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

–    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.

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

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

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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