J. theor. Biol. (1995) 175, 373–388

0022–5193/95/150373+16 $12.00/0 7 1995 Academic Press Limited

George Price’s Contributions to Evolutionary Genetics

S A. F

Department of Ecology and Evolutionary Biology, University of California, Irvine,CA 92717, U.S.A.

(Received on 9 November 1994, Accepted in revised form on 14 February 1995)

George Price studied evolutionary genetics for approximately seven years between 1967 and 1974. Duringthat brief period Price made three lasting contributions to evolutionary theory; these were: (i) the PriceEquation, a profound insight into the nature of selection and the basis for the modern theories of kinand group selection; (ii) the theory of games and animal behavior, based on the concept of theevolutionarily stable strategy; and (iii) the modern interpretation of Fisher’s fundamental theorem ofnatural selection, Fisher’s theorem being perhaps the most cited and least understood idea in the historyof evolutionary genetics. This paper summarizes Price’s contributions and briefly outlines why, towardthe end of his painful intellectual journey, he chose to focus his deep humanistic feelings and sharp,analytical mind on abstract problems in evolutionary theory.

7 1995 Academic Press Limited

Introduction

At the age of 44, George Price quit his engineering jobat IBM and took up evolutionary genetics. Pricemoved to the Galton Laboratories at London in 1967,where C. A. B. Smith, Weldon Professor of Biometry,provided him with space and encouragement. Pricemade three lasting contributions to the conceptualstructure of evolutionary genetics during the briefperiod between 1970 and 1973. He died in 1975.

(i) Price’s (1970, 1972a) first contribution was thePrice Equation, a formal method for the hierarchicalanalysis of natural selection. Hamilton (1970) used thisequation in his later studies to construct the foundationfor modern analyses of kin selection and groupselection. The equation has also been used to clarify adiverse array of conceptual issues, from quantitativegenetics to the abstract properties of natural selection.

(ii) Price was the first to show that many unsolvedpuzzles in animal behavior could be understood byapplying the logic of game theory. Price’s workstimulated Maynard Smith’s interest, which eventuallyled Maynard Smith to develop his concept of theEvolutionarily Stable Strategy (ESS; Maynard Smith& Price, 1973) and to write the classic book Evolutionand the Theory of Games (Maynard Smith, 1982).

(iii) Price (1972b) was the first to crack the mostobscure and misunderstood of R. A. Fisher’s manyoracles: the fundamental theorem of natural selection.This theorem is perhaps the most widely quoted resultin evolutionary theory, yet Price demonstratedconvincingly that the consensus interpretation of thetheorem strongly contradicts the lessons Fisher himselfdrew from the theorem. Fisher stated on severaloccasions that the average fitness of a population is ameaningless quantity, and theories such as Wright’sadaptive topography that emphasize average fitnesscreate a strongly misleading view of evolutionaryprocess.

Many biologists have heard of one or another ofthese contributions. Few people are aware that eachwas made by a single man. It must be unusual in thehistory of science for someone, without professionalexperience, to take up a field while in his fortiesand make significant contributions to the theoreticalfoundations of that field.

In the first sections of this paper I focus onPrice’s contributions to evolutionary genetics and thestudy of natural selection. In the final section I providea few details about his life and the factors thatmotivated him to choose particular problems inevolutionary biology.

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The Price Equation

. . . a good notation has a subtlety and suggestivenesswhich at times make it seem almost like a live teacher.(Bertrand Russell)

The Price Equation was a new mathematicalformulation for evolutionary change (Price, 1970,1972a). It may seem strange that one could write downa truly new equation that describes evolution. Afterall, evolution is simply the change over time insome characteristics of a population. The brillianceof the Price Equation is that it adds nothing to thefundamental simplicity of evolutionary change but, bymaking a few minor rearrangements and changes innotation, the equation provides an easier and morenatural way to reason about complex problems.

The Price Equation relies on a degree ofmathematical abstraction that is rare in evolutionarygenetics. Indeed, Price believed that his work was animportant step toward a general theory of selec-tion. He introduced his manuscript, ‘‘The Nature ofSelection,’’ (Price, 1995) with:

A model that unifies all types of selection (chemical,sociological, genetical, and every other kind ofselection) may open the way to develop a general‘Mathematical Theory of Selection’ analogous tocommunication theory . . . Selection has been studiedmainly in genetics, but of course there is much moreto selection than just genetical selection . . . yet, despitethe pervading importance of selection in science andlife, there has been no abstraction and generalisationfrom genetical selection to obtain a general selectiontheory and general selection mathematics.

I begin with simple genetical applications that use areduced form of the equation and move slowly towardthe full equation. At the end of this section I discuss thegeneral properties of the equation and the idea thatthere can be a theory of selection that unifies differentproblems such as trial-and-error learning, chemicalcrystallization, and linguistic evolution.

The reduced form of the equation that can be usedfor many simple problems is

wDz=Cov(w, z)=bwzVz , (1)

where w is fitness and z is a quantitative character. Theequation shows that the change in the average valueof a character, Dz, depends on the covariance betweenthe character and fitness or, equivalently, theregression coefficient of fitness on the charactermultiplied by the variance of the character. Thisequation was discovered independently by Robertson(1966), Li (1967) and Price (1970).

Because fitness itself is a quantitative character, onecan let the character z in eqn (1) be equivalent to fitness,

w. Then the regression, bww , is 1, and the variance, Vw ,is the variance in fitness. Thus the equation shows thatthe change in mean fitness, Dw, is proportional to thevariance in fitness,Vw . The fact that the change inmeanfitness depends on the variance in fitness is usuallycalled ‘‘Fisher’s fundamental theorem of naturalselection,’’ although that is not what Fisher (1958)really meant. Price himself clarified Fisher’s theorem ina fascinating paper that I shall discuss later (Price,1972b).

Robertson (1968) named eqn (1) the ‘‘secondarytheorem of natural selection’’ as an extension to whatis usually called Fisher’s fundamental theorem. Thisgeneral covariance equation has an unspecified errorthat can be influenced by nonlinear genetic interactions(dominance, epistasis), the mating system, meioticdrive and a variety of other factors. Crow & Nagylaki(1976) provide an elegant summary of the relationshipbetween traditional population genetics and thecovariance equation. The full Price Equation(described later) has an additional term that, togetherwith the covariance, always provides an exact and fulldescription for evolutionary change.

The Price Equation has played an important rolein work on kin selection (Hamilton, 1970, 1975; Wade,1980; Seger, 1981; Uyenoyama, 1988). The equationitself cannot reduce the inherent complexity of models,but the simple covariance relationship between acharacter and fitness provides a compact way to see theessential features of social evolution. Recent papershave summarized the importance of Price’s work inthis field (Grafen, 1985; Wade, 1985; Taylor, 1988a, b,1989; Queller, 1992a), so I present here only a briefexample and a few historical comments.

I illustrate the problem by summarizing the first partof Queller’s (1992a) model. I use the covarianceequation, wDg=Cov(w, g), where g is the breedingvalue that determines the level of altruism. One canwrite the least-squares multiple regression that predictsfitness, w, as

w=a+gbwg·g'+g'bwg'·g+e,

where g' is the average g value of an individual’s socialneighbors, a is a constant, and e is the residual whichis uncorrelated with g and g'. The b are partialregression coefficients that elegantly summarize costsand benefits: bwg·g' is the effect an individual’s breedingvalue has on its own fitness in the presence of neighborsg'—the cost of altruism, and bwg'·g is the effect of anindividual’s breeding value on the fitness of itsneighbors—the benefit of altruism. Substituting into

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the covariance equation and solving for the conditionunder which wDgq0 yields Hamilton’s rule

bwg·g'+bg'gbwg'·gq0,

where bg'g is the regression coefficient of relatedness(Hamilton, 1972). Thus Queller’s model shows thatthe covariance equation provides a simple way toreason about the effects of relatedness on socialbehavior.

Hamilton (1964a, b), in his original formulation ofkin selection, described genetic similarity in terms ofgenes identical by descent. Hamilton (1970) reformu-lated kin selection by explicit derivation from the PriceEquation; this derivation is often regarded as the firstmodern theoretical treatment of inclusive fitness(Grafen, 1985). Price’s covariance equation shows thatwhat matters is not common ancestry, but statisticalassociations between the genotypes of donor andrecipient.

One interesting consequence of treating relatednessas a statistical association is that relatedness can benegative, leading to selection for spiteful behavior.That new insight was a key point in Hamilton (1970:1218):

Previously I showed that the average geneticalrelatedness of interacting individuals is an importantfactor in the evolution of social adaptations. In themodel, selfishness within certain limits was readilyaccounted for; spite [harm to self in order to harmanother more] did not seem possible. But another lineof reasoning shows that spite can be selected [whenrelatedness is negative]. Independently, using his newformulation of natural selection in a more generalanalysis, Dr G. R. Price reached the same conclusion.

Spite can be favored because the product of negativerelatedness and a negative benefit to a recipient (harm)is positive, thus benefit multiplied by relatedness canoutweigh the cost.

One possible case of spite occurs in the flourbeetle Tribolium (Wade & Beeman, 1994). The Medeaallele in a heterozygous mother apparently stimulatesa mechanism that kills all offspring lacking a copyof the allele. There are two possible explanations forthe maintenance of the Medea allele. First, the allelemay increase its number of copies in offspring bykilling competitors for limited resources. Thisexplanation requires that destroyed zygotes arereplaced with new, successful zygotes carrying theallele. In this case the allele is ‘‘selfish’’ because itenhances its own reproduction at the expense of acompetitor.

The second possibility is that there is no replacementof killed zygotes and the number of copies of theMedeaallele does not increase by this mechanism. Medea can

still spread in this case because it reduces the numberof copies of the alternative allele, thereby increasing itsown frequency in the next generation. This is a formof spite, where the Medea allele uses kin recognitionto destroy individuals that are negatively related toitself. Hurst (1991) has discussed spite in systems ofcytoplasmic incompatibility which have propertiessimilar to the Medea allele.

The original formulation of kin selection relied onthe probability of identity by descent to derive thedegree of relatedness (Hamilton, 1964a, b). Price’scovariance equation shows that what matters is notcommon ancestry, but statistical associations betweenthe genotypes of donor and recipient. Those asso-ciations often arise because individuals that live neareachother tend tohave commonancestors. But naturalselection is indifferent to the cause of the statisticalassociations, and negative statistical associations favorspite. Covariance is the only proper way to think aboutthe role of genetic relatedness in evolutionary biology.Price was the first to see this. His work provided thebasis for Hamilton’s (1970) reformulation of kinselection in terms of covariance, which can properly becalled the modern theory of kin selection.

The covariance equation provides an approximatedescription for evolutionary change that is useful formany applications. That equation was discoveredindependently by Robertson (1966), Li (1967) andPrice (1970). Price’s unique contribution is a moregeneral equation that is an exact, complete descriptionof evolutionary change under all conditions. The fullequation is not just a more accurate form of thecovariance equation. It adds considerable insight intomany evolutionary problems by partitioning selec-tion into meaningful components. In this section Iderive the full equation, and in the following sectionssummarize some applications.

Here is the derivation. Let there be a population (set)where each element is labeled by an index i. Thefrequency of elements with index i is qi , and eachelement with index i has some character, zi . One canthink of elements with a common index as forminga subpopulation that makes up a fraction qi of the totalpopulation. No restrictions are placed on howelements may be grouped.

A second (descendant) population has frequencies q'iand characters z'i . The change in the average charactervalue, z, between the two populations is

Dz=Sq'i z'i −Sqizi . (2)

Note that this equation applies to anything thatevolves, since z may be defined in any way. For

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example, zi may be the gene frequency of entities i,and thus z is the average gene frequency in thepopulation, or zi may be the square of a quantitativecharacter, so that one can study the evolution ofvariances of traits. Applications are not limited topopulation genetics. For example, zi may be theabundance of a particular chemical compound ingalaxy i.

The queerness of the Price Equation comes from theway it associates entities from two populations, whichare typically called the ancestral and descendantpopulations (see Fig. 1). The value of q'i is not obtainedfrom the frequency of elements with index i in thedescendant population, but from the proportion of thedescendant population that is derived from theelements with index i in the parent population. If wedefine the fitness of element i as wi , the contribution tothe descendant population from type i in the parentpopulation, then q'i =qiwi /w, where w is the meanfitness of the parent population.

The assignment of character values z'i also usesindices of the parent population. The value of z'i is theaverage character value of the descendants of index i.Specifically, for an index i in the parent population, z'iis obtained by weighting the character value of eachentity in the descendant population by the fraction ofthe total fitness of i that it represents (Fig. 1). Thechange in character value for descendants of i is definedas Dzi=z'i −zi .

Equation (2) is true with these definitions for q'i andz'i . We can proceed with the derivation by a fewsubstitutions and rearrangements:

Dz=Sqi (wi /w)(zi+Dzi )−Sqizi

=Sqi (wi /w−1)zi+Sqi (wi /w)Dzi

which, using standard definitions from statistics forcovariance (Cov) and expectation (E), yields the PriceEquation,

wDz=Cov(wi , zi )+E(wiDzi ). (3)

The two terms may be thought of as changes due toselection and transmission, respectively. The covari-ance between fitness and character value gives thechange in the character caused by differentialreproductive success. The expectation term is a fitnessweighted measure of the change in character valuesbetween ancestor and descendant. The full equationdescribes both selective changes within a generationand the response to selection (cf. Wade, 1985).

The covariance term in eqn (3) would normally bewritten without subscripted variables as Cov(w, z).The reason for the subscripts is additional clarity whenthe equation is used to expand itself:

wDz = Cov(wi , zi ) + Ei{Covj (wj·i , zj·i ) + Ej (wj·iDzj·i )},(4)

F. 1. Example of a selective system using the notation of the Price Equation. The initial population, the left column of beakers, is dividedinto subpopulations indexed by i, where qi is the fraction of the total population in the i-th subpopulation. In this drawing, the two differentkinds of transmissible material, solid and striped, are in separate subpopulations initially, but that is not necessary. Each subpopulationexpresses a character value (phenotype), zi . Any arbitrary rule can be used to assign trait values. Selection describes the changes in the quantitiesof the transmissible materials, where the primes on symbols denote the next time period. Thus q'i =qiwi /w is the proportion of the descendantpopulation derived from the i-th subpopulation of the initial population. The transmissible material may be redistributed to new groupingsduring or after the selective processes. The q'j·i are the fractions of the i-th parental subpopulation, after selection, that end up in the j-thdescendant subpopulation, thus Sjq'j·i=1. The new mixtures in the j-th descendant subpopulations express trait values yj according to whateverarbitrary rules are in effect. This allows full context-dependence (non-additivity) in the phenotypic expression of the transmissible material.Descendant trait values are assigned to the original subpopulations by weighting the contributions of those subpopulations, z'i =Sjq'j·iyj . Thusthe average trait value in the descendant population is z'=Siq'i z'i .

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where E and Cov are taken over their subscripts whenthere is ambiguity, and j ·i are subsets of the group iwith members that have index j. The partition of i intosubgroups j is arbitrary. This recursive expansion ofthe E term in eqn (3) shows that transmission is itselfan evolutionary event that can be partitioned intoselection among subgroups and transmission of thosesubgroups. The expansion of the trailing expectationterm can continue until no change occurs duringtransmission.

In later sections I shall return to the generalproperties of the equation itself. But first I show somesimple applications in evolutionary genetics.

The hierarchical expansion of the Price Equationhas been used to study group selection and sex ratio(Hamilton, 1979; Nunney, 1985; Frank, 1986a,1987a). The relationship between group selection andkin selection has been a controversial aspect of thisfield (Wilson, 1983; Grafen, 1984; Queller, 1991,1992b). Here, I give a simple derivation of the standardLocal Mate Competition model (Hamilton, 1967) thatshows the equivalence of kin and group selection. Myderivation is somewhat more general than usualbecause, by using the Price Equation, any pattern ofinteraction among relatives (population structure) isallowed rather than the standard sib-interactionmodels.

Let the frequency of males produced by a mother ber=g+ge, where g is the number of sex ratio alleles inthe mother each with additive effect e. The sex ratioin the neighborhood in which mating and matecompetition occur is r'=g+g'e, where g' is the averagenumber of alleles with effect e among mothers whenweighted for each mother’s contribution to theneighborhood.

The fitness (number of grandprogeny) of a femalewith g sex ratio alleles is wg=r(1−r')/r'+(1−r), andthe fitness of a neighborhood with g' as its averagenumber of sex ratios alleles is wg'=2(1−r') (Hamilton,1967).

Rewriting eqn (4) in the notation of this problem

wDg=Cov(wg', g')+Eg'{Covg (wg , g ·g')}

=bwg'Vg'+bwg.g'Vg·g' ,

where Vg' is the genotypic variance among groups andVg·g' is the average genotypic variance within groups.The term bwg' is the regression of fitness on groupgenotype, and bwg·g' is the regression of fitness onindividual genotype given the group genotype, g'.

The standard method to find the EvolutionarilyStable Strategy (ESS) sex ratio is to solve (dwDg/de)=0 when evaluated at e=0 (Hamilton, 1967;Maynard Smith 1982). This yields the solution(Hamilton, 1979)

r*=(1/2)(1−Vg'/Vg ),

where Vg=Vg'+Vg·g' is the total genotypic variance inthe population. This form shows that a bias away fromthe Fisherian 1/2 occurs only when there is geneticvariance among neighborhoods, Vg'. This supports theclaim that group selection causes biased sex ratios instructured populations. However, one can also write

Vg'

Vg=

Cov(g', g')Cov(g, g)

=Cov(g, g')Cov(g, g)

=bg'g ,

where bg'g is the regression coefficient of relatedness ofinclusive fitness theory. Thus the sex ratio bias iscaused by a nonzero average relatedness amongneighbors. The definitions of group and kin selectiongiven here are clearly equivalent. The regressioncoefficient form has been particularly useful for solvingsex ratio and dispersal problems with complexinteractions among kin (Frank, 1986a, b, 1987a, b, c;Taylor, 1988a, b, 1989). In fact, the Price Equation hasbeen the only general way to study sex ratio evolutionfor arbitrary population structures. Traditionalpopulation genetic methods require special, complexequations for each particular set of assumptions abouthow relatives interact.

In the next few sections I describe technical aspectsof the Price Equation. These issues are important forunderstanding the role of the Price Equation in formalevolutionary theory. Readers who are more interestedin Price’s contribution to particular biologicalproblems may wish to skip ahead to the section on ESStheory.

Suppose we wish to study the evolution of acharacter, z.We are given all of the information neededto calculate the covariance and expectation in eqn (3).Thus we know the average value, z, at the start, andwe can calculate the change in the average value afterone time period, Dz. There is a problem, however, if wewish to study the continued evolution of this traitthrough time. The Price Equation gives us back theaverage character value in the next time period, but notthe information needed to calculate the covariance andexpectation term to apply the equation again.

We cannot follow the continued evolution of zthrough time (Barton & Turelli, 1987) or, put anotherway, we lack the information to achieve dynamicsufficiency in our analysis (Lewontin, 1974). Several

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authors have noted this problem and have concludedthat the Price Equation lacks dynamic sufficiency,whereas other methods of analysis achieve dynamicsufficiency for the same problem (Crow & Nagylaki,1976; Grafen, 1985; Queller, 1992a). This is aconfusing summary of the problem because it is partlytrue and partly false.

The true part is that one can apply the PriceEquation to a problem that lacks dynamic sufficiency.It is not true, however, that dynamic sufficiency is aproperty that can be ascribed to the Price Equation—this equation is simply a mathematical tautology forthe relationship among certain quantities of popu-lations. Instead, dynamic sufficiency is a property ofthe assumptions and information provided in aparticular problem, or added by additional assump-tions contained within numerical techniques such asdiffusion analysis or applied quantitative genetics.

The conditions under which an evolutionary systemis dynamically sufficient can be seen from the PriceEquation. For convenience assume Dzi=0 so that onlythe covariance term of eqn (3) need be examined.Initially we require z, w, and wz to calculate Dz becauseCov(w, z)=wz−w z. We now have z after one timestep, but to use the Price Equation again we also needCov(w, z) in the next time period. This requiresequations for the dynamics of w and wz , which can beobtained by substituting either w or wz for z in eqn (3);recall that z can be used to represent any quantity, sowe can substitute fitness, w, or the product of fitnessand character value, wz, for z. If we ignore theexpectation term, the dynamics of wz are given by

wDwz=Cov(w, wz)=w2z−w wz .

Changes in the covariance over time depend on thedynamics of wz , which in turn depends on w2z , whichdepends on w3z , and so on. Similarly, the dynamics ofw depend on w2, which depends on w3, and so on.Dynamic sufficiency requires that higher moments canbe expressed in terms of the lower moments (Barton &Turelli, 1987).

The full Price Equation, eqn (3), with both thecovariance and expectation terms, has been used oftento study the hierarchical decomposition of selectionwithin and among groups (e.g. Price, 1972a; Hamilton,1975, 1979; Wade, 1980, 1985; Arnold & Fistrup, 1982;Ohta, 1983; Frank, 1985, 1986a, b, c, 1987a, b, 1992,1994a, b; Grafen, 1985; Nunney, 1985; Heisler &Damuth, 1987; Breden, 1990; see the section above onhierarchical analysis and sex ratio).

Hierarchical decomposition is clearly one importantuse of the equation, but the equation itself is not limited

to these applications. Another interpretation of thecovariance and expectation terms in eqn (3) is apartition between selection and transmission. I brieflyillustrate this point with a model that balancesselection among adults, the covariance term, againstmutational changes that occur during transmission,the expectation term (Frank & Slatkin, 1990).

One useful aspect of the Price Equation in thesemutation-selection models is that any character valuecan be analyzed. Instead of studying the change in acharacter with value zi , we can transform the charactervalue and study, for example, the change in the n-thpower of the character, zn

i . The Price Equation thengives the change in zn, which is the n-th non-centralmoment

wDzn=Cov(wi , zni )+E(wiDzn

i ).

This set of equations for the moments zn, n=1, 2, . . . ,is a complete description for the evolutionarydynamics of the character z for any system ofselection, mutation, mating and inheritance. Themost interesting aspect of the Price Equation is the wayin which mutation, non-random mating, dominance,and epistasis come into the transmission (expectation)term by changing Dzn

i , the phenotypic differencesbetween parent and offspring (Frank & Slatkin,1990). At equilibrium all of the moments must berelated by

Cov(wi , zni )=−E(wiDzn

i ).

If we normalize the population such that theequilibrium mean, z, is zero, then this equilibriumequation for n=2 describes how selection affects thevariance. If selection is stabilizing toward theequilibrium mean of zero, then the covariance termdescribes the rate at which the variance is reducedbecause of the negative association between fitness andthe squared distance of character values from theoptimum. This reduction in the variance caused byselection must be balanced by the rate at whichmutation adds variance to the population byincreasing, relative to the optimum, the squareddistance of the offspring relative to their parents. Thissimple equation provides easy calculation for manyinteresting mutation-selection problems, includingnonrandom mating and nonadditive genetics (Frank &Slatkin, 1990).

The dynamics of genetic variability are difficult tostudy for many assumptions. The problem with theseequations of mutation-selection dynamics is that thechanges in each moment, zn, depend on changes inhigher-order moments—the equations lack dynamicsufficiency without additional assumptions (Barton &Turelli, 1987). This lack of dynamic sufficiency is a

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property of mutation-selection dynamics and not ofthe Price Equation (see above; Frank & Slatkin, 1990).The value of the Price Equation is that, by addingnothing except clear notation, the forces that act ongenetic variability are decomposed into naturalcomponents and the underlying difficulties of theanalysis are made explicit. Different approaches mayprovide better tools for numerical calculation in somecases (Turelli & Barton, 1990).

The Price Equation is a very strange mathematicalrelation when compared with other formalisms inevolutionary biology. The equation is much moreabstract than the usual applied mathematics ofevolution; it simply suggests a way to map members ofone set to members of a second set. In this section Ispeculate about why this minimalism seems sopowerful. My purpose is to call attention to what Ibelieve is a deep problem of a purer sort than is typicalin biology.

The Price Equation has been the only way to studygeneral problems of kin interactions. Other methodsrequire special assumptions about which relativesinteract. Why is Price’s simple mapping so successfulfor kin selection problems? Kin selection requires thatone think of interactions between individuals asinteractions between sets of alleles, where there is somestatistical relation between the sets (relatedness). Inaddition, groups of individuals may interact. Theconsequences of group structure depend on thestatistical associations within the group comparedwith the associations among the group means. ThePrice Equation provides a natural way to thinkabout hierarchical decomposition (species, group,individual, gene) and statistical association at varioushierarchical levels. Although most applications havebeen to groups of relatives, species-level selection(Arnold & Fistrup, 1982) and community-level selec-tion (Frank, 1994a) can be studied in an elegant way.

The Price Equation also provides insight intofundamental problems such as dynamic sufficiencythat require a minimal description of how selectionworks. In the case of mutation-selection balance asimple partition between selection among adults andmutational changes in transmission must be analyzed.This partition follows the natural separation betweenselection and transmission in the two terms of theequation.

There is a beauty in the equation’s spareness anddescriptive power. Claims for mathematical beautyrarely impress biologists, however. I have sometimesheard the question: What problems can the Priceequation solve that cannot be solved by other

methods? The answer is, of course, none, because thePrice Equation is derived from, and is no more than,a set of notational conventions. It is a mathematicaltautology.

What is the practical value of the equation? The firststeps in using the equation are often quite difficultbecause one has to match the problem to the strangenotation. This requires labeling individuals, genotypesor groups in a nonstandard way. Once the rightstructure is found, solving problems seems very naturalboth algebraically and biologically. The gain is inforcing one, right at the start, to look for the strangetwist that makes the solution inevitable. The PriceEquation works well because it provides nothing morethan a way to fit a problem to the fundamen-tal properties of evolutionary change. Naturalselection is a statistical process, and the Price Equationforces a statistical description of problems.

Price recognized that his equation was a step towarda more general theory of selection. At the start of thissection I quoted part of the introduction to Price’s(1995) manuscript ‘‘The Nature of Selection.’’ Here isthe full text of the opening paragraph from thatmanuscript:

Selection has been studied mainly in genetics, but ofcourse there is much more to selection than justgenetical selection. In psychology, for example,trial-and-error learning is simply learning by selection.In chemistry, selection operates in a recrystallisationunder equilibrium conditions, with impure andirregular crystals dissolving and pure, well-formedcrystals growing. In paleontology and archaeology,selection especially favours stones, pottery, and teeth,and greatly increases the frequency of mandiblesamong the bones of hominid skeletons. In linguistics,selection unceasingly shapes and reshapes phonetics,grammar, and vocabulary. In history we see politicalselection in the rise of Macedonia, Rome, andMuscovy. Similarly, economic selection in privateenterprise systems causes the rise and fall of firms andproducts. And science itself is shaped in part byselection, with experimental tests and other criteriaselecting among rival hypotheses.

Price was not the first to note the generality ofselection. Karl Popper is the founder of modernphilosophical analyses of learning and knowledge asextensions of the general properties of biologicalselection. Donald Campbell has contributed substan-tially to this philosophy during the past 20 years (e.g.Campbell, 1974). This work was summarized in arecent book by Plotkin (1993). The philosophical workfocuses mainly on the question of whether allknowledge is necessarily the outcome of selectiveprocesses, hence the label for this work of

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‘‘Evolutionary Epistemology’’ coined by Campbell(1974).

Price was not particularly concerned with thesephilosophical questions. Instead he sought a generalformulation of selection that could be applied to anyproblem. He hoped that a formal theory would beuseful in other fields in the sameway thatmathematicalgenetics became the foundation for analyses ofgenetical selection. Here are the second and thirdparagraphs from ‘‘The Nature of Selection’’:

And yet, despite the prevading importance of selectionin science and life, there has been no abstraction andgeneralisation from genetical selection to obtain ageneral selection theory and general selectionmathematics. Instead, particular selection problemsare treated in ways appropriate to particular fields ofscience. Thus one might say that ‘selection theory’ isa theory waiting to be born—much as communicationtheory was fifty years ago. Probably the main lack thathas been holding back any development of a generalselection theory is lack of a clear concept of the generalnature or meaning of ‘selection’. That is what thispaper is about.

Let us pursue a little further the analogy with com-municationtheory.Probably thesinglemost importantprerequisite for Shannon’s famous 1948 paper on ‘‘AMathematical Theory of Communication’’ was thedefinitionof ‘information’ givenbyHartley in1928, forit was impossible to have a successful mathematicaltheory of communication without having a clearconcept of the commodity ‘information’ that acommunication system deals with. Hartley gave whathe described as a ‘‘physical as contrasted withpsychological’’ definition of information, whichomitted all considerations of the meaningfulness ofmessagesbutmeasuredattributes relevant to thedesignof communication systems. Similarly, for developmentof a useful mathematical theory of selection, one needsa physical rather than psychological definition ofselection, which excludes psychological factors ofpreferences and decision making. It is my hope that theconcept of selection proposed in this paper willcontribute to the future development of ‘selectiontheory’ as helpfully as Hartley’s concept of informationcontributed to Shannon’s communication theory.

Price then describes abstract properties of selectionin terms of mappings between pre-selection andpost-selection sets (populations). [The full text ofPrice’s The Nature of Selection is published as anaccompanying paper in this issue (Price, 1995)]. In myopinion the Price Equation itself is the closest anyonehas come to a general, abstract theory of selection.Price avoids formal theory in ‘‘The Nature ofSelection,’’ but clearly bases his presentation onconcepts that he learned while studying the PriceEquation. It remains for others to decide howmuch thePrice Equation can aid in building a general theory ofselection. I close by quoting Price’s first and lastparagraph from the final section of his paper.

When Shannon’s ‘‘Mathematical Theory of Com-munication’’ appeared in 1948, many scientists musthave felt surprise to find that at so late a date there hadstill remained an opportunity to develop sofundamental a scientific area. Perhaps a similaropportunity exists today in respect to ‘selectiontheory’. If we compare the high level of communi-cation technology reached fifty years ago [1922] withthe very disappointing results usually reachednowadays in computer simulations of evolution (forexample, as described by Bossert [1967]), and if wenote the degree of understanding of communicationsystems shown in the 1928 papers of Nyquist andHartley and then consider that it took another twentyyears before Shannon’s [1948] paper appeared, we canreasonably predict that much difficult work will berequired before an interesting and useful ‘‘Mathemati-cal Theory of Selection’’ can be developed. Theremainder of this paper contains suggestions forreaders who may wish to consider working on thisproblem themselves . . .Consideration of questions such as these, though interms of abstract models rather than genes orcontinents, should lead to deepening understanding ofselection such that in time someone will have theinsight to take a very large step forward like that takenby Shannon in 1948.

Game Theory and Evolutionarily Stable Strategies

(ESS)

Maynard Smith (1972: vii–viii) credits George Pricefor introducing game theory analysis to the study ofanimal behavior:

The essay on ‘Game theory and the evolution offighting’ was specially written for this book. I wouldprobably not have had the idea for this essay if I hadnot seen an unpublished manuscript on the evolutionof fighting by Dr George Price, now working in theGalton Laboratory at University College, London.Unfortunately, Dr Price is better at having ideas thanat publishing them. The best I can do therefore is toacknowledge that if there is anything in the idea, thecredit should go to Dr Price and not to me.

The above quote should not be taken too literally,in that we clearly owe our current understanding ofevolution and the theory of games to Maynard Smith(1982). The point is that George Price was not justanother person who happened to think aboutevolution and game theory at the time when the fieldwas taking shape. Rather, he was one of the first to seethe theory in broad outline, and his ideas directlyinfluenced Maynard Smith and others who developedthe field. In this section I briefly summarize how ideaschanged from the late 1960s to the early 1970s, andPrice’s role in that change. I also mention thecontributions of John Price (not related), who in 1969published the first clear description of how gametheory reasoning could be used to analyze ritualizedbehavior.

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The puzzle is why animals often settle fights in aritualized way rather than inflicting serious or deadlywounds. Maynard Smith & Price (1973) introduce theproblem with these examples:

. . . in many snake species the males fight each other bywrestling without using their fangs. In mule deer(Odocoileus hemionus) the bucks fight furiously butharmlessly by crashing or pushing antlers againstantlers, while they refrain from attacking when anopponent turns away, exposing the unprotected side ofits body. And in the Arabian oryx (Oryx leucoryx) theextremely long, backward pointing horns are soinefficient for combat that in order for two males tofight they are forced to kneel down with their headsbetween their knees to direct their horns forward . . .

The accepted explanations for the conventional natureof contests is that if no conventional methods existed,many individuals would be injured, and this wouldmilitate against the survival of the species . . . Thedifficultywith this type of explanation is that it appearsto assume the operation of ‘‘group selection’’.

Although group or species level selection was nolonger an acceptable explanation for behavioralevolution in the late 1960s, no one had yet formulateda successful theory of ‘‘limited war’’ based onindividual advantage. Hamilton (1971) had recognizedthe problem and developed a game theory analysis toexplain ritualized settlement of conflict. However, hisemphasis was on the genetic relatedness of contestantsand the reduced conflict that may occur among kin.Thus his explanation required that groups besufficiently genetically differentiated to favor kin-selected altruism among group members, a conditionthat often does not hold.

It is not clear whether George Price or John Pricewas the first to develop a theory to explain ritualizedsettlement of conflict based on a model of individualselection. John Price laid out the problem and itssolution with admirable clarity in a paper published in1969:

It is easy to see the advantage of yielding behaviour tothe species as a whole, but what is the advantage to theindividual who yields? Assuming the distribution ofyielding behaviour in the population to be continuous,then it is likely that at one end of the distribution wewill find individuals who do not yield at all. Thesenon-yielders will win all their ritual agonisticencounters with yielders, and since there is clearbiological advantage in being the victor in a ritualagonistic encounter, we must explain why it is thatyielding behaviour has not been bred out of thepopulation, even if it managed to get established in thefirst place. This is not likely to be a simple problem, butsome of the reasons may be briefly summarized asfollows:

(1) The disadvantage of being a yielder is counterbal-anced by the likely mortality when two non-yielders

meet each other. Thus it is advantageous to be a yielderwhen everyone else is a non-yielder, and to be anon-yielder when everyone else is a yielder. Thisdependence of the advantage of one’s phenotype onthe phenotypes of the rest of the population isanalogous to the situation with mimetic butterflies andtends towards the maintenance of variation in thepopulation.

These quotes concisely summarize the main ideasof evolutionary game theory. J. S. Price wrotethese lines in a paper that developed an evolutionarytheory of psychological depression (see also J. S.Price, 1991). George Price had arrived at the samesolution. He wrote, in a grant proposal that I describelater:

A paper entitled ‘‘Antlers, intraspecific combat, andaltruism’’ was accepted by Nature on 7 February 1969(provided that it is shortened). This gives particularattention to the problem, recognised by Darwin(Descent, Chapt. 17), that deer antlers are developedat great cost to the animal and yet are highly inefficientweapons for inflicting injury on an opponent similarlyarmed (since branching antlers are effective shieldsagainst other branching antlers, though they wouldnot protect against unbranched antlers projectingforward).

For some reason, Price never resubmitted themanuscript. Fortunately, John Maynard Smith, whorefereed the paper for Nature, understood Price’sinsights and used them to develop evolutionary gametheory into an active field of research. It would beinteresting to read Price’s original thoughts on thistopic in his ‘‘Antlers’’ paper, but I have not been ableto find a copy of the manuscript.

It is always difficult, in retrospect, to see theoriginality and insight of a simple idea. In this casefrequency-dependent individual advantage explainedwhat was, at that time, the long-standing puzzle ofritualized settlement of conflict. Fisher had, in 1930,used a frequency dependent model to explain theevolution of the sex ratio. Hamilton’s (1967) sex ratiomodel of local mate competition extended Fisher’sapproach, and introduced the first modern gametheory analysis of adaptation. In particular, Hamiltondeveloped the idea of an ‘‘unbeatable strategy’’, which,if adopted by all members of the population, cannot bebeaten by any individual using a different strategy.This provides a method for finding the frequencydependent equilibrium, where the frequency depen-dence is of the sort described in J. S. Price’s quoteabove.

Although Hamilton (1967) had developed theformal analysis of behavior with the game theorymethods that are still used today, he did not seethe application of these ideas to the problem ofritualized behavior (Hamilton, 1971). In a recent letter

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Hamilton (personal communication) wrote:

J. S. Price is definitely quite a different person andunrelated [to George Price]. I can remember Georgetalking about him from time to time. The Galton Labhad contacts with the Maudsley psychiatric hospital inSouth London and George may have visited there orcome in contact with John Price or the ‘‘other Price’’as he sometimes called him. It may have been readingthat article by him [J. S. Price 1969] that startedGeorgethinking about whether contact was really so‘‘ritualised’’ and so ultimately to his ESS idea. I myselfwas fairly happy with ‘‘ritualised submission’’ at thetime as my paper in the Man and Beast Symposiumshows [Hamilton 1971], andonly gradually came to seeunder George’s reiteration what he was on about, andthere could be a resolution basically akin to my‘‘unbeatable’’ sex ratio strategy.

The publication of Maynard Smith & Price (1973)is the end of the story as far as George Price isconcerned. As Maynard Smith noted in the quote atthe top of this section, Price stimulated his interestedin the field (see also Maynard Smith, 1976). By 1973Price had apparently lost interest in the subject and inpublishing his work. He had turned intensely religious,and left his mark on Maynard Smith & Price (1973) byinsisting that, in the Hawk–Dove game, the word‘‘dove’’ not be used because of its religious significance.Thus that particular paper analyzes the Hawk–Mousegame, the only instance of that game in the literature.

Fisher’s Fundamental Theorem Made Clear

Fisher’s Fundamental Theorem of Natural Selec-tion is probably the most widely quoted theorem inevolutionary genetics. Under the usual interpretationthe theorem is believed to say that the rate of increasein the mean fitness of a population is equal to thepopulation’s additive genetic variance for fitness. Thusnatural selection causes a continual increase in themean fitness of a population.

This interpretation of the theorem is true only whenthe population mates randomly and there is nodominance or epistasis. This very limited scopecontrasts sharply with Fisher’s (1930, 1941, 1958)claims that his theorem is exact for all conditions, thatit is similar in power to the second law of thermo-dynamics, and that it holds the supreme positionamong the biological sciences.

While most authors accepted the standard interpret-ation and its many exceptions, others saw thecontradiction with Fisher’s bold claims and tried to getat his meaning. For example, Kempthorne (1957),Crow & Kimura (1970), and Turner (1970) all sawsomething deeper in the theorem, but they could notfathom Fisher’s derivation and meaning, and theirown equations fell far short of proving a result with the

generality or depth Fisher claimed. By 1970 all authorshad abandoned the theorem as Fisher had intended it,except for Edwards (1967) who thought that Fisher’stheorem may indeed be correct and important if onlywe could understand what he meant (Edwards, 1994).

Price (1972b) solved the problem by proving thetheorem as Fisher intended. This is an entertainingpaper which can still be read with ease (see also Ewens,1989; Frank & Slatkin, 1992; Edwards, 1994). Here Ibriefly outline Price’s key insight, speculate as to whyPrice succeeded where the best minds in populationgenetics failed, andmentionwhere thework in this fieldis currently heading.

Price noted that Fisher partitioned the total changein fitness into two components. To show this I firstwrite the total change in fitness as

Dw=w'=E '−w =E, (5)

where primes denote one time step or instant into thefuture, w=E is mean fitness when measured in thecontext of a particular environment, E, and Dw is thetotal change in fitness which everyone had assumedwas the object of Fisher’s analysis. However, Fisher’stheorem is not concerned with the total evolutionarychange, which depends at least as much on changes inthe environment as it does on natural selection.Instead, Fisher partitioned the total change into

Dw=(w'=E−w =E)+(w'=E '−w'=E).

Fisher called the first term the change in fitness causedby natural selection because there is a constant frameof reference, the initial environmental state E. TheFundamental Theorem states that the change in fitnesscaused by natural selection is equal to the additivevariance in fitness. Fisher referred to the second termas the change caused by the environment, or moreoften, as the change caused by the deterioration of theenvironment, to stress that this term is often negativebecause natural selection increases fitness but the totalchange in fitness is usually close to zero. Densitydependence is one simple way in which adaptiveimprovements in organismal efficiency must bebalanced by greater competition for resources(deterioration in the biotic environment). Thus, in onemodel presented by Fisher (1941), ‘‘Intense selectiveactivity is shown to be compatible with an entireabsence of change in the average survival value of thepopulation’’.

Price showed that the Fundamental Theorem can beproved if one adheres to Fisher’s rather queerdefinitions. In particular, Fisher focused on the firstterm of eqn (5), which Ewens (1989) has called thepartial change in fitness caused by natural selection.This partial change uses Fisher’s measure for the

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average effect of a gene substitution while holdingall aspects of the environment, E, constant. Theenvironment includes the initial gene frequencies, thusevolutionary change in fitness is measured whileholding gene frequencies constant. The effect ofchanging gene frequency on fitness appears in thesecond, environmental, term.

Why was Price able to see what Fisher meant whereothers had failed? Part of the answer is certainly Price’sgreat ability to describe deep and general properties ofnatural selection with simple equations. In addition,Price may have been predisposed by his earlier workbecause Fisher’s view depended on two propertiesshared with the Price Equation: a partition of totalchange into components and strict use of unconven-tional definitions.

Now that Price has shown us what Fisher reallymeant, we can ask whether the theorem is as deep anduseful as Fisher claimed. Price’s own view was that thetheorem was interesting mathematically but of littlepractical value. Ewens (1989) echoed this view aftergivinga clearderivationof the theorembasedonPrice’spaper. Perhaps the strongest case against the theoremis that Fisher himself never seemed to use it beyonddeveloping some rather vague heuristical conclusionsabout how competition causes the ‘‘environment’’ todeteriorate (Frank & Slatkin, 1992).

The case against Fisher is not closed, however.Ewens (1992) has recently shown a relationshipbetween the Fundamental Theorem and somefairly deep optimization principles that have beenused successfully in mathematical genetics. Fisher’s‘‘average effect of a gene substitution’’ provides thekey link between the Fundamental Theorem and theoptimality principle that ‘‘of all gene frequencychanges which lead to the same partial increasein mean fitness as the natural selection genefrequency changes, the natural selection valuesminimize a generalized distance measure betweenparent and daughter gene frequency values’’ (Ewens,1992). This view fits nicely with Fisher’s writingsbecause Fisher repeatedly emphasized the import-ance of average effects in both the FundamentalTheorem and in general aspects of mathematicalgenetics.

What was Price trying to do?

In this section I briefly summarize what is known ofPrice’s life. I then return to the problem of how he cameto work on abstract properties of natural selection,altruism and ritualized behavior.

Although I have come across some facts abouthis life in the late 1960s and early 1970s from hisCV (Table 1), correspondence and unpublished

T 1George Price’s Curriculum Vitae in 1974

Personal Data Age 51, divorced, U. S. citizen, admitted to permanent residence in the U.K. (work permit not needed).Education S.B. in chemistry, University of Chicago, 1943 Ph.D. in chemistry, University of Chicago, 1946Employment

1944–46 Manhattan Project (atom bomb project) research on uranium analysis at the University of Chicago.1946–48 Instructor in chemistry at Harvard and consultant to Argonne National Laboratory.1950–57 Research Associate in medicine, University of Minnesota, working on fluorescence microscopy, liver perfusion, etc.

1957–61 Trying to write book, NO EASY WAY, (first for Harper’s, later for Doubleday) on what the United States shoulddo about Russia and China, while supporting myself as a freelance magazine article writer and a subcontract technicalwriter. (The book was never finished: the world kept changing faster than I could write about it!)

1961–62 Consultant to IBM on graphic data processing.1962–67 IBM employee in Poughkeepsie and Kingston, New York. Started as a Market Planner helping in the design phase

of System/360. Final work was on mathematical optimisation carried out through simulation of private enterprisemarket mechanisms. Programming was in FAP (Fortran Assembly Program) on the 7094.

1967–68 Reading and writing in London on evolutionary biology, while living on savings.1968–74 Research in mathematical genetics, on academic staff of University College London (under Professor Cedric Smith

in the Department of Human Genetics and Biometry). Final position was Associate Research Fellow. The researchinvolved much FORTRAN programming (both IBM/360 Mod 65 and CDC 6600) plus some PL/I and FORMAC.(Left because I felt that the sort of theoretical mathematical genetics I was doing wasn’t very relevant to humanproblems, and I wanted to change to economics.)

1974 Worked June 14th to August 17th as a night office cleaner for a contract cleaning firm. (This work was undertakenfor reasons having something to do with Christianity. I was considered to be slow but unusually dependable, so thatafter a while the supervisor did not bother to inspect my work. Left because my reason for wanting a night job nolonger held.)

1975 [From a brief article in a January 15, 1975 edition of Sennet (Careers Supplement), a London student’s newspaper,entitled Jesus ‘hot-line’]: A prominent genetics researcher at University College Hospital gave up everything,including his life for his religious beliefs St. Pancras Coroner’s Court was told last week. Dr. George Price gave awayall his money, clothes and possessions to homeless alcoholics and left his flat in Bloomsbury to live as a squatterin Drummond Street, Kentish Town. It was there that he was found dead. A respected scientific researcher, Dr. Pricewas convinced that he had a ‘‘hot line to Jesus’’.

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manuscripts, I do not have a clear impression of theman. He ended in great sadness and poverty, confidentin the clarity of his mind and his insights in biology, buttormented by the feeling that he had failed tocontribute in any significant way to easing humansuffering.Hewas increasingly religious toward the end,when he gave all his money and possessions tohomeless alcoholics andwent to live among the poorestsquatters of London. But he continued to work andwrite with clarity. His last plans were to take upeconomics, because he hoped to find in economics abetter way to match his analytical power to problemsof humanity.

In Price’s last years he also wrote a detailed analysisof theapparent contradictions in thegospelsof theNewTestament. This is a subject with a long history ofbiblical scholarship,where the goal is to resolve contra-dictions about the timing of the crucifixion and resur-rectionofJesus.Price’sworkonthis subject ispresentedin an unpublished manuscript ‘‘The twelve days ofEaster’’. I cannot judge the quality of his scholarship,but this paper is written with the same clarity andprecision of his work in evolutionary biology.

I return now to the limited goal of understand-ing Price’s contributions to evolutionary genetics.Specifically, why was he interested in abstract prop-erties of natural selection? How was that interestassociated with his contributions to problems ofcombat, altruism and game theory?

I have found only limited clues. Price wrote athoughtful review of the relation between science andsupernatural phenomena in the mid 1950s (Price, 1955,1956). This work shows his keen analytical mind andhis interest in fundamental processes. His interest heremay also reflect a tension between his restless search formeaning, in terms of human interests and values, andhis tenacious belief that all hypotheses must besubjected to cold, analytic scrutiny.

The next major project that I know of is hisinterest in the arms race and the cold war in the late1950s (see Table 1). This interest may have set the stagefor his later work on game theory and the resolutionof conflict by ritualized behavior.

The hints from his pre-genetics work provide onlyvague clues about Price’s goals in evolutionarybiology. The best evidencewehave comes fromhis owngrant proposal to the Science Research Council ofGreat Britain. The proposal was written in 1969, afterPrice had spent approximately two years studyingevolution. In the remainder of this section I brieflysummarize the contents of that proposal. I list each ofthe ten topic headings in the proposal, with a concisedescription of the main points. Generally, I will avoidcommentary; the purpose is to make available this

information in the hope that someone will look moredeeply into his life and his work.

I quote the first section in full.

The main purpose of the work is to develop improvedtechniques for making inferences about hominidevolution in the Pleistocene going beyond what isdirectly shown by fossils and artifacts. It is felt that themost fruitful way to begin is by developing (a) newmathematical treatments of evolution under con-ditions of complex social interactions, and (b) moresimple and transparent mathematical genetics modelsthat can provide rules-of-thumb for qualitative orsemi-quantitative reasoning. One important benefitfrom emphasizing a mathematical approach is thatthis should help to protect against biasing effects ofemotional prejudices about human nature and humanancestry. Also, a mathematical approach gives anadvantage in exposition; to cite Haldane: when one isfaced with a difficulty or controversy in science, ‘‘anounce of algebra is worth of a ton of verbal argument’’(obituary by Maynard Smith, Nature, 206, 239(1965)).

As these mathematical tools become available, theywill be applied to specific problems of humanevolution—though in the early stages of work,emphasis will be on development rather thanapplication. In addition, since it is likely that much ofthe mathematical work will also have broadapplicability to evolutionary biology (especially inrelation to social animals), it is planned to apply themathematical models to some ethological problems.

Price introduces the idea of evolutionary stability ofbehavior in the context of alternative hypoth-eses about the mating system of Pleistocene humans.

As I discussed in the earlier section on Evolu-tionarily Stable Strategies, Price’s genetical approachto the evolutionary stability of behavioral systems wasa novel way to think about the problem. Pricesummarized the approach in this way:

We now concentrate attention on behaviour. A systemof behaviour that is close to a selective maximum willbe called a genetically optimal behaviour system(‘‘optimal’’ in the sense of maximizing the frequencyof an individual’s genes in the next few generations ofthe local population). For a genetically optimalbehaviour system to be stable, the main requirementis that children should tend to behave like theirparents. Even though we have little knowledge of howcultural and genetical inheritance interacted inPleistocene hominids, let us assume that this conditionof parent-child resemblance held. Then a sudden largesaltation in behaviour was comparatively improbableduring that period (though such have occurred inhistorical times), and small changesmoving away fromthe peak tended to be corrected against because theywere disadvantageous.

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The point is that we can reject any hypothesis for amating system if it does not satisfy the condition ofbeing locally stable against invasion by slightlydifferent individual behaviors. Price also noticed thatall the complexities of genetic and cultural inheri-tance may often be reduced to the issue ofparent–offspring correlation when one is trying toanalyze the evolutionary stability of a behavioralsystem.

Price then suggested two alternative hypothesesabout Pleistocene mating systems to illustrate hismethod of reasoning. These alternatives were notmeant to be a list of plausible hypotheses, but werepresented to ‘‘save space’’ in the context of a grantproposal.

System I is one of total cooperation, withpromiscuous, non-competitive mating and coopera-tive rearing of the young. Price concluded that: ‘‘Thiswill be a genetically optimal system under thecondition that any deficiency in cooperation isretaliated against by physical punishment and/orwithholding benefits. Then an individual wouldincrease his or her fitness both by cooperating withothers and thereby avoiding punishment, and byhelping to punish others who are deficient incooperation and thereby causing them to cooperate.’’

System II is one of group cooperation principally inhunting by adult males, and of individual or familyaction in most other behaviours. Cooperation in grouphunting was maintained by reciprocal exchange offood among males and punishment of significantnon-cooperation, as in System I. After hunting spoilswere divided among men, the distribution of food towomen and children was a matter of individual choiceby each man.

Price concluded, after further commentary, thatSystem II is probably near to the truth, with each maleusually having a single wife. This justifies, later in theproposal, the need to study more fully sexual selectionin monogamous mating systems.

Price provided in this section an outline of theor-etical work needed to create a foundation forevolutionary analyses of human behavior. Most of theimportant theoretical advances in the study of socialbehavior are foreshadowed here. A complete grasp ofthe fundamental problems listed here remains wellbeyond current understanding. I quote the section infull.

From the example [of human mating systems] it can beseen that mathematical understanding of several typesof biological phenomena would be useful. First, sinceearly man presumably lived in groups, and because of

the importance of understanding the relative signifi-cance of individual and group selection in humandescent, better understanding of group selectionwould be desirable. (Even merely for the purpose ofrejecting group selection as an explanatory mechan-ism, it would be helpful to understand it better. But itseems likely that group selection should not be entirelyrejected as a factor in human evolution.) A secondneed is to work out details of reciprocity andpunishment systems that make it individuallyadvantageous to act in group- and species-benefitingways, bymeans of transfers of benefit andharmamongindividuals on a basis of degree of cooperation. A thirdneed is better understanding of nepotism effects,assortative mating, and other behaviour involvingtransfer of benefit or harm among individuals on abasis of degree of genetical similarity. A fourth needis investigation of the mathematics of sexual selectionunder the postulated conditions of individual familieswith permanent mating. A fifth need is work on theinteractions of cultural and genetical inheritance—since the period under consideration was one of majorcultural advances. A sixth need is better understandingof conditions for stability in evolutionary trends. Aseventh need is simplemathematicalmodels relating tosuch basic matters as the speed of evolutionarychanges.

These needs lead to the specific plans to be discussedin the sections that follow.

This is standard population genetics. Price citesKimura, Wright and others on drift-migrationdynamics.

Price noted that a different kind of group selectionfrom that traditionally studied by populationgeneticists may be important in evolutionary studies ofbehavior. He cited Hamilton’s analysis of meiotic driveof the Y against the X. A gene enhancing drive of theY spreads in the population, but causes an excess ofmales and a decline in the population size becausefemales become rare. Thus selection against driveoccurs at the group level. He also mentioned differentsystems of territoriality that could be genetically stable(roughly, an ESS) within populations but that wouldhave different consequences for the success of thegroup. If the observed state corresponds to the stablestate with higher group productivity, then Pricesuggested that group selection may partly explain theobserved pattern.

HerePrice talks about hiswork that eventually led tothe theory of Evolutionarily Stable Strategies (ESS, seeabove). First he noted that cooperative behavior canbe maintained if group members tend to punish

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uncooperative individuals, thereby making coopera-tive behaviour individually advantageous. He thendiscussed his paper that addressed apparentlygroup-benefiting behaviors in terms of individualadvantage:

A paper entitled ‘‘Antlers, intraspecific combat, andaltruism’’ was accepted by Nature on 7 February 1969(provided that it is shortened). This gives particularattention to the problem, recognised by Darwin(Descent, Chapt. 17), that deer antlers are developedat great cost to the animal and yet are highly inefficientweapons for inflicting injury on an opponent similarlyarmed (since branching antlers are effective shieldsagainst other branching antlers, though they wouldnot protect against unbranched antlers projectingforward). Also the paper gives briefer treatment toseveral other problems, such as the remarkablecooperative behaviour of the African hunting dogs.

Price then mentioned some references fromanthropology and sociology, and that his specifictheory has much in common with game theory modelsof limited war strategy. (The Nature paper was neverpublished. See the section above on

.)

Price followed Darwin in citing the probableimportance of sexual selection in human evolution.Issues concern ‘‘attractiveness’’, ‘‘fashion’’, and therelation between resources (e.g. food) and the geneticalevolution of the mating system. But Price was mainlyconcerned with showing, formally, how sexualselection could have affected the evolution of humansbecause, as Price states, ‘‘Many critics, beginning withWallace even before the Descent was finished (letter toDarwin, 29 May 1864), have questioned whethersexual selection could occur under known or assumedconditions of primitive human life.’’

Price described in some detail how he planned tostudy a formal model of sexual selection. His modelhad characters for behavior or anatomy, preferencesfor the opposite sex, a paternal rating for each malebased onhis character values, and amaternal rating foreach female based on her character values. Ratings arebased on utilitarian characters rather than those thatmerely affect appearance. He recognized the import-ance of sex-limited expression, suggesting an examplemodel with characters that vary in their expression inmales and females.

—

Price noted that his previous model, in whichpreferences are rigidly related to characters, wouldprobably lead to a Fisher runaway process. But

Fisher’s process runs away only in its intermediatestages. It is slow when getting started and towardthe end when natural selection begins effectively toslow sexual selection (Price cites O’Donald’s 1969simulations).

These rate changes in character evolution suggestedto Price that females might compare old and youngmales to determine which traits are changing mostrapidly and, therefore, are most strongly correlatedwith fitness. If females chose the most rapidly changingtraits in this way, they would be preferring what iscoming into fashion, a strategy similar to economicspeculation.

I don’t know of any work that has followed thisline of thought, and probably for good reason. Theidea may work formally, but whether females couldactually assess rate of character change in a useful wayseems doubtful.

The work Price eventually published on the PriceEquation and Fisher’s fundamental theorem aredescribed in full detail.

I quote the final paragraphs in full.

This completes the discussion of contemplatedprojectswhere plans are sufficientlywell-formulated topermit detailed explanation. Other areas where it ishoped to accomplish something useful include theproblem of the interaction of cultural and geneticalinheritance (which on page 4 was treated in a quitecrude, inadequate way). Also it is hoped that theinvestigations of social interaction effects may lead tothe finding of very broad generalizations. Forexample, the cases discussedwhere individual selectiondecreases group fitness are closely and deeplyanalogous to economic effects recently discussed byHardin in a paper entitled ‘‘The tragedy of thecommons’’ (Science, 162, 1243 (1968)); and the samegeneral types of mechanisms can be used in biologicalsystems and in human economic systems to make itadvantageous to the individual (in a genetical sense orin an economic sense) to act in a way that benefits thegroup. It is hoped that other parallels betweeneconomics and genetics can be found.

To sum up: The general plan of the work is thatmathematical tools will be developed specifically inorder to handle problems of human evolution, butwhen developed they will be applied to both humanand animal evolution problems. Initially emphasis willbe on developing the tools; later it will increasingly beon applying them.

Conclusions

George Price had original, fundamental insights intoevolutionary biology. The Price Equation had a direct

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and profound influence on W. D. Hamilton’s work onkin selection and on the subsequent theor-etical development of social evolution. Price was thefirst to see that ritualized behavior, game theoryand evolutionary genetics belonged together in acoherent vision of behavioral evolution. Price’s visionstarted Maynard Smith’s work on evolution and thetheory of games, which is among the most influentialdevelopments of the past few decades.

These past achievements are sufficient reasons tostudyPrice’s life andwork.Butmyown curiosity aboutPrice’s work is kept alive by the Price Equation. Inevolutionary genetics, the equation continues toprovide fresh insight into many difficult problems. Andthe future may show that this rich and poorlyunderstood equation is indeed the key to a broadertheory of selection that transcends population genetics.

W. D. Hamilton has, over the past 15 years, helped me inmany ways to understand more about evolutionary geneticsand about George Price. Warren Ewens changed this paperfrom a vague idea to an actual manuscript by his steadyencouragement and his insight into the esoteric details ofFisher’s fundamental theorem. Jon Seger made characteristi-cally insightful suggestions for revision. I received helpfulcomments from N. H. Barton, R. M. Bush, J. F. Crow, T.Nusbaum, M. Ridley, P. D. Taylor and M. J. Wade. I owespecial thanks to L. D. Hurst for his editorial assistance andfor his infective enthusiasm about George Price.

REFERENCES

A, A. J. & F, K. (1982). The theory of evolution bynatural selection: a hierarchical expansion. Paleobiology 8,

113–129.B, N. H. & T, M. (1987). Adaptive landscapes, genetic

distance and the evolution of quantitative characters. Genet. Res.49, 157–173.

B, W. (1967). Mathematical optimization: are there abstractlimits on natural selection? In: Mathematical Challenges to theNeo-Darwinian Interpretation of Evolution (Moorhead, P. S. &Kaplan, M. M., eds) pp. 35–40. Philadelphia: Wistar InstitutePress.

B, F. (1990). Partitioning of covariance as a method forstudying kin selection. Trends Ecol. Evol. 5, 224–228.

C, D. T. (1974). Evolutionary epistemology. In: ThePhilosphy of Karl Popper, Volume 1 (Schilpp, P. A., ed.)pp. 413–463. LaSalle, IL. Open Court Press.

C, J. F. & K, M. (1970). An Introduction to PopulationGenetics Theory. Minneapolis, MN: Burgess.

C, J.F.&N,T. (1976). The rate of change of a charactercorrelated with fitness. Am. Nat. 110, 207–213.

E, A. W. F. (1967). Fundamental theorem of naturalselection. Nature, Lond. 215, 537–538.

E, A. W. F. (1994). The fundamental theorem of naturalselection. Biol. Rev. 69, 443–474.

E,W.J. (1989). An interpretation and proof of the fundamentaltheorem of natural selection. Theor. Popul. Biol. 36, 167–180.

E, W. J. (1992). An optimizing principle of natural selection inevolutionary population genetics.Theor.Popul.Biol. 42, 333–346.

F, R. A. (1930). The Genetical Theory of Natural Selection.Oxford: Clarendon.

F, R. A. (1941). Average excess and average effect of a genesubstitution. Ann. Eugenics 11, 53–63.

F,R.A. (1958). The Genetical Theory of Natural Selection, 2ndedn. New York: Dover.

F, S.A. (1985). Hierarchical selection theory and sex ratios. II.On applying the theory, and a test with fig wasps. Evolution 39,

949–964.F, S. A. (1986a). Hierarchical selection theory and sex ratios I.

General solutions for structured populations. Theor. Popul. Biol.29, 312–342.

F, S. A. (1986b). The genetic value of sons and daughters.Heredity 56, 351–354.

F, S. A. (1986c). Dispersal polymorphisms in subdividedpopulations. J. theor. Biol. 122, 303–309.

F, S. A. (1987a). Demography and sex ratio in social spiders.Evolution 41, 1267–1281.

F, S. A. (1987b). Variable sex ratio among colonies of ants.Behav. Ecol. Sociobiol. 20, 195–201.

F, S. A. (1992). A kin selection model for the evolution ofvirulence. Proc. R. Soc. Lond. B 250, 195–197.

F, S. A. (1994a). The genetics of mutualism: the evolution ofaltruism between species. J. theor. Biol. 170, 393–400.

F, S. A. (1994b). Kin selection and virulence in the evolutionof protocells and parasites. Proc. R. Soc. Lond. B 258, 153–161.

F, S.A.&S,M. (1990). The distribution of allelic effectsunder mutation and selection. Genet. Res. 55, 111–117.

F, S. A. & S, M. (1992). Fisher’s fundamental theoremof natural selection. Trends Ecol. Evol. 7, 92–95.

G, A. (1984). Natural selection, kin selection and groupselection. In: Behavioural Ecology (Krebs, J. R. & Davies, N. B.,eds), pp. 62–84. Oxford: Blackwell Scientific Publications.

G, A. (1985). A geometric view of relatedness. Oxf. Surv.evol. Biol. 2, 28–89.

H, W. D. (1964a). The genetical evolution of socialbehaviour. I. J. theor. Biol. 7, 1–16.

H, W. D. (1964b). The genetical evolution of socialbehaviour. II. J. theor. Biol. 7, 17–52.

H, W. D. (1967). Extraordinary sex ratios. Science 156,

477–488.H, W. D. (1970). Selfish and spiteful behaviour in an

evolutionary model. Nature, Lond. 228, 1218–1220.H, W. D. (1971). Selection of selfish and altruistic behavior

in some extreme models. In: Man and Beast: Comparative SocialBehavior (Eisenberg, J. F. & Dillon, W. S., eds) pp. 57–91.Washington, DC: Smithsonian Press.

H, W. D. (1972). Altruism and related phenomena, mainlyin social insects. A. Rev. ecol. Syst. 3, 193–232.

H, W. D. (1975). Innate social aptitudes of man: anapproach from evolutionary genetics. In: Biosocial Anthropology(Fox, R., ed.) pp. 133–155. New York: Wiley.

H, W. D. (1979). Wingless and fighting males in fig waspsand other insects. In: Reproductive Competition and SexualSelection in Insects (Blum, M. S. & Blum, N. A., eds) pp. 167–220.New York: Academic Press.

H, R. V. L. (1928). Transmission of information. Bell Syst.Tech. J. 7, 617.

H, I.L.&D, J. (1987). A method for analyzing selectionin hierarchically structured populations. Am. Nat. 130, 582–602.

H, L. D. (1991). The evolution of cytoplasmic incompatibilityor when spite can be successful. J. theor. Biol. 148, 269–277.

K, O. (1957). An Introduction to Genetic Statistics. NewYork: Wiley.

L, R. C. (1974). The Genetic Basis of Evolutionary Change.New York: Columbia University Press.

L, C. C. (1967). Fundamental theorem of natural selection. Nature,Lond. 214, 505–506.

M S, J. (1972). On Evolution. Edinburgh: EdinburghUniversity Press.

MS, J. (1976). Evolution and the theory of games. Am.Sci. 64, 41–45.

M S, J. (1982). Evolution and the Theory of Games.Cambridge: Cambridge University Press.

M S, J. & P, G. R. (1973). The logic of animalconflict. Nature, Lond. 246, 15–18.

. . 388

N, L. (1985). Female-biased sex ratios: individual or groupselection? Evolution 39, 349–361.

N,H. (1928). Certain topics in telegraph transmission theory.Trans. A.I.E.E. 47, 617–644.

O’D, P. (1969). ‘‘Haldane’s dilemma’’ and the rate of naturalselection. Nature, Lond. 221, 815–816.

O, K. (1983). Hierarchical theory of selection: the covarianceformula of selection and its applications. Bull. Biometr. Soc. Jpn4, 25–33.

P, H. C. (1994). Darwin Machines and the Nature ofKnowledge. Cambridge, MA. Harvard University Press.

P, G. R. (1955). Science and the supernatural. Science 122,

359–367.P,G.R. (1956). Where is the definitive experiment? Science 123,

17–18.P, G. R. (1970). Selection and covariance. Nature, Lond. 227,

520–521.P, G. R. (1971). Extension of the Hardy–Weinberg law to

assortative mating. Ann. hum. Genet. 34, 455–458.P, G. R. (1972a). Extension of covariance selection math-

ematics. Ann. hum. Genet. 35, 485–490.P, G. R. (1972b). Fisher’s ‘fundamental theorem’ made clear.

Ann. hum. Genet. 36, 129–140.P, G. R. (1995). The nature of selection. J. theor. Biol. 175,

389–396 [written circa 1971].P,G.R.&S,C.A.B. (1972). Fisher’sMalthusian parameter

and reproductive value. Ann. hum. Genet. 3, 1–7.P, G. R. (unpublished). Proposal to the Science Research

Council. (May 1969).P, G. R. (unpublished). Antlers, intraspecific combat, and

altruism. [In another unpublished manuscript, Price stated thatthis paper on ritualized fighting was ‘‘accepted by Nature on 7February 1969 (provided that it is shortened).’’The paper wasnever published.]

P,G.R. (unpublished). The twelve days of Easter. (Written circa1972–73).

P, J. S. (1969). The ritualization of agonistic behaviour as adeterminant of variation along the neuroticism/stability dimen-sion of personality. Proc. R. Soc. Med. 62, 37–40.

P, J. S. (1991). Change or homeostasis? A systems theoryapproach to depression. Br. J. Med. Psychol. 64, 331–344.

Q, D. C. (1991). Group selection and kin selection. TrendsEcol. Evol. 6, 64.

Q,D.C. (1992a). A general model for kin selection. Evolution46, 376–380.

Q, D. C. (1992b). Quantitative genetics, kin selection, andgroup selection. Am. Nat. 139, 540–558.

R, A. (1966). A mathematical model of the culling processin dairy cattle. Anim. Prod. 8, 95–108.

R, A. (1968). The spectrum of genetic variation. In:Population Biology and Evolution (Lewontin, R. C., ed.), pp. 5–16.Syracuse University Press.

S, J. (1981). Kinship and covariance. J. theor. Biol. 91,

191–213.S, C. E. (1948). A mathematical theory of communication.

Bell Syst. Tech. J. 27, 379–423, 623–656.T, P. D. (1988a). Inclusive fitness models with two sexes.

Theor. Popul. Biol. 34, 145–168.T, P. D. (1988b). An inclusive fitness model for dispersal of

offspring. J. theor. Biol. 130, 363–378.T, P. D. (1989). Evolutionary stability in one-

parameter models under weak selection. Theor. Popul. Biol. 36,

125–143.T, P. D. (1990). Allele-frequency change in a class-structured

population. Am. Nat. 135, 95–106.T, M. & B, N. H. (1990). Dynamics of poly-

genic characters under selection. Theor. Popul. Biol. 38,

1–57.U, M. K. (1988). On the evolution of genetic

incompatibility systems: incompatibility as a mechanism for theregulation of outcrossing distance. In: The Evolution of Sex(Michod,R.E.&Levin, B.R., eds) pp. 212–232. Sunderland,MA:Sinauer Associates.

W, M. J. (1980). Kin selection: its components. Science 210,

665–667.W, M. J. (1985). Soft selection, hard selection, kin selection, and

group selection. Am. Nat. 125, 61–73.W, M. J. & B, R. W. (1994). The population

dynamics of maternal-effect selfish genes. Genetics 138,

1309–1314.W, D. S. (1983). The group selection controversy: history and

current status. A. Rev. ecol. Syst. 14, 159–187.