What do Radamacher Complexity and mean field approximations have to do with kin and group selection models in evolutionary biology? And what might consideration of the Radamacher Complexity of kin and group selection models suggest to us about cooperation and competition in human groups? Here are tentative answers to what seem like important questions that nobody is asking.
Group Selection in Evolutionary Biology
The Puzzle of Altruism
Curious students learning the theory of natural selection for the first time have been known to ask: "What about altruism?" What explains how animals, and especially humans, sometimes make a willing sacrifice of their own self-interest for others? Charles Darwin himself acknowledged in the Origin of Species that this is a "special difficulty, which at first appeared ... insuperable, and actually fatal to my whole theory." Darwin's own answer, using neuter or sterile female insects (like worker ants or bees) as an example, was as follows:
We can see how useful their production may have been to a social community of insects, on the same principle that the division of labour is useful to civilised man. As ants work by inherited instincts and by inherited tools or weapons, and not by acquired knowledge and manufactured instruments, a perfect division of labour could be effected with them only by the workers being sterile; for had they been fertile, they would have intercrossed, and their instincts and structure would have become blended. And nature has, as I believe, effected this admirable division of labour in the communities of ants, by the means of natural selection. But I am bound to confess, that, with all my faith in this principle, I should never have anticipated that natural selection could have been efficient in so high a degree, had not the case of these neuter insects convinced me of the fact.
Origin of the Species, Chapter VII (last paragraph before Summary) (1859).
The Kin Selection Theory Solution
Darwin's emphasis on "inherited instincts" vs. "acquired knowledge", which enforced "a perfect division of labour" (my italics) that "could be effected ... only by the workers being sterile" was in 1964 formalized by William D. Hamilton into a mathematical rule:
$$rB > C$$
Where:
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