Emergence of cooperation through coevolving time scale in spatial prisoner’s dilemma. Zhihai Rong ( 荣智海 ) [email protected] 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 - PowerPoint PPT Presentation
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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.
1( )
1 exp[( ) / ]i ji j
W s sf f
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Some key quantities to characterize the cooperative behaviors
Frequency of cooperators: fc
The extinction threshold of defectors/cooperators:bc1 and bc2
player2
player1
C D
C 1 0
D 0
:1 2
b
PD b
AllD
AllC
C & Dcoexist
DHUDHU Donghua UniversityDonghua UniversityMonomorphic time scale
a↗fc ↗ Optimal fc occurs at p=0.1 for a=0.9p1, C is frequently exploited by D.
P0, 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
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DHUDHU Donghua UniversityDonghua UniversityPolymorphic time scaleThe 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 ≈0.712
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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, the loser i has 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?
1( )
1 exp[( ) / ]i ji j
W s sf f
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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, rD
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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
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a
High time scale C High time scale DLow 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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β =0.1, increasing α(punishment)
(α,β)=(0.1,0.1)
(α,β)=(0.9,0.1)
α↗, fc↗Feedback mechanism for
C/D:Winner Cfc↗fintess↗
Winner Dfc↘fintess↘α↗, their losing D neighbors
have greater chance to becoming C, hence cooperation is promoted.
b=1.05
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a
High time scale C High time scale DLow 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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(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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Coevolution of Teaching activity
A. Szolnoki and M. Perc, New J. Phys. 10 (2008) 043036A. Szolnoki,et al.,Phys.Rev.E 80(2009) 021901
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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, wywy+Δw.A highly inhomogeneous distribution of influence may emerge.
1( )
1 exp[( ) / ]x y yx y
W s s wP P
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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, the loser i has to shorten its lifetime: pi=min(pi*α,pmax)
pmin=0.1 and pmax=1.0
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The extinction threshold of cooperators, rD
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The extinction threshold of cooperationFor loser:α↗
For winner: βmidThe 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
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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.