American Institute of Mathematical Sciences

July 2018, 23(5): 1917-1937. doi: 10.3934/dcdsb.2018187

On arbitrarily long periodic orbits of evolutionary games on graphs

 1 Center for Dynamics & Institute for Analysis, Dept. of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany 2 Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 30614, Pilsen, Czech Republic

* Corresponding author

Received  April 2017 Revised  August 2017 Published  May 2018

A periodic behavior is a well observed phenomena in biological and economical systems. We show that evolutionary games on graphs with imitation dynamics can display periodic behavior for an arbitrary choice of game theoretical parameters describing social-dilemma games. We construct graphs and corresponding initial conditions whose trajectories are periodic with an arbitrary minimal period length. We also examine a periodic behavior of evolutionary games on graphs with the underlying graph being an acyclic (tree) graph. Astonishingly, even this acyclic structure allows for arbitrary long periodic behavior.

Citation: Jeremias Epperlein, Vladimír Švígler. On arbitrarily long periodic orbits of evolutionary games on graphs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1917-1937. doi: 10.3934/dcdsb.2018187
References:
 [1] G. Abramson and M. Kuperman, Social games in a social network, Physical Review E, 63 (2001), 030901. doi: 10.1103/PhysRevE.63.030901. [2] B. Allen and M. A. Nowak, Games on graphs, EMS Surv. Math. Sci., 1 (2014), 113-151. doi: 10.4171/EMSS/3. [3] M. Broom and J. Rychtář, Game-Theoretical Models in Biology 1st edition, CRC Press, Taylor & Francis Group, 2013. [4] J. T. Cox, R. Durrett and E. A. Perkins, Voter Model Perturbations and Reaction Diffusion Equations, vol. 349 of Astérisque, Société Mathématique de France, 2013. [5] O. Durán and R. Mulet, Evolutionary prisoner's dilemma in random graphs, Physica D: Nonlinear Phenomena, 208 (2005), 257-265. doi: 10.1016/j.physd.2005.07.005. [6] J. Epperlein, S. Siegmund and P. Stehlík, Evolutionary games on graphs and discrete dynamical systems, Journal of Difference Equations and Applications, 21 (2015), 72-95. doi: 10.1080/10236198.2014.988618. [7] J. Epperlein, S. Siegmund, P. Stehlík and V. Švígler, Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics, Discrete and Continuous Dynamical Systems- Series B, 21 (2016), 803-813. doi: 10.3934/dcdsb.2016.21.803. [8] J. Epperlein and V. Švígler, Periodic orbits of an evolutionary game on a tree, https://figshare.com/articles/6-periodic_orbit_of_an_evolutionary_game_on_a_tree/5110981, June 2017, DOI: 10.6084/m9.figshare.5110981. [9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179. [10] B. J. Kim, A. Trusina, P. Holme, P. Minnhagen, J. S. Chung and M. Y. Choi, Dynamic instabilities induced by asymmetric influence: Prisoners' dilemma game in small-world networks, Physical Review E, 66 (2002), 021907. doi: 10.1103/PhysRevE.66.021907. [11] C. Marr and M.-T. Hütt, Outer-totalistic cellular automata on graphs, Physics Letters A, 373 (2009), 546-549. doi: 10.1016/j.physleta.2008.12.013. [12] N. Masuda and K. Aihara, Spatial prisoner's dilemma optimally played in small-world networks, Physics Letters A, 313 (2003), 55-61. doi: 10.1016/S0375-9601(03)00693-5. [13] M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Belknap Press of Harvard University Press, 2006. [14] M. A. Nowak and R. M. May, Evolutionary games and spatial chaos, Nature, 359 (1992), 826-829. doi: 10.1038/359826a0. [15] M. Tomochi, Defectors' niches: Prisoner's dilemma game on disordered networks, Social Networks, 26 (2004), 309-321. doi: 10.1016/j.socnet.2004.08.003.

show all references

References:
 [1] G. Abramson and M. Kuperman, Social games in a social network, Physical Review E, 63 (2001), 030901. doi: 10.1103/PhysRevE.63.030901. [2] B. Allen and M. A. Nowak, Games on graphs, EMS Surv. Math. Sci., 1 (2014), 113-151. doi: 10.4171/EMSS/3. [3] M. Broom and J. Rychtář, Game-Theoretical Models in Biology 1st edition, CRC Press, Taylor & Francis Group, 2013. [4] J. T. Cox, R. Durrett and E. A. Perkins, Voter Model Perturbations and Reaction Diffusion Equations, vol. 349 of Astérisque, Société Mathématique de France, 2013. [5] O. Durán and R. Mulet, Evolutionary prisoner's dilemma in random graphs, Physica D: Nonlinear Phenomena, 208 (2005), 257-265. doi: 10.1016/j.physd.2005.07.005. [6] J. Epperlein, S. Siegmund and P. Stehlík, Evolutionary games on graphs and discrete dynamical systems, Journal of Difference Equations and Applications, 21 (2015), 72-95. doi: 10.1080/10236198.2014.988618. [7] J. Epperlein, S. Siegmund, P. Stehlík and V. Švígler, Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics, Discrete and Continuous Dynamical Systems- Series B, 21 (2016), 803-813. doi: 10.3934/dcdsb.2016.21.803. [8] J. Epperlein and V. Švígler, Periodic orbits of an evolutionary game on a tree, https://figshare.com/articles/6-periodic_orbit_of_an_evolutionary_game_on_a_tree/5110981, June 2017, DOI: 10.6084/m9.figshare.5110981. [9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179. [10] B. J. Kim, A. Trusina, P. Holme, P. Minnhagen, J. S. Chung and M. Y. Choi, Dynamic instabilities induced by asymmetric influence: Prisoners' dilemma game in small-world networks, Physical Review E, 66 (2002), 021907. doi: 10.1103/PhysRevE.66.021907. [11] C. Marr and M.-T. Hütt, Outer-totalistic cellular automata on graphs, Physics Letters A, 373 (2009), 546-549. doi: 10.1016/j.physleta.2008.12.013. [12] N. Masuda and K. Aihara, Spatial prisoner's dilemma optimally played in small-world networks, Physics Letters A, 313 (2003), 55-61. doi: 10.1016/S0375-9601(03)00693-5. [13] M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Belknap Press of Harvard University Press, 2006. [14] M. A. Nowak and R. M. May, Evolutionary games and spatial chaos, Nature, 359 (1992), 826-829. doi: 10.1038/359826a0. [15] M. Tomochi, Defectors' niches: Prisoner's dilemma game on disordered networks, Social Networks, 26 (2004), 309-321. doi: 10.1016/j.socnet.2004.08.003.
Regions of admissible parameters $\mathcal{P}$ with normalization $a = 1, d = 0$
Example of the graph $\mathcal{G}$ with parameters $p = 5$, $q = 3$, $r = 2$ and $s = 4$. Cooperators are depicted by full circles
Example of the graph $\mathcal{G}$ with parameters $p = 5, o = 4, s = 6, r = 1, q = 2$ with strategy vector $X(4)$. Cooperators are depicted by full circles. Note, that this graph exhibits periodic behavior as described in Section 3.2 for $(a, b, c, d) = (1, 0.45, 1.24, 0)$
Regions of parameters $o, q$ satisfying the inequalities (15)and (16). The regions are depicted for $(a, b, c, d) = (1, -0.45, 1.35, 0)$ and $p = 10$
Example of the graph constructed in the proof of Theorem 4.1 with an initial condition. The cooperators are depicted by filled black circles, defectors by white ones. The parameters are $r = 3, q = 6$
The example from Section 4.2
The example from Section 4.2
The example from Section 4.2
The example from Section 4.2
The example from Section 4.2
The example from Section 4.2
Illustration of the local situation in Lemma 4.2. Generally, nothing can be stated about the behavior of the cooperating neighbor on the left
Illustration of the local situation in Lemma 4.3
The function $f$ governing the shrinking and expansion of cooperation among the special vertices for $q = 8$
Development of the number of cooperators for the evolutionary game on the tree $\mathcal{G}$ in Section 4 with $r = 3$ and game theoretic parameters $(a, b, c, d) = (1, 0.7, 2, 0)$. On the left the tree has depth $q = 6$, on the right $q = 9$
 [1] Jeremias Epperlein, Stefan Siegmund, Petr Stehlík, Vladimír  Švígler. Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 803-813. doi: 10.3934/dcdsb.2016.21.803 [2] Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic & Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187 [3] Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981 [4] Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics & Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013 [5] William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics & Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485 [6] Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002 [7] Jeremias Epperlein, Stefan Siegmund. Phase-locked trajectories for dynamical systems on graphs. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1827-1844. doi: 10.3934/dcdsb.2013.18.1827 [8] Stefan Siegmund, Petr Stehlík. Preface: Special issue on dynamical systems on graphs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : ⅰ-ⅲ. doi: 10.3934/dcdsb.201805i [9] Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91 [10] Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33 [11] W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351 [12] Andrzej Swierniak, Michal Krzeslak. Application of evolutionary games to modeling carcinogenesis. Mathematical Biosciences & Engineering, 2013, 10 (3) : 873-911. doi: 10.3934/mbe.2013.10.873 [13] P.E. Kloeden, Desheng Li, Chengkui Zhong. Uniform attractors of periodic and asymptotically periodic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 213-232. doi: 10.3934/dcds.2005.12.213 [14] Yves Achdou, Victor Perez. Iterative strategies for solving linearized discrete mean field games systems. Networks & Heterogeneous Media, 2012, 7 (2) : 197-217. doi: 10.3934/nhm.2012.7.197 [15] B. Coll, A. Gasull, R. Prohens. On a criterium of global attraction for discrete dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (3) : 537-550. doi: 10.3934/cpaa.2006.5.537 [16] Jean-Luc Chabert, Ai-Hua Fan, Youssef Fares. Minimal dynamical systems on a discrete valuation domain. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 777-795. doi: 10.3934/dcds.2009.25.777 [17] Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187 [18] Mostafa Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont, Olivier Goubet. Discrete Schrödinger equations and dissipative dynamical systems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 211-227. doi: 10.3934/cpaa.2008.7.211 [19] Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential trichotomy of dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2929-2962. doi: 10.3934/dcds.2014.34.2929 [20] Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039

2016 Impact Factor: 0.994