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© 2002 IBM Corporation Donghua University Emergence of cooperation through coevolving time scale in spatial prisoner’s dilemma Zhihai Rong ( 荣智海 ) Donghua University 4th China-Europe Summer School on Complexity Science, Shanghai

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DHU Donghua University 2 Acknowledgements Dr. Zhi-Xi Wu Dr. Wen-Xu Wang Dr. Petter Holme Zhi-Xi Wu, Zhihai Rong & Petter Holme, Phys.Rev.E,036106,2010 Zhihai Rong, Zhi-Xi Wu & Wen-Xu Wang, Phys.Rev.E,026101, 2010

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DHU Donghua University 3 阿豺折箭 戮力一心 阿豺有子二十人。阿豺谓曰： “ 汝 等各奉吾一支箭。 ” 折之地下。俄 而命母弟慕利延曰： “ 汝取一支箭 折之。 ” 慕利延折之。又曰： “ 汝取 十九支箭折之。 ” 延不能折。阿豺 曰： “ 汝曹知否？单者易折，众则 难摧，戮力一心，然后社稷可固 ！ ” —— 《魏书 吐谷浑传》

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DHU Donghua University 4 Cooperation: the basis of human societies Robert Boyd and Sarah Mathew, A Narrow Road to Cooperation, SCIENCE,2007

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DHU Donghua University 5 Prisoner ’ s dilemma ( 囚徒困境,PD) Cooperator: help others at a cost to themselves. Defector: receive the benefits without providing help. Whatever opponent does, player does better by defecting … CD C(-2,-2)(-5,-1) D(-1,-5)(-3,-3)

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DHU Donghua University 6 Some rules for evolutions cooperation Nowak MA (2006). Five rules for the evolution of cooperation. Science Kin selection: relative Hamilton, J. Theor. Biol.7 (1964) Direct reciprocity: unrelated individuals Tit for tat(TFT): nice, punishing, forgiving, but for noise… Axelrod & Hamilton, Science 211, (1981) Win stay, lost shift(WSLS) Nowak, Sigmund, Nature 364, (1993) Indirect reciprocity: reputation Nowak, Sigmund, Nature 437 (2005). Network reciprocity

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DHU Donghua University 7 Spatial Game Theory M. Nowak and R. May, Evolutionary games and spatial chaos, Nature 1992 Each player x occupying a site on a network playing game with neighbors and obtaining payoff: P x (t) updating rule( replicator dynamics): select a neighbor and learn its behavior with probability ~ f(P y (t)-P x (t))

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DHU Donghua University 8 Evolutionary games on graphs G. Szabo&G. Fath, Evolutionary games on graphs, Phys. Rep. 446, 2007 Cooperator frequency fc Game Rule Selection rule Best take over Random Preferential … PD,SG,SH,UG,PGG, Rock-paper-scissors… Evolutionary Rule Structure & property Replacement rule replicator dynamics W(x y) =f(P y -P x ) Fermi dynamics: W(x y)=(1+exp(x-y/κ)) -1 Win stay, lost shift Memory … Lattice, random graph, small-world, scale-free… , γ, r k, CC, community

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DHU Donghua University 9 Diversity of lifetime (time scale) C.Roca, J.Cuesta, A.S á nchez (2006),Physical review letters, vol.97, pp Z.X.Wu, Z.H.Rong, P.Holme (2009), Physical Review E, vol.80, pp The interaction time scale — how frequently the individuals interact with each other The selection time scale — how frequently they modifies their strategies The selection time scale is slower than the interaction time scale, the player has a finite lifetime. Individuals local on a square lattice. The fitness of i at t-th generation: f i (t)=af i (t-1)+(1-a)g i, where -- g i is the payoff of i -- a characterizes the maternal effects. With probability p i, an individual i is selected to update its strategy: where κ characterizes the rationality of individuals, and is set as 1/p i is the lifetime of i ’ s current strategy, f(0)=1.

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DHU Donghua University 10 Some key quantities to characterize the cooperative behaviors Frequency of cooperators: fc The extinction threshold of defectors/cooperators: b c1 and b c2 AllD AllC C & D coexist

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DHU Donghua University Monomorphic time scale a ↗ fc ↗ Optimal fc occurs at p=0.1 for a=0.9 p 1, C is frequently exploited by D. P 0, Ds around the boundary have enough time to obtain a fitness high enough to beat Cs. Coherence resonance M. Perc, New J. Phys. 2006,M. Perc & M. Marhl,New J. Phys J. Ren, W.-X. Wang, & F. Qi, Phys. Rev. E 75,

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DHU Donghua University Polymorphic time scale The leaders are the individual with low p the followers are the individual with high p. v% of individuals ’ p are 0.1, and others ’ p are 0.9. v=0.5, a=0.9, b=1.1, fc ≈0.7 12

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DHU Donghua University 13 Coevolving time scale Z.H.Rong, Z.X. Wu, W.X.Wang, Emergence of cooperation through coevolving time scale in spatial prisoner's dilemma, submitted to Physical Review E, 82, , 2010 “ win-slower, lose-faster ” rule: i updates its strategy by comparing with neighbor j with a different strategy with probability If i successfully resists the invasion of j, the winner i is rewarded by owing longer lifetime: p i =p i -β, where β is reward factor If i accepts j's strategy, the loser i has to shorten its lifetime: p i =p i +α, where α is punishment factor 0.1 ≤ p i ≤1.0, initially p i =1.0, κ=0.01 What kind of social norm parameters (α,β) can promote the mergence of cooperation?

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DHU Donghua University 14 a High time scale C(p>0.5) High time scale D(p>0.5) Low time scale C (p≤0.5) Low time scale D(p ≤0.5) (α, β)=(0.0,0.1)(α, β)=(0.2,0.1) (α, β)=(0.9,0.1) Long-term C cluster (α, β)=(0.9,0.05) short-term C cluster (α, β)=(0.9,0.9) Long-term D cluster The extinction threshold of cooperators, r D

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DHU Donghua University 15 α=0, increasing β(reward) Initially p=1, p min =0.1 High time scale C High time scale D Low time scale C Low time scale D t=100t=50000

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DHU Donghua University 16 a High time scale C High time scale D Low time scale C Low time scale D (α, β)=(0.0,0.1)(α, β)=(0.2,0.1) (α, β)=(0.9,0.1)

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DHU Donghua University 17 β =0.1, increasing α(punishment) (α,β)=(0.1,0.1) (α,β)=(0.9,0.1) α ↗, f c ↗ Feedback mechanism for C/D: Winner C fc ↗ fintess ↗ Winner D fc ↘ fintess ↘ α ↗, their losing D neighbors have greater chance to becoming C, hence cooperation is promoted. b=1.05

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DHU Donghua University 18 a High time scale C High time scale D Low time scale C Low time scale D (α, β)=(0.0,0.1)(α, β)=(0.2,0.1) (α, β)=(0.9,0.1) (α, β)=(0.9,0.05) (α, β)=(0.9,0.9)

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DHU Donghua University 19 (α,β)=(0.9,0.1) α =0.9, increasing β(reward) (α,β)=(0.9,0.9) (α,β)=(0.9,0.05)

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DHU Donghua University Coevolution of Teaching activity A. Szolnoki and M. Perc, New J. Phys. 10 (2008) A. Szolnoki,et al.,Phys.Rev.E 80(2009) The player x will adopt the randomly selected neighbor y ’ s strategy with: w x characterizes the strength of influence (teaching activity) of x. The leader with w x 1. Each successful strategy adoption process is accompanied by an increase in the donor ’ s teaching activity: If y succeeds in enforcing its strategy on x, w y w y +Δw. A highly inhomogeneous distribution of influence may emerge.

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DHU Donghua University Multiplicative “ win-slower, lose-faster ” “ win-slower, lose-faster ” rule: i updates its strategy by comparing with neighbor j with a different strategy: If i successfully resists the invasion of j, the winner i is rewarded by owing longer lifetime: p i =max(p i /β, p min ) If i accepts j's strategy, the loser i has to shorten its lifetime: p i =min(p i *α,p max ) p min =0.1 and p max = The extinction threshold of cooperators, r D

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DHU Donghua University 22 The extinction threshold of cooperation For loser:α ↗ For winner: β mid The additive-increase /multiplicative-decrease (AIMD) algorithm in the TCP congestion control on the Internet Jacobson, Proc. ACM SIGCOMM' 88 The extinction threshold of cooperators, r D

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DHU Donghua University Conclusions The selection time scale is slower than the interaction time scale. Both the fixed and the coevolving time scale. “win-slower, lose-faster ” rule The potential application in the design of consensus protocol in multi-agent systems. 23

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DHU Donghua University 24 东华大学 东华大学位于上海松江区，原名中国纺织大学，是国家 教育部所属的 211 全国重点大学，也是我国首批具有博士、 硕士、学士三级学位授予权的大学之一。 信息学院现有 “ 控制理论与控制工程 (90) ” 和 “ 模式识别与 智能系统 (02) ” 2 个博士点以及 7 个硕士点， “ 控制科学与 工程 (03) ” 一级学科博士后流动站，拥有 “ 教育部数字化 纺织服装技术工程研究中心 ” 。 信息学院现有教职工近 120 人，其中校特聘教授 2 人，长 江特聘讲座教授 1 人，博士生导师 16 人，具有正高级职称 25 人，副高级职称 41 人。

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DHU Donghua University 25 THANKS! Discussing Rong Zhihai ( 荣智海 ) ： Department of Automation, DHU

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