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individual that plays
i
against a randomly drawn opponent faces every strategy
present in the population with the associated frequency with which that strategy
occurs in the population. This is formally identical to this individual playing against
an opponent who plays a mixed strategy
x
(
t
). The associated population average
payoff is
u
(
x
,
x
)
¼ Σ
i
x
i
*
u
(
i
,
x
).
The frequency of strategy
i
changes to the degree that the expected payoff
u
(
i
,
x
)
differs from the population average payoff
u
(
x
,
x
). If
u
(
i
,
x
) is greater than
u
(
x
,
x
),
the number of individuals playing
i
in the next period will grow more than the
population average. If
u
(
i
,
x
) is smaller than
u
(
x
,
x
), the number of individuals
playing
i
in the next period will shrink more than the population average. This
relative growth is assumed to be linearly proportional to the difference between
strategy payoff and the population average payoff.
4
Consequently, the continuous
RD is specified as follows:
d
x
i
d
t
¼
½
ui
ðÞ
x
ux
ðÞ
x
x
i
Weibull 1995
ð
p
72
Þ
(5.1)
;
;
;
:
That is, the change in
x
i
's population share is determined by
x
i
's current
population share and the difference between its expected payoff and the population
average payoff.
Through analysis of a phase diagram of these dynamics, convergent trajectories,
stable states and regular changes can be identified. Under the biological interpreta-
tion, regular changes identify the temporal predominance of certain traits in the
population, while stable states identify results of adaptation of organisms to their
environment.
4 The Biological RD
The RD was first derived in the late 1970s and quickly became the most prominent
model of evolutionary dynamics in EGT.
5
The RD is derived from EGT by
implicitly presupposing EGT to describe an underlying biological mechanism.
The core idea of EGT in biology is that organisms often find themselves in strategic
situations, in which the fitness-relevant outcome of their behaviour at a certain time
depends on the behaviour of the other organisms in the population at that time. The
fitness of an organism thus is influenced by the frequency of behaviour in that
population. Consequently, there is a systematic relationship between the kind of
4
The relation between proliferation and payoffs characterises different classes of selection
dynamics. While a linear relation characterises the RD, wider classes are characterised by payoff
positivity and payoff monotonicity, respectively (Weibull
1995
, pp. 139-152). Yet the RD, which
takes payoffs to represent fitness differences, is the most prominent selection dynamic in EGT and
therefore will be discussed here.
5
For a historical survey, see Gr¨ne-Yanoff (
2011a
).
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