Influence of Strategy Continuity on Cooperation in Spatial Prisoner’s Dilemma Games with Migrating Players
ZHAO Xiaowei1,2, XIA Haoxiang1†1. Institute of Systems Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China; 2. School of Software Technology, Dalian University of Technology, Dalian 116620, Liaoning, China
The phenomenon of cooperation is prevalent in both nature and human society. In this paper a simulative model is developed to examine how the strategy continuity influences cooperation in the spatial prisoner’s games in which the players migrate through the success-driven migration mechanism. Numerical simulations illustrate that the strategy continuity promotes cooperation at a low rate of migration, while impeding cooperation when the migration rate is higher. The influence of strategy continuity is also dependent on the game types. Through a more dynamic analysis, the different effects of the strategy continuity at low and high rates of migration are explained by the formation, expansion, and extinction of the self-assembled clusters of “par-tial-cooperators” within the gaming population.
 Dawkins R. The Selfish Gene[M]. New York: OUP, 1976.
 Olson M. The Logic of Collective Action: Public Goods and the Theory of Groups[M]. Cambridge: Harvard University Press, 1971.
 Pennisi E. How did cooperative behavior evolve?[J]. Science, 2005, 309(5731):93.
 Axelrod R. The evolution of cooperation[J]. Basic Books, 1984, 211(1) :227-229.
 Smith J M. Evolution and the Theory of Games[M]. Cam-bridge: Cambridge University Press, 1982.
 György S, Gábor F. Evolutionary games on graphs[J]. Physics Reports, 2007, 446(4-6):97-216.
 Bahbouhi J E, Moussa N. Prisoner’s dilemma game model for e-commerce[J]. Applied Mathematics & Computation, 2017, 292(1):128-144.
 Tanimoto J. How does resolution of strategy affect network reciprocity in spatial prisoner’s dilemma games?[J]. Applied Mathematics & Computation, 2017, 301:36-42.
 Nowak M A. Five rules for the evolution of cooperation [J]. Science, 2006, 314(5805):1560-1563.
 Ye D, Zhang M. A self-adaptive strategy for evolution of cooperation in distributed networks[J]. Computers IEEE
Transactions on, 2015, 64(4):899-911.
 Li P, Duan H. Evolution of cooperation driven by incremental learning[J]. Physica A Statistical Mechanics & Its Applications, 2015, 419:14-22.
 Chen W, Wu T, Li Z, et al. Coevolution of aspirations and cooperation in spatial prisoner’s dilemma game[J]. Journal of Statistical Mechanics Theory & Experiment, 2015, 2015(1): 1-13.
 Shen C, Lu J, Shi L. Does coevolution setup promote cooperation in spatial prisoner’s dilemma game?[J]. Applied Mathematics & Computation, 2016, 290(C): 201-207.
 Vainstein M, Silva A, Arenzon J. Does mobility decrease cooperation?[J]. Journal of Theoretical Biology, 2007, 244 (4): 722-728.
 Helbing D, Yu W. The outbreak of cooperation among success-driven individuals under noisy conditions[J]. Proceedings of the National Academy of Sciences of the United States of America, 2009, 106(10): 3680-3685.
 Meloni S, Buscarino A, Fortuna L, et al. Effects of mobility in a population of prisoner’s dilemma players[J]. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2009, 79(2): 067101.
 Jiang L L, Wang W X, Lai Y C, et al. Role of adaptive migration in promoting cooperation in spatial games[J]. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2010, 81(2): 036108.
 Yang H X, Wu Z X, Wang B H. Role of aspiration-induced migration in cooperation[J]. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2010, 81(6 Pt 2): 065101.
 Yang H X, Wang B H, et al. Universal role of migration in the evolution of cooperation[J]. Science Bulletin, 2011, 56 (34):3693-3696.
 Cong R, Wu B, Qiu Y, et al. Evolution of cooperation driven by reputation-based migration[J]. Plos One, 2012, 7(5): e35776.
 Buesser P, Tomassini M, Antonioni A. Opportunistic migration in spatial evolutionary games [J]. Physical Review E Statistical Nonlinear & Soft Matter Physics, 2013, 88(1): 042806.
 Zhang C, Zhang J, Xie G. Evolution of cooperation among game players with non-uniform migration scopes[J]. Chaos Solitons & Fractals, 2014, 59(2): 103-111.
 Antonioni A, Tomassini M, Sánchez A. Short-range mobility and the evolution of cooperation: An experimental study [J]. Scientific Reports, 2015, 5: 10282.
 Li Y, Ye H. Effect of migration based on strategy and cost on the evolution of cooperation[J]. Chaos Solitons & Fractals, 2014, 76: 156-165.
 Li Y, Ye H, Zhang H. Evolution of cooperation driven by social-welfare-based migration[J]. Physica A Statistical Mechanics & Its Applications, 2016, 445: 48-56.
 Harrald P G, Fogel D B. Evolving continuous behaviors in the Iterated Prisoner’s Dilemma[J]. Biosystems, 1996, 37 (1-2): 135-145.
 Killingback T, Doebeli M. The continuous prisoner’s dilemma and the evolution of cooperation through reciprocal altruism with variable investment[J]. American Naturalist, 2002, 160(4): 421-438.
 Tanimoto J. A study on a difference of dynamics between discrete and continuous strategies of a multi player game having linear payoff structure[J]. IPSJ Journal, 2007, 48(7): 2372-2376.
 Zhong W, Kokubo S, Tanimoto J. How is the equilibrium of continuous strategy game different from that of discrete strategy game?[J]. Biosystems, 2012, 107(2): 88-94.
 Kishimoto N, Kokubo S, Tanimoto J. Combination of continuous and binary strategies enhances network recipro- city in a spatial prisoner’s dilemma game[J]. Chaos Solitons & Fractals, 2013, 56(56): 83-90.
 Zhong W, Liu J, Zhang L. Evolutionary dynamics of continuous strategy games on graphs and social networks under weak selection [J]. Biosystems, 2013, 111(2): 102-110.
 Silva M M, Costa A P C S, Gusmão A P H D. Continuous cooperation: A proposal using a fuzzy multicriteria sorting method [J]. International Journal of Production Economics, 2014, 151(1): 67-75.