While writing a recent short note on Richard Dawkins and kin selection, I looked through my previous posts on the subject, and found what I thought was a blunder in an old post from 2004. To avoid misleading anyone who came across it in a search, I deleted it from the archive. But on further reflection I have concluded that there was no blunder after all…
My original post was a critique of an argument used by an anthropologist against the importance of kin selection in human social evolution. I did not mention the anthropologist’s name, and I do not now recall it, so I will refer to him simply as ‘Anthropologist’. Anthropologist’s argument was, in essence, that kin selection cannot be important because (a) for relatives beyond the closest it is too weak to be effective, and (b) in the human species (unlike, say, ants) an individual has few close relatives.
As it happens, I am sceptical about the importance of kin selection in human social evolution, and I agree with Anthropologist that kin selection is too weak to be important in relationships more distant than (roughly) uncle-nephew. (I will not consider Anthropologist’s second point, that there are too few close relatives.) According to Hamilton’s Rule, the value (to the relevant gene in the donor) of providing benefits to a relative is proportional to the coefficient of relatedness, and this declines rapidly with each extra degree of remoteness in the relationship: 1/2 for siblings, 1/8 for first cousins, 1/32 for second cousins and so on (or half of these figures if the relationship is through a single common ancestor, and not a pair). The problem with giving benefits to distant relatives is not just that beyond first cousins the relatedness is weak in absolute terms, but that it will usually be possible to give the benefit to a closer relative. Other things being equal, it is better to give a benefit to a sibling than a nephew, a nephew than a cousin, and so on. The circumstances in which it is advantageous to give benefits to a distant relative are probably rare.
So I agree with Anthropologist that kin selection in favour of helping distant relatives is unlikely to be important. But Anthropologist went beyond this qualitative conclusion, and attempted to quantify the importance of kin selection at each degree of relatedness in a way that I originally believed – and now believe again – is fallacious.
The importance of kin selection at any given degree of relatedness depends on the number of relatives helped and the inclusive fitness benefit of helping them. The number of relatives actually helped is an empirical matter, but it is possible to set an approximate upper limit to the number of relatives in any given degree who may be helped. In a stable or slowly changing population, each individual (or monogamous pair) will on average have 2 offspring, and the number of descendants after n generations will be approximately 2^n. Anthropologist takes this as his figure for the number of relatives in any given degree that can be helped. Even if we only count the descendants of a single ancestor, this is not strictly correct. For each individual ancestor, only about half of its descendants after n generations will have the most distant degree of relatedness to each other (e.g. second cousins, if they are descended from a common great-grandparent), while the other half will be more closely related (e.g. first cousins), because they have a more recent common ancestor. But it is true that on average the number of relevant descendants approximately doubles with each generation.
The selective value to a donor of giving benefits to a particular relative in any given degree depends on the coefficient of relatedness, or some similar measure, as used in Hamilton’s Rule. This may be calculated by tracing the connection between two relatives through their nearest common ancestor or ancestors, and taking a factor of 1/2 for each link in the chain. For example, for second cousins linked through a single great-grandparent, there are three links in the chain up from one cousin to the great-grandparent, and another three links in the chain down to the other cousin, so the coefficient of relatedness is (1/2)^6 = 1/64. More generally, if n is the number of steps back to the common ancestor, the relatedness is (1/2)^2n. Relatedness therefore declines by a factor of 1/4 for each unit increase in n.
So far, so good. But here comes the problematic part. Anthropologist calculates the (potential) importance of kin selection in each degree of relatedness by multiplying the estimated number of descendents from a single ancestor (or pair) by the coefficient of relatedness appropriate to that degree of relatedness. As already noted, one of these numbers increases by a factor of 2 with each generation, while the other declines by a factor of 1/4. Since 2 x 1/4 = 1/2, Anthropologist concludes that the potential aggregate value of helping relatives in any given degree is halved with each step of distance in relatedness, and rapidly becomes negligible.
In my 2004 post I raised several objections of detail to Anthropologist’s calculations, which I will not repeat here. But my more fundamental objection was that Anthropologist had taken into account only the relatives descended from a single pair of ancestors, for example second cousins descended from a single pair of great-grandparents. This underestimates the number of relatives an individual is likely to have in any given degree, since he or she may have such relatives by several different lines of descent. For example, excluding inbreeding, an individual will have 8 great-grandparents, and may have second-cousins descended from any of them. When this is taken into account, the number of relatives in any given degree increases in the same ratio as genetic relatedness declines. With each step back, the number of ancestors doubles (ignoring inbreeding) which exactly cancels out the alleged decline in the importance of kin selection.
Why then on re-reading my post did I think I had blundered?
In dealing with kin selection it is essential to take a gene’s-eye-view. If we focus on the particular gene of interest, kin selection only promotes it by increasing the fitness of genes that are identical by descent (IBD) with it. Since IBD genes are, by definition, derived by replication from a single common ancestor’s gene, IBD genes can only be found in the descendants of that ancestor, such as the descendants of a common great-grandparent. Other individuals may be equally closely related to the donor in the conventional sense – e.g. they may also be second-cousins – but their genes cannot be IBD with the gene of interest. The relevant number of relatives – those who can possess IBD copies of the gene of interest – is therefore confined to the descendants of a single common ancestor. On re-reading my post I thought I must have overlooked this, and that Anthropologist was right to count only the descendants of a single ancestor (or pair).
But on further reflection, I think I was right first time. (Indeed, on checking my old working papers I find that I considered this objection and dismissed it, unfortunately without mentioning it in my post.)
What is relevant to kin selection is the probability that a relative will have a copy of the gene that is IBD with the copy possessed by the actor. This probability is conventionally measured by the coefficient of relatedness, r. But probabilities are relative to the statistical ‘population’ under consideration, and this depends on the information available to us. For example, if all we know about an individual is that he is a white American male, and we wish to know the probability that he will die within a year, the relevant population consists of all white American males. If on the other hand we also know that he has been diagnosed with cancer, then the relevant population consists of all white American males who have been diagnosed with cancer. In the case of relatedness, the usual calculation of r assumes that the ancestral source of a gene is unknown, and that it may with equal probability have come from any ancestor at the appropriate distance. If on the other hand we know (or assume) that the source is some particular ancestor, then the usual calculation of r is not appropriate to determine the probability that another descendant of that ancestor has inherited the same gene. If that ancestor is definitely the source of the gene in one descendant, then the probability that it is IBD in another descendent is simply 1/2^n, where n is the number of generations from the common ancestor. It is easy to see that this probability is greater than r (as normally calculated) by a factor of 2^n.
We may therefore legitimately measure the potential importance of kin selection among relatives of a given degree in two ways. Either we may take account of all the relatives in that degree descended from all ancestors at the appropriate distance back, and multiply their number by the usual r, or we may take the number of relatives descended from a particular ancestor and multiply by (2^n)r. The result by both methods is the same. What we cannot legitimately do is to take the descendants of a single ancestor (or pair) and multiply their number by the usual r, as done by Anthropologist.
In my original post I made various more technical points. The only one worth mentioning here is that a randomly selected individual is more likely to come from a large, flourishing lineage than from a small, declining one. He certainly does not come from an extinct one! The average number of relatives in a given degree, for a randomly selected individual in the present generation, may therefore be significantly larger than the simple 2^n formula would suggest. However, I doubt that this consideration is enough to make kin selection in favour of distant relatives a major factor in human evolution. I agree with Anthropologist that this is unlikely, even if I disagree with the reasoning by which he reached this conclusion.
