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

Peter Bednarik; Josef Hofbauer

doi:10.3934/jdg.2017005

Article Contents

2017, Volume 4, Issue 1: 75-86. Doi: 10.3934/jdg.2017005

This issue Previous Article Control systems of interacting objects modeled as a game against nature under a mean field approach Next Article

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: March 31, 2016

Revised: November 30, 2016

Published: March 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 34A36, 91A22; Secondary: 34A60, 39A28.

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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

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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 $e$

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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)

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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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References

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