多人雪堆演化博弈在愿景驱动规则下的扩展平均丰度函数
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摘要
随着博弈理论的深入研究,演化博弈模型在许多社会现象和经济问题的分析中得到了广泛应用.演化博弈模型将策略更新规则引入状态转移方程,得到相应的马尔可夫链,从而研究种群演化状态.当此马尔可夫链无吸收态时,采用平均丰度函数来研究种群演化状态.采用策略更新规则中的愿景驱动规则,通过分析马尔可夫链的平稳分布导出了扩展平均丰度函数.同时,通过将多人演化博弈模型应用于雪堆演化博弈中,得到了多人雪堆演化博弈模型的扩展平均丰度函数.采用数字分析的方式,计算分析了相应参数对平均丰度函数的影响,结合具体案例研究了参数变化如何影响企业在博弈中的行为.研究表明可以通过改变相关参数来提高合作者的占比,这一结论为在实际应用中如何调控相应参数以促进合作指出了方向.
With further study of game theory, evolutionary game model has been widely used in the analysis of many social phenomena and economic problems. In evolutionary game model, strategy updating rule is introduced into the state transition equation, and the corresponding Markov chain is obtained to study the evolutionary state of population. When the Markov chain has no absorbing states, the average abundance function is used to study the evolutionary state of population. The extended average abundance function is derived by analyzing the stationary distribution of Markov chain using aspiration driven rule in the strategy updating rules. At the same time, by applying the multi-player evolutionary game model to snowdrift evolutionary game, extended average abundance function of multi-player snowdrift evolutionary game model is obtained. By means of numerical analysis, the influence of corresponding parameters on the average abundance function is calculated and analyzed, and how the variation of parameters affects the behavior of enterprises in the game is studied with a specific case. The research shows that the proportion of cooperators will be increased by changing relevant parameters, and this conclusion points out the direction for how to regulate the corresponding parameters to promote cooperation in practical application.
With further study of game theory, evolutionary game model has been widely used in the analysis of many social phenomena and economic problems. In evolutionary game model, strategy updating rule is introduced into the state transition equation, and the corresponding Markov chain is obtained to study the evolutionary state of population. When the Markov chain has no absorbing states, the average abundance function is used to study the evolutionary state of population. The extended average abundance function is derived by analyzing the stationary distribution of Markov chain using aspiration driven rule in the strategy updating rules. At the same time, by applying the multi-player evolutionary game model to snowdrift evolutionary game, extended average abundance function of multi-player snowdrift evolutionary game model is obtained. By means of numerical analysis, the influence of corresponding parameters on the average abundance function is calculated and analyzed, and how the variation of parameters affects the behavior of enterprises in the game is studied with a specific case. The research shows that the proportion of cooperators will be increased by changing relevant parameters, and this conclusion points out the direction for how to regulate the corresponding parameters to promote cooperation in practical application.
引文
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[18] Xu K, Li K, Cong R, et al. Cooperation guided by the coexistence of imitation dynamics and aspiration dynamics in structured populations[J]. Europhysics Letters, 2017, 117(4):48002.
[19] Claussen J C, Traulsen A. Non-Gaussian fluctuations arising from finite populations:Exact results for the evolutionary Moran process[J]. Physical Review E:Statistical Nonlinear&Soft Matter Physics, 2005, 71(2):025101.
[20]金帅,杜建国,盛昭翰.区域环境保护行动的演化博弈分析[J].系统工程理论与实践,2015, 35(12):3107-3118.Jin S, Du J G, Sheng Z H. Evolutionary game analysis of regional environmental protection programme[J].Systems Engineering—Theory&Practice, 2015, 35(12):3107-3118.
[21]潘峰,西宝,王琳.基于演化博弈的地方政府环境规制策略分析[J].系统工程理论与实践,2015, 35(6):1393-1404.Pan F, Xi B, Wang L. Analysis on environmental regulation strategy of local government based on evolutionary game theory[J]. Systems Engineering—Theory&Practice, 2015, 35(6):1393-1404.
[22]焦建玲,陈洁,李兰兰,等.碳减排奖惩机制下地方政府和企业行为演化博弈分析[J].中国管理科学,2017(10):140-150.Jiao J L, Chen J, Li L L, et al. A study of local governments'and enterprises'actions in the carbon emission mechanism of subsidy or punishment based on the evolutionary game[J]. Chinese Journal of Management Science,2017(10):140-150.
[23]姚洪心,吴伊婷.绿色补贴、技术溢出与生态倾销[J].管理科学学报,2018, 21(10):47-60.Yao H X, Wu Y T. Green subsidy, technology spillover and eco-dumping[J]. Journal of Management Sciences in China, 2018, 21(10):47-60.
[24]郭道燕,陈红,龙如银.消费端碳交易市场中政府初始碳配额分配策略研究[J].中国人口·资源与环境,2018, 28(4):43-54.Guo D Y, Chen H, Long R Y. The allocation strategy of government for initial carbon allowance in downstream carbon trading market[J]. China Population, Resources and Environment, 2018, 28(4):43-54.
[2] Hauert C, Doebeli M. Spatial structure often inhibits the evolution of cooperation in the snowdrift game[J].Nature, 2004, 428(6983):643-646.
[3] Doebeli M, Hauert C. Models of cooperation based on the Prisoner's Dilemma and the Snowdrift game[J]. Ecology Letters, 2005, 8(7):748-766.
[4] Taylor C, Fudenberg D, Sasaki A, et al. Evolutionary game dynamics in finite populations[J]. Bulletin of Mathematical Biology, 2004, 66(6):1621-1644.
[5] Chen Y S, Yang H X, Guo W Z. Aspiration-induced dormancy promotes cooperation in the spatial prisoner's dilemma games[J]. Physica A:Statistical Mechanics&its Applications, 2017, 469(5):625—630.
[6] Du J, Wu B, Wang L. Aspiration dynamics and the sustainability of resources in the public goods dilemma[J].Physics Letters, 2016, 380(16):1432-1436.
[7] Allen B, Tarnita C E. Measures of success in a class of evolutionary models with fixed population size and structure[J]. Journal of Mathematical Biology, 2014, 68(1-2):109-143.
[8] Zhang L, Ying L, Zhou J, et al. Fixation probabilities of evolutionary coordination games on two coupled populations[J]. Physical Review E, 2016, 94(3):032307.
[9] Souza M O, Pacheco J M, Santos F C. Evolution of cooperation under N-person snowdrift games[J]. Journal of Theoretical Biology, 2009, 260(4):581-588.
[10] Du J, Wu B, Altrock P M, et al. Aspiration dynamics of multi-player games in finite populations[J]. Journal of the Royal Society Interface, 2014, 11(94):20140077.
[11] Du J, Wu B, Wang L. Aspiration dynamics in structured population acts as if in a well-mixed one[J]. Scientific Reports, 2015, 5:8014.
[12] Liu X, He M, Kang Y, et al. Aspiration promotes cooperation in the prisoner's dilemma game with the imitation rule[J]. Physical Review E, 2016, 94(1):012124.
[13] Wu B, Traulsen A, Gokhale C S. Dynamic properties of evolutionary multi-player games in finite populations[J].Games, 2013, 4(2):182-199.
[14] Wu B, Gokhale C S, Wang L, et al. How small are small mutation rates?[J]. Journal of Mathematical Biology,2012, 64(5):803-827.
[15] Sui X, Cong R, Li K, et al. Evolutionary dynamics of N-person snowdrift game[J]. Physics Letters, 2015,379(45-46):2922-2934.
[16] Santos M D, Santos F C, Pacheco J M. Collective evolutionary dynamics and spatial reciprocity under the N-person snowdrift game[C]//International Conference on Bio-Inspired Models of Network, Information, and Computing Systems, 2010.
[17] Pena J, Wu B, Traulsen A. Ordering structured populations in multiplayer cooperation games[J]. Journal of the Royal Society Interface, 2016, 13(114):20150881.
[18] Xu K, Li K, Cong R, et al. Cooperation guided by the coexistence of imitation dynamics and aspiration dynamics in structured populations[J]. Europhysics Letters, 2017, 117(4):48002.
[19] Claussen J C, Traulsen A. Non-Gaussian fluctuations arising from finite populations:Exact results for the evolutionary Moran process[J]. Physical Review E:Statistical Nonlinear&Soft Matter Physics, 2005, 71(2):025101.
[20]金帅,杜建国,盛昭翰.区域环境保护行动的演化博弈分析[J].系统工程理论与实践,2015, 35(12):3107-3118.Jin S, Du J G, Sheng Z H. Evolutionary game analysis of regional environmental protection programme[J].Systems Engineering—Theory&Practice, 2015, 35(12):3107-3118.
[21]潘峰,西宝,王琳.基于演化博弈的地方政府环境规制策略分析[J].系统工程理论与实践,2015, 35(6):1393-1404.Pan F, Xi B, Wang L. Analysis on environmental regulation strategy of local government based on evolutionary game theory[J]. Systems Engineering—Theory&Practice, 2015, 35(6):1393-1404.
[22]焦建玲,陈洁,李兰兰,等.碳减排奖惩机制下地方政府和企业行为演化博弈分析[J].中国管理科学,2017(10):140-150.Jiao J L, Chen J, Li L L, et al. A study of local governments'and enterprises'actions in the carbon emission mechanism of subsidy or punishment based on the evolutionary game[J]. Chinese Journal of Management Science,2017(10):140-150.
[23]姚洪心,吴伊婷.绿色补贴、技术溢出与生态倾销[J].管理科学学报,2018, 21(10):47-60.Yao H X, Wu Y T. Green subsidy, technology spillover and eco-dumping[J]. Journal of Management Sciences in China, 2018, 21(10):47-60.
[24]郭道燕,陈红,龙如银.消费端碳交易市场中政府初始碳配额分配策略研究[J].中国人口·资源与环境,2018, 28(4):43-54.Guo D Y, Chen H, Long R Y. The allocation strategy of government for initial carbon allowance in downstream carbon trading market[J]. China Population, Resources and Environment, 2018, 28(4):43-54.