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:
[1]

G. Abramson and M. Kuperman, Social games in a social network,, Phys. Rev. E, 63 (2001).  doi: 10.1103/PhysRevE.63.030901.  Google Scholar

[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.  Google Scholar

[3]

R. Axelrod, "The Evolution of Cooperation,", Revised Edition, (2006).  doi: 10.1126/science.7466396.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. R. Dong, Dynamic evolution with limited learning information on a small-world network,, Commun. Theor. Phys., 54 (2010), 578.   Google Scholar

[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).   Google Scholar

[13]

R. Durrett, "Random Graph Dynamics,", Cambridge University Press, (2007).   Google Scholar

[14]

D. Easley and J. Kleinberg, " Networks, Crowds, and Markets: Reasoning about a Highly Connected World,", Cambridge University Press, (2010).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. Granovetter, The strength of weak ties,, Am. J. Sociol., 78 (1973), 1360.   Google Scholar

[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.   Google Scholar

[19]

C. Hauert and G. Szabo, Game theory and physics,, Am. J. Phys., 73 (2005), 405.  doi: 10.1119/1.1848514.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Maynard-Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982).  doi: 10.1016/0022-5193(74)90110-6.  Google Scholar

[27]

S. Milgram, The small-world problem,, Psychol. Today, 2 (1967), 60.  doi: 10.1037/e400002009-005.  Google Scholar

[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.  Google Scholar

[29]

M. E. J. Newman, "Networks: An Introduction,", Oxford University Press, (2010).  doi: 10.1093/acprof:oso/9780199206650.001.0001.  Google Scholar

[30]

M. A. Nowak and R. M. May, Evolutionary games and spatial chaos,, Nature, 359 (1992), 826.  doi: 10.1038/359826a0.  Google Scholar

[31]

M. A. Nowak, "Evolutionary Dynamics: Exploring the Equations of Life,", Harvard University Press, (2006).   Google Scholar

[32]

M. A. Nowak, "Super Cooperators: Altruism, Evolution, and Why We Need Each Other to Succeed,", Free Press, (2012).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[36]

G. Szabo and J. Vukov, Cooperation for volunteering and partially random partnerships,, Phys. Rev. E, 69 (2004).  doi: 10.1103/PhysRevE.69.036107.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

A. L. Traud, P. J. Mucha and M. A. Porter, Social structure of Facebook networks,, Physica A, 391 (2012), 4165.   Google Scholar

[42]

J. von Neumann and O. Morgenstern, "Theory of Games and Economic Behavior,", Princeton University Press, (1944).   Google Scholar

[43]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440.   Google Scholar

[44]

D. J. Watts, "Small Worlds: The Dynamics of Networks, Between Order and Randomness,", Princeton University Press, (1999).   Google Scholar

[45]

G. S. Wilkinson, Reciprocal food sharing in the vampire bat,, Nature, 308 (1984), 181.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

R. Axelrod, "The Evolution of Cooperation,", Revised Edition, (2006).  doi: 10.1126/science.7466396.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. R. Dong, Dynamic evolution with limited learning information on a small-world network,, Commun. Theor. Phys., 54 (2010), 578.   Google Scholar

[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).   Google Scholar

[13]

R. Durrett, "Random Graph Dynamics,", Cambridge University Press, (2007).   Google Scholar

[14]

D. Easley and J. Kleinberg, " Networks, Crowds, and Markets: Reasoning about a Highly Connected World,", Cambridge University Press, (2010).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

M. Granovetter, The strength of weak ties,, Am. J. Sociol., 78 (1973), 1360.   Google Scholar

[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.   Google Scholar

[19]

C. Hauert and G. Szabo, Game theory and physics,, Am. J. Phys., 73 (2005), 405.  doi: 10.1119/1.1848514.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Maynard-Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982).  doi: 10.1016/0022-5193(74)90110-6.  Google Scholar

[27]

S. Milgram, The small-world problem,, Psychol. Today, 2 (1967), 60.  doi: 10.1037/e400002009-005.  Google Scholar

[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.  Google Scholar

[29]

M. E. J. Newman, "Networks: An Introduction,", Oxford University Press, (2010).  doi: 10.1093/acprof:oso/9780199206650.001.0001.  Google Scholar

[30]

M. A. Nowak and R. M. May, Evolutionary games and spatial chaos,, Nature, 359 (1992), 826.  doi: 10.1038/359826a0.  Google Scholar

[31]

M. A. Nowak, "Evolutionary Dynamics: Exploring the Equations of Life,", Harvard University Press, (2006).   Google Scholar

[32]

M. A. Nowak, "Super Cooperators: Altruism, Evolution, and Why We Need Each Other to Succeed,", Free Press, (2012).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[36]

G. Szabo and J. Vukov, Cooperation for volunteering and partially random partnerships,, Phys. Rev. E, 69 (2004).  doi: 10.1103/PhysRevE.69.036107.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

A. L. Traud, P. J. Mucha and M. A. Porter, Social structure of Facebook networks,, Physica A, 391 (2012), 4165.   Google Scholar

[42]

J. von Neumann and O. Morgenstern, "Theory of Games and Economic Behavior,", Princeton University Press, (1944).   Google Scholar

[43]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks,, Nature, 393 (1998), 440.   Google Scholar

[44]

D. J. Watts, "Small Worlds: The Dynamics of Networks, Between Order and Randomness,", Princeton University Press, (1999).   Google Scholar

[45]

G. S. Wilkinson, Reciprocal food sharing in the vampire bat,, Nature, 308 (1984), 181.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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