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Emergence of cooperation through coevolving time scale in spatial prisoner’s dilemma

Emergence of cooperation through coevolving time scale in spatial prisoner’s dilemma. Zhihai Rong ( 荣智海 ) rongzhh@gmail.com Donghua University 2010.08@The 4th China-Europe Summer School on Complexity Science, Shanghai. Acknowledgements . Dr. Zhi-Xi Wu Dr. Wen-Xu Wang Dr. Petter Holme

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Emergence of cooperation through coevolving time scale in spatial prisoner’s dilemma

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  1. Emergence of cooperation through coevolving time scale in spatial prisoner’s dilemma Zhihai Rong (荣智海) rongzhh@gmail.com Donghua University 2010.08@The 4th China-Europe Summer School on Complexity Science, Shanghai

  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

  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》

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

  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…

  6. Some rules for evolutions cooperationNowak 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

  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: Px(t) • updating rule( replicator dynamics): select a neighbor and learn its behavior with probability ~ f(Py(t)-Px(t))

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

  9. Diversity of lifetime (time scale)C.Roca, J.Cuesta, A.Sánchez (2006),Physical review letters, vol.97, pp.158701.Z.X.Wu, Z.H.Rong, P.Holme (2009), Physical Review E, vol.80, pp.36106. • 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: fi(t)=afi(t-1)+(1-a)gi, where -- gi is the payoff of i -- a characterizes the maternal effects. • With probability pi, an individual i is selected to update its strategy: where κ characterizes the rationality of individuals, and is set as 0.01. • 1/pi is the lifetime of i’s current strategy, f(0)=1.

  10. Some key quantities to characterize the cooperative behaviors • Frequency of cooperators: fc • The extinction threshold of defectors/cooperators: bc1 and bc2 AllD C & D coexist AllC

  11. 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. 2006 • J. Ren, W.-X. Wang, & F. Qi, Phys. Rev. E 75,2007

  12. 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

  13. Coevolving time scaleZ.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, 026101 , 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: pi=pi-β, where β is reward factor • If i accepts j's strategy, theloser ihas to shorten its lifetime: pi=pi+α, where α is punishment factor • 0.1 ≤ pi≤1.0, initially pi=1.0, κ=0.01 • What kind of social norm parameters (α,β) can promote the mergence of cooperation?

  14. a The extinction threshold of cooperators, rD (α, β)=(0.9,0.9) Long-term D cluster • 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)=(0.9,0.1) Long-term C cluster (α, β)=(0.9,0.05) short-term C cluster (α,β)=(0.0,0.1) (α,β)=(0.2,0.1)

  15. α=0, increasing β(reward) • Initially p=1, pmin=0.1 • High time scale C High time scale D • Low time scale C Low time scale D t=100 t=50000

  16. a • High time scale C High time scale D • Low time scale C Low time scale D (α, β)=(0.9,0.1) (α,β)=(0.0,0.1) (α,β)=(0.2,0.1)

  17. α↗, fc↗ • 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. β =0.1, increasing α(punishment) b=1.05 (α,β)=(0.1,0.1) (α,β)=(0.9,0.1)

  18. a (α, β)=(0.9,0.9) • High time scale C High time scale D • Low time scale C Low time scale D (α, β)=(0.9,0.1) (α, β)=(0.9,0.05) (α,β)=(0.0,0.1) (α,β)=(0.2,0.1)

  19. (α,β)=(0.9,0.9) α =0.9, increasing β(reward) (α,β)=(0.9,0.1) (α,β)=(0.9,0.05)

  20. Coevolution of Teaching activityA. Szolnoki and M. Perc, New J. Phys. 10 (2008) 043036A. Szolnoki,et al.,Phys.Rev.E 80(2009) 021901 • The player x will adopt the randomly selected neighbor y’s strategy with: • wx characterizes the strength of influence (teaching activity) of x. The leader with wx 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, wywy+Δw. • A highly inhomogeneous distribution of influence may emerge.

  21. 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: pi=max(pi/β, pmin) • If i accepts j's strategy, theloser ihas to shorten its lifetime: pi=min(pi*α,pmax) • pmin=0.1 and pmax=1.0 The extinction threshold of cooperators, rD

  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, rD

  23. 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.

  24. 东华大学http://cist.dhu.edu.cn/index.asp • 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。

  25. THANKS! Discussing Rong Zhihai (荣智海)：rongzhh@gmail.com Department of Automation, DHU

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