American Institute of Mathematical Sciences

January  2017, 4(1): 75-86. doi: 10.3934/jdg.2017005

Discretized best-response dynamics for the Rock-Paper-Scissors game

 1 International Institute for Applied Systems Analysis (IIASA), Schlossplatz 1, A-2361 Laxenburg, Austria 2 Department of Economics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria 3 Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

Received  April 2016 Revised  December 2016 Published  December 2016

Discretizing a differential equation may change the qualitative behaviour drastically, even if the stepsize is small. We illustrate this by looking at the discretization of a piecewise continuous differential equation that models a population of agents playing the Rock-Paper-Scissors game. The globally asymptotically stable equilibrium of the differential equation turns, after discretization, into a repeller surrounded by an annulus shaped attracting region. In this region, more and more periodic orbits emerge as the discretization step approaches zero.

Citation: Peter Bednarik, Josef Hofbauer. Discretized best-response dynamics for the Rock-Paper-Scissors game. Journal of Dynamics & Games, 2017, 4 (1) : 75-86. doi: 10.3934/jdg.2017005
References:

show all references

References:
Best response regions $R_i$ of the Rock-Paper-Scissors game separated by line segments $\ell_i$
Constructing the outer boundary for the attractor
The outer triangle $\Delta_q$ is constructed such that the $\omega$-limits of all orbits must be inside of it. The inner triangle $\Delta_p$ contains the set of points which do not have a pre-image under $F$. Thus, the region bounded by the two green triangles attracts all orbits, except the constant one at $e$
Periodic orbits of periods $3n$, which exist for $h<h_n$, are shown for $n \leq 5$. The red curves correspond to the inner and outer demarkation of the attractor calculated in section 2 and $h_k$ are numerical solutions to the equation corresponding to (26)
Periodic orbits of various periods together with their (numerically calculated) respective basins of attraction, for various values of the stepsize $h$. Red is the basin of attraction for period 3, dark red for period 6, light green: 9, green: 12, yellow 15, olive 18 and blue 21. The inner and outer triangles $\Delta_p$ and $\Delta_q$ are also shown (gray lines)
 [1] Włodzimierz Bąk, Tadeusz Nadzieja, Mateusz Wróbel. Models of the population playing the rock-paper-scissors game. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 1-11. doi: 10.3934/dcdsb.2018001 [2] Gunter Neumann, Stefan Schuster. Modeling the rock - scissors - paper game between bacteriocin producing bacteria by Lotka-Volterra equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 207-228. doi: 10.3934/dcdsb.2007.8.207 [3] Josef Hofbauer, Sylvain Sorin. Best response dynamics for continuous zero--sum games. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 215-224. doi: 10.3934/dcdsb.2006.6.215 [4] Luca Dieci, Timo Eirola, Cinzia Elia. Periodic orbits of planar discontinuous system under discretization. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2743-2762. doi: 10.3934/dcdsb.2018103 [5] Viktor L. Ginzburg, Başak Z. Gürel. On the generic existence of periodic orbits in Hamiltonian dynamics. Journal of Modern Dynamics, 2009, 3 (4) : 595-610. doi: 10.3934/jmd.2009.3.595 [6] Marian Gidea, Yitzchak Shmalo. Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6123-6148. doi: 10.3934/dcds.2018264 [7] Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537 [8] Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177 [9] Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505 [10] Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949 [11] Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451 [12] Yingxiang Xu, Yongkui Zou. Preservation of homoclinic orbits under discretization of delay differential equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 275-299. doi: 10.3934/dcds.2011.31.275 [13] Fernando Jiménez, Jürgen Scheurle. On the discretization of nonholonomic dynamics in $\mathbb{R}^n$. Journal of Geometric Mechanics, 2015, 7 (1) : 43-80. doi: 10.3934/jgm.2015.7.43 [14] Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331 [15] Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005 [16] Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 [17] Haitao Song, Weihua Jiang, Shengqiang Liu. Virus dynamics model with intracellular delays and immune response. Mathematical Biosciences & Engineering, 2015, 12 (1) : 185-208. doi: 10.3934/mbe.2015.12.185 [18] Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109 [19] Jorge Rebaza. Bifurcations and periodic orbits in variable population interactions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2997-3012. doi: 10.3934/cpaa.2013.12.2997 [20] Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057

Impact Factor: