Skip to main content
main-content
Top

Hint

Swipe to navigate through the articles of this issue

Published in: Dynamic Games and Applications 3/2021

02-11-2020

Periodic Attractor in the Discrete Time Best-Response Dynamics of the Rock-Paper-Scissors Game

Authors: José Pedro Gaivão, Telmo Peixe

Published in: Dynamic Games and Applications | Issue 3/2021

Login to get access
share
SHARE

Abstract

The Rock-Paper-Scissors (RPS) game is a classic non-cooperative game widely studied in terms of its theoretical analysis as well as in its applications, ranging from sociology and biology to economics. In this work, we show that the attractor of the discrete time best-response dynamics of the RPS game is a finite union of periodic orbits. Moreover, we also describe the bifurcations of the attractor and determine the exact number, period and location of the periodic orbits.

To get access to this content you need the following product:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 69.000 Bücher
  • über 500 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Testen Sie jetzt 15 Tage kostenlos.

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 50.000 Bücher
  • über 380 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




Testen Sie jetzt 15 Tage kostenlos.

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 58.000 Bücher
  • über 300 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Testen Sie jetzt 15 Tage kostenlos.

Appendix
Available only for authorised users
Footnotes
1
In fact, the smallest period is \(3\left( \left\lfloor \frac{\log (a/b)}{\log (1-\varepsilon )}\right\rfloor +1\right) \sim \frac{3\log (b/a)}{\varepsilon }\) as \(\varepsilon \rightarrow 0\).
 
2
The inverse of the supergolden ratio.
 
3
For every \(x\in {\mathbb {R}}\), \(x\le \lfloor x\rfloor <x+1\) and \(x-1<\lceil x \rceil \le x\).
 
Literature
1.
go back to reference Andrea G, Josef H (1995) Fictitious play, shapley polygons, and the replicator equation. Games Econ Behav 11(2):279–303 MathSciNetCrossRef Andrea G, Josef H (1995) Fictitious play, shapley polygons, and the replicator equation. Games Econ Behav 11(2):279–303 MathSciNetCrossRef
2.
go back to reference Bednarik P, Hofbauer J (2017) Discretized best-response dynamics for the rock-paper-scissors game. J Dyn Games 4(1):75–86 MathSciNetCrossRef Bednarik P, Hofbauer J (2017) Discretized best-response dynamics for the rock-paper-scissors game. J Dyn Games 4(1):75–86 MathSciNetCrossRef
3.
go back to reference Benaïm M, Hofbauer J, Sorin S (2012) Perturbations of set-valued dynamical systems, with applications to game theory. Dyn Games Appl 2(2):195–205 MathSciNetCrossRef Benaïm M, Hofbauer J, Sorin S (2012) Perturbations of set-valued dynamical systems, with applications to game theory. Dyn Games Appl 2(2):195–205 MathSciNetCrossRef
4.
go back to reference Brown GW (1951) Iterative solution of games by fictitious play. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, New York, pp 374–376 Brown GW (1951) Iterative solution of games by fictitious play. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, New York, pp 374–376
5.
go back to reference Bruin H, Deane Jonathan HB (2009) Piecewise contractions are asymptotically periodic. Proc Am Math Soc 137(4):1389–1395 MathSciNetCrossRef Bruin H, Deane Jonathan HB (2009) Piecewise contractions are asymptotically periodic. Proc Am Math Soc 137(4):1389–1395 MathSciNetCrossRef
6.
go back to reference Cason TN, Friedman D, Hopkins E (2013) Cycles and instability in a rock-paper-scissors population game: a continuous time experiment. Rev Econ Stud 81(1):112–136 MathSciNetCrossRef Cason TN, Friedman D, Hopkins E (2013) Cycles and instability in a rock-paper-scissors population game: a continuous time experiment. Rev Econ Stud 81(1):112–136 MathSciNetCrossRef
7.
go back to reference Catsigeras E, Guiraud P, Meyroneinc A, Ugalde E (2016) On the asymptotic properties of piecewise contracting maps. Dyn Syst 31(2):107–135 MathSciNetCrossRef Catsigeras E, Guiraud P, Meyroneinc A, Ugalde E (2016) On the asymptotic properties of piecewise contracting maps. Dyn Syst 31(2):107–135 MathSciNetCrossRef
8.
9.
go back to reference Gaivão JP (2018) Asymptotic periodicity in outer billiards with contraction. Ergodic Theory Dyn Syst 1–16 Gaivão JP (2018) Asymptotic periodicity in outer billiards with contraction. Ergodic Theory Dyn Syst 1–16
10.
11.
12.
go back to reference Hopkins E, Seymour RM (2002) The stability of price dispersion under seller and consumer learning. Int Econ Rev 43(4):1157–1190 MathSciNetCrossRef Hopkins E, Seymour RM (2002) The stability of price dispersion under seller and consumer learning. Int Econ Rev 43(4):1157–1190 MathSciNetCrossRef
15.
go back to reference Maskin E, Tirole J (1988) A theory of dynamic oligopoly, II: price competition, kinked demand curves, and edgeworth cycles. Econometrica 56(3):571–599 MathSciNetCrossRef Maskin E, Tirole J (1988) A theory of dynamic oligopoly, II: price competition, kinked demand curves, and edgeworth cycles. Econometrica 56(3):571–599 MathSciNetCrossRef
17.
18.
go back to reference Noel Michael D (2007) Edgeworth price cycles: evidence from the Toronto retail gasoline market. J Ind Econ 55(1):69–92 CrossRef Noel Michael D (2007) Edgeworth price cycles: evidence from the Toronto retail gasoline market. J Ind Econ 55(1):69–92 CrossRef
19.
go back to reference Nogueira A, Pires B (2015) Dynamics of piecewise contractions of the interval. Ergodic Theory Dyn Syst 35(7):2198–2215 MathSciNetCrossRef Nogueira A, Pires B (2015) Dynamics of piecewise contractions of the interval. Ergodic Theory Dyn Syst 35(7):2198–2215 MathSciNetCrossRef
20.
go back to reference Nogueira A, Pires B, Rosales RA (2014) Asymptotically periodic piecewise contractions of the interval. Nonlinearity 27(7):1603–1610 MathSciNetCrossRef Nogueira A, Pires B, Rosales RA (2014) Asymptotically periodic piecewise contractions of the interval. Nonlinearity 27(7):1603–1610 MathSciNetCrossRef
21.
go back to reference Nogueira A, Pires B, Rosales RA (2018) Topological dynamics of piecewise \(\lambda \)-affine maps. Ergodic Theory Dyn Syst 38(5):1876–1893 MathSciNetCrossRef Nogueira A, Pires B, Rosales RA (2018) Topological dynamics of piecewise \(\lambda \)-affine maps. Ergodic Theory Dyn Syst 38(5):1876–1893 MathSciNetCrossRef
23.
go back to reference Semmann D, Krambeck H-J, Milinski M (2003) Volunteering leads to rock-paper-scissors dynamics in a public goods game. Nature 425(6956):390–393 CrossRef Semmann D, Krambeck H-J, Milinski M (2003) Volunteering leads to rock-paper-scissors dynamics in a public goods game. Nature 425(6956):390–393 CrossRef
24.
go back to reference Szolnoki A, Mobilia M, Jiang L-L, Szczesny B, Rucklidge AM, Perc M (2014) Cyclic dominance in evolutionary games: a review. J R Soc Interface 11(100):20140735 CrossRef Szolnoki A, Mobilia M, Jiang L-L, Szczesny B, Rucklidge AM, Perc M (2014) Cyclic dominance in evolutionary games: a review. J R Soc Interface 11(100):20140735 CrossRef
25.
go back to reference Wang Zhijian X, Bin ZH-J (2014) Social cycling and conditional responses in the rock-paper-scissors game. Sci Rep 4:5830 EP CrossRef Wang Zhijian X, Bin ZH-J (2014) Social cycling and conditional responses in the rock-paper-scissors game. Sci Rep 4:5830 EP CrossRef
Metadata
Title
Periodic Attractor in the Discrete Time Best-Response Dynamics of the Rock-Paper-Scissors Game
Authors
José Pedro Gaivão
Telmo Peixe
Publication date
02-11-2020
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 3/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-020-00371-y

Other articles of this Issue 3/2021

Dynamic Games and Applications 3/2021 Go to the issue

Premium Partner