Choose altruism

Reading Yann Klementidis today, I caught a reference to a paper by Traulsen and Nowak in PNAS, titled "Evolution of cooperation by multilevel selection." The paper involves simulations of a large metapopulation split into small groups of individuals performing Prisoner's Dilemma-type games.

There is a long-standing tradition of comparing group selection with kin selection [citations redacted], and, often, the distinction between these two approaches is blurred. Our present model can be interpreted as describing purely cultural evolution: Groups consist of genetically unrelated individuals, and successful groups attract new individuals, which learn the strategies of others in the same group. For this interpretation, kin selection seems to be inappropriate. But our model can also be interpreted as describing genetic evolution, in which case, the members of the same group could be said to be more related than members of different groups, and the machinery of kin selection might apply. It would be interesting to see how the mathematical methods of kin selection can be used to derive our central results given by Eqs. 1-3 and what assumptions are needed for such a derivation. The problem is that the typical methods of kin selection are based on traditional considerations of evolutionary stability, which are not decisive for games in finite populations (Traulsen and Nowak 2006:10954, emphasis added).

The contrast between the bolded sentences is an interesting twist: the same model can describe systems with entirely different transmission modes. A purely cultural altruistic trait can be fixed or eliminated based on its performance in groups of individuals, based on its benefits, costs, group sizes and movement of alternative traits among groups. Or, an allele can be fixed or eliminated based on its benefits, costs, group sizes, and movement of alleles (which is a simple function of the movement of individuals) among groups.

Some other factors come in also, such as what, exactly, groups do when they get big -- do they fission or just cast off individuals? Also, the total number of groups makes a difference. But at the boundary, as shown below, the ratio of benefits to costs necessary to select for cooperation is minimized when migration is minimal, when the number of groups is very large, and when the sizes of groups are small.

In summary, we have presented a minimalist model of multilevel selection that allows the analytic calculation of a critical benefit-to-cost ratio of the altruistic act required for the evolution of cooperation. If b/c > 1 + n/m, then a single cooperator has a fixation probability that is greater than the inverse of the population size, and a single defector has a fixation probability that is less than the inverse of the population size. Hence, this simple condition ensures that selection favors cooperators and opposes defectors. The condition holds in the limit of weak
selection and rare group splitting. The parameters n and m denote the maximum group size and the number of groups. If we
include migration, the fundament al condition becomes b/c > 1 + z + n/m, where z is the average number of migrants arising from one group during its lifetime. These simple conditions have to hold for the group selection of altruistic behavior (ibid.)

Of course, the benefit to any individual is conditional on the proportion of cooperators in the group, so that the crucial problem is getting a cooperator trait to a high enough frequency for groups to benefit from it. Smaller groups help (by making it more likely for a group to have a high proportion of cooperators by chance), as do lots of groups (by giving many opportunities for high-cooperator groups to be thrown together by chance). On the same score, more migration makes things more difficult, because it decreases the chance of homogeneous groups. Hence the necessary b/c ratio increases with greater z and greater n, and decreases with greater m.

Group selection and its relationship (or lack of relationship) to altruism are well-worn topics. As far as genetics go, the conditions for selection to favor a selfless variant are well-known. So someone knowledgeable about genetics who reads this article may notice the same thing about the equation that I did: where is the r?

For those not in the know, r is the coefficient of relatedness -- how likely are two individuals to share genes. r increases within groups with greater inbreeding. It increases among relatives with greater degrees of kinship (r between you and a full sibling or parent is 0.5, a half sibling, grandparent or uncle is 0.25, and a full first cousin is 0.125, etc.). And it determines the genetic benefits of altruistic behavior -- a cost to your own reproduction can be paid by double the benefit to the reproduction of your full sibling, or eight times the benefit to your first cousins, and so on. In a population genetic perspective, group selection pays if your relatedness to the other members of your group is high, so that your sacrifice benefits your own genes indirectly through other individuals who are in fact your relatives. This happens when migration between groups is low and groups are small. In other words, an allele can only increase in frequency if it promotes behaviors that increase the number of copies of it in the population. This can't happen if it helps other individuals indiscriminately, but it can happen if it tends to help individuals who have other copies of the same allele, through inbreeding and metapopulation structure.

So, where is the r here? The answer is that it is factored into the b -- the benefit of the behavior. The authors make the model general, and in doing so eliminate mention of the kinship element of group selection on alleles. Then, they mention that their model applies to alleles under conditions of "genetic evolution", considering the level of relatedness within groups as a factor impacting benefits. Genetic evolution is a special case of the model.

But to me the topic of much greater interest is that the model shows the relative ease with which altruism might be learned. Selfless behaviors will work to the extent that their benefit/cost ratio exceeds 1 + z + n/m. That's a pretty simple calculus. In the real world, these values take on a bit more complexity -- z not only represents "migration", it also represents the chance of learning some alternative behavior (like deception). The number of people effectively in a group might vary by behavior -- you might share food with a small group, but share information with a large group, for example. So there are some mental details to work out for any given behavior. But it seems like a long ways from an insuperable goal, and it lends itself quite easily to piecewise extension -- adding a new mode of cooperation here or there, or dropping them when they don't pay off.

It means that humans can learn and coordinate their activities as groups without undergoing group selection on alleles. And it means that the kinds of alleles that humans needed to become cooperators were the kind that made them smarter, or at least better able to calculate costs and benefits of cooperation.

References:

Traulsen A, Nowak MA. 2006. Evolution of cooperation by multilevel selection. Proc Nat Acad Sci USA (online before print). Abstract