Download Group selection and kin selection: formally equivalent approaches

Opinion
Group selection and kin selection:
formally equivalent approaches
James A.R. Marshall
Department of Computer Science/Kroto Research Institute, University of Sheffield, Sheffield, S3 7HQ, UK
Inclusive fitness theory, summarised in Hamilton’s
rule, is a dominant explanation for the evolution of
social behaviour. A parallel thread of evolutionary theory holds that selection between groups is also a
candidate explanation for social evolution. The mathematical equivalence of these two approaches has long
been known. Several recent papers, however, have
objected that inclusive fitness theory is unable to deal
with strong selection or with non-additive fitness
effects, and concluded that the group selection framework is more general, or even that the two are not
equivalent after all. Yet, these same problems have
already been identified and resolved in the literature.
Here, I survey these contemporary objections, and
examine them in the light of current understanding
of inclusive fitness theory.
Approaches to understanding social evolution
The theory of inclusive fitness, formalised by Hamilton
[1,2], was arguably the most fundamental advance in
understanding evolution since Darwin. Hamilton’s
breakthrough was to realise that natural selection acts
not only on genes according to their effect on the fitness of
their bearers, but also according to the fitness change
they effect on genetic relatives containing copies of the
same gene [1,2]. Inclusive fitness theory has had great
success in explaining diverse aspects of social evolution
(e.g. [3–8]).
Inclusive fitness theory is not the only attempt to understand social evolution, however. An alternative perspective is to argue that natural selection acts on
groups, and those groups whose members are more frequently pro-social outcompete those with fewer pro-social
members. Although the theory of group selection had
inauspicious origins, as proponents talked of ‘benefit to
the group’ while neglecting the importance of ‘benefit to the
individual’ [9], since Hamilton’s original work a contemporary version of the group selection perspective has been
shown, in fact, to be simply a different viewpoint on the
same process as that described by inclusive fitness theory
[10]. Despite this, several recent papers argue that the two
are distinct processes [11] (but see [12–18]), or that inclusive fitness theory is vulnerable to limitations that do not
trouble group selection [11,19,20]. As a result, some
authors argue that group selection should now become
the predominant explanation for the evolution of social
behaviour [11].
Corresponding author: Marshall, J.A.R. (James.Marshall@sheffield.ac.uk).
Historical roots
The roots of the theories of social evolution discussed here
extend back into at least the mid-19th century. Over these
many years, concepts and terminology have shifted, even
when used by the same authors. This section summarises
these roots, which are discussed at greater length else-
Glossary
Altruism: donation of aid to another individual or individuals, such that their
lifetime individual fitness is increased while the lifetime individual fitness of
the donor is decreased (see ‘Benefit’ and ‘Cost’).
Benefit: the lifetime individual fitness increment resulting from receipt of aid
(see ‘Altruism’).
Cost: the lifetime individual fitness decrement resulting from donation of aid
(see ‘Altruism’).
Fecundity: the total production of potentially reproductive offspring of an
individual (cf. ‘Fitness’).
Fitness: the long-term descendants of an individual over evolutionary time (cf.
‘Fecundity’).
Group selection: a partitioning of selective forces into between-group and
within-group components. Group selection theory is mathematically equivalent to inclusive fitness theory.
Hamilton’s rule: a summary prediction of the direction of selection on some
social trait, according to inclusive fitness theory, taking account of genetic
relatedness, and costs and benefits of interactions defined in terms of fitness.
Inclusive fitness theory: the theory of natural selection extended to deal with
inclusive fitness. The direction of selection according to inclusive fitness theory
is often summarised in Hamilton’s rule. Inclusive fitness theory is mathematically equivalent to group selection theory.
Inclusive fitness: the total fitness of an individual owing to the effects of their
own actions on their own individual fitness, additionally taking account of the
effects of the action of an individual on the individual fitnesses of other
population members, weighted by the genetic relatedness of the focal
individual to them. Inclusive fitness is mathematically equivalent to personal
fitness.
Individual fitness (or direct fitness): the fitness of an individual calculated in
terms of direct descendants.
Kin selection: a popular term for inclusive fitness theory, originating from early
explanations of the theory in terms of relatedness estimated from pedigree.
Payoff: the outcome of a social interaction in some currency that is
proportional to changes in individual fecundity. Care must be taken in making
evolutionary predictions based on using payoff as a proxy for fitness.
Personal fitness (or neighbour-modulated fitness): the total fitness of an
individual owing to the effects of their own actions on their own individual
fitness, as well as the effects owing to the actions of other population
members. Personal fitness is mathematically equivalent to inclusive fitness.
Price equation: a general equation for describing change in terms of selective
and other forces. When applied to modelling evolution, the Price equation
considers the association between possession of a trait, and individual
fecundity.
Relatedness: a measure of the genetic similarity between two or more
individuals, often described in terms of pedigree, but more correctly
considered as genetic association howsoever caused.
Strong selection: selection on a trait having a large average effect on the
fitness of the bearer (see ‘Weak selection’).
Weak selection: selection on a trait having a small average effect on the fitness
of the bearer in a population; for example, owing either to a small effect when
expressed relative to other fitness components, or to rare expression (cf.
‘Strong selection’).
0169-5347/$ – see front matter ß 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tree.2011.04.008 Trends in Ecology and Evolution, July 2011, Vol. 26, No. 7
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Trends in Ecology and Evolution July 2011, Vol. 26, No. 7
where [21–23], but also settles on contemporary definitions
of the key concepts to be discussed here.
What is group selection?
Even allowing for the earliest glimmerings of inclusive
fitness theory during the ‘modern synthesis’, group selection can lay claim to a much longer intellectual pedigree.
Darwin himself discussed the possibility of selection at the
level of the tribe favouring altruism [24]. Models of early
proposals for group selection, based on differential survival
and reproduction of isolated groups, led some to conclude
that group selection was unlikely to be important, owing to
its susceptibility to non-altruist cheats invading groups of
altruists [25]. Others considered that group-level benefits
arising from individual selection could occur, but that
adaptations exclusively for the benefit of the group were
unlikely [9]. With these apparently definitive analyses,
group selection receded as an explanation for social evolution.
It was not long, however, until new approaches
showed how, for a population subdivided into groups of
interacting individuals, selection can be decomposed into
within and between-group components [26], and altruism
can be favoured when between-group selection for altruistic groups exceeds within-group selection against altruists [10,27]. Given that most models of group selection to
date are of this type [21], one of the main aims here is
to summarise the relationship between this contemporary conception of group selection, and inclusive fitness
theory.
What is inclusive fitness theory?
Inclusive fitness theory is a generalisation of classical
Darwinian theory that deals with social behaviour,
achieved by extending Darwin’s concept of individual fitness to also take account of the effects of the actions of an
individual on the fitness of others they interact with [1,2].
The fitness effects on others are weighted according to their
relatedness to the focal individual, because close relatives
are more likely to share genes for the social behaviour of
interest than are distant relatives. The problem of how to
define relatedness is actually nontrivial, but before dealing
Box 1. Eleven misunderstandings of inclusive fitness theory
More than 30 years ago, Richard Dawkins listed ‘twelve misunderstandings of kin selection’ [91]. Today, eleven mostly different
misunderstandings seem worth addressing (Table I). Many of these misunderstandings have a long history but, for simplicity, only the most
recent re-statements of them are cited.
Table I. Inclusive fitness theory: misunderstandings and reality
Misunderstanding
Inclusive fitness theory is Hamilton’s rule [11,19]
Inclusive fitness theory is the Price equation [20]
Inclusive fitness theory requires weak selection
and rare mutants [11,20]
Inclusive fitness theory requires fitness
additivity [11,19]
Inclusive fitness theory requires pairwise
interactions [11,19]
Inclusive fitness theory is not dynamically
sufficient [20]
Inclusive fitness is different from personal fitness [11]
Inclusive fitness theory should separate fitness effects
of traits from population structure, and so fails when
social interactions are non-additive [11,17,19]
‘Relatedness’ is pedigree [20,73,74]
Fecundity is fitness [11,19,20]
The group selection and kin selection methodologies
are not equivalent [11,73,74]
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Reality
Hamilton’s rule is a summary of the direction of selection based on inclusive
fitness. Inclusive fitness theory does not preclude more sophisticated analyses
of selection [32,71]
The Price equation is one popular approach to deriving Hamilton’s rule and
studying inclusive fitness [92]. Other approaches, such as population genetics
and evolutionary game theory, are equally applicable [1,31,32,42,65,93]
These are required only to estimate relatedness from pedigree [39],
for Hamilton’s rule to accommodate certain kinds of non-additive fitness
effects [39,58] and for simplifying inclusive fitness models [86,94,95].
Appropriate formulations of Hamilton’s rule relax these requirements [32] (Box 3)
Inclusive fitness can be decomposed into direct and indirect effects even with
non-additivities, although non-additive effects might need to be divided
between interactants (e.g. [31]). Hamilton’s rule can directly accommodate
most forms of non-additivity [32] (Box 3)
Inclusive fitness theory is valid for arbitrary interaction group size [32] (Box 3)
Given that inclusive fitness theory can be studied with various modelling
approaches, one need only chose a methodology appropriate for a particular
model to satisfy dynamic sufficiency [32] (see ‘Inclusive fitness theory is the
Price equation’)
Personal fitness is mathematically equivalent to inclusive fitness, although
only inclusive fitness is meaningful in evolutionary terms (Box 4)
Even for non-social traits, frequency-dependent selection introduces a
relationship between population structure and fitness effects. This relationship
between population structure and fitness effects of traits is a feature of
frequency-dependent selection, and is not peculiar to inclusive fitness theory
‘Relatedness’ can be estimated from pedigree under certain assumptions,
but what matters for inclusive fitness theory is genetic identity, not identity
by descent (Box 3)
This misunderstanding is often unstated and probably unrecognised,
and arises only in the analysis of particular models in terms of Hamilton’s
rule (e.g. [11,51,52])
As this article and other authors have summarised, ‘group selection’
and ‘kin selection’ are equivalent views of the same evolutionary process
[10,31,32,57,63,70,96]
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Trends in Ecology and Evolution July 2011, Vol. 26, No. 7
with that it is perhaps best to summarise the concept of
inclusive fitness in Hamilton’s own words:
Inclusive fitness may be imagined as the personal
fitness which an individual actually expresses in its
production of adult offspring as it becomes after it has
been first stripped and then augmented in a certain
way. It is stripped of all components which can be
considered as due to the individual’s social environment, leaving the fitness which he would express if
not exposed to any of the harms or benefits of that
environment. This quantity is then augmented by
certain fractions of the quantities of harm and benefit
which the individual himself causes to the fitnesses of
his neighbours. The fractions in question are simply
the coefficients of relationship appropriate to the
neighbours whom he affects. [1]
Hamilton described these ‘coefficients of relationship’, or
relatedness, as they would be inferred from pedigree analysis; for example, ‘one-half for sibs, one-quarter for half-sibs,
one eighth for cousins. . .’ [1]. In doing so, Hamilton conceded
the need for weak selection. Yet, he also clearly had in mind
that genetic association rather than kinship was important,
when he described a mechanism for conditionally directing
altruism towards other bearers of an altruist trait, that
subsequently came to be known as ‘greenbeard altruism’
[28]. Despite these subtleties, Hamilton’s theory was quickly
labelled as the theory of ‘kin selection’ [25]. Perhaps because
of this, many contemporary authors appear to understand
relatedness in inclusive fitness theory solely in terms of
pedigree, rather than in genetic terms (Box 1).
Although inclusive fitness is the central concept of
inclusive fitness theory, it also gave rise to a powerful
predictive tool that came to be known as Hamilton’s rule.
This follows very directly and simply from the idea of
inclusive fitness and the observation that, for a particular
social trait to receive positive selection, its inclusive fitness
effect must be positive. That is, for some behaviour having
a fitness cost c to the individual, and conferring a fitness
benefit b on a recipient whose relatedness to the focal
individual is r, then that behaviour will be selected whenever rb–c >0. Various arrangements of this inequality are
possible (e.g. Equation I, Box 3), and all are referred to as
Hamilton’s rule. The rule is a useful summary of selection,
but arguably it is too frequently identified as inclusive
fitness theory, rather than as part of inclusive fitness
theory, as discussed below.
The generality of inclusive fitness theory
Much of the work in refining and generalising inclusive
fitness theory has used the Price equation, a powerful and
general approach to understanding selection and evolution
[29]. Although the Price equation is not the only way to
understand inclusive fitness theory (Box 1), it has proved
both useful and influential. The Price equation shows that
the intergenerational population-level change in some
trait is given by (Equation 1):
DEðGÞ / CovðG; WÞ þ EðWDGÞ
(1)
where G and W are random variables for, respectively, the
value of the trait in question and individual fecundity (see
Glossary), and D is the change from one generation to the
next. If there is no systematic transmission bias for the
trait in question, so that on average offspring inherit traits
faithfully from their parents, the second term is zero. When
this is the case, the evolutionary change is entirely captured by the first term of Equation 1 and, in particular, a
trait G will increase in frequency from one generation to
the next if (Equation 2):
CovðG; WÞ > 0
(2)
Equation 2 can be used to derive a general version of
Hamilton’s rule in terms of fitness costs and benefits
(e.g. [30–32], Box 2), as well as extensions of Hamilton’s
rule specified in terms of conditionally expressed traits and
non-additive payoffs [33,34] (Box 3).
The generality of modern inclusive fitness theory is
discussed below, with reference to the specific objections
raised by recent critics [11,19,20]. An excellent review
of this generality is also to be found in [32]. Further
Box 2. The group selection and kin selection approaches are formally equivalent
The equivalence between the group selection and kin selection
approaches is demonstrated here using the Price equation (Equation
1, main text) [29], although other techniques can be applied (e.g. [70]).
The Price equation can easily be used to derive Hamilton’s rule [1,2]
as follows. First, fecundity is written as (Equation I):
W ¼ G 0 B GC
(B2.I)
where G is the dose in an individual of the altruist gene (e.g. G = 0 or
G = 1 for a single-locus bi-allelic haploid trait), G0 is the total frequency
of the gene in their social partners (which could depend on group size,
denoted N), and C and B are the costs and benefits of altruism,
respectively, in terms of changes in fecundity (potentially reproductive
offspring in the next generation). Now, given that the altruist trait is
favoured whenever Cov(G,W) >0 [Equation 2 (main text), assuming no
transmission bias], this inequality can be rearranged for the fitness
defined in Equation I to give Equation II:
CovðG; G 0 Þ=VarðGÞ > EðC Þ=EðBÞ
(B2.II)
which is Hamilton’s rule with relatedness defined as the genetic
association (formally, a regression coefficient) between an individual
and their social partners [39,82], and averaged costs and benefits [33]
(Box 3).
The link between Hamilton’s rule and decomposition of selection
into between-group and within-group parts is achieved by returning
to the starting point of the previous derivation, Equation 2 (main text).
One simply notes that, by conditioning on a third random variable for
group size (N), Cov(G,W) can be rewritten using the law of total
covariance [83] as Equation III:
CovðG; W Þ ¼ CovðEðGjNÞ; EðW jNÞÞ þ EðCovðG; W jNÞÞ
(B2.III)
This is a partitioning of selection into between-group (first term) and
within-group (second term) components. The expectations and
covariances are all weighted by group size, however, making clear
that this is a different viewpoint on selection acting at the level of
individuals [21]. A similar partitioning, but for constant group sizes,
is given in [84]. The relationship between Hamilton’s rule and the
group selection viewpoint is thus formally established using the
Price equation. The equivalence is valid for arbitrary strength of
selection, as well as for non-additive payoffs, as described in the
main text.
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Box 3. Hamilton’s rule with non-additive payoffs
Hamilton’s original rule [1,2] was expressed as (Equation I):
r >k
(B3.I)
where r is relatedness and k is the cost–benefit ratio c/b in terms of
fitness. A contemporary approach to accounting for non-additive
interactions in Hamilton’s rule is to define fitness costs and benefits
as the partial-regression coefficients that give a best-fit linear model of
fitness [31]. Under this approach, fitness is modelled with a linear
regression (Equation II):
Ŵ ¼ EðW Þ þ bW ;GG0 ðG EðGÞÞ þ bW ;G0 G ðG 0 EðGÞÞ
(B3.II)
where bW,XY is a partial regression coefficient of W on X, holding Y
constant. When these partial regression coefficients are chosen to
2
minimise the unexplained variance of the linear model EðŴ W Þ ,
then costs and benefits of altruism are derived that take account of
payoff non-additivity, and these can be used to recover Hamilton’s rule
in the form of Equation I, with relatedness defined in terms of genetic
association [31].
An alternative approach is to extend Hamilton’s rule with extra
parameters capturing deviation from payoff additivity. This might be
required when non-additive effects accrue during the lifetime of an
individual [39,58]. An influential extension of Hamilton’s rule [30,33]
can be written as Equation III:
CovðG;DÞ
CovðG; P 0 Þ EðC Þ CovðG;P Þ
>
EðBÞ
CovðG; PÞ
r b c þ m d>0
(B3.IV)
where relatedness r, benefit b and deviation from additivity d, as well
as m, are all vectors of moments (mean, variance, skewness, etc.)
derived in terms of Taylor expansions of payoff functions. When
payoffs are additive, then Hamilton’s rule is recovered with relatedness defined as a genetic regression coefficient (Equation II, Box 2)
[34].
(B3.III)
refinements of the theory to deal with more realistic evolutionary questions have also been made (e.g. [35]), but are
not discussed here.
Arbitrary group size
To many, it might seem unnecessary to point out that
inclusive fitness theory is applicable to interactions within
groups of arbitrary size, yet it has recently been asserted
that inclusive fitness can only deal with interactions within
pairs of individuals [11,19]. It is true that inclusive fitness
theory is most simply explained in terms of a single trait
having fitness effects on the actor and a single recipient,
and the relatedness between the two [1,2], and is frequently presented in the context of a two-player payoff matrix.
However, it is a misunderstanding to think this means that
inclusive fitness theory necessarily can only deal with
pairwise interactions, as the quote from [1] given above
illustrates (also see Box 1). The focus on pairwise interactions stems from asking the simplest possible question
about social evolution; when should one individual sacrifice
its own fitness to aid another? Yet such a question can
easily be extended to whether an individual should sacrifice its fitness to aid fellow members of its group, as
formalised in the public-goods game [36], and inclusive
fitness theory can describe such situations [30,37,38]. Box
2 presents Hamilton’s rule in terms of interactions within
groups with more than two members.
Calculating relatedness
Originally, relatedness, r, was discussed in terms of pedigree by Hamilton and others. This contributed to the
labelling of Hamilton’s theory as ‘kin selection’ [25]. Given
that estimation of relatedness from pedigree requires certain assumptions, namely weak selection acting on a
rare gene [39], this has led to the misunderstanding that
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where two things have happened: first, an additional fitness variable D
has been introduced to capture deviations from payoff additivity and,
second, fecundity has been written in terms of phenotypes rather than
genotypes, to allow conditionally expressed traits to be modelled
without the need to derive fitness costs and benefits, as in Equation
II. This phenotypic version of Hamilton’s rule can be related back to the
version with genetic relatedness (e.g. Equation II in Box 2) by interpreting the effects of conditionally expressed traits on the behaviour of
others in terms of indirect genetic effects and indirect genetic relatedness [85].
Still others have extended Hamilton’s rule to accommodate
arbitrary payoff structures by generalising it to use distributions of
relatedness, cost and benefit [34]. In this frame work, Hamilton’s rule
becomes (Equation IV):
inclusive fitness theory requires weak selection [11,20]
(Box 1). In fact, what matters is genetic similarity between
interacting individuals, a fact recognised by Hamilton
himself when describing conditionally cooperative traits
that came to be known as greenbeards [2,28,40]. It was
subsequently recognised that, in models of social evolution,
this could be captured with an assortment parameter
describing the increased tendency for individuals to interact with their own type [41], obviating the need for weak
selection. The derivation of Hamilton’s rule in terms of
genetic association between individuals (Equation II, Box
2) also dispenses with pedigree, and with the need for weak
selection to estimate relatedness from it.
Fitness effects and non-additivity
Further confusion over the applicability of Hamilton’s rule
arises when it is not realised that benefits and costs are
specified in terms of fitness, and instead arbitrary payoffs
proportional to fecundity are used [42,43]. There is a
crucial difference between potentially reproductive offspring, and the correct definition of evolutionary fitness
in terms of long-term descendants [44–46]. There is also a
need, as pointed out in Hamilton’s original presentation of
inclusive fitness [1], to take account of all the effects of the
actions of an individual on the fitness of its relatives.
Solutions to this problem are either to recalculate fitness
costs and benefits in Hamilton’s rule taking account of all
the effects of a social behaviour, including increased competition over future reproduction [47], to recalculate relatedness in terms of the local competitive neighbourhood
[48,49] or to separate the effects of selection and demography [50]. Recognising such problems resolves apparent
paradoxes, such as different population structures with
seemingly the same relatedness having drastically different evolutionary outcomes [11,51,52]. These developments
Opinion
of inclusive fitness theory were motivated by results on
selection for altruism in constant-size populations with
local interactions [47,53,54], but inclusive fitness theory
has also been extended to deal with selection in expanding
populations (surveyed in [55]).
A final complication in the applicability of Hamilton’s
rule comes when non-additive fitness effects are admitted.
If effects are additive then, in the notation of Equation II
(Box 2), simultaneous provision and receipt of altruism will
sum to E(B)–E(C). However, many realistic social interactions are positively or negatively non-additive. For some
time since Hamilton’s rule was first proposed, biologists
have argued that, because of this, the rule is heuristic and
often incorrect, whereas the group selection approach is
more useful (e.g. [56]). Yet, it has also long been recognised
that the two are in fact vulnerable to the same difficulties
with non-additivity [57]. Others have argued that Hamilton’s rule is able to deal with non-additivity if cost and
benefits are correctly defined as average fitness effects
across the population [39,58] except where non-additivities
accrue to the same individuals over their lifetime, in which
case an additional assumption of weak selection is required
[39,58]. Very recently, it has again been claimed that
inclusive fitness theory is valid only for additive interactions [11,19]. Box 3 shows that Hamilton’s rule accommodates non-additivity when the correct definition of fitness
costs and benefits is used, and presents classical and more
recent generalisations of Hamilton’s rule in terms of payoffs. All of these generalisations are valid for arbitrary
selection strength. Some authors appear unconvinced by
this approach, feeling that inclusive fitness theory and
Hamilton’s rule should separate population structure from
the fitness effects of the actions of individuals [17]. It is not
clear where this impression should have come from, for
even in the case of non-social but frequency-dependent
traits, the fitness effect of a gene will be a function of
population structure; this is not a feature peculiar to
inclusive fitness theory, but rather to frequency-dependent
selection itself.
Arbitrary selection strength
It has recently been claimed that inclusive fitness theory is
applicable only in situations where selection is weak
[11,20]. In fact, this is a misunderstanding of inclusive
fitness theory (Box 1). Although certain aspects of inclusive
fitness theory have historically required weak selection,
namely estimating relatedness from pedigree and dealing
with non-additive fitness effects, these have all been
addressed by development of the theory, as summarised
above. Box 3 presents formulations of Hamilton’s rule that
are valid for arbitrary selection strength.
Two perspectives on the same process
Almost since the first presentation of inclusive fitness
theory, it has been recognised that the group selection
and inclusive fitness viewpoints are equivalent [10]. Despite this, some efforts have recently been made to find
models of social evolution that can be understood in terms
of group selection, but not in terms of inclusive fitness
theory [59,60]. These purported examples, as well as others
presented in neither group selection nor inclusive fitness
Trends in Ecology and Evolution July 2011, Vol. 26, No. 7
terms [51,61], have repeatedly been successfully rederived
in terms of inclusive fitness theory [52,62–65]. Inclusive
fitness models have also been analysed in terms of group
selection [66–68]. Current arguments for the nonequivalence of group selection and inclusive fitness theory seem to
rest on misunderstandings that inclusive fitness theory is
applicable only to pairwise and additive interactions, under weak selection (Box 1). In fact, as the previous section
summarised, inclusive fitness theory is subject to none of
these limitations; neither is the equivalence between the
theories of inclusive fitness and group selection contingent
on them, as this section demonstrates.
Just as it was used to derive Hamilton’s rule, the Price
equation can also be used to derive a group, or multilevel,
selection perspective on evolution (Box 2). The usual approach to doing this is to consider the average trait and the
average fitness in a group, and calculate the covariance of
these as the first term of the Price equation (Equation 1).
This is the between-group selection that, when considering
altruism, favours altruists. Then, for the within-group
selection, the covariance between individual trait and
individual fitness within groups (which disfavours altruists), is simply substituted into the second, transmission
bias, term of Equation 1 [10,21,26,69]. It might be, however, that this substitution step has lead to mistrust of the
equivalence between the inclusive fitness and group selection viewpoints [11], yet neither is necessary. Box 2 presents a simple proof of the formal equivalence between the
two perspectives, for arbitrary group size, strength of
selection, and non-additivity of payoff. The equivalence
has also been demonstrated in other ways, such as using
population genetical approaches, although these can require assumptions of weak selection [70]. Given that the
Price equation approach used in Box 2 is valid for arbitrary
traits and arbitrary strengths of selection, the equivalence
it shows is very general. Also, because relatedness is
defined as a genetic correlation, there is no requirement
for weak selection there either (see above). The only matter
open to debate is whether Hamilton’s rule is always valid
in its classical form; of particular interest is whether it is
valid for non-additive fitness effects. In fact, Hamilton’s
rule might have to be extended for certain kinds of nonadditivity [39,58] (Box 3), but this will always result in a
corresponding change in the group selection formulation. It
has long been known that, for non-additive payoffs, both
the inclusive fitness and group selection viewpoints fail to
achieve a clean separation between fitness effects (response to selection) and genetic effects (heritability), for
the same reasons [57]. Thus, despite assertions to the
contrary [11], the group selection and inclusive fitness
viewpoints have long been known to be equivalent perspectives on the same process [10], even for strong selection
and non-additive interactions [57].
Tools are not concepts
As well as the objections addressed above, several other
criticisms of inclusive fitness theory have been made. In
particular, it has been claimed that inclusive fitness theory
is not a dynamical theory of evolution. This objection argues
that inclusive fitness theory can thus only answer questions
about whether a given behaviour is favoured on average or
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Box 4. Personal fitness: inclusive fitness in disguise
In [11], a model for the evolution of eusociality is presented entirely in
terms of personal fitness, along with the claim that this means
inclusive fitness is not involved. In fact, Hamilton himself proposed
precisely the same approach to modelling social evolution as part of
inclusive fitness theory, under the name of ‘neighbour-modulated
fitness’ [1]. This approach was used in early models of interactions
between relatives [41] and in generalising Hamilton’s rule, where it
was shown to be equivalent to the inclusive fitness accounting
approach [30,33]. Today, the term ‘personal fitness’ is generally used,
and it is recognised as being equivalent to inclusive fitness [32], as well
as being increasingly popular among theorists owing to the frequent
difficulty of identifying all the consequences of a social behaviour (as
illustrated in the main text) [87]. Thus models using personal fitness,
such as that in [11], are readily interpreted in inclusive fitness terms (in
fact, the authors themselves concede this equivalence in the supplementary information for their paper [11]). Given the equivalence of the
approaches, one might ask why the inclusive fitness concept is needed
at all, and might claim that social evolution can be explained entirely in
terms of classical Darwinian fitness, extended to include the effects of
the behaviours of others on the fitness of an individual [11]. The
not, but cannot answer questions such as the probability of
fixation of a particular trait, or its expected evolutionary
trajectory, unlike a ‘full’ model of selection in groups [11,20].
This misunderstanding (Box 1) arises from a confusion
between concepts and the tools used to reason about them.
In particular, a prevalent misunderstanding seems to conflate inclusive fitness theory with Hamilton’s rule, and with
the Price equation (Box 1). In fact, many different
approaches to modelling inclusive fitness have been pursued, such as population genetics (e.g. [1,31,35,42,65]) and
evolutionary game theory (e.g. [38,41]). Different tools require different assumptions and, hence, trade generality
against predictive power [71]. The Price equation is very
general as a description of selection, and it is true that it is
not dynamically sufficient, except when applied to certain
types of model [31,72]. Yet richer inclusive fitness models
can achieve greater predictive power at the expense of
reduced generality. In this, the comparison between a caricature of inclusive fitness theory in terms of the Price
equation, and a detailed dynamical or even population
genetical model is unfair. Similarly, it is no fairer to say
that inclusive fitness theory can only predict when a trait is
favoured based on Hamilton’s rule than it is to say group
selection theory is limited to determining whether betweengroup selection exceeds within-group selection, or even that
selection theory in general can only predict whether a trait
receives positive or negative selection. Inclusive fitness,
similar to group fitness and classical fitness, is a concept
that can be analysed with a variety of different tools. Conflating tools with concepts can only cause confusion.
The evolution of eusociality
The primary motivation for recent proposals favouring
group selection over inclusive fitness has been to explain
the evolution of eusociality in social insects [73,74]. In
response, it has been pointed out that this informal
group-selection explanation is really inclusive fitness theory in disguise [75,76]. Since then, a formal mathematical
treatment of the group selection explanation for the evolution of eusociality has been presented [11]. It is illuminating to consider the details of this model in two regards.
330
answer is that, although mathematically the two are equivalent, in
evolutionary terms they are very different. Hamilton introduced the
concept of inclusive fitness as a new quantity that selection should
maximise, as a replacement for classical fitness [1,2]. Formal justification for the concept of natural selection leading to the maximisation of
inclusive fitness has also been developed [88,89]. Informally, the
reason inclusive fitness, rather than personal fitness, is maximised by
natural selection is because an individual actor only has control over
their own inclusive fitness, but does not have full control over their
personal fitness. This is because the inclusive fitness is calculated in
terms of the effects of the behaviours of an individual on their own
direct fitness, and on that of their genetic relatives (see discussion in
main text) and, because the behaviour of an actor is under their own
control, so is their inclusive fitness. By contrast, only part of the
personal fitness of an actor is due to their own behaviour and, hence,
under their own control, with the remainder the result of social
partners over whom they might have no influence. Despite the
aforementioned mathematical equivalence, because only inclusive
fitness is under the control of an individual, it is this quantity that they
should ‘act as if to maximise’ [88,90].
First, the authors formulate their model entirely in terms
of personal fitness (which they refer to as ‘direct fitness’),
and claim that, as a result, inclusive fitness cannot be at
work. However, it has long been recognised that personal
fitness and inclusive fitness are equivalent mathematical
treatments of social evolution and, in fact, the authors
concede this in the supplementary information for their
paper [11]; thus, personal fitness is inclusive fitness in
disguise, but only inclusive fitness can be subject to natural
selection (Box 4). Second, the model presented puts unrealistic restrictions on the set of strategies considered.
Specifically, the model in [11] considers only the following
two strategies for virgin queens: leave the nest to potentially mate and found a colony, or stay at the home nest to
raise sisters. The third, free-rider strategy, to which all
altruistic systems are susceptible, is not included: that is,
virgin queens that play the role of workers are not permitted to raise their own offspring in the model, by which they
would enjoy personal fitness advantages both from individual reproduction and from the safety of the nest. The
employment of this strategy by workers is observed in
many species of social insects [77], so its omission is
puzzling. Furthermore, theory predicts, and empirical data
agree, that the extent to which this strategy is suppressed
by workers themselves does indeed depend crucially on
within-nest relatedness [4,5,7,78].
Conclusion
Drawing false dichotomies is not helpful for the theory of
social evolution [18]. The debate over the equivalence
between the group selection and inclusive fitness viewpoints should have already ended and, arguably, recent
attempts to restart it have only caused further confusion
[79–81]. In The Modern Synthesis, J.S. Huxley wrote that
‘as in the case of Mark Train, the reports [of the death of
Darwinism] seem to have been greatly exaggerated.’ Today, the same could be said of inclusive fitness theory.
Acknowledgements
I gratefully acknowledge discussions with, and comments from, Kevin
Foster, Andy Gardner, Samir Okasha and Geoff Wild, as well as the
comments of two anonymous reviewers.
Opinion
Trends in Ecology and Evolution July 2011, Vol. 26, No. 7
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