Inclusive fitness errors and divorce in birds

In Darwin’s theory of evolution by natural selection the organism was king – natural selection would act to favour the traits that helped an organism to survive and reproduce. Whilst this idea explained so much of the adaptation and apparent design that we observe in the natural world, it also presented a problem. Why do organisms such as workers in social insect colonies sacrifice their own reproduction to help the queen reproduce and raise her offspring? The solution to this came from the work of Bill Hamilton, later clarified and promoted by Richard Dawkins in The Selfish Gene. Hamilton showed how the seemingly paradoxical altruistic behaviour of organisms (like workers helping the queen) made sense if we considered the genes involved. The great insight was to show how natural selection can favour an organism to sacrifice its own reproduction if it can provide benefits to related individuals, with whom the organism has a predictable chance of sharing genes. In short, a gene for altruism can spread even if it causes the individual carrying it to sacrifice its reproduction, as long as those effects are outweighed by the benefits the altruistic act provides to a relative, modified by the degree to which the altruistic individual shares genes with the beneficiary. This logic forms the basis of the famous ‘Hamilton’s rule’, which shows that a gene for a social behaviour such as altruism will spread if rb-c>0, where c and b are the costs and benefits of the social behaviour on the actor and recipient respectively, and r is the relatedness of the actor to the recipient. As a simple example, imagine that I can pay a cost c=1 to provide a benefit b=3 to another individual. This behaviour will be favoured if rb-c>0, or 3r-1>0 in this case, which simplifies to r>1/3. In this case, the behaviour would be favoured if the recipient was a brother (r=0.5) but not a half-brother (r=0.25).

Hamilton’s ‘inclusive fitness’ theory that is captured simply by Hamilton’s rule has been extremely useful in explaining so much of the complex cooperation (and conflict) that we see in nature, partly because of the simple logic of Hamilton’s rule, and the basic idea of individuals valuing other individuals according to the degree of relatedness between them (e.g. Haldane’s idea of jumping into a river and sacrificing yourself if you can save 2 siblings, or 4 half-siblings, or 8 cousins etc.). However, the simple verbal logic can sometimes lead you astray. A recent series of papers by Caro (2016), Bebbington & Kingma (2017), and Levin (2019) provide a great example of this, which I will discuss here.


Imagine a species of bird which is monogamous and produces one offspring per year, which both parents work to feed at the nest. The chick signals its desire for food, and could demand more food than it really needs to provide the maximum benefit to itself, or could demand less. By demanding less food than it could, the chick is paying a cost, but the consequences of doing so are to increase the chance of the parents surviving to breed the following year – producing a sibling (r=0.5) which shares genes with our altruistic chick. As such, we could imagine a scenario where the costs (to the chick) and benefits (of allowing the parents to produce a chick the following year) are such that this behaviour is favoured by natural selection.


With the basic scenario now set-up, we can ask interesting questions about what would happen if we changed certain aspects of our scenario to fit in with what we know about the kinds of behaviours we see in birds in the real world. Imagine first that our chick pays the same cost to be prudent with demands for food, and provides the same benefit of survival odds to the parents, but that the female has mated with multiple male (as is the very common in birds). The consequence of this multiple-mating would be that the offspring produced might be only a half-sibling (r=0.25) of our altruistic chick. Consequently, the altruism in the chick may no longer be favoured.

So far, our verbal arguments of the logic of Hamilton’s rule have served us well, and everyone would agree with the analysis shown here for these scenarios. However, what if we make a different change to our assumption about how the birds pair up. Imagine now that the scenario is the same, expect that our bird parents now ‘divorce’ after they have raised our altruistic chick, and (thanks to the altruism of our chick not demanding too much food) both survive to reproduce the following season with new partners. Would this effect the chance of altruism being favoured?

We might answer this problem by considering how the altruistic act of our chick is now helping the mother to survive and reproduce the following year – producing a half-sibling of our chick with her new partner (r=0.25). The chick’s altruism also helps the father survive and reproduce – producing a half sibling with his new partner (r=0.25). Helping its parents produce two extra half-siblings (in the case of divorce) is equivalent to helping the parents produce one extra sibling (if they stay together), so we would argue that the divorce has made no difference to whether the trait is favoured. At first glance it is hard to fault this explanation: it sounds logical, and it sounds like it has correctly interpreted the effects of producing siblings versus half siblings. When I was a student just becoming interested in social evolution, I’m sure I would have completely accepted this logic. However, this answer is wrong*, and I think it’s a really good example of where the simple logic of Hamilton’s rule can mislead us.


To explain why the idea that divorce wouldn’t make a difference in whether altruism is favoured is wrong, we can go back to what Hamilton himself pointed out about his own theory. The critical point is that inclusive fitness isn’t about calculating all of the offspring produced by relatives, rather it is about those which are caused by the behaviour. This is important because what we are aiming to do is calculate whether a rare mutant allele for altruism will spread in our population, which will depend on the effects that are specifically caused by that allele, not on the overall fitness of the individual’s relatives. In order to make this calculation, we have to be very careful about what effects can actually be ascribed as being due to the behaviour of the allele. Going back to our scenario, we can illustrate this. Without divorce, our altruistic chick is giving a benefit (let’s call it B) to each parent, and those benefits are combining (i.e. B+B=2B) to help those parents produce one extra offspring (a sibling of our chick) the following year. With divorce, our altruistic chick is giving a benefit (B) to each parent, and each parent is apparently able to use that to produce one extra offspring (a half-sibling) the following year. The error is that without the divorce we are saying that the parent only needs B to produce an offspring the following year, whereas with divorce we said they needed 2B to produce an offspring. Of course, the extra B comes from the new partner of the mother and father – but those effects are not caused by the altruism of our chick, and so aren’t part of the fitness effect of the trait. Instead, they are part of the fitness effect of some other trait, perhaps a different (competing) allele in a different chick. As such, we can only attribute half of the production of the half-sibling as an effect of the altruism gene. To count the extra B as part of the fitness effect of altruism would be to commit a so-called ‘double counting’ error, where a fitness effect has been attributed to multiple different genes. This error becomes obvious in formal mathematical treatments of inclusive fitness theory, but can often be obscured in seemingly sound logic of verbal arguments based on the simple idea of Hamilton’s rule. I think that the example of the consequences of ‘divorce’ on selection for altruism provide a compelling example of this, and one which could be very instructive to people learning about social evolution

*It is important to note that the argument is wrong here for the example we are considering – which is very much focused on birds, for which parental care is hugely important and constitutes a major part of the investment required to successfully raise offspring. If there was no parental care, and the important step in producing offspring was simply surviving to produce a successful gamete, then divorce wouldn’t make a difference. This is because the benefit provided to the parent by the offspring (B) would be sufficient to produce the half-sibling, which could be fully attributed as an effect of the altruism gene in our focal chick. In this way, the answer to the question of whether divorce matters for altruism depends very much on the biology of the system in question – another important aspect that should always be considered. However, in the case of birds it is very clear that the limiting factor is the investment of the parents in feeding the chick, not simply the production of gametes.