Journal of Mathematical Biology manuscript No.(will be inserted by the editor)

Foundations of a mathematical theory of darwinism

Charles J. K. Batty · Paul Crewe · AlanGrafen · Richard Gratwick

Received: date / Accepted: date

Abstract This paper pursues the ‘formal darwinism’ project of Grafen, whoseaim is to construct formal links between dynamics of gene frequencies and op-timization programmes, in very abstract settings with general implications forbiologically relevant situations. A major outcome is the definition, within wideassumptions, of the ubiquitous but problematic concept of ‘fitness’. This paperis the first to present the project for mathematicians. Within the framework ofoverlapping generations in discrete time and no social interactions, the currentmodel shows links between fitness maximization and gene frequency change ina class-structured population, with individual-level uncertainty but no uncer-tainty in the class projection operator, where individuals are permitted to ob-serve and condition their behaviour on arbitrary parts of the uncertainty. Theresults hold with arbitrary numbers of loci and alleles, arbitrary dominanceand epistasis, and make no assumptions about linkage, linkage disequilibriumor mating system. An explicit derivation is given of Fisher’s FundamentalTheorem of Natural Selection in its full generality.

Keywords Formal darwinism · reproductive value · fitness maximization ·Price Equation

Mathematics Subject Classification (2010) 28B99 · 49N99 · 60J99 ·92D15

This work is part of the ‘Formal Darwinism’ project funded by St John’s Research Centre,St John’s College, Oxford, grant to CJKB and AG.

Charles J. K. Batty · Paul Crewe · Alan Grafen · Richard GratwickSt John’s College, Oxford, OX1 3JPTel.: +44(0)1865 280146E-mail: richard.gratwick@sjc.ox.ac.uk

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1 Introduction

This is the first paper to present the formal darwinism project for mathemati-cians. It presents all the results of Grafen (2002, 2006b) completely anew, butalso combines into a single model all their separate features, correcting someerrors and making explicit all needed assumptions. While the earlier papershave relevant discussion of the biological motivation and background, froma mathematical point of view, the current paper should be regarded as thestarting point of the formal darwinism project.

The first explicit links of this kind between gene dynamics and optimiza-tion were made by Grafen (2002), for a population evolving in discrete timewith non-overlapping generations, considering the e!ects of uncertainty (e.g.climate). The starting point for this work is the Price Equation, which recordshow the frequency of a given allele changes from the parent generation to theo!spring generation. Grafen (2006b) considers the more complicated case ofa population with a class structure, e.g. age, ploidy, etc., and considers theproblem of rigorously defining Fisher’s notion of reproductive value (Fisher1930), still in discrete time, and without the e!ects of uncertainty. As partof this Grafen derives versions of the Price Equation for a class-structuredpopulation. Given that this equation is the starting point for proving linksto optimization in Grafen (2002), it seems natural to investigate what corre-sponding optimization results we can derive in the case of a class-structuredpopulation.

This paper provides analogous results in the class-structured case to thefundamental links proved in Grafen (2002). To avoid the apparently substan-tial and more sophisticated problems inherent in a Markov process includinguncertainty (by which a general expression for reproductive value and henceour maximand would be defined) we have to make some assumptions so thatthere is no population-level e!ect of uncertainty, i.e. that the total o!springdistribution over classes is independent of the state of nature. A current lineof investigation in the formal darwinism project is to allow uncertainty to in-teract more substantially with the population over time. Neither do we allowany social interactions between individuals; the full extension of the project inthe context of inclusive fitness, begun in Grafen (2006a), is the next challenge.

As a preliminary to proving the links to optimization, we re-derive the PriceEquation for a class-structured population, following the original derivationin Grafen (2006b), but making some changes for clarity and correctness. Ourgeneral results are illustrated by an examination of the case of a finite age-structured population, which has traditionally been analyzed by means ofso-called Leslie matrices.

The final contribution we make is to provide (in discrete time) an explicitstatement, and a complete and rigorous derivation, of Fisher’s celebrated Fun-damental Theorem of Natural Selection (Fisher 1930), placing the theoremwithin our wider optimization programme, which seems to formalize mathe-matically its original and natural conceptual context.

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2 Biological motivation

It has been a frequent and recurrent theme in biology since Darwin that naturalselection has a tendency to result in the maximization of something. HerbertSpencer’s phrase ‘survival of the fittest’ led to the term ‘fitness’ being usedfor whatever that relevant quantity might be. Many significant advances inbiology, as well as a number of recurrent debates, hinge on what exactly fitnessis, who or what should be regarded as doing the maximizing, and how themaximization can be formalized. These ideas can be found and pursued intextbooks such as Davies et al (2012).

Many empirical biologists today base research projects and paradigms onthe idea that natural selection leads to individual organisms acting as if max-imizing their ‘inclusive fitness’, subject to the physical, physiological and in-formational constraints on the development and behaviour of an individual.An intermediate level of biological theory, represented by the theory of evo-lutionarily stable strategies (Maynard Smith 1982) and inclusive fitness the-ory (originated by Hamilton 1964), simply assumes that individuals act asif maximizing their Darwinian fitness (roughly, lifetime number of o!spring)or their inclusive fitness (a more sophisticated concept that recognizes socialbehaviour), and studies topics of interest on that basis. However, the mostfundamental level of biological theory, mathematical population genetics, haslong been resistant to the idea that any useful maximization principle can bederived from the known processes of gene frequency change, which are mod-elled using di!erence or di!erential equations.

From a mathematical point of view, the obvious candidate principles arethose familiar to students of dynamical systems, such as Lyapunov functionsand gradient functions, and the general conclusion of the literature is thatonly under rather special and not very interesting conditions are populationgenetic systems of a kind that will admit these kinds of functions (Ewens2004). The main premiss of the current paper is that the empirical biologist’sindividual-based idea of fitness-maximization can be done justice only in amore sophisticated setting, in which the equations of motion are taken as fun-damental in representing gene frequency change, an optimization program isconstructed from those equations of motion in which the implicit decision-takeris the individual organism, and then links are proved between the equilibriumconcepts of the equations of motion on the one hand, and of the optimizationprogram on the other. The maximand of the optimization program plays therole of fitness, and a major interest lies in the nature of that maximand, andin how tightly the maximand is defined by the structures imposed on it. Thestronger the links between the equilibrium concepts, the more constrained isthe nature of the maximand, and so the more precisely the concept of fitnessis defined.

In the present work, the population of individuals is assumed to be dividedinto classes, such as sexes and/or size and/or age. There are discrete overlap-ping generations, so that in a formal sense an individual may have a specialkind of asexual o!spring that is itself surviving to the next period, as well

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4

as contribute gametes to new individuals. The population may be finite orinfinite, as may the set of classes. An individual takes an action in each periodwhich a!ects the o!spring it leaves in the next period. So does its class, andthe ‘state of nature’, which is drawn from a set of possible outcomes of ‘actionsby nature’, with a given probability distribution. Thus the o!spring distribu-tion across classes produced by an individual depends on its own action andon uncertainty, but we impose the restriction that it does not depend on theactions of others — social behaviour is therefore excluded. A further restric-tion is that the uncertainty is assumed not to a!ect the class demographics,that is, the projection operator from classes to classes has no stochastic com-ponent. The action taken by an individual depends on its phenotype, whichwe assume is determined by its genotype, and also by informational cues thatmake it possible for an individual to condition its choice of action on aspectsof the uncertainty.

The aim is to prove a fitness-maximization principle at a very high levelof generality. So far as gene frequencies are concerned, we employ the covari-ance selection mathematics of Price (1970), though not the generalization inPrice (1972a), and very weak assumptions, which allow us not to say anythingabout mating systems, linkage, or some other potential complications. On theoptimization side, we consider the equilibrium concepts of an optimizationprogram in which an individual organism is the decision-taker. The programrepresents a sophisticated individual, who has a prior distribution over all therelevant uncertainties, and who updates this distribution in the light of infor-mation received. For an individual to solve this program implies that the priordistribution is correct, and that the updating is optimal Bayesian. The linksare of two kinds. Three results make an assumption related to how individualsfare in the optimization program, and draw conclusions about gene frequencychange. A fourth result makes an assumption about gene frequency change,and draws a conclusion that individuals each solve the optimization program.

The first explicit mathematical fitness-maximization principle in biologywas the Fundamental Theorem of Natural Selection of Fisher (1930), andthe current work can be viewed as an extension and generalization of thistheorem. Fisher’s theorem has been notoriously hard to understand, and hisarguments are famously opaque. The early rejection (endorsed and reviewedby Ewens 1979) by mathematical population geneticists of the theorem mustnow be read in the context of the exposition of the theorem and proofs byPrice (1972b). The current view in that discipline, as represented by Ewens(2004), is that the fundamental theorem is true, and Fisher’s proofs are validgive or take some minor typographical and other errors (see Lessard (1997)for a careful modern derivation), but that the meaning of the theorem remainsobscure. Recent papers focus on the meaning (Okasha 2008; Ewens 2011) butcome to no firm conclusion. The present paper contributes to the debate onfitness-maximization by generalizing the Fundamental Theorem of NaturalSelection to include arbitrary classes rather than just age classes, and to allowfor arbitrary uncertainty at an individual level; and by making fully explicitthe nature of the optimization in the conceptual scheme. Our presentation

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5

should indicate that interest in the theorem is not merely historical: whereasmuch of the literature on the theorem seems wholly motivated by solving theenigma of what one man meant by one theorem, and the result is left tostand alone and justify itself, we, contrastingly, demonstrate that it sits verynaturally alongside related results about the optimizing behaviour of naturalselection.

The high level of generality sought in this paper and in the formal dar-winism project more generally is important for two reasons. First, the fewerthe assumptions, the more widely the framework will apply as a meta-model:that is, the assumptions of other models will not contradict those made here.Our results then o!er an optimization interpretation of the results of thoseprevious models. Application as a meta-model also explains one significanceof dealing with infinite populations, which are admittedly hard to find in na-ture. The assumptions we do make are so weak as to be what populationgeneticists call ‘dynamically insu"cient’ (Lewontin 1974), that is, they use alot of information about the parental generation and calculate only one pieceof information about the o!spring generation, so one cannot ‘crank the han-dle’ and repeat the process. One virtue is that the framework can apply as ameta-model to models with di!erent detailed assumptions about mating sys-tems and genetic architecture. Second, biologists today read Darwin (1859)and agree with his conclusions for the reasons he gave. If Darwin’s argumentsare generally valid, then there must be a formal framework in which they canbe expressed. Darwin did not concern himself with continuous versus discretetime, ploidy levels, or whether generations were overlapping or not, and hisarguments apply regardless: so should ours. This encompassing of Darwin’sargument within a formal framework will reduce the scope for misunderstand-ing and misinterpretation. The broad aim of the paper and the project isto justify a fitness-maximization principle for understanding the outcome ofnatural selection, which will involve defining fitness, in as wide a setting aspossible. Indeed, Fisher’s theorem is best viewed as showing that the changein the mean of a quantity equals the variance of that same quantity, and itis natural to regard that quantity as what increases under natural selectionwherever possible. Significantly, that quantity, whose exact nature is too tech-nical to explain in this section, is intimately related to the maximand in theoptimization programme of our Theorems 2 to 5.

3 Notation and concepts

We set out to define an extremely general model of a biological populationthat has an arbitrary class-structure, with genotypes in an arbitrary set, andphenotypes in an arbitrary set. Each class may have its own ploidy (number ofhaploid sets in the genome). There will be arbitrary environmental uncertaintythat, together with its phenotype, a!ects the number of o!spring each indi-vidual has. The way phenotype a!ects o!spring number is important. Eachindividual possesses partial knowledge of the uncertainty (it observes a ‘cue’),

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6

and the phenotype dictates how the value of that cue in turn determines anaction. The action, together with the whole uncertainty and an individual’sclass, determines its number of o!spring. No restrictions are placed on howgenotype determines phenotype. Thus, the model is extremely general: its chiefrestrictions are that it is in discrete time, the environmental uncertainty doesnot a!ect the class-to-class projection at a population level, and an individual’so!spring number is not a!ected by phenotypes other than its own. Formally,this model is a simultaneous generalization of two previous papers (Grafen2002, 2006b), using a single consistent mathematical argument that replacesless sophisticated, and in places erroneous, arguments.

We firstly outline the very basic structure of how we shall describe a class-structured population reproducing under the e!ects of environmental uncer-tainty. Our aim is generality, and so we impose only the minimum mathemati-cal structure required, and we make only the weakest assumptions necessary toensure the discussion makes sense. This has the consequence that some tech-nical e!ort has to be made to attain results which are unremarkable in morefamiliar settings, for example being able to exchange the order of integration.Since our aim is mathematical rigour, it is appropriate to demonstrate thatthese results can be formally justified, and that our conclusions are indeedvalid in this generality.

Let (I, I, µI) and (!,O, ") be probability spaces, representing the pop-ulation and states of nature respectively. We remark that no assumption ismade on the cardinality of I, or !. For notational ease, we use subscripts forfunctions from I and superscripts for functions from !. Since context shall notalways make things clear, we shall throughout the paper fully notate spaces ofmeasurable and integrable functions, including the relevant domain, #-algebra,measure, and co-domain, e.g. L1(I, I, µI ;R) is the space of functions from Ito R, integrable with respect to the measure µI on the #-algebra I.

Let X be a compact Hausdor! space equipped with the usual Borel #-algebra X , and let $ : I ! X be measurable. X is the space of classes, andthus $ is the map allocating each individual to a class. We let #($) denotethe #-algebra on I generated by $, i.e. generated by the set of pre-images{$!1(Y ) : Y " X}.

We let d : I ! N denote the ploidy of the individuals, and we assume thatd " L1(I,#($), µI ;N), so in particular d is measurable with respect to #($).A ploidy-weighted probability measure µI on (I, I) is then defined by

µI(J) =

!"

I

di µI(di)

#!1"

J

di µI(di).

Evidently a set J " I is µI -null if and only if it is µI -null. Expectations andcovariances over I shall always be taken with respect to this ploidy-weightedmeasure µI . Expectations with respect to i " I or % " ! shall be denoted EI

and E! respectively, and CI shall denote the corresponding covariance over I.We assume individuals to produce o!spring as measures over classes, i.e.

as elements of M(X), the space of signed finite measures on (X,X ), equipped

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7

with the usual norm and Borel #-algebra. This is a Banach space, and thuswe may in principle integrate functions taking values in this space, using theBochner integral. We let M+(X) denote the subset of positive measures, inwhich the o!spring distributions will of course lie. (We do not demand thatthese o!spring distributions are probability measures.)

The technical work of section 4 below is devoted to deriving a Price equa-tion in this context, in the style of Grafen (2006b), but with the added gener-alization to environmental uncertainty. The generality of the set-up describedhere means that the manipulations required of population, class, and uncer-tainty are rather delicate, and the argument must be pursued with great care.The argument given here provides both a generalization of the correspondingPrice equation of Grafen (2006b), and a more sensitive handling of the di"cultmathematical structures used.

We now begin to articulate the decision structure for individuals, so thatour model is of an individual in possession of some (possibly none and possiblycomplete) information about environmental uncertainty. In order to move to-wards regarding individuals as facing the same decision, we also define a spacefor how the environmental uncertainty can a!ect an individual. This allowsus to have a function, which is the same function for all individuals, to mapfrom action and uncertainty (and class) to number of o!spring. This reductionfrom the whole population to a single implicit decision taker is a key theme ofthe development. The set of possible phenotypes available to each individualis also defined as a central part of the decision structure for an individual.

Following Grafen (2002), we further suppose we have measurable spacesR, U , and A, where we shall not notate the associated #-algebras. R denotesthe observable local environment on which individual behaviour can be con-ditioned, U denotes the set of chance events from the point of view of theindividual that are determined by the state of nature % " !, which mayrepresent events experienced by the whole population and events a!ectingindividuals separately (we thus combine in one notation what was notatedseparately in Grafen (2002)). A denotes the space of actions which may betaken and which in turn (partly) determine the o!spring produced.

A measurable function is understood to be a function between measurablespaces such that pre-images of elements of the target space #-algebra are el-ements of the domain #-algebra. All subspaces, product spaces, and functionspaces shall be equipped with the usual #-algebras generated under these oper-ations; in particular function spaces are equipped with the smallest #-algebrawhich makes each evaluation map measurable (see Kechris 1995, §10.B). Weshall let Q # AR consist of the measurable functions q : R ! A and Q denotethe induced #-algebra on Q.

For each individual i " I let Si be a subset of the class of measurablefunctions from R to A (‘strategies’). Thus each individual has a set of possi-ble ways to react to any given local environment. We suppose the followingfunctions are measurable:

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– r : I $ ! ! R, representing the information about the local environmentavailable to individual i in state of nature %;

– u : I$! ! U , uncertainty, in principle a!ecting both individuals separatelyand the population as a whole;

– a : I $ R ! A, the phenotype or strategy, specifying the action taken byindividual i in local environment r; moreover we assume that ai " Si for alli " I, i.e. that the realized phenotype is indeed an admissible phenotype;and

– w : A$X$U ! M+(X), the o!spring distribution, depending on action,class, and chance events.

A purely technical assumption we shall require in the proofs is that for allI " I of full µI -measure and all E " X $Q (the product #-algebra on X$Q),

$

x " X :%

{x}$ {ai : i " I % $!1({x})}&

% E &= ''

" X , (1)

i.e. this set is a measurable subset of X . Thus the set of those classes whichcontain some individual playing one of a given measurable set of strategies isitself measurable. This is somewhat reminiscent of the assumptions used inmeasurable selection theorems, e.g. that of Kuratowski-Ryll-Nardzewski (seeWagner 1977), and its role will be in precisely this kind of context. We notethat it is satisfied trivially if the set of classes X is finite or countable.

We now condense our notation a little: we define w "(

(i,")"I#!

)

M+(X)Si

*

byw"

i (q) = d!1i w(q(r"i ),$i, u

"i ),

for q " Si. So w"i (Si) # M+(X) is the set of all possible o!spring distributions

per haploid set of individual i in state of nature %, when considering all admis-sible strategies for that individual. Note that the partial maps % (! w"

i (q) forfixed individual i " I and strategy q " Si and i (! w"

i (ai) for fixed % " ! aremeasurable (see Rudin 1966, Theorem 7.5). The need to take averages in theo!spring generation, which is described (only) in terms of measures, means weneed some theory of vector integration. We use that of the Bochner integral,which is the most powerful, and bears the strongest resemblance to the morefamiliar Lebesgue integral (see Diestel and Uhl 1977). We make some impor-tant assumptions about the o!spring distribution, the first three of which arepurely technical and simply guarantee that we may pursue our argument ingreat generality:

– that the function (i,%) (! w"i (ai) is strongly measurable;

– that the function % (! w"i (q) is Bochner integrable for all i " I and q " Si;

– that the function i (! w"i (ai) is Bochner integrable for all % " !;

– that the total o!spring distribution W" := EI [w"i (ai)] does not in fact

depend on %, we therefore notate this by just W ; and– that µX := $#µI ) W . This is the assertion that classes may not be aban-

doned except by µI -null sets of individuals: a positive distribution of theparental generation on some set of classes Y implies a positive distributionon Y of the o!spring.

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9

We record here two simple facts about Bochner integration which we shallneed. The first is essentially a version of Fubini’s theorem, stating that inte-grating a function with respect to an average measure is the same as averagingover the integrals over individual measures.

Lemma 1 Let (Y,Y, µY ) be a measure space, and let (Z,Z) be a measurablespace. Let m : Y ! M+(Z) be Bochner integrable, m :=

+

Ym(y)µY (dy) "

M+(Z), and f : Z ! [0,*] be measurable.Then the function

Y + y (!

"

Z

f(z)m(y)(dz)

is measurable, and"

Z

f(z) m(dz) =

"

Y

!"

Z

f(z)m(y)(dx)

#

µY (dy). (2)

Remark 1 We do not assume that the function f is integrable; thus (2) alsoapplies in the case that the integrals are infinite. The result will not extendin this generality to arbitrary real-valued (i.e. possibly negative-valued) func-tions, since in this case the argument would involve the indefinite term *,*.

Proof The measurability result can be seen by returning to the definition ofthe integral, and using the measurability of the map taking measures to theirevaluation on some fixed set, and the algebra of measurable functions.

Equation (2) can be seen by routine approximation of the integral by simplefunctions. -.

Lemma 2 Let (Y,Y, µY ) be a finite measure space, (Z,Z) be a measurablespace, m : Y ! M(Z) be Bochner integrable, E " Z.

Then y (! m(y)(E) is integrable and!"

Y

m(y)µY (dy)

#

(E) =

"

Y

m(y)(E)µY (dy).

Proof This is a trivial consequence of the fact that m (! m(E) is a boundedlinear operator from M(Z) to R, and the result of Hille (Diestel and Uhl1977, Chapter 2, Theorem 2.6) that Bochner integration commutes with closedoperators. -.

Finally, it is unusual in population genetic models to leave the connectionunspecified between genotype and phenotype, but it is one of the notable fea-tures of the covariance selection mathematics of Price (1970). We can obtaina su"cient purchase on that connection by using two population genetic con-cepts. A p-score is an arbitrary weighted sum of allele frequencies, and is thusa linear functional on the set of genotypes: by proving results for an arbitraryp-score, we manage to say something significant about selection of genotypesin general. If a phenotype is a real number, such as height, we can find the

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10

p-score that best predicts the phenotype across the whole population. Some-times it will be useful to discuss those predicted phenotypes, which are knownin biology as additive genetic values (of that given trait).

Thus we formally introduce the concepts of additive genetic value and p-scores. Suppose each individual i " I has at most N / 1 loci, and at mostn / 1 rival alleles for each locus. Then the space Rn#N of real-valued n $N matrices G can be regarded as containing all the matrices of genotypesof individuals: the entry gk,l of the matrix G representing the genotype ofindividual i is the number of alleles k at locus l of i. Let g : I ! Rn#N be themap assigning each individual its genotype. A p-score is a function p : I ! R

representing an additive genetic trait (Grafen 2000), i.e. a linear combinationof allele frequencies. Hence the following definition.

Definition 1 (p-score and additive genetic trait) Let p " L$(I, I, µI ;R).Then p is a p-score, i.e. represents an additive genetic trait, if there exists alinear map & : Rn#N ! R such that pi = &(d!1

i gi). Let P denote the subspaceof L$(I, I, µI ;R) comprising these p-scores; P is then a finite-dimensionalsubspace of L$(I, I, µI ;R), of dimension K 0 N $ n, say.

Using the Gram-Schmidt process, we can construct a basis for P consistingof functions {pl}Kl=1 such that

"

I

(pl)i(pl!)i µI(di) =

,

1 l = l%

0 otherwise.

To define the additive genetic value of an arbitrary integrable function ofindividuals, we simply project onto the set of p-scores, P. We choose ourcoe"cients so that the kernel of this projection lies in the pre-annihilator ofthe subspace P, with the usual identification of dual spaces, or rather thatthe average over all individuals carrying any given allele of the additive geneticvalue of a trait is equal to the same average of the trait itself.

Definition 2 (Additive genetic value) Let f " L1(I, I, µI ;R). Then theadditive genetic value of f , agv(f) " P, is given by

agv(f) =K-

l=1

!"

I

(fi)(pl)i µI(di)

#

pl.

This is then a p-score, and represents that component of f which is heritable.We have thus defined a bounded linear operator agv : L1(I, I, µI ;R) ! P.Evidently then f = agv(f) + (f , agv(f)), where by choice of the pl, we seethat for any p " P,

"

I

(fi , agv(f)i)pi µI(di) = 0. (3)

This equation also assures us that agv does not depend on the choice of ba-sis for P, for suppose two bases give two definitions, agv1 and agv2, say,

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11

both of which satisfy equation (3). Then for all f " L1(I, I, µI ;R), sinceagv1(f), agv2(f) " P, applying (3) gives that"

I

(agv1(f)i , agv2(f)i)2 µI(di)

=

"

I

(agv1(f)i)2 µI(di), 2

"

I

agv1(f)iagv2(f)i µI(di) +

"

I

(agv2(f)i)2 µI(di)

=

"

I

fiagv1(f)i µI(di),

"

I

fiagv1(f)i µI(di)

,

"

I

fiagv2(f)i µI(di) +

"

I

fiagv2(f)i µI(di)

= 0,

and hence that agv1(f) = agv2(f) as elements of P.

4 Reproductive Value and the Price Equation

The Price Equation represents gene frequency change in our argument, but inits original form does not admit uncertainty or class-structure. This sectionderives a suitable Price Equation, simultaneously generalizing the uncertaintyof Grafen (2002) and the class-structure of Grafen (2006b). The Markov theoryallows us to weight over the classes to obtain a single average change in p-scorefrom one time period to the next. The central property needed is that two setsof weights are the same: those used to average across classes to obtain anaverage change in mean p-score on the left hand side; and those used to obtaina single measure of reproductive success for each individual, averaging acrossthe classes of its o!spring, to include in the right hand side. The original ideaof using the leading eigenvector of a class-to-class transition matrix goes backto Taylor (1990, 1996), and may be detected in embryo form in the famoussex ratio argument of Fisher (1930).

4.1 Links to Markov Theory

In this section we apply our assumption that, while uncertainty may a!ectthe fitnesses of individuals, it does not a!ect class-to-class projection at apopulation level. We do this by defining a Markov process for arbitrary %, andthen assuming that there exists an associated invariant measure that does notdepend on the choice of %. Further work aims to remove this assumption. Theinvariant measure represents the ‘reproductive value’ of a subset of classes: thatis, if we take a su"ciently distant generation and choose an allele at random,what is the probability that its ancestor today is present in an individual thatbelongs to a class in that subset?

Given all our data, we can follow the methods of Grafen (2006b) for eachstate of nature % " !. To find an invariant measure and the appropriate

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12

weightings for classes, we must first understand how the class distribution ofthe population changes, given the pattern of o!spring production describedabove.

Lemma 3 Let % " !, and let f : I ! R be such that i (! fiw"i (ai) as a map

from I to M(X) is Bochner integrable.Then EI [fiw"

i (ai)] ) W .

Proof This is trivial using Lemma 2 since i (! w"i (ai) maps into positive

measures. -.

Therefore for such a measure EI [fiw"i (ai)], the Radon-Nikodym derivative

with respect to W , ddW

EI [fiw"i (ai)], is defined. Applying this remark to char-

acteristic functions of sets of individuals sharing classes, we can define a (dis-crete time) Markov process with state space X by defining the probabilitytransition function P" : X $ X ! [0, 1] by

P"(x,A) =

.

d

dW

.

"

$"1(A)w"

i (ai) µI(di)

//

(x).

P"(x,A) is then well-defined W -almost everywhere, and represents the pro-portionate contribution by parents belonging to a class in A to o!spring ofclass x.

Following Rosenblatt (1971), we can use P"(·, ·) to define a linear functionalT" : L$(X,X , µX ;R) ! L$(X,X , µX ;R) by defining

(T"f)(x) =

"

X

f(y)P"(x, dy).

This is the average of f over all those parents contributing o!spring to class x,and therefore represents how values on classes may be traced back through ageneration. T" is clearly well-defined since P"(x, ·) ) µX , and for W -almostevery x " X , we have that

|(T"f)(x)| 0 1f1L#(X,X ,µX ;R).

Since µX ) W , we see that this exceptional set is µX -null, i.e. T"f "L$(X,X , µX ;R) indeed. We shall find the following alternative expressionfor T"f useful when deriving the Price equation.

Lemma 4 Let f " L$(X,X , µX ;R).Then

(T"f)(x) =d

dW

!"

I

(f 2 $)iw"i (ai) µI(di)

#

(x) (4)

for W -almost every x " X.

Proof This follows by routine approximation of the integral by simple func-tions, using the vectorial version of the dominated convergence theorem (Di-estel and Uhl 1977, Chapter 2, Theorem 2.3). -.

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13

Associated with any such process is the notion of an invariant measure, i.e. ameasure '" " M+(X) such that

"

X

P"(x,A) '"(dx) = '"(A)

for all A " X , or, equivalently,"

X

f(x) '"(dx) =

"

X

(T"f)(x) '"(dx)

for all f " L$(X,X , µX ;R). Such a weighting then appropriately balancesthe reproductive outputs of classes so that average values across classes arepreserved from parental to o!spring generation.We remark that since P"(x,A)is only well-defined for W -almost every x " X , we can a priori only integrateP"(x,A) with respect to a measure '" if '" ) W . Thus we assume that anydiscussion of invariant measures is restricted to such absolutely continuousmeasures, so that the derivative d%!

dW is well-defined.We assume that an invariant measure exists for each % " !; we refer

to Grafen (2006b) for a discussion of the assumptions required to guaranteethis. We further assume that there exists an invariant measure '" which is infact independent of %, thus we can just write ' for this invariant measure.

This weighting of classes based on reproductive output is the key ingredientin our definition of fitness of individuals. Given the invariant measure ' , wecan now define ‘fitness operator’ F" "

(

i"I [0,*]Si by, for i " I, setting

Si + q (! F"i (q) :=

"

X

d'

dW(x) w"

i (q)(dx).

For each individual i " I, this is a function of strategy, and is an appropriatelyweighted average of the individual’s o!spring when playing each given strategy.

For this and the following subsection we consider a fixed element p "L$(I, I, µI ;R). We emphasize that at this stage this is an arbitrary elementof the space, and not in general a p-score, and assigns (bounded) numbers toindividuals in a manner not necessarily determined by genotype. We definethe class-average value of p by defining X -measurable function ( : X ! R asthe Radon-Nikodym derivative with respect to µX of the measure on X givenby Y (!

+

$"1(Y ) pi µI(di). We see that ( satisfies

(( 2 $) = E[p|$],

as elements of L1(I, I, µI ;R). Note that properties of conditional expectationsimply that ( " L$(X,X , µX ;R), since p " L$(I, I, µI ;R).

Fix % " !. Since p " L$(I, I, µI ;R), the correspondingly weighted o!-spring function pw"(a) is Bochner integrable and EI [piw"

i (ai)] ) W byLemma 3. We can define the class average of this weighted o!spring func-tion, (" " L1(X,X ,W ;R), by

(" =d

dW(EI [piw

"i (ai)]). (5)

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14

We expand later on the precise interpretation of this value and under whatfurther assumptions it becomes a quantity of greater relevance, e.g. when itbecomes a useful estimate for the class average value of p in the followinggeneration. In the next lemma we make the observation that this o!springclass average is, like the parental class average, a bounded function.

Lemma 5 With (" as defined above, in fact (" " L$(X,X ,W ;R).

Proof Let c > 0 and use Lemma 2 to see that

cW (((")!1((c,*))) 0

"

(&!)"1((c,$))("(x)W (dx)

= EI [piw"i (ai)](((

")!1((c,*)))

=

"

I

piw"i (ai)(((

")!1((c,*))) µI(di)

0 1p1L#(I,I,µI ;R)

"

I

w"i (ai)(((

")!1((c,*))) µI (di)

= 1p1L#(I,I,µI ;R)

!"

I

w"i (ai) µI(di)

#

(((")!1((c,*)))

= 1p1L#(I,I,µI ;R)W (((")!1((c,*))).

Hence if W (((")!1((c,*))) &= 0, we see that c 0 1p1L#(I,I,µI ;R). Similarly wecan show that

,cW (((")!1((,*,,c)) / ,1p1L#W (((")!1((,*,,c))),

and hence again that if W (((")!1((,*,,c))) &= 0 then c 0 1p1L#(I,I,µI ;R).Combining these results and taking the contrapositive, we see that if c >1p1L#(I,I,µI ;R), then W ((|(" |)!1((c,*))) = 0. In other words, we infer that1("1L#(X,X ,W ;R) 0 1p1L#(I,I,µI ;R). -.

4.2 The Price Equation

It turns out that the properties of the invariant measure discussed in the pre-vious subsection are those needed to construct a suitable Price Equation. Thearguments of this subsection apply for a fixed % " !. We therefore suppressthe superfluous % superscript from the notation.

The definition of ( in (5), linearity of Radon-Nikodym di!erentiation, andLemma 4 imply that

( =d

dW(EI [piwi(ai)]) = T( +

d

dW(EI [(pi , E[p|$]i)wi(ai)]) .

Using the above equation, the invariance of the class-weighting ' , and thechange of variables formula for Radon-Nikodym derivatives (Halmos 1950,

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15

§32, Theorem B), we see that

"

X

(((x), ((x)) '(dx) =

"

X

!

d

dW(EI [(pi , E[p|$]i)wi(ai)])

#

'(dx)

=

"

X

d'

dW(x)EI [(pi , E[p|$]i)wi(ai)] (dx). (6)

Note that since W and ' are positive measures, d%dW : X ! [0,*). Integration

with respect to signed measures is understood as in Rudin (1966, §6.18), thatis, as an integral with respect to the ‘polar decomposition’ of the measure, thusit is defined in terms of a usual integral with respect to a positive measure.

Our plan is to interchange the order of integration in (6) so that the expres-sion becomes an average over individuals of an integrand involving a weightedaverage over classes of o!spring measures, and thus looks rather more like anintegral involving the fitness of individuals as defined above. This manipu-lation is not immediate, but is permitted by the following technical lemma,which is a similar statement to the quasi-Fubini result of Lemma 1, but nowinvolves in general non-positive weights on the measures wi(ai), correspondingto our desire to allow p-scores to be defined via arbitrary allelic weightings.This makes the argument a little more delicate, so we include some details ofthe proof.

Lemma 6 Let f " L$(I, I, µI ;R), g " L1(X,X , µX ; [0,*)), and all othernotation be as above.

Then

"

X

g(x)

!"

I

fiwi(ai) µI(di)

#

(dx) =

"

I

fi

!"

X

g(x) wi(ai)(dx)

#

µI(di). (7)

Proof Define the weighted o!spring measureWf " M(X) byWf = EI [fiwi(ai)].The result is easily seen to be true, via Lemma 2, for measurable simple

functions. Let sn : X ! [0,*) be a sequence of measurable simple functionssuch that sn(x) 3 g(x). We easily see by the monotone convergence theoremthat for i " I,

"

X

g(x) (fiwi(ai))(dx) = fi

"

X

g(x) wi(ai)(dx).

Now, for i " I, we have that

0

0

0

0

fi

"

X

sn(x) wi(ai)(dx)

0

0

0

0

0 1f1L#(I,I,µI ;R)

"

X

g(x) wi(ai)(dx),

where by Lemma 1,

"

I

!"

X

g(x) wi(ai)(dx)

#

µI(di) =

"

X

g(x)W (dx) < *

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16

since g " L1(X,X ,W ; [0,*)). So i (!+

Xg(x) wi(ai)(dx) is µI -integrable, and

thus by the dominated convergence theorem,

limn&$

"

I

fi

!"

X

sn(x) wi(ai)(dx)

#

µI(di) =

"

I

fi

!"

X

g(x) wi(ai)(dx)

#

µI(di).

(8)

This argument shows that the limits behave as required for the right-hand sideof our required expression.

We recall from the definition of integration with respect to a signed measurethat there exists some X -measurable function ) : X ! {±1} such that dWf =) d|Wf |, thus

"

X

g(x)Wf (dx) =

"

X

g(x))(x) |Wf |(dx)

and"

X

sn(x)Wf (dx) =

"

X

sn(x))(x) |Wf |(dx)

for all n / 1. Now we observe that for arbitrary Y " X , use of Lemma 2 showsthat

|Wf (Y )| 0 1f1L#(I,I,µI ;R)W (Y ),

and hence"

X

g(x)Wf (dx) 0 1f1L#(I,I,µI ;R)

"

X

g(x)W (dx),

since g / 0. So g " L1(X,X ,Wf , [0,*)), and thus g) " L1(X,X , |Wf |; [0,*)).Since |)(x)| = 1 for µX -almost every x " X , the choice of sn implies that

|sn(x))(x)| 0 |g(x))(x)| ,

so we can used the dominated convergence theorem to see that

limn&$

"

X

sn(x))(x) |Wf |(dx) =

"

X

g(x))(x) |Wf |(dx).

Hence by definition

limn&$

"

X

sn(x)Wf (dx) =

"

X

g(x)Wf (dx).

So, using the result on measurable simple functions and (8), we see that"

X

g(x)Wf (dx) = limn&$

"

X

sn(x)Wf (dx)

= limn&$

"

I

!"

X

sn(x) fiwi(ai)(dx)

#

µI(di)

=

"

I

fi

!"

X

g(x) wi(ai)(dx)

#

µI(di),

as required. -.

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17

Applying this result with fi = pi , E[p|$]i and g(x) = d%dW

(x), we recall (6)and the definition of Fi(ai) and see now that"

X

(((x) , ((x)) '(dx) =

"

I

(pi , E[p|$]i)

!"

X

d'

dW(x) wi(ai)(dx)

#

µI(di)

= EI [(pi , E[p|$]i)Fi(ai)]. (9)

We now wish to reintroduce and average over the states of nature % " !which have hitherto been fixed.

Fixing both an individual i " I and an admissible strategy q " Si, Lemma 1implies that % (! F"

i (q) is measurable. So we may define the expected fitnessfunction F "

(

i"I [0,*]Si by

Fi(q) = E! [F"

i (q)] ,

or, equivalently, by (2),

Fi(q) =

"

X

d'

dW(x)E! [w"

i (q)](dx). (10)

This is the expected fitness of an individual i " I playing strategy q " Si, andthe latter expression shows it to be given by an appropriately-weighted averageof the expected contributions to o!spring classes, when playing strategy q.

In this general context we must check that the fitness of the individualswith their realized phenotypes is a well-defined finite number. As above, themap (i,%) (! F"

i (ai) is measurable, and hence by the classical Fubini Theo-rem (Rudin 1966, Theorem 7.8), i (! Fi(ai) is measurable. Fubini’s Theoremfor Bochner integrals (see Dunford and Schwartz 1958, §III.11.9 Theorem 9)implies that

EI [E![w"

i (ai)]] = E![EI [w

"i (ai)]].

Hence we note, using (10) and (2) once more, that

0 0

"

I

Fi(ai) µI(di) =

"

I

!"

X

d'

dW(x)E! [w"

i (ai)](dx)

#

µI(di)

=

"

X

d'

dW(x)EI [E

! [w"i (ai)]](dx)

=

"

X

d'

dW(x)E! [EI [w

"i (ai)]](dx)

=

"

X

d'

dW(x)E! [W ](dx)

=

"

X

d'

dW(x)W (dx)

= '(X)

< *.

Thus i (! Fi(ai) is a function in L1(I, I, µI ;R), and hence 0 0 Fi(ai) < * forµI -almost every i " I.

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18

The final step is to define the expected value of our class-average value(" in the following generation. Radon-Nikodym di!erentiation is an isometryfrom {µ " M(X) : µ ) W} to L1(X,X ,W ;R), so the function % ! (" isstrongly measurable. Moreover, for each % " !, we see using standard factsabout Bochner integrals (see Diestel and Uhl 1977, Chapter 2, Theorem 2.4)that

1

1

1

1

d

dWEI [piw

"i (ai)]

1

1

1

1

L1(X,X ,W ;R)

0 1EI [piw"i (ai)]1M(X)

0 EI

2

1piw"i (ai)1M(X)

3

0 1p1L#(I,I,µI ;R)W (X),

and therefore that % (! (" is Bochner integrable. We may therefore define theexpected value ( " L1(X,X ,W ;R) by

( = E! [("].

Using once more that Radon-Nikodym di!erentiation is an isometry, we usethe result of Hille (Diestel and Uhl 1977, Chapter 2, Theorem 2.6) to see thatwe can commute it with the expectation and conclude that

( = E!

4

d

dWEI [piw

"i (ai)]

5

=d

dWE! [EI [piw

"i (ai)]]. (11)

Having established these expected values of our variables, we use expres-sion (9), linearity of expectation and Fubini’s theorem to see that the expecteddi!erence in the class-average values calculated in the two generations can berelated to the p-score with which we began and expected fitness:

"

X

(((x) , ((x)) '(dx) = E!

4"

X

(("(x), ((x)) '(dx))

5

= EI

2

(pi , E[p|$]i)E![F"

i (ai)]3

= EI [(pi , E[p|$]i)Fi(ai)] (12)

= EI [pi(Fi(ai), E[F (a)|$]i)] , (13)

where the conditional expectation of F (a) is understood here and elsewhere(including references to the corresponding covariance) to refer to the functioni (! Fi(ai).

We remark that at this point, with the definitions given and properties as-sumed, these equations are not necessarily susceptible to intelligible interpre-tation, despite our suggestive notation, since they do not in general representthe change from one generation to the next of a comparable quantity. When,however, the function p : I ! R is a p-score, i.e. represents an additive genetictrait and is thus an allele frequency or linear combination of allele frequen-cies, then as discussed above p is the composition of a map on an underlyingspace of genotypes and a map specifying each individual’s genotype. Thus inprinciple corresponding values may be computed in the o!spring generation.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

19

Assuming perfect transmission, i.e. no mutation, fair meiosis, and no gameticselection (Grafen 2000), precisely by definition of being an additive genetictrait, the expected mean value by class of this trait in the o!spring generationis given by the expectation of the average over the gametic contributions (Fal-coner 1981). However, (11) implies that for all Y " X ,

"

Y

((x)W (dx) = E! [EI [piw

"i (ai)]](Y ).

The right-hand side is precisely the expected average of the gametic contri-butions to the set of classes Y . Since ( " L1(X,X ,W ; [0,*)) is the uniqueelement satisfying this equation for all Y " X , we see that ( therefore rep-resents the expected mean value by class of the additive genetic trait in theo!spring generation, W -almost everywhere. Since ' ) W , any discrepancieson W -null sets are lost when weighted by reproductive value ' . Thus the equa-tions we derived above record the change in the mean value by class of thisadditive genetic trait from the parental to the o!spring generation, weightedby reproductive value. We record these remarks in the following theorem.

Theorem 1 (The Price Equation) Supposing perfect transmission, the ex-pected change in the mean value of an additive genetic trait, weighted by re-productive value, is given by

"

X

(((x) , ((x)) '(dx) = EI [(pi , E[p|$]i)Fi(ai)]

= CI [pi , E[p|$]i, Fi(ai)]

= EI [CI [(p, F (a))|$]i]

= EI [pi(Fi(ai), E[F (a)|$]i)] (14)

= EI [pi(agv(F (a), E[F (a)|$])i)], (15)

where p " P represents the parental values of the trait.

Proof The interpretation of the left-hand side of (15) is discussed above. Theequality of the first four expressions on the right-hand side follows by standardproperties of conditional expectations (see Billingsley 1995, Theorem 34.3).Moving from (14) to (15) is possible in this situation because by assumptionp is a p-score, and therefore we can apply equation (3). -.

Equation (15) is the final version of the Price Equation in the situation of aclass-structured population with uncertainty, when this uncertainty does nota!ect the class distribution of o!spring at the population level.

The Price Equation thus represents for us how a special class-weightedmean of any arbitrary weighted sum of allele frequencies changes from thisgeneration to the next. The class-weights do not depend on the allele-weights,and individual expected fitnesses on the right hand depend on the class-weightsbut not on the allele-weights. Thus we obtain some purchase on changes inour arbitrary space of genotypes by knowing how every arbitrary weighted

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20

sum of allele frequencies changes. The genetic side can be linked to pheno-types at an individual level because the genetic changes depend only on theexpected individual fitnesses. These abstract connections make the most ofthe phenotype-genotype separation inherent in the original covariance selec-tion mathematics of Price (1970) and, remarkably, allow links to be made tofitness-maximization ideas without any further specification of how genotypedetermines phenotype.

4.3 Comparison with Grafen (2006b)

Before pursuing the implications of the Price Equation for fitness-maximization,we conclude this section by observing that our approach is indeed consistentwith that of Grafen (2006b). To see this we reduce our setup to the cases ex-amined there. Thus we suppose there is no e!ect of uncertainty, so ! = {%},and the o!spring distributions have no explicit dependence on class. Rather,the situation Grafen considers is where the o!spring distribution depends onlyon the individual. We may capture this situation by considering the measurew(i) in the notation of Grafen (2006b) to be given in our notation as w(ai).Let Y " X . We calculate the class reproductive value of Y , using our defini-tions and concepts, by regarding our fitness Fi(ai) as a Fisherian per-capitareproductive value per ploidy at the level of the individual. So our calculationis, using Lemma 1, change of variables in Radon-Nikodym derivatives, and theinvariance of ' :

"

$"1(Y )Fi(ai) µI(di) =

"

$"1(Y )

!"

X

d'

dW(x)w(ai)(dx)

#

µI(di)

=

"

X

d'

dW(x)EI [(1$"1(Y ))iw(ai)](dx)

=

"

X

d

dW(EI [(1$"1(Y ))iw(ai)])(x) '(dx)

=

"

X

d

dW(EI [(1Y 2 $)iw(ai)])(x) '(dx)

=

"

X

T1Y (x) '(dx)

=

"

X

1Y (x) '(dx)

= '(Y ).

This is precisely what Grafen considers to be class reproductive value. We can,then, for example, quickly recover Fisher’s sex ratio argument (after Grafen2006b, §8.1). In this situation we have a space of two classes, X = {M,F}say, representing the sexes. The assumption of equal male and female con-tribution to o!spring is precisely the assumption that EI [(1$"1(M))iw(ai)] =

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

21

EI [(1$"1(F ))iw(ai)] as measures on X , so, arguing as above,

"

$"1(M)Fi(ai) µI(di) =

"

X

d'

dW(x)EI [(1$"1(M))iw(ai)](dx)

=

"

X

d'

dW(x)EI [(1$"1(F ))iw(ai)](dx)

=

"

$"1(F )Fi(ai) µI(di).

Thus already our work seems to support the interpretation of Fisher’s no-tion of reproductive value as an evolutionary maximand. We shall uncoverfurther connexions to Fisher’s work as we formalize the fitness-maximizationconsequences of the Price Equation in the following sections.

5 Optimization

The work of this section is to construct an optimization program at the samelevel of generality as the Price Equation of the previous section. The instru-ment will be phenotype, the constraint set will be the set of possible pheno-types, and, most significantly from a biological point of view, the maximandwill be expected fitness. This definition of what fitness-maximization meansstands in contrast to the way population geneticists have in the past attemptedto represent the biologist’s sense of fitness-maximization, namely in terms ofnatural structures on dynamical systems such as Lyapunov functions and gra-dient functions (Ewens 2004). The contrast is immediately clear: both of thosemathematical concepts are functions from the space of gene-frequencies to thereal line, rather than from the set of possible phenotypes; it may also be notedthat working with those concepts requires dynamic su"ciency, which the cur-rent framework lacks. Grafen (2002) and Grafen (2006a) provided a paralleloptimization program including uncertainty, while Grafen (2006b) was unableto provide one with class structure: this section provides both simultaneously.Thus even providing an optimization program is a technical advance, and ithas biological significance in defining what fitness-maximization means.

It is worth remembering that a strategy, shortly to be defined, is a map-ping from environmental cues to actions, and thus the individual is regardedas making decisions in the face of partial knowledge about uncertainty. Solv-ing the program will therefore imply acting as if in possession of a correctprior distribution over the whole uncertainty, and as if performing appropri-ate Bayesian updating of that prior in the light of the partial informationreceived. The maximand is a probability-weighted arithmetic mean over theuncertainty, and so this fitness-maximization does not exhibit the risk aversionor bet-hedging that appears in many biological discussions of uncertainty (forreasons best explained by Frank and Slatkin 1990): that di!erence arises be-cause in this paper fitness is defined as relative to the population mean in eachgiven state of nature. (The recent series of papers by Frank (2011a,b, 2012a,b,c,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

22

2013a,b) presents a modern discussion of bet-hedging and many other relevanttopics.) The advantage of the relative definition is precisely that we can oper-ate at the current very high level of generality. The bet-hedging models are all,in comparison, very special cases, in that they assume some definite geneticarchitecture, and they do not focus on individual fitness maximization.

In moving from the population genetic model to the optimization program,there are two steps of note. The whole population of the genetic model mustbe reduced to the single implicit decision-taker of the optimization. In orderto do this, an assumption must be made to ensure that individuals withinthe same class are, in some suitable sense, equivalent. Nothing assumed up tonow prevents the population being in two separate halves with quite di!erentselection pressures. First noticed by Grafen (2002), this kind of assumptionis an essential part of any general argument linking population genetics tofitness maximization. Its precise form could be of interest in applications, indetermining whether fitness-maximization ideas can be applied or not. Perhapseven more importantly, requiring mathematical rigour and proofs allowed usto find and articulate this once-latent assumption, and also allows us to beconfident that there are no further assumptions waiting to be uncovered.

The definitions and results of the rest of the paper follow those of Grafen(2002), but are extended to include simultaneously uncertainty and the divi-sion of the population into classes.

Definition 3 (Pairwise exchangeability) We say the assumption of pair-wise exchangeability holds if

– for all measurable functions v : A$X $ U ! R, the function

(i, q, c) (! " ({% " ! : v (q(r"i ),$i, u"i ) 0 c}) ,

mapping from I$Q$R to [0, 1], is measurable with respect to the product#-algebra #($)$Q$ B, where B denotes the usual Borel #-algebra on R;and

– for all x " X , Sj = Sk whenever j, k " $!1({x}).

The significance lies in the stipulation of the #-algebra #($) on I with re-spect to which measurability is asserted. Technical reasons arising from thesubtleties of integration and measurability on product spaces demand that theassumption is stated in this somewhat elaborate form, but the content of theassertion lies in the reference to #($), rather than to I. Roughly, the assertionis that any pair of individuals of the same class have a symmetric distributionof chance events, and the collection of admissible strategies is the same. Thisis thus the natural generalization of the corresponding assumption in Grafen(2002, §4).

This can also be read as a comment on how the class structure is defined, instipulating that classes are not so broad as to allow within them individualswhich face on average wholly di!erent situations under the e!ects of environ-mental uncertainty. The question of how appropriate and workable the classstructure is arises again later (see definition 7), where a tension in the opposite

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23

direction is revealed, towards classes not being too small. We therefore find itmore informative to retain pairwise exchangeability as an explicit assumptionrather than incorporate it more implicitly into the fundamental properties ofthe class structure, and we thereby admit the possibility of a structured pop-ulation in which the assumption fails. For concreteness, suppose we have apopulation in which males and females have di!erent sets of possible actions,and in which juveniles di!er in survivorship from adults in each sex. Then itis possible that the assumption holds when we model the population with afull age and sex class structure. However, if we modelled with only age classesor with only sex classes, the assumption would fail. We further remark that,as indicated in the statements, some of our results hold without this assump-tion. As an aside, the issues of correctness of class structure, and whether suchquestions can be precisely formulated, are intriguing, particularly in relationto the interaction of class and genotype, and warrant further attention.

The important consequence of pairwise exchangeability is captured in thefollowing lemma, in which we see that under this assumption, within classes,expected fitness is a function only of strategies, not of individuals. Again, tech-nical reasons demand that the result is stated more subtly, but the essence isthe blindness to individual di!erences within classes, in expectation, of fit-ness. This allows us to pass from a class of individuals each playing their ownstrategy to a single implicit decision maker in each class.

Lemma 7 Suppose we have pairwise exchangeability.Then the map

6

i"I

({i}$ Si) + (i, q) (! Fi(q),

assigning an expected fitness to an individual playing a strategy admissible forthat individual, is measurable with respect to the induced product #-algebra#($) $Q. As in the definition of pairwise exchangeability, the significance ofthe statement lies in the assertion of measurability with respect to the #-algebra#($).

Proof By (10) and the definition of the integral, it su"ces to show that foreach set of classes Y " X the map

(i, q) (! E! [w"

i (q)](Y )

is measurable with respect to #($) $Q. So fix a set Y " X . Using Lemma 2and, for example, Kingman and Taylor (1966, Theorem 11.4) we have for eachindividual i " I and strategy q " Si that

E! [w"

i (q)](Y ) = E! [w"

i (q)(Y )] =

"

R

" ({% " ! : w"i (q)(Y ) 0 c}) L

1(dc),

(16)where the final integral is with respect to the usual one-dimensional Lebesguemeasure L 1 on R. That this final expression is measurable as a function of(i, q) with respect to #($)$Q follows by pairwise exchangeability and Fubini’sTheorem. -.

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24

Definition 4 (Optimization program) Let

S # {s : I $R ! A : s is measurable and si " Si for all i " I}

contain the realized phenotype allocation a and satisfy the following substitu-tion condition: if s, t " S and J " I, then the function st,J "

(

i"I Si definedby

(st,J)i(r) =

,

si(r) i /" J

ti(r) i " J

also lies in S. Thus substituting a di!erent admissible specification of pheno-types on a certain subset of individuals is also admissible. Let k " I. Weconsider the optimization program of maximizing Fk(sk) over all s " S,where s : k (! sk " Sk. We say s " S is a solution for k in relation to Sif Fk(sk) / Fk(sk) for all s " S.

Remark 2 The substitution condition on the choice set S precisely stipulatesthat the strategies available to any one individual do not depend on the strate-gies played by any other set of individuals, and is implied by our underlyingbiological assumption of lack of social interaction.

We now move on to two key concepts relating to the Price Equation. Theyare carefully constructed to be weak enough to be usable despite our lack ofmodel connecting phenotype to genotype, but strong enough to allow mean-ingful links to be made to optimization. Scope for selection almost means thatextant gene frequencies do change in expectation — in fact it says there isa possible constructible p-score that would change in expectation, where weallow ourselves to construct a ‘possible’ p-score by assigning an arbitrary realnumber to each individual (in a measurable way, of course).

Definition 5 (Scope for selection) We say there is no scope for selectionwhenever the expected change in any class average value is zero, i.e. whenever

"

X

(((x) , ((x)) '(dx) = 0

for all p " L$(I, I, µI ;R) (recall that ( and (" depend on p). Thus thereis scope for selection when there are di!erences in fitness that could causean allele frequency to change. We emphasize that this definition discussesbehaviour for arbitrary essentially bounded functions p, not just p-scores.

This is not an analogue of the first part of the ESS definition of Maynard Smithand Price (1973), nor of a first-order condition in simple optimization. Notethat the condition is that no mean p-score and so no gene frequency changes,and that nothing is said about genotype frequencies. This is an inevitable con-sequence of our abstract setting, and has consequences for the interpretationof the links to be proved later on.

The second condition considers a counter-factual case in which some of thephenotypes in the population are replaced with a new phenotype, and asks

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25

whether, supposing the individuals with altered phenotype each had one copyof a new allele, that allele would spread in expectation. There is potential forselection if there is a possible phenotype for which the answer is yes. Thusscope for selection is about standing variation in phenotypes, and potentialfor selection is about whether a new phenotype would spread if caused by arare dominant mutation.

Definition 6 (Potential for positive selection) Consider an alternativeset of strategies s " S and a subset of individuals J " I. We define a hypo-thetical rival set of admissible strategies as,J " S by

(as,J )i(r) =

,

ai(r) i /" J

si(r) i " J.

Then as,J " S represents individuals in J swapping to strategies given bys, and indeed is itself admissible and lies in S precisely by our substitutionassumption on S.

We say there is no potential for positive selection in relation to S if for alls " S and all J " I, we have

EI [(d!1i (1J )i , E[(d!1

1J)|$]i)Fi((as,J)i)] 0 0.

Note that the condition is satisfied trivially, with equality holding, if µI(J) = 0.We make the stipulation that the substitution of strategy on the set J a!ectsonly the evaluation of the function F on points i " J , despite the fact thatF depends by definition on W , which is in general a di!erent measure underthis substitution.

The purpose of this definition is to enable us to discuss the possibility of amutant allele invading the population on some set of individuals J , and therebyaltering the strategy ai of these individuals to si. The defining inequality comesof course from the relation (12). The function i (! d!1

i (1J )i is the p-scoreobtained by allocating weight 1 to the mutant allele and 0 to all others: ittherefore represents the presence of one copy of the mutant allele in preciselythose individuals i " J . Thus there is potential for selection when a raremutant, producing with dominance some admissible phenotype, would initiallyincrease in density in the population. The restriction of rarity arises becauseof the implicit assumption that no individual possesses more than one copy ofthe mutant gene.

6 Links between gene dynamics and optimization

Having established the situation, we prove analogous version of the four links tobe found in Grafen (2002). These comprise three implications for gene dynam-ics based on optimization assumptions, and one implication for optimizationbased on assumptions about gene dynamics. As the instrument and constraintset of the optimization are more or less copied over from the population ge-netic assumptions, the focus of interest is the maximand. In particular, it is

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26

important to show that the maximand has many of the properties a biologistwould wish ‘fitness’ or ‘expected fitness’ to have. From the point of view ofits construction, it is an average over environmental uncertainty of a class-weighted sum of o!spring numbers, which is biologically reasonable. From thepoint of view of links to genetics, an unattainable dream would be to showthat exactly when each individual maximizes its fitness, the population geneticsystem itself is in equilibrium. Our level of abstraction prevents us obtainingsuch a definite result, but it is in any event untrue, for example in simplecases such as over-dominance in a diploid population (Allison 1954). Our aimis therefore to prove the strongest possible results in that direction with thetwin aims of showing that there are close ties between fitness-maximizationand gene-frequency change, and that our definition of fitness is essentiallyunique, though this latter point will not be pursued formally in the currentpaper. The interpretation of the theorems will be discussed in the followingsection.

Theorem 2 Suppose we have pairwise exchangeability, and suppose for someset of individuals I " I of full µI-measure that a is a solution for i in relationto S for every i " I.

Then there is no scope for selection and no potential for positive selectionin relation to S.

Proof The trick is to exploit the assumption of pairwise exchangeability toinfer from the assumption of maximization that the expected fitness of therealized phenotypes is equal to its class average. By Lemma 7 and the Doob-Dynkin Lemma (see Rao 2004, §3.1 Theorem 8), there is a measurable functionH : X $Q ! [0,*] such that

Fi(q) = H($i, q)

for all i " I and q " Si.Fix x " X and consider j, k " $!1({x}) % I. Note that by pairwise ex-

changeability aj " Sk and ak " Sj , so it makes sense to evaluate Fk(aj) andFj(ak). Then

Fj(aj) = H($j , aj) = H($k, aj) = Fk(aj) 0 Fk(ak),

since a is a solution for k in relation to S. Swapping the roles of j and k we getthe reverse inequality, thus Fk(ak) = Fj(aj). Hence for any Borel set B # R,

$

i " I : Fi(ai) " B'

=$

i " I : ($i, ai) " H!1(B)'

=$

i " I :%

{$i}$ {aj : j " I % $!1({$i})}&

%H!1(B) &= ''

= I % $!1%$

x " X :%

{x}$ {aj : j " I % $!1({x})}&

%H!1(B) &= ''&

.

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27

Since H is measurable, this final line is the restriction to I of the $-pre-imageof a measurable subset of X , by our assumption (1), which therefore lies in#($) by definition. Hence i (! Fi(ai) is measurable with respect to #($) as amap from I. By definition, i (! E[F (a)|$]i is measurable with respect to #($).Furthermore, by definition of conditional expectation, and since I\I is µI -null,we see for any Y " X that

"

I'$"1(Y )Fi(ai) µI(di) =

"

$"1(Y )Fi(ai) µI(di)

=

"

$"1(Y )E[F (a)|$]i µI(di)

=

"

I'$"1(Y )E[F (a)|$]i µI(di).

Since Y is arbitrary, this implies (see for example Halmos 1950, §25 Theo-rem E) that Fi(ai) = E[F (a)|$]i for µI -almost every i " I, and hence for µI -almost every i " I. Thus the expression (13) is zero for any p " L$(I, I, µI ;R).

For the second assertion of the theorem, fix s " S and J " I. Note thatthe first argument gives us in particular that

EI [(d!1i (1J )i , E[d!1

1J |$]i)Fi(ai)] = EI [d!1i (1J)i(Fi(ai), E[F (a)|$]i)] = 0,

using properties of conditional expectation, and recalling that ploidy d is mea-surable with respect to #($). So again using properties of conditional expec-tation, we see that, since Fi((as,J )i) = Fi(ai) for i /" J ,

EI

2

(d!1i (1J)i , E[d!1

1J |$]i)Fi((as,J)i)3

= EI

2

d!1i ((1J )i , E[1J |$]i)(Fi((as,J )i), Fi(ai))

3

(17)

=

!"

I

di µI(di)

#!1"

J

d!1i ((1J )i , E[1J |$]i) (Fi((as,J )i), Fi(ai)) di µI(di)

=

!"

I

di µI(di)

#!1"

J

(1, E[1J |$]i) (Fi((as,J )i), Fi(ai)) µI(di)

0 0, (18)

where the final inequality follows since E[1J |$]i 0 1 µI -almost everywhere andFi((as,J )i) 0 Fi(ai) for i " I because a is a solution for i in S by assumption,and µI(J\I) = µI(J\I) = 0. -.

We must now consider on what subsets of individuals we can consider a hypo-thetical mutant allele invading, when discussing potential for positive selection.

Definition 7 (Invadable sets) Let

#($) = {J " I : J = K4N for some K " #($) and some null set N " I},

where K4N = K\N 5 N\K is the usual set-theoretic symmetric di!erence.We shall say that J " I of positive µI -measure is invadable if there exists

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28

K " I\#($) of positive µI -measure such that µI(K\J) = 0. In particular anyset J " I\#($) of positive µI -measure is invadable. We shall further say thatour phenotype specification a : I $R ! A is invadable if the set

{i " I : a is not a solution for i in relation to S}

is either µI -null or contains an invadable subset.

Invadable sets and their existence are determined by the nature of the classmap $, as the remarks below indicate. Class is intuitively intended to be away of dividing a large population of individuals into smaller groups: naivelyone would expect each class to be populated by many individuals, in whichcase invadable sets are in abundance. In the generality pursued here, where norestriction is placed on the cardinality of either the population or the set ofclasses, such informal remarks have little meaning, but the principle underlyingthis discussion is that the condition of a set being invadable should not, undera natural and useful class allocation, be a restrictive one.

The technical point of this definition is that an invadable set contains asubset K for which the defining condition of potential for positive selectionis not trivially zero. Restricting attention to invadable sets ensures that thepresence or absence of potential for positive selection is indeed determined bythe evaluation of the maximand.

Remark 3 The crucial consequence of K not lying in #($) is that then it isnot the case that E[1K |$] = 1 µI -almost everywhere on K.

Remark 4 Removing a µI -null set from an invadable set leaves an invadableset, and supersets of invadable sets are invadable.

Remark 5 We note the following special cases:

(5.1) J " I is invadable if

0 < µI(J) < inf7

µI($!1({x})) : x " X such that µI($

!1({x})) > 08

.

(5.2) If $!1({x}) is an atom of µI for all x " X , then no set is invadable.(5.3) If I is finite, and singletons have positive µI -measure, then invadable

sets exist if and only if $ is not injective.

Theorem 3 Suppose that there exists a set of individuals I " I of full µI-measure such that Fi(ai), E[F (a)|$]i = 0 for each i " I, but that the set

J = {i " I : a is not a solution for i in relation to S}

is of positive µI-measure.Then there is no scope for selection. However, assuming pairwise exchange-

ability and that a is invadable, there is potential for positive selection in relationto S.

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29

Proof For the first assertion, we first observe that the corresponding assertionof Theorem 2 in fact required only that Fi(ai) , E[F (a)|$]i = 0 µI -almosteverywhere. In that situation this was implied by optimality of each a for µI -almost every individual i, but here it is given explicitly by assumption. So thesame proof applies.

For the second assertion, by the assumption of invadability, we can choosesetsK, K " I of positive µI -measure with K # K such that (1K)i,E[1K |$]i >0 for each i " K, and µI(K\J) = 0.

We know that, since a " S is not a solution in relation to S for eachi " K % J , we can choose s " S and for each i " K % J a number *i > 0 suchthat Fi(ai) + *i < Fi(si). Since i (! Fi(si) is measurable, the map i (! *i canbe assumed to be measurable. By the substitution assumption on S, as,K " S.Thus for i " K % J , we have by choice of s that

Fi((as,K)i) > Fi(ai) + *i. (19)

Since there is no scope for selection, we again have the identity (17), and sowe can argue, using (19) and the fact that µI(K\J) = 0, as follows:

EI

2

(d!1i (1K)i , E[d!1

1K |$]i)Fi((as,K)i)3

=

!"

I

di µI(di)

#!1"

K

((1K)i , E[1K |$]i) (Fi((as,K)i), Fi(ai)) µI(di)

/

!"

I

di µI(di)

#!1"

K'J

((1K)i , E[1K |$]i)*i µI(di)

/

!"

I

di µI(di)

#!1"

K'J

((1K)i , E[1K |$]i)*i µI(di)

> 0,

since the choice of K implies that the integrand is strictly positive at eachpoint of K % J , which is a set of positive µI -measure. -.

Theorem 4 Suppose there exists a set of individuals J " I of positive µI-measure such that Fi(ai), E[F (a)|$]i &= 0 for each i " J .

Then there is scope for selection.

Proof Define p : I ! R by pi = Fi(ai) , E[F (a)|$]i. Then we do not know apriori that p defines an essentially bounded function on I. However, since it isonly important for this proof that the function is non-zero on a set of positivemeasure, we can truncate the function if necessary and assume without lossof generality that p " L$(I, I, µI ;R). Then (13) implies, for this definition ofp, that"

X

(((x), ((x)) '(dx) = EI [(Fi(ai), E[F (a)|$]i) (Fi(ai), E[F (a)|$]i)]

/

"

J

(Fi(ai), E[F (a)|$]i)2 µI(di)

> 0. -.

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30

Theorem 5 Suppose we have pairwise exchangeability and that a is invad-able, and suppose there is no scope for selection and no potential for positiveselection in relation to S.

Then there exists a set of individuals I " I of full µI-measure such that ais a solution for i in relation to S for each i " I.

Proof By the contrapositive to Theorem 4, since there is no scope for selectionwe know that Fi(ai), E[F (a)|$]i = 0 for µI -almost every i " I.

Given this, the contrapositive to the second assertion of Theorem 3 givesthe required result. -.

A glance at the proof of Theorem 4 prompts us to record the following impor-tant consequence of our work.

Theorem 6 (Fisher’s Fundamental Theorem of Natural Selection (Dis-crete Time)) Let p " P be defined by

pi = agv(F (a), E[F (a)|$])i.

Then"

X

(((x), ((x)) '(dx)

=

"

I

!

agv(F (a) , E[F (a)|$])i , EI [agv(F (a), E[F (a)|$])]

#2

µI(di),

recalling that

(($i) = E[(agv(F (a), E[F (a)|$])|$]iand

((x) = E!

4

d

dWEI [(agv(F (a), E[F (a)|$])iw

"i (ai)]

5

represent the (expected) mean values by class of agv(F (a),E[F (a)|$]) in theparental and the o!spring generations respectively.

That is: the expected change in the mean of the additive genetic value of thedeviation of the expected fitness from the class mean, weighted by reproductivevalue, is equal to the (unweighted) variance of the additive genetic value of thedeviation of the expected fitness from the class mean.

Proof We first recall the definition of p-scores, and observe that if we choosea locus 1 0 L 0 N , this corresponds to choosing the column (gk,L)nk=1 ofthe genotype matrix G = (gk,l) " Rn#N . The total number of alleles at thislocus in any individual must equal the ploidy of that individual, thus when weconsider the linear map &L : Rn#N ! R defined by summing the entries of thecolumn of the matrix G = (gk,l) " Rn#N , i.e.

&L(G) =n-

k=1

gk,L,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

31

we see that for all i " I,&L(gi/di) = 1.

By definition, since &L : Rn#N ! R is linear, this—the constant function—isa p-score. Hence we can apply (3) to see that

EI [agv(F (a), E[F (a)|$])i] = EI [1 · (agv(F (a), E[F (a)|$])i)]

= EI [Fi(ai), E[F (a)|$]i]

= 0.

With the definition of p " P made in the statement, we now apply the PriceEquation (15) to obtain"

X

(((x), ((x)) '(dx)

=

"

I

(agv(F (a), E[F (a)|$])i)2 µI(di)

=

"

I

!

agv(F (a), E[F (a)|$])i , EI [agv(F (a), E[F (a)|$])i]

#2

µI(di),

as required. -.

7 Interpretation, significance, and context of our results

Theorems 2 to 5 are the four links between gene frequency change and op-timization, parallel to the four links proved in previous work (Grafen 2002,2006a). If all individuals in the population solve the optimization program,then the expected change in every gene frequency equals zero, and no pheno-type, if caused by a rare dominant mutant, would cause that mutant to spread.The strengths of this result are that it applies to all gene frequencies, whetherthey a!ect any given trait or not, and whether they a!ect fitness or not; thatit therefore applies to all weighted sums of allele frequencies and so to theadditive genetic value of every quantitative trait; that fitness is a property ofthe individual and in particular is the same whichever gene frequency is beingconsidered; that it applies to a class-structured population, with evolutionar-ily appropriate class-weights that are used both to aggregate the mean p-scoreacross classes when considering the change in mean p-score, and to provideclass-weights for o!spring in the evaluation of the fitness of an individual; andthat the class-weights are the same no matter which class the parent belongsto. Notable features of the result are that it shows that the expected change ingene frequencies equals zero, but in any given state of nature the gene frequen-cies may indeed change; that it says expected gene frequencies don’t change,but this does not imply that expected genotype frequencies do not change; thatonce gene or genotype frequencies have changed from one generation to thenext, there is no guarantee, or even reason to expect, that the class-weights willremain the same; hence the way of evaluating fitness is quite likely to change

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32

from one period to the next. Theorem 3 is a diminished form of the previoustheorem, which supposes that each individual attains the same value of themaximand, but does not solve it, and then while no gene-frequency changesin expectation, there is a phenotype which, if produced by a rare dominantmutant, would spread in expectation. Theorem 4 supposes individuals attaindi!erent values of the maximand, and merely asserts that the expected changein each gene frequency equals its covariance with fitness. This is a restatementof the Price Equation, and the purpose of presenting it in this form is thatthe previous two theorems would hold if the maximand were replaced with amonotonic increasing function of the maximand, and this theorem holds onlyfor the expected fitness itself (up to addition of a constant). So including thistheorem ensures that the links presented do constrain as strongly as possiblethe nature of the maximand of an optimization program that can take partin the set of links. The final link is Theorem 5, which reverses the directionof inference, and is in one way the most important. It states that if there isno expected change in gene frequency, and if there is no phenotype which,if produced by a rare dominant mutant, would cause that mutant to spread,then each individual solves the optimization program: it is the only theoremto proceed from gene frequencies to optimization, and is the central resultthat contributes to establishing the reasonableness of the adaptationist view-point, namely that we should expect organisms to be optimally adapted. Ofcourse, there are many questions, which cannot be considered here, about theextent to which this result does justify adaptationism, but the very generalnature of the whole framework, and the particularities of how ‘adaptiveness’and in particular fitness have to be defined, represent major advances in ourunderstanding of adaptationism and its connection to genetics.

Theorem 6 is a generalization of the Fundamental Theorem of Natural Se-lection of Fisher (1930), except that we remain in discrete time and do notpass to a continuous time limit as Fisher did. The current status of the fun-damental theorem in the literature is discussed by Okasha (2008) and Ewens(2011), and the best previous technical treatment is due to Lessard (1997).These modern authors have discussed what the theorem states, and whetherit is true, but the consensus has been that no-one knows whether it has bi-ological significance. Its inclusion here is intended to settle that question inthe a"rmative. The formal darwinism project is the first attempt to constructa formal justification of individual fitness maximization ideas since the fun-damental theorem, and this paper shows that a wide range of abstract andperhaps arcane concepts have to be introduced to do the job properly. It turnsout that the fundamental theorem is readily expressed and proved in terms ofthose concepts, in a much more explicit way than Fisher was able to prove it,and in a more general way than more recent derivations.

Integrating our work with the important literature on the fundamental the-orem (for example Price 1972b; Ewens 1989; Frank and Slatkin 1992; Edwards1994; Lessard 1997; Okasha 2008; Ewens 2011) must remain a task for the fu-ture. Here we indicate three significant contributions of the current work tounderstanding the theorem. First, Fisher has been criticized because his verbal

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33

statement of the theorem (‘the rate of increase in fitness of any organism atany time is equal to its genetic variance in fitness at that time’) di!er from histechnical statement. However, a statement that the mean change in X equalsthe variance in X is obviously closely tied to the idea of X being maximized.Our version is of such a form, and it is not surprising that Fisher would wish themost accessible form of his theorem to display what he clearly believed to beits more important implication. Further, the exact sense in which the theoremprovides a maximand, in light of the technical qualifications of the theorem,is precisely the subject of Theorems 2 to 5. Second, the reason for the dis-crepancy between verbal and technical statements is that so much in Fisher’sargument has not been made explicit. We have made the whole argument ex-plicit, and in view of the length and complexity of the argument, it is notsurprising that Fisher did not do so, perhaps from a combination of inability(tracking classes and reproductive values and gene frequencies is notationallycumbersome and hard work; further, most of our mathematical references arefrom later than 1930) and unwillingness (the book was for a lay audience, andit may be doubted how much of our current argument would have been appre-ciated even by contemporary scientific readers, of Fisher (1941), for example).Third, some obstacles to understanding are now explained and removed. Forexample, Price (1972b)’s first technical move is to assert that the left handside of the theorem is not in fact the change in fitness, but only that part ofthe change in fitness due to changes in gene frequencies. The meaning of thisqualification has played a major role in discussions of the meaning and signif-icance of the theorem (Okasha 2008; Ewens 2011): our version shows that thisqualification is made because, while fitness itself does not admit a result of theform ‘change in mean of X equals variance in X ’, the additive genetic value offitness does. This settles in a very precise technical way what the qualificationmeans, and provides a wholly understandable reason for Fisher to make it.

We regard the mathematical framework of this paper as a set of ideasabout which Fisher had strong and correct intuitions, but which he managedto articulate only in small part, and about which he drew a most significantconclusion: that population genetics could exhibit the design-making natureof Darwinian natural selection, demonstrate that it was constantly at work ina very general setting, and make precise what quantity was the appropriatemeasure of goodness of design. Fisher rightly regarded the fundamental theo-rem of natural selection as providing the fundamental link between Darwin’sargument that natural selection brought about adaptation, on the one hand,and population genetics, on the other.

On the dust-jacket of the 1999 variorum edition of Fisher’s book, W.D.Hamilton writes “In some ways some of us have overtaken Fisher; in many,however, this brilliant, daring man is still far in front”. In showing exactlyhow the fundamental theorem relates to fitness-maximization, and that thefull argument is even today at the boundaries of mathematical biology, wehave taken a significant step towards “catching up with Fisher”.

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34

8 An example with an age-structured population

We present an example to show our various theorems at work, and choosethe classic case of an age-structured population for which Fisher first defined‘reproductive value’ and proved his fundamental theorem. The results of thecurrent paper would have allowed us to have a sexual population and to studysex ratio simultaneously with survival-fertility tradeo!, thus uniting Fisher’soriginal uses of reproductive value. However, for simplicity, and for the histor-ical interest of exhibiting implications of the original form of the fundamentaltheorem, we have reserved that more advanced example for the future. In par-ticular, we note that versions of the theorem that omit age structure, and sohave no need to involve reproductive value, miss out a fundamental feature ofthe fundamental theorem.

We suppose I is a finite population, and our classes are K + 1 age classesfor some K " N, comprising ages 0 to K. We assume the population to beof constant ploidy one, and to be asexual. We shall consider the set of localenvironments to be the set of possible amounts of some resource available toeach individual without competition: thus R = [0,*). Thus being in environ-ment r is interpreted as having r resources available. These resources are to beinvested entirely in reproduction and/or survival. We shall assume each indi-vidual i " I requires bi resources to produce each o!spring, where bi " (0,*).We shall identify the phenotype of an individual with the choice the individ-ual makes of how best to spend her resources, which, since we demand thatresources are exhausted between reproduction and survival, is the choice ofhow to distribute the resources between o!spring production and attemptedsurvival. Chance events shall represent how many of the produced o!spring ofan individual will survive to the next census point, when they will be of age 0,and whether an individual herself will survive to the next age class. Thus ourphenotype space A is given by ([0,*)$ [0,*)), the first coordinate represent-ing energy devoted to o!spring, the second how much energy the individualinvests in survival.

We thus have a function r : I $ ! ! [0,*) determining how much re-source individual i " I finds available in the state of nature % " !. Theset of admissible phenotypes for an individual i is then the set of functionsq : [0,*) ! [0,*)$ [0,*) which must satisfy, writing q = (q1, q2), the rela-tions

biq1(r) + q2(r) = r, and (20)

q2(r) = 0 for all r " [0,*) if $i = K. (21)

The second condition states that no individual attempts to survive beyondage K.

The chance events of the state of nature % influencing the individual i arethen represented by u : I $ ! !

)

(N 5 {0})[0,$) $ {0, 1}[0,$)*

, the first termtelling us, for each x " [0,*), how many surviving o!spring are producedgiven the investment of x resources in o!spring production, and the second

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35

term telling us, given the individual devotes r " [0,*) resources to survival,whether the parent individual survives (1) or dies (0). It is reasonable toassume that the function (u"

i )2 : [0,*) ! {0, 1} is monotone increasing: themore e!ort an individual puts into survival, the more likely she is to survive;and that (u"

i )2(0) = 0: individuals do not survive if they do not try to, so inparticular no individual survives beyond age K.

O!spring are then produced as follows:

w"i (ai) = (u"

i )1((ai)1(r"i ))+0 + (u"

i )2((ai)2(r"i ))+$i+1. (22)

Define ,"i := (u"

i )1((ai)1(r"i )), the number of surviving o!spring, and -"

i :=(u"

i )2((ai)2(r"i )), which determines survival of the individual.

The o!spring distribution over ages may then be captured by consideringthe conditional expectations ,"(k) := E[,"

i |$i = k] and -"(k) := E[-"i |$i =

k]. Supposing the parental age distribution to be given by the vector v =(v0, . . . , vK)T , the o!spring distribution, in state of nature %, is given by thevector w = (w0, . . . , wK)T where

w0 =K-

k=0

,"(k), and (23)

wk = -"(k , 1) for k / 1. (24)

We assume, as in the general argument, that these coe"cients are independentof %. In particular, then, -"(k) is independent of % for each k / 0, andthus may be written as -(k). Furthermore, by linearity, the situation can becaptured by considering the coe"cients ,(k) := E! [,"(k)] and writing

L =

9

:

:

:

:

:

;

,(0) ,(1) · · · · · · ,(K)-(0) 0 · · · · · · 00 -(1) · · · · · · 0...

.... . .

...0 . . . 0 -(K , 1) 0

<

=

=

=

=

=

>

(25)

and noting that

Lv = w. (26)

L is then the so-called Leslie matrix associated with demographic processes(Leslie 1945, 1948; Lewis 1942). The left eigenvector then gives us the per-capita reproductive value in the sense of Fisher — an observation consistentwith the more general assertions in Grafen (2006b) that per-capita reproduc-tive value is found as an eigenvector of the adjoint of the forward process transi-tion operator. We therefore denote such a vector ' = ('0, . . . , 'K)T . Our defini-tion of the fitness of an individual playing strategy q : [0,*) ! [0,*)$ [0,*)then amounts to

Fi(q) = E![(u"

i )1(q1(r"i ))'0 + (u"

i )2(q2(r"i ))'$i+1],

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36

in particularFi(ai) = E

![,"i '0 + -"

i '$i+1].

We define ,i := E! [,"i ] and -i := E![-"

i ]. Thus

Fi(ai) = ,i'0 + -i'$i+1.

The situation is then analyzed by considering when this number is at a maxi-mum. Individuals of age K cannot choose to survive, thus there is no choice ofphenotype available to them; so we concentrate on individuals in age classesk for 0 0 k < K. Fix such an individual i " I. Then

F"i (q) = (u"

i )1(q1(r"i ))'0 + (u"

i )2(q2(r"i ))'$i+1

= (u"i )1(q1(r

"i ))'0 + (u"

i )2(r"i , biq1(r

"i ))'$i+1.

For an admissible strategy q, define A"i (q) := (u"

i )1(q1(r"i )) and B"

i (q) :=(u"

i )2(q2(r"i )). Taking expectations over states of nature we see that

Fi(q) = E! [(u"

i )1(q1(r"i ))'0 + (u"

i )2(r"i , biq1(r

"i ))'$i+1]

=

"

{""!:B!i(q)=0}

A"i (q)'0 "(d%) +

"

{""!:B!i(q)=1}

(A"i (q)'0 + '$i+1) "(d%)

= '0

"

!

A"i (q) "(d%) + '$i+1"({% " ! : B"

i (q) = 1}).

The first summand is the expected number of o!spring weighted by theirreproductive value; the second is the probability of survival, weighted by re-productive value of an individual in the next age class.

Let a strategy q be fixed, and consider another strategy q = (q1, q2). Recallthat the choice to be made is the value of q1(r"i ) " [0, b!1

i r"i ]. Suppose q1 <q1, i.e. playing the strategy q means having fewer o!spring. Since (u"

i )2 ismonotone increasing, and B"

i (q) = (u"i )2(r

"i , biq1(r"i )), we then have that

"({B"i = 1}) 0 "({B"

i (q) = 1}). So Fi(q) / Fi(q) if and only if

'0

"

!

A"i (q) "(d%) + "({B"

i (q) = 1})'$i+1 / '0

"

!

A"i (q) "(d%)

+ "({B"i (q) = 1})'$i+1,

equivalently

("({B"i (q) = 1}), "({B"

i (q) = 1}))'$i+1 / '0

"

!

(A"i (q),A"

i (q)) "(d%).

That is, for q to be a better strategy, the increased chance of survival, weightedby the reproductive value of a surviving individual, must exceed the change inexpected number of o!spring, weighted by the reproductive value of o!spring.Of course, if having more o!spring leads to a lower expected number of sur-viving o!spring (e.g. if then limited resources are shared between too manyinfants), then this is trivial.

Assuming pairwise exchangeability, this then rigorously justifies our intu-ition for this example. Theorems 2 and 3 together assert that the population

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37

is at an evolutionary equilibrium if and only if every individual maximizesthe value Fi(ai). This number is the reproductive value-weighted sum of theexpected number of o!spring and the survival probability of the individual.The above analysis of the maximand then shows that evolutionary equilibriumis attained precisely when each individual reacts to each environment in sucha way that the expected contribution to the following generation, in terms ofreproductive value, is at a maximum.

This is a simple but fitting example with which to end. Fisher proved hisfundamental theorem in this setting, and concluded that in general we shouldexpect individuals to maximize their fitness. Here we explicitly articulate thatoptimization argument in an example, and note that the fundamental theo-rem shows how survival and reproduction trade o! in the calculation of fitness,through the use of reproductive value. Notice that the individual is regardedas making only the decision relevant to its current age as opposed to all life-decisions simultaneously, and that the fecundity-survival trade-o! operatesthrough varying the chance of survival to the next age period. The value ofsurviving is obtained through knowing the reproductive value of an individualin the next age class. Thus reproductive value in an age-structured populationis the expected future reproductive value of an individual of a given age. If wehad extended the set of classes to include condition, incorporating body weightand health, at each age, then we could have modelled more complex trade-o!s in which producing more o!spring reduced one’s condition, and in whichreproductive value would presumably be an increasing function of conditionwithin each age class.

Our general results extend Fisher’s theorem by permitting an arbitraryclass-structure, explicitly incorporating uncertainty, allowing each class to haveits own ploidy and, in particular, by fully articulating the meaning of fitness-maximization. Future generalizations may further permit social behaviour,time to be continuous or discrete, and random variation in class-to-class pro-jection at the population level along with the demographic stochasticity thatimplies.

Acknowledgements The authors wish to thank Alain Goriely and two anonymous refereesfor numerous helpful comments on earlier versions of this manuscript, and the participants ofthe reading seminar on Fisher (1999) held in the St John’s College Research Centre in 2012,in particular organizer Jean-Baptiste Grodwohl, for the opportunity to explore Fisher’s workand the topics central to this paper.

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