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Control systems of interacting objects modeled as a game against nature under a mean field approach
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 |
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.
References:
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P. Bednarik, Discretized Best-Response Dynamics for Cyclic Games Diplomarbeit (Master thesis), University of Vienna, Austria, 2011. Google Scholar |
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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. |
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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. |
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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. |
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A. Gaunersdorfer and J. Hofbauer,
Fictitious play, Shapley polygons, and the replicator equation, Games and Economic Behaviour, 11 (1995), 279-303.
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Social stability and equilibrium, Econometrica, 59 (1991), 859-867.
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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. |
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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. |
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J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.
doi: 10.1017/CBO9781139173179.![]() ![]() |
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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.![]() |
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W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, 2010.
![]() |
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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. |
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. |
[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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