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January 05, 2005

Limits to Hamilton's Rule

There are often references to Hamilton’s Rule, for example in the context of so-called ‘ethnic nepotism’. Some time ago I promised/threatened to comment on Hamilton’s Rule more fully, so here is my offering…

In considering an altruistic form of behaviour (actions that reduce the reproductive fitness of the actor while increasing the reproductive fitness of the recipient), Hamilton’s Rule states that such behaviour is not expected to evolve by natural selection unless br > c, where b is the fitness benefit to the recipient, c is the fitness cost to the actor, and r is the appropriate measure of relationship between them. I discussed the appropriate measure of relationship here.

The Rule also has an important application to selfish behaviour (actions that increase the reproductive fitness of the actor while reducing the reproductive fitness of the recipient). This time, if b is the fitness benefit to the actor, and c is the fitness cost to the recipient, then we do not expect the behaviour to evolve by natural selection unless b > cr. So, for example, it would not make biological sense for a man to kill more than two of his brothers (r = ½), or eight of his first cousins (r = 1/8), even if by doing so he saved his own life (other things being equal). This acts as a biological restraint on selfishness, but evidently not a very tight one! To adapt the famous example of J. B. S. Haldane, if you can save your life by drowning seven of your cousins, you should push them out of the lifeboat.

At first sight the Rule has a beautiful simplicity, but this is deceptive. Some of the possible problems were described by Dawkins in his article ’Twelve misunderstandings of kin selection’, which I discussed here, but Dawkins was writing from the perspective of an enthusiast for kin selection. I suspect that kin selection has been oversold, and that it is seldom important except in the special case of parents and offspring. The main point of kin selection theory was to explain altruism, but it has not been shown that altruism is widespread in nature, apart from parental care. The main apparent exceptions are among social insects and humans, who are, for different reasons, very peculiar animals. Human altruism (in the everyday sense) is a complex phenomenon. I don’t claim to fully understand it, and I won’t discuss it in depth here.

So here are some reservations about the Rule:

1. Hamilton’s Rule (HR) is essentially negative. It tells you what should not evolve, but not what will evolve. In logical terms, it states a necessary but not a sufficient condition. It would be a serious misunderstanding to suppose that just because a certain behaviour would be consistent with HR, then that behaviour is expected to evolve. Hamilton’s general positive principle is that we expect inclusive fitness to be maximised. Many different behaviour patterns might be consistent with HR, but the only one we expect to evolve is the one for which inclusive fitness is greatest. A useful principle in identifying this is mentioned by Dawkins: other things being equal, any altruistic benefit should be given to the closest relative available, and none at all to any more distant relatives.

This can be slightly generalised. Assuming that the costs to the actor are equal, but that the benefits to each potential beneficiary may be different, then the benefit should go to the relative for whom br is greatest. For completeness, we can regard the actor himself as a potential beneficiary with r = 1. But r diminishes rapidly as we go from the actor to more and more distant relatives, so the corresponding b would have to increase equally rapidly to make altruism towards distant relatives worthwhile. This is highly improbable: in general there is no reason why distant relatives should benefit more than close ones from a given action! And the actor himself is usually by far the closest ’relative’ - twice as close as even a full sibling. [Parental care is again a special case, because of the asymmetry of age, size, self-sufficiency, etc. The evolution of parental care is discussed in many texts, e.g. Clutton-Brock.] As I pointed out here, the expected number of relatives in each degree of relatedness increases in proportion as relatedness diminishes, but this is irrelevant unless there is some way of giving the same benefit to many different relatives without increasing the cost to the actor at the same time. Altruism towards distant relatives would only be favoured if the actor has a surplus of resources so large that both he and his close relatives would experience sharply diminishing returns (in terms of fitness) if they used all the resources themselves, and this situation must be very rare in nature. (Human hunters who have killed a large animal might be among the few exceptions.)

It is true that if there are no close relatives available, it would be ’better’ to give assistance to a distant relative than to an unrelated stranger, but it would usually be even better for the actor to keep the resources for himself, since the actor is always ‘available‘! Natural selection does not produce Boy Scouts, eager to do good deeds. So I think the general conclusion is that we do not expect altruism to evolve by natural selection towards relatives other than offspring (even siblings) except in unusual circumstances. If such altruism is observed, it may be a by-product of actions aimed primarily at offspring. For example, alarm calls aimed at offspring may benefit other relatives at no extra cost, or at a small cost outweighed by the benefits. In an earlier note I suggested that courage in battle might be another example. However, I doubt that this is the true explanation of courage. In primitive warfare (EEA conditions) the warriors are usually risk averse - not to say cowardly - the preferred tactics being to catch the enemy in a surprise attack by superior numbers (see Lawrence Keeley: War before Civilization). In the heat of actual combat, warriors may be brave to the point of recklessness, but this could well be a ‘selfish’ adaptation improving their chances of survival - in a fight, he who hesitates is lost.

The above discussion is ‘adaptationist’, in the sense that it explores what we would ideally expect to evolve by natural selection. For many reasons actual behaviour may be maladaptive or sub-optimal. But at least it is a useful starting point to be clear what an adaptationist account predicts. Some prevalent ideas (e.g. ethnic nepotism theory) seem to assume as a matter of theory that Hamilton’s Rule predicts widespread altruism towards relatives - even distant relatives - when in fact it does nothing of the kind.

2. The next major qualification about HR is that the Rule tacitly assumes that r is greater than the average for the relevant population. [See Note 1.] In his original formulation Hamilton referred to ‘a relative of the altruist, therefore having an increased chance of having the gene’ (Narrow Roads, vol. 1, p.7, italics added). The assumption in the italicised passage is invalid if the relative is an average member of the population. If bearers of an altruistic gene merely dispense altruism at random, then the fitness benefit will be enjoyed by altruistic and non-altruistic genes according to their existing proportions in the population, whereas the cost will fall only on those individuals who bear the gene. The frequency of the gene within the relevant population will therefore fall. If the population is fixed (or growing more slowly than the frequency of the gene is falling), then the gene will be eliminated. I think that the same is true if altruists dispense altruism specifically to recipients whose relatedness to them is average for the population - see Note 2.

It is theoretically possible that the fitness benefits to the recipients of altruism in the local population would be large enough to increase the size of the population faster than the decline in frequency of the gene within the population. In this case the gene would survive, and might even increase in absolute numbers. But one would expect the population before long either to run out of resources for growth, or to clash with other populations. This raises the possibility of inter-group selection, which is best analysed in terms of Price’s Equation. The familiar problem with inter-group selection is that it can only work if between-group variance in the frequency of the altruistic gene remains sufficiently high, despite gene flow between the groups, and despite the decline in within-group frequency. The fall in within-group frequency also means that, other things being equal, the amount of altruism per recipient will fall, so that the rate of population increase will slow down. And within groups, the genes for indiscriminate altruism would always be liable to be squeezed out by more discriminating mutations (whether they originate within the group or enter it by gene flow). For these reasons I doubt that the possibility of inter-group selection of altruism is much more than an academic curiosity. Hamilton himself seems to have had a fondness for inter-group selection among humans (see e.g. Narrow Roads, vol. 1, p.222), but I don’t think he faced up to the problem of maintaining between-group variance despite gene flow. He may have overlooked the importance of capture of females, which is almost universal in primitive warfare, and leads to substantial gene flow even between hostile tribes (Keeley: War before Civilization, p.86, etc.)

3. A third reservation is that HR neglects the effects of selection within pedigrees, as Hamilton noted from the outset. For example, an altruist will probably have fewer than one-in-eight altruistic cousins, because the altruists among his uncles and aunts will have fewer children than the non-altruists. This becomes more important as the chain of common ancestry linking actor and recipient becomes longer. If actor and recipient have inherited genes from a common ancestor 100 generations ago, then the fitness cost of the altruistic gene itself cannot safely be neglected. The intervening ancestors may also have been subject to different selective regimes for other reasons.

4. As Hamilton also pointed out, HR does not apply to very recent mutations, since they have not yet had time to spread to the appropriate relatives. (E.g. if you inherit a gene that mutated in your grandfather, you will not share it with any relative more distant than first cousin.) In the case of a new mutation causing its bearer to dispense altruism to distant relatives, it may be eliminated before there are any copies of the gene in such relatives to receive the benefit. In contrast, a gene that causes an individual to be selfish towards distant relatives will give him an immediate fitness benefit and will not be subject to counter-selection for many generations. Selfishness and altruism are therefore asymmetrical with respect to the time factor: as the old saying goes, ’a lie runs around the world before the truth can get its boots on’.

5. The same level of r may be obtained in different ways, e.g. r = ¼ applies equally to half-siblings, uncle-nephew, grandparent-grandchild, and double-first cousins (offspring of brother and sister from one family mated to sister and brother from another family). More complicated relationships may build up similar levels of r, given enough time, depending on the size of the population, migration rates, etc. But these different relationships cannot all be treated as equivalent for the purposes of HR. Suppose for example that within the British Isles two randomly chosen individuals have a coefficient of relationship of 1/8 (equal to that of outbred first cousins by the usual reckoning) due to common ancestry within the last 50 generations (about 1500 years). Does this mean we would expect them to show the same degree of altruism that they would towards their actual first cousins? Not at all, because (a) the chains of ancestry connecting them are long, so that gene frequencies will have been affected by selection; (b) altruism towards mere random members of the population is not selected for; and (c) if the ’baseline’ level of relatedness is 1/8, the relatedness of actual first cousins will be higher than 1/8, so we would still expect them to be preferred over strangers.

6. There have been attempts to apply a version of HR to interactions between members of widely separated populations, such as humans from different continents. Since such interactions were rare until recently, it is unlikely that behavioural traits have evolved in response to them, but it may still be of interest to consider how HR would apply in such a context. For this purpose, the relevant measure of relationship would have to take account of the genetic differences between and within groups. Over time, separated populations will evolve differences in gene frequencies due to drift and selection. Any given gene in one population is therefore more likely to be identical to a gene in another member of the same population than in a different population. A coefficient of ’ethnic kinship’ can be devised, to measure the similarity between individuals within the same populations, as compared with other populations. It is then argued that this is the coefficient that would be relevant in the application of HR to interactions between populations. It is further suggested that it should be the basis for political and ethical decisions involving different ethnic groups. The latter suggestion goes beyond biology and I won’t consider it here. [Note 3.] But on a purely biological level it seems to me misconceived, as it combines in one package all the problems I have already discussed. There is one further serious defect. The so-called coefficient of ’ethnic kinship’ is based on average differences in frequency of a large number of different genes in the populations to be compared. However, what is relevant to HR is the coefficient applicable to a particular altruistic or selfish gene, not the average for all genes. In the case of genes identical by recent descent, the appropriate coefficient can be calculated from pedigrees and is the same for all genes (subject to reservation 3 above). But in the case of widely separated populations there is no reason to suppose that the coefficient of ’ethnic kinship’ gives any useful guide to the relative frequencies of an altruistic (or selfish) gene in those populations. Since it is, by assumption, not a selectively neutral gene, the part of the coefficient of ’ethnic kinship’ which arises from genetic drift is irrelevant (unless populations are very small). Any difference in the frequency of the gene would be due to differences in selective factors between the populations concerned, and these are likely to be matters such as local group size, family structure, and mating patterns, rather than factors which differ systematically from one geographical area to another.

In summary, I doubt that Hamilton’s Rule can usefully be applied except in the simplest cases. I also doubt that there is much altruism in nature other than parental care. [Note 4.] (Again, I am not talking about human altruism in the everyday sense.)

I do not deny the importance of kinship as such in human society and cultural evolution. In outline, my view is that human life in the EEA was potentially a war of all against all, to obtain access to hunting territory and other resources (including females). For both offensive and defensive purposes, people needed allies, and networks of kin (by blood or marriage) provided a ready-made system of alliances. Inclusive fitness might help tip the balance in favour of alliance between relatives, but proximity and familiarity in themselves would lead individuals to favour kin as allies in preference to strangers. As anthropologists have emphasised, in primitive societies individuals will seldom encounter people on a day-to-day basis who are not related to them in some way. The scope for surveillance of behaviour, and reciprocal rewards and punishments, within a local group of kin networks would strengthen trust within the group.

Nor do I doubt that humans have evolved psychological adaptations for group living, such as a general liking for human company, and a tendency to distinguish between ’in-group’ and ’out-group’. When groups of unrelated individuals are brought together, as in schools or military units, they develop bonds of group loyalty, and hostility towards other groups, which probably have some evolutionary basis. Such traits could be adaptive quite apart from kin selection. In a ‘war of all against all’, anyone who is not within the scope of group surveillance, rewards and punishments must be regarded as a threat. It would therefore also be adaptive for individuals to conform to group customs and fashions, etc, to ensure that they are not treated as enemies of the group. Psychological phenomena such as guilt, pride, fairness, sympathy and generosity have evolved in this context and are probably unique to humans, though they may have faint precursors in other social mammals. I don’t think we yet have a satisfactory understanding of human morality, though every time I read Darwin’s chapters on the subject in the Descent of Man I am impressed by the depth and subtlety of his analysis. As Darwin concludes: ‘Ultimately our moral sense or conscience becomes a highly complex sentiment - originating in the social instincts, largely guided by the approbation of our fellow-men, ruled by reason, self-interest, and in later times by deep religious feelings, and confirmed by instruction and habit’ (Descent of Man, 2nd edition, chapter 5).

Note 1: For most purposes the relevant population should be regarded as the group of individuals with whom the actor has a significant probability of interacting - in the EEA probably a local group of a few hundred people. In some circumstances a wider population could be relevant - e.g. if members of the local group can migrate freely into the surrounding population. In this case the relevant ’background’ level of relationship would be somewhat below that of the local group itself.

Note 2: Consider the effects of an altruistic gene on the actor and a recipient whose relatedness to him is average for the population. (By this I mean the average relatedness to him, i.e. (Sum ri)/N, where ri is the relatedness of the i’th individual to the actor, and N is the population size excluding the actor. Usually (Sum ri)/N will be much the same as the average relatedness of one randomly selected individual in the population to another.) The genes of the recipient can be divided into two portions: those that are i.b.d. to the altruistic gene in the actor (and therefore themselves altruistic), and those that are not i.b.d. to that gene. It is customary to assume that the proportion of altruistic genes in the non-i.b.d. part of the genome is the same as the proportion in the gene pool as a whole, though this cannot be strictly accurate (unless the proportion is 100%) as the i.b.d. genes themselves are part of the gene pool. But making this approximate assumption, we can disregard the effect of the altruism on the non-i.b.d. part of the recipient’s genome, as any increase in his fitness (number of offspring) will not affect the proportions of the gene in the population derived from this part of his genome. And if we consider the i.b.d. part of his genome, his expected number of genes i.b.d. to the actor is proportional to r, his relatedness to the actor - in a diploid, it is equal to twice the coefficient of kinship between them. But since his relatedness to the actor is (by assumption) average for the population, his expected number of i.b.d. altruistic genes is also average for the population. The effect of the fitness benefit on a recipient with average r to the actor is therefore neutral w.r.t. the gene frequency in the population, since any increase in the number of his offspring only increases the number of people with the average frequency of the gene, and this cannot affect the average itself. In contrast, the fitness cost to the actor definitely reduces the frequency of the altruistic gene. So the net effect is negative. But I am not sure how to reconcile this with the reasoning underlying HR itself, which seems to show that the altruistic gene always increases if br > c, whatever the value of r. I think the answer is that the frequency may fall even if the absolute number increases. (And if there is a firm ceiling to the population, any increase in fitness of the recipients of altruism must be offset by a reduction in the fitness of others, who, by assumption, have on average the same frequency of altruistic genes.) Another complication is that there is a non-negligible probability that the actor will be inbred. If average relatedness of two individuals in the population is r, then the actor’s parents will also have average relatedness of r (assuming random mating), and a coefficient of kinship r/2. The actor himself will therefore have an average coefficient of inbreeding of r/2. But this means that the appropriate measure of relatedness for the purposes of HR will not be r but r/(1+r/2), which is smaller than r. On this basis, the altruism gene will not ‘break even’ until br/(1+r/2) = c.

Note 3: this paragraph is aimed at some of Frank Salter’s ideas on ’genetic interests’. I expect to return to this subject.

Note 4: Evidence of altruism towards siblings, etc, should be examined critically. Undeniably there are cases among both birds and mammals (e.g. African hunting dogs) where non-breeding individuals help feed and care for their siblings or their siblings’ offspring. But this only counts as altruism if it reduces their prospects of breeding in their own right, which is not always the case. Among African hunting dogs, for example, the dominant female in the pack prevents all other females from breeding, so their only alternatives (apart from helping raise the dominant’s offspring) are to take over the dominant role (which involves a hazardous fight), or to leave the pack and hunt on their own. In these circumstances helping raise their siblings or other close relatives may be ’making the best of a bad job’.

Related posts:
Ethnic Genetic Interests: Part 2
Ethnic Genetic Interests
Interracial Marriage: Salter's fallacy
Dawkins on Kin Selection
On Genetic Interests
Dissin' Dawkins
Green Beard and Ethnic Nepotism

Posted by David B at 02:57 AM