# American Institute of Mathematical Sciences

2013, 10(5&6): 1381-1398. doi: 10.3934/mbe.2013.10.1381

## Prisoner's Dilemma on real social networks: Revisited

 1 East Tennessee State University, Department of Mathematics and Statistics, Box 70663, Johnson City, TN 37614-1701, United States 2 East Tennessee State University, Department of Mathematics and Statistics, Institute for Quantitative Biology, Box 70663, Johnson City, TN 37614-1701, United States

Received  August 2012 Revised  October 2012 Published  August 2013

Prisoner's Dilemma is a game theory model used to describe altruistic behavior seen in various populations. This theoretical game is important in understanding why a seemingly selfish strategy does persist and spread throughout a population that is mixing homogeneously at random. For a population with structure determined by social interactions, Prisoner's Dilemma brings to light certain requirements for the altruistic strategy to become established. Monte Carlo simulations of Prisoner's Dilemma are carried out using both simulated social networks and a dataset of a real social network. In both scenarios we confirm the requirements for the persistence of altruism in a population.
Citation: Sharon M. Cameron, Ariel Cintrón-Arias. Prisoner's Dilemma on real social networks: Revisited. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1381-1398. doi: 10.3934/mbe.2013.10.1381
##### References:
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Zhang, Emergence of synchronization induced by the interplay between two prisoner's dilemma games with volunteering in small-world networks,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.032103. [8] X. J. Chen and L. Wang, Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.017103. [9] F. Chen, A mathematical analysis of public avoidance behavior during epidemics using game theory,, J. Theor. Biol., 302 (2012), 18. doi: 10.1016/j.jtbi.2012.03.002. [10] X. H. Deng, Y. Liu and Z. G. Chen, Memory-based evolutionary game on small-world network with tunable heterogeneity,, Physica A, 389 (2010), 5173. doi: 10.1016/j.physa.2010.08.004. [11] L. R. Dong, Dynamic evolution with limited learning information on a small-world network,, Commun. Theor. Phys., 54 (2010), 578. [12] W. B. Du, X. B. Cao, L. Zhao and H. Zhou, Evolutionary games on weighted Newman-Watts small-world networks,, Chinese Phys. Lett., 26 (2009). [13] R. Durrett, "Random Graph Dynamics,", Cambridge University Press, (2007). [14] D. Easley and J. Kleinberg, " Networks, Crowds, and Markets: Reasoning about a Highly Connected World,", Cambridge University Press, (2010). [15] V. M. Eguiluz, M. G. Zimmermann, C. J. Cela-Conde and M. S. Miguel, Cooperation and the emergence of role differentiation in the dynamics of social networks,, Am. J. Sociol., 110 (2005), 977. doi: 10.1086/428716. [16] F. Fu, L. H. Liu and L. Wang, Evolutionary prisoner's dilemma on heterogeneous Newman-Watts small-world network,, Eur. Phys. J. B, 56 (2007), 367. doi: 10.1140/epjb/e2007-00124-5. [17] M. Granovetter, The strength of weak ties,, Am. J. Sociol., 78 (1973), 1360. [18] J. Y. Guan, Z. X. Wu, Z. G. Huang and Y. H. Wang, Prisoner's dilemma game with nonlinear attractive effect on regular small-world networks,, Chinese Phys. Lett., 23 (2006), 2874. 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A, 313 (2003), 55. doi: 10.1016/S0375-9601(03)00693-5. [25] A. Mayer and S. L. Puller, The old boy (and girl) network: Social network formation on university campuses,, J. Public Econom., 92 (2008), 328. doi: 10.1016/j.jpubeco.2007.09.001. [26] J. Maynard-Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982). doi: 10.1016/0022-5193(74)90110-6. [27] S. Milgram, The small-world problem,, Psychol. Today, 2 (1967), 60. doi: 10.1037/e400002009-005. [28] J. F. Nash, Equilibrium points in $n$-person games,, Proc. Natl. Acad. Sci. USA, 36 (1950), 48. doi: 10.1073/pnas.36.1.48. [29] M. E. J. Newman, "Networks: An Introduction,", Oxford University Press, (2010). doi: 10.1093/acprof:oso/9780199206650.001.0001. [30] M. A. Nowak and R. M. May, Evolutionary games and spatial chaos,, Nature, 359 (1992), 826. doi: 10.1038/359826a0. [31] M. A. Nowak, "Evolutionary Dynamics: Exploring the Equations of Life,", Harvard University Press, (2006). [32] M. A. Nowak, "Super Cooperators: Altruism, Evolution, and Why We Need Each Other to Succeed,", Free Press, (2012). [33] H. Ohtsuki, C. Hauert, E. Lieberman and M. A. Nowak, A simple rule for the evolution of cooperation on graphs and social networks,, Nature, 441 (2006), 502. doi: 10.1038/nature04605. [34] M. Perc, Double resonance in cooperation induced by noise and network variation for an evolutionary prisoner's dilemma,, New J. Phys., 8 (2006). doi: 10.1088/1367-2630/8/9/183. [35] E. Shim, L. A. Meyers and A. P. Galvani, Optimal H1N1 vaccination strategies based on self-interest versus group interest,, BMC Public Health, 11 (2011). [36] G. Szabo and J. Vukov, Cooperation for volunteering and partially random partnerships,, Phys. Rev. E, 69 (2004). doi: 10.1103/PhysRevE.69.036107. [37] C. L. Tang, W. X. Wang, X. Wu and B. H. Wang, Effects of average degree on cooperation in networked evolutionary game,, Eur. Phys. J. B, 53 (2006), 411. doi: 10.1140/epjb/e2006-00395-2. [38] M. Tomochi, Defectors' niches: Prisoner's dilemma game on disordered networks,, Soc. Networks, 26 (2004), 309. doi: 10.1016/j.socnet.2004.08.003. [39] X. Thibert-Plante and L. Parrott, Prisoner's dilemma and clusters on small-world networks,, Complexity, 12 (2007), 22. doi: 10.1002/cplx.20182. [40] A. L. Traud, E. D. Kelsic, P. J. Mucha and M. A. Porter, Comparing community structure to characteristics in online collegiate social networks,, SIAM Rev., 53 (2011), 526. doi: 10.1137/080734315. [41] A. L. Traud, P. J. Mucha and M. A. Porter, Social structure of Facebook networks,, Physica A, 391 (2012), 4165. [42] J. von Neumann and O. Morgenstern, "Theory of Games and Economic Behavior,", Princeton University Press, (1944). [43] D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440. [44] D. J. Watts, "Small Worlds: The Dynamics of Networks, Between Order and Randomness,", Princeton University Press, (1999). [45] G. S. Wilkinson, Reciprocal food sharing in the vampire bat,, Nature, 308 (1984), 181. [46] Z. X. Wu, X. J. Xu, Y. Chen and Y. H. Wang, Spatial prisoner's dilemma game with volunteering in Newman-Watts small-world networks,, Phys. Rev. E, 71 (2005). doi: 10.1103/PhysRevE.71.037103. [47] Z. X. Wu, X. J. Xu and Y. H. Wang, Prisoner's dilemma game with heterogeneous influential effect on regular small-world networks,, Chinese Phys. Lett., 23 (2006), 531. [48] Q. Z. Xia, X. H. Liao, W. Li and G. Hu, Enhance cooperation by catastrophic collapses of rich cooperators in coevolutionary networks,, EPL-Europhys. Lett., 92 (2010). doi: 10.1209/0295-5075/92/40009. [49] L. X. Zhong, D. F. Zheng, B. Zheng, C. Xu and P. M. Hui, Networking effects on cooperation in evolutionary snowdrift game,, Europhys. Lett., 76 (2006), 724. doi: 10.1209/epl/i2006-10323-2.

show all references

##### References:
 [1] G. Abramson and M. Kuperman, Social games in a social network,, Phys. Rev. E, 63 (2001). doi: 10.1103/PhysRevE.63.030901. [2] E. Ahmed and A.S. Elgazzar, On local prisoner's dilemma game with Pareto updating rule,, Int. J. Mod. Phys. C, 11 (2000), 1539. doi: 10.1142/S0129183100001334. [3] R. Axelrod, "The Evolution of Cooperation,", Revised Edition, (2006). doi: 10.1126/science.7466396. [4] C. T. Bauch, A. P. Galvani and D. J. Earn, Group interest versus self-interest in smallpox vaccination policy,, P. Natl. Acad. Sci. USA, 100 (2003), 10564. doi: 10.1073/pnas.1731324100. [5] D. M. Boyd and N. B. Ellison, Social network sites: Definition, history, and scholarship,, J. Computer-Mediated Comm., 13 (2008), 210. doi: 10.1109/EMR.2010.5559139. [6] A. Cassar, Coordination and cooperation in local, random and small world networks: Experimental evidence,, Game Econ. Behav., 58 (2007), 209. doi: 10.1016/j.geb.2006.03.008. [7] Y. Chen, S. M. Qin, L. C. Yu and S. L. Zhang, Emergence of synchronization induced by the interplay between two prisoner's dilemma games with volunteering in small-world networks,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.032103. [8] X. J. Chen and L. Wang, Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game,, Phys. Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.017103. [9] F. Chen, A mathematical analysis of public avoidance behavior during epidemics using game theory,, J. Theor. Biol., 302 (2012), 18. doi: 10.1016/j.jtbi.2012.03.002. [10] X. H. Deng, Y. Liu and Z. G. Chen, Memory-based evolutionary game on small-world network with tunable heterogeneity,, Physica A, 389 (2010), 5173. doi: 10.1016/j.physa.2010.08.004. [11] L. R. Dong, Dynamic evolution with limited learning information on a small-world network,, Commun. Theor. Phys., 54 (2010), 578. [12] W. B. Du, X. B. Cao, L. Zhao and H. Zhou, Evolutionary games on weighted Newman-Watts small-world networks,, Chinese Phys. Lett., 26 (2009). [13] R. Durrett, "Random Graph Dynamics,", Cambridge University Press, (2007). [14] D. Easley and J. Kleinberg, " Networks, Crowds, and Markets: Reasoning about a Highly Connected World,", Cambridge University Press, (2010). [15] V. M. Eguiluz, M. G. Zimmermann, C. J. Cela-Conde and M. S. Miguel, Cooperation and the emergence of role differentiation in the dynamics of social networks,, Am. J. Sociol., 110 (2005), 977. doi: 10.1086/428716. [16] F. Fu, L. H. Liu and L. Wang, Evolutionary prisoner's dilemma on heterogeneous Newman-Watts small-world network,, Eur. Phys. J. B, 56 (2007), 367. doi: 10.1140/epjb/e2007-00124-5. [17] M. Granovetter, The strength of weak ties,, Am. J. Sociol., 78 (1973), 1360. [18] J. Y. Guan, Z. X. Wu, Z. G. Huang and Y. H. Wang, Prisoner's dilemma game with nonlinear attractive effect on regular small-world networks,, Chinese Phys. Lett., 23 (2006), 2874. [19] C. Hauert and G. Szabo, Game theory and physics,, Am. J. Phys., 73 (2005), 405. doi: 10.1119/1.1848514. [20] C. Hauert and L. A. Imhof, Evolutionary games in deme structured, finite populations,, J. Theor. Biol., 299 (2012), 106. doi: 10.1016/j.jtbi.2011.06.010. [21] B. J. Kim, A. Trusina, P. Holme, P. Minnhagen, J. S. Chung and M. Y. Choi, Dynamic instabilities induced by asymmetric influence: Prisoner's dilemma game in small-world networks,, Phys. Rev. E, 66 (2002). [22] K. Lewis, J. Kaufman, M. Gonzalez, M. Wimmer and N. A. Christakis, Tastes, ties, and time: A new (cultural, multiplex, and longitudinal) social network dataset using Facebook.com,, Social Networks, 30 (2008), 330. [23] T. Lenaerts, J. M. Pacheco and F. C. Santos, Evolutionary dynamics of social dilemmas in structured heterogenous populations,, P. Natl. Acad. Sci. USA, 109 (2006), 3490. [24] N. Masuda and K. Aihara, Spatial prisoner's dilemma optimally played in small-world networks,, Phys. Lett. A, 313 (2003), 55. doi: 10.1016/S0375-9601(03)00693-5. [25] A. Mayer and S. L. Puller, The old boy (and girl) network: Social network formation on university campuses,, J. Public Econom., 92 (2008), 328. doi: 10.1016/j.jpubeco.2007.09.001. [26] J. Maynard-Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982). doi: 10.1016/0022-5193(74)90110-6. [27] S. Milgram, The small-world problem,, Psychol. Today, 2 (1967), 60. doi: 10.1037/e400002009-005. [28] J. F. Nash, Equilibrium points in $n$-person games,, Proc. Natl. Acad. Sci. USA, 36 (1950), 48. doi: 10.1073/pnas.36.1.48. [29] M. E. J. Newman, "Networks: An Introduction,", Oxford University Press, (2010). doi: 10.1093/acprof:oso/9780199206650.001.0001. [30] M. A. Nowak and R. M. May, Evolutionary games and spatial chaos,, Nature, 359 (1992), 826. doi: 10.1038/359826a0. [31] M. A. Nowak, "Evolutionary Dynamics: Exploring the Equations of Life,", Harvard University Press, (2006). [32] M. A. Nowak, "Super Cooperators: Altruism, Evolution, and Why We Need Each Other to Succeed,", Free Press, (2012). [33] H. Ohtsuki, C. Hauert, E. Lieberman and M. A. Nowak, A simple rule for the evolution of cooperation on graphs and social networks,, Nature, 441 (2006), 502. doi: 10.1038/nature04605. [34] M. Perc, Double resonance in cooperation induced by noise and network variation for an evolutionary prisoner's dilemma,, New J. Phys., 8 (2006). doi: 10.1088/1367-2630/8/9/183. [35] E. Shim, L. A. Meyers and A. P. Galvani, Optimal H1N1 vaccination strategies based on self-interest versus group interest,, BMC Public Health, 11 (2011). [36] G. Szabo and J. Vukov, Cooperation for volunteering and partially random partnerships,, Phys. Rev. E, 69 (2004). doi: 10.1103/PhysRevE.69.036107. [37] C. L. Tang, W. X. Wang, X. Wu and B. H. Wang, Effects of average degree on cooperation in networked evolutionary game,, Eur. Phys. J. B, 53 (2006), 411. doi: 10.1140/epjb/e2006-00395-2. [38] M. Tomochi, Defectors' niches: Prisoner's dilemma game on disordered networks,, Soc. Networks, 26 (2004), 309. doi: 10.1016/j.socnet.2004.08.003. [39] X. Thibert-Plante and L. Parrott, Prisoner's dilemma and clusters on small-world networks,, Complexity, 12 (2007), 22. doi: 10.1002/cplx.20182. [40] A. L. Traud, E. D. Kelsic, P. J. Mucha and M. A. Porter, Comparing community structure to characteristics in online collegiate social networks,, SIAM Rev., 53 (2011), 526. doi: 10.1137/080734315. [41] A. L. Traud, P. J. Mucha and M. A. Porter, Social structure of Facebook networks,, Physica A, 391 (2012), 4165. [42] J. von Neumann and O. Morgenstern, "Theory of Games and Economic Behavior,", Princeton University Press, (1944). [43] D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440. [44] D. J. Watts, "Small Worlds: The Dynamics of Networks, Between Order and Randomness,", Princeton University Press, (1999). [45] G. S. Wilkinson, Reciprocal food sharing in the vampire bat,, Nature, 308 (1984), 181. [46] Z. X. Wu, X. J. Xu, Y. Chen and Y. H. Wang, Spatial prisoner's dilemma game with volunteering in Newman-Watts small-world networks,, Phys. Rev. E, 71 (2005). doi: 10.1103/PhysRevE.71.037103. [47] Z. X. Wu, X. J. Xu and Y. H. Wang, Prisoner's dilemma game with heterogeneous influential effect on regular small-world networks,, Chinese Phys. Lett., 23 (2006), 531. [48] Q. Z. Xia, X. H. Liao, W. Li and G. Hu, Enhance cooperation by catastrophic collapses of rich cooperators in coevolutionary networks,, EPL-Europhys. Lett., 92 (2010). doi: 10.1209/0295-5075/92/40009. [49] L. X. Zhong, D. F. Zheng, B. Zheng, C. Xu and P. M. Hui, Networking effects on cooperation in evolutionary snowdrift game,, Europhys. Lett., 76 (2006), 724. doi: 10.1209/epl/i2006-10323-2.
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