Inclusive fitness is the personal fitness which an individual actually expresses in its production of adult offspring as it becomes after it has been first stripped and then augmented in a certain way.

It is stripped of all components which can be considered as due to the individual’s social environment, leaving the fitness which he would express if not exposed to any of the harms or benefits of that environment. This quantity is then augmented by certain fractions of the quantities of harm and benefit which the individual himself causes to the fitnesses of his neighbours. The fractions in question are simply the coefficients of relationship appropriate to the neighbours whom he affects : unity for clonal individuals, one-half for sibs, one-quarter for halfsibs, one-eighth for cousins, . . . and finally zero for all neighbours whose relationship can be considered negligibly small.

Actually, in the preceding mathematical account we were not concerned with the inclusive fitness of individuals as described here but rather with certain averages of them which we call the inclusive fitnesses of types. But the idea of the inclusive fitness of an individual is nevertheless a useful one.

Just as in the sense of classical selection we may consider whether a given character expressed in an individual is adaptive in the sense of being in the interest of his personal fitness or not, so in the present sense of selection we may consider whether the character or trait of behaviour is or is not adaptive in the sense of being in the interest of his inclusive fitness.

3. Three Special Cases

Equation (2) may be written 6Rf-6R’ Api = pied+. . . (3) t That is, it is aimed “ uphill “: that it need not be at all directly towards the local maximum is well shown in the classical example illustrated by Mulholland & Smith (1959).

Now ST: = c (&z,),~ is the sum and 6Rb = 1 r(&,.), is the first moment about r = 0 of ;he array of effects {BU,,,,}ij cauie by the genotype GiGi; it appears that these two parameters are sufficient to fix the progress of the system under natural selection within our general approximation. Let SRf. rt = sj (ST; # 0); and let 6Rj rFj = gT39 (6Tj # 0). (4) (5) These quantities can be regarded as average relationships or as the first moments of reduced arrays, similar to the first moments of probability distributions. We now consider three special cases which serve to bring out certain important features of selection in the model. (a) The sums 6TG differ between genotypes, the reduced first moment r” being common to all. If all higher moments are equal between genotypes, that is, if all arrays are of the same “shape”, this corresponds to the case where a stereotyped social action is performed with differing intensity or frequency according to genotype. Whether or not this is so, we may, from equation (4) substitute r’6TG for SR: in equation (3) and have Api = pi? 6T;.-dT:. 1+6T;. ’ Comparing this with the corresponding equation of the classical model, da,.-da.. APi=Pi l+sa . . . we see that placing genotypic effects on a relative of degree r’ instead of reserving them for personal fitness results in a slowing of selection progress according to the fractional factor I. If, for example, the advantages conferred by a “classical” gene to its carriers are such that the gene spreads at a certain rate the present result tells us that in exactly similar circumstances another gene which conferred similar advantages to the sibs of the carriers would progress at exactly half this rate. Jn trying to imagine a realistic situation to fit this sort of case some concern may be felt about the occasions where through the probabilistic nature of things the gene-carrier happens not to have a sib, or not to have one suitably placed to receive the benefit. Such possibilities and their frequencies of reali- 10 W. D. HAMILTON zation must, however, all be taken into account as the effects (b~+,), etc., are being evaluated for the model, very much as if in a classical case allowance were being made for some degree of failure of penetrance of a gene. (b) The reduced first moments rz differ between genotypes, the sum 6T” being common to all. From equation (4), substituting rG8p for 6R; in equation (3) we have Ap, = pi $ (r:. - 4.). But it is more interesting to assume 6a is also common to all genotypes. If so it follows that we can replace o by ’ in the numerator expression of equation (3). Then, from equation (5), substituting r,“,6T” for aRig, we have Api = Pi~(rp.-rp.). Hence, if a giving-trait is in question (6T” positive), genes which restrict giving to the nearest relative (r: greatest) tend to be favoured; if a takingtrait (6T” negative), genes which cause taking from the most distant relatives tend to be favoured. If all higher reduced moments about r = r; are equal between genotypes it is implied that the genotype merely determines whereabouts in the field of relationship that centres on an individual a stereotyped array of effects is placed. With many natural populations it must happen that an individual forms the centre of an actual local concentration of his relatives which is due to a general inability or disinclination of the organisms to move far from their places of birth. In such a population, which we may provisionally term “viscous”, the present form of selection may apply fairly accurately to genes which affect vagrancy. It follows from the statements of the last paragraph but one that over a range of different species we would expect to find givingtraits commonest and most highly developed in the species with the most viscous populations whereas uninhibited competition should characterize species with the most freely mixing populations. In the viscous population, however, the assumption of random mating is very unlikely to hold perfectly, so that these indications are of a rough qualitative nature only. (c) Si$ = 0 for all genotypes. .*. 6TFj = -aaij THE GENETICAL EVOLUTION OF SOCIAL BEHAVIOUR. I for all genotypes, and from equation (5) 11 SR,“j = -8aijri”j. Then, from equation (3), we have Api = pi(SRy.-6Rf:) = pi{(6ai.+6Ri”)-(6a..+6R.q)} = pi(6Ui,(l- rp.)-6a..(l- rp.)). Such cases may be described as involving transfers of reproductive potential. They are especially relevant to competition, in which the individual can be considered as endeavouring to transfer prerequisites of survival and reproduction from his competitors to himself. In particular, if rb = r” for all genotypes we have Api = pi(l-r”)(&i.-da,,). Comparing this to the corresponding equation of the classical model (equation (6) ) we see that there is a reduction in the rate of progress when transfers are from a relative. It is relevant to note that Haldane (1923) in his first paper on the mathematical theory of selection pointed out the special circumstances of competition in the cases of mammalian embryos in a single uterus and of seeds both while still being nourished by a single parent plant and after their germination if they were not very thoroughly dispersed. He gave a numerical example of competition between sibs showing that the progress of genefrequency would be slower than normal, In such situations as this, however, where the population may be considered as subdivided into more or less standard-sized batches each of which is allotted a local standard-sized pool of reproductive potential (which in Haldane’s case would consist almost entirely of prerequisites for pre-adult survival), there is, in addition to a small correcting term which we mention in the short general discussion of competition in the next section, an extra overall slowing in selection progress. This may be thought of as due to the wasting of the powers of the more fit and the protection of the less fit when these types chance to occur positively assorted (beyond any mere effect of relationship) in a locality; its importance may be judged from the fact that it ranges from zero when the batches are indefinitely large to a halving of the rate of progress for competition in pairs. 4. Artificialities of the Model When any of the effects is negative the restrictions laid upon the model hitherto do not preclude certain situations which are clearly impossible 12 W. D. HAMILTON from the biological point of view. It is clearly absurd if for any possible set of gene-frequencies any aiT turns out negative; and even if the magnitude of 6aij is sufficient to make a; positive while 1 +e,g is negative the situation is still highly artificial, since it implies the possibility of a sort of overdraft on the basic unit of an individual which has to be made good from his own takings. If we call this situation “improbable” we may specify two restrictions : a weaker, e; > - 1, which precludes “improbable” situations; and a stronger, eiT > - 1, which precludes even the impossible situations, both being required over the whole range of possible gene-frequencies as well as the whole range of genotypes. As has been pointed out, a formula for e: can only be given if we have the arrays of effects according to a double coefficient of relationship. Choosing the double coefficient (cz, cl) such a formula is e$ = c’ [CZ Dev (6a,,, CJij + $cl f.Dev (aac2, CJi. + Dev (aac2, .,I. j)l + 5 T c2, Cl where Similarly Dev (8ac2,cl)ij = (dacz, dij - (8ac2, J., etc. e:j = Co ["]+dTP., the self-effect (ha,, ,Jij being in this case omitted from the summations. The following discussion is in terms of the stronger restriction but the argument holds also for the weaker; we need only replace . by ’ throughout. If there are no dominance deviations, i.e. if (6arei.)ij = ${(6arei.)ii +(6a,,i,)jj} for all in’ and rel., it follows that each ij deviation is the sum of the i. and the j. deviations. In this case we have et = c’ rDev(6aJjj+6TT.. Since we must have e,: = 6T.y, it is obvious that some of the deviations must be negative. Therefore 6T.T > - 1 is a necessary condition for e; > - 1. This is, in fact, obvious when we consider that 6T.q = - 1 would mean that the aggregate of individual takings was just sufficient to eat up all basic units exactly. Considering that the present use of the coefficients of relationships is only valid when selection is slow, there seems little point in attempting to derive mathematically sufficient conditions for the restriction to hold; THE GENETICAL EVOLUTION OF SOCIAL BEHAVIOUR. I 13 intuitively however it would seem that if we exclude over- and underdominance it should be sufficient to have no homozygote with a net taking greater than unity. Even if we could ignore the breakdown of our use of the coefficient of relationship it is clear enough that if 6T.y approaches anywhere near - 1 the model is highly artificial and implies a population in a state of catastrophic decline. This does not mean, of course, that mutations causing large selfish effects cannot receive positive selection; it means that their expression must moderate with increasing gene-frequency in a way that is inconsistent with our model. The “killer” trait of Paramoecium might be regarded as an example of a selfish trait with potentially large effects, but with its only partially genetic mode of inheritance and inevitable density dependance it obviously requires a selection model tailored to the case, and the same is doubtless true of most “social” traits which are as extreme as this. Really the class of model situations with negative neighbour effects which are artificial according to a strict interpretation of the assumptions must be much wider than the class which we have chosen to call “improbable”. The model assumes that the magnitude of an effect does not depend either on the genotype of the effectee or on his current state with respect to the prerequisites of fitness at the time when the effect is caused. Where takingtraits are concerned it is just possible to imagine that this is true of some kinds of surreptitious theft but in general it is more reasonable to suppose that following some sort of an encounter the limited prerequisite is divided in the ratio of the competitive abilities. Provided competitive differentials are small however, the model will not be far from the truth; the correcting term that should be added to the expression for Api can be shown to be small to the third order. With giving-traits it is more reasonable to suppose that if it is the nature of the prerequisite to be transferable the individual can give away whatever fraction of his own property that his instincts incline him to. The model was designed to illuminate altruistic behaviour; the classes of selfish and competitive behaviour which it can also usefully illuminate are more restricted, especially where selective differentials are potentially large. For loci under selection the only relatives to which our metric of relationship is strictly applicable are ancestors. Thus the chance that an arbitrary parent carries a gene picked in an offspring is +, the chance that an arbitrary grandparent carries it is ), and so on. As regards descendants, it seems intuitively plausible that for a gene which is making steady progress in genefrequency the true expectation of genes i.b.d. in a n-th generation descendant will exceed y, and similarly that for a gene that is steadily declining in frequency the reverse will hold. Since the path of genetic connection with a 14 W. D. HAMILTON simple same-generation relative like a half-sib includes an “ascending part” and a “descending part” it is tempting to imagine that the ascending part can be treated with multipliers of exactly 4 and the descending part by multipliers consistently more or less than 4 according to which type of selection is in progress. However, a more rigorous attack on the problem shows that it is more difficult than the corresponding one for simple descendants, where the formulation of the factor which actually replaces i is quite easy at least in the case of classical selection, and the author has so far failed to reach any definite general conclusions as to the nature and extent of the error in the foregoing account which his use of the ordinary coefficients of relationship has actually involved. Finally, it must be pointed out that the model is not applicable to the selection of new mutations. Sibs might or might not carry the mutation depending on the point in the germ-line of the parent at which it had occurred, but for relatives in general a definite number of generations must pass before the coefficients give the true-or, under selection, the approximate-expectations of replicas. This point is favourable to the establishment of takingtraits and slightly against giving-traits. A mutation can, however, be expected to overcome any such slight initial barrier before it has recurred many times. 5. The Model Limits to the Evolution of Altruistic and Selfish Behaviour With classical selection a genotype may be regarded as positively selected if its fitness is above the average and as counter-selected if it is below. The environment usually forces the average fitness a.. towards unity; thus for an arbitrary genotype the sign of 6aij is an indication of the kind of selection. In the present case although it is T.: and not R.‘f that is forced towards unity, the analogous indication is given by the inclusive fitness effect 6R$, for the remaining part, the diluting effect SSij, of the total genotypic effect ST; has no influence on the kind of selection. In other words the kind of selection may be considered determined by whether the inclusive fitness of a genotype is above or below average. We proceed, therefore, to consider certain elementary criteria which determine the sign of the inclusive fitness effect. The argument applies to any genotype and subscripts can be left out. Let 6T” = kiia. (7) According to the signs of 6a and 6T” we have four types of behaviour as set out in the following diagram: THE GENETICAL EVOLUTION OF SOCIAL BEHAVIOUR. I 15 gains ; 6a +ve Individual loses ; 6a -ve Neighbours gain ; 6T” +ve lose; 6T” -ve k +ve Selected k -ve Altruistic behaviour ? k -ve Selfish behaviour ? k +ve

Counterselected The classes for which k is negative are of the greatest interest, since for these it is less obvious what will happen under selection. Also, if we regard fitness as like a substance and tending to be conserved, which must be the case in so far as it depends on the possession of material prerequisites of survival and reproduction, k -ve is the more likely situation. Perfect conservation occurs if k = - 1. Then 6T’ = 0 and T’ = 1: the gene-pool maintains constant “volume” from generation to generation. This case has been discussed in Case (c) of section 3. In general the value of k indicates the nature of the departure from conservation. For instance, in the case of an altruistic action ]k] might be called the ratio of gain involved in the action: if its value is two, two units of fitness are received by neighbours for every one lost by an altruist. In the case of a selfish action, Ikj might be called the ratio of diminution: if its value is again two, two units of fitness are lost by neighbours for one unit gained by the taker. The alarm call of a bird probably involves a small extra risk to the individual making it by rendering it more noticeable to the approaching predator but the consequent reduction of risk to a nearby bird previously unaware of danger must be much greater.7 We need not discuss here just how risks are to be reckoned in terms of fitness : for the present illustration it is reasonable to guess that for the generality of alarm calls k is negative but jkj > 1. How large must Ikl be for the benefit to others to outweigh the risk to self in terms of inclusive fitness ?

t The alarm call often warns more than one nearby bird of course-hundreds in the case of a flock-but since the predator would hardly succeed in surprising more than one in any case the total number warned must be comparatively unimportant.

6R’ = 6R” + da = r”6T”+6a from (5) = Ga(kr” + 1) from (7).

Thus of actions which are detrimental to individual fitness (6a -ve) only those for which -k > $ will be beneficial to inclusive fitness (6R* + ve). This means that for a hereditary tendency to perform an action of this kind to evolve the benefit to a sib must average at least twice the loss to the individual, the benefit to a half-sib must be at least four times the loss, to a cousin eight times and so on. To express the matter more vividly, in the world of our model organisms, whose behaviour is determined strictly by genotype, we expect to find that no one is prepared to sacrifice his life for any single person but that everyone will sacrifice it when he can thereby save more than two brothers, or four half-brothers, or eight first cousins . . . Although according to the model a tendency to simple altruistic transfers (k = - 1) will never be evolved by natural selection, such a tendency would, in fact, receive zero counter-selection when it concerned transfers between clonal individuals. Conversely selfish transfers are always selected except when from clonal individuals.

As regards selfish traits in general (6a +ve, k - ve) the condition for a benefit to inclusive fitness is -k < LO. Behaviour that involves taking too much from close relatives will not evolve. In the model world of genetically controlled behaviour we expect to find that sibs deprive one another of reproductive prerequisites provided they can themselves make use of at least one half of what they take; individuals deprive half-sibs of four units of reproductive potential if they can get personal use of at least one of them; and so on. Clearly from a gene’s point of view it is worthwhile to deprive a large number of distant relatives in order to extract a small reproductive advantage.