1
Pair Approximation
Some notation to get started: πA (kA ) = kA a + (k − kA )b, πB (kA ) = kA c + (k − kA )d represent the payoffs
for A and B types with kA neighbours of type A; fA = 1 − w + wπA , fB = 1 − w + wπB denotes the fitness
of A and B types with w indicating the selection strength.
The population fitness is given by N = pA fA (qA|A k) + (1 − pA )fB (qA|B k) where pA denotes the frequency of A types in the population and qX|Y indicates the conditional probability that the neighbour of an Y
pA
type is of type X. Note that qA|B = 1−p
(1 − qA|A ).
A
1.1
Birth-death updating
Note: naively, we may assume that an A type is selected for reproduction with probability N1 pA fA (qA|A k)
and replaces a B type neighbours with probability 1 − qA|A . However, this neglects that the selected A may
have no B neighbours. Indeed, only when selecting an A type with at least one B type neighbours a change
in the strategy composition of the population may arise. An A type with at least one B type neighbour is
selected with probability N1 pA fA (qA|A (k − 1)). This can be easily verified by considering the sum over all
possible configurations of the reproducing A type:
T
+
k
X
1
k
1
k − kA
kA
:
pA fA (kA )
qA|A
= pA fA (qA|A (k − 1))(1 − qA|A ), (1)
(1 − qA|A )k−kA
N
kA
k
N
kA =0
A
where kA denotes the number of A type neighbours and the term k−k
denotes the probability that one of the
k
B types gets replaced. Analogously, we obtain the probability that a B type with at least one A type neighbour
is selected for reproduction and replaces one of its A neighbours:
T
−
k
X
k
1
1
kA
kA
(1 − pA )fB (kA )
qA|B
= (1 − pA )fB (1 + qA|B (k − 1))qA|B
:
(1 − qA|B )k−kA
N
kA
k
N
kA =0
(2)
=
1
pA fB (1 + qA|B (k − 1))(1 − qA|A ).
N
(3)
In each update the frequency of A types changes at a rate proportional to 1/N where N is the population
size, just as for the death-birth process. However, the rate of change of the frequency of A − A links, pAA ,
requires more careful attention. First, consider an A that has successfully replaced a B neighbour. This B
neighbour has one A neighbour (the reproducing individual) and qA|B (k − 1) further A types among its k − 1
other neighbours. Hence pAA increases at a rate proportional to 2(1 + qA|B (k − 1))/(N k), where N k/2
indicates the normalization, i.e. the total number of links in the population. Similarly, consider a B type that
has successfully replaced an A neighbour. This decreases pAA at a rate 2qA|A (k − 1)/(N k) because the A
neighbour has one B neighbour (the reproducing individual) and qA|A (k − 1) neighbours of type A among its
k − 1 other neighbours.
With this we can now derive the rate equations for the change in A type frequency, pA , and frequency
of A − A links. In order to keep the expressions more manageable we introduce the abbreviations ϕA =
fA (qA|A (k − 1)), ϕB = fB (1 + qA|B (k − 1)):
1
pA (1 − qA|A )(ϕA − ϕB )
N
1
2
= pA (1 − qA|A ) ϕA (1 + qA|B (k − 1)) − ϕB qA|A (k − 1) .
N
k
ṗA =
ṗAA
1
(4)
(5)
Note that the common factor N1 pA (1 − qA|A ) can be omitted as it merely represents a non-linear rescaling of
time but does not affect the qualitative predictions of the pair approximation.
Changes in the frequency of A − A links is easily translated into changes of the conditional probability
qA|A , which measures the local density of A types, by considering the following:
q̇A|A
d
=
dt
pAA
pA
=
ṗAA pA − pAA ṗA
1
=
ṗAA − ṗA qA|A
2
pA
pA
(6)
and hence we obtain:
ṗA ≈ ϕA − ϕB
2
ϕA (1 + qA|B (k − 1)) − ϕB qA|A (k − 1) − qA|A (ϕA − ϕB ) .
q̇A|A ≈
k
1.1.1
(7)
(8)
Weak selection, w 1
In the limit of weak selection, let us consider the leading order terms in ṗA and q̇A|A using ϕA = 1 − w +
wπA (qA|A (k − 1)), ϕB = 1 − w + wπB (1 + qA|B (k − 1)). Note that the zeroth -order cancels in ṗA but not
in q̇A|A :
ṗA ≈ w πA (qA|A (k − 1)) − πB (1 + qA|B (k − 1)) + o(w2 )
2
q̇A|A ≈ 1 + qA|B (k − 1) − qA|A (k − 1) ,
k
(9)
(10)
which results in a separation of time scales. Local equilibration happens much faster than global frequency
changes. Hence we can solve for the equilibrium q̇A|A = 0 and use the local equilibrium to analyze the slow
dynamics. The local equilibrium yields
qA|A − qA|B =
1
,
k−1
(11)
just as in the death-birth process. Equipped with this, we obtain
ṗA ≈ w πA (qA|A (k − 1)) − πB (qA|A (k − 1))
(12)
= w(k(b − d) + qA|A (k − 1)(a − b − c + d))
(13)
= w(k(b − d) + (1 + (k − 2)pA )(a − b − c + d)).
(14)
Finally, in the case of the donation game with a = β − γ, b = −γ, c = β, and d = 0 we get
ṗA = − wkγ,
(15)
which is always negative and hence the frequency of cooperators decreases and extinction is inevitable. This
is in contrast to the death-birth process where local clustering can support cooperators.
2