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:
[1]

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

[2]

M. BenaimJ. 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. CasonD. 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. SemmannH. 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. WangB. Xu and H. Zhou, Social cycling and conditional responses in the Rock-PaperScissors game, Scientific Reports, 4 (2014), 5830. doi: 10.1038/srep05830.

show all references

References:
[1]

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

[2]

M. BenaimJ. 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. CasonD. 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. SemmannH. 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. WangB. Xu and H. Zhou, Social cycling and conditional responses in the Rock-PaperScissors game, Scientific Reports, 4 (2014), 5830. doi: 10.1038/srep05830.

Figure 1.  Best response regions $R_i$ of the Rock-Paper-Scissors game separated by line segments $\ell_i$
Figure 2.  Constructing the outer boundary for the attractor
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 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 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)
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