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January 29, 2005

Group Selection can work...just

I’ve frequently been sceptical about the plausibility of group selection - for example here - so I am honour-bound to report a model which makes a form of group selection somewhat more credible.

The model, due to Henry Harpending and Alan Rogers, dates from 1987, but doesn’t seem to be as well known as it might be. For example, it is not cited in Sober and Wilson’s Unto Others (1998), or in the recent symposium volume Genetic and Cultural Evolution of Cooperation (ed. P. Hammerstein) (2003).

For more details….

The model is set out in the paper: Henry Harpending and Alan Rogers, 'On Wright's mechanism for intergroup selection’, Journal of Theoretical Biology, 1987, vol. 127, pp.51-61. As the title suggests, it is a development of ideas originally sketched by Sewall Wright, though with important modifications.

A further analysis and critique of the model is in M. Gilpin and B. Taylor, ‘Comment on Harpending and Roger’s [sic] model of intergroup selection’, Journal of Theoretical Biology, 1988, vol. 135, pp.131-135.

The key elements of the model are:

1. The population is divided into a large number of small groups.

2. All groups are of equal size n. (Values of n from around 5 to 50 are considered.)

3. The individuals in the groups are either ‘altruistic’ or ‘selfish’. This is determined by a haploid gene. (The haploid assumption is a convenient simplification.)

4. During any time interval, altruistic individuals have a higher probability of death than a selfish individual. The probabilities are in the ratio 1:1+c, which does not vary between groups. There is therefore a real cost to altruism.

5. The size of each group is fixed at n. Groups cannot grow larger or smaller.

6. All groups produce a surplus of births over deaths. Some or all of the surplus goes into a ‘migrant pool’.

7. Whenever an individual dies, it is immediately replaced, either from the migrant pool or from a birth within the group. The probability of replacement from the migrant pool is m, where m can range from 0 to 1. A value of 1 would mean that all deaths are replaced by migrants. Since there are more births than deaths within groups, and the total population within groups is fixed, not all migrants can enter groups. The leftovers die.

8. The size of each group’s surplus varies according to the proportion of altruists in the group. This is the beneficial effect of altruism - the ’group effect’. It is assumed that the size of the group effect is a linear function of the proportion of altruists in the group.

With these assumptions, H & R derive an equation to quantify the change in the proportion of altruists in the migrant pool compared with the population from which the migrants come. [Note 1.] The equation shows that provided there is some variance in the proportion of altruists in different groups, then the migrant pool will have a higher proportion of altruists than the preceding average for the groups. This is not surprising, as the ‘group effect’ means that groups with a higher than average proportion of altruists contribute disproportionately to the migrant pool.

H&R explore the effects of different values of the parameters by several methods of calculation and simulation. There are four key variables: the cost of altruism c; the group effect g; the size of the groups n; and the ’migration rate’ m. H&R find that the population is mixed (polymorphic) for altruism and selfishness when c is 0.03 (i.e. mortality of altruists 3 percent higher than selfish individuals); g is 1 (implying that pure altruist groups have a surplus twice as large as pure selfish groups); m is .2 (i.e. 20 percent of replacements are drawn from the migrant pool); and n is 25. They then illustrate the effects of doubling or halving each parameter, while the others are held constant. On this basis if n is halved, altruism is fixed; if it is doubled selfishness is fixed. If c is halved (to 0.015), altruism is fixed; if it is doubled (to 0.06) selfishness is fixed. If g is halved, selfishness is either fixed or in a large majority; if it is doubled, altruism is either fixed or in a large majority. If m is halved, altruism and selfishness are polymorphic; if it is doubled, altruism is either fixed or polymorphic.

Gilpin and Taylor confirm H&R’s results, and give an analysis of the effects of simultaneously varying group size and the cost of altruism. This shows that groups have to be very small - with only around 5 members - if altruism is to survive when the cost is above 10 percent.

The results of varying the parameters are not surprising except for the effect of raising the ’migration rate’. Migration tends to reduce the variance of the frequency of altruists in different groups, so it might be expected that raising m would make altruism less likely to succeed. However, H&R find that over the range of m from .2 to 1, changing the level of m makes little difference to the success of altruism. They attribute this to the presence of two counterbalancing effects: higher migration reduces between-group variance, but in this model the benefits of altruism are spread only through migration, so higher migration helps ensure its success. Gilpin and Taylor also point out that m is not strictly a migration rate, but only the probability that vacant places will be filled by migrants. Groups are therefore not swamped by migrants, however high the surplus going into the migrant pool.

Since the model only works if there is variance in the frequency of altruists, and migration tends to reduce this, it is of interest to consider how variance is maintained. It seems to be agreed (see Hamilton, Narrow Roads, vol.1, p.335) that purely random assortment of altruists afresh in each generation does not produce sufficient variance for group selection to work, at least without synergistic fitness benefits. In H&R’s model the group effect is simply proportional to the frequency of altruists, so the benefit is not synergistic. However, the groups are not completely ’reshuffled’ in each generation. I presume this gives some scope for higher levels of variance to be achieved by genetic drift. If there were no selection and no migration, all groups would eventually become fixed either for altruism or selfishness, in roughly equal numbers, at which point variance would reach a limit. But I confess I don’t fully understand what is going on in the model. Altruism seems to survive even when m = 1, in which case altruists dying within groups are only replaced by other altruists at the average frequency of altruists in the migrant pool. I don’t see how sufficient levels of variance are sustained in these circumstances. Gilpin and Taylor also seem to see a problem with this, but don’t discuss it in depth.

Opinions will differ on the importance of H&R’s model in biological reality. H&R themselves think their results ‘cast doubt on the primacy of genic selfishness as a tenet of Darwinian theory’, and suggest that group selection could be important in hominid evolution. Gilpin and Taylor on the other hand think the model has 'a very limited range of applicability to the biological world’.

There are two aspects to this question: the range of parameters for which the model works, and the plausibility of the model itself. It seems clear that it works only in fairly special circumstances: small groups, low individual cost of altruism, and large group benefit of altruism. The groups cannot be much larger than 25 individuals (including offspring), the cost cannot be more than about 5% of normal fitness, and the benefit must be at least a 50% increase in surplus offspring. It may be thought inherently unlikely that there are many altruistic traits that would simultaneously have such low costs and high benefits. This may however be an unfair way of putting it. In H&R’s model the ’group effect’ consists in a higher group surplus of births over deaths, and this does not necessarily require a large increase in average fitness. A surplus of, say, 10% instead of 5% would be sufficient to give g = 1.

On the plausibility of the model itself, it would be unreasonable merely to complain that it is oversimplified, because all models are. However, as Gilpin and Taylor point out, in several respects the simplifications work in favour of group selection, which would therefore be less plausible if the model were made more realistic. Notably, if some groups were much larger than others, selfishness would be favoured. It would also be favoured if groups were allowed some growth, rather than being constant in size. G&T also make an obscurely worded point about sex balance. I am not sure what they mean, but it does seem that any large imbalance in the sex ratio of migrants would produce difficulties for the model. In many species it is common for all young males (or occasionally females, as in chimpanzees) to disperse from their natal group (presumably as an evolved adaptation to prevent inbreeding). In these cases migration would probably be on a larger scale than allowed in the model. Migrants would not just be replacing the dead, but also replacing migrants of the same sex leaving their natal group. It is not clear if the model would still work in these circumstances.

So overall I’m inclined to say ‘close, but no cigar’. However, the model does have the merit that it represents genuine group selection. Unlike some models, there is no suggestion that altruists associate preferentially together, or that benefits go to relatives more than to non-relatives. In practice, of course, groups as small as 25 (including offspring) are likely to contain close relatives, so if altruism is found in such groups it would require very careful examination to distinguish between group and kin selection. The crucial test would be whether the behaviour violates Hamilton’s Rule; if it does, but nevertheless flourishes, an explanation by group selection may be called for, though one would also need to consider reciprocal altruism and other indirect benefits as possible explanations.

Note 1

If the frequency of altruists in the i’th group is Pi, the surplus of the i’th group is a+bPi, where a and b are constants. The group effect g is defined as the ratio b/a, so b = ag. S indicates summation over the i’s. With these definitions, it follows that the frequency of altruists in the migrant pool is S[Pi(a+bPi)]/S(a+bPi). (Note that in the denominator the constant a has to be added once for each group.) H&R then say that this is equal to P+ [ gV/(1+gP)], where P is the mean frequency of altruists in the population from which the migrants come, and V is the variance of the frequency in the different groups. H&R do not show how the equation is derived. To a mathematician it may be obvious, but for the benefit of plodders like myself I offer the following derivation:

First, note that if there are N groups, each with fixed size n, the total number of altruists in all groups is SPin = NPn, and therefore SPi = NP. The variance V of the Pi’s can be shown to be S(Pi^2)/N - P^2 (I omit proof of this - it is a standard result). It follows by simple rearrangement that SPi^2 = N(V+P^2).

Substituting equivalent expressions where appropriate in the formula S[Pi(a+bPi)]/S(a+bPi), it follows that the frequency of altruists in the migrant pool can be expressed as [NPa + S(PibPi)]/[Na + bSPi] = [NPa + bS(Pi^2)]/[Na + bSPi] = [NPa + Nb(V+P^2)]/N(a+bP). Cancelling the common factor N and substituting ag for b gives [Pa + ag(V+P^2)]/(a+agP) = [Pa+agV+agP^2]/(a+agP). Cancelling the common factor a and rearranging we get [P(1+gP) + gV]/(1+gP) = P + gV/(1+gP). QED.

Posted by David B at 07:51 AM