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 and 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. [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. [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. [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. [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. [6] I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230. [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. [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. [9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9781139173179. [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. [11] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982.  doi: 10.1017/CBO9780511806292. [12] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, 2010. [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. [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.

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References:
 [1] P. Bednarik, Discretized Best-Response Dynamics for Cyclic Games Diplomarbeit (Master thesis), University of Vienna, Austria, 2011. [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. [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. [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. [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. [6] I. Gilboa and A. Matsui, Social stability and equilibrium, Econometrica, 59 (1991), 859-867.  doi: 10.2307/2938230. [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. [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. [9] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9781139173179. [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. [11] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982.  doi: 10.1017/CBO9780511806292. [12] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, 2010. [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. [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.
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)
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