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HG
R
>
T
S
>
P
SG
T
>
R
S
>
P
(a) & (c)
<
Φ
CD
Φ
CC
R
T
(a)
(b) & (c)
(
a
)
&
(
d
)
(
b
)
&
(
d
)
>
Φ
CD
Φ
CC
R
T
(b)
(d)
&
(b)
(c)
&
(b)
(d)
&
(a)
(c)
&
(a)
<
Φ
CD
Φ
DD
P
S
(c)
(
a
)
&
(
c
)
(
b
)
&
(
c
)
SH
R
>
T
P
>
S
PD
T
>
R
P
>
S
(a) & (d)
>
Φ
CD
Φ
DD
(d)
P
S
(b) & (d)
Fig. 16.2. When linking dynamics occurs much faster than strategy dynamics, the nature of the
game being played changes. The arrows indicate the conditions under which a game located at
the arrow start is transformed into a game located at the arrow end.
For example,
CD
can be increased by reducing the death rate of CD links,
CD
.
With increasing
CD
, the condition is fullled at some point. At the transition
point, C is either transformed into a Nash equilibrium or loses this property. An
equivalent transition for D is given by the condition
P
S
<
CD
DD
=
C
2
D
+
DD
C
D
+
CD
:
(16.12)
D
However, the conditions are not entirely independent, since at least two parameters
have to be varied. Usually, it is enough to vary the three link-death rates and x
the link-birth rates to observe these transitions. It is also worth mentioning that,
in coordination games, the transformation can change risk dominance.
16.2.4. Comparable timescales
As we have shown, active linking can lead to a wide range of scenarios that eec-
tively change the character of the game. However, the analytical results have been
obtained assuming time scale separation. Figure 16.3 shows the results of numerical
simulations for a gradual change of the time scale ratio. Deviations from the ana-
lytical predictions are limited to a single order of magnitude. In other words, the
time scale separation is not a very strong assumption and remains valid for a much
wider range of parameters than expected. Even for moderate active linking, our an-
alytical results are recovered, i.e. they hold when self-organising network structures
and the evolutionary game dynamics on the network are intimately entangled. Hav-
ing identied the relevance of time scale separation in a minimal model of linking
dynamics, we now turn to more complex linking dynamics based on local rules.
