese)[高佳 2012 博士学位论文 (西安:西安电子科技大学)]
[13] Hauert C, Traulsen A, Brandt H, Nowak M A, Sigmund K 2007 Science 316 1905
[14] Dreber A, Rand D G, Fudenberg D, Nowak M A 2008 Nature 452 348
[15] Helbing D, Szolnoki A, Perc M, Szab G 2010 New J. Phys. 12 083005
[16] Helbing D, Szolnoki A, Perc M, Szab G 2010 Phys. Rev. E 81 057104
[17] Ohtsuki H, Iwasa Y, Nowak M A 2009 Nature 457 79
[18] Szolnoki A, Szab G, Perc M https://journals.aps.org/pre/abstract/10.1103/PhysRevE.83.036101 2011 Phys. Rev. E 83 036101
[19] Hauert C, Doebeli M 2004 Nature 428 643
[20] Santos F C, Santos M D, Pacheco J M 2008 Nature 454 213
[21] Szab G, Tőke C https://journals.aps.org/pre/abstract/10.1103/PhysRevE.58.69 1998 Phys. Rev. E 58 69
[22] Szab G, Hauert C 2002 Phys. Rev. Lett. 89 118101
[23] Szab G, Vukov J, Szolnoki A 2005 Phys. Rev. E 72 047107
[24] Wang W X, Ren J, Chen G, Wang B H 2006 Phys. Rev. E 74 056113
[25] Gao K, Wang W X, Wang B H 2007 Physica A 380 528
[26] Liu Y K, Li Z, Chen X J, Wang L 2009 Chin. Phys. Lett. 26 377
[27] Liu Y K 2010 Ph. D. Dissertation (Xi'an:Xidian University) (in Chinese)[刘永奎2010博士学位论文(西安:西安电子科技大学)]
[28] Yang B, Li X T, Chen W, Liu J, Chen X S 2016 Commun. Theor. Phys. 66 439
[29] McKelvey R D, Palfrey T R 1995 Games Econ. Behav. 10 6
[30] He J T 2006 Behavioral Economics The Economic Behavior of China (Beijing:China Financial Economic Publishing House) p79 (in Chinese)[贺京同 2006 行为经济学与中国经济行为 (北京:中国财政经济出版社) 第 79 页]
[31] Zhuang Q, Di Z R, Wu J S https://doi.org/10.1371/journal.pone.0105391 2014 PloS One 9 e105391
[32] Fisher M E 1967 Rep. Prog. Phys. 30 615
[33] Privman V 1990 Finite Size Scaling and Numerical Simulation of Statistical Systems (Singapore:World Scientific) p223
[34] Landau D P, Binder K 2014 A Guide to Monte Carlo Simulations in Statistical Physics (United Kingdom:Cambridge University Press) p82
[35] Binder K, Heermann D 2010 Monte Carlo Simulation in Statistical Physics An Introduction (Berlin Heidelberg:Springer-Verlag) p56
[36] Binder K, Landau D P https://journals.aps.org/prb/abstract/10.1103/PhysRevB.30.1477 1984 Phys. Rev. B 30 1477
[37] Lee J, Kosterlitz J M 1991 Phys. Rev. B 43 3265
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