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May  2016, 21(3): 803-813. doi: 10.3934/dcdsb.2016.21.803

Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics

Jeremias Epperlein 1, , Stefan Siegmund 1, , Petr Stehlík 2, and  Vladimír  Švígler 2,

1. 

Center for Dynamics & Institute for Analysis, Dept. of Mathematics, Technische Universitat Dresden, 01062, Dresden, Germany, Germany
2. 

Dept. of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitni 8, 30614 Pilsen, Pilsen, Czech Republic, Czech Republic

Received  July 2015 Revised  November 2015 Published  January 2016

Cooperative behaviour is often accompanied by the incentives to defect, i.e., to reap the benefits of others' efforts without own contribution. We provide evidence that cooperation and defection can coexist under very broad conditions in the framework of evolutionary games on graphs under deterministic imitation dynamics. Namely, we show that for all graphs there exist coexistence equilibria for certain game-theoretical parameters. Similarly, for all relevant game-theoretical parameters there exists a graph yielding coexistence equilibria. Our proofs are constructive and robust with respect to various utility functions which can be considered. Finally, we briefly discuss bounds for the number of coexistence equilibria.
Keywords: game theory, equilibrium, Evolutionary games on graphs, cooperation., coexistence.
Mathematics Subject Classification: 05C90, 37N25, 37N40, 91A2.
Citation: Jeremias Epperlein, Stefan Siegmund, Petr Stehlík, Vladimír  Švígler. Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 803-813. doi: 10.3934/dcdsb.2016.21.803
References:
[1]

R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys., 74 (2002), 47-97. doi: 10.1103/RevModPhys.74.47.

[2]

M. Broom and J. Rychtář, An analysis of the fixation probability of a mutant on special classes of non-directed graphs, Proc. Royal. Soc., Ser. A, 464 (2008), 2609-2627. doi: 10.1098/rspa.2008.0058.

[3]

M. Broom, C. Hadjichrysanthou, J. Rychtář and B. T. Stadler, Two results on evolutionary processes on general non-directed graphs, Proc. Royal. Soc., Ser. A, 466 (2010), 2795-2798. doi: 10.1098/rspa.2010.0067.

[4]

T. Clutton-Brock, Cooperation between non-kin in animal societies, Nature, 462 (2009), 51-57. doi: 10.1038/nature08366.

[5]

J. Epperlein, S. Siegmund and P. Stehlík, Evolutionary games on graphs and discrete dynamical systems, J. Difference Eq. Appl., 21 (2015), 72-95. doi: 10.1080/10236198.2014.988618.

[6]

W. D. Hamilton, The genetical evolution of social behaviour, J. Theor. Biol., 7 (1964), 1-16. doi: 10.1016/0022-5193(64)90038-4.

[7]

C. Hauert and M. Doebeli, Spatial structure often inhibits the evolution of cooperation in the snowdrift game, Nature, 428 (2004), 643-646. doi: 10.1038/nature02360.

[8]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc., 40 (2003), 479-519. doi: 10.1090/S0273-0979-03-00988-1.

[9]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, 1988.

[10]

J. Libich and P. Stehlík, Monetary policy facing fiscal indiscipline under generalized timing of actions, Journal of Institutional and Theoretical Economics, 168 (2012), 393-431. doi: 10.1628/093245612802920962.

[11]

J. Maynard Smith, The theory of games and the evolution of animal conflicts, Journal of Theoretical Biology, 47 (1974), 209-221. doi: 10.1016/0022-5193(74)90110-6.

[12]

M. A. Nowak, Five rules for the evolution of cooperation, Science, 314 (2006), 1560-1563. doi: 10.1126/science.1133755.

[13]

M. A. Nowak and R. M. May, Evolutionary games and spatial chaos, Nature, 359 (1992), 826-829. doi: 10.1038/359826a0.

[14]

H. Ohtsuki and M. A. Nowak, Evolutionary games on cycles, Proc. R. Soc. B, 273 (2006), 2249-2256. doi: 10.1098/rspb.2006.3576.

[15]

H. Ohtsuki and M. A. Nowak, Evolutionary stability on graphs, Journal of Theoretical Biology, 251 (2008), 698-707. doi: 10.1016/j.jtbi.2008.01.005.

[16]

G. Szabó and G. Fáth, Evolutionary games on graphs, Phys. Rep., 446 (2007), 97-216. doi: 10.1016/j.physrep.2007.04.004.

show all references

References:
[1]

R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys., 74 (2002), 47-97. doi: 10.1103/RevModPhys.74.47.

[2]

M. Broom and J. Rychtář, An analysis of the fixation probability of a mutant on special classes of non-directed graphs, Proc. Royal. Soc., Ser. A, 464 (2008), 2609-2627. doi: 10.1098/rspa.2008.0058.

[3]

M. Broom, C. Hadjichrysanthou, J. Rychtář and B. T. Stadler, Two results on evolutionary processes on general non-directed graphs, Proc. Royal. Soc., Ser. A, 466 (2010), 2795-2798. doi: 10.1098/rspa.2010.0067.

[4]

T. Clutton-Brock, Cooperation between non-kin in animal societies, Nature, 462 (2009), 51-57. doi: 10.1038/nature08366.

[5]

J. Epperlein, S. Siegmund and P. Stehlík, Evolutionary games on graphs and discrete dynamical systems, J. Difference Eq. Appl., 21 (2015), 72-95. doi: 10.1080/10236198.2014.988618.

[6]

W. D. Hamilton, The genetical evolution of social behaviour, J. Theor. Biol., 7 (1964), 1-16. doi: 10.1016/0022-5193(64)90038-4.

[7]

C. Hauert and M. Doebeli, Spatial structure often inhibits the evolution of cooperation in the snowdrift game, Nature, 428 (2004), 643-646. doi: 10.1038/nature02360.

[8]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc., 40 (2003), 479-519. doi: 10.1090/S0273-0979-03-00988-1.

[9]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, 1988.

[10]

J. Libich and P. Stehlík, Monetary policy facing fiscal indiscipline under generalized timing of actions, Journal of Institutional and Theoretical Economics, 168 (2012), 393-431. doi: 10.1628/093245612802920962.

[11]

J. Maynard Smith, The theory of games and the evolution of animal conflicts, Journal of Theoretical Biology, 47 (1974), 209-221. doi: 10.1016/0022-5193(74)90110-6.

[12]

M. A. Nowak, Five rules for the evolution of cooperation, Science, 314 (2006), 1560-1563. doi: 10.1126/science.1133755.

[13]

M. A. Nowak and R. M. May, Evolutionary games and spatial chaos, Nature, 359 (1992), 826-829. doi: 10.1038/359826a0.

[14]

H. Ohtsuki and M. A. Nowak, Evolutionary games on cycles, Proc. R. Soc. B, 273 (2006), 2249-2256. doi: 10.1098/rspb.2006.3576.

[15]

H. Ohtsuki and M. A. Nowak, Evolutionary stability on graphs, Journal of Theoretical Biology, 251 (2008), 698-707. doi: 10.1016/j.jtbi.2008.01.005.

[16]

G. Szabó and G. Fáth, Evolutionary games on graphs, Phys. Rep., 446 (2007), 97-216. doi: 10.1016/j.physrep.2007.04.004.

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