Stochastic evolution dynamic model of Moran process with selection difference and its application
WANG Xianjia1,2, GU Cuiling2, ZHAO Jinhua1, HE Qilong3
1. School of Economics and Management, Wuhan University, Wuhan 430072, China; 2. Institute of Systems Engineering, Wuhan University, Wuhan 430072, China; 3. Business School, Zhengzhou University, Zhengzhou 450001, China
Abstract:We studied 2×2 symmetric games of finite population in which individuals who choose different strategies have different selection intensities. The frequency-dependent Moran process with different selection intensities was established and the fixation probability of each strategy was obtained by diffusion approximation under weak selection. The fixation probability is not only related to the population size and game matrix, but also to the differential selection intensities. The conditions that natural selection is beneficial to the fixation of strategies and that the evolutionary stability strategy ( ESS_N) satisfies were analyzed by comparing the fixation probability with that under neutral selection. In prisoner's dilemma, coexistence and coordination games, the relationships between the fixation probabilities and the selection intensities and population size were obtained by numerical analysis, and the relationships between fixation time and different selection intensities were obtained by simulation analysis. Finally, the stochastic evolution dynamic model of Moran process with selection difference was applied to the problem of strategic evolution of third party logistics enterprises. The fixation probability that the third party logistics enterprises choose to participate supply chain finance and the conditions under which the participation strategy becomes an evolutionary stable strategy were obtained. The influence of each parameter on the game behavior of the third party logistics enterprises were analyzed by numerical analysis or simulation analysis. Our research enriches stochastic evolutionary game theory based on the Moran Process.
王先甲, 顾翠伶, 赵金华, 何奇龙. 选择差异下Moran过程的随机博弈模型及其应用[J]. 系统工程理论与实践, 2020, 40(5): 1193-1209.
WANG Xianjia, GU Cuiling, ZHAO Jinhua, HE Qilong. Stochastic evolution dynamic model of Moran process with selection difference and its application. Systems Engineering - Theory & Practice, 2020, 40(5): 1193-1209.
[1] Smith J M, Price G R. The logic of animal conflict[J]. Nature, 1973, 246(5427): 15-18. [2] Nowak M A, Sigmund K. Evolutionary dynamics of biological games[J]. Science, 2004, 303(5659): 793-799. [3] Smith J M. Evolution and the theory of games[M]. Cambridge: Cambridge University Press, 1982. [4] 王先甲, 何奇龙, 全吉. 基于复制动态的消费者众筹策略演化动态[J]. 系统工程理论与实践, 2017, 37(11): 2812-2820. Wang X J, He Q L, Quan J. Evolutionary dynamics of consumers' crowd funding strategies based on replicator dynamics[J]. Systems Engineering—Theory & Practice, 2017, 37(11) : 2812-2820. [5] 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. [6] 王先甲, 何奇龙, 全吉, 等. 基于Moran过程的消费者众筹策略演化动态[J]. 运筹与管理, 2017, 26(11): 105-110. Wang X J, He Q L, Quan J, et al. Evolutionary dynamics of consumers' crowdfunding strategies based on Moran process[J]. Operations Research and Management Science, 2017, 26(11): 105-110. [7] 柴彩春, 肖条军, 许甜甜. 基于Moran过程的制造商生产策略演化动态[J]. 系统工程理论与实践, 2015, 35(9): 2262-2270. Chai C C, Xiao T J, Xu T T. Evolutionary dynamics of manufacturers production strategies based on Moran process[J]. Systems Engineering—Theory & Practice, 2015, 35(9): 2262-2270. [8] Taylor P D, Jonker L B. Evolutionary stable strategies and game dynamics[J]. Mathematical Biosciences, 1978, 40(1-2): 145-156. [9] Cabrales A. Stochastic replicator dynamics[J]. International Economic Review, 2000, 41(2): 451-481. [10] Hofbauer J, Sigmund K. Evolutionary game dynamics[J]. Bulletin of the American Mathematical Society, 2011, 69(4): 479-519. [11] Cressman R, Vickers G T. Spatial and density effects in evolutionary game theory[J]. Journal of Theoretical Biology, 1997, 184(4): 359-369. [12] Hutson V C L, Vickers G T. Travelling waves and dominance of ESS's[J]. Journal of Mathematical Biology, 1992, 30(5): 457-471. [13] Kandori M, Mailath G J, Rob R. Learning, mutation, and long run equilibria in games[J]. Econometrica, 1993, 61(1): 29-56. [14] Amir M, Berninghaus S K. Another approach to mutation and learning in games[J]. Games and Economic Behavior, 1996, 14(1): 19-43. [15] Nowak M A, Sasaki A, Taylor C, et al. Emergence of cooperation and evolutionary stability in finite populations[J]. Nature, 2004, 428(6983): 646-650. [16] Altrock P M, Traulsen A. Fixation times in evolutionary games under weak selection[J]. New Journal of Physics, 2008, 11(1): 013012. [17] Wang X J, Gu C L, Zhao J H, et al. Evolutionary game dynamics of combining the imitation and aspiration-driven update rules[J]. Physical Review E, 2019, 100(2): 022411. [18] Wang X J, Gu C L, Quan J. Evolutionary game dynamics of the Wright-Fisher process with different selection intensities[J]. Journal of Theoretical Biology, 2019, 465(21): 17-26. [19] Taylor C, Fudenberg D, Sasaki A, et al. Evolutionary game dynamics in finite populations[J]. Bulletin of Mathematical Biology, 2004, 66(6): 1621-1644. [20] Wild G, Traulsen A. The different limits of weak selection and the evolutionary dynamics of finite populations[J]. Journal of Theoretical Biology, 2007, 247(2): 382-390. [21] Lieberman E, Hauert C, Nowak M A. Evolutionary dynamics on graphs[J]. Nature, 2005, 433(7023): 312-316. [22] Ohtsuki H, Nowak M A. Evolutionary games on cycles[J]. Proceedings Biological Sciences, 2006, 273(1598): 2249-2256. [23] Altrock P M, Traulsen A, Nowak M A. Evolutionary games on cycles with strong selection[J]. Physical Review E, 2017, 95(2): 022407. [24] Traulsen A, Nowak M A, Pacheco J M. Stochastic dynamics of invasion and fixation[J]. Physical Review E, 2006, 74(1): 011909. [25] Altrock P M, Gokhale C S, Traulsen A. Stochastic slowdown in evolutionary processes[J]. Physical Review E, 2010, 82(1): 011925. [26] 李玉英. 基于生灭过程的策略进化动态[D]. 南京: 南京航空航天大学, 2009. Li Y X. Evolutionary dynamics for strategies based on birth-death processes[D]. Nanjing: Nanjing University of Aeronautics and Astronautics, 2009. [27] Liu X S, Pan Q H, Kang Y B, et al. Fixation times in evolutionary games with the Moran and Fermi processes[J]. Journal of Theoretical Biology, 2015, 387: 214-220. [28] Liu X S, Pan Q H, Kang Y B, et al. Fixation probabilities in evolutionary games with Moran process and Fermi process[J]. Journal of Theoretical Biology, 2015, 364: 242-248. [29] Wang X J, Gu C L, Lü S J, et al. Evolutionary game dynamics of combining the Moran and imitation processes[J]. Chinese Physics B, 2019, 28(2): 020203. [30] 邹世辰, 王慧强, 冯光升, 等. 基于Moran过程的认知中继网络多策略信任演化模型[J]. 小型微型计算机系统, 2014, 35(10): 2209-2214. Zou S C, Wang H Q, Feng G S, et al. Multi-strategy trust evolution model for cognitive relay network based on Moran process[J]. Journal of Chinese Computer Systems, 2014, 35(10): 2209-2214. [31] Eigen M. Selforganization of matter and the evolution of biological macromolecules[J]. Die Naturwissenschaften, 1971, 58(10): 465-523. [32] Zhang S, Chen T, Zhou M. The relationship between the heritability of a ratio and selection intensity[J]. Journal of Genetics and Genomics, 1994, 21(2): 112-117. [33] 高雷阜, 毕玲玲. 具有选择差异的随机博弈进化动力系统[J]. 生物数学学报, 2015, 30(1): 161-167.Gao L F, Bi L L. Stochastic game evolutionary dynamics with different selection intention[J]. Journal of Biomathematics, 2015, 30(1): 161-167. [34] Traulsen A, Pacheco J M, Imhof L A. Stochasticity and evolutionary stability[J]. Physical Review E, 2006, 74(1): 021905. [35] 胡跃飞. 供应链金融——极富潜力的全新领域[J]. 中国金融, 2007(22): 38-39. Hu Y F. Supply chain finance: A new field with great potential[J]. China Finance, 2007(22): 38-39. [36] Randall W S, Theodore Farris M. Supply chain financing: Using cash-to-cash variables to strengthen the supply chain[J]. International Journal of Physical Distribution and Logistics Management, 2009, 39(8): 669-689. [37] Gomm M L. Supply chain finance: Applying finance theory to supply chain management to enhance finance in supply chains[J]. International Journal of Logistics Research and Applications, 2010, 13(2): 133-142. [38] 李小莉, 辛玉红. 基于供应链金融的中小企业信贷市场演化分析[J]. 运筹与管理, 2017, 26(10): 101-105.Li X L, Xin Y H. Evolution analysis of SMEs' credit market based on supply chain finance[J]. Operations Research and Management Science, 2017, 26(10): 101-105. [39] 毛晟栋, 朱其特. 基于供应链金融的第三方物流进化博弈分析[J]. 世界科技研究与发展, 2016, 38(4): 882-886. Mao S D, Zhu Q T. Evolutionary game analysis of Third-party logistics based on supply chain finance[J]. World Sci-Tech R&D, 2016, 38(4): 882-886. [40] 盛鑫, 陈长彬. 政府行为对供应链金融业务协同发展的影响——基于演化博弈论的研究[J]. 技术经济与管理研究, 2019(2): 81-85. Sheng X, Chen C B. The impact of government behavior on the coordinated development of financial business with supply chain-based on the evolutionary game theory[J]. Journal of Technical Economics and Management, 2019(2): 81-85. [41] Nowak M A. Evolutionary dynamics: Exploring the equations of life[M]. Cambridge: Harvard University Press, 2006. [42] Traulsen A, Claussen J C, Hauert C. Coevolutionary dynamics: From finite to infinite populations[J]. Physical Review Letters, 2005, 95(23): 238701. https://doi.org/10.1103/PhysRevLett.95.238701. [43] Ewens W J. Mathematical population genetics: Theoretical introduction[M]. Berlin: Springer-Verlag, 2004. [44] 全吉. 有限理性下的演化博弈与合作机制研究[D]. 武汉: 武汉大学, 2012. Quan J. Study on Evolutionary games and cooperation mechanism within the framework of bounded rationality[D]. Wuhan: Wuhan University, 2012. [45] Antal T, Scheuring I. Fixation of strategies for an evolutionary game in finite populations[J]. Bulletin of Mathematical Biology, 2006, 68(8): 1923-1944. [46] Traulsen A, Hauert C. Stochastic evolutionary game dynamics reviews of nonlinear dynamics and complexity[M]. New York: Wiley, 2009. [47] 孙思乐. 第三方物流企业参与金融物流案例研究[D]. 北京: 对外经济贸易大学, 2015. Sun S L. Research on involvement of 3PL company in financial logistics[J]. Beijing: University of International Business and Economics, 2015. [48] 王先甲, 夏可. 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[J]. 系统工程理论与实践, 2019, 39(5): 1128-1136. Wang X J, Xia K. Extended average abundance function of multi-player snowdrift evolutionary game under aspiration driven rule[J]. Systems Engineering—Theory & Practice, 2019, 39(5): 1128-1136.