Rock_ Paper_ Scissors_ Lizard_ Spock_.pdf

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					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	
               •  Where	
  we	
  number	
  states:	
  
                                        –    Rock(1)	
  
1	
  2	
  3	
  4	
  1	
                 –    Paper(2)	
  
1	
  2	
  5	
  4	
  1	
                 –    Scissors	
  (3)	
  
1	
  2	
  5	
  3	
  1	
                 –    Spock(4)	
  
                                        –    Lizard(5)	
  
1	
  2	
  5	
  3	
  4	
  1	
     •  (use	
  	
  0’s	
  	
  for	
  vacancies)	
  
1	
  5	
  4	
  2	
  3	
  1	
  
1	
  5	
  4	
  1	
               •  Easier	
  logically	
  to	
  use	
  RPSSL	
  
                                    although	
  its	
  oXen	
  pronounced	
  
1	
  5	
  3	
  1	
                  verbally	
  as	
  	
  RPSLS!	
  
1	
  5	
  3	
  4	
  1	
  
2	
  3	
  4	
  2	
               •  Gives	
  us	
  Twelve	
  cycles	
  
                                        Two	
  5-­‐cycles’s	
  
2	
  5	
  4	
  2	
                      Five	
  4-­‐cycles’s	
  
2	
  5	
  3	
  4	
  2	
                 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!	
  

				
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