Rock, Paper, Scissors, Lizard, Spock! Cycles, complexity and emergence in spa;al game models Prof Ken Hawick, March 2011 Complexity Ques;ons •  High informa;on content from macroscopic paGern of many microscopically simple (ruled) individuals. •  eg from spa;al game of par;cipant players •  Can Ask: – 
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What causes it? What parameters control it? (phase transi;ons?) How sensi;ve is it? Is it unavoidable/inevitable? Does it emerge spontaneously? Games & Game Theory •  Games like: –  Prisoner Dilemma (payoff dilemma) –  Rock, Paper, Scissors (has a cycle) –  Rock, Paper, Scissors, Lizard, Spock (longer cycle) •  Can have soXware agents play: –  Iterated Games (tournaments, with memory) –  Spa;al Games (many players on a mesh) Spa;al Games • 
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Arrange paGern of players eg on a mesh Each plays against its local neighbours Ini;alise completely randomly and uniformly Define a ;mestep for the whole system as: For all player agents 1. 
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Pick a player at random Pick a neighbour at random Pick a game process at random Play according to the rules of that process Rock, Paper, Scissors •  The rules form a single 3-­‐cycle: •  Scissors cuts paper •  Paper covers rock •  Rock bluntens scissors Rock, Paper, Scissors, Lizard, Spock! Actor Jim Parsons exposi;ng RPSLS, as the “Sheldon Cooper” character in: “The Lizard-­‐Spock Expansion” of the TV Series “The Big Bang Theory” Season 2, 2008, directed by Mark Cendrowski. hGp://www.youtube.com/watch?
v=iapcKVn7DdY Rock, Paper, Scissors, Lizard, Spock! • 
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Scissors cuts paper Paper covers rock Rock crushes lizard Lizard poisons Spock Spock smashes scissors Scissors decapitates lizard Lizard eats paper Paper disproves Spock Spock vapourizes rock Rock bluntens scissors Actually… •  RPSLS, aGributed to Sam Kass: hGp://www.samkass.com/theories/RPSSL.html •  And see also: –  Zhang, G.-­‐Y.; Chen, Y.; Qi, W.-­‐K. & Qing, S.-­‐M. Four-­‐state rock-­‐paper-­‐scissors games in constrained Newman-­‐WaGs networks, Phys. Rev. E, American Physical Society, 2009, 79, 062901 –  Reichenbach, T., Mobilia, M., Frey, E.: Mobility promotes and jeopardizes biodiversity in rock-­‐paper-­‐
scissors games, Nature 448 (2007) 1046–1049 RPSLS, (RPSSL) 5-­‐cycle gives rise to: 1 2 3 1 1 2 3 4 1 1 2 5 4 1 1 2 5 3 1 1 2 5 3 4 1 1 5 4 2 3 1 1 5 4 1 1 5 3 1 1 5 3 4 1 2 3 4 2 2 5 4 2 2 5 3 4 2 •  Where we number states: – 
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Rock(1) Paper(2) Scissors (3) Spock(4) Lizard(5) •  (use 0’s for vacancies) •  Easier logically to use RPSSL although its oXen pronounced verbally as RPSLS! •  Gives us Twelve cycles Two 5-­‐cycles’s Five 4-­‐cycles’s Five 3-­‐cycles’s Try a Simple Case First •  Ignore the RPSLS star rela;onships •  Just focus on the single longest (outer) cycle •  What does this give rise to? •  Suprisingly complex spa;al structure •  Mul;phasic layers -­‐ as it turns out Simple Cyclic Model Red(1), Yellow(2), Blue(3), Green(4), Cyan(5) Where are the “vacancies” ? Some Nomenclature • 
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Q is number of states = 5 + 1 (for vacancies) Formulate Model in terms of rate equa;ons Tradi;onal to use Greek leGers for the rates Diffusion: epsilon Reproduc;on: sigma Selec;on: mu & alpha Cyclic Selec;on & Reproduc;on Diffusion Generalising to arbitrary Q Q=3,4,5,6,7,8,9,10,11,12 Vacancies for Q=3,4,5,6,7,8 What to Measure? Measuring against Time Single run: 256x256, 2048 steps Averaged over 1000 Runs 1) The error bars are present but too small to see… 2) Note the tendency to reach (dynamic) equilibrium values Long Term Frac;on of Vacancies Note the different fluctua;ons for Odd Q Frac;on of Like-­‐Like Spa;al Bonds Note the interleaving of the high mobility values Long Term Frac;on of Neutral bonds Contras;ng behaviour for odd and even-­‐Q (even-­‐Q plays the game beGer!) Some Preliminary Conclusions • 
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So there are interes;ng symmetries Interleaving of the curves Dras;c difference between even and odd Q Vacancies play an important part •  But that was only single cycle simplifica;on… Puvng “Spock vapourises rock…” in •  Use mu for the outer cycle rate •  Use alpha for the inner cycle •  Parameter varia;on experiments to see what happens… Vary inner cycle reac;on(alpha, mu=1) Selec;on 1 (mu) & 2 (alpha) Three Dimensions •  Qualita;vely similar behaviour as in 2D •  More work to simulate •  Small system too liable to ex;nc;ons •  May need to adjust diffusion rate to slow down cf 2D case 3d System 40x40x40 – Way too small Summary • 
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Layers of “my enemy’s enemy is my friend” Symmetry -­‐ cycles can be reversed Spa;al Complexity Growth – looks like a power law Decay & Ex;nc;ons Vacancies and rate equa;on formula;on works •  The RPSLS model has it all! –  and maybe even universality ….? What Next? •  Complete mu-­‐alpha parameter scans •  Fit power laws to parameters •  Growth dependency is power law or logarithmic or ??? •  3D model will take longer, but now know where to look •  Suspect dimensional dependence •  Small-­‐World and damaged lavce varia;ons… Further Informa;on •  hGp://complexity.massey.ac.nz •  Complex Domain Layering in Even-­‐Odd Cyclic State Rock-­‐Paper-­‐
Scissors Game Simula;ons, K.A.Hawick, January 2011, submiGed to IASTED Modelling & Simula;on MS’11. •  Roles of Space and Geometry in the Spa;al Prisoners' Dilemma, K.A.Hawick and C.J.Scogings, Proc. IASTED Interna;onal Conference on Modelling, Simula;on and Iden;fica;on, 12-­‐14 October 2009, Beijing, China. •  Defensive Spiral Emergence in a Predator-­‐Prey Model, K.A.Hawick, C.J.Scogings and H.A.James, Complexity Interna;onal, Vol 12, 2008, PP 37. Live Long and Prosper!