Rock,&Paper,&Scissors,&Lizard,&Spock!& - Massey …complexity.massey.ac.nz/talks/rpsls-talk.pdf ·...
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Rock, Paper, Scissors, Lizard, Spock! Cycles, complexity and emergence in spa;al game models Prof Ken Hawick, March 2011
Transcript of Rock,&Paper,&Scissors,&Lizard,&Spock!& - Massey …complexity.massey.ac.nz/talks/rpsls-talk.pdf ·...
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: – 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
• 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. Pick a player at random 2. Pick a neighbour at random 3. Pick a game process at random 4. 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!
• 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: – 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
• 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
• 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
• 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!
