Articles

THE AVERAGE ABUNDANCE FUNCTION WITH MUTATION OF THE MULTI-PLAYER SNOWDRIFT EVOLUTIONARY GAME MODEL

  • Ke XIA ,
  • Xianjia WANG
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  • 1. School of Economics, Zhengzhou University of Aeronautics, Zhengzhou 450000, China;
    2. School of Economics and Management, Wuhan University, Wuhan 430072, China
Ke XIA,E-mail:xkforeducation@163.com;Xianjia WANG,E-mail:wangxj@whu.edu.cn

Received date: 2019-11-07

  Revised date: 2020-04-08

  Online published: 2021-04-06

Supported by

This research was supported by the National Natural Science Foundation of China (71871171, 72031009).

Abstract

This article explores the characteristics of the average abundance function with mutation on the basis of the multi-player snowdrift evolutionary game model by analytical analysis and numerical simulation. The specific field of this research concerns the approximate expressions of the average abundance function with mutation on the basis of different levels of selection intensity and an analysis of the results of numerical simulation on the basis of the intuitive expression of the average abundance function. In addition, the biological background of this research lies in research on the effects of mutation, which is regarded as a biological concept and a disturbance to game behavior on the average abundance function. The mutation will make the evolutionary result get closer to the neutral drift state. It can be deduced that this affection is not only related to mutation, but also related to selection intensity and the gap between payoff and aspiration level. The main research findings contain four aspects. First, we have deduced the concrete expression of the expected payoff function. The asymptotic property and change trend of the expected payoff function has been basically obtained. In addition, the intuitive expression of the average abundance function with mutation has been obtained by taking the detailed balance condition as the point of penetration. It can be deduced that the effect of mutation is to make the average abundance function get close to 1/2. In addition, this affection is related to selection intensity and the gap. Secondly, the first-order Taylor expansion of the average abundance function has been deduced for when selection intensity is sufficiently small. The expression of the average abundance function with mutation can be simplified from a composite function to a linear function because of this Taylor expansion. This finding will play a significant role when analyzing the results of the numerical simulation. Thirdly, we have obtained the approximate expressions of the average abundance function corresponding to small and large selection intensity. The significance of the above approximate analysis lies in that we have grasped the basic characteristics of the effect of mutation. The effect is slight and can be neglected when mutation is very small. In addition, the effect begins to increase when mutation rises, and this effect will become more remarkable with the increase of selection intensity. Fourthly, we have explored the influences of parameters on the average abundance function with mutation through numerical simulation. In addition, the corresponding results have been explained on the basis of the expected payoff function. It can be deduced that the influences of parameters on the average abundance function with mutation will be slim when selection intensity is small. Moreover, the corresponding explanation is related to the first-order Taylor expansion. Furthermore,the influences will become notable when selection intensity is large.

Cite this article

Ke XIA , Xianjia WANG . THE AVERAGE ABUNDANCE FUNCTION WITH MUTATION OF THE MULTI-PLAYER SNOWDRIFT EVOLUTIONARY GAME MODEL[J]. Acta mathematica scientia, Series B, 2021 , 41(1) : 127 -163 . DOI: 10.1007/s10473-021-0108-2

References

[1] Matsen F A, Nowak M A. Win-stay, lose-shift in language learning from peers.. Proceedings of the National Academy of Sciences of the United States of America, 2004, 101(52):18053-18057
[2] Chen X, Fu F, Wang L. Promoting cooperation by local contribution under stochastic win-stay-lose-shift mechanism. Physica a Statistical Mechanics & its Applications, 2008, 387(22):5609-5615
[3] Chen X, Wang L. Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game. Physical Review E, 2008, 77:017103
[4] Liu Y, Chen X, Zhang L, et al. Win-stay-lose-learn promotes cooperation in the spatial prisoner's dilemma game. Plos One, 2012, 7(2):e30689
[5] Zhou L, Wu B, Vasconcelos V V, et al. Simple property of heterogeneous aspiration dynamics:Beyond weak selection. Physical Review E, 2018, 98(6):062124
[6] Du J. Redistribution promotes cooperation in spatial public goods games under aspiration dynamics. Applied Mathematics and Computation, 2019, 363:124629
[7] Perc M, Gomez-Gardenes J, Szolnoki A, et al. Evolutionary dynamics of group interactions on structured populations:a review. Journal of the Royal Society Interface, 2013, 10(80):20120997
[8] Szolnoki A, Perc M. Impact of critical mass on the evolution of cooperation in spatial public goods games. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2010, 81(5):57101
[9] Perc M, Jordan J J, Rand D G, et al. Statistical physics of human cooperation. Physics Reports-Review Section of Physics Letters, 2017, 687:1-51
[10] Yang H X, Wu Z X, Wang B H. Role of aspiration-induced migration in cooperation. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2010, 81(2):65101
[11] Lin H, Yang D P, Shuai J W. Cooperation among mobile individuals with payoff expectations in the spatial prisoner's dilemma game. Chaos Solitons & Fractals, 2011, 44(1):153-159
[12] Peng Y, Wang X, Lu Q, et al. Effects of aspiration-induced adaptation and migration on the evolution of cooperation. International Journal of Modern Physics C, 2014, 25(7):1450025
[13] Chen X, Fu F, Wang L. Could feedback-based self-learning help solve networked prisoner's dilemma?. IEEE Conference on Decision and Control, 2009:1526-1531
[14] Zeng W, Li M, Feng N. The effects of heterogeneous interaction and risk attitude adaptation on the evolution of cooperation. Journal of Evolutionary Economics, 2017, 27(3):435-459
[15] Zhang J, Fang Y P, Du W B, et al. Promotion of cooperation in aspiration-based spatial prisoner's dilemma game. Physica a Statistical Mechanics & its Applications, 2011, 390(12):2258-2266
[16] Liu Y, Chen X, Wang L, et al. Aspiration-based learning promotes cooperation in spatial prisoner's dilemma games. EPL, 2011, 94(6):60002
[17] Platkowski T, Bujnowski P. Cooperation in aspiration-based N-person prisoner's dilemmas. Phys Rev E Stat Nonlin Soft Matter Phys, 2009, 79(2):036103
[18] Platkowski T. Enhanced cooperation in prisoner's dilemma with aspiration. Appl Math Lett, 2009, 22(8):1161-1165
[19] Roca C P, Helbing D. Emergence of social cohesion in a model society of greedy, mobile individuals. Proc National Acad Sci USA, 2011, 108(28):11370-11374
[20] Platkowski T. Aspiration-based full cooperation in finite systems of players. Appl Math Comput, 2015, 251:46-54
[21] Feng X, Wu B, Wang L. Voluntary vaccination dilemma with evolving psychological perceptions. Journal of Theoretical Biology, 2018, 439:65-75
[22] Wu B, Zhou L. Individualised aspiration dynmics:Calculation by proofs. PLOS Computational Biology, 2018, e1006035
[23] Wakano J Y, Yamamura N. A simple learning strategy that realizes robust cooperation better than Pavlov in Iterated Prisoners' Dilemma. Journal of Ethology, 2001, 19(1):1-8
[24] Tang C, Wang Y, Cao L, et al. Towards the role of social connectivity and aspiration level on evolutionary game. European Physical Journal B, 2013, 86(1):1-6
[25] Wu Z X, Rong Z. Boosting cooperation by involving extortion in spatial prisoner's dilemma games. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2014, 90(6):62102
[26] Rong Z H, Zhao Q, Wu Z X, et al. Proper aspiration level promotes generous behavior in the spatial prisoner's dilemma game. European Physical Journal B, 2016, 89(7):166
[27] Du J, Wu B, Wang L. Aspiration dynamics in structured population acts as if in a well-mixed one. Scientific Reports, 2015, 5:8014
[28] Du J, Wu B, Wang L. Aspiration dynamics and the sustainability of resources in the public goods dilemma. Physics Letters A, 2016, 380(16):1432-1436
[29] Wang Z. Heterogeneous aspirations promote cooperation in the prisoner's dilemma game. Plos One, 2010, 5(12):e15117
[30] Zhang H F, Liu R R, Wang Z, et al. Aspiration-induced reconnection in spatial public goods game. EPL, 2011, 94(1):73-79
[31] Du J, Wu B, Wang L. Evolutionary game dynamics of multi-agent cooperation driven by self-learning. 9th Asian Control Conference, 2013
[32] Li M, Jia C X, Liu R R, et al. Emergence of cooperation in spatial public goods game with conditional participation. Physica a Statistical Mechanics & its Applications, 2013, 392(8):1840-1847
[33] Li Z, Yang Z, Wu T, et al. Aspiration-based partner switching boosts cooperation in social dilemmas. Plos One, 2014, 9:e978666
[34] Chen W, Wu T, Li Z, et al. Coevolution of aspirations and cooperation in spatial prisoner's dilemma game. Journal of Statistical Mechanics-Theory and Experiment, 2015:P01032
[35] Liu X, He M, Kang Y, et al. Aspiration promotes cooperation in the prisoner's dilemma game with the imitation rule. Physical Review E, 2016,94(1):012124
[36] Xu K, Li K, Cong R, et al. Cooperation guided by the coexistence of imitation dynamics and aspiration dynamics in structured populations. EPL, 2017, 117(4):48002
[37] Willensdorfer M, Nowak M A. Mutation in evolutionary games can increase average fitness at equilibrium. Journal of Theoretical Biology, 2005, 237(4):355-362
[38] Eriksson A, Lindgren K. Cooperation driven by mutations in multi-person Prisoner's Dilemma. Journal of Theoretical Biology, 2005, 232(3):399-409
[39] Traulsen A, Claussen J C, Hauert C. Stochastic differential equations for evolutionary dynamics with demographic noise and mutations. Physical Review E, 2012, 85(4):041901
[40] Fudenberg D, Imhof L A. Imitation processes with small mutations. Journal of Economic Theory, 2006, 131(1):251-262
[41] Fudenberg D, Nowak M A, Taylor C, et al. Evolutionary game dynamics in finite populations with strong selection and weak mutation. Theoretical Population Biology, 2006, 70(3):352-363
[42] Antal T, Nowak M A, Traulsen A. Strategy abundance in 2×2 games for arbitrary mutation rates. Journal of Theoretical Biology, 2009, 257(2):340-344
[43] Wu B, Gokhale C S, Wang L, et al. How small are small mutation rates?. Journal of Mathematical Biology, 2012, 64(5):803-827
[44] Antal T, Traulsen A, Ohtsuki H, et al. Mutation-selection equilibrium in games with multiple strategies. Journal of Theoretical Biology, 2009, 258(4):614-622
[45] Tarnita C E, Antal T, Nowak M A. Mutation-selection equilibrium in games with mixed strategies. Journal of Theoretical Biology, 2009, 261(1):50-57
[46] Helbing D, Szolnoki A, Perc M, et al. Defector-accelerated cooperativeness and punishment in public goods games with mutations. Physical Review E, 2010, 81(5):057104
[47] Allen B, Traulsen A, Tarnita C E, et al. How mutation affects evolutionary games on graphs. Journal of Theoretical Biology, 2012, 299(SI):97-105
[48] Kaiping G A, Jacobs G S, Cox S J, et al. Nonequivalence of updating rules in evolutionary games under high mutation rates. Physical Review E, 2014, 90(4):042726
[49] Wu X, Zhu H. Fast maximum likelihood estimation of mutation rates using a birth-death process. Journal of Theoretical Biology, 2015, 366:1-7
[50] Mobilia M. Oscillatory dynamics in rock-paper-scissors games with mutations. Journal of Theoretical Biology, 2010, 264(1):1-10
[51] Uehara T, Iwasa Y. Global mutations and local mutations have very different effects on evolution, illustrated by mixed strategies of asymmetric binary games. Journal of Theoretical Biology, 2010, 262(2):223-231
[52] Kelly J K. Mutation-selection balance in mixed mating populations. Journal of Theoretical Biology, 2007, 246(2):355-365
[53] de Oliveira V M, Campos P. Dynamics of fixation of advantageous mutations. Physica A-Statistical Mechanics and Its Applications, 2004, 337(3/4):546-554
[54] Sasaki A, Nowak M A. Mutation landscapes. Journal of Theoretical Biology, 2003, 224(2):241-247
[55] Golding I, Drossel B, Shapira Y, et al. A quantitative study of the dynamics of adaptive mutation appearance. Physica A, 2001, 294(1/2):195-212
[56] Wilke C O, Ronnewinkel C. Dynamic fitness landscapes:expansions for small mutation rates. Physica A-Statistical Mechanics and Its Applications, 2001, 290(3/4):475-490
[57] Manneville J B, Bassereau P, Ramaswamy S, et al. Active membrane fluctuations studied by micropipet aspiration. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2001, 64(2):021908
[58] Napel S. Aspiration adaptation in the ultimatum minigame. Games & Economic Behavior, 2003, 43(1):86-106
[59] Bettendorff-Bakman D E, Schmid P, Lunkenheimer P P, et al. Diastolic ventricular aspiration:A mechanism supporting the rapid filling phase of the human ventricles. Journal of Theoretical Biology, 2008, 250(4):581-592
[60] Yakushkina T, Saakian D B. New versions of evolutionary models with lethal mutations. Physica AStatistical Mechanics and Its Applications, 2018, 507:470-477
[61] Ferretti L, Schmiegelt B, Weinreich D, et al. Measuring epistasis in fitness landscapes:The correlation of fitness effects of mutations. Journal of Theoretical Biology, 2016, 396:132-143
[62] Wu B, Traulsen A, Gokhale C S. Dynamic Properties of Evolutionary Multi-player Games in Finite Populations. Games, 2013, 4(2):182-199
[63] Gokhale C S, Traulsen A. Evolutionary games in the multiverse. Proc National Acad Sci USA, 2010, 107(12):5500-5504
[64] Lessard S. On the Robustness of the extension of the one-third law of evolution to the multi-player game. Dynamic Games and Applications, 2011, 1(3/SI):408-418
[65] Wang J, Wu B, Ho D W C, et al. Evolution of cooperation in multilevel public goods games with community structures. EPL, 2011, 93(5):650-667
[66] Wu B, Zhou D, Wang L. Evolutionary dynamics on stochastic evolving networks for multiple-strategy games. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2011, 84(2):046111
[67] Arnoldt H, Timme M, Grosskinsky S. Frequency-dependent fitness induces multistability in coevolutionary dynamics. Journal of the Royal Society Interface, 2012, 9(77):3387-3396
[68] Du J, Wu B, Altrock P M, et al. Aspiration dynamics of multi-player games in finite populations. Journal of the Royal Society Interface, 2014, 11(94):20140077
[69] Szolnoki A, Perc M. Group-size effects on the evolution of cooperation in the spatial public goods game. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2011, 84(4):47102
[70] Chen X, Szolnoki A, Perc M, et al. Impact of generalized benefit functions on the evolution of cooperation in spatial public goods games with continuous strategies. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2012, 85(6):066133
[71] Wang W, Ren J, Chen G, et al. Memory-based snowdrift game on networks. Physical Review E, 2006, 74(5):06113
[72] Souza M O, Pacheco J M, Santos F C. Evolution of cooperation under N-person snowdrift games. Journal of Theoretical Biology, 2009, 260(4):581-588
[73] Santos M D, Santos F C, Pacheco J M. Collective evolutionary dynamics and spatial reciprocity under the N-person snowdrift game. International Conference on Bio-Inspired Models of Network, Information, and Computing Systems, 2010
[74] Doebeli M, Hauert C. Models of cooperation based on the Prisoner's Dilemma and the Snowdrift game. Ecology Letters, 2005, 8(7):748-766
[75] Hauert C, Doebeli M. Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature, 2004, 428(6983):643-646
[76] Santos M D, Pinheiro F L, Santos F C, et al. Dynamics of N-person snowdrift games in structured populations. Journal of Theoretical Biology, 2012, 315:81-86
[77] Pena J, Wu B, Traulsen A. Ordering structured populations in multiplayer cooperation games. Journal of the Royal Society Interface, 2016, 13(114):20150881
[78] McAvoy A, Hauert C. Structure coefficients and strategy selection in multiplayer games. Journal of Mathematical Biology, 2016, 72(1/2):203-238
[79] Wang L, Du J M. Evolutionary game theoretic approach to coordinated control of multi-agent systems. Journal of Systems Science and Mathematical Sciences, 2016, 36:302-318
[80] Sui X, Wu B, Wang L. Multiple tolerances dilute the second order cooperative dilemma. Physics Letters A, 2017, 381(45):3785-3797
[81] Du J, Tang L. Evolution of global contribution in multi-level threshold public goods games with insurance compensation. Journal of Statistical Mechanics Theory & Experiment, 2018, 2018(1):013403
[82] Lin Y T, Yang H X, Wu Z X, et al. Promotion of cooperation by aspiration-induced migration. Physica A-Statistical Mechanics & its Applications, 2011, 390(1):77-82
[83] Masuda N, Nakamura M. Numerical analysis of a reinforcement learning model with the dynamic aspiration level in the iterated Prisoner's dilemma. Journal of Theoretical Biology, 2011, 278(1):55-62
[84] de Oliveira V M, Campos P. Topological effects of fitness landscapes on the process of fixation of advantageous mutations. Physica A-Statistical Mechanics and its Applications, 2005, 346(3/4):451-458
[85] Bonadonna P, Zanotti R, Melioli G, et al. Effects of aspiration on public cooperation in structured populations. Physica A-Statistical Mechanics & its Applications, 2012, 391(15):4043-4049
[86] Tanabe S, Masuda N. Evolution of cooperation facilitated by reinforcement learning with adaptive aspiration levels. Journal of Theoretical Biology, 2012, 293:151-160
[87] Tanimoto J, Nakata M, Hagishima A, et al. Spatially correlated heterogeneous aspirations to enhance network reciprocity. Physica A-Statistical Mechanics & its Applications, 2012, 391(3):680-685
[88] Amaral M A, Wardil L, Perc M, et al. Stochastic win-stay-lose-shift strategy with dynamic aspirations in evolutionary social dilemmas. Physical Review E, 2016, 94(3):032317
[89] Chen Y S, Yang H X, Guo W Z. Aspiration-induced dormancy promotes cooperation in the spatial Prisoner's Dilemma games. Physica A-Statistical Mechanics & its Applications, 2017, 469:625-630
[90] Samudrala N, Jin N, Sarfati R, et al. Mechanical stability of particle-stabilized droplets under micropipette aspiration. Phys Rev E, 2017, 95(1):012805
[91] Yang Z, Li Z, Wu T, et al. Effects of payoff-related velocity in the co-evolutionary snowdrift game. Physica A-Statistical Mechanics & its Applications, 2014, 393(1):304-311
[92] Zheng D F, Yin H P, Chan C H, et al. Cooperative behavior in a model of evolutionary snowdrift games with N-person interactions. EPL, 2007, 80(1):18002
[93] Wu T, Fu F, Wang L. Coevolutionary dynamics of aspiration and strategy in spatial repeated public goods games. New Journal of Physics, 2018, 20(6):063007
[94] Wang B, Pei Z, Wang L. Evolutionary dynamics of cooperation on interdependent networks with Prisoner's Dilemma and Snowdrift Game. EPL, 2014, 107:58006
[95] Chen X J, Wang L. Effects of cost threshold and noise in spatial snowdrift games with fixed multi-person interactions. EPL, 2010, 90(3):1303-1324
[96] Traulsen A, Hauert C, De Silva H, et al. Exploration dynamics in evolutionary games. Proc National Acad Sci USA, 2009, 106(3):709-712
[97] Chen X, Szolnoki A, Perc M. Competition and cooperation among different punishing strategies in the spatial public goods game. Phys Rev E Stat Nonlin Soft Matter Phys, 2015, 92(1):012819
[98] Attila S, Matja P. Competition of tolerant strategies in the spatial public goods game. New Journal of Physics, 2016, 18(8):083021
[99] Csercsik, David. Competition and Cooperation in a Bidding Model of Electrical Energy Trade. Networks & Spatial Economics, 2016, 16(4):1043-1073
[100] Du J. Insurance optimizes complex interactive and cooperative behaviors in public goods games. Plos One, 2018, 13(5):e197574
[101] Szolnoki A, Chen X. Competition and partnership between conformity and payoff-based imitations in social dilemmas. New Journal of Physics, 2018, 20(9):093008
[102] Du J, Wang B. Evolution of global cooperation in multi-level threshold public goods games with income redistribution. Frontiers in Physics, 2018, 6:67
[103] Li K, Cong R, Wu T, et al. Social exclusion in finite populations. Physical Review E, 2015,91(4):042810
[104] Quan J, Yang X, Wang X. Spatial public goods game with continuous contributions based on Particle Swarm Optimization learning and the evolution of cooperation. Physica A-Statistical Mechanics and its Applications, 2018, 505:973-983
[105] Quan J, Liu W, Chu Y, et al. Stochastic dynamics and stable equilibrium of evolutionary optional public goods game in finite populations. Physica A-Statistical Mechanics and its Applications, 2018, 502:123-134
[106] Dong Y, Xu H, Fan S. Memory-based stag hunt game on regular lattices. Physica a Statistical Mechanics and its Applications, 2019, 519:247-255
[107] Chen W, Wu T, Li Z, et al. Evolution of fairness in the mixture of the ultimatum game and the dictator game. Physica A-Statistical Mechanics and its Applications, 2019, 519:319-325
[108] Yang G, Zhu C, Zhang W. Adaptive and probabilistic strategy evolution in dynamical networks. Physica A-Statistical Mechanics and its Applications, 2019, 518:99-110
[109] Gao J, Li Z, Cong R, et al. Tolerance-based punishment in continuous public goods game. Physica AStatistical Mechanics & its Applications, 2012, 391(16):4111-4120
[110] Sun D, Kou X. Punishment effect of prisoner dilemma game based on a new evolution strategy rule. Mathematical Problems in Engineering, 2014, 2014:108024
[111] Traulsen A, Pacheco J M, Nowak M A. Pairwise comparison and selection temperature in evolutionary game dynamics. Journal of Theoretical Biology, 2007, 246(3):522-529
[112] Lessard S, Ladret V. The probability of fixation of a single mutant in an exchangeable selection model. Journal of Mathematical Biology, 2007, 54(5):721-744
[113] Altrock P M, Traulsen A. Fixation times in evolutionary games under weak selection. New Journal of Physics, 2009, 11:013012
[114] Ashcroft P, Altrock P M, Galla T. Fixation in finite populations evolving in fluctuating environments. Journal of the Royal Society Interface, 2014, 11(100):20140663
[115] Zhang L, Ying L, Zhou J, et al. Fixation probabilities of evolutionary coordination games on two coupled populations. Physical Review E, 2016, 94(3):032307
[116] Wang B, Chen X, Wang L. Probabilistic interconnection between interdependent networks promotes cooperation in the public goods game. Journal of Statistical Mechanics Theory & Experiment, 2012, 2012(11):11017
[117] Li Z, Gao J, Suh I H, et al. Degree-based assignation of roles in ultimatum games on scale-free networks. Physica a Statistical Mechanics & its Applications, 2013, 392(8):1885-1893
[118] Yang Z, Li Z, Wu T, et al. Effects of adaptive dynamical linking in networked games. Phys Rev E Stat Nonlin Soft Matter Phys, 2013, 88(1):42128
[119] Cong R, Wu T, Qiu Y Y, et al. Time scales in evolutionary game on adaptive networks. Physics Letters A, 2014, 378(13):950-955
[120] Zhao J, Luo C, Zheng Y. Evolutionary dynamics of the cooperation clusters on interdependent networks. Physica A-Statistical Mechanics and its Applications, 2019, 517:132-140
[121] De Oliveira V M, Campos P R A. The emergence of division of labor in a structured response threshold model. Physica A-Statistical Mechanics and its Applications, 2019, 517:153-162
[122] Hu K, Guo H, Geng Y, et al. The effect of conformity on the evolution of cooperation in multigame. Physica A-Statistical Mechanics and its Applications, 2019, 516:267-272
[123] Pinsky M L. Species Coexistence through competition and rapid evolution. Proc National Acad Sci USA, 2019, 116(7):2407-2409
[124] Sui X, Cong R, Li K, et al. Evolutionary dynamics of N-person snowdrift game. Physics Letters A, 2015, 379(45/46):2922-2934
[125] Wang X, Lv S, Quan J. The evolution of cooperation in the prisoner's dilemma and the snowdrift game based on particle swarm optimization. Physica A-Statistical Mechanics and its Applications, 2017, 482:286-295
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