Downloaded from http://rstb.royalsocietypublishing.org/ on July 12, 2017
Evolutionary dynamics of n-player games
played by relatives
Hisashi Ohtsuki
rstb.royalsocietypublishing.org
Research
Cite this article: Ohtsuki H. 2014 Evolutionary dynamics of n-player games played by
relatives. Phil. Trans. R. Soc. B 369: 20130359.
http://dx.doi.org/10.1098/rstb.2013.0359
One contribution of 14 to a Theme Issue
‘Inclusive fitness: 50 years on’.
Department of Evolutionary Studies of Biosystems, School of Advanced Sciences, The Graduate University for
Advanced Studies (SOKENDAI), Shonan Village, Hayama, Kanagawa 240-0193, Japan
One of the core concepts in social evolution theory is kin selection. Kin selection provides a perspective to understand how natural selection operates when
genetically similar individuals are likely to interact. A family-structured population is an excellent example of this, where relatives are engaged in social
interactions. Consequences of such social interactions are often described in
game-theoretical frameworks, but there is a growing consensus that a naive
inclusive fitness accounting with dyadic relatedness coefficients are of limited
use when non-additive fitness effects are essential in those situations. Here,
I provide a general framework to analyse multiplayer interactions among relatives. Two important results follow from my analysis. First, it is generally
necessary to know the n-tuple genetic association of family members when
n individuals are engaged in social interactions. However, as a second
result, I found that, for a special class of games, we need only measures of
lower-order genetic association to fully describe its evolutionary dynamics. I
introduce the concept of degree of the game and show how this degree is related
to the degree of genetic association.
Subject Areas:
evolution, theoretical biology
Keywords:
kin selection, game theory, relatedness,
family group, cooperation
Author for correspondence:
Hisashi Ohtsuki
e-mail: [email protected]
1. Introduction
Kin selection theory describes evolutionary consequences of social interactions
that occur between individuals, especially between those who are genetically or
phenotypically correlated such as relatives [1–8]. One of its major contributions
to evolutionary biology is that the theory revealed the possibility of evolution of
self-sacrificing cooperation towards kin [9–16]. In the simplest case, this fact is
beautifully demonstrated by an inclusive fitness accounting, namely that such a
cooperation evolves if the benefit of cooperation weighted by the relatedness to
the beneficiary exceeds its cost, famously known as Hamilton’s rule [1,17,18].
Game theory, on the other hand, often presupposes a more complex type of
interaction than a simple unilateral donation of cooperation [19–21]. The virtue
of game theory lies in that it can describe social interactions where one’s
pay-off not only depends on your own action but also, often non-additively, on
partners’ ones. For example, in the traditional hawk–dove game, the advantage
of choosing an aggressive strategy, hawk, very much depends on the strategy
of the co-player; the outcome of a hawk–hawk interaction is detrimental owing
to the fighting cost, whereas the outcome of a hawk–dove interaction is the
most profitable for the hawk player, because he wins all the resources [22,23].
Therefore, an evolutionary consequence of game interactions between relatives is not obvious [24–27]. There is a large amount of literature indicating that
a simple inclusive-fitness accounting as provided by Hamilton [1] does not
always work for complex interactions [2,3,10,20,25,28–40] (but see [7]), and it
has been shown that this is especially true when we study games with discrete
strategies (i.e. pure hawk or pure dove) rather than with continuous ones (i.e.
mixed strategies of hawk and dove) [41–43]. Some literature suggests that
higher-order genetic association of more than two individuals is necessary to
fully describe evolutionary game dynamics [15,30,31,44 –48].
Despite analytical difficulty, social interaction among kin is ubiquitous in
biology. Recent developments in microbiology revealed that microbes are engaged
in various kinds of social interactions [49–57]. An essential feature is that those
microbes often clonally reproduce and therefore they are inevitably genetically
& 2014 The Author(s) Published by the Royal Society. All rights reserved.
Downloaded from http://rstb.royalsocietypublishing.org/ on July 12, 2017
I assume an infinitely large population of haploid individuals.
Each individual adopts one of the two strategies, A or B, that
are genetically determined and genetically inherited from a
parent. I can phrase these assumptions in terms of cultural
evolution; in this case, one’s strategy is culturally transmitted.
At each time step, groups of n-players are formed in order
for them to play an n-player game. The pay-off of an A-player
when k (0 k n 2 1) of the other n 2 1 players in the group
choose A (and therefore n 2 1 2 k other players choose B) is
denoted by ak. Similarly, the pay-off of a B-player when k (0 k n 2 1) of the other n 2 1 players in the group choose A is
given by bk. In other words, a list of a0 , . . . , an1 and
b0 , . . . , bn1 defines a single game.
The most important assumption of my model is that
the formation of groups of n-individuals is not random but
assortative, often favouring kin (i.e. nepotistic). This means,
for example, that relatives are more likely to come together
than expected by chance. I do not explicitly specify an
actual process of assortative group formation, but for later
use in my analysis I need to know the average property of
such groups with respect to common ancestry. More specifically, imagine infinitely many groups of n-individuals that
are formed through a certain process of assortative grouping.
Then, randomly take ‘ (1 ‘ n) different individuals from
a randomly sampled group and ask who shares a common
ancestor with whom. Let u‘!m be the probability that those
‘ different individuals have m (1 m ‘) different origins
in an ancestral population (theoretically speaking, infinitely)
many generations before. One trivial case is m ¼ 1, which
means that all the ‘ individuals share a single common ancestor. Another trivial case is m ¼ ‘, where no two individuals
share a common ancestor. Because I do not assume any
mutations in strategy during reproduction, players sharing
a common ancestor always share the same strategy. Therefore, the probabilities u‘!m (1 m ‘ n) measure the
extent of genetic association in the group of n individuals.
After group formation, those n-players in the same group
are involved in the n-player game described above, and each
player gains an individual pay-off, f. This pay-off is translated
into its fecundity, 1 þ df, where one is the baseline value and d
can be any positive constant as long as everyone’s fecundity is
guaranteed to be strictly positive. Then, each player produces
multiple gametes in proportion to their fecundity, and the
next generation is formed by either asexual or sexual reproduction. In the case of asexual reproduction, gametes directly grow
into adults. For sexual reproduction, two (or possibly more)
(a) Calculation of pay-offs
Here, I calculate the game pay-off of a focal player. For that purpose, for 0 k n 2 1, an indicator variable, fk, is defined. This
new variable fk is equal to one if the number of A-strategists
among its n 2 1 group members is exactly equal to k, and otherwise zero. For example, fn21 can be easily constructed as
fn1 ¼ p1 . . . pn1 (the product of p1 to pn21), suggesting that
all n21 group members of the focal player adopt strategy A if
and only if p1 ¼ ¼ pn1 ¼ 1. In appendix A, I show
that each fk (0 k n 2 1) is expressed as a polynomial of
p1 , . . . , pn1 .
With the help of fk, the game pay-off of a focal player is
given by
f0 ¼
n1
X
{ p0 ak þ (1 p0 )bk }fk :
(2:1)
k¼0
That is, the focal player gains the pay-off of ak if and only if p0 ¼
1 ( player adopts strategy A) and fk ¼ 1 (exactly k other players
adopt strategy A). Similarly, the focal player gains the pay-off
of bk if and only if 1 2 p0 ¼ 1 ( player adopts strategy B) and
fk ¼ 1 (exactly k other players adopt strategy A).
(b) Evolutionary dynamics
The change in the frequency of A-strategists in the population
is described by the Price equation [63]
w Dp ¼ Cov[ p0 , w0 ]:
(2:2)
Here, w0 is the expected fitness of a focal individual, defined
as the probability of the focal individual surviving a single
generation plus the expected number of his offspring that
successfully replaced adults in the previous generation. w represents the average fitness in the population. The covariance
above is defined by a hypothetical sampling of a focal player
from the population [47,64]; in such a hypothetical sampling,
randomness arises in whom I sample as a focal individual,
so the indicator variable and the fitness of a focal individual can
be regarded as random variables (and therefore this covariance
is conditional on a given state of the population).
2
Phil. Trans. R. Soc. B 369: 20130359
2. Model
gametes are chosen randomly, they fuse, and all but one
gamete instantly vanish to make haploid individuals in the
next generation. Generations can be either non-overlapping
or overlapping. I assume that a fraction, s, of adult individuals
in the previous generation survive and that newborns compete
on an equal basis for the vacancies that arise owing to the death
of adults in the previous generation. Quite importantly,
I assume that such a competition among newborns occurs
globally, so no local regulation of the population is assumed.
This assumption is critical for my results, especially the one
in §5, which I will discuss later. It will soon be shown that
the parameter s does not affect the results qualitatively.
Of central importance is to see how the overall frequency
of A-strategists in the population, which is denoted by p in
my paper, changes over time. To see this, I use a relative, not
absolute, indexing of players in the population. More specifically, to any given focal player, I give the index 0, and to his
n 2 1 group members, I give indices of 1, 2, . . . , n21, respectively. To conveniently express the strategies of those players, I
introduce an indicator variable, pi (0 i n 2 1), that takes a
value of either 1 or 0; if player i adopts strategy A, then its indicator variable is pi ¼ 1, and if B, then its value is pi ¼ 0.
rstb.royalsocietypublishing.org
highly related to each other. The reason for such an assortment
is mainly due to spatial structure and limited dispersal [58,59].
As another instance, in larger animals such as primates, we
often find nepotistic cooperation among kin [60–62].
A goal of this paper was to clarify the evolutionary game
dynamics when players are related to each other. A milestone
work in this direction is that of Grafen [25], who studied a
hawk–dove game played between two relatives. However,
real social interactions do not necessarily involve only two individuals. As seen from examples above, in many situations, more
than two relatives are simultaneously involved in a single social
interaction. This motivates me to extend an existing two-player
game model to an n-player one to see its evolutionary game
dynamics.
Downloaded from http://rstb.royalsocietypublishing.org/ on July 12, 2017
From my life cycle assumptions, the fitness of a focal
player is given as
1 þ d f0
,
1 þ df
(2:3)
where f represents the average pay-off in the whole
population. Combining equations (2.1)– (2.3), I arrive at the
main result (see appendix A for its detailed derivation)
n1 X
n1
n1
(1 s)d X
‘k ‘
Dp ¼
(1)
1 þ df k¼0 ‘¼k
k
‘
(2:4)
where r‘ (1 ‘ n) is the probability with which all different ‘ players who are randomly sampled from a randomly
sampled group of n-players adopt strategy A (I set r0 ¼ 1).
Remember that the probability with which those ‘ different
players have m different origins is given by u‘!m. In such a
case, I assume that those m different ancestral lineages are
independent, and therefore that the probability with which
all of those m lineages adopted strategy A in their ancestral
state is given by pm. Therefore, I obtain
r‘ ¼
‘
X
u‘!m pm
(1 ‘ n):
(2:5)
r1 ¼ p,
r2 ¼ rp þ (1 r) p2 ,
(3:2)
[36,66]. Note that the second expression in equation (3.2) is
the regression definition of relatedness given in Hamilton
[66]. Grafen [25] studied a general two-strategy game where
the pay-offs of A matched with A, A matched with B, B
matched with A and B matched with B were respectively
given by a, b, c, d. In my terminology, it corresponds to
(a0, a1, b0, b1) ¼ (b, a, d, c). Applying equation (2.4) to this
specific example leads to
Dp / (1 r1 )r2 a þ (1 r1 )(r1 r2 )b r1 (r1 r2 )c
r1 (1 2r1 þ r2 )d
¼ p(1 p)[(1 r)(a b c þ d)p þ (b d) þ r(a b)],
(3:3)
which correctly predicts the position of equilibrium described
in eqn (6) in Grafen [25].
m¼1
Hereafter, I call r‘ ‘genetic association of degree ‘’ or
‘‘-tuple genetic association’, because it measures genetic
similarity of ‘ different individuals.
Two trivial cases are useful in facilitating our understanding of the main result, equation (2.4). The first baseline case
lacks any genetic association, where we assume that ‘ different players always have ‘ different origins, so r‘ ¼ p‘ follows.
Substituting this in equation (2.4) gives us
Dp ¼
(1 s)d
p(1 p)( fA fB ),
1 þ df
(2:6)
where fA and fB are average pay-offs of an A-player and a
B-player calculated based on a binomial sampling as
n1 X
n1
pk (1 p)n1k ak ,
fA ¼
k
k¼0
(2:7)
n1 X
n1
n1k
k
p (1 p)
bk ,
fB ¼
k
k¼0
[65] (see appendix B for the derivation). The second case
assumes that all n individuals in a group are clonal, leading
to r0 ¼ 1 and r‘ ¼ p for ‘ 1. In this case, equation (2.4)
becomes
Dp ¼
(1 s)d
p(1 p)(an1 b0 ),
1 þ df
(2:8)
(see appendix B for the derivation), which makes sense
because an A-player always interacts with n21 A-players
( pay-off ¼ an21), whereas a B-player always interacts with
n21 B-players ( pay-off ¼ b0).
3. Two-player games
The case of n ¼ 2 goes back to the analysis of two-player
games between relatives by Grafen [25]. He assumed that
the average relatedness between two-game players was
given by r. In my terminology, this r corresponds to the probability with which a randomly sampled pair of players
4. Three-player games
(a) Linear public goods game
Imagine a public goods game with three players, where strategy A corresponds to cooperation and B to defection. A
cooperator pays the cost of C to contribute to the public
goods, whereas a defector pays nothing. I assume that the
net benefit of cooperation is given by B times the total
number of cooperators, which is, in turn, equally divided
by three players, irrespective of their contribution. This
game is called a linear public goods game [67]. The pay-off
of this game is given by
0
1
1
2
C þ B C þ B C þ B
a0 a1 a2
B
C
3
3
(4:1)
¼@
A:
1
2
b0 b1 b2
B
B
0
3
3
Applying equation (2.4) to the game (equation (4.1))
leads to
B
Dp / r1 (1 r1 )C þ (r1 þ 2r2 2r21 r1 r2 ) :
3
(4:2)
Observe that the triplet association, r3, is absent from
equation (4.2) but the dyadic association, r2, is present.
Substituting us in equation (3.1) gives us
B
(4:3)
Dp / p(1 p) C þ (1 þ 2r) :
3
Note that the expression inside the square brackets in
equation (4.3) can be understood intuitively in terms of
inclusive fitness, that is, the benefit others received can
be interpreted as if the focal player received that benefit,
only after it is weighted by the genetic relatedness to
those others. An interpretation is as follows. If I cooperate,
I pay C but that increases the net benefit of public goods
by B, benefiting each of the three individuals in the
group by B/3. My relatedness to self is 1, and my relatedness to each of the other two is r, so the benefit B/3
should be weighted by 1 þ 2r.
Phil. Trans. R. Soc. B 369: 20130359
[(1 r1 )r‘þ1 ak r1 (r‘ r‘þ1 )bk ],
leading to
3
rstb.royalsocietypublishing.org
w0 ¼ s þ (1 s)
engaged in the two-player game shares a common ancestor.
Therefore, us of these two players are given by
u1!1
1
¼
,
(3:1)
u2!1 u2!2
r 1r
Downloaded from http://rstb.royalsocietypublishing.org/ on July 12, 2017
(b) Volunteer’s dilemma
Dp / (1 r1 )( Cr1 þ Br3 ):
(4:5)
The condition for evolution of cooperation is, therefore,
r3 C
. ,
r1 B
(4:6)
which reproduces the result of van Veelen [28, p. 597] and
also reproduces eqn (14) in Gardner et al. [7]. The condition
(4.6) explicitly contains the triplet association, r3. Therefore,
I need to specify the probabilities of three different players
having m (1 m 3) origins, namely u3!m, to derive its
evolutionary dynamics.
Suppose that the probability u2!1 (relatedness) is r and
that u3!1 (called triplet relatedness by Ohtsuki [30]) is s.
These two probabilities measure the chance that two (or
three) individuals share a unique common ancestor. In this
case, I have
0
1 0
1
u1!1
1
@ u2!1 u2!2
A ¼ @r 1 r
A, (4:7)
u3!1 u3!2 u3!3
s 3r 3s 1 3r þ 2s
because there is a trivial relationship between us (see appendix C). Then, the condition (4.6) is rewritten as
s þ 3(r s)p þ (1 3r þ 2s) p2 .
C
:
B
(4:8)
To see how (4.8) reveals the condition for evolution of
cooperation between relatives, consider sexual reproduction
and imagine that three full-sibs are engaged in the volunteer’s dilemma, equation (4.4). It is easy to see that the
coalescent probabilities of them are given by r ¼ 1/2 and
s ¼ 1/4. Therefore, condition (4.8) becomes
1 þ 3p C
. :
4
B
(4:9)
On the other hand, if I consider three half-sibs who share a
mother but never share a father, then I have r ¼ 1/4 and
s ¼ 1/8, arriving at
1 þ 3p þ 4 p2 C
. :
B
8
(4:10)
Most importantly, a naive heuristics does not work here:
that the left-hand side of (4.10) should be a half of the lefthand side of (4.9), because half-sibs are half as related
as full-sibs. Rather, we observe that those left-hand sides
are qualitatively different because they show different
dependence on p.
By contrast, the pay-offs of the volunteer’s dilemma, equation
(4.4), can never be described by linear functions, but I need a
quadratic function, as
8
>
< ak ¼ B k 2 B k C
2
2
(5:2)
>
: b ¼ 0:
k
For a general n-player game, the pay-offs, {ak }k¼0,...,n1 , can
be described uniquely by a polynomial of k of degree n 2 1 or
smaller, so can be the pay-offs, {bk }k¼0,...,n1 . The largest degree
of those two polynomials I shall define as the degree of the
game. For example, both polynomials in equation (5.1) are of
degree one, so the degree of the linear public goods game,
equation (4.1), is 1. By contrast, those polynomials in equation
(5.2) are of degree two and zero, respectively, so the degree of
the volunteer’s dilemma game, equation (4.4), is 2. The minimal and maximal degrees of n-player games are 0 and n 2 1,
respectively. Hereafter, I occasionally use the term ‘linear
game’ to mean a degree-1 game.
In appendix D, I have proved that evolutionary dynamics of an n-player game of degree d can be described
with genetic association of degree up to d þ 1; in other
words, rd þ 1 is necessary to describe evolutionary game
dynamics but higher ones such as rdþ2 , . . . are not. In
short, the degree of nonlinearity of a game determines how
high a genetic association is needed.
It is worth clarifying the difference between a degree-1
game and an ‘additive game’ [30], also known as a game of
‘(generalized) equal gains from switching’ [28]. Additivity is a
useful concept in classifying social interactions, and the effect
of deviation from additivity has been studied in some previous
literature [30,67]. For my n-player, two-strategy game framework, additivity is defined in the following way. First, if the
focal player adopts strategy A (or B), we think that it contributes
to his pay-off by a0 (or b0). Second, if a player other than the
focal player adopts strategy A (or B), we think that it contributes
to the pay-off of the focal player by a (or b). Additivity means
that the pay-off of the focal player is given by the sum of those
contributions from the focal player and from the n 2 1 other
players. In this case, the pay-off is written as
(
ak ¼ a0 þ ka þ (n 1 k)b
(5:3a)
bk ¼ b0 þ ka þ (n 1 k)b:
or
(
ak ¼ [a0 þ (n 1)b] þ k(a b)
bk ¼ [b0 þ (n 1)b] þ k(a b):
(5:3b)
Phil. Trans. R. Soc. B 369: 20130359
Setting C ¼ 1 and B ¼ 2, leads to the game analysed in
Gardner et al. and van Veelen [7,28]. Applying equation
(2.4) to this game leads to
One might wonder why r3 is absent from the evolutionary
dynamics of the linear public goods game, equation (4.1), but
is present in its counterpart of the volunteer’s dilemma game,
equation (4.4), although both are three-player games. The
answer lies in the nature of these games. As we can guess
from its name, the pay-offs of the linear public goods game,
equation (4.1), can be described by linear functions of k, as
8
B
B
>
>
a
k
þ
C
¼
k
<
3
3
(5:1)
>
B
>
: b ¼ k:
k
3
4
rstb.royalsocietypublishing.org
Here, I consider another three-player game. Similar to the previous example, strategies A and B correspond to cooperation
and defection, respectively. A cooperator pays the cost, C,
whereas a defector pays nothing. The difference between the
public goods game and this game is that the benefit of
cooperation, 3B, arises if and only if all the three players
cooperate; in that case this benefit is equally distributed. Let
me call this game the volunteer’s dilemma after Archetti et al.
[67] (see also [68]). The pay-off matrix of this game is given by
a0 a1 a2
C C C þ B
¼
:
(4:4)
b0 b1 b2
0
0
0
5. Relationship between the degree of the game
and degree of genetic association
Downloaded from http://rstb.royalsocietypublishing.org/ on July 12, 2017
6. Conclusion
In this paper, I have derived evolutionary game dynamics when
family members of size n are engaged in a social interaction.
The genetic structure of those family members is described by
probabilities u‘!m, which tell us the chance that ‘ different individuals have m different origins. In other words, we need to
know the coalescent tree structure of those individuals. Genetic
association between two individuals is often measured by a
relatedness coefficient, but I have shown that such dyadic
relationship is, generally speaking, not sufficient to exactly tell
the resulting evolutionary dynamics. To see why, imagine a
task that can be completed only by full cooperation of all n members in the group. In such a case, the chance of success is equal to
the chance of everyone being a cooperator, and therefore the
probability with which those n individuals are simultaneously
cooperators, namely n-tuple genetic association, is important.
As one can see in this example, the pay-off in a multiplayer
game is usually determined by joint effects of one’s strategy
and others’ strategies. That is an intuitive reason of why
higher-order genetic association is important. By contrast, for
two-players games, one’s pay-off is determined by one’s own
strategy and the partner’s strategy, so a dyadic genetic association is enough to describe the evolutionary dynamics unless
local competition is assumed (see eqn (15) in Lehmann &
Keller [43] for example, and see also the discussion below).
It is not trivial to describe the kin structure of groups of
more than two individuals. For example, McElreath & Boyd
[72, p. 162] describe an n-player kin-biased group formation,
assuming that the probability with which a colleague of an
A-player adopting the same strategy is r þ (1 2 r)p, and that
the probability with which two colleagues of an A-player simultaneously adopting strategy A is its square, [r þ (1 2 r)p]2.
However, the latter argument is generally wrong, because the
conditional probability with which the second colleague adopting strategy A, given that both the first colleague and the focal
individual adopt strategy A, is not equal to r þ (1 2 r)p if we
take triplet genetic association into account.
One might wonder why I have not taken a more straightforward way to derive the evolutionary dynamics than considering
genetic association, u. In fact, van Veelen [69] provided a general
machinery to analyse the n-player, two-strategy games studied
here. He explicitly considered distribution of groups that differ
in the number of A-players. Let us denote by qk the frequency of
groups with exactly k many A-players and (n 2 k) many Bplayers. In that case, the evolutionary dynamics, equation
(2.4), is expressed in terms of qk (0 k n). However, it is not
5
Phil. Trans. R. Soc. B 369: 20130359
where strategies A and B represent cooperation and defection, respectively. The pay-offs B and C, respectively, represent
the benefit and the cost of cooperation, and the pay-off D
represents the synergistic effect of mutual cooperation.
obvious how measures of genetic association are mapped to
those frequencies, q. Therefore, I have given a different
expression of the same evolutionary dynamics in terms of genetic association to facilitate applications to family–group
interactions. In studying interactions of three full-sibs, for
example, it is much easier to specify their genetic association
(r ¼ 1/2 and s ¼ 1/4, as in §4) than to explicitly calculate the
frequencies of different groups, (q0, q1, q2, q3).
I have not explicitly modelled how assortative group formation favouring kin is done, in order to make the scope
of application of my result as wide as possible. One of the mechanisms of such assortment is via kin recognition [45,46,73],
where individuals discriminate others by using genetic or phenotypic markers. Understanding the interplay between social
interaction and kin recognition is important, and extending
the current model in such a direction should further enrich
our understanding of social evolution based on nepotism.
Another plausible mechanism of assortative group formation is population viscosity often via limited dispersal
[58,59,74–76]. Wright’s island model [77] and its variants are
typical ones in population genetics [30,44,45,78,79], and others
prefer network formulations [80–82]. In any case, limited dispersal causes endogenous clustering of individuals with a
common ancestry, thus those relatives are likely to interact.
However, in these models, the effect of kin competition is not
negligible, because limited dispersal hinders the transport of
genes to the global gene pool, thus relatives must compete
with each other over reproduction [38,75]. Such an effect is not
incorporated in the fitness function, equation (2.3), because I
have assumed global competition. Peña et al. recently studied
n-player games in Wright’s island model (J. Peña 2014, personal
communication), which incorporate not only the primary effect
of social interaction on evolutionary dynamics but also its
secondary effect realized as local kin competition in demes.
The concept of degree of games introduced in my paper
will be very useful to elucidate the connection between nonlinearity of games and degree of genetic association involved
in the evolutionary dynamics. The reason why, in some cases,
dyadic genetic relatedness, r, is sufficient for describing evolutionary dynamics is now clear, because the game is of degree
one (i.e. the pay-offs ak and bk are linear in k). However, the
main result in §5 that connects a game of degree d and genetic
association of degree d þ 1 strongly depends on the specific
assumption of my model that population regulation occurs
only globally. In other words, my model focuses on the primary
effect of social interaction on the fecundity of interactants themselves, but not on the secondary effect that neighbours of those
interactants receive. If one assumes the presence of local competition, such as in the Wright’s island model or in network
models, then a game of degree d would require higher genetic
association than degree d þ 1 for its evolutionary dynamics to
be fully described. An example is given in Nowak et al. [29],
which finds that evolutionary game dynamics of a non-additive
two-player game (its degree is d ¼ 1 except for degenerated
cases) played in the Wright’s island model needs not only
dyadic genetic association (order 2) but also triplet genetic
association (order 3) for its full description.
Although higher-order genetic association is important,
there exists an exception where dyadic genetic association
(dyadic relatedness, r) is enough to fully describe evolution;
that is, when continuous strategies (i.e. mixed strategies) rather
than discrete ones are discussed. In this case, an evolvable trait
can be the probability of using strategy A, which ranges from
rstb.royalsocietypublishing.org
Therefore, an additive game is a degree-1 game (except for
the case of a ¼ b) with the special property that the slopes
of ak and bk (as linear functions of k) are the same.
A good example of an additive degree-1 game is the linear
public goods game, equation (4.1). By contrast, a good example
of a non-additive degree-1 game is the prisoner’s dilemma
game with synergism [20,30,37,43,47,69–71], given as
a0 a1
C C þ B þ D
¼
(D = 0),
(5:4)
0
B
b0 b1
Downloaded from http://rstb.royalsocietypublishing.org/ on July 12, 2017
zero to one. Studying the evolution of continuous strategies in
n-player games is another direction of future study.
for their kind invitation to this special issue, and thank two anonymous
referees for their insightful comments. My special thanks go to Jorge
Peña who provided me with simple proofs in appendices B and D.
Funding statement. The author acknowledges support from JSPS
"
¼E
KAKENHI grant no. 23870009.
"
E
k¼0
n1
X
#
p0 ak fk E[ p0 ]
k¼0
n1
X
(A 3)
#
{ p0 ak þ (1 p0 )bk }fk :
k¼0
Appendix A. Deriving the main result
S,{1,...,n1},jSj¼k
i[S
iS
‘¼k
where the symbol S , {1, . . . , n 1}, jSj ¼ k means that S is a
size-k subset of {1, . . . , n 1}. The summation is taken over all
such subsets.
The intuition behind equation (A 1) is very simple. The
number of A-players being k suggests that k out of n 2 1 players
have their indicator variable, pi, being one and that the others
have their indicator variable, pi, being zero. The latter is equivalent to the variable, 1 2 pi, being one. Therefore, a summand,
Q
Q
i[S pi
iS (1 pi ), equals one if and only if all the members
of subset S adopt strategy A and all the others adopt strategy B,
otherwise, the summand equals zero. Summing it over all
subsets of size k gives fk, as in equation (A 1).
A direct calculation combining equations (2.1)–(2.3) leads to
(1 s)d
Dp ¼
Cov[ p0 , f0 ]
1 þ df
"
#
n1
X
(1 s)d
¼
{ p0 ak þ (1 p0 )bk }fk :
Cov p0 ,
1 þ df
k¼0
Note that fk, calculated as equation (A 1), is a symmetric
polynomial with respect to p1 , . . . , pn1 . From a well-known
result in algebra, such a symmetric polynomial can be rewritten in terms of elementary symmetric polynomials. In fact, it
is not difficult to see that
n1
X
‘
fk ¼
( 1)‘k
s‘ ,
(A 4)
k
(A 2)
The following two identities are useful: Cov[X, Y] ¼
E[XY] E[X]E[Y] and p20 ¼ p0 (remember that p0 is either 1
or 0). Here, E[ ] represents an expectation, the randomness
holds, where s‘ is the elementary symmetric polynomial of
degree ‘;
8
s0 ¼ 1
>
>
>
>
>
s1 ¼ p1 þ þ pn1
>
>
>
<
s2 ¼ p1 p2 þ þ pn2 pn1
(A 5)
>
>
.
>
>
..
>
>
>
>
:
sn1 ¼ p1 pn1 :
Appendix E shows
8
n1
>
>
r‘þ1
>
< E[ p0 s‘ ] ¼
‘
>
n1
>
>
: E[s‘ ] ¼
r‘ :
‘
(A 6)
Combining equations (A 3), (A 4) and (A 6) leads to the
main result, equation (2.4).
Appendix B. Two trivial cases
I rewrite equation (2.4) by swapping the order of sums as
n1 X
‘
‘
n1
(1 s)d X
[(1 r1 )r‘þ1 ak r1 (r‘ r‘þ1 )bk ]
( 1)‘k
1 þ df ‘¼0 k¼0
k
‘
(
(
"
)
)#
n1 ‘
‘
X
X
n1
‘
‘
(1 s)d X
¼
(1 r1 )r‘þ1
ak r1 (r‘ r‘þ1 )
bk :
( 1)‘k
( 1)‘k
1 þ df ‘¼0
‘
k
k
k¼0
k¼0
Dp ¼
For mathematical convenience, I introduce the ‘th forward difference of the sequence {ak }k¼0,..., n1 evaluated at
k ¼ 0, which is denoted by D‘ a0 , as follows. The first forward
difference is defined as
Dak ; akþ1 ak :
(B 2)
Given the definition of the (‘ 2 1)th forward difference (‘ 2),
the ‘th forward difference is recursively defined as
D‘ ak ; D‘1 akþ1 D‘1 ak
(B 3)
0
(we set D a0 ; a0 ). It is known in discrete algebra that
‘
X
‘
D‘ a0 ¼
( 1)‘k
a
k k
k¼0
(B 4)
(B 1)
holds [83], so equation (B 1) is simplified as
n1 (1 s)d X
n1
[(1 r1 )r‘þ1 D‘ a0 r1 (r‘ r‘þ1 )D‘ b0 ]:
Dp ¼
‘
1 þ df ‘¼0
(B 5)
When genetic correlation is absent we have r‘ ¼ p‘ , and
therefore (B 5) becomes
n1 n1
(1 s)d X
[(1 p)p‘þ1 D‘ a0 p( p‘ p‘þ1 )D‘ b0 ]
Dp ¼
1 þ df ‘¼0
‘
n1 X
n1
(1 s)d
¼
[p‘ D‘ a0 p‘ D‘ b0 ]:
p(1 p)
1 þ df
‘
‘¼0
(B 6)
Phil. Trans. R. Soc. B 369: 20130359
First, I write down fk in terms of p1 , . . . , pn1 . The following expression equals one, if and only if the number of
A-strategists among n 2 1 group members of a focal player
is k (0 k n 2 1), and otherwise equals zero:
!
X
Y Y
fk ¼
pi
(1 pi ) ,
(A 1)
6
rstb.royalsocietypublishing.org
Acknowledgements. I thank Stuart West, Andy Gardner and Ashleigh Griffin
of which arises from a random sample of an individual
from the population. Using these, I arrive at
"
#
n1
X
Cov p0 ,
{p0 ak þ (1 p0 )bk }fk
Downloaded from http://rstb.royalsocietypublishing.org/ on July 12, 2017
(B 7)
k¼0
‘¼0
(B 8)
Using the identity equation (B 7) (with p ¼ 1) in equation (B 8)
leads to equation (2.8) in the main text.
Appendix C. Number of free parameters
The extent of assortment during kin group formation is qualitatively measured by probabilities u‘!m (1 m ‘ n).
However, there exists trivial relation among those us. This
section clarifies such underlying dependency.
Imagine a group of n individuals. Those players may have
different origins; some may share a common ancestor, and
others may not. If we classify those n-players in terms of
their common ancestors, then we can obtain a partition
of them. Let Fn!{n1 ,...,nm } be the probability with which those
n-players have m different origins, with n1 players sharing a
common ancestor, n2 players sharing a different common
ancestor, and so on (n1 n2 nm ).
For example, for n ¼ 4, we have five different partitions of four players as f4g, f3,1g, f2,2g, f2,1,1g, f1,1,1,1g
and therefore we can define five probabilities, F4!f4g,
F4!f3,1g, F4!f2,2g, F4!f2,1,1g and F4!f1,1,1,1g, that sum up to
unity. It is not difficult to see that from these five probabilities
we can calculate u‘!m (1 m ‘ 4), because knowing how
n-players split is enough to tell into how many groups their
subset of ‘ players split. In fact, us are given by
1
1 0
0
1
1
1
1
1
u1!1
B u2!1 C B 1 1/2 1/3 1/6 0 C
C
C B
B
B u2!2 C B 0 1/2 2/3 5/6 1 C0
1
C
C B
B
F4!{4}
C
B u3!1 C B 1 1/4
0
0
0
CB F4!{3,1} C
C B
B
B u3!2 C B 0 3/4
C
B
1
1/2 0 C
CB F4!{2,2} C: (C 1)
C¼B
B
CB
B u3!3 C B 0
C
0
0
1/2
1
C@ F4!{2,1,1} A
C B
B
C
B u4!1 C B 1
0
0
0
0
C F4!{1,1,1,1}
C B
B
B u4!2 C B 0
1
1
0
0C
C
C B
B
@ u4!3 A @ 0
0
0
1
0A
0
0
0
0
1
u4!4
Therefore, the degree of freedom among us is five minus one,
that is, four.
E[ p0 s1 ] ¼
¼
should be fully described by P2 2 1 ¼ 1 parameter. To be more
precise, for n ¼ 2, it is enough to specify F2!f2g. Setting
F2!f2g ¼ r leads to F2!f1,1g ¼ 1 2 r, and therefore u2!1 ¼ r
and u2!2 ¼ 1 2 r follow, yielding equation (3.1) in the main text.
Similarly, the matrix
0
1
u1!1
@ u2!1 u2!2
A
(C 3)
u3!1 u3!2 u3!3
should fully be described by P3 2 1 ¼ 2 parameters. To see
this, consider the three probabilities, F3!f3g, F3!f2,1g and
F3!f1,1,1g. From these, the probabilities u‘!m (‘ different
samples having m origins) are given by
1
1 0
0
1
1 1
u1!1
B u2!1 C B 1 1=3 0 C0
1
C
C B
B
F3!{3}
B u2!2 C B 0 2=3 1 C
C@ F3!{2,1} A:
C B
B
(C 4)
B u3!1 C ¼ B 1
0 0C
C F3!{1,1,1}
C B
B
@ u3!2 A @ 0
1 0A
0
0 1
u3!3
Setting F3!f3g ¼ s and F3!f3g þ F3!f2,1g/3 ¼ r leads to
equation (4.7) in the main text.
Appendix D. Degree theorem
Here, I prove the claim made in §5. It is known that if ak
is a polynomial in k of degree d, then its dth forward difference is non-zero and its mth (m . d) forward difference is
zero, namely
Dd ak = 0,
Dm ak ¼ 0 (m . d)
(D 1)
[83]. From equations (B 5) and (D 1), it is obvious that genetic
association of order up to dþ1 appears, but none of the higher
association appear in equation (B 5). It is also easy to see that
the two rdþ1s in equation (B 5) never cancel out, because one
has the factor 12 r1, whereas the other has the factor r1.
Appendix E. Proof of equation (A 6)
Below, I prove equation (A 6) for n ¼ 4 and ‘ ¼ 1, but a
similar reasoning covers all n and ‘.
For that purpose, I use an absolute indexing of players
rather than use relative ones. To be more precise, let p (i,j ) be
the indicator variable of jth player (j ¼ 1, . . . , n) in ith
group. Under this notation, an expectation under the relative
indexing in equation (A 6) turns to
1 X 1 (i,1) (i,2)
[ p ( p þ p(i,3) þ p(i,4) ) þ p(i,2) ( p(i,1) þ p(i,3) þ p(i,4) ) þ p(i,3) ( p(i,1) þ p(i,2) þ p(i,4) ) þ p(i,4) ( p(i,1) þ p(i,2) þ p(i,3) )]
nG i 4
1 X 1 (i,1) (i,2)
[ p p þ p(i,1) p(i,3) þ p(i,1) p(i,4) þ p(i,2) p(i,3) þ p(i,2) p(i,4) þ p(i,3) p(i,4) ] ,
3
|{z}
nG i 6
41
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
¼
1
¼r1þ1
(E 1)
7
Phil. Trans. R. Soc. B 369: 20130359
(see [84]) in equation (B 6) leads to equation (2.6) in the
main text.
Next, I consider the case where all n individuals in a group
are clonal. In this case, we have r0 ¼ 1 and r‘ ¼ p (‘ 1).
Substituting these in equation (B 5) leads to
"(
)
#
n1 X
n1
(1 s)d
‘
(1 p)pD a0 p(1 p)b0
Dp ¼
1 þ df
‘
‘¼0
"(
)
#
n1 X
n1
(1 s)d
‘
¼
D a 0 b0 :
p(1 p)
1 þ df
‘
‘¼0
Generally speaking, the total number of ways to partition
a given integer n into a sum of positive integers is called ‘the
partition number of n’, and here I denoted it by Pn. For
example, (P1, P2, P3, P4, P5, P6,. . . ) ¼ (1, 2, 3, 5, 7, 11,. . . ).
The argument above suggests that the degree of freedom
among u‘!m (1 m ‘ n) is Pn 2 1. Therefore, the matrix
u1!1
(C 2)
u2!1 u2!2
rstb.royalsocietypublishing.org
Using the identity
n1 n1 X
X
n1
n1
p‘ D‘ a0 ¼
pk (1 p)n1k ak
‘
k
Downloaded from http://rstb.royalsocietypublishing.org/ on July 12, 2017
where nG is the total number of groups. Similarly, I obtain
¼
1 X 1 (i,2)
[( p þ p(i,3) þ p(i,4) ) þ ( p(i,1) þ p(i,3) þ p(i,4) ) þ ( p(i,1) þ p(i,2) þ p(i,4) ) þ ( p(i,1) þ p(i,2) þ p(i,3) )]
nG i 4
1 X 1 (i,1)
[ p þ p(i,2) þ p(i,3) þ p(i,4) ] :
3
|{z}
nG i 4
41
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
¼
1
(E 2)
¼r1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Hamilton WD. 1964 The genetical evolution of
social behaviour, I & II. J. Theor. Biol. 7, 1–52.
(doi:10.1016/0022-5193(64)90038-4)
Michod RE, Hamilton WD. 1980 Coefficients of
relatedness in sociobiology. Nature 288, 694–697.
(doi:10.1038/288694a0)
Queller DC. 1992 A general model for kin selection.
Evolution 46, 376–380. (doi:10.2307/2409858)
Crozier RH, Pamilo P. 1996 Evolution of social insect
colonies: sex allocation and kin selection. Oxford, UK:
Oxford University Press.
Taylor PD, Frank SA. 1996 How to make a kin
selection model. J. Theor. Biol. 180, 27 – 37.
(doi:10.1006/jtbi.1996.0075)
Frank SA. 1998 Foundations of social evolution.
Princeton, NJ: Princeton University Press.
Gardner A, West SA, Wild G. 2011 The genetical
theory of kin selection. J. Evol. Biol. 24, 1020 –
1043. (doi:10.1111/j.1420-9101.2011.02236.x)
Frank SA. 2013 Natural selection. VII. History and
interpretation of kin selection theory. J. Evol. Biol.
26, 1151 –1184. (doi:10.1111/jeb.12131)
Hamilton WD. 1972 Altruism and related
phenomena, mainly in social insects. Annu. Rev.
Ecol. Syst. 3, 193–232. (doi:10.1146/annurev.es.03.
110172.001205)
Michod RE. 1982 The theory of kin selection. Annu.
Rev. Ecol. Syst. 13, 23 –55. (doi:10.1146/annurev.es.
13.110182.000323)
Foster KR, Wenseleers T, Ratnieks FL. 2006 Kin
selection is the key to altruism. Trends Ecol. Evol.
21, 57 –60. (doi:10.1016/j.tree.2005.11.020)
Lehmann L, Keller L. 2006 The evolution of
cooperation and altruism. A general framework and
classification of models. J. Evol. Biol. 19, 1365 –
1378. (doi:10.1111/j.1420-9101.2006.01119.x)
Nowak MA. 2006 Five rules for the evolution of
cooperation. Science 314, 1560–1563. (doi:10.
1126/science.1133755)
West SA, Griffin AS, Gardner A. 2007 Evolutionary
explanations for cooperation. Curr. Biol. 17, R661 –
R672. (doi:10.1016/j.cub.2007.06.004)
Gardner A, West SA. 2010 Greenbeards. Evolution
64, 25 –38. (doi:10.1111/j.1558-5646.2009.
00842.x)
Lehmann L, Rousset F. 2010 How life history and
demography promote or inhibit the evolution of
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
helping behaviours. Phil. Trans. R. Soc. B 365,
2599 –2617. (doi:10.1098/rstb.2010.0138)
Grafen A. 1985 Hamilton’s rule OK. Nature 318,
310 –311. (doi:10.1038/318310a0)
Bourke AFG. 2011 Principles of social evolution.
Oxford, UK: Oxford University Press.
Trivers RL. 1971 The evolution of reciprocal altruism.
Q. Rev. Biol. 46, 35 –57. (doi:10.1086/406755)
Queller DC. 1985 Kinship, reciprocity and synergism
in the evolution of social behavior. Nature 318,
366 –367. (doi:10.1038/318366a0)
Fletcher JA, Zwick M. 2006 Unifying the theories of
inclusive fitness and reciprocal altruism. Am. Nat.
168, 252 –262. (doi:10.1086/506529)
Maynard Smith J, Price GR. 1973 The logic of
animal conflict. Nature 246, 15 –18. (doi:10.1038/
246015a0)
Maynard Smith J. 1982 Evolution and the theory of
games. Cambridge, UK: Cambridge University Press.
Maynard Smith J. 1978 Optimization theory in
evolution. Annu. Rev. Ecol. Syst. 9, 31– 56. (doi:10.
1146/annurev.es.09.110178.000335)
Grafen A. 1979 The hawk-dove game played
between relatives. Anim. Behav. 27, 905 –907.
(doi:10.1016/0003-3472(79)90028-9)
Michod RE. 1980 Evolution of interactions in familystructured populations; mixed mating models.
Genetics 96, 275–296.
Roze D, Rousset F. 2004 The robustness of
Hamilton’s rule with inbreeding and dominance: kin
selection and fixation probabilities under partial sib
mating. Am. Nat. 164, 214–231. (doi:10.1086/
422202)
van Veelen M. 2009 Group selection, kin selection,
altruism and cooperation: when inclusive fitness is
right and when it can be wrong. J. Theor. Biol. 259,
589 –600. (doi:10.1016/j.jtbi.2009.04.019)
Nowak MA, Tarnita C, Wlison EO. 2010 The
evolution of eusociality. Nature 466, 1057–1062.
(doi:10.1038/nature09205)
Ohtsuki H. 2010 Evolutionary games in Wright’s
island model: kin selection meets evolutionary
game theory. Evolution 64, 3344 –3353. (doi:10.
1111/j.1558-5646.2010.01117.x)
Taylor PD. 2013 Inclusive and personal fitness in
synergistic evolutionary games on graphs. J. Theor.
Biol. 325, 76 –82. (doi:10.1016/j.jtbi.2013.02.002)
32. Cavalli-Sforza LL, Feldman MW. 1978 Darwinian
selection and ‘altruism’. J. Theor. Biol. 14,
263–281.
33. Orlove MJ, Wood CL. 1978 Coefficients of
relationship and coefficients of relatedness in kin
selection: a covariance form for the Rho formula.
J. Theor. Biol. 73, 679–686. (doi:10.1016/00225193(78)90129-7)
34. Seger J. 1981 Kinship and covariance. J. Theor. Biol.
91, 191– 213. (doi:10.1016/0022-5193(81)90380-5)
35. Karlin S, Matessi C. 1983 Kin selection and altruism.
Proc. R. Soc. Lond. B 219, 327 –353. (doi:10.1098/
rspb.1983.0077)
36. Grafen A. 1985 A geometric view of relatedness.
Oxford Surv. Evol. Biol. 2, 28 –89.
37. Queller DC. 1992 Quantitative genetics, inclusive
fitness, and group selection. Am. Nat. 139,
540–558. (doi:10.1086/285343)
38. Queller DC. 1994 Genetic relatedness in viscous
populations. Evol. Ecol. 8, 70 –73. (doi:10.1007/
BF01237667)
39. Rousset F, Billiard S. 2000 A theoretical basis for
measures of kin selection in subdivided populations:
finite populations and localized dispersal. J. Evol.
Biol. 13, 814–825. (doi:10.1046/j.1420-9101.2000.
00219.x)
40. Rousset F. 2004 Genetic structure and selection in
subdivided populations. Princeton, NJ: Princeton
University Press.
41. Wild G, Traulsen A. 2007 The different limits of
weak selection and the evolutionary dynamics of
finite populations. J. Theor. Biol. 247, 382– 390.
(doi:10.1016/j.jtbi.2007.03.015)
42. Traulsen A. 2010 Mathematics of kin- and groupselection: formally equivalent? Evolution 64,
316–323. (doi:10.1111/j.1558-5646.2009.00899.x)
43. Lehmann L, Keller L. 2006 Synergy, partner choice
and frequency dependence: their integration into
inclusive fitness theory and their interpretation in
terms of direct and indirect fitness effects. J. Evol.
Biol. 19, 1426– 1436. (doi:10.1111/j.1420-9101.
2006.01200.x)
44. Lehmann L, Rousset F, Roze D, Keller L. 2007
Strong-reciprocity or strong-ferocity? A population
genetic view of the evolution of altruistic
punishment. Am. Nat. 170, 21 –36. (doi:10.1086/
518568)
Phil. Trans. R. Soc. B 369: 20130359
References
rstb.royalsocietypublishing.org
E[s1 ] ¼
8
Downloaded from http://rstb.royalsocietypublishing.org/ on July 12, 2017
71. Gardner A, West SA, Barton N. 2007 The relation
between multilocus population genetics and social
evolution. Am. Nat. 169, 207–226. (doi:10.1086/
510602)
72. McElreath R, Boyd R. 2007 Mathematical models of
social evolution: a guide for the perplexed. Chicago,
IL: University of Chicago Press.
73. Lehmann L, Feldman MW, Rousset F. 2009 On the
evolution of harming and recognition in finite
panmictic and infinite structured populations.
Evolution 63, 2896– 2913. (doi:10.1111/j.15585646.2009.00778.x)
74. Queller DC. 1992 Does population viscosity promote
kin selection? Trends Ecol. Evol. 7, 322–324.
(doi:10.1016/0169-5347(92)90120-Z)
75. Taylor PD. 1992 Altruism in viscous populations: an
inclusive fitness model. Evol. Ecol. 6, 352 –356.
(doi:10.1007/BF02270971)
76. Wilson DS, Pollock GB, Dugatkin LA. 1992
Can altruism evolve in purely viscous populations?
Evol. Ecol. 6, 331–341. (doi:10.1007/BF02270969)
77. Wright S. 1931 Evolution in Mendelian populations.
Genetics 16, 97 –159.
78. Taylor PD, Irwin A, Day T. 2000 Inclusive fitness in
finite deme structured and stepping-stone
populations. Selection 1, 83 –93.
79. Irwin A, Taylor PD. 2001 Evolution of altruism in
stepping-stone populations with overlapping
generations. Theor. Popul. Biol. 60, 315 –325.
(doi:10.1006/tpbi.2001.1533)
80. Ohtsuki H, Hauert C, Lieberman E, Nowak MA. 2006
A simple rule for the evolution of cooperation on
graphs and social networks. Nature 441, 502 –505.
(doi:10.1038/nature04605)
81. Lehmann L, Keller L, Sumpter D. 2007 The evolution
of helping and harming on graphs: the return of the
inclusive fitness effect. J. Evol. Biol. 20, 2284–2295.
(doi:10.1111/j.1420-9101.2007.01414.x)
82. Santos FC, Santos MD, Pacheco JM. 2008 Social
diversity promotes the emergence of cooperation in
public goods games. Nature 454, 213 –216.
(doi:10.1038/nature06940)
83. Goldberg S. 1958 Introduction to difference
equations: with illustrative examples from
economics, psychology, and sociology. New York, NY:
John Wiley & Sons.
84. Phillips GM. 2003 Interpolation and approximation
by polynomials. New York, NY: Springer.
9
Phil. Trans. R. Soc. B 369: 20130359
57. Fukuyo M, Sasaki A, Kobayashi I. 2012 Success of a
suicidal defense strategy against infection in a
structured habitat. Sci. Rep. 2, 238. (doi:10.1038/
srep00238)
58. Kümmerli R, Griffin AS, West SA, Buckling A,
Harrison F. 2009 Viscous medium promotes
cooperation in the pathogenic bacteria
Pseudomonas aeruginosa. Proc. R. Soc. B 276,
3531 –3538. (doi:10.1098/rspb.2009.0861)
59. Kümmerli R, Jiricny N, Clarke LS, West SA, Griffin AS.
2009 Limited dispersal, budding dispersal and
cooperation: an experimental study. Evolution 63,
939–949. (doi:10.1111/j.1558-5646.2008.00548.x)
60. Trivers RL. 1985 Social evolution. Menlo Park, CA:
Benjamin/Cummings Publishing Co.
61. Silk JB. 2002 Kin selection in primate groups.
Int. J. Primatol. 23, 849– 875. (doi:10.1023/
A:1015581016205)
62. Silk JB. 2009 Nepotistic cooperation in nonhuman
primate groups. Phil. Trans. R. Soc. B 364,
3243 –3254. (doi:10.1098/rstb.2009.0118)
63. Price GR. 1970 Selection and covariance. Nature
227, 520 –521. (doi:10.1038/227520a0)
64. van Veelen M. 2005 On the use of the Price
equation. J. Theor. Biol. 237, 412–426. (doi:10.
1016/j.jtbi.2005.04.026)
65. Gokhale CS, Trataulsen A. 2010 Evolutionary
games in the multiverse. Proc. Natl Acad. Sci.
USA 107, 5500–5504. (doi:10.1073/pnas.
0912214107)
66. Hamilton WD. 1970 Selfish and spiteful behavior in
an evolutionary model. Nature 228, 1218–1220.
(doi:10.1038/2281218a0)
67. Archetti M, Scheuring I, Hoffman M, Frederickson
ME, Pierce NE, Yu DW. 2011 Economic game theory
for mutualism and cooperation. Ecol. Lett. 14,
1300 –1312. (doi:10.1111/j.1461-0248.2011.
01697.x)
68. Diekmann A. 1985 Volunteer’s dilemma. J. Conflict
Resolut. 29, 605–610. (doi:10.1177/0022002785
029004003)
69. van Veelen M. 2011 The replicator dynamics with n
players and population structure. J. Theor. Biol. 276,
78 –85. (doi:10.1016/j.jtbi.2011.01.044)
70. Wenseleers T. 2006 Modelling social evolution: the
relative merits and limitations of a Hamilton’s rulebased approach. J. Evol. Biol. 19, 1419–1422.
(doi:10.1111/j.1420-9101.2006.01144.x)
rstb.royalsocietypublishing.org
45. Rousset F, Roze D. 2007 Constraints on the origin
and maintenance of genetic kin recognition.
Evolution 61, 2320 –2330. (doi:10.1111/j.15585646.2007.00191.x)
46. Antal T, Ohtsuki H, Wakeley J, Taylor P, Nowak MA.
2009 Evolution of cooperation by phenotypic
similarity. Proc. Natl Acad. Sci. USA 106,
8597–8600. (doi:10.1073/pnas.0902528106)
47. Ohtsuki H. 2012 Does synergy rescue the evolution
of cooperation? An analysis for homogeneous
populations with non-overlapping generations.
J. Theor. Biol. 307, 20 –28. (doi:10.1016/j.jtbi.
2012.04.030)
48. Taylor PD, Maciejewski W. 2012 An inclusive fitness
analysis of synergistic interactions in structured
populations. Proc. R. Soc. B 279, 4596 –4603.
(doi:10.1098/rspb.2012.1408)
49. Crespi BJ. 2001 The evolution of social behavior in
microorganisms. Trends Ecol. Evol. 16, 178–183.
(doi:10.1016/S0169-5347(01)02115-2)
50. Rainey PB, Rainey K. 2003 Evolution of cooperation
and conflict in experimental bacterial populations.
Nature 425, 72 –74. (doi:10.1038/nature01906)
51. Velicer GJ, Yu YTN. 2003 Evolution of novel
cooperative swarming in the bacterium Myxococcus
xanthus. Nature 425, 75 –78. (doi:10.1038/
nature01908)
52. Keller L, Surette MG. 2006 Communication in
bacteria: an ecological and evolutionary perspective.
Nat. Rev. Microbiol. 4, 249–258. (doi:10.1038/
nrmicro1383)
53. Kerr B, Neuhauser C, Bohannan BJM, Dean AM.
2006 Local migration promotes competitive restraint
in a host-pathogen ‘tragedy of the commons’.
Nature 442, 75 –78. (doi:10.1038/nature04864)
54. West SA, Griffin AS, Gardner A, Diggle SP. 2006
Social evolution theory for microorganisms. Nat.
Rev. Microbiol. 4, 597– 607. (doi:10.1038/
nrmicro1461)
55. Diggle SP, Griffin AS, Campbell GS, West SA. 2007
Cooperation and conflict in quorum-sensing
bacterial populations. Nature 450, 411–414.
(doi:10.1038/nature06279)
56. Brown SP, West SA, Diggle SP, Griffin AS. 2009
Social evolution in micro-organisms and a Trojan
horse approach to medical intervention strategies.
Phil. Trans. R. Soc. B 364, 3157–3168. (doi:10.
1098/rstb.2009.0055)