自我质疑机制下公共物品博弈模型的相变特性

杨波; 范敏; 刘文奇; 陈晓松

doi:10.7498/aps.66.196401

摘要

公共物品博弈是研究群体相互作用的经典模型，广泛用于解释自私个体间合作的涌现和保持.本文从理论分析和蒙特卡罗模拟两个方面研究了二维正方格子上一个有偿惩罚机制下随自我质疑更新规则演化的公共物品博弈模型的相变特性.理论分析方面，将公共物品博弈模型转化为一个外场不为零的铁磁Ising模型.通过有效能量发现：不存在惩罚时，个体间的耦合强度为零，体系只有外场作用；存在惩罚时，个体间包含最近邻、次近邻和第三近邻相互作用且外场不为零.蒙特卡罗模拟方面，首先验证了理论分析的正确性，然后对公共物品博弈模型相关的一级相变和二级相变进行了有限尺度标度分析.研究发现：1）蒙特卡罗模拟所得结果与类Ising模型分析结果完全吻合；2）相比二维Ising模型，公共物品博弈的二级相变临界指数发生了变化；3）公共物品博弈的一级相变与二维Ising模型相同.

关键词:

Abstract

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.

Keywords:

作者及机构信息

1.
昆明理工大学数据科学研究中心, 昆明 650500;

2.
昆明理工大学理学院, 昆明 650500;

3.
中国科学院理论物理研究所, 理论物理前沿重点实验室, 北京 100190;

4.
中国科学院大学物理科学学院, 北京 100049

通信作者: 杨波, yangbo@kmust.edu.cn

基金项目: 昆明理工大学引进人才科研启动基金项目（批准号：KKSY201607047）和国家自然科学基金（批准号：61573173）资助的课题.

Authors and contacts

1.
Data Science Research Center, Kunming University of Science and Technology, Kunming 650500, China;

2.
Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China;

3.
Institute of Theoretical Physics, Key Laboratory of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China;

4.
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China}

Corresponding author: Yang Bo, yangbo@kmust.edu.cn

Funds: Scientific Research Foundation for Introduced Scholars, Kunming University of Science and Technology(Grant No. KKSY201607047) and the National Natural Science Foundation of China (Grant No. 61573173).

参考文献

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施引文献

[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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