The mistaken view that the inclusive fitness of an organism is a weighted sum of the reproductive successes of all its relatives everywhere, who have ever lived, and who will ever live, is extremely common. Although Hamilton is not responsible for the errors of his followers, this may be one reason why many people have so much difficulty dealing with the concept of inclusive fitness, and it may provide a reason for abandoning the concept at some time in the future. There is yet a fifth meaning of fitness, which was designed to avoid this particular difficulty of inclusive fitness, but which has difficulties of its own.
Fit the Fifth
Fitness[5] is ‘personal fitness’ in the sense of Orlove (1975, 1979). It can be thought of as a kind of backwards way of looking at inclusive fitness. Where inclusive fitness[4] focuses on the effects the individual of interest has on the fitness[3] of his relatives, personal fitness[5] focuses on the effects that the individual’s relatives have on his fitness[3]. The fitness[3] of an individual is some measure of the number of his offspring or descendants. But Hamilton’s logic has shown us that we may expect an individual to end up with more offspring then he could rear himself, because his relatives contribute towards rearing some of his offspring. An animal’s fitness[5] may be briefly characterized as ‘the same as his fitness[3] but don’t forget that this must include the extra offspring he gets as a result of help from his relatives’
The advantage, in practice, of using personal fitness[5] over inclusive fitness[4] is that we end up simply counting offspring, and there is no risk of a given child’s being mistakenly counted many times over. A given child is a part of the fitness[5] of his parents only. He potentially corresponds to a term in the inclusive fitness[4] of an indefinite number of uncles, aunts, cousins, etc., leading to a danger of his being counted several times (Grafen 1979; Hines & Maynard Smith 1979).
Inclusive fitness[4] when properly used, and personal fitness[5], give equivalent results. Both are major theoretical achievements and the inventor of either of them would deserve lasting honour. It is entirely characteristic that Hamilton himself quietly invented both in the same paper, switching from one to another with a swiftness that bewildered at least one later author (Cassidy 1978, p. 581). Hamilton’s (1964a) name for fitness[5] was ‘neighbour-modulated fitness’. He considered that its use, though correct, would be unwieldy, and for this reason he introduced inclusive fitness[4] as a more manageable alternative approach. Maynard Smith (1982) agrees that inclusive fitness[4] is often easier to use than neighbour-modulated fitness[5], and he illustrates the point by working through a particular hypothetical example using both methods in turn.
Notice that both these fitnesses, like ‘classical’ fitness[3], are firmly tied to the idea of the individual organism as ‘maximizing agent’. It is only partly facetiously that I have characterized inclusive fitness as ‘that property of an individual organism which will appear to be maximized when what is really being maximized is gene survival’ (Dawkins 1978a). (One might generalize this principle to other ‘vehicles’. A group selectionist might define his own version of inclusive fitness as ‘that property of a group which will appear to be maximized when what is really being maximized is gene survival’!)
Historically, indeed, I see the concept of inclusive fitness as the instrument of a brilliant last-ditch rescue attempt, an attempt to save the individual organism as the level at which we think about natural selection. The underlying spirit of Hamilton’s (1964) papers on inclusive fitness is gene selectionist. The brief note of 1963 which preceded them is explicitly so: ‘Despite the principle of “survival of the fittest” the ultimate criterion that determines whether G will spread is not whether the behavior is to the benefit of the behaver but whether it is to the benefit of the gene G …’ Together with Williams (1966), Hamilton could fairly be regarded as one of the fathers of gene selectionism in modern behavioural and ecological studies:
A gene is being favored in natural selection if the aggregate of its replicas forms an increasing fraction of the total gene pool. We are going to be concerned with genes supposed to affect the social behavior of their bearers, so let us try to make the argument more vivid by attributing to the genes, temporarily, intelligence and a certain freedom of choice. Imagine that a gene is considering the problem of increasing the number of its replicas and imagine that it can choose between causing purely self-interested behavior by its bearer A (leading to more reproduction by A) and causing ‘disinterested’ behavior that benefits in some way a relative, B [Hamilton 1972].
Having made use of his intelligent gene model, Hamilton later explicitly abandons it in favour of the inclusive fitness effect of an individual on the propagation of copies of his genes. It is part of the thesis of this book that he might have done better to have stuck by his ‘intelligent gene’ model. If individual organisms can be assumed to work for the aggregate benefit of all their genes, it doesn’t matter whether we think in terms of genes working to ensure their survival, or of individuals working to maximize their inclusive fitness. I suspect that Hamilton felt more comfortable with the individual as the agent of biological striving, or perhaps he surmised that most of his colleagues were not yet ready to abandon the individual as agent. But, of all the brilliant theoretical achievements of Hamilton and his followers, which have been expressed in terms of inclusive fitness[4] (or personal fitness[5]), I cannot think of any that could not have been more simply derived in terms of Hamilton’s ‘intelligent gene’, manipulating bodies for its own ends (Charnov 1977).
Individual-level thinking is superficially attractive because individuals, unlike genes, have nervous systems and limbs which render them capable of working in obvious ways to maximize something. It is therefore natural to ask what quantity, in theory, they might be expected to maximize, and inclusive fitness is the answer. But what makes this so dangerous is that it, too, is really a metaphor. Individuals do not consciously strive to maximize anything; they behave as if maximizing something. It is exactly the same ‘as if’ logic that we apply to ‘intelligent genes’. Genes manipulate the world as if striving to maximize their own survival. They do not really ‘strive’, but my point is that in this respect they do not differ from individuals. Neither individuals nor genes really strive to maximize anything. Or, rather, individuals may strive for something, but it will be a morsel of food, an attractive female, or a desirable territory, not inclusive fitness. To the extent that it is useful for us to think of individuals working as if to maximize fitness, we may, with precisely the same licence, think of genes as if they were striving to maximize their survival. The difference is that the quantity the genes may be thought of as maximizing (survival of replicas) is a great deal simpler and easier to deal with in models than the quantity individuals may be thought of as maximizing (fitness). I repeat that if we think about individual animals maximizing something, there is a serious danger of our confusing ourselves, since we may forget whether we are using ‘as if’ language or whether we are talking about the animals consciously striving for some goal. Since no sane biologist could imagine DNA molecules consciously striving for anything, the danger of this confusion ought not to exist when we talk of genes as maximizing agents.
It is my belief that thinking in terms of individuals striving to maximize something has led to outright error, in a way that thinking in terms of genes striving to maximize something would not. By outright error, I mean conclusions that their perpetrators would admit are wrong after further reflection. I have documented these errors in the section labelled ‘Confusion’ of Dawkins (1978a), and in Dawkins (1979a, especially Misunderstandings 5, 6, 7 and 11). These papers give detailed examples, from the published literature, of errors which, I believe, result from ‘individual-level’ thinking. There is no need to harp on them again here, and I will just give one example of the kind of thing I mean, without mentioning names, under the title of the ‘Ace of Spades Fallacy’.
The coefficient of relationship between two relatives, say grandfather
and grandson, can be taken to be equivalent to two distinct quantities. It is often expressed as the mean fraction of a grandfather’s genome that is expected to be identical by descent with that of the grandson. It is also the probability that a named gene of the grandfather will be identical by descent with a gene in the grandson. Since the two are numerically the same, it might seem not to matter which we think in terms of. Even though the probability measure is logically more appropriate, it might seem that either measure could be used for thinking about how much ‘altruism’ a grandfather ‘ought’ to dispense to his grandson. It does matter, however, when we start thinking about the variance as well as the mean.
Several people have pointed out that the fraction of genome overlap between parent and child is exactly equal to the coefficient of relationship, whereas for all other relatives the coefficient of relationship gives only the mean figure; the actual fraction shared might be more and it might be less. It has been said, therefore, that the coefficient of relationship is ‘exact’ for the parent/child relationship, but ‘probabilistic’ for all others. But this uniqueness of the parent/child relationship applies only if we think in terms of fractions of genomes shared. If, instead, we think in terms of probabilities of sharing particular genes, the parent/child relationship is just as ‘probabilistic’ as any other.
This still might be thought not to matter, and indeed it does not matter until we are tempted to draw false conclusions. One false conclusion that has been drawn in the literature is that a parent, faced with a choice between feeding its own child and feeding a full sibling exactly the same age as its own child (and with exactly the same mean coefficient of relationship), should favour its own child purely on the grounds that its genetic relatedness is a ‘sure thing’ rather than a ‘gamble’. But it is only the fraction of genome shared that is a sure thing. The probability that a particular gene, in this case a gene for altruism, is identical by descent with one in the offspring is just as chancy as in the case of the full sibling.
It is next tempting to think that an animal might try to use cues to estimate whether a particular relative happens to share many genes with itself or not. The reasoning is conveniently expressed in the currently fashionable style of subjective metaphor: ‘All my brothers share, on average, half my genome, but some of my brothers share more than half and others less than half. If I could work out which ones share more than half, I could show favouritism towards them, and thereby benefit my genes. Brother A resembles me in hair colour, eye colour and several other features, whereas brother B hardly resembles me at all. Therefore A probably shares more genes with me. Therefore I shall feed A in preference to B.’
That soliloquy was supposed to be spoken by an individual animal. The fallacy is quickly seen when we compose a similar soliloquy, this time to be spoken by one of Hamilton’s ‘intelligent’ genes, a gene ‘for’ feeding brothers: ‘Brother A has clearly inherited my gene colleagues from the hair-colour department and from the eye-colour department, but what do I care about them? The great question is, has A or B inherited a copy of me? Hair colour and eye colour tell me nothing about that unless I happen to be linked to those other genes.’ Linkage is, then, important here, but it is just as important for the ‘deterministic’ parent/offspring relationship as for any ‘probabilistic’ relationship.
The fallacy is called the Ace of Spades Fallacy because of the following analogy. Suppose it is important to me to know whether your hand of thirteen cards contains the ace of spades. If I am given no information, I know that the odds are thirteen in fifty-two, or one in four, that you have the ace. This is my first guess as to the probability. If somebody whispers to me that you have a very strong hand in spades, I would be justified in revising upwards my initial estimate of the probability that you have the ace. If I am told that you have the king, queen, jack, 10, 8, 6, 5, 4, 3 and 2, I would be correct in concluding that you have a very strong hand in spades. But, so long as the deal was honest, I would be a mug if I therefore placed a bet on your having the ace! (Actually the analogy is a bit unfair here, because the odds of your having the ace are now three in forty-two, substantially lower than the prior odds of one in four.) In the biological case we may assume that, linkage aside, knowledge of a brother’s eye colour tells us nothing, one way or the other, about whether he shares a particular gene for brotherly altruism.
There is no reason to suppose that the theorists who have perpetrated the biological versions of the Ace of Spades Fallacy are bad gamblers. It wasn’t their probability theory they got wrong, but their biological assumptions. In particular, they assumed that an individual organism, as a coherent entity, works on behalf of copies of all the genes inside it. It was as if an animal ‘cared’ about the survival of copies of its eye-colour genes, hair-colour genes, etc. It is better to assume that only genes ‘for caring’ care, and they only care about copies of themselves.
I must stress that I am not suggesting that errors of this kind follow inevitably from the inclusive fitness approach. What I do suggest is that they are traps for the unwary thinker about individual-level maximization, while they present no danger to the thinker about gene-level maximization, however unwary. Even Hamilton has made an error, afterwards pointed out by himself, which I attribute to individual-level thinking.
The problem arises in Hamilton’s calculation of coefficients of relationship, r, in hymenopteran families. As is now well known, he made brilliant use of the odd r values resulting from the haplodiploid sex-determining system of Hymenoptera, notably the curious fact that r between sisters is ¾. But consider the relationship between a female and her father. One half of the female’s genome is identical by descent with that of her father: the ‘overlap’ of her genome with his is ½, and Hamilton correctly gave ½ as the coefficient of relationship between a female and her father. The trouble comes when we look at the same relationship the other way round. What is the coefficient of relationship between a male and his daughter? One naturally expects it to be reflexive, ½ again, but there is a difficulty. Since a male is haploid, he has half as many genes as his daughter in total. How, then, can we calculate the ‘overlap’, the fraction of genes shared? Do we say that the male’s genome overlaps with half his daughter’s genome, and therefore that r is ½? Or do we say that every single one of the male’s genes will be found in his daughter, and therefore r is 1?
Hamilton originally gave ½ as the figure, then in 1971 changed his mind and gave 1. In 1964 he had tried to solve the difficulty of how to calculate an overlap between a haploid and a diploid genotype by arbitrarily treating the male as a kind of honorary diploid. ‘The relationships concerning males are worked out by assuming each male to carry a “cipher” gene to make up his diploid pair, one “cipher” never being considered identical by descent with another’ (Hamilton 1964b). At the time, he recognized that this procedure was ‘arbitrary in the sense that some other value for the fundamental mother–son and father–daughter link would have given an equally coherent system’. He later pronounced this method of calculation positively erroneous and, in an appendix added to a reprinting of his classic paper, gave the correct rules for calculating r in haplodiploid systems (Hamilton 1971b). His revised method of calculation gives r between a male and his daughter as 1 (not ½), and r between a male and his brother as ½ (not ¼). Crozier (1970) independently corrected the error.
The problem would never have arisen, and no arbitrary ‘honorary diploid’ method would have been called for, had we all along thought in terms of selfish genes maximizing their survival rather than in terms of selfish individuals maximizing their inclusive fitness. Consider an ‘intelligent gene’ sitting in the body of a male hymenopteran, ‘contemplating’ an act of altruism towards a daughter. It knows for certain that the daughter’s body contains a copy of itself. It does not ‘care’ that her genome contains twice as many genes as its present, male, body. It ignores the other half of her genome, secure in the knowledge that when the daughter reproduces, making grandchild
ren for the present male, it, the intelligent gene itself, has a 50 per cent chance of getting into each grandchild. To the intelligent gene in a haploid male, a grandchild is as valuable as an ordinary offspring would be in a normal diploid system. By the same token, a daughter is twice as valuable as a daughter would be in a normal diploid system. From the intelligent gene’s point of view, the coefficient of relationship between father and daughter is indeed 1, not ½.
Now look at the relationship the other way round. The intelligent gene agrees with Hamilton’s original figure of ½ for the coefficient of relationship between a female hymenopteran and her father. A gene sits in a female and contemplates an act of altruism towards the father of that female. It knows that it has an equal chance of having come from the father or from the mother of the female in which it sits. From its point of view, then, the coefficient of relationship between its present body and either of the two parent bodies is ½.
The same kind of reasoning leads to an analogous non-reflexiveness in the brother–sister relationship. A gene in a female sees a sister as having ¾ of a chance of containing itself, and a brother as having ¼ of a chance of containing itself. A gene in a male, however, looks at the sister of that male and sees that she has ½ a chance of containing a copy of itself, not ¼ as Hamilton’s original cipher gene (‘honorary diploid’) method gave.