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to well-mixed systems. In this case, the xation probability can be approximated
by
erf[
1
] erf[
0
]
erf[
N
] erf[
0
]
;
C
=
(16.7)
q
u
(ku + v) [Traulsen et al. (2006)].
We have 2u = RS T + P and 2v = R + SN TN + T.
For u ! 0, we have
C
= (1 e
2v
)=(1 e
2vN
).
If strategy dynamics is fast, the linking dynamics only becomes relevant in states
where the system can no longer evolve from strategy dynamics alone, but changing
the topology allows to escape from these states.
where erf(x) is the error function and
k
=
16.2.3.2. Fast linking dynamics
Whenever W 1 linking dynamics is fast enough to ensure that the network will
reach a steady state before the next strategy update takes place. At the steady
state of the linking dynamics, the average payos of C and D individuals are given
by
C
= R
CC
(N
C
1) + S
CD
N
D
(16.8)
and
D
= T
CD
N
C
+ P
DD
(N
D
1) :
(16.9)
Note that the eective number of interactions of cooperators and defectors can
become very dierent if
CC
DD
or vice versa. Comparing Eqs. (16.8) and
(16.9) to Eqs. (16.4) and (16.5) suggests that the linking dynamics introduces a
simple transformation of the payo matrix. We can study standard evolutionary
game dynamics using the modied payo matrix
C D
C R
CC
S
CD
D T
CD
d
DD
C D
C R
0
S
0
=
:
(16.10)
D T
0
P
0
Consequently, linking dynamics can change the nature of the game [Pacheco et al.
(2006b)]. So far, we have only shown this in the limit where linking dynamics is
much faster than strategy dynamics (W 1). However, the result is expected
to hold even when the two time scales are comparable (see below and also Refs.
[Pacheco et al. (2006)b,a].
In general, all generic transformations are possible, as illustrated in Fig. 16.2.
The transition points can be determined as follows: Strategy C is a Nash equilibrium
for R > T. This property changes to R
0
< T
0
when
2
C
+
CC
R
T
<
CD
CC
=
D
C
D
+
CD
:
(16.11)
C
