[yorwba]

> It would be absolutely incorrect to say that "all of the next generation will be offspring of the more variable subpopulation B1".

No, that's exactly correct. The example is set up so that no individual from B2 gets to mate with A. Therefore, the next generation only consists of offspring of B1. (I think you understand that, since you say "sex A is choosing most desirable mates who happened to be part of the subpopulation B1".) B1 is the more variable subpopulation. Therefore, the sentence "all of the next generation will be offspring of the more variable subpopulation B1" is correct.

I suspect you are interpreting it as making a statement about the subpopulation B11 (of the subpopulation B1) that actually gets to mate. But if you split B1 into the more desirable B11 and the less desirable B12, then you can't even talk about their variability relative to B2, because the paper only defines comparisons of variability for distributions with the same median.

So the sentence "all of the next generation will be offspring of the more variable subpopulation B11" is not only incorrect but meaningless. Although they seem to be talking about the same individuals (and "all of the next generation will be offspring of the subpopulation B11" is true) they are talking about different populations, and "more variable" can only be applied meaningfully to B1.

In terms of your fruit example, the correct translation would be "All fruit in Sack 2 came from Sack 1, which had high variability." without any implications about variability of Sack 2.

Heritability of variability is only introduced on page 6: " it will be assumed that the pace of evolution is negligible compared to the pace of reproduction, so the two subpopulations remain distinct, with offspring distributed the same way as the parent subpopulation". In other words, although three subpopulations B11, B12 and B2 can be clearly distinguished, only membership in B1 or B2 is actually heritable, with B11's offspring either in B11 or B12. If B2 ever got to reproduce, it would have B2 offspring, while B12 would produce B11 or B12. So although B12 never reproduces, it never dies out due to offspring of B11.

Of course the conditions in those examples are contrived and that makes the conclusions almost trivial (a caveat that is noted in the paper: "The precise formal definitions and assumptions made here are clearly not applicable in real-life scenarios, and thus the contribution here is also merely a general theory intended to open the discussion to further mathematical modeling and analysis") but it is certainly free from egregious and basic errors.

[brenschluss]

This is helpful, thank you.

> If B2 ever got to reproduce, it would have B2 offspring, while B12 would produce B11 or B12.

Okay, this is clear regarding the paper’s assumption of heritability of variability. Assuming that variability is a completely static characteristic innate to a population and perfectly heritable seems to me such a simplistic assumption as render most of the conclusions ineffective or correct only within a narrow set of assumptions.

After all, every GA would never work if this was the case.

[brenschluss]

Actually, I change my statement upon rereading the paper: If offspring is "distributed the same way as the parent subpopulation", then B12 would only produce B12.

In other words, if subpopulation B11 is chosen by A, then B11 would produce B11, this reducing the variability, not increasing it.

[yorwba]

The paper only recognizes two subpopulations with their respective probability distributions. I only introduced B11 and B12 to be able to talk about different outcomes within the B1 subpopulations. Otherwise you could just choose any arbitrary grouping (e.g. mixing B11 and B12) and get completely different results depending on how you do it.

Maybe it's less ambiguous if we talk about machines which spit out a randomly sized ball when you press a lever. After collecting a certain number of balls from the machines, you determine what proportion of the largest p% is from each machine and replace all balls by selecting from the machines according to their proportion.

If you have a machine M1 producing very large balls and very small ones in equal quantities, as well as M2 producing medium-sized balls, then the prediction is that the proportion of balls coming from M1 will increase if p < 50% and decrease otherwise.

In that model, although you can clearly group the very large balls as B11, there is no machine M11 only producing those balls. Both very large and very small balls will, if they are in the top p%, require you to press the lever on M1. The size of the balls only influences which machine will be chosen, not what kind of balls will be coming out of that machine.

In genetic terms, B1 and B2 carry different alleles of the same gene and have different genotypes, while B11 and B12 are different phenotypes possible for B1. If B11 were to produce only B11, that would be inheritance of phenotype, i.e. Lamarckian evolution, which is also interesting, but less relevant for real-world genetics.