4) “the complexity of biological phenomena is an argument for the use of mathematical methods rather than against it. In the case of a simple phenomenon we may hope to understand it without the use of mathematics, by simple inspection. But in a complex case we are left hopeless without mathematics.”

This is helpful! If allele helps those who are IBS, it's clear how this spreads, but I guess I just don't get why we expect IBD (a statement about average effects only) to matter unless there is an assumption of this kind (ex: allele is moderately frequent to begin with rather than rare mutant).

I'm not sure what (a) means biologically, but (b) seems to me to imply that we assume such mutations should arise (infinitely!) many times in the same mutational background - otherwise, why is a statement about expected/average behaviour meaningful given that we only observe a single realisation?

Maybe I'm being obtuse, but isn't it still true that this benefit is either (a) over time, conditioned on non-extinction of the allele, or (b) over realizations, which is some sort of assumption about how often such alleles arise via mutation (?) if we are looking within a species.

Why do we think these averages should tell us something meaningful about any given gene? Populations are typically not very large, and neither are clutch sizes so why is the average a useful metric?

I'm likely missing something, but why do the averages matter? Doesn't "average" rely on a LLN argument, either over crosses (if these genotypes meet many times, you share the gene with your sib more often than not) or over clutch size (if you have many sibs, you share the gene with half of them)?

*Maybe* if there were a large number of crosses of each type in every generation, LLN can kick in and ensure that IBD helps an individual estimate IBS.

This feels like a very strong, unjustified assumption given how wild populations of multicellular organisms typically work. What am I missing here?

So why is IBD relevant?

Naively, I would think that for helping to spread, either individuals should be able to detect IBS (= some sort of greenbeard) rather than IBD. In particular, this means you ignore siblings/pedigree and only look for the green beard, so relatedness can be entirely ignored.

the precise way to say "more likely" really depends on a law of large numbers (LLN) argument. In particular, LLN says that *on average*, half the siblings of a helper will be helpers, *given sufficiently many realisations*. This says nothing about what happens if your number of offspring is small.

Ignoring other assortment processes (ex: spatial structure/limited dispersal), this means that helping spreads if you can detect which individuals are identical by state (IBS) at the helping locus. While we like to say a sibling is "more likely" to be IBD (and thus IBS) than a random individual,

For simplicity, let's fix n=1 and say you believe that one locus models of "a gene for helping" (or similar) are plausible.

Classic theory suggests that if helping is costly, a mutant with the helping allele will spread if helpers can preferentially direct interactions towards other helpers.