Presentation slides for the following two papers (mainly (1)): (1) Masuda. Proceedings of the Royal Society B: Biological Sciences, 274, 1815-1821 (2007). (2) Masuda and Aihara. Physics Letters A, 313, 55-61 (2003).

1. ParNcipaNon
costs
dismiss
the
advantage
of
heterogeneous
networks
in
evoluNon
of
cooperaNon
Naoki
Masuda
(University
of
Tokyo)
Ref:
Masuda.
Proc.
R.
Soc.
B,
274,
1815-­‐1821
(2007).
Also
see
Masuda
&
Aihara,
Phys.
LeK.
A,
313,
55-­‐61
(2003).

2. Prisoner’s
Dilemma
Opponent
Cooperate
Defect
Cooperate
(3,
3)
(0,
5)
Defect
(5,
0)
(1,
1)
Self
unique
Nash
equilibrium
C
D C
CD D
CD C
D
D
D D
CD D
DD C
D
D
D D
DD D
DD D
D

3. Mechanisms
for
cooperaNon
•
•
•
•
•
•
•
Kin
selecNon
(Hamilton,
1964)
Direct
reciprocity
(Trivers,
1971;
Axelrod
&
Hamilton
1981)
•
Iterated
Prisoner’s
dilemma
Group
selecNon
(Wilson,
1975;
Traulsen
&
Nowak,
2006)
SpaNal
reciprocity
(Axelrod,
1984;
Nowak
&
May,
1992)
Indirect
reciprocity
(Nowak
&
Sigmund,
1998)
•
Image
scoring
Network
reciprocity
(Lieberman,
Hauert
&
Nowak,
2005;
Santos
et
al.,
2005,
2006;
Ohtsuki
et
al.,
2006)
Others
(punishment?,
voluntary
parNcipaNon
etc.)

4. Iterated
Prisoner’s
Dilemma
• Players
randomly
interact
with
others
• Discount
factor
w
(0
≤
w
≤
1)
to
specify
the
prob.
that
the
next
game
is
played
in
a
round
A
plays
C
D
D
C
D
C
B
plays
C
C
D
C
D
D
A
gets
3
5
1
3
1
0
A’s
accumulated
payoff
=
3
+
5w
+
1w2
+
3w3
+
1w4
+
0w5
+
…
acNon
C
D
C
(3,
3)
(0,
5)
D
(5,
0)
(1,
1)

5. • SelecNon
based
on
accumulated
payoff
aher
each
round
• Replicator
dynamics
• Best-­‐response
dynamics
• Nice,
retalitatory,
and
forgiving
strategies
(e.g.
Tit-­‐for-­‐Tat)
are
generally
strong
(but
not
the
strongest).

6. SpaNal
Prisoner’s
Dilemma
(Axelrod,
1984;
Nowak
&
May,
1992)
• e.g.
square
laice
• Either
cooperator
or
C:24 C:18 D:24 D:12
defector
on
each
vertex
• Each
player
plays
against
all
(4
or
8)
neighbors.
C:24
C:21
C:12
D:16
C:24
C:21
C:12
D:16
C:21
C:15
D:20
D:16

7. •
•
•
Successful
strategies
propagate
aher
one
generaNon.
Result:
Cs
form
“clusters”
to
resist
invasion
by
Ds.
Note:
2-­‐neighbor
CA.
More
complex
than
1-­‐neighbor
dynamics
such
as
spin
(opinion)
dynamics
and
disease
dynamics

8. PD
on
the
WaKs-­‐Strogatz
small-­‐world
network
(Masuda
and
Aihara,
Phys.
LeK.
A,
2003)
acNon
C
D
C
(1,
1)
(0,
T)
D
(T,
0)
(0,
0)
p
=
0
p:
small
p
≈
1
1
0.8
0.6
%C
0.4
Note:
The
degrees
of
all
the
nodes
are
the
same
regardless
of
the
rewiring.
0.2
p=0
p=0.01
p=0.9
0
1
1.5
T
2
2.5

9. large
p
1
•
1-­‐dim
ring
•
QualitaNvely
the
0.8
0.6
%C
0.4
same
results
for
2-­‐dim
networks
small
p
1
small
p
0.2
0
0
100
200
generation
T=1.1
300
400
1
0.8
0.8
0.6
%C
0.4
0.6
%C
0.4
0.2
0.2
0
large
p
0
100
200
generation
T=1.7
300
400
large
p
0
small
p
T=3.0
0
100
200
generation
300
400

10. more
C
slow
Clustering
helps
cooperaNon
Small
distance
(L)
accelerates
whatever
propagaNon
more
D
fast

11. Social
dilemma
games
on
scale-­‐free
networks?

12. Scale-­‐free
networks
promote
cooperaNon
acNon
Cooperate
Defect
Cooperate
(1,
1)
(S,
T)
Defect
(T,
S)
(0,
0)
Regular
RG
(or
the
complete
graph)
1
(a)
no
dilemma
snowdrih
S
1
(b)
S
0
0
stag
hunt
-1
scale-­‐free
0
PD
1
T
2
-1
0
1
T
2
Originally
by
Santos,
Pacheco
&
Lenaerts,
PNAS
2006

13. Two
assumpNons
underlie
the
enhanced
coopraNon
in
SF
nets

14. AssumpNon
1:
addiNve
payoff
scheme
• AddiNve:
add
the
payoffs
gained
via
all
the
neighbors
• Nowak,
Bonhoeffer
&
May
1994;
Abramson
&
Kuperman
2001;
Ebel
&
Bornholdt,
2002;
Ihi,
Killingback
&
Doebeli
2004;
Durán
&
Mulet,
2005;
Santos
et
al.,
2005;
2006;
Ohtsuki
et
al.,
2006
• Average:
divide
the
summed
payoffs
by
the
number
of
neighbors
• Kim
et
al.,
2002;
Holme
et
al.,
2003;
Vukov
&
Szabó,
2005;
Taylor
&
Nowak,
2006

15. acNon
Cooperate
Defect
Cooperate
(3,
3)
(0,
5)
Defect
(5,
0)
(1,
1)
10
90
v1
k 1 = 100
v2
k2 = 2
=C
=D
payoff
scheme
addiNve
3×10+0×90
=
30
5×2
=
10
average
(3×10+0×90)/100
=
0.3
(5×2)/2
=
5

16. • Average
payoff
diminishes
cooperaNon
in
heterogeneous
networks
(Santos
&
Pacheco,
J.
Evol.
Biol.,
2006).

17. AssumpNon
2:
PosiNvely
biased
payoffs
•
C
D
C
a
b
D
c
d
(
Originate
from
the
translaNon
invariance
of
replicator
dynamics
!
!
(
⇡C =
⇡D =
axC + bxD
cxC + dxD
h⇡i = ⇡C xC + ⇡D xD
xC =
˙
xD =
˙
xC (⇡C
xD (⇡D
✓
a b
c d
◆
✓
◆
h b h →
parNcipaNon
h d h
cost
✓
◆ ✓
◆
a b
ka kb
!
→
Nme
rescaling
c d
kc kd
◆
◆ ✓
✓
a b
a+k b+k
!
c d
c
d
!
a
c
h⇡i)
h⇡i)
acNon
C
D
acNon
C
D
C
(1,
1)
(0,
T)
C
(1,
1)
(S,
T)
D
(T,
0)
(0,
0)
D
(T,
S)
(0,
0)

18. Reason
for
enhanced
cooperaNon
•
Hubs
earn
more
than
leaves.
•
C
on
hubs
(with
at
least
some
C
neighbors)
are
stable.
•
C
spreads
from
hubs
to
leaves.
10
acNon
C
D
C
(1,
1)
(0,
T)
D
(T,
0)
(0,
0)
90
v1
k 1 = 100
v2
k2 = 2
=C
=D

19. Payoff
matrix
is
not
invariant
on
heterogeneous
networks
C
D
C
a
b
D
c
d
(
!
!
(
⇡C =
⇡D =
✓
h⇡i = ⇡C xC + ⇡D xD
xC =
˙
xD =
˙
xC (⇡C
xD (⇡D
✓
!
h⇡i)
h⇡i)
acNon
C
D
C
(1,
1)
(0,
T)
D
(T,
0)
(0,
0)
a
c
◆
h b h →
parNcipaNon
h d NG
h
cost
✓
◆ ✓
◆
a b
ka kb
!
✔ →
Nme
rescaling
c d
kc kd
◆
◆ ✓
✓
a b
a+k b+k
!
NG
c d
c
d
axC + bxD
cxC + dxD
a b
c d
◆

20. Our
assumpNons
• AddiNve
payoff
scheme
• Introduce
the
parNcipaNon
cost
• Do
numerics
• N
=
5000
players
• Each
player
plays
against
all
the
neighbors.
• Replicator-­‐type
update
rule:
player
i
copies
player
j’s
strategy
with
prob
(πj-­‐πi)/[max(ki,kj)
*
(max
possible
payoff
–
min
possible
payoff)]

21. Simplified
prisoner’s
dilemma
regular
random
net
(a)
3
1
cf
0.75
2
0.5
1
0.25
0
acNon
C
D
C
(1-h,
1-h) (-h,
T-h)
D
(T-h,
-h)
0
h
0.7
1 T 1.3
scale-­‐free
net
1.6
(-h,
-h)
3
(roughly
separated)
regimes
Strong
influence
of
iniNal
cnds
due
1
to
long
transients
(a)
3
2
Somewhat
reduced
cooperaNon
h
1
2
Enhanced
cooperaNon
(prev
results) 3
0
0.7
1 T 1.3
1.6

22. T
=
1.5
(b)
h
=
0
100
50
h
=
0.2
h
=
0.23
0
h
=
0.24
h
=
0.25
-50
h
=
0.5
0
50
h
=
0.3
100
150
# neighbors
200
h
=
0.2 h
=
0
# flips
generation payoff
(a)
h
=
0.5
50
h
=
0.23
h
=
0.3
h
=
0.24
h
=
0.25
0
10
100
# neighbors
Strategy
spreads
from
stubborn
leaves
to
hubs.
h
1
0
0.7
1 T 1.3
1.6
2
From
hub
cooperators
to
leaves.
2
1
From
leaves
to
hubs.
PD
payoff
structure
is
most
relevant.
(a)
3
3

23. General
matrix
game
• Homogeneous
(in
degree)
→
2
parameters
(S
and
T)
• e.g.
well-­‐mixed,
square
laice,
regular
random
graph
• Heterogeneous
→
3
parameters
(S,
T,
and
h)
• e.g.
ER
random
graph,
scale-­‐free
• PD,
snowdrih
game,
hawk-­‐dove
game
included
acNon
C
D
C
(1-h,
1-h) (S-h,
T-h)
D
(T-h,
S-h)
(-h,
-h)

24. consistent
with
Santos
et
al.,
PNAS
(2006)
Regular
RG
(h
=
0)
1
(a)
SF
(h
=
0.5)
snowdrih
0
1
(b)
1
(c)
S
no
dilemma
S
S
0
stag
hunt
-1
SF
(h
=
0)
0
PD
1
T
1
2
-1
0
1
0
SF
(h
=
1)
1
T
2
SF
(h
=
2)
1
(d)
-1
2
0
1
1
cf
1
(e)
0.75
S
S
0
-1
0.5
0
2
0
1
T
2
-1
3
0
1
T
2
0.25
0
T
2

25. Thoughts
about
the
payoff
bias
• Naturally
understood
as
the
parNcipaNon
cost
• Payoffs
may
be
negaNve
in
many
pracNcal
situaNons.
• Environmental
problems?
• InternaNonal
relaNons?
• When
one
is
‘forced’
to
play
games

26. Conclusions
• Games
with
parNcipaNon
costs
on
networks
• More
C
for
small
parNcipaNon
cost
h
(previous
work).
• Networks
determine
dynamics
for
small
and
large
h.
• Payoff
matrix
is
most
relevant
for
intermediate
h.
• Think
twice
about
the
use
of
simplified
PD
payoff
matrices.