American Institute of Mathematical Sciences

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

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

Peter Bednarik 1,2, and  Josef Hofbauer 3,

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.

Keywords: Best response dynamics, discretization, periodic orbits, rock-paper-scissors game.
Mathematics Subject Classification: Primary: 34A36, 91A22; Secondary: 34A60, 39A28.
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:
[1]

P. Bednarik, Discretized Best-Response Dynamics for Cyclic Games Diplomarbeit (Master thesis), University of Vienna, Austria, 2011. Google Scholar

[2]

M. Benaim, J. Hofbauer and S. Sorin, Perturbations of set-valued dynamical systems, with applications to game theory, Dynamic Games and Applications, 2 (2012), 195-205.  doi: 10.1007/s13235-012-0040-0.  Google Scholar

[3]

G. W. Brown, Iterative solution of games by fictitious play, in Activity analysis of production and allocation (ed. T. C. Koopmans), Wiley, New York, (1951), 374–376.  Google Scholar

[4]

T. N. Cason, D. Friedman and E. D. Hopkins, Cycles and instability in a rock-paper-scissors population game: A continuous time experiment, The Review of Economic Studies, 81 (2014), 112-136.  doi: 10.1093/restud/rdt023.  Google Scholar

[5]

A. Gaunersdorfer and J. Hofbauer, Fictitious play, Shapley polygons, and the replicator equation, Games and Economic Behaviour, 11 (1995), 279-303.  doi: 10.1006/game.1995.1052.  Google Scholar

[6]

I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230.  Google Scholar

[7]

J. Hofbauer, Deterministic evolutionary game dynamics, in Evolutionary Game Dynamics (ed. K. Sigmund), Proceedings of Symposia in Applied Mathematics, 69, Amer. Math. Soc. (2011), 61–79. doi: 10.1090/psapm/069/2882634.  Google Scholar

[8]

J. Hofbauer and G. Iooss, A Hopf bifurcation theorem for difference equations approximating a differential equation, Monatshefte für Mathematik, 98 (1984), 99-113.  doi: 10.1007/BF01637279.  Google Scholar

[9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[10]

J. Hofbauer and S. Sorin, Best response dynamics for continuous zero-sum games, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 215-224.   Google Scholar

[11] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982.  doi: 10.1017/CBO9780511806292.  Google Scholar
[12] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, 2010.   Google Scholar
[13]

D. Semmann, H. J. Krambeck and M. Milinski, Volunteering leads to rock-paper-scissors dynamics in a public goods game, Nature, 425 (2003), 390-393.  doi: 10.1038/nature01986.  Google Scholar

[14]

Z. Wang, B. Xu and H. Zhou, Social cycling and conditional responses in the Rock-PaperScissors game, Scientific Reports, 4 (2014), 5830.  doi: 10.1038/srep05830.  Google Scholar

show all references

References:
[1]

P. Bednarik, Discretized Best-Response Dynamics for Cyclic Games Diplomarbeit (Master thesis), University of Vienna, Austria, 2011. Google Scholar

[2]

M. Benaim, J. Hofbauer and S. Sorin, Perturbations of set-valued dynamical systems, with applications to game theory, Dynamic Games and Applications, 2 (2012), 195-205.  doi: 10.1007/s13235-012-0040-0.  Google Scholar

[3]

G. W. Brown, Iterative solution of games by fictitious play, in Activity analysis of production and allocation (ed. T. C. Koopmans), Wiley, New York, (1951), 374–376.  Google Scholar

[4]

T. N. Cason, D. Friedman and E. D. Hopkins, Cycles and instability in a rock-paper-scissors population game: A continuous time experiment, The Review of Economic Studies, 81 (2014), 112-136.  doi: 10.1093/restud/rdt023.  Google Scholar

[5]

A. Gaunersdorfer and J. Hofbauer, Fictitious play, Shapley polygons, and the replicator equation, Games and Economic Behaviour, 11 (1995), 279-303.  doi: 10.1006/game.1995.1052.  Google Scholar

[6]

I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230.  Google Scholar

[7]

J. Hofbauer, Deterministic evolutionary game dynamics, in Evolutionary Game Dynamics (ed. K. Sigmund), Proceedings of Symposia in Applied Mathematics, 69, Amer. Math. Soc. (2011), 61–79. doi: 10.1090/psapm/069/2882634.  Google Scholar

[8]

J. Hofbauer and G. Iooss, A Hopf bifurcation theorem for difference equations approximating a differential equation, Monatshefte für Mathematik, 98 (1984), 99-113.  doi: 10.1007/BF01637279.  Google Scholar

[9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[10]

J. Hofbauer and S. Sorin, Best response dynamics for continuous zero-sum games, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 215-224.   Google Scholar

[11] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982.  doi: 10.1017/CBO9780511806292.  Google Scholar
[12] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, 2010.   Google Scholar
[13]

D. Semmann, H. J. Krambeck and M. Milinski, Volunteering leads to rock-paper-scissors dynamics in a public goods game, Nature, 425 (2003), 390-393.  doi: 10.1038/nature01986.  Google Scholar

[14]

Z. Wang, B. Xu and H. Zhou, Social cycling and conditional responses in the Rock-PaperScissors game, Scientific Reports, 4 (2014), 5830.  doi: 10.1038/srep05830.  Google Scholar

Figure 1.  Best response regions $R_i$ of the Rock-Paper-Scissors game separated by line segments $\ell_i$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Constructing the outer boundary for the attractor
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  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 $e$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  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)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  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)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004
[2]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176
[3]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019
[4]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072
[5]

Hongyu Cheng, Shimin Wang. Response solutions to harmonic oscillators beyond multi–dimensional brjuno frequency. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020222
[6]

He Zhang, John Harlim, Xiantao Li. Estimating linear response statistics using orthogonal polynomials: An RKHS formulation. Foundations of Data Science, 2020, 2 (4) : 443-485. doi: 10.3934/fods.2020021
[7]

Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021026
[8]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[9]

David W. K. Yeung, Yingxuan Zhang, Hongtao Bai, Sardar M. N. Islam. Collaborative environmental management for transboundary air pollution problems: A differential levies game. Journal of Industrial & Management Optimization, 2021, 17 (2) : 517-531. doi: 10.3934/jimo.2019121
[10]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051
[11]

Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021027
[12]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032
[13]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347
[14]

Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025
[15]

Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028
[16]

Zhongbao Zhou, Yanfei Bai, Helu Xiao, Xu Chen. A non-zero-sum reinsurance-investment game with delay and asymmetric information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 909-936. doi: 10.3934/jimo.2020004
[17]

Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087
[18]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[19]

Tinghua Hu, Yang Yang, Zhengchun Zhou. Golay complementary sets with large zero odd-periodic correlation zones. Advances in Mathematics of Communications, 2021, 15 (1) : 23-33. doi: 10.3934/amc.2020040
[20]

Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang. Periodic consensus in network systems with general distributed processing delays. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2021002

 Impact Factor: 

Tools

Metrics

  • PDF downloads (65)
  • HTML views (143)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences