Jonathan Birch 

Hamilton's two conceptions of social fitness 
 
Article (Published version) 
(Refereed) 
 
 
 

 

Original citation: 
Birch, Jonathan (2016) Hamilton's two conceptions of social fitness. Philosophy of Science, 83 
(5). pp. 848-860. ISSN 0031-8248 
 
DOI: 10.1086/687869 
 
© 2016 Philosophy of Science Association 
 
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Hamilton’s Two Conceptions
of Social Fitness

Jonathan Birch*y

Hamilton introduced two conceptions of social fitness, which he called neighbor-modulated
fitness and inclusive fitness. Although he regarded them as formally equivalent, a re-
analysis of his own argument for their equivalence brings out two important assump-
tions on which it rests: weak additivity and actor’s control. When weak additivity breaks
down, neither fitness concept is appropriate in its original form. When actor’s control breaks
down, neighbor-modulated fitness may be appropriate, but inclusive fitness is not. Yet I
argue that, despite its more limited domain of application, inclusive fitness provides a dis-
tinctively valuable perspective on social evolution.

1. Introduction. W. D. Hamilton is rightly celebrated as the progenitor of
modern social evolution theory. This symposium marks the fiftieth anniver-
sary of the publication of his seminal article, “The Genetical Evolution of So-
cial Behaviour” (1964). It is an article bursting with ideas, many of which
were hugely innovative at the time. Perhaps the best known is the principle
now called Hamilton’s rule, which states that a social behavior will be favored
by natural selection if and only if rb > c, where c is the fitness cost to the or-
ganism that performs the behavior, b is the fitness benefit the trait confers
on another organism, and r is the coefficient of relatedness.

I have discussed this idea elsewhere (Birch 2014a; Birch and Okasha
2015), but here I want to focus on two other major innovations. For in the

*To contact the author, please write to: Department of Philosophy, Logic and Scientific
Method, London School of Economics and Political Science, Houghton Street, London,
WC2A 2AE; e-mail: j.birch2@lse.ac.uk.

yThis article is based on my contribution to the symposium “50 Years of Inclusive Fit-
ness” held at the 2014 PSA meeting in Chicago, IL, November 6–9. I am very grateful to
my fellow contributors (Ullica Segerstrale, Patrick Forber, Rory Smead, Dave Queller,
and Samir Okasha) and also to Andrew Buskell, Ellen Clarke, James Marshall, and a
reading group at Australian National University. This work was supported by a Philip
Leverhulme Prize from the Leverhulme Trust.

Philosophy of Science, 83 (December 2016) pp. 848–860. 0031-8248/2016/8305-0018$10.00
Copyright 2016 by the Philosophy of Science Association. All rights reserved.

848



same article, Hamilton introduced two alternative ways of thinking about fit-
ness in the context of social evolution. He called them inclusive fitness and
neighbor-modulated fitness (Hamilton 1964, 5–6), and they continue to be the
most commonly used fitness concepts in social evolution research.

Hamilton chose to focus on developing the inclusive fitness approach,
and this continues to be the better known of the two. By the mid-1990s, how-
ever, the neighbor-modulated fitness approach had inconspicuously grown
into a full-blown rival framework (Taylor and Frank 1996; Frank 1998), and
in recent years it has become the preferred methodology of many social evo-
lution theorists (Taylor, Wild, and Gardner 2007; Wenseleers, Gardner, and
Foster 2010; Frank 2013).

This has led to discussion of the relationship between the two fitness con-
cepts. The key questions are: When are they equivalent for the purpose of cal-
culating gene frequency change? When do they come apart? And when the
choice between them is not forced by considerations of accuracy, which fit-
ness concept is preferable? Here I revisit some of Hamilton’s early papers in
order to bring his own work to bear on these questions. In short, I will argue
that although the neighbor-modulated fitness concept has a wider domain of
application than the inclusive fitness concept, the latter remains distinctively
valuable in those cases to which it does apply.

2. The Conceptual Contrast. To understand the difference between the
two fitness concepts, consider two perspectives on what happens when al-
truism evolves because of genetic relatedness between organisms. One is to
view relatedness as a source of correlated interaction: when r is high, bear-
ers of the genes for altruism are more likely to interact with organisms who
express that same gene, and hence are more likely to receive the benefits of
altruism. Thus, bearers of the genes for altruism may have greater reproduc-
tive success, on average, than nonbearers. The other is to view relatedness
as a source of indirect reproduction: when r is high, recipients provide ac-
tors with an indirect means of securing genetic representation in future gen-
erations. Thus, the genes for altruism may spread if the representation an al-
truist secures through helping its relatives exceeds the representation it loses
by sacrificing a fraction of its own reproduction success.

The first perspective is captured in Hamilton’s neighbor-modulated fit-
ness approach, which analyzes the correlations between an individual’s ge-
notype and its social neighborhood in order to predict when these correla-
tions will give bearers of the genes for altruism greater reproductive output,
on average, than nonbearers (Hamilton 1964; Taylor and Frank 1996; Frank
1998). The second perspective is captured in Hamilton’s inclusive fitness ap-
proach, which adds up all the fitness effects causally attributable to a social
actor, weighting each component by a coefficient of relatedness, in order to

HAMILTON’S SOCIAL FITNESS CONCEPTIONS 849



calculate the net effect of a social behavior on the actor’s genetic represen-
tation in the next generation (Hamilton 1964; Frank 1998; Grafen 2006).

To preempt some misunderstandings, I should explain what these fitness
concepts are not. First, neighbor-modulated fitness is not simply a new name
for classical individual fitness. Neighbor-modulated fitness assumes that an
individual’s reproductive success can be decomposed into a sum of compo-
nents, each attributable to a particular neighborhood phenotype, plus a “base-
line” component that is independent of what these neighbors do. Since the
classical Darwinian fitness concept does not make any such assumption about
the causal structure of fitness, it would be incorrect to simply equate the two
(cf. Marshall 2015, 57–58).

Second, inclusive fitness is not simply an organism’s classical individual fit-
ness plus the classical individual fitness of its relatives, with the latter weighted
by relatedness. This was never Hamilton’s conception, and he avoided it with
good reason. As Grafen (1982, 1984) emphasizes, any adequate fitness con-
cept must be such that if bearers of one allele are, on average, fitter than bear-
ers of another allele, then the former should be selected. A simple weighted
sum violates this constraint, essentially because it allows the same offspring
to be counted multiple times, once in computing the fitness of its parents, and
again (and again, and again . . .) in computing the fitness of any collateral rel-
atives. This multiple counting means that organisms with “bushier” family
trees can be much “fitter” than organisms with more sparse family trees, even
though the bushiness of one’s family tree makes no difference in itself to the
future representation of one’s genes in a population.

What Hamilton (1964) saw from the outset is that inclusive fitness must
be defined in a way that avoids multiple counting. His solution was to as-
sume that every organism’s reproductive output can be written as a sum of
components, each causally attributable to the behavior of a specific actor.
Given this assumption, we can make sure that each component is counted
once and only once, by insisting that each component counts only toward
the fitness of the actor who was causally responsible for it. As Hamilton him-
self put it:

Inclusive fitness may be imagined as the personal fitness which an individ-
ual actually expresses in its production of adult offspring as it becomes af-
ter it has been stripped and augmented in a certain way. It is stripped of
all components which can be considered as due to the individual’s social en-
vironment, leaving the fitness 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 individ-
ual himself causes to the fitness of his neighbours. The fractions in question
are simply the coefficients of relationship. (1964, 8)

850 JONATHAN BIRCH



Inclusive fitness is thus an inherently causal notion: a weighted sum of the fit-
ness components for which a given social actor is causally responsible.

Correlated interaction and indirect reproduction sound like very different
processes, and neighbor-modulated and inclusive fitness sound like very dif-
ferent concepts. Despite this, the two are often considered formally equiv-
alent, in the sense that they reliably yield identical predictions regarding the
direction of gene frequency change (Wenseleers et al. 2010; Gardner, West,
and Wild 2011; Queller 2011; Marshall 2015). Yet there have always been
dissenters from the consensus. For example, Maynard Smith (1983) contrasted
“the exact ‘neighbour-modulated fitness’ approach” with “the more intui-
tive ‘inclusive fitness’ method” (315). He advocated inclusive fitness on the
grounds that he considered it easier to apply, but he thought it less accurate.
More recently, Frank has advocated the neighbor-modulated approach, claim-
ing that “inclusive fitness is more limited and more likely to cause confusion”
(2013, 1172).

Hamilton (1964) claimed that the direction of selection can be calculated
using either fitness concept, but he provided no formal argument for their
equivalence. He did, however, include such an argument in his paper on self-
ishness and spite (Hamilton 1970). The 1970 paper is quite brilliant: drawing
on the work of Price (1970), Hamilton articulates clearly and concisely the
basic insights he had presented in a rather dense way in earlier work. But per-
haps the argument is a little too concise. As I will show, it leaves important
assumptions unarticulated.

3. Hamilton’s Argument Reconsidered. Here I reconstruct Hamilton’s
(1970) argument for the formal equivalence of his two fitness concepts.1 In
doing so, I want to draw attention to two assumptions Hamilton leaves im-
plicit, since they point to important limitations of this equivalence.2

Consider a finite population of N numbered individuals. Let W toti repre-
sent the total reproductive success of the ith individual (“the recipient”),
and let sij represent the additive effect of the social behavior of the jth indi-
vidual (“the actor”) on the reproductive success of the ith individual.

We should pause here to consider the meaning of the sij. Hamilton (1970)
simply glosses the sij as “additive effects.” However, I suggest that, to do jus-
tice to Hamilton’s explicitly causal conception of social fitness, we should in-
terpret sij in explicitly causal terms, as the causal effect of the jth individual on

1. Frank (1998) also reconstructs and discusses Hamilton’s model, although without
drawing attention to its assumptions.

2. Readers wishing to avoid mathematical details may skip this section, but please note
the two key assumptions stated at the end of it.

HAMILTON’S SOCIAL FITNESS CONCEPTIONS 851



the reproductive success of the ith individual. Roughly, it is the amount by
which the ith individual’s reproductive success would have differed had it
never interacted with the jth individual.

We can now return to the model. Let rij represent the coefficient of relat-
edness. More precisely, let rij 5 Cov(qi, qj)=Var(qj), where qi is the recip-
ient’s individual gene frequency for a particular allele (i.e., its total number
of copies of the allele divided by its ploidy) and qj is the actor’s individual
gene frequency. This is equivalent to the simple regression (across all inter-
acting pairs) of qi on qj.

We can express W toti as a sum of additive causal effects attributable to
distinct social actors, plus a term representing its baseline nonsocial fitness
(ai), plus a residual component (eWi) that represents deviations from fitness
additivity:

W toti 5 ai 1 o
j

sij 1 eWi: (1)

Let us define the neighbor-modulated fitness of the ith individual as its
total reproductive success as predicted by this additive causal model, ne-
glecting the residual eWi:

(NMF)

W NMFi 5 ai 1 o
j

sij:

If the residuals are all zero (i.e., eWi 5 0 for all i), then an individual’s neighbor-
modulated fitness can be equated with its reproductive success. If eWi ≠ 0,
then neighbor-modulated fitness can still be used to calculate gene frequency
change accurately, provided the residuals do not covary with any genes. How-
ever, if these residuals do covary with genes, neighbor-modulated fitness may
mislead. At this point, a substantive assumption of “additivity” is required.
We must assume that the causal structure of social interaction is such that eWi
is either zero or, if nonzero, irrelevant to the direction and magnitude of gene
frequency change.

Now let us define the inclusive fitness of the jth individual as its baseline
fitness (aj) plus the sum of all the additive causal effects (sij) for which it is
responsible, weighted in each case by the coefficient of relatedness rij:

(IF)

W IFj 5 aj 1 o
i

sijrij:

We can now ask: under what conditions does Cov(W NMFi , qi) equal
Cov(W IFj , qj)? This is the crucial question as regards the “formal equiva-

852 JONATHAN BIRCH



lence” of the two fitness concepts. It turns out that, given one further impor-
tant assumption, these quantities are equal.

First, we use our formal definitions of NMF and IF to split each covariance
into a nonsocial and social component:

Cov(W NMFi , qi) 5 Cov(ai, qi)
zfflfflfflfflfflffl}|fflfflfflfflfflffl{nonsocial

1 Cov ojsij, qi
� �

:
zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{social

(2)

Cov(W IFj , qj) 5 Cov(aj, qj)
zfflfflfflfflfflffl}|fflfflfflfflfflffl{nonsocial

1 Cov oisijrij, qj
� �zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{social

: (3)

The nonsocial component is the same in both cases (since i and j are being
used to label elements of the same set, the difference in indices is merely no-
tational). Hence, only the equivalence of the two social components needs to
be established. Following Hamilton, let us call these components (respec-
tively) the “neighbor-modulated fitness effect” and the “inclusive fitness ef-
fect.” Hamilton further simplifies matters by assuming a to be a constant (of
value 1), so that the nonsocial component is zero in both cases, but this as-
sumption is dispensable to the argument.

The neighbor-modulated fitness effect can be rewritten as

Cov o
j

sij, qi

 !
5 o

j

1

N oi
(qi 2 q)sij)

� �
: (4)

Now note that, from the definition of relatedness as the simple regression of
qi on qj, it follows that

qi 2 q 5 rij(qj 2 q) 1 eqi, (5)

where eq denotes the extent to which the recipient’s actual genotype deviates
from the regression prediction. Assume now that

Cov o
j

sij, eqi

 !
5 0: (6)

This key assumption, which Hamilton (1970) makes implicitly, amounts
to assuming that the recipient’s individual gene frequency predicts its so-
cial fitness only via correlations with actors and not via any other pathway
(e.g., via conferring an ability on the recipient to make better use of the help
of others). This entitles us to substitute rij(qj 2 q) for (qi 2 q) in equation 4,
yielding

HAMILTON’S SOCIAL FITNESS CONCEPTIONS 853



Cov o
j

sij, qi

 !
5 o

i

1

N oj
(qj 2 q)rijsij)

( )
: (7)

The right-hand side of (7) can now be rewritten once again as a covariance:

Cov o
j

sij, qi

 !
5 Cov o

i

sijrij, qj

� �
: (8)

Comparing this result to (2) and (3), we see that

Cov(W NMFi , qi) 5 Cov(W
IF
j , qj), (9)

as we hoped to prove.
We can now see that Hamilton’s (1970) argument for the formal equiv-

alence of neighbor-modulated and inclusive fitness relies on two implicit as-
sumptions, both of which amount to assumptions of uncorrelated residuals:

Additivity: Deviations from fitness additivity (eWi) are either zero or, if non-
zero, irrelevant to the direction and magnitude of gene frequency change.

Actor’s control: The recipient’s genotype predicts its social fitness only via
its correlation with actor genotypes and not via any other pathway (e.g., by
enabling it to make better use of the help received).

Hamilton is not alone in relying on these assumptions. More recently, Grafen
(2006, 543–49) has provided an argument for formal equivalence that im-
proveson Hamilton’s: inparticular, it accommodatesuncertainty, and it accom-
modates the various social “roles” an actor can occupy. Nevertheless, Grafen’s
argument still relies on the assumptions of additivity and actor’s control. The
only difference is that, while Hamilton left these assumptions implicit, Grafen
makes them explicit.3

4. Actor’s Control and Additivity. Actor’s control points to one impor-
tant qualification of Hamilton’s equivalence result. If the recipient’s geno-
type predicts the fitness effects it receives in ways that are not fully explained
by correlations with actor genotypes, the result will be a situation in which
Cov(ojsij, eq) ≠ 0. This is a situation in which neighbor-modulated fitness
remains valid but in which inclusive fitness could lead to errors.

Such a situation may seem hard to visualize. But all it needs is for there
to be some genotype that, in addition to disposing an organism to express a

3. Frank (1998) derives a qualified equivalence result that holds under similar assump-
tions in a framework that incorporates class structure (discussed in Birch 2013).

854 JONATHAN BIRCH



social behavior, affects its ability to receive the benefit of that behavior when
expressed in others. Consider, for example, a genotype that disposes its bearer
to produce an alarm call. In so doing, it reveals the organism’s location to
nearby predators, adversely affecting its ability to benefit from the alarm calls
of others. In this scenario, the benefit of receiving an alarm call for a recipient
does not just depend on properties of the actor. It also depends on whether the
recipient has itself produced an alarm call.

Let us turn now to additivity. We should distinguish here between strong
and weak varieties. If the deviation from additivity eWi is zero for all individ-
uals, then we can say that the structure of social interaction is strongly ad-
ditive. If eWi is nonzero for at least some individuals but makes no difference
to changes in gene frequency, we can say that the structure of social inter-
action is weakly additive. Either way, we are talking about a property of the
causal structure of social interaction in a population, not a property of any par-
ticular organism or gene.

It is clear that real social interactions frequently violate strong additivity.
For recall what it requires: that an individual’s social fitness can be written,
without remainder, as a sum of components, each reflecting the causal influ-
ence of a particular social actor. This is unlikely to be the case when coopera-
tion takes the form of collaborative tasks involving multiple actors, because
task structure tends to create situations in which the total payoff cannot be
expressed as a sum of components, each corresponding to the difference made
by a single actor’s contribution (Birch 2012).

Weak additivity, however, is compatible with substantial deviations from
the additive causal model. Its tenability in any particular case depends on
whether these deviations covary with genes. This makes the empirical status
of weak additivity difficult to assess. We are often in a position to know em-
pirically that a social interaction violates strong additivity, since this depends
only on the causal structure of the interaction, but we are less often in a po-
sition to know whether the deviations from strong additivity covary with any
genes. I will not try to settle this empirical question here. Note, however, that
failures of weak additivity are clearly possible in principle whenever there are
deviations from strong additivity. We should therefore be cautious about as-
suming weak additivity when strong additivity is violated.

These considerations broadly support the view that, as Grafen puts it,
“the assumption of additivity . . . is not in general a realistic assumption.
In many applications, non-additivity is an important part of the problem”
(2006, 543; see also Queller 1985, 2011; Marshall 2015). What does this
mean for our two conceptions of social fitness and for the relationship be-
tween them?

The immediate challenge is not to the formal equivalence of the two fit-
ness concepts but rather to the generality of both. Hamilton formulated both
fitness concepts in terms of an additive causal model. If the model is inap-

HAMILTON’S SOCIAL FITNESS CONCEPTIONS 855



propriate in some biological scenario, then both fitness concepts as Hamil-
ton originally conceived them are inappropriate in that scenario.

Yet it would be wrong to conclude that the two fitness concepts are on a
par when it comes to accommodating deviations from additivity. A key dif-
ference is that neighbor-modulated fitness, because it does not assume ac-
tor’s control, has more leeway for accommodating effects that depend on
the behavior of multiple actors. Neighbor-modulated fitness requires that
an individual’s fitness can be expressed as a sum of effects attributable to
properties of its social neighborhood, but it does not require that each prop-
erty is controlled by a single actor. This means that, as Queller (1985, 2011)
has shown, we can augment the basic additive causal model with “synergis-
tic effects” that depend in complex ways on the phenotypes of multiple ac-
tors (see also Marshall 2015, 66–67). This extended version of the neighbor-
modulated fitness approach can handle cases in which deviations from the
additive causal model arise from synergistic interactions.

By contrast, the inclusive fitness concept relies fundamentally on the as-
sumption that each fitness effect can be attributed to a single controlling ac-
tor, whose inclusive fitness it counts toward. Since synergistic effects are not
controlled by any single actor, there is no principled answer to the question
of whose inclusive fitness they promote. In these contexts, inclusive fitness,
as Hamilton conceived it, is no longer a well-defined property of an individ-
ual organism.

Thus, the two fitness concepts are threatened in different ways by failures
of weak additivity. Put simply: for neighbor-modulated fitness, the problem
is a technical one that can usually be surmounted by expanding the causal
model of fitness. For inclusive fitness, however, the problem runs deeper, for
it is a problem of a conceptual nature. The reassignment of fitness components
to controlling actors that Hamilton envisaged is no longer possible when the
additive causal model fails.

5. Inclusive Fitness, Adaptation, and Selection-For. The bottom line is
that, although neighbor-modulated and inclusive fitness are equivalent in
their predictions when weak additivity and actor’s control obtain, the inclu-
sive fitness concept has a more restricted domain of application. One might
take this as an argument for abandoning the inclusive fitness concept. Yet
I maintain that, for all its disadvantages, the inclusive fitness concept pro-
vides a distinctively valuable perspective on social evolution. This is because
it provides a perspective from which we can make sense of altruistic (or in-
deed spiteful) behaviors as adaptations that have been selected-for, rather than
as traits that were selected merely because they correlated with favorable so-
cial neighborhoods (cf. Gardner 2009; West and Gardner 2013).

To make sense of this idea, let us briefly revisit Sober’s (1984) selection-
for/selection-of distinction: “To say that there is selection for a given property

856 JONATHAN BIRCH



means that having that property causes success in survival and reproduction.
But to say that a given object was selected [or ‘selected-of’] is merely to say
that the result of the selection process was to increase the representation of
that kind of object” (100). As Sober observes, it seems intuitively important
to some trait’s being an adaptation that it has been selected-for and not merely
selected-of (197).

Within this framework, we can see the dilemma that altruistic (or spiteful)
traits presented to biologists before Hamilton. On the one hand, these traits ap-
parently could not be adaptations, because they make no causal contribution
to their bearer’s success in survival and reproduction. On the other hand, if
they were not adaptations, their existence seemed to defy explanation.

Hamilton’s two conceptions of social fitness resolve this dilemma in subtly
different ways. Neighbor-modulated fitness offers one resolution: the prop-
erty of being an altruist is selected not because it causally promotes its bear-
ers’ fitness but because, in populations with the right kind of structure, it sys-
tematically correlates with receiving the benefits of altruism in others. On this
picture, it is the extrinsic property of having an advantageous social neigh-
borhood that is selected-for, whereas having the trait oneself is selected only
because it correlates with this extrinsic property. If an individual could sup-
press its own altruism without altering its social environment, it would increase
its fitness by doing so. From a neighbor-modulated fitness perspective, then,
we can see why the property of being an altruist is sometimes selected, but it
is only ever selected-of.

By contrast, the inclusive fitness approach offers a resolution that puts
selection-for back at the center of the picture. From an inclusive fitness per-
spective, a social trait that detracts from the actor’s viability or fecundity may
still contribute causally to its fitness, and evolve for that reason, if the benefits
of expressing the trait fall systematically on genetic relatives. For this reason,
I suggest, the inclusive fitness concept provides the more satisfying resolu-
tion to the dilemma. It shows how being an altruist can be an adaptation, not
just a correlate of having a favorable social environment.

6. Inclusive Fitness Maximization. A different way of arguing for the dis-
tinctive value of inclusive fitness is to argue that organisms maximize this
quantity, in a certain sense of the term (Grafen 2006). Hamilton (1964, 1) him-
self made such a claim, writing that populations satisfying the assumptions
of his model “should tend to evolve behavior such that each organism appears
to be attempting to maximize its inclusive fitness.”

Pinning down the sense of “maximization” at stake here is crucial in order
to evaluate Hamilton’s claim, since there are senses of the term on which in-
clusive fitness is clearly not maximized (Birch 2016). In economics, it is com-
mon to model humans as “maximizing agents” who make strategic choices
that, within a set of feasible options, maximize a quantity known as utility.

HAMILTON’S SOCIAL FITNESS CONCEPTIONS 857



Along similar lines, behavioral ecologists often assume that organisms behave
in ways that maximize, within a set of feasible options, their inclusive fitness.
Following Grafen (1984, 1999), we can call this an individual-as-maximizing-
agent analogy.

What enables inclusive fitness to play this role as the putative “maxi-
mand” of animal behavior is its focus on which actors control which phe-
notypes. Because an actor’s inclusive fitness is a weighted sum of the fitness
effects for which it is causally responsible, we can put ourselves in the po-
sition of the actor and ask: “How should I behave, in order to maximize my
expected inclusive fitness?” Since this quantity is under the actor’s control,
this can serve as an informal route to predictions of how we should expect
an organism to behave. By contrast, we cannot usefully ask the same question
with regard to neighbor-modulated fitness, because an individual’s neighbor-
modulated fitness contains components over which it may have no control. All
we can do is put ourselves in the position of a recipient and ask: “What ge-
notypes are correlated with good outcomes, as far as my neighbor-modulated
fitness is concerned?” But this heuristic is less intuitive, because consider-
ations of causation and control are replaced by considerations of statistical
auspiciousness (cf. Gardner 2009; Marshall 2015).

Hamilton’s claim, as quoted above, appears to invoke an individual-as-
maximizing-agent analogy. However, Hamilton provided no formal argument
for its validity. What he actually showed was that, within his one-locus model,
the mean inclusive fitness of the population increases until a local maximum
is reached. Population geneticists have constructed numerous counterexam-
ples to this sort of mean fitness maximization (reviewed in Birch 2016), so
this feature of Hamilton’s model cannot be considered a general truth about
the operation of natural selection.

Grafen’s (2006, 2014) “Formal Darwinism project” can be regarded as a
sophisticated attempt to vindicate the individual-as-maximizing-agent anal-
ogy that Hamilton verbally gestured toward. It aims to do this by forging links
between formal representations of gene frequency change and optimal strat-
egy choice. In a nutshell, Grafen aims to prove that “natural selection always
changes gene frequencies in the direction of increasing inclusive fitness; and
that a population genetic equilibrium in which no feasible mutations can spread
implies that the individuals in the population are each acting so as to maximize
their inclusive fitness” (2006, 543).

I have criticized the Formal Darwinism project elsewhere, and I cannot
do justice to this complex topic here (Birch 2014b, 2016). In short, Grafen
does prove what he aimed to prove, given a very specific and unorthodox
understanding of the concept of “equilibrium.” But the “equilibrium” con-
cept that features in his links, defined in terms of “scope” and “potential” for
selection, is neither necessary nor sufficient for a population-genetic equilib-
rium in the usual sense. The true relationship between population-genetic

858 JONATHAN BIRCH



equilibria and inclusive fitness maxima is much more complicated than
Grafen’s links initially suggest.

Despite my doubts about Formal Darwinism, I remain convinced that the
inclusive fitness concept remains valuable. One reason is that a heuristic
need not be completely or even mostly reliable in order to warrant its con-
tinued use. If the individual-as-maximizing-agent analogy sometimes gen-
erates fruitful hypotheses, as it surely has done, then this provides a prag-
matic justification for its use as a method of hypothesis generation. We do
not need to put empirical projects on hold while we wait for a theoretical ar-
gument to reassure us that the hypotheses thus generated will be correct.

More fundamentally, however, we can reject the idea that organisms in
any sense maximize their inclusive fitness and yet retain the idea that social
traits are selected because they causally contribute to this quantity. This milder
claim is already enough to make inclusive fitness valuable. For it allows us
to see how social traits that detract from their bearers’ classical fitness can
be selected-for, not just selected-of, and it thus allows us to see how such traits
can be adaptations, regardless of whether they maximize inclusive fitness.

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