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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,
International Institute for Applied Systems Analysis (IIASA), Schlossplatz 1, A-2361 Laxenburg, Austria
Department of Economics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
Received April 2016 Revised December 2016 Published December 2016
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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
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Figure 1. Best response regions $R_i$ of the Rock-Paper-Scissors game separated by line segments $\ell_i$
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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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