Altruism in Persistent Groups

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There is one last loose end to tie up before concluding my series on George Price.

In a previous post I discussed the meaning of altruism in biology, and the distinction between strong and weak altruism. With strong altruism the altruist obtains no benefit from its own actions, whereas with weak altruism it does, though less than other members of a group to which it belongs.

W. D. Hamilton showed that strong altruism cannot evolve if altruism is randomly distributed in each generation. By varying his key assumptions several authors have put forward models in which a population is divided randomly into groups, yet altruism still increases in frequency. For example, groups may be formed randomly, but the aggregate benefit of altruism increases disproportionately with the number of altruists in the group.

A more surprising result is that strong altruism may evolve if groups are formed randomly, but allowed to persist for more than one generation. A natural first suspicion is that this is simply due to kin selection. In small groups which persist for several generations, many of the members in later generations are likely to be close relatives. However, an ingenious recent study by Fletcher and Zwick shows that in some circumstances strong altruism can evolve even if benefits to close relatives are excluded.

The general principle behind Fletcher and Zwick’s model can be illustrated with an example. Suppose a large (theoretically infinite) population is divided into groups of 4 members. Let the proportion of altruists in the population be 20%. Altruists and non-altruists are assigned to groups randomly. Groups may therefore have 0, 1, 2, 3 or 4 altruists. The expected proportions of these types of group in the population, and the proportions of all altruists belonging to each type, can be worked out using elementary probability theory. For example, with the parameters just stated, the number of groups which contain a single altruist, as a proportion of all groups, is just over 40%, while the proportion of all altruists belonging to such groups is just over 51%. Of course there are also groups – in fact around 40% of them – containing no altruists at all.

For simplicity we assume that reproduction is asexual. As a baseline for fitness assume that a non-altruist in a group with no altruists has 1 offspring. An altruist incurs a cost of c and distributes a total benefit b to the other members of its group, who share it equally. In a group with only 1 altruist the fitness of the altruist will be 1 – c, while the fitness of each of the 3 non-altruists will be 1 + b/3. In a group with 2 altruists each altruist will have a fitness of 1 – c + b/3 (since it obtains a benefit b/3 from the other altruist) and each non-altruist will have a fitness of 1 + 2b/3. In a group with 3 altruists each altruist will have a fitness of 1 – c + 2b/3 and the non-altruist will have a fitness of 1 + b. In a group with 4 altruists, each altruist will have a fitness of 1 – c + b.

At the end of the first generation, in the ‘mixed’ groups (on average) the proportion of altruists will have declined, since non-altruists are fitter than altruists within the group. It can also be shown that the proportion of altruists in the total population declines, as we would expect from Hamilton’s result. If all groups were now broken up and their members assigned randomly to new groups, as in Hamilton’s model, the proportion of altruists would decline again in the next generation, and so on indefinitely. It is also easy to see that as the frequency of altruists in the population declines, the proportion of all altruists belonging to single-altruist groups will increase (and more generally, the average number of fellow-altruists in a group will fall). Since altruists get less of the benefit of altruism in each successive generation, altruism is doomed.

The crucial difference in Fletcher and Zwick’s model is that the offspring produced by each group are allowed to stay together for at least one more generation. Altruists in relatively altruist-rich groups have more offspring than those in altruist-poor ones, since they receive fitness benefits from their fellow-altruists. If all offspring stay in their parental groups the average number of fellow-altruists in a group will therefore rise. This contrasts with the position if the offspring were assigned randomly to new groups. It follows that in the Fletcher-Zwick model there is a departure from random assortment, and a positive association of altruists in the second and any subsequent generations. It is this positive association which in principle allows even strong altruism to escape from Hamilton’s negative result. If altruists are positively associated, then the benefits of altruism within the total population go disproportionately to altruists, even though within a mixed group non-altruists receive more benefits than altruists. All that is needed is for groups to have some persistence over more than one generation, which seems a very modest and realistic requirement. (Of course it is also necessary for b to be greater than c, otherwise even in a group consisting entirely of altruists their fitness will be below the baseline.) Fletcher and Zwick show that their finding is still valid if various modifications are made to their model. Notably, altruism towards close relatives (kin selection) can be excluded, groups consisting entirely of altruists can be excluded, and a moderate amount of migration between groups can be allowed, yet altruism can still evolve (though less easily). It is also possible for altruism to evolve when the altruistic gene is initially rare (of the order of 1 in 1000), provided the benefits are high enough.

The Fletcher-Zwick model is a notable contribution to the debate on group selection, and may clarify some otherwise puzzling cases [see Note 1].

Now for some reservations. The Fletcher-Zwick process only works if groups are very small, if the collective benefit of altruism is large compared to the cost, or both. Fletcher and Zwick consider mainly groups between 2 and 5 in size, and even for these tiny groups, the collective benefit has to be around 5 times the cost for altruism to survive when the relevant gene is already common in the population. The collective benefit has to be much larger if altruism is to grow from initial rarity. But if groups are very small, and persist over two or more generations, we would expect many of the members (after the first generation) to be close relatives, and the growth or survival of altruism would be assisted by kin selection. Kin selection is a stronger mechanism than the Fletcher-Zwick process, as it allows altruism to evolve even when the benefit-cost ratio is relatively low (as low as 2:1 for full siblings), and is as effective when a gene is rare as when it is common (provided it is not a brand new mutation). With kin selection groups may also persist indefinitely without altruism losing its selective advantage, whereas with the Fletcher-Zwick process a periodical reassortment is necessary. It may therefore be that in nature there are few circumstances where the Fletcher-Zwick process would be a major factor in the evolution of altruism.

Note 1
The Fletcher-Zwick process may help explain a result which I found puzzling a few years ago. A paper by Harpending and Rogers described a model of group selection in which altruism could evolve despite an apparently random distribution of altruists. In their model groups are formed randomly but persist for more than a generation. The group size is fixed, and in every generation there are more births than deaths. The surplus goes into a ‘migrant pool’. Deaths are replaced either by births within the group or, with a specified probability, by a migrant. Groups may contain altruists and non-altruists. The birth-rate of all members of the group increases in proportion to the number of altruists (the beneficial effect of altruism), but altruists have a higher death-rate in any given period of time (the cost of altruism).

At the time when I commented on this model I did not appreciate the distinction between weak and strong altruism. On considering it again now, it seems that it involves weak altruism, since the birth rate of all members of a group rises with the number of altruists; altruists therefore enhance their own birth rate. With weak altruism it is accepted that altruism can evolve even with a random distribution of altruism in each generation. This may therefore account for Harpending and Rogers’s results. But in their model groups also have a partial persistence for more than one generation. The Fletcher-Zwick process may therefore also come into play. I still don’t fully understand what is going on, but at least it is now less puzzling that altruism can evolve despite the initial random distribution.


  1. How does this relate to Axelrod’s work? The models, as described, seem weird and primitive to me (and not just in the sense that all simple models are weird and primitive). Presumably because I am wholly ignorant of the literature, I don’t get why these models are even interesting. If someone asked me to rationalize seemingly altruistic behavior on evolutionary grounds, for sure the toolbox I would pick up is something like Axelrod’s. In that world, “groups” are just sets of individual organisms which have a high probability of future interaction, and the rationale for altruism is that altruism today will be reciprocated in a future interaction (if the other organism is playing a “nice” strategy). This avoids the rigid line-drawing exercise to delineate groups. It also points towards key supporting features of the environment. Environments (or traits) which create lots of opportunities for future interaction or which help organisms recognize other individual organisms or which help organisms recognize other nice players support cooperation. Furthermore, the reciprocal altruism stuff is old now.

    Why am I wrong?

  2. Thanks for commenting. Personally I don’t think the Fletcher-Zwick model is likely to be important in nature, for the reasons I have sketched at the end of my post, but it is theoretically interesting because it had been widely if rather loosely assumed that strong altruism cannot evolve in randomly formed groups. If a physicist showed that water can sometimes flow up hill, you would be interested even if in practice it turns out to be a very rare event. (Actually, I think it could happen, with water on a very gentle slope next to a very high mountain. The local gravity anomaly could – in theory – pull the water up the slope.)

    As to Axelrod, Fletcher and Zwick do reference a paper by Axelrod and Hamilton, but don’t say much about it. I can’t really offer an informed opinion about the game-theoretical approach, I just have a gut feeling that it has got detached from biological reality, especially since the death of Maynard Smith. The models get more and more elaborate, but not more robust to changes in the assumptions.

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