408 工程科学学报，第44卷.第3期 目前最后通牒博弈的研究均是在恒定环境下 [16]Debarre F.Fidelity of parent-offspring transmission and the 进行.而策略演化动力学与环境相互影响而形成 evolution of social behavior in structured populations.J Theor 的反馈在生物和社会系统中普遍存在，同时环境 Biol,2017,420:26 [17]McAvoy A,Fraiman N,Hauert C,et al.Public goods games in 反馈已被证实可以促进合作的演化.则环境反馈 populations with fluctuating size.Theor Popul Biol,2018,121: 如何影响公平行为的涌现是一个值得深入探讨和 72 研究的科学问题 [18] Szabo G,Borsos I.Evolutionary potential games on lattices.Phys Rep,2016,624:1 参考文献 [19]Lessard S,Soares C D.Frequency-dependent growth in class- [1] Barabasi A L,Albert R.Emergence of scaling in random structured populations:Continuous dynamics in the limit of weak networks.Science,1999,286(5439):509 selection.J Math Biol,2018,77(1):229 [2] Wang L.Wu T,Zhang Y L.Feedback mechanism in [20]Smith J M,Price G R.The logic of animal conflict.Nature,1973, coevolutionary games.Control Theory Appl,2014,31(7):823 246(5427):15 (王龙，吴特，张艳玲.共演化博弈中的反馈机制.控制理论与 [21]Khoo T,Fu F,Pauls S.Spillover modes in multiplex games: 应用，2014,31(7)：823) Double-edged effects on cooperation and their coevolution.Sci [3] Gallo E,Yan C.The effects of reputational and social knowledge Rep,2018,8:6922 on cooperation.Proc Natl Acad Sci USA,2015,112(12):3647 [22]Wang Q,He N R,Chen X J.Replicator dynamics for public [4] Li X,Cao L.Largest Laplacian eigenvalue predicts the goods game with resource allocation in large populations.Appl emergence of costly punishment in the evolutionary ultimatum Math Comput,2018,328:162 game on networks.Plys Rev E,2009,80(6):066101 [23]Perc M,Jordan J J,Rand D G,et al.Statistical physics of human [5] Doebeli M,Ispolatov Y,Simon B.Point of view:Towards a cooperation.Phys Rep,2017,687:1 mechanistic foundation of evolutionary theory.Elife,2019,6: [24]Hilbe C,Martinez-Vaquero L A,Chatterjee K,et al.Memory-n e23804 strategies of direct reciprocity.Proc Natl Acad Sci USA.2017. [6] Goswami A,Gupta R,Parashari G S.Reputation-based resource 114(18):4715 allocation in P2P systems:A game theoretic perspective./EEE [25] Cheng D Z,Liu Z Q.Application of semi-tensor product of Commmn Lett,2017,21(6):1273 matrices to finite games.Control Theory Appl,2019,36(11): [7] Hilbe C,Wu B,Traulsen A,et al.Cooperation and control in 1812 multiplayer social dilemmas.Proc Natl Acad Sci USA,2014, (程代展，刘泽群.有限博弈的矩阵半张量积方法.控制理论与 111(46):16425 应用.2019,36(11)：1812) [8] Tan S L,Wang Y N.Graphical Nash equilibria and replicator [26] Deng Z H,Nian X H.Distributed generalized Nash equilibrium dynamics on complex networks.IEEE Trans Neural Networks seeking algorithm design for aggregative games over weight- Learn Syst,2020,31(6:1831 balanced digraphs.IEEE Trans Neural Networks Learn Syst, [9] Julou T,Mora T,Guillon L,et al.Cell-cell contacts confine 2019.30(3):695 public goods diffusion inside pseudomonas aeruginosa clonal [27]Zhang R,Zhu Q Y.A game-theoretic approach to design secure microcolonies.Proc Natl Acad Sci USA,2013,110(31):12577 and resilient distributed support vector machines.IEEE Trans [10]Qi S,Footer O,Camerer C F,et al.A collaborator's reputation Neural Networks Learn Syst,2018,29(11):5512 can bias decisions and anxiety under uncertainty.J Neurosci, [28]Chotibut T,Nelson D R.Population genetics with fluctuating 2018,38(9:2262 population sizes.J Stat Phys,2017,167(3-4):777 [11]Feng C,Luo Y J,Krueger F.Neural signatures of fairness-related [29]Constable G W,Rogers T,McKane A J,et al.Demographic noise normative decision making in the ultimatum game:A coordinate- can reverse the direction of deterministic selection.Proc Narl based meta-analysis.Humn Brain Mapp,2015,36(2):591 Acad Sci US4,2016,113(32):4745 [12]Duradoni M,Paolucci M,Bagnoli F,et al.Faimess and trust in [30]Liu Y,Zhang J.An B.et al.A simulation framework for virtual environments:The effects of reputation.Future Internet, measuring robustness of incentive mechanisms and its 2018,10(6:50 implementation in reputation systems.Auton Agents Multi Agent [13]Jeong HC,Oh S Y,Allen B,et al.Optional games on cycles and S31,2016,30(4):581 complete graphs.J Theor Biol,2014,356:98 [31]Zhang Y L,Liu A Z,Sun C Y.Development of several studies on [14]Melamed D,Harrell A,Simpson B.Cooperation,clustering,and indirect reciprocity and the evolution of cooperation.Acta Autom assortative mixing in dynamic networks.Proc Natl Acad Sci Sim,2018,44(1):1 S4,2018,115(5):951 (张艳玲，刘爱志，孙长银.间接互惠与合作演化的若干问题研 [15]Adami C,Schossau J,Hintze A.Evolutionary game theory using 究进展.自动化学报，2018,44(1)：1) agent-based methods.Phys Life Rev,2016,19:1 [32]Rong Z H,Xu X R,Wu Z X.Experiment research on the目前最后通牒博弈的研究均是在恒定环境下 进行. 而策略演化动力学与环境相互影响而形成 的反馈在生物和社会系统中普遍存在，同时环境 反馈已被证实可以促进合作的演化. 则环境反馈 如何影响公平行为的涌现是一个值得深入探讨和 研究的科学问题. 参    考    文    献 Barabási  A  L,  Albert  R.  Emergence  of  scaling  in  random networks. Science, 1999, 286（5439）: 509 [1] Wang  L,  Wu  T,  Zhang  Y  L.  Feedback  mechanism  in coevolutionary games. Control Theory Appl, 2014, 31（7）: 823 （王龙, 吴特, 张艳玲. 共演化博弈中的反馈机制. 控制理论与 应用, 2014, 31（7）：823） [2] Gallo E, Yan C. The effects of reputational and social knowledge on cooperation. Proc Natl Acad Sci USA, 2015, 112（12）: 3647 [3] Li  X,  Cao  L.  Largest  Laplacian  eigenvalue  predicts  the emergence  of  costly  punishment  in  the  evolutionary  ultimatum game on networks. Phys Rev E, 2009, 80（6）: 066101 [4] Doebeli  M,  Ispolatov  Y,  Simon  B.  Point  of  view:  Towards  a mechanistic  foundation  of  evolutionary  theory. Elife,  2019,  6: e23804 [5] Goswami A, Gupta R, Parashari G S. Reputation-based resource allocation  in  P2P  systems:  A  game  theoretic  perspective. IEEE Commun Lett, 2017, 21（6）: 1273 [6] Hilbe  C,  Wu  B,  Traulsen  A,  et  al.  Cooperation  and  control  in multiplayer  social  dilemmas. Proc Natl Acad Sci USA,  2014, 111（46）: 16425 [7] Tan  S  L,  Wang  Y  N.  Graphical  Nash  equilibria  and  replicator dynamics  on  complex  networks. IEEE Trans Neural Networks Learn Syst, 2020, 31（6）: 1831 [8] Julou  T,  Mora  T,  Guillon  L,  et  al.  Cell-cell  contacts  confine public  goods  diffusion  inside  pseudomonas  aeruginosa  clonal microcolonies. Proc Natl Acad Sci USA, 2013, 110（31）: 12577 [9] Qi  S,  Footer  O,  Camerer  C  F,  et  al.  A  collaborator's  reputation can  bias  decisions  and  anxiety  under  uncertainty. J Neurosci, 2018, 38（9）: 2262 [10] Feng C, Luo Y J, Krueger F. Neural signatures of fairness-related normative decision making in the ultimatum game: A coordinate￾based meta-analysis. Hum Brain Mapp, 2015, 36（2）: 591 [11] Duradoni M, Paolucci M, Bagnoli F, et al. Fairness and trust in virtual  environments:  The  effects  of  reputation. Future Internet, 2018, 10（6）: 50 [12] Jeong H C, Oh S Y, Allen B, et al. Optional games on cycles and complete graphs. J Theor Biol, 2014, 356: 98 [13] Melamed D, Harrell A, Simpson B. Cooperation, clustering, and assortative  mixing  in  dynamic  networks. Proc Natl Acad Sci USA, 2018, 115（5）: 951 [14] Adami C, Schossau J, Hintze A. Evolutionary game theory using agent-based methods. Phys Life Rev, 2016, 19: 1 [15] Débarre  F.  Fidelity  of  parent-offspring  transmission  and  the evolution  of  social  behavior  in  structured  populations. J Theor Biol, 2017, 420: 26 [16] McAvoy A, Fraiman N, Hauert C, et al. Public goods games in populations  with  fluctuating  size. Theor Popul Biol,  2018,  121: 72 [17] Szabó G, Borsos I. Evolutionary potential games on lattices. Phys Rep, 2016, 624: 1 [18] Lessard  S,  Soares  C  D.  Frequency-dependent  growth  in  class￾structured populations: Continuous dynamics in the limit of weak selection. J Math Biol, 2018, 77（1）: 229 [19] Smith J M, Price G R. The logic of animal conflict. Nature, 1973, 246（5427）: 15 [20] Khoo  T,  Fu  F,  Pauls  S.  Spillover  modes  in  multiplex  games: Double-edged  effects  on  cooperation  and  their  coevolution. Sci Rep, 2018, 8: 6922 [21] Wang  Q,  He  N  R,  Chen  X  J.  Replicator  dynamics  for  public goods  game  with  resource  allocation  in  large  populations. Appl Math Comput, 2018, 328: 162 [22] Perc M, Jordan J J, Rand D G, et al. Statistical physics of human cooperation. Phys Rep, 2017, 687: 1 [23] Hilbe C, Martinez-Vaquero L A, Chatterjee K, et al. Memory-n strategies  of  direct  reciprocity. Proc Natl Acad Sci USA,  2017, 114（18）: 4715 [24] Cheng  D  Z,  Liu  Z  Q.  Application  of  semi-tensor  product  of matrices  to  finite  games. Control Theory Appl,  2019,  36（11）: 1812 （程代展, 刘泽群. 有限博弈的矩阵半张量积方法. 控制理论与 应用, 2019, 36（11）：1812） [25] Deng Z H, Nian X H. Distributed generalized Nash equilibrium seeking  algorithm  design  for  aggregative  games  over  weight￾balanced  digraphs. IEEE Trans Neural Networks Learn Syst, 2019, 30（3）: 695 [26] Zhang R, Zhu Q Y. A game-theoretic approach to design secure and  resilient  distributed  support  vector  machines. IEEE Trans Neural Networks Learn Syst, 2018, 29（11）: 5512 [27] Chotibut  T,  Nelson  D  R.  Population  genetics  with  fluctuating population sizes. J Stat Phys, 2017, 167（3-4）: 777 [28] Constable G W, Rogers T, McKane A J, et al. Demographic noise can  reverse  the  direction  of  deterministic  selection. Proc Natl Acad Sci USA, 2016, 113（32）: 4745 [29] Liu  Y,  Zhang  J,  An  B,  et  al.  A  simulation  framework  for measuring  robustness  of  incentive  mechanisms  and  its implementation in reputation systems. Auton Agents Multi Agent Syst, 2016, 30（4）: 581 [30] Zhang Y L, Liu A Z, Sun C Y. Development of several studies on indirect reciprocity and the evolution of cooperation. Acta Autom Sin, 2018, 44（1）: 1 （张艳玲, 刘爱志, 孙长银. 间接互惠与合作演化的若干问题研 究进展. 自动化学报, 2018, 44（1）：1） [31] [32] Rong  Z  H,  Xu  X  R,  Wu  Z  X.  Experiment  research  on  the · 408 · 工程科学学报，第 44 卷，第 3 期