Adaptive co-evolution of strategies and network leading to optimal cooperation level in spatial prisoner's dilemma game
Zhang Ji-Qian^{a} , Chen Han-Shuang^{b} , Xin Hou-Wen^{b} , Hou Zhong-Huai^{c}
^{a} College of Physics and Electronic Information,
Anhui Normal University, Wuhu 241000, China; ^{b} Department of Chemical Physics, University of Science
and Technology of China, Hefei 230026, China; ^{c} Department of Chemical Physics, University of Science
and Technology of China, Hefei 230026, China;Hefei
National Laboratory for Physical Sciences at Microscale,
University of Science and Technology of China, Hefei 230026, China

Abstract We study evolutionary prisoner's dilemma game on adaptive
networks where a population of players co-evolves with their
interaction networks. During the co-evolution process, interacted
players with opposite strategies either rewire the link between them
with probability $p$ or update their strategies with probability
$1-p$ depending on their payoffs. Numerical simulation shows that
the final network is either split into some disconnected communities
whose players share the same strategy within each community or forms
a single connected network in which all nodes are in the same
strategy. Interestingly, the density of cooperators in the final
state can be maximised in an intermediate range of $p$ via the
competition between time scale of the network dynamics and that of
the node dynamics. Finally, the mean-field analysis helps to
understand the results of numerical simulation. Our results may
provide some insight into understanding the emergence of cooperation
in the real situation where the individuals' behaviour and their
relationship adaptively co-evolve.
Key words ：
prisoner's dilemma game
adaptive network
co-evolution
cooperation
Received: 30 July 2009
Published: 15 May 2010
:
01.75.+m
(Science and society)
02.50.Le
(Decision theory and game theory)
02.50.Cw
(Probability theory)
02.60.Cb
(Numerical simulation; solution of equations)
89.75.Hc
(Networks and genealogical trees)
87.23.Cc
(Population dynamics and ecological pattern formation)

Fund: Project supported by the National
Natural Science Foundation of China (Grant No.~20873130), the
Graduate Innovation Fund of USTC.

[1]
Albert R and Barab\'{asi A-L 2002 Rev. Mod. Phys. 74 47
[2]
Newman M E J 2003 SIAM Review 45 167
[3]
Dorogovtsev S N and Mendes J F F 2002 Adv. Phys. 51 1079
[4]
Boccaletti S, Latora V, Moreno Y, Chavez M and Hwang D U 2006 Phys. Rep. 424 175
[5]
Watts D J and Strogatz S J 1998 Nature 393 440
[6]
Barab\'{asi A-L and Albert R 1999 Science 286 509
[7]
Barab\'{asi A-L, Albert R and Jeong H 1999 Physica A 272 173
[8]
Zhou C and Kurths J 2006 Phys. Rev. Lett. 96 164102
[9]
Holme P and Newman M E J 2006 Phys. Rev. E 74 056108
[10]
Gil S and Zanette D H 2006 Phys. Lett. A 356 89
[11]
Vazquez F, Egu\'{\hiluz V M and Miguel M S 2008 Phys. Rev. Lett. 100 108702
[12]
Nardini C, Kozma B and Barrat A 2008 Phys. Rev. Lett. 100 158701
[13]
Benczik S Z, Schmittmann B and Zia R K P 2008 Europhys. Lett. 82 48006
[14]
Ebel H and Bornholdt S 2002 Phys. Rev. E 66 056118
[15]
Zimmermann M G, Egu\'{\hiluz V M and Miguel M S 2004 Phys. Rev. E 69 065102(R)
[16]
Zimmermann M G and Egu\'{\hiluz V M 2005 Phys. Rev. E 72 056118
[17]
Egu{\hiluz V M, Zimmermann M G, Cela-Conde C J and Miguel M S 2005 Am. J. Sociol. 110 977
[18]
Pacheco J M, Traulsen A and Nowak M A 2006 Phys. Rev. Lett. 97 258103
[19]
Santos F C, Pacheco J M and Lenaerts T 2006 PLOS Comput. Biol. 2 E 140
[20]
Fu F, Chen X, Liu L and Wang L 2007 Physica A 383 651
[21]
Fu F, Hauert C, Nowak M A and Wang L 2008 Phys. Rev. E 78 026117
[22]
Ren J, Wu X, Wang W X, Chen G and Wang B H 2006 arXiv: physics/0605250v2
[23]
Suzuki R, Kato M and Arita T 2008 Phys. Rev. E 77 021911
[24]
Tanimoto J 2007 Phys. Rev. E 76 021126
[25]
Szolnoki S, Perc M and Danku Z 2008 Europhys. Lett. 84 50007
[26]
Szolnoki A and Perc M 2008 arXiv:0812.1122
[27]
Guan J Y, Wu Z X and Wang Y H 2007 Chin. Phys. 16 3566
[28]
Gross T, Dommar D'Lima C J and Blasius B 2006 Phys. Rev. Lett. 96 208701
[29]
Frasca M, Buscarino A, Rizzo A, Fortuna L and Boccaletti S 2006 Phys. Rev. E 74 036110
[30]
Gross T and Blasius B 2007 arXiv: 0709.1858v2
[31]
Maynard Smith J 1982 Evolution and the Theory of Games (Cambridge, UK: Cambridge University Press)
[32]
Colman A M 1995 Game Theory and its Applications in the Social and Biological Sciences (Oxford: Butterworth-Heinemann)
[33]
Hofbauer J and Sigmund K 1998 Evolutionary Games and Population Dynamics (Cambridge, UK: Cambridge University Press)
[34]
Axelrod R and Hamilton W D 1981 Science 211 1390
[35]
Axelrod R 1984 The Evolution of Cooperation (New York: Basic Books)
[36]
Nowak M and May R M 1992 Nature (London) 359 826
[ 36a]Nowak M and May R M 1993 Int. J. Bifurcation Chaos Appl. Sci. Eng. 3 35
[37]
Szab\'{o G and Toke C 1998 Phys. Rev. E 58 69
[38]
Hauert C and Doebeli M 2004 Nature 428 643
[39]
Abramson G and Kuperman M 2001 Phys. Rev. E 63 030901
[40]
Kim B J, Trusina A, Holme P, Minnhagen P, Chuang J S and Choi M Y 2002 Phys. Rev. E 66 021907
[41]
Fu F, Liu L H and Wang L 2007 Eur. Phys. J. B 56 367
[42]
Tang C L, Wang W X, Wu X and Wang B H 2006 Eur. Phys. J. B 53 411
[43]
Wu Z X, Xu X J, Chen Y and Wang Y H 2005 Phys. Rev. E 71 036107
[ 43a]{Wu Z X, Guan J Y, Xu X J and Wang Y H 2007 Physica A 379 672
[44]
Szolnoki A, Perc M and Szab\'{o G 2008 Eur. Phys. J. B 61 505
[45]
Santos F C and Pacheco J M 2005 Phys. Rev. Lett. 95 098104
[46]
Liu Y K, Li Z, Chen X J and Wang L 2009 Chin. Phys. Lett. 26 048902
[47]
Gao K, Han X P, Wang B H and Yang H X 2008 Chin. Phys. B 17 2759
[48]
Chen X J, Li Z, Li Y K and Wang L 2009 Chin. Phys. B 18 2623
[49]
Szab\'{o G and F\'{ath G 2007 Phys. Rep. 446 97
[50]
Ren J, Wang W X and Qi F 2007 Phys. Rev. E 75 045101(R)
[51]
Wu Z X, Xu X J, Huang Z G, Wang S J and Wang Y H 2006 Phys. Rev. E 74 021107
[52]
Rong Z H, Li X and Wang X F 2007 Phys. Rev. E 76 027101
[53]
Portugali J 2000 Self-Organization and the City (Berlin: Springer)
[54]
Dunbar R, Knight C and Power C (Eds.) 1999 The Evolution of Culture. An Interdisciplinary View (New Brunswick: Rutgers Univ. Press)

[1]
Zhen-Zhou Lei, Bo-Yang Liu, Ying Yi, Hong-Yi Dai, Ming Zhang. On fairness, full cooperation, and quantum game with incomplete information [J]. Chin. Phys. B, 2018, 27(3): 30202-030202.
[2]
Ma Ya-Yan, Wang Bao-Yun. Improving the secrecy rate by turning foes to allies: An auction scheme [J]. Chin. Phys. B, 2015, 24(9): 90209-090209.
[3]
Hao Dong, Rong Zhi-Hai, Zhou Tao. Zero-determinant strategy：An underway revolution in game theory [J]. Chin. Phys. B, 2014, 23(7): 78905-078905.
[4]
Song Yu-Rong, Jiang Guo-Ping, Gong Yong-Wang. Epidemic propagation on adaptive coevolutionary networks with preferential local-world reconnecting strategy [J]. Chin. Phys. B, 2013, 22(4): 40205-040205.
[5]
Hua Da-Yin. Hysteresis behavior and nonequilibrium phase transition in a one-dimensional evolutionary game model [J]. Chin. Phys. B, 2013, 22(4): 40512-040512.
[6]
Wang Xi-Peng, Jiang Luo-Luo, Wang Bing-Hong. Effects of information asymmetry on cooperation in the prisoners’ dilemma game [J]. Chin. Phys. B, 2012, 21(7): 70210-070210.
[7]
Fang Xiang-Sheng, Zhu Ping, Liu Run-Ran, Liu En-Yu, Wei Gui-Yi. A 2-stage strategy updating rule promotes cooperation in the prisoner's dilemma game [J]. Chin. Phys. B, 2012, 21(10): 108702-108702.
[8]
Zhao Lin, Zhou Xin, Liang Zhi, Wu Jia-Rui. Integrating the environmental factor into the strategy updating rule to promote cooperation in evolutionary games [J]. Chin. Phys. B, 2012, 21(1): 18701-018701.
[9]
Li Ke-Ping, Fan Hong-Qiang. Evolving laws of human cooperative behaviour [J]. Chin. Phys. B, 2010, 19(9): 90101-090101.
[10]
Liu Yong-Kui, Li Zhi, Chen Xiao-Jie, Wang Long. Payoff-based accumulative effect promotes cooperation in spatial prisoner's dilemma [J]. Chin. Phys. B, 2010, 19(9): 90203-090203.
[11]
Xu Xiu-Lian, Fu Chun-Hua, Liu Chun-Ping, He Da-Ren. Entropy description of a cooperation－competition system [J]. Chin. Phys. B, 2010, 19(6): 60501-060501R.
[12]
Guan Jian-Yue, Wu Zhi-Xi, Huang Zi-Gang, Wang Ying-Hai. Cooperation influenced by the correlation degree of two-layered complex networks in evolutionary prisoner's dilemma games [J]. Chin. Phys. B, 2010, 19(2): 20203-020203.
[13]
Du Wen-Bo, Cao Xian-Bin, Yang Han-Xin, Hu Mao-Bin. Evolutionary prisoner's dilemma on Newman--Watts socialnetworks with an asymmetric payoff distribution mechanism [J]. Chin. Phys. B, 2010, 19(1): 10204-010204.
[14]
Liu Yong-Kui, Li Zhi, Chen Xiao-Jie, Wang Long. Evolutionary prisoner's dilemma game on highly clustered community networks [J]. Chin. Phys. B, 2009, 18(7): 2623-2628.
[15]
Yang Han-Xin, Gao Kun, Han Xiao-Pu, Wang Bing-Hong. Evolutionary snowdrift game on heterogeneous Newman--Watts small-world network [J]. Chin. Phys. B, 2008, 17(8): 2759-2763.