An Introduction to Sociobiology: Inclusive Fitness and the Core Genome

Herbert Gintis

June 29, 2013

The besetting danger is ...mistaking part of the truth for the whole...in every one of the leading controversies...both sides were in the right in what they affirmed, though wrong in what they denied John Stuart Mill, On Coleridge, 1867 A Mendelian populationhas a common gene pool, whichis itscollective or corporate genotype. Theodosius Dobzhansky, Cold Springs Harbor Symposium, 1953. The interaction between regulator and structural genes... [reinforces] the concept that the genotype of the individual is a whole. Ernst Mayr, Populations, Species and Evolution, 1970

Abstract This paper develops inclusive fitness theory with the aim of clarifying its appropriate place in sociobiological theory and specifying the associated principles that render it powerful. The paper introduces one new concept, that of the core genome. Treating the core genome as a unit of selection solves problems concerning levels of selection in evolution.

1 Summary

Sociobiology is the study of biological interaction, both intragenomic, among loci in the genome, and intergenomic, among individuals in a reproductive popula- tion (Gardner et al. 2007). William Hamilton (1964) extended the theory of gene frequencies developed in the first half of the Twentieth century (Crow and

I would like to thank Samuel Bowles, Eric Charnov, Steven Frank, Michael Ghiselin, Peter Godfrey-Smith, David Haig, David Queller, Laurent Lehmann, Samir Okasha, Peter Richerson, Joan Roughgarden, Elliot Sober, David Van Dyken, Mattijs van Veelen and Edward O. Wilson for advice in preparing this paper.

1 Kimura 1970, B¨urger 2000, Provine 2001) to deal with such behavior. Hamil- ton’s rule has a simplicity that belies its considerable analytical power. But it is virtually powerless unless supplemented by other principles that are far less well understood. My purpose here is to delineate the proper place of Hamilton’s rule in sociobiological theory, and articulate these additional principles. I have kept the mathematical arguments self-contained and rather transparent, relegating equation-heavy arguments to Section 4 and Appendices A and B, all of which may be skipped without loss of continuity. There is only one new concept offered in this paper, that of the core genome. Treating the core genome as a replicator reconceptualizes the role of groups in evolution, and resolves chronic divisive issues concerning levels of selection. Using Hamilton’s rule, Alan Grafen has shown that under extremely general conditions, genes can be modeled as maximizers of inclusive fitness (Grafen 2000, 2006ab), which supports the concept of the selfish gene developed by George Williams (1966) and Richard Dawkins (1976). The simplicity of inclusivefitness theory is purchased at a price: it applies only to the behavior of alleles at a single locus, without the slightest influence from or upon, alleles at otherloci of thegenome. In essence, inclusive fitness theory models a direct relationship between a single genetic locus and the behavior of individual who carry the genome of which this locus is a tiny part. Because the various loci in the genome are highly interdependent, and social behavior generally involves the interaction of gene networks comprising many loci, inclusive fitness theory is not a full theory of gene frequency dynamics. Inclusive fitness theory would be dramatically strengthened if individualfitness were a weighted average of allele frequencies in the genome, for then maximiza- tion of inclusive fitness at the level of the single locus would extend directly to individual fitness maximization (Frank 1997, Grafen 2006a). However, it is well known that reproductive populations do not maximize fitness (Moran 1964, Akin 1982, 1987), and if individualfitness were a weightedaverage of allele frequencies, then Price’s equation could be used to show that populations do maximize fitness (Appendix B). Thus individual fitness cannot generally be treated as a weighted average of allele frequencies. It follows that the notion that inclusive fitness theory implies that individuals maximize inclusive fitness is incorrect. Inclusive fitness theory can be extended to complex social processes governed by networksof genes by employingthe so-called phenotypic gambit (Grafen 1984). The phenotypic gambit assumes that the behavior under study is governed by a single locus in the genome. With this assumption, Hamilton’s rule directly implies that behavior is the product of inclusive fitness maximization. More generally, total organismal fitness might be partitioned into several parts, each governed by distinct genetic networks,so that the phenotypicgambit could be applied separately in each

2 part, each covering a single behavioral complex, such as beak and wing shape, mating rituals, patterns of brood care, foraging strategies, predator avoidance, and territorial signaling. While the phenotypic gambit is extremely useful, it is, in our present state of knowledge, merely a heuristic device for generating plausible hypotheses. This ex- plains why the notion that individualsmaximize inclusive fitness is false in general but often offers powerful insight into social behavior. Yet it fails, for instance, in species that exhibit a social division of labor in which tasks are distributed across multiple participants (castes) with different inclusive fitness maximands.1 Without the phenotypic gambit, inclusive fitness theory has not the slightest power to account for the fact that complex metazoan species appear to be the prod- uct of design (Dawkins 1996); i.e., that the various loci in a successful organism generally cooperate harmoniously in enhancing the fitness of their carriers. Indeed the assumption of additivity across loci (frequency independence) that permits in- clusive fitness theory to extend beyond a single locus is precisely equivalent to the assumption that genes at distinct loci neither cooperate nor conflict. As we show below, the inclusive fitness conditions for success at a locus are distinct from the conditions for the success of its carriers and because genes are utterly selfish, each maximizes its inclusive fitness without regard for the fitness of its carriers. Although at times this leads to the fixation of deleterious mutations (Burt and Trivers 2006) and dysfunctional social norms (Edgerton 1992), for the most part such antisocial alleles are suppressed through the actions of regulatory networks of genes at other loci—phenomena that cannot be modeled within inclu- sive fitness theory (Burt and Trivers 2006, Ratnieks et al. 2006). If we add a Harmony Principle that asserts that organisms are evolutionarily successful to the extent that they suppress antisocial alleles, inclusive fitness theory becomes very strong. Much of the intuitive appeal of inclusive fitness theory lies in the tacit acceptance of the Harmony Principle. However, the mechanisms un- derlying the Harmony Principle are still largely unknown, and are likely be under- stood through bioengineering and social theory in addition to population biology (Maynard Smith and Szathm´ary 1997, Crespi 2001, Strassmann and Queller 2004, Queller and Strassmann 2009, Bowles and Gintis 2011, Wilson 2012). Recognizing that the genome as a whole is responsible for ensuring the co- operative interaction of genes in the genome as well as individuals in a social group leads us to reassess the role of the genome as a unit of selection. Richard

1The general equilibrium model of economic theory (Arrow and Hahn 1971) shows that indi- vidual utility maximization of all economic actors can be achieved when all actors face a common set of equilibrium prices, and with minimal assumptions, these common prices are dynamically sta- ble (Gintis and Mandel 2012). There is no known equivalent of a price system in other biological systems.

3 Dawkins’ (1976) replicator/vehicle distinction suggests that individuals, and a for- tiori groups, have no evolutionary permanence, but are merely carriers for the re- productive population’s genetic material. Individuals cannot adapt genetically be- cause they are destroyed by death and groups are destroyed by dissolution. Genetic material, however, can be faithfully reproduced across generations. Hence genetic material is subject to the evolutionary forces of reproduction, mutation, and se- lection (Lewontin 1974). Dawkins argues that the gene is the only replicator in diploid organisms, because larger units, such as chromosomes, are “torn apart” in each generation through meiosis and crossover. We suggest in Section 8, however, that a large fraction of the genome, which we call the core genome, has precisely such an identity across generations. The core genome is thus a replicator. The core genome codes for biochemical interactions among loci, as well as the characteristic environment of a social species and the characteristic social relations and signaling patterns among its individual members. For instance, in a territorial species, the behaviors signaling territorial boundaries and those underlying respect for such boundaries are coded in the species’ core genome (Gintis 2007). The is- sue of individual vs. group selection vanishesaccording to this conception, because both individuals and groups are phenotypic effects of the core genome. When the core genome codes for a subdivided population, and the result is evolutionarily successful, we can speak of group selection, with the understanding that selection is for a pattern of population subdivision, not selection among competing groups. Similarly, individual selection is not selection among individuals, but rather selec- tion at the level of the core genome for particular individual phenotypic traits. It is well known that whether one carries out the appropriate accounting re- lationship at the level of the gene, the individual carrier, or a social group, the answers must be the same (Lehmann et al. 2007, Queller 1992, Dugatkin and Reeve 1994, Sober and Wilson 1994, Kerr and Godfrey-Smith 2002, Wilson and Wilson 2007). The reason this is the case is that all coincide with accounting at the level of the core genome.

2 The Deceptive Simplicity of Hamilton’s Rule

Classical genetics did not model cases in which individuals sacrifice on behalf of non-offspring, such as sterile workers in an insect colony (Wheeler 1928), cooper- ative breeding in birds (Skutch 1961), and altruistic behavior towards stranger in humans (Darwin 1871). This problem was addressed by William Hamilton (1963, 1964ab, 1970, 1975), who noted that if a gene favorable to helping other indi- viduals is likely to be present in the recipient of a helping act, then the gene can increase its frequency in the population even if the helping act reduces the fitness

4 of its carrier. More precisely, Hamilton proposed the condition br >c (1) for the growthof the gene inthe population. The standardexplanation for (1) is that if a donor loses fitness c in helping a recipient who gains fitness b, the net effect on the allele for helping will be the probability r that the recipient has the focal allele, which is the degree of relatedness of the recipient to the donor, times the fitness benefit b, minus the fitness cost c. Hamilton called this inclusive fitness theory. Subsequent research supported many of Hamilton’s predictions (Maynard Smith and Ridpath 1972;Brown, 1974;West-Eberhard 1975; Trivers and Hare 1976). This standard justification of (1) is specious, however, as has been pointed out by many, including famously Sherwood Washburn (1978), who writes: All members of a species share more than 99% of their genes, so why shouldn’t selection favour universal altruism? (p. 415) The problem with the standard justification is that there is no functional rela- tionship between the probability that the recipient shares the focal allele with the donor and the degree of genetic relatedness r between donor and recipient. For instance, if the focal allele is very common in the population, the probability of the recipient having a copy of the allele will be close to unity, even if donor and recipient are unrelated. Expositorsoften add the qualification that the copy of the focal allele in the re- cipient must be identical by descent. However, the identity by descent qualification is irrelevant. Hamilton’srule deals with the current distribution of gene frequencies and the current fitness costs and benefits; how they came to be what they are does not enter into the analysis. Moreover, most identical alleles are identical by de- scent since the probability of two unrelated but identical alleles being maintained in the population is small (McElreath and Boyd 2006). Finally, the recipient allele need not be identical, by descent or otherwise. It must only produce the same pro- tein and RNA products, or otherwise have the same phenotypic effects as the focal allele. This shows that identity by descent is not important for Hamilton’s rule. Richard Dawkins (1979) treats Washburn’s criticism as the fifth of his Twelve Misunderstandings of Kin Selection, arguing that “this misconception arises not from Hamilton’s own mathematical formulation but from oversimplified secondary sources to which Washburn refers” (p. 191). Dawkins neither offers nor refers to an accessible derivation that avoids this and other common misinferences. Indeed, there is to my knowledge no clear exposition of Hamilton’s argument retaining his degree of generality. Yet this degree of generality must be maintained, and even expanded, if we are to evaluate properly the place of Hamilton’s rule in sociobio- logical theory. We offer such a derivation in the following two sections.

5 3 A Sociobiological Derivation of Hamilton’s Rule

We first derive Hamilton’s rule in the simpler case where the genome is haploid. The diploid version is developed in Section 4. Suppose there is a focal allele at a locus of the genome that leads a ‘donor’ carrier X to incur a fitness cost c while bestowing a fitness gain b distributed over a set of ‘recipients’ Y. Suppose also that the focal allele has a social fitness effect2 ˇ impacting on all the alleles at the focal locus, or equivalently, on the carrier (West et al. 2007). The case ˇ>0, which we call pollution, occurs in tragedy of the commons situations (Wenseleers and Ratnieks 2004), such as when the focal allele is an outlaw gene that depletes a protein used in chemical processes by somatic cells (Noble 2011). The case ˇ <0, which we call a public good, occurs in a promoter or suppressor gene, such in a parasite where the focal allele suppresses an allele at another locus that encourages such rapid growth that it kills its host prematurely (Frank 1996).3 Finally, suppose the focal allele steals an amount ˛  0 of fitness from other alleles at the focal locus, which it transfers to itself and the members of Y. We will call the thieving effect of the focal allele. Examples of this phenomenon include outlaw genes (Burt and Trivers 2006), cases in which a worker bee redirects brood- ing care from non-relative to relative larvae in an insect colony, and nepotism in human society. Hamilton (1964) did not consider this case. It is quite unrelated to the notion of a spiteful gene, where b;r < 0 (Bourke 2011). For a clear if oversimplified example, consider an obligately sterile worker in a colony who cares for larvae. Suppose there is a neutral allele at a focal locus controlling feeding that leads the worker to extend her services independent from the degree of relatedness between herself and the larva. Then consider an invading greenbeard allele that leads the worker to feed only larvae that have copies of itself. In this case c D 0 because the worker is sterile, and r D 1 because the recipients all have the greenbeard allele. Suppose the worker tends to m larvae and increases their fitness by a total amount f by her care, so each larvae receives f =m fitness increments from the neutral allele. Let q be the fraction of greenbeard alleles among the m larvae served by the worker. Then the greenbeard allele leads the worker to redirect ˛ D f.1 q/ fitness units from non-kin to kin. Now b D ˛, so Hamilton’s rule for the expansion of the thieving allele becomes ˛ > ˛, which is true.

2This is my term and notation. I develop Hamilton’s argument, with notation closer to his own, in Gintis (2013). 3The social fitness effect ˇ is missing in standard expositions of Hamilton’s rule. This omission is curious because, as we will see, the phenomenon is perhaps the most surprising and important implications of Hamilton’s (1964a) argument.

6 If the transfer of fitness is efficient, the growth of the thieving allele will not affect population fitness. However, it is found empirically that eusocial colony workers do not favor their own close kin (Nonacs 2011a), which suggests that worker kin recognition has been suppressed by genes at other loci, which suggests that favoring kin in this situationis costly to the colony. It iseasy to see that if only a fraction  of the fitness stolen from non-relative larvae is received by relative larvae, then the success of the greenbeard allele will reduce the fitness of colony members by ˛.1 q/. To return to the derivation of Hamilton’s rule, suppose the frequency of the focal allele in the population is q, and the mean frequency of copies of this allele inY is p.4 Then if the size of the population is n, which we assume is sufficiently large that we may safely replace frequency distributions over the population by their expected values, then there are qn donors, and population size n0 in the next period will be n0 D n.1 ˇq C q.b c//, so

n D n0 n D qn.b c ˇ/; (2) while the number of focal alleles will be qn.1 ˇq C pb c C ˛.1 q//. Thus we have qn.1 ˇq C pb c C ˛.1 q// q D q >0; (3) n C n which simplifies to p q b > c ˛: (4) 1 q  à This is the most general expression for Hamilton’s rule in the haploid case. We may refer to p as the degree of genetic assortment, between donor and recipient. When p > q, we will say that there is positive assortment in the population, and we call the reverse inequality negative assortment. We will term a focal allele altruistic if c>0 and self-serving if c<0. We will also term the allele prosocial if b c ˇ>0 and antisocial if b c ˇ<0. The reason for this designationis that (2) shows that the fitness of the carrier is enhanced by the success of the focal allele exactly when b c ˇ>0.5 The above derivation of Hamilton’s rule allows us to answer Washburn (1978). If the focal allele is fixed in the population, so p D q D 1, then (4) implies that q D 0 for any combination of signs of b and c. However, the focal allele is altruistic and satisfies Hamilton’s rule only if br>c>0. Thus the fixation of a gene does not imply that it is altruistic. Moreover (4) shows that we must measure

4By a copy of an allele I mean an allele that produces the same protein or RNA sequencesas the allele, or that otherwise produces the same phenotypic effects as the allele. 5Van Veelen (2009) and Bourke (2011) offer slightly different terminology.

7 the degree of assortment as a deviation from the population mean frequency of the allele, as suggested by Dawkins (1979). Thus however close the focal allele is to fixation (q  1), the allele need not be altruistic. Note that an altruistic allele with no social fitness effects (i.e., for which ˇ D 0) and that satisfies Hamilton’s rule (3) is prosocial if assortment is positive and antisocial if assortment is negative. To see this, assuming (4) and writing b.p q/=.1 q/ D c C d for some d >0, we have .c C d/.1 q/ c.1 p/ C d.1 q/ b c ˇ D c D : p q p q The final expression is clearly positive for positive assortment and negative for negative assortment, because the numerator is always positive (c.1p/;d.1q/ > 0).

3.1 Hamilton’s Rule and Genealogical Relatedness

What is the connection between our formulation of Hamilton’s rule (4) and the standard expression (1)? Suppose all individuals in Y have the same genealogical relatedness r to the donor. Then each recipient’s allele at the focal locus is identical to that of the donor by descent from a common ancestor with probability r, and otherwise is an allele randomly chosen from the population. For instance, suppose donor and recipient are full siblings in an outbred popu- lation, and the donor has the focal allele, which is inherited from a common parent. Then recipient has the focal allele identically by descent provided the recipient’s al- lele was also inherited from the same parent. This occurs with probability r D 1=2, assuming Mendelian segregation at the focal locus. If the recipient inherited the other parent’s gene at this locus, it will be the focal allele with probability q. This is an example of the general principle that, if the probability p that the re- cipient has the focal allele by common descent is r, and otherwise, with probability 1 r, the recipient has the focal allele with probability q, then

p D r C .1 r/q: (5)

Substituting this expression for p into (4), we get the expanded Hamilton’s rule

br >c ˛: (6)

It is permissible to use this form of Hamilton’s rule even in the non-genealogical case by simply defining r D .pq/=.1q/,solongas it alwayskeptinmind that r is now not a representation of relatedness, and can take any value from q=.1 q/ to 1.

8 More generally, and closely following Hamilton (1964), suppose there is an array fqj jj D 1;:::;kg of types of recipients, where recipients of type j have a copy of the focal allele with probability qj . Let pj be the probabilitythat the donor meets a recipient of type j , and let p0 D 1 j pj , which we interpret as the probability of the recipient being a random member of the population. Then we define P p D pj qj ; (7) Xj which is the expected probability of the recipient having the focal allele. We may then use this value of p in equation (4). In the special case that recipients of type j all have genealogical relatedness rj with the donor, and then any recipient has the focal allele with probability rj and with probability 1 rj has the focal allele with some probability sj , so qj D rj C.1rj /sj . If sj D q (i.e., there is no inbreedingin Y), then pj .rj C.1rj /q/ is the probability that the recipient is of type j and has the focal allele. We then have

p D pj .rj C .1 rj /q/ C p0q (8) j X D pj rj .1 q/ C q (9) Xj D r C .1 r/q; (10) as is required by the genealogical interpretation (5). This however, is an extremely special case, because the outbreeding assumption sj D q is not implied by the assumption of outbreeding for the whole population, and is likely to fail when the members of Y have some special social characteristics, such as being family, clan, or colony members. It is reasonable to call fqj ; pj g, where qj is the prevalence of the focal allele th in the j recipient type, and pj is the probability that the donor will interact with this recipient type, the social structure of the population. The numerical arrays fqj g and fpj g are not defined at the level of the focal locus, but at the level of the distri- bution of the genome in the population, which codes for how individuals seek out specific environments and how, within these environments, they associate assorta- tively, adopting particular mating patterns, embracing particular rituals and signals, favoring certain patterns of offspring care, and participating in certain forms of so- cial collaboration, in ways giving rise to a particular social structure. Inclusive fitness thus presupposes a general type of social dynamics. While the simple inequality br >c at first sight appears to connect genealog- ical relatedness, costs, and benefits at the level of a single locus, in fact a correct

9 derivation of the inequality reveals a complex social structure underlying each of the three terms.

3.2 Hamilton’sRule does notImplytheProsocialityof aSuccessful Allele

The change in populationsize induced by the focal allele is given by (2). Note that the social fitness effect ˇ does not occur in Hamilton’s rule, either in the general form (4) or the genealogical form (6), but it appears prominently in the expression for the fitness effect of the focal allele on the reproductive population. Hamilton (1964) calls ˇ a dilution effect precisely because it affects the rate but not the direction of change in the focal allele. The thieving effect ˛ does not appear in the population effect because it is assumed to be an efficient transfer among alleles. Because the social fitness effect for an allele has no bearing on the allele’s ca- pacity to evolve, even an altruistic allele that benefits relatives, and with positive assortment, can be successful according to Hamilton’s rule, and yet may be antiso- cial. This is because b;c >0 and b > c do not imply b c ˇ>0. In the same vein, a thieving allele (˛>0,ˇ D 0) that satisfies Hamilton’s generalized rule (6) can be antisocial (b < c) so longas b c C ˛>0. Thus inclusive fitness theory does not imply that successful alleles will tend to be prosocial. Consider the following supplementary principle: Harmony Principle: An evolutionarily successful genome suppresses most anti- social alleles, either by masking the products of an antisocial allele or promoting the fitness of a prosocial alternative to the antisocial allele. Helene Cronin (2007, p. 14–15) nicely expresses the Harmony Principle, writ- ing:

Among genes all is selfishness, every gene out for its own replication. But from conflict can come forth harmony; the very selfishness of genes can give rise to cooperation. For among the potential resources that genes can exploit is the potential for cooperation with other genes. And, if it pays to cooperate, natural selection will favour genes that do so.

Note that this assumption says nothing about the success of genomes that tol- erate or promote a high degree of conflict among individual carriers. Promoting conflict among carriers often affords the genome a considerable degree of evo- lutionarily success (e.g., lekking in birds), although unsuppressed intragenomic conflict appears usually to compromise organismal fitness (Burt and Trivers 2006). Similarly, the genome may promote conflict among the groups for which it codes,

10 as an instrument of the evolutionary success of the genome. Examples are wars and raiding between ant colonies or hunter-gather human groups. The Harmony Principle implies that negative assortment will be relatively rare, so Hamilton’s rule plus Harmony Principle imply that a successful allele will be prosocial. Moreover, in this case an altruistic allele is more likely to evolve the higher the degree of positive assortment. Thus we have Theorem 1. Assuming the Harmony Principle, Hamilton’s rule implies that a suc- cessful species will tend to exhibit positiveassortment,and successful alleles within such species will tend to be prosocial. The Harmony Principle explains the ubiquityof kin selection, because the mechan- ics of mating and offspring care often render kin recognition a reliable source of positive assortment. The Harmony Principle is one of the deepest principles of sociobiology, and one whose biochemical and social mechanisms are least well understood. We elab- orate upon this point below.

4 Hamilton’s Rule in the Diploid Case

This section presents a diploid version of the analysis. This is often termed the regression approach in the literature, but there is in fact no statistical estimation in- volved in the derivation (Michod and Hamilton 1980). It will be of interest mainly to population biologists. Consider a reproductivepopulation X with individuals fXi 2 Xji D 1;:::;ng. Suppose the genome has a diploid autosomal locus with two alleles, s (selfish) which leads to a behavior that does not affect the fitness of other individuals, and a (altruistic), which leads its carrier Xi to incur an increased fitness cost ci over that of the selfish allele, and to bestow fitness benefit bi distributed over a subset Yi of recipients. Suppose in addition that the altruistic allele has a social fitness effect ˇ (pollution when ˇ>0 or a public good when ˇ<0) on both alleles. This cost may be intragenomic, borne by the carrier, or intergenomic, distributed over the population in some arbitrary manner. Hamilton (1964a) assumes the social fitness effect is distributed uniformly over the genome. This is a significant limitation of his analysis because intragenomi- cally, meiotic drive and other forms of segregation distortion, and socially, altru- istic acts that are purchased in part by reducing the fitness of non-relatives, which we may call thieving effects, are of extreme importance, although the Harmony Principle suggests that natural selection will limit their observed frequency. We can represent these thieving effects as transfers of fitness ˛ from non-relatives to relatives.

11 Standard expositionsof Hamilton’s rule take Yi to be an individual. This, how- ever, is a restrictive assumption because in many social species individuals inter- act in groups where it is difficult to apportion the benefit bi among the various participants. Moreover, as we shall see, Hamilton’s rule does not depend on this assumption. i The genotypic value Xg of Xi at the focal locus, the frequency of the focal allele at this locus, is 0, 1/2, and 1 for genotypes ss, sa, and aa, respectively. The i phenotypic value Xp of Xi is 0, h, or 1 according as Xi is ss and never confers the benefit, is sa and confers the benefit with intensity h, or is aa and confers the benefit with intensity one. Here h can have any value, positive or negative, but if the allele effects are additive, then h D 1=2. Because there are 2n alleles at the i i focal locus in the population, the frequency of a is qa D i Xg =n. Let Yg be the mean genotype of members of Yi . Pi The fitness cost to Xi in the current period is thus ci Xp, and the fitness gain to i the recipients Yi is bi Xp . The population in the next period is then

n.1 ˇqa C .b c/xp/ (11)

i i where xp D i Xp =n is the mean phenotypeof the population, b D i bi Xp=xp is the mean benefit, and c D c X i =x is the mean cost. Note that because the P i i p p P thieving effect ˛ is a within-population fitness transfer, it does not appear in (11). The number of donor alleles inP the next period is

i i i i nqa.1 ˇqa C ˛.1 qa// C bi Xp Yg ci XpXg : i i X X The increase in the frequency of the donor allele in the next period, writing the y i mean genotype of recipients as qa D i Yg=n, is then given by P i i i i nqa.1 ˇqa C ˛.1 qa// C i bi XpYg i ci XpXg qa D n.1 ˇqa C .bP c/xp / P i i y bi X Y nbxpqa C nqa˛.1 qa/ i p g P n.1 ˇqa C .b
c/xp / i i y ci X X ncxp qa C nbxp.qa qa / i p g D P n.1 ˇqa C .b
c/xp/ b c y cov.X ; Yg/ cov.X ; Xg/ C ˛var.Xp/ bxp.qa qa / p p ; (12) 1 ˇqa C .b c/xp

b c i i where Xp and Xp are the variables bi Xp and ci Xp , respectively, and Xg is a binomial variable, so var.Xp/ D nqa.1 qa/. Note that the expression (12) is

12 positive, assuming weak selection, when

b y cov.Xp ; Yg/ C ˛var.Xp/ bxp.qa qa / c > 1: (13) cov.Xp ; Xg / This inequality is the most general form of Hamilton’s rule, including both social fitness and thieving effects. If we assume donors distribute benefits that are, on y average, independent from the allelic composition at the focal locus, i.e., qa D qa then (13) becomes

b c cov.Xp ; Yg/ C ˛var.Xp/ > cov.Xp ; Xg /: (14)

If we further assume that bi D b and ci D c for all individuals i D 1;:::;n, we get the expression:

b cov.X ; Y / C ˛ var.X / p g p > c: (15) cov.Xp; Xg/

Finally, if the effect of the altruisticallele is additive, so h D 1=2, then (15) becomes cov.X ; Y / b p g > c ˛: (16) var.Xg / This is a standard expression for Hamilton’s rule (Michod and Hamilton 1980), except we have taken into account the thieving effect ˛ (and the pollution/public good effect ˇ, which does not appear in Hamilton’s rule). More generally, for arbitrary h, we have br >crp ˛; (17) where cov.X ; Y / r D p g var.Xg / p is the regression coefficient of Yg on Xp , and r is the regression coefficient of Xp on Xg : cov.X ; X / rp D p g : var.Xg / It should be clear that, while we use mathematical terminology from statistical estimation theory, no statistical estimation is in fact involved. To illustrate the increased generality of the form (14) of Hamilton’s rule, sup- pose the reproductive population is partitioned into social castes fZj  Xjj D 1;:::;mg, where caste j has frequency zj in the population,and suppose members j of the same caste j have the same costs cj and benefits bj . Let Y be the weighted

13 j sum of fYi jXi 2 Z g, where each individual is weighted by the number of times the individual appears in the sum. Then we can write (14) as

m j j j j cj .bj cov.Zp ; Yg / cj cov.Zp ;Zg/ C ˛ var.Xp/>0: (18) D jX1  Á Equation (18) shows that in general the social structure of the population allows a caste to be fundamentally altruistic in the sense that its net costs of helping exceed the net benefits that the caste contributes to the population. Because the inclusive fitness of caste j is

j j j j bj cov.Zp ; Yg / cj cov.Zp ;Zg/<0 (19) it is then clear that caste j members would maximize their inclusive fitness by simply refusing to contribute to the social process. This shows that in a caste social structure, individuals do not maximize their inclusive fitness. Of course, if castes are genetically determined, then the partition fzj jj D 1;:::;mg will be variable across periods and a fundamentally altruistic caste will become extinct in the long run. However, if castes are determined by developmental conditions (e.g., feeding in eusocial insects or socialization in humans), fundamentally altruistic castes can be maintained in the long run.

4.1 The Sociobiological Dynamics of Hamilton’s Rule

The mapping Xi ! Yi , which we have taken as given, reflects the social structure of the reproductive population. This mapping does not presume any particular set of social relations of kinship, which is why we suggest that kin selection is in general an inappropriate description of inclusive fitness dynamics. Note that if the frequency of the a allele in the population does not affect the fitnesses of alleles at other loci in the genome, then the a allele will move to fixation in the population if Hamilton’s rule is satisfied, and will become extinct if the reverse inequality is satisfied. Ultimately, the focal locus will be heterozygous with zero probability. With frequency dependence, when the focal allele becomes prevalent in the population, if b c>0, so the allele is beneficial to its carriers, there will be no selection at the level of the genome for genes that suppress the a allele at the focal locus, so the a allele will still move to fixation in the population. When the focal allele is prevalent and b c <0, there will be natural selection at other loci for genes that either alter the sociobiological mapping Xi ! Yi or otherwise suppress the a allele at the focal locus, so that Hamilton’s rule no longer holds for the antisocial allele. This is the essence of the Harmony Principle (Section 3).

14 Of course there may be no likely mutation that suppresses an anti-social a allele, in which case the antisociality reflected in the behavior induced by the a allele will become ubiquitous in the population. natural selection does not guarantee optimality. This phenomenon also represents a plausible counterexample to Fisher’s Fun- damental Theorem (Ewens 1969, Price 1972, Frank and Slatkin 1992, Edwards 1994, Frank 1997): as an antisocial allele moves to fixation, the average fitness of population members declines. Some population biologists save Fisher’s theo- rem by calling this a transmission effect, and insisting that natural selection always produces fitness-enhancing gene frequency changes (Edwards 1994, Frank 1997, Gardner et al. 2011). This interpretation of natural selection should be avoided because it is arbitrary and difficult to understand for those who are not experts in population biology. It follows that Hamilton’s rule is useful only in charting short-term genetic dy- namics. Weak selection and additivity across loci are extremely powerful analytical tools, but in the long run changes in gene frequency at one locus are likely to in- duce compensatory and synergistic changes at other loci. Indeed, the very mapping Xi ! Yi on which Hamilton’s rule is based is itself coded in the core genome of the reproductive population, and hence in the long run is modified in the course of evolutionary selection and adaptation.

4.2 Altruism Among Relatives

A relative is a person “allied by blood...a kinsman” (Biology Online). The ar- gument to this point has nothing to do with genealogy, and hence says nothing about altruism among family members. This is an attractive property of our ex- position because in a highly social species, individuals interact frequently with non-relatives. It remains to determine the exact relationship between the sociobiological con- ception (15) and the genealogical conception of relatedness. We follow Michod and Hamilton (1980), except that we assume the population is outbred at the focal locus. Suppose that each Yi is an individual recipient, and all recipients have the same genealogical relationship to their donors (e.g., Yi is a sibling of Xi ). Let pxyzw be the joint distribution of genotypes xy for donor and zw for recipient x x x where x;y;z;w 2 fs; ag. Let pss, pas, and paa be the marginal distributionof the genotypes ss, sa, and aa for the donor (i.e., the fraction of these genotypes in the y y y population), and similarly for pss, pas, and paa for the recipient. We have x x xp D hpas C paa;

15 y y yp D hpas C paa; x because pas is the fraction of sa genotypes, their phenotypic value is h, and paa is the fraction of aa genotypes, which have phenotypic value one. Also,

x pas D2qnqa (20) x 2 paa Dqa (21)

To derive (20), note that either the paternal allele is s with probability qn D 1 qa and the second is a with probability qa, or else the paternal allele is a with probability qa and the second is s with probability qn. The second equation is derived in a similar manner. We thus have

2 xp D 2hqnqa C qa (22) 2 yp D 2hqnqa C qa (23)

Note that

1 x x xg D =2pas C paa D qa 1 y y yg D =2pas C paa D qa:

To derive cov.Xg ; Xp/, note that

i i x x XpXg =n D hpas =2 C paa i X 2 D hqnqa C qa

x x Given the values of pas and paa from equations (20) and (21), and after alge- braic simplification, we find

cov.Xp ; Xg/ D qnqa˛=2; (24) where ˛ D 2.h C qa.1 2h/: (25) Also,

cov.yg xp/ D hpsasa=2 C hpsaaa C paasa=2 C paaaa yg xp:

Now let p11 be the probability Xi and Yi share both alleles at the focal locus identically by descent, let p10 be the probability the share one allele at the focal

16 locus identically by descent, and let p00 be the probability they share neither allele identically by descent. then we have

2 2 pasas D2qnqap11 C qnqap10 C 4qnqap00 (26) 2 3 pasaa Dqaqnp10 C 2qnqap00 (27) 2 3 paaas Dqnqap10 C 2qnqap00 (28) 2 3 4 paaaa Dqap11 C qap10 C qap00: (29)

If we define fXY as the probability that a random allele in Xi and a random allele in Yi are identical by descent, then

fXY D p11=2 C p10=4: (30) Then a little algebra shows that the r in Hamilton’s rule is given by

cov.XpYg / r D D 2fXY : (31) cov.XpXg / Note that r is then the expected number of copies of the focal allele in the recipient. Consider, for instance, the case of siblings. The two share the same allele from the father with probability 1/2, and similarly for the mother. therefore p11 D 1=4, p10 D 1=2, and p00 D 1=4. Substitutingthese values in (26), we get cov.Y ; X / 1 r D g p D : (32) cov.Xg ; Xp/ 2 Thus the sociobiological definition of relatedness and the genealogical definition coincide.

4.3 The Haploid Form of Sociobiological Relatedness

We now show that the fraction in haploid expression of Hamilton’s rule (4) is pre- cisely the sociobiologicaldefinition of relatedness. In this case, s is the selfish gene and a is the altruistic gene at the focal locus. The variance of Xg is now

var.Xg / D qnqa and 2 cov.YgXp / D pqa qa so cov.Y X / p q g p D a ; var.Xg / 1 qa which is equivalent to (4).

17 5 The Sociobiology of Hamilton’s Rule: An Example

The point of this example is to show that Hamilton’srule in principle has no neces- sary relationship with genealogy or kin selection, but rather is an expression of the social structure of the reproductive population. This model is based on Hamilton (1975), where the author develops a similar model for the same purpose. Consider a population in which groups of size n form in each period. In each group individuals can cooperate by incurring a fitness cost c>0 that bestows a fitness gain b that is shared equally among all group members. Individuals who do not cooperate (defectors) receive the same share of the benefit as cooperators, but do not pay the cost c and do not generate the benefit b. Let pcc be the expected fraction of cooperating neighbors in a group if a focal individual is a cooperator, and let pcd be the expected fraction of cooperating neighbors if the focal individual is a defector. Then the payoff to a cooperator is c D bpcc c, and the payoff to a defector is d D bpcd . The condition for the cooperative allele to spread is then c d D b.pcc pcd / c>0, or b.pcc pcd />c: (33)

Now pcc is the probability that a cooperator will meet another cooperator in a ran- dom interaction in a group, so we can define the relatedness r between individuals, following (5), by pcc D r C .1 r/q; (34) where q is the mean frequency of cooperation in the population. If we write pdd D 1pcd for the probability that a defector meets another defector, then we similarly can write pdd D r C .1 r/.1 q/; (35) since 1 q is the frequency of defectors in the population. Then we have

pcc pcd D r C .1 r/q .1 .r C .1 r/.1 q/// (36) D r: (37)

Substituting in (33), we recover Hamilton’s rule, br >c. Of course, if group formation is random, then pcc D pcd so r D 0 and Hamil- ton’s Rule cannot hold. However, to illustrate the importance of social structure, suppose each group is formed by k randomly chosen individuals who then each raises a family of n=k clones of itself. We need not assume parents interact with their offspring, or that siblings interact preferentially with each other. There is no kin selection.

18 At maturity, the parents die and the resulting n individuals interact, but do not recognize kin. In this case a cooperator surely has k 1 other cooperators (his sibs) in his group, and the other n k individuals are cooperators with probability q. Thus k 1 n k k.1 q/ 1 p D C q D q C : cc n n n Similar reasoning, replacing q by 1 q gives

kq 1 p D 1 q C : dd n Then k 2 r D p p D p 1 C p D ; cc cd cc dd n so Hamilton’s rule will hold when k 2 br D b > c: n  à Note that the related recipients are all clones of the donor, with relatedness unity, although the r in Hamilton’s rule is .k 2/=n. The inclusive fitness inequality is accurate here, but kin selection as defined above is inoperative in this model: the altruistic behavior is more likely to spread when the number of families n=k in a group is small. This model suggests that the interesting question from the point of view of sociobiology is how the genome of the species manages to induce individuals to aggregate in groups of size n and to limit family size to n=k, so that the benefits of cooperation (b c) can accrue to the population. This is a true miracle of Nature.

6 The Selfish Gene

A central insight of inclusive fitness theory was stated by George Williams (1966) and developed in Dawkins’ (1976) selfish gene theory, where the author observes:

Any gene that behaves in such a way as to increase its own survival chances in the gene pool at the expense of its alleles will... tend to survive.

Alan Grafen has strengthened this asserting by showing that, under extremely gen- eral conditions, genes can be modeled as maximizers of inclusive fitness (Grafen 2002, 2006a, 2006b), Genes, then, are utterly selfish and do not reflect the interests of their carriers.

19 The Selfish Gene: Theallelesat a geneticlocusare selfishinthesense that the con- ditions for their evolutionary success are generically distinct from the conditions for the evolutionary success of alleles at every other locus and for the evolutionary success of their carriers. Note that I do not include in this definition the notion that the gene is the only, or the only important, replicator or unit of selection.

7 The Phenotypic Gambit

The genome of a multicellular organism includes a myriad of interdependent gene networks. The phenotypic gambit is a procedure that permits modeling a behavior that may be extremely complex at the genetic level in a straightforward manner. In the words of Alan Grafen (1984),

The phenotypic gambit is to examine the evolutionary basis of a char- acter as if the very simplest genetic system controlled it: as if there were a haploid locus at which each distinct strategy was represented by a distinct allele, as if the payoff rule gave the number of offspring for each allele, and as if enough mutation occurred to allow each strat- egy the chance to invade.

Adoptingthe phenotypicgambit vastly increases the power of Hamilton’s rule, pro- viding a direct analytical relationship between biochemical processes at the level of a single genetic locus and macro-level social behavior. The analytical justification of the phenotypicgambit is, to my knowledge, unknown, but it often produces use- ful results that are empirically supported. Often, however, it does not. We therefore must treat the phenotypic gambit with a degree circumspection. I conjecture that the phenotypic gambit is rigorously justified when behavior is governed by several interacting loci precisely when the interactions are additive, but I have checked this out only for two loci.

8 The Core Genome as Replicator

In exploring the implicationsof the selfish gene concept, Dawkins (1976) proposed the replicator/distinction. A replicator for Dawkins is a biological entity that, ab- stracting from mutation and selection, induces biochemical processes leading to the production of accurate copies of itself. A vehicle, or in David Hull’s (1980) terminology, an interactor, by contrast, is an ephemeral carrier of replicators that is destroyed and replaced in each generation by new vehicles. Population biology

20 suggests that genes are replicators, while organisms and social groups are vehi- cles. Evolution (replication, mutation, selection) operates on replicators, not on vehicles. The logic of the replicator/vehicle distinction is persuasive. However, Richard Dawkins, following George Williams (1966), uses the distinction to conclude, in- correctly, that in species that reproduce through meiosis and recombination, the genome itself is a purely temporary association of genes, and thus that only the gene is the appropriate replicator (Dawkins 1976, 1982a). Dawkins argues that the genome, like its vehicles, is torn apart and reassembled in each generation. He observes (Dawkins 1982b, p. 47) that a replicator must have a

low rate of spontaneous, endogenous change, if the selective advan- tage of its phenotypic effects is to have any significant evolutionary effect.

However, he asserts, incorrectly, that a long piece of chromosome

will quantitatively disqualify itself as a potential unit of selection, since it will run too high a risk of being split by crossing over in any generation.

In fact, the set of genes that are fixed in the genome of a reproductive popula- tion are preserved in just the same sense as the individual gene. Indeed, the set of fixed loci are more stable over time than the individual gene because the latter can be broken up by crossover at heterozygous loci, and both the gene and the set of fixed genes are subject to mutations that reduce their accurate reproduction across time. The recombination rate, which “tears apart” the genome, is typically of the same order of magnitude as the mutation rate, which “tears apart” the gene (Rosenberg and Nordborg 2002). The central position of gene regulatory networks in promoting intragenomic cooperation in virtually all living creatures and interge- nomic cooperation (interactions among individuals) in social species (Johnson and Linksvayer 2010, Linksvayer et al. 2011) would thus be impossibleif the members of such networks were not substantively fixed in the genome. Inadditiontothe fixed loci inthegenome of a reproductivepopulation, synony- mous alleles (alleles whose differences are base substitutions that entail identical protein, enzyme, and regulatory products) can be identified as a fixed synonymous set that changes composition across generations but does not alter phenotypic ef- fects. Moreover, non-synonymous alleles that have fitness neutral, or near-neutral, phenotypic effects (e.g., tail length or eye color), can be identified as fixed fitness neutral sets that are highly stable across generations despite their somewhat la- bile internal composition. For instance, body size may be fitness independent over

21 some range, and many genes interact to produce a phenotypic body size that is generally in the fitness-neutral range. The frequency distribution of these genes in the core genome is determined by natural selection and unchanged by meiosis and crossover. In addition, if the set of alleles at a particular locus have equal fitness but have distinct phenotypic effects, and if this set is preserved across generations, the alleles are likely to be alternative strategies in a Nash equilibrium among loci, each being a fitness enhancing best response to the probability distribution of the other loci in the genome. For example, a population equilibrium can sustain a positive fraction of altruistic and selfish alleles, or alleles promoting aggressive vs. docile behavior, under certain conditions. Similarly, loci that protect carriers against frequency-dependent variations in environmental conditions, including that of bacterial and viral enemies, can be maintained in a polyallelic state as a means of species-level risk reduction. Another example is the interaction of suppressor genes and their targets, where the fitness of the suppressor depends on a positive frequency of target genes. We can call such sets of complex alternative alleles mixed strategy sets. Leffler (2013) document such a set stabilized by balancing selection at least since the primate-hominin split. Finally, heterozygote advantage involves a pair of alleles that maintain positive frequency despite the fitness cost to homozygous carriers. We may call this an overdominance set. Additional fea- tures arise in dealing with sex-linked genes, including maternal-paternal conflict, but these also can be identified as characteristics of the species that are conserved across many generations. Core Genome: The core genome of a sexually reproducing reproductive popula- tion is the union of its fixed loci, its fixed synonymous sets of loci, its fixed fitness- neutral sets, its mixed strategy sets, its overdominance sets, and its characteristic sex-linked gene complements. The core genome is reproduced across generations in the same sense as the gene, and has the added feature of representing the essential phenotypic character- istics of a reproductive population. The core genome is subject to selection through the usual mechanisms of gene duplication and inversion, as well as the emergence of a mutant gene that replaces a fixed gene in the core genome. In a reproduc- tive population in genetic equilibrium, it is plausible to posit that the core genome will comprise virtually the whole genome because if alternative alleles persist at a locus, all must have equal fitness, and hence fit into one of the above categories (Grafen 2006a). The notion of the core genome as replicator is supported by gene sequencing data that show that all but a small fraction of genes in a complex metazoan organ- ism are shared by all individuals (Frazer and Elnitski et al. 2003). The core genome may include alternative alleles with virtually identical fitnesses, but that express

22 phenotypic features of sociobiological importance, such as racial and ethnic fea- tures in humans. However, the typical phenotypic characteristics of the species, including biochemistry, physiology, independent assortment of chromosomes in meiosis and social behavior, are conserved despite crossover and meiosis. The core genome is thus a replicator subject to the laws of natural selec- tion. A more nuanced conception of the core genome would include, in addi- tion to germline DNA sequences, cytoplasmic genetic material and even non-living and inorganic information forms that are reliably subject to mutation, replication, and selection. Individuals, social groups, as well as their extended phenotypes (Dawkins 1982a), constructed niches (Odling-Smee et al. 2003), and culture in hu- mans (Lumsden and Wilson 1981, Cavalli-Sforza and Feldman 1981, Boyd and Richerson 1985, Gintis 2011, Bowles and Gintis 2011) are vehicles for the core genome of a reproductive population of complex multicellular organisms.

9 Levels of Selection

The core genome is Ernst Mayr’s (1963), species concept and TheodosiusDobzhan- sky’s (1953) collective genotype concept translated from systematics to population genetics. Sociobiology is best viewed as beginning with Hamilton’s rule and the core genome and dealing with interaction among alleles at a locus, among loci in the genome, among individuals in a species, and among species in an ecological setting. The defining characteristics of a species include not only the physiology of zygote formation and the strategic interaction in mating, but all key social inter- actions that comprise the life history of the organism. All such characteristics are coded in the core genome, despite the fact that every allele at every locus is utterly selfish, blind to the interests and needs of the genome. The core genome codes for kin selection in many species, including charac- teristic patterns of mating, offspring care, and social interactions among close ge- nealogical kin. As we have seen, when the core genome codes for complex social interaction within small subsets of the population, we may speak of group selec- tion. Group Selection: Group selection occurs when the core genome of a reproductive population specifies a typical social structure in which individuals aggregate into many small groups in which within-group social interactions are high and across- group interactions are relatively low. The concept of core genome as replicator clears up the confusion surrounding group vs. individual selection. Neither the individual nor the group is a unit of selection, so there can be no question as to their relative value as units of selec- tion. Nor is there any gain to be had by evaluatingthe relative importance of genes

23 and core genomes as units of selection: sociobiology is the study of the interac- tion between utterly selfish genes (Dawkins 1976) and the core genome, with its ubiquitous “parliament of genes” (Leigh 1977) as units of selection. Group selection does not necessarily involve competition among groups or group extinctions. Group selection is selection for groups, not selection among groups. By this I mean that organisms that aggregate into large numbers of small groups with high frequencies of within-group interactions and low frequency of across-group interactions evolve when the fitness of the core genome is enhanced by this method of facilitating social cooperation. Exactly how this works on bio- chemical and socio-structural levels is of extreme importance, but does not involve units of selection beyond the gene and the core genome. For instance, many thrips species form galls in which they lay eggs, providing a social structure inhabited for a time by the next generation of thrips (Crespi et al. 1997). What we must under- stand is the evolutionary advantage of gall formation. Competition among galls is simply adaptation at the level of the core genome. Where the core genomes in several species jointly code for interspecific inter- actions, we can speak of ecological selection. Examples include predator-prey dynamics, inter-specific cooperation, and the dynamics of persistent ecological systems. In ecological systems, communities evolve as species interact via pre- dation, competition, synergistic exchange, and mutualism. A species that can sur- vive in many communities is all else equal more likely to speciate than one that can live in few communities. Depending on the relative time scales, evolution can be dominated by community evolution as much as or even more than speci- ation (Johnson and Stinchcombe 2007). For instance, species that have the ca- pacity to survive in many different communities may survive under conditions of rapid climatic change, when community composition changes faster than specia- tion (Coope 1994).

10 The Inclusive Fitness Debate

A half century has passed since the clash between population geneticist J. B. S. Haldane and systematist Ernst Mayr over “beanbag genetics” (Mayr 1963, Haldane 2008[1964], Rao and Nanjundiah 2011). Never resolved, popularized in the 1990’s debate involving Richard Dawkins, John Maynard Smith, Daniel Dennett, and Stephen Jay Gould (Dawkins 1985, Maynard Smith 1995, Dennett 1996, Gould 1997, Sterelny 2003), this dispute recently exploded with Martin Nowak, Carina Tarnita, and Edward O. Wilson’s (2010) attack on inclusive fitness theory. Many prominent evolutionary biologists counterattacked with scathing reviews and let- ters to the journal Nature, one bearing 154 signatures (Abbot et al. 2011, Boomsma

24 et al. 2011, Strassmann et al. 2011, Rousset and Lion 2011, Dawkins 2012). There are no apparent methodological differences between the two sides. Both engage in mathematical model-building, both accept the same basic principles of population biology, and both include talented naturalists with a strong dedication to the empirical substantiation of their theories. Moreover, there appears to be no empirical evidence that would resolve the dispute. Indeed, most participants have asserted, at one time or another, that the differences are a matter of taste. Some may stress that Hamilton’s rule gives the right answer, while others stress that is a case of group selection in which cooperators are handicapped by contrast with defectors in every group, but groups with many cooperators do better than groups with few cooperators. Price’s equation may be deployed to this end (Price 1970, Hamilton 1975, Frank 1997), but the resulting calculations cannot, in this case or any other of which I am aware, conflict with the simple argument based on the distribution of gene frequencies in the population (Wade 1978, Queller 1992, Dugatkin and Reeve 1994, Sober and Wilson 1994, Kerr and Godfrey-Smith 2002, Lehmann et al. 2007, Wilson and Wilson 2007). One consistent distinction characterizes the two sides in the dispute. Critics of kin selection tend to specialize in the study of highly complex social species, such as eusocial insects and Homo sapiens, where the key to organismal success depends on genes and social organization suppressing the favoring of kin in coop- erative interactions. Examples are the lack of kin recognition in insects that prac- tice cooperative rearing (Nonacs 2011b), and the widespread cooperation among strangers and the suppression of nepotism in humans (Bowles and Gintis 2011). The following are suggestions for a harmonization of perspectives.

10.1 The Meaning of ‘Kin Selection’

William Hamilton’s early work in inclusive fitness focused on the role of genealog- ical kinship in promoting prosocial behavior. Hamilton writes, in his first full pre- sentation of inclusive fitness theory (Hamilton 1964, p. 19): In the hope that it may provide a useful summary, we therefore hazard the following generalized unrigorous statement of the main principle that has emerged from the model. The social behaviour of a species evolves in such a way that in each distinct behaviour-evoking situation the individual will seem to value his neighbours’ fitness against his own according to the coefficients of relationship appropriate to that situation. Because of this close association between inclusive fitness and the social relations among genealogical relatives, John Maynard Smith (1964) called Hamilton’s the-

25 ory kin selection, by which he meant that individuals are predisposed to sacrifice on behalf of highly related family members. Hamilton’s bold generalization is as interesting when it fails as when it suc- ceeds. It fails, for instance, in fair meiosis in diploid organisms, where an al- lele does not distort segregation by favoring copies of itself. This occurs not by biochemical necessity but rather through the suppression of favoritism at regula- tory loci (Nur and Brett 1985, Haig and Grafen 1991, Burt and Trivers 2006). In addition, social cooperation in many species is effective because it involves the suppression of behavior that favors genealogical relatives, such as the lack of ge- netic kin recognition in multiply-mated eusocial colonies (Queller and Strassmann 1998, Nonacs 2011a), and the tempering of kin-motivated altruism by commitment to larger social goals and public morality in humans (Boyd et al. 2010, Bowles and Gintis 2011). I venture an equally speculative generalization: Hamilton’s descrip- tion is progressively less accurate as the social complexity of the species increases. A decade after Hamilton’s seminal inclusive fitness papers, motivated by new empirical evidence and Price’s equation (Price 1970), Hamilton (1975, p. 337) revised his views, writing: Kinship should be considered just one way of getting positive regres- sion of genotype...the inclusive fitness concept is more general than kin selection. Hamilton illustrates his new position with a model similar to that of Section 5 of this paper. Modern inclusive fitness theory has adopted the Hamilton-Price regression in- terpretation of Hamilton’s rule (Queller 1992, Frank 1997, Section 4) but popular expositions continue to equate inclusive fitness with kin selection. Thus West et al. (2011) write: Hamilton’s inclusive fitness (kin selection) theory explains how altru- istic cooperation can be favoured between relatives. (p. 237)

A similar identificationis prevalent in the technicalliterature. For instance, through- out his authoritative presentation of sexual allocation theory, Stuart West (2009), identifies inclusive fitness with kin selection in several places and never distin- guishes between the two terms at any point in the book. This idiosyncratic identification is a source of confusion, because for most so- ciobiologists, kin selection remains, as conceived by Maynard Smith (1964), a social dynamic based on close genealogical association:

By kin selection I mean the evolution of characteristics which favour the survival of close relatives of the affected individual.

26 The Wikipedia definition is similar:

Kin selection is the evolutionary strategy that favours the reproductive success of an organism’s relatives, even at a cost to the organism’s own survival and reproduction....Kin selection is an instance of inclusive fitness.

Much confusion would be avoided if kin selection were reserved for the socio- biological notion of behavior that is fitness-enhancing due to differential associa- tion based on the recognition of kin through social nature of family life: mating, care of offspring, and privileged interactions based on genealogy. For a typical example of the confusion fostered by the many proponents of inclusive fitness theory who equate inclusive fitness and kin selection, consider Nowak et al. (2010, p. 1060):

Inclusive fitness theorists have pointed to resulting close pedigree re- latedness as evidence for the key role of kin selection in the origin of eusociality, but as argued here and elsewhere, relatedness is bet- ter explained as the consequence rather than the cause of eusociality. Grouping by family can hasten the spread of eusocial alleles, but it is not a causative agent.

Inclusive fitness theory, however, does not assert that kinship is a necessary causal agent in all social behavior. The Nowak et al. (2010)assessment of inclusivefitness is properly a critique of kin selection as an adequate explanation of eusociality. This confusion is, however, understandable given the tendency of many population biologists to equate the two theories.

10.2 Maximization of Inclusive Fitness

Inclusive fitness theory does not imply that either individuals or groups maximize inclusive fitness. Hamilton’s rule depends on weak selection, in the sense that over a single period, the frequency distribution of alleles at other loci that affect the focal allele’s success, b,c,r,˛, and ˇ in our models, do not change in response to change in the frequency of alleles at thefocal locus.Thisweak selection assumption is reasonable, but in tracking the long-run history of alleles at the focal locus, even abstracting from mutation, duplication, translocation and the like, interaction with other loci cannot be assumed away. In the long run, alleles at each locus evolve by reacting selfishly to the current distribution of alleles at all other loci. This type of complex mutual interaction has been studied in a two locus setting (Moran 1964, Akin 1982). Akin (1982) shows that interaction between loci can generate

27 cyclical orbits of the sort encountered in predator-prey models (May 1974), which precludes alleles increasing in frequency by maximizing inclusive fitness (or any other global index of fitness). A typical affirmation of inclusive fitness maximization in this larger sense is that of West et al. (2011), who write: Inclusive fitness is...our modern interpretation of Darwinian fitness in its most general form, explaining both the process and purpose of adaptation (Grafen, 2007b, 2009). The process is that genes or traits which lead to an increase in inclusive fitness will be favoured...The purposeis that individualsshould appear as if they have been designed to maximize their inclusive fitness. However, both the key strength and key limitation of inclusive fitness theory are that the theory analyzes the dynamics of individual loci. The notion that genes that enhance the fitness of their carriers will tend to proliferate is exactly the Harmony Principle. To claim that this principle follows from an analysis of the dynamics at single genetic loci is incorrect, and statements to the contrary, as in the case of West et al. (2011), are bound to confuse readers not versed in the technical litera- ture, inducing them to infer that inclusive fitness theory must be a more ambitious doctrine than it actually is. Moreover, the phrase “individuals should appear as if they have been designed to maximize their inclusive fitness” is inaccurate. Its most straightforward inter- pretation is that individualsappear optimally designed to help themselves and their close genealogical kin, which is true in many species, but false in complex social organisms, including eusocial insects and Homo sapiens. For instance, in the research on cooperation in humans, my colleagues and I account for altruism assumingthat individualsmaximize inclusivefitness, applying the phenotypic gambit, but individuals in multi-family groups do not appear to be designedfor this purposeat all (Gintiset al. 2005,Bowles and Gintis2011). Rather, they appear to be designed to help unrelated others and to make personal sacrifices on behalf of individuals in cooperative groups of unrelated individuals. Even given an extended notion of inclusive fitness at the level of carriers, only loosely and impressionistically related to the notion as developed in Hamilton’s rule, and even given the sociobiological definition of relatedness, the conditions under which individuals maximize inclusive fitness are very narrow. West et al. (2011) write: Grafen. . . [has shown] the mathematical equivalence between the dy- namics of gene frequency change and the purpose represented by an optimisationprogram which uses an “individualas maximising agent” analogy.

28 Grafen’s analysis is indeed remarkable (Grafen 1999, 2002, 2006a), but it as- sumes the absence of frequency dependence even in his most general formula- tion (see also Metz et al. 2008, Gardner and Wild 2011, Gardner, West and Wild 2011). Nor is this issue simply a formality. With frequency dependence, as we have seen, we expect individuals to engage in complex strategic interac- tions involving the adoption of mutual best responses leading to a Nash, or more likely, a correlated equilibrium (Aumann 1987, Maynard Smith 1982, Hammer- stein and Reichert 1988, Taylor 1989, Weibull 1995, Nowak 2006, Traulsen and Nowak 2006, Gintis 2009). Moreover, the equilibriumcan be neutrally stable, sup- porting cyclical behavior in which no individual characteristic is determined by fitness maximization (Akin 1982, Vega-Redondo and Hasson 1993).

10.3 Inclusive Fitness and the Social Division of Labor

In a species in which cooperation is organized into morphological or socially strat- ified castes, the conditions of fitness maximization of members of one caste must generally be purchased by a reduction in the inclusive fitness of other castes (e.g. queens and workers in a eusocial species, or males and females in dioecious species). Hence the maximization of inclusive fitness as a principle for explaining behavior must logically and necessarily fail. Nevertheless West et al. (2011) write:

The traditional Darwinian view struggled to explain many coopera- tive social behaviours, with the most famous example being the ster- ile worker caste in eusocial insect species, the ants, bees, wasps, and termites. . . Hamilton (1964) incorporated indirect fitness effects into a genetical theory of social evolution and showed that the characters favoured by natural selection are those which improve the individual’s “inclusive fitness,” which is the sum of its direct and indirect fitness.

But Hamilton (1964) did not show that “the characters favoured by natural selection are those which improve the individual’s inclusive fitness,” and did not even claim to. In his paper, Hamilton (1964a) states (p. 8)

For an important class of genetic effects where the individual is sup- posed to dispense benefits to his neighbours, we have formally proved that the average inclusive fitness in the population will always in- crease. (p. 8)

Of course for another class of genetic effects, as Hamilton’s stresses and his calcu- lations clearly show, average fitness does not increase. The former class is indeed

29 much more “important” than the latter, but only by virtue of the Harmony Princi- ple.6 Similarly, despite innumerable popular accounts, inclusive fitness theory does not show that workers of a eusocial species maximize inclusive fitness, and indeed there is no evidence that they do. In Appendix A we derive the equations for sex allocation in a eusocial colony based on genetics, assuming that either the queen or the workers control the sex ratio, and show that it is not the same as the result using an inclusive fitness argument. However, the differences are not large, so the inclusive fitness approximation is surprisingly accurate. Empirical studies suggest that sex allocation in such colonies in general conforms to neither queen nor worker control, and we currently lack analytical model of the relative power of queen and workers, or of sex ratios given the relative power of the two. Let us call a colony advanced eusocial if, in a queenright colony, the workers lack functional ovaries, or worker reproduction is effectively suppressed by polic- ing (Frank 1995) or queen manipulation (Reeve and Keller 1997). In his critique of inclusive fitness theory, Edward O. Wilson (2012) asserts that

the workers of each colony [are] produced as phenotypic extensions of the mother queen...The defending worker is part of the queen’s phenotype, as teeth and fingers are part of your own phenotype.

If we accept the phenotypic gambit, this observation is accurate. Workers in such a setting have no evolutionary existence at all, because mutations in the worker genome cannot enter the species’ gene pool. As we explain in Appendix A, when the only instrument available to the queen for brood care is a daughter specially modified for this purpose, she cannot achieve her fitness maximizing sex ratio of reproductive offspring, but this is most insightfully seen as a bioengineering prob- lem, not a social conflict. It is dramatic to describe the prosocial behavior of sterile workers and soldier castes as altruistic and sacrificial, but it is misleading to do so. Such castes have no fitness to sacrifice. Similarly, the notion of conflict between obligately sterile workers and the queen has no evolutionary standing. However, the phenotypic gambit is not applicable to this case, because the net- work of genes that controls the relative frequency of male and female zygotes does not interact additively with the network of genes dictating the relative allocation of resources by workers invested in raising males and female reproductives.

6Hamilton’s use of “average inclusive fitness in the population” is itself questionable. If Hamilton means the average inclusive fitness over all loci, then assuming weak selection this will unambigu- ously increase in all cases. Because there is no other plausible definition of “inclusive fitness of a population,” I conclude that the adjective “inclusive” should simply be dropped from Hamilton’s statement to render it meaningful and accurate.

30 10.4 The Fallacy of Composition

What is true for each genetic locus in the genome is not necessarily true of the genome as a whole. Imputing a behavior characteristic of each part to the whole itself is the fallacy of composition. The tendency to commit this fallacy is espe- cially easy in inclusive fitness theory, which inexorably links the dynamics of a single genetic locus with the phenotypic behavior of an individual organism. Thus it is extremely tempting to infer from the fact that each genetic locus is utterly selfish that the genome necessarily promotes selfishness in its carriers. Similarly, it is tempting to infer from Hamilton’s rule, which characterizes a genetic locus, that individuals necessarily favor kin over non-kin. However obviously fallacious such reasoning, it has often been committed by “gene’s-eye view” versions of pop- ulation genetics. William Hamilton himself, in his 1975, p. 346, argues that the fact that interactions among unrelated individuals becomes very common with in- creased social complexity in human society, “civilization probably slowly reduces its altruism of all kinds.” I know of no evidence that this is the case, and much evidence to the contrary (Elias 1969, 1982, Pinker 2011). Consider also that, in The Selfish Gene (1976), Richard Dawkins asserts

We are survival machines—robot vehicles blindly programmed to pre- serve the selfish molecules known as genes...Let us try to teach gen- erosity and altruism, because we are born selfish.

Similarly, in The Biology of Moral Systems (1987 , p. 3), Richard Alexander asserts that

ethics, morality, human conduct, and the human psyche are to be un- derstood only if societies are seen as collections of individuals seeking their own self-interest.”

More poetically, Michael Ghiselin (1974, p. 3) writes:

No hint of genuine charity ameliorates our vision of society, once sen- timentalism has been laid aside. What passes for cooperationturns out to be a mixture of opportunism and exploitation...Scratch an altruist, and watch a hypocrite bleed.

In fact, the role of altruism, moral precepts, empathy, and other-regarding prefer- ences in general lie at the heart of our evolutionary success as a species (Gintis et al. 2005, Fehr and Gintis 2007, Bowles and Gintis 2011, Wilson 2012). Selfish genes do not necessarily create selfish carriers.

31 10.5 Complexity

Multicellular organisms are complex dynamic adaptive systems with several lev- els of emergent properties (Maynard Smith and Szathm´ary 1997, Morowitz 2002). This complexity is due to the nature of the genome, in which there are many inter- acting particles (loci) with roughly similar structure, but heterogeneous in their de- tails (Holland 1986, Mitchell 2009). Modeling such systems requires sophisticated techiques of which our understanding remains relatively primitive, and modeling promoter and suppressor genes, even in the simplest cases, quickly takes us from population biology to bioengineering and complex social theory. To say that biological organisms are complex has no explanatory value and does not imply the futility of analytical modeling. Rather, it suggests that un- derstanding multicellular organisms benefits from tools not needed in modeling the behavior of classical dynamical systems, including laboratory and field exper- iments, agent-based simulation, evolutionary game theory, and even thick ecologi- cal description. In a dynamic context, we must employ analytical tools that recog- nize the fact that the rules of the game themselves, as inscribed in the genomes of the players, will evolve according to adaptive principles (Levin 2009, Akc¸ay and Roughgarden 2011, Akc¸ay and van Cleve 2012). For instance, the rate of recom- bination in the genome can evolve to counteract collusion among mutant alleles (Haig and Grafen 1991). Tension between the analytical and descriptive poles in sociobiology would dissolve were their respective proponents to recognize the complexity of the socio- biological enterprise. Albert Einstein was asked many years ago why we humans are smart enough to harness atomic energy but not smart enough to contain a potentially catastrophic nuclear arms race. He replied, “This is simple, my friend. It is because politics is more difficult than physics.” We can as easily say that metazoan biology is more difficult than physics. The reason is that the simplifying assumptions used in physics very often give very close to the right answer, enough so that we can use the laws of physics to create a world of stunning technological complexity. Yet even the simplest metazoan, with a few hundred genes, is astronomically more complicated than any of our engineering accomplishments.

11 Conclusion

Sociobiologyis the study of how natural selection leads utterly selfish genes to par- ticipate in cooperative microbial communities, metazoan organisms, and complex societies. Inclusive fitness theory is a powerful framework for tracking changes in allele frequencies at a genetic locus across generations. Because the theory

32 analyzes the dynamics at a single genetic locus under weak selection, it is not sufficiently broad to model dynamics at the level of the genome. The most impor- tant implication of inclusive fitness theory is that genes are selfish in the sense of Dawkins (1976): the criteria for the direction of selection at a genetic locus are distinct from the criteria for the direction of selection for its carriers, and hence on the fitness of its carriers. If we add the Harmony Principle, which is a general observation concerning the cooperative nature of the genome, operating by virtue of the fact that a genome that successfully suppresses antisocial genes has an evolutionary advantage over a genome that does not, then the predictive value of Hamilton’s rule becomes con- siderable. In particular, it then follows that high genetic relatedness increases the range over which cooperative interactions within the genome can evolve, and an evolutionarily successful allele must be prosocial. The Harmony Principle is a plausible regularity because the genomes of suc- cessful organisms manage to elicit cooperative behavior from a set of genes each of which is utterly selfish, their individualinclusivefitnesses depending on factors that never include the evolutionary success of their carriers. How this is accomplished lies at the heart of sociobiological theory, is ill-understood, and its understanding requires sustained forays into biochemical and socio-structural theory. The most important tool for handling the extreme complexity of the metazoan genotype is the phenotypic gambit, which models complex regulatory and expres- sive networks as though the behavior they regulate were controlled by alleles at a single locus. The phenotypic gambit plus Hamilton’s rule become powerful an- alytical allies in modeling social behavior. However, little is known as to when applying the phenotypic gambit is a plausible simplificationand when it is not. Explaining gene frequency change requires that inclusive fitness theory be with models of the interaction of structural and regulatory genes. I have suggested that the notion of the core genome as replicator may facilitate this process of integra- tion, by directing attention away from the dynamics at individual genetic loci and the dynamics of higher level social groupings, whose character is in fact a product of the complex organization of the core genome.

Appendix A: Haploid Sex Allocation

The following simple example, based on Charnov (1978), clarifies the meaning of conflict at the gene level in a sociobiological setting, and shows how the res- olution of this conflict deviates from the model of conflict based on divergence of genetic “interests” and inclusive fitness maximization as the conflict-resolving process. The genetic analysis in this case is sufficiently simple that a complete

33 gene-level analysis is possible with the assumption that either the queen or the workers control the relative allocation of resources devoted to the production of male and female reproductives. Consider a eusocial haplodiploid species where each colony has one queen, singly mated and mated for life. The workers, all female, raise the colony’s brood, which consists of male and female eggs due to queen oviposition, and male eggs due to worker oviposition. Workers cannot produce female zygotes because there are no males in the population except during the mating season. We assume first that the queen controls the proportion of female and male reproductives, according to a preference that, however complexly regulated in the queen’s genome, can be represented by a single locus with alleles a and A, subject to Medelian segregation, the mutant allele A being dominant. We assume an aa queen prefers a female-male sex ratio of r, whereas an Aa or AA queen prefers a sex ratio rO. We suppose that aa and a are fixed in the population and investigate the conditions for the A allele to invade (Charnov 1978). More precisely, we seek to specify a value of r that cannot be invaded by a distinct sex ratio rO preferred by a mutant. We denote the colonies by xyz, where x;y;z 2 fa;Ag, with xy being the queens type and z being her mate’s type. Because the mutant is rare, we can ignore colonies with more than one mutant type, and we can ignore females of type AA. This assumption is purely for ease of exposition, and cannot change the results, provided the mutant is sufficiently rare. The three remaining types are then aaa, Aaa, and aaA. Let nxyz be the number of colonies of type xyz. We assume the colony has one unit of resources to expend on raising reproduc- tives, so an aa queen devotes a fraction r resources to gynes (female reproductives) and 1 r to males (all of which are reproductive), while an Aa queen devotes rO and 1 r O to gynes and males, respectively. We normalize the colony size, which we assume the same for all colonies, to unity. Let sf and sm be the expected num- ber of gynes and males, respectively, that survive to reproductive age per unit of resource devoted to their care. We assume all surviving gynes found new colonies, and all surviving males compete equally successfully for mating opportunities. We also assume that in each period a colony has a probability p of persisting until the next period, and workers produce a fraction q of new males. In the current period, an aaa colony produces r gynes of type aa and 1 r males of type a, so the population produces naaarsf new gynes of type aa. An aaa colony passes naaa.1 r/sm males of type a and naaa.p C rsf / gynes of type aa to the next generation of reproductives. If  is the fraction of A males at mating time, then this gives rise to naaa.p C rsf /.1 / colonies of type aaa and naaarsf  colonies of type aaA. The aaA colony produces Aa gynes, with r resources devoted to gynes, and

34 resources .1 r/ to males, all of which are of type a. Thus there are naaArsf new Aa gynes and naaA.1r/sm new males in the populationfrom aaA colonies. The queen produces a fraction .1 q/ of the males, all of which are a. The workers, which are Aa, produce a fraction q of the males, half of which are a and half are A. Thus aaA colonies contribute naaA.1 r/qsm=2 males of type A and naaArsf new Aa gynes to the next generation. The Aaa colony produces half aa and half Aa gynes and a fraction rO are de- voted by the Aa queen to new gynes and .1 r/ O to males. The gynes are half aa and half Aa, while the males are half a and half A. Thus Aaa colonies contribute nAaa.p Crs O f =2/ gynes of type Aa and nAaa.1 r/s O m=2 males of type A to the next generation, a fraction q of which are from worker’s eggs. If n0 is the next period frequency, we thus have 0 naaa D .p C rsf /naaa (A1) 0 naaA D pnaaA C rsf naaa (A2) 0 rsO n D n rs C p C f n ; (A3) Aaa aaA f 2 Aaa  à where  is the fraction of aa new gynes that mate with A males. The number of new males of type a is naaa.1r/sm. Thenew A males consist of those produced in Aaa and aaA colonies. In an Aaa colony, the queen produces a fraction .1 q/ of the males, half of which are A, so nAaa.1 q/.1 r/s O m=2 males are thus produced. The workers produce a fraction q of males, giving

nAaa.2.1 q/ C q/.1 r/s O m=4/ D nAaa.1 r/.2 O q/sm=4 new A males from Aaa colonies. In aaA colonies the queen devotes a fraction r to the production of males, but only the Aa workers, who produce a fraction r of the males, produce A males. This gives naaA.1 r/qsm=2 type A males. The ratio of new A males to new a males is then n .1 r/.2 O q/s =4 n q.1 r/s  D Aaa m C aaA m (A4) naaa.1 r/ 2naaa.1 r/sm n .1 r/.2 O q/ n q D Aaa C aaA (A5) 4naaa.1 r/ 2naaa We thus have 0 naaa D .p C rsf /naaa (A6) 0 rs q rs .1 r/.2 O q/ n D p C f n C f n (A7) aaA 2 aaA 4.1 r/ Aaa  Á 0 rsO f n D rs naaA C p C nAaa; (A8) Aaa f 2 Â Ã

35 Equation (A6) shows that when all reproductives are of type aa and a, the popula- tion grows at rate p C rsf  1. The second and third equations are not linked with the first, and show the fate of an invasion of the populationby a small number of mutant gynes of type Aa and males of type A. We can depict the dynamics of the invasion by a matrix equation

0 rsf q rsf .1Or/.2q/ naaA p C 2 4.1r/ naaA D (A9) 2 0 3 0 1 2 3 rsO f nAaa rs p C nAaa B f 2 C 4 5 @ A 4 5 The matrix in (A9) has positive entries, so it has a maximal real eigenvalue that represents the growth rate of the dynamical system (Elaydi 1999). If the incum- bent population is incapable of being invaded, then the value of rO at which this eigenvalue is maximized must be r. After some rather tedious calculations, we find that the maximal eigenvalue occurs for rO D r when r satisfies the equation

.2 q/.1 2r/s f D 0; (A10) 2.3 q/.1 r/ which implies r D 1=2. This prediction of equal investment in males and females, first stated by Fisher (1915) for diploid species, is valid for haplodiploid as well, and does not depend on the fraction q of workers’s sons. The inclusive fitness analysis of this model is much simpler (Trivers and Hare 1976). The queen is related 1/2 to her daughter queens, and each daughter queen produces one gyne, .1 q/ males, and q=2 males via her workers, for a total reproductive value of 1 C .1 q/ C q=2 D 1 C .1 C p/=2. The queen is related .1 q/ C q=2 D .2 q/=2 to a male reproductive, and the male’s reproductive value per inseminated queen is the queen herself plus one half the number of males produced by her workers, which is q=2. The males reproductive value is thus .1 C q=2/x, where x is the ratio of gynes to males in the reproductive population. The population equilibrium occurs when the relatedness times reproductive value for males and gynes are equal: q q 1 q 1 1 C x D 2 ; (A11) 2 2 2 2  Á Á  Á giving a sex ratio of 4 q x D : (A12) 4 q2 This differs from the correct ratio of x D 1 by at most about 7%.

36 Now let us assume that the workers rather than the queen control the allo- cation of resources to the reproductives. The aaA colony produces Aa work- ers, Aa gynes, and a males. The workers now devote rO resources to gynes, and .1 r/ O to males. Thus there are naaArsO f new Aa gynes and naaA.1 r/s O m new males in the population from aaA colonies. The aaA colonies thus contribute naaA.1 r/qs O m=2 males of type A and naaArsO f new Aa gynes to the next gen- eration. The Aaa colony produces half aa workers and half Aa workers, so the average workers devotes r D .r Cr/=2 O resources to producing gynes and .1 r/ to producing males. The gynes are half aa and half Aa, while the males are half a and half A. Thus Aaa colonies contribute nAaa.p C rsf =2/ gynes of type Aa and nAaa.1 r/sm=2 males of type A to the next generation, a fraction q of which are from worker’s eggs. If n0 is the next period frequency, we thus have 0 naaa D .p C rsf /naaa (A13) 0 naaA D pnaaA C rsf naaa (A14) 0 rsf n D p C nAaa C naaArsO ; (A15) Aaa 2 f  à where  is the fraction of aa new gynes that mate with A males. The new A males still consist of those produced in Aaa and aaA colonies. In an Aaa colony, the queen produces a fraction .1q/ of the males, half of whichare A, so nAaa.1 q/.1 r/sm=2 are thus produced. The workers produce a fraction q of males, half of the workers are Aa , half of their male offspring are A, giving

nAaa.2.1 q/ C q/.1 r/sm=4/ D .1 r/.2 q/sm=4 new A males from Aaa colonies. In aaA colonies, the workers are Aa and half of them produce the fraction q of A males, or naaA.1 r/qs O m=2A males. The ratio of new A males to new a males is then .1 r/.2 q/=4 naaAq  D nAaa C (A16) naaa.1 r/ 2naaa.1 r/ n .1 r/.2 q/ n q.1 r/ O D Aaa C aaA (A17) 4naaa.1 r/ 2naaa.1 r/ We thus have 0 naaa D .p C rsf /naaa (A18) 0 rsf q.1 r/ O rsf .1 r/.2 q/ n D p C naaA C nAaa (A19) aaA 2.1 r/ 4.1 r/  à 0 rsf n D p C nAaa Crs O naaA; (A20) Aaa 2 f  Ã

37 Equation A6 shows that when all reproductives are of type aa and a, the population grows at rate p C rsf 1. The second and third equations are not linked with the first, andshowthefate ofan invasionof thepopulationby a small number of mutant gynes of type Aa and males of type A. We can depict the dynamics of the invasion by a matrix equation

0 rsf q.1Or/ rsf .1r/.2q/ naaA p C 2.1r/ 4.1r/ naaA D (A21) 0 0 1 2 3 rsf 2 3 nAaa rsfO p C nAaa B 2 C 4 5 @ A 4 5 The matrix in (A21) has positive entries, so it has a maximal real eigenvalue that represents the growth rate of the dynamical system (Elaydi 1999). If the incumbent population is incapable of being invaded, then the value of rO at which this eigen- value is maximized must be r, in which case the eigenvalue must equal, pCrsf 1, the growth rate of the incumbent population. After some rather tedious calculations, we find that the maximal eigenvalue occurs for rO D r when r satisfies the equation

6 8r C q.2r 3/ sf D 0: (A22) 4.3 q//.1 r//

Solving for the value of r that cannot be invaded, we find 6 3q r D : (A23) 8 2q

This shows that when the queen produces all the males, the worker’s desired sex ratio is r D 3=4, while if the workers produce the males, the ratio is r D 1=2. The inclusive fitness analysis of this model is similar (Trivers and Hare 1976). The workers are related 3/4 to their sisters, and related .1 q/=4 C .3=8/q D .2 C q/=8 to their brothers. Equation (A11) now becomes

2 C q q 3 q 1 C x D 2 ; (A24) 8 2 4 2 Â Ã  Á  Á giving a sex ratio of 1 3.4 q/ r D D : (A25) 1 C x 16 C q.1 q/ This differ from the actual value (A23) by at most 2.5%.

38 Appendix B: Fitness is not an Additive Function of Gene Frequencies

Consider a large reproductive populationof size N , with individuals i D 1;:::;N . Consider a focal locus in the genome, assumed diploid, with alleles j D 1;:::;J , and let zi D k if individual i has k 2 f0;1;2g copies of a focal allele j at the focal locus. Then z D i zi =N is the frequency of the focal allele in the population. Let wi =w be the number of offspring of individual i, where w D i wi =N is an adjustment factor setP to maintain population size at N in the next generation. If we assume Medelian segregation, the expected number of copies of theP focal allele in the offspring of individual i will equal zi , so the change in the expected number of copies of the focal allele in the next generation will be given by

N N wi wz D w zi zi (B1) D w D ! Xi 1 Xi 1 N D zi .wi w/: (B2) D Xi 1 n Because iD1.wi w/ D N w N w D 0; we can rewrite the above equation as P N wz D .wi w/.zi z/: (B3) iD1 X This is known as Price’s equation for a single genetic locus assuming Medelian segregation. Suppose now that z represents some additive function of gene frequencies at an arbitrary number of genetic loci in the genome. We number the loci by l D 1;:::;L, and let zijl be the number of copies of allele j at locus l in individual i. Then we can write zi as L jl zi D ajl zijl ; D D Xl 1 jX1 where jl is the number of alleles at locus l, and ajl is the weight we wish to assign to allele j at locus l. An argument parallel to that used to derive (B3) that z satisfies (B3). Now suppose individual fitness is an additive function of the frequency distri- bution of genes at the various loci in the genome. Then we can write zi D wi in (B3), getting N 2 ww D .wi w/ : (B4) D Xi 1 39 This expression is unambiguously positive unless all individual fitness are equal to one another, in which case fitness w is a maximum. R. A. Fisher (Fisher 1930) called this fact the Fundamental Theorem of Natural Selection, writing The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time. (p. 37) In fact, of course it is well known that reproductive populations do not maximize inclusive fitness (Moran 1964, Akin 1982, 1987). Fisher’s Fundamental Theorem was widely questioned until Price (1972), and later Edwards (1994), reinterpreted the theorem as applying to the “additive part” of fitness. Using the above notation, we can write individual fitness wi in the form

L jl wi D ajl zijl C ıi Á gi C di D D Xl 1jX1 for arbitrary fajl g. If we then choose the fajl g to minimize the mean square de- 2 viation i ıi from additivity, and define the variance in the resulting gi as the additive variance in fitness, then we can perfectly accurately state (Edwards 1994, p. 450) asP The rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequen- cies is exactly equal to its genetic variance in fitness at that time. Of course, the choice of weights for the additive expression is purely a heuristic device without theoretical motivation, and the weights themselves will change over time. The latter fact is particularly important because we know from Akin (1982) that in a multilocus setting, no additive function of gene frequencies with constant weights can in general monotonically increase. For many purposes the partition of total individual fitness into an additive com- ponent and a non-additive residual is extremely useful (Frank 1997, Gardner et al. 2011), but the terminology used by population biologists to describe this parti- tion is sometimes infelicitous and misleading if we treat the core genome, as op- posed to individual genetic loci, as the unit of evolutionary selection. Frank (1997, p 1712) writes: ...total change in the character can be partitioned into two compo- nents. The first is the direct effect of natural selection in changing the frequency of the predictor variables. . . The second component is the difference in the contribution of each predictor variable in the context of the changed population, the fidelity of transmission.

40 According to this terminology, deviations from additivity are imperfections: natu- ral selection acts only on gene frequencies, although as with R. A. Fisher, natural selection can be thwarted when the “fidelity of transmission” is imperfect. This identification of natural selection and adaptation with changes in gene frequencies and all frequency-dependent changes in genomic organization as bio- logical transmission effects that do not involve natural selection has been generally accepted by population biologists, as explained by Gardner et al. (2011, p. 1038– 1039):

We...explicitly separate the selection and transmission components of evolutionary change, and setting aside the transmission effect and focusing upon the selection effect.

The authors further explain that

Natural selection is the part of evolutionary change in heritable char- acters that is driven by the differential reproductive success of in- dividual organisms...It is not the sole factor in evolution, nor is it necessarily the major driver of genetic change. However, it receives much attention because it is the part of evolutionary change that gen- erates biological adaptation, and hence leads organisms to appear de- signed....naturalselectiononly ‘sees the heritablecomponent of traits. (p. 1020–1021)

By ‘heritable component’ the authors mean the additive component in fitness, and it is a nominal tautology that natural selection ‘sees’ only this component, because this is how the authors define natural selection. According to this definition, nat- ural selection barely ‘sees’ the core genome of an reproductive population at all, as most of the loci in the core genome are passed from generation to generation without alteration in relative frequencies. For instance, all of the characteristics we use to identify a particular species of duck (such as feather patterns, mating calls, breeding behavior, bill color and width), according to this terminology, are “trans- mitted” from parent to offspring, but are not the subject of natural selection. The overwhelmingly main source of additive variance is therefore purifying selection, through with mutations other random genetic innovations within a coadapted com- plex of tightly linked genes are either eliminated or, rarely, replace an allele in the core genome. In fact, however, there is a strong reason for rejecting the description of additive and non-additive components in fitness in terms of ‘selection’ and ‘transmission’in its own terms. The non-additive component includes the intragenomic interactions that turn a chaotic, fragmented, and utterly selfish agglomeration of genetic loci

41 into a functional biological entity of synergisticallyoperating parts. To say that this is just ‘transmission’ of variable ‘fidelity’ rather than natural selection is incorrect.

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