Erika Matthews: Zach and I went over Day’s ideas as presented by Will in the PowerPoint that Will provided to us in that lecture a month prior. And because of that, we didn’t cover all of Vox’s math and all of Vox’s ideas as they exist across all of his blog posts, all of his AI-written books, and all of his AI-written non-peer reviewed articles on Zenodo. And so because of that, some of his fans as well as he himself cried a lot about how we didn’t read his book.
Indeed. I was going to simply ignore the first video, but after reading this in the transcript of the second video, I decided that beginning by responding to some of the criticism from the first one would serve as an excellent illustration for why it’s necessary to not only read the specific work one intends to criticize, but all of the related works as well. I think it’s a little strange that Erika imagines I would shed any tears in response to critics not reading the book, when all their not-reading accomplishes is to explain why their attempts to criticize it are so inept that it makes them look not only innumerate, but downright illiterate as well.
So here are the scientific credentials for the two Evo-Avengers in the first video:
Dr. Zach Hancock: Assistant Professor, Augusta University. PhD in Ecology and Evolutionary Biology, Texas A&M. Post-doc in statistical population genetics at the University of Michigan. Specialization: population genetics, quantitative genetics.
Erika Matthews: PhD candidate in biological anthropology at the University of Arkansas. Masters of Research degree in Primate Biology, Behavior and Conservation from the University of Roehampton with a BSA in Pre-Professional Animal Science and minors in Anthropology and in Biology from the University of Arkansas. Specialization: Canine morphology, sexual dimorphism, Miocene apes.
OVERLAPPING GENERATIONS AND d, THE SELECTIVE TURNOVER COEFFICIENT
Matthews: (47:20) It’s probably a good idea to take a course in population genetics before you insist that you have debunked the entire field. What Zach has just pointed out here is something that is very basic within a population genetics course but something which Vox Day seems completely oblivious to. There are models which can handle overlapping generations or sequential generations with replacement. So let me just make sure I’m getting this right. There’s more than one model and some of them don’t account for overlapping generations. Idealized theoretical ones, or perhaps applied to very specific situations, do not. And then there are ones which do. Is that a fair way of summarizing that?
Hancock: I can phrase it even simpler in that there are two classic models. There’s the Wright-Fisher model and the Moran model. And in its classic sense, the Wright-Fisher model does not have overlapping generations. But you can equate the two by setting the population size, the N value in the Wright-Fisher model, to an effective population size. And you can scale that effective population size by the slowdown of drift due to overlapping generations.
Matthews: So even even the Wright-Fisher model can can account for overlapping generations within itself. I’m sorry for interrupting. I’m just I’m trying to act as the audience stand-in, and also I want to clarify this for myself. So the Wright-Fisher model can account for these overlapping generations, just by changing the effective population size or iterating it over time. But then you also have a separate model which specifically can deal with that too. And these are both considered classic models.
Hancock: Yeah, that builds it from the base up, right? So the artificial model can be adjusted to account for it with by basically changing N to Ne. The Moran model was built from the ground up with the idea of overlapping generations. Both of which are really well characterized statistically. Both of which have been reviewed forward in time as well as backward in time. Backward in time being like coalescent theory. We understand the coalescent dynamics of both of these models. We know what the site frequency spectrum should look like. We know fixation probabilities in both of these models. These are extremely well-characterized models.
Matthews: Excellent! I love that! Cool, that makes sense!
Dr. Hancock is correct about what the Wright-Fisher and Moran models do. And they are well-characterized. Even so, it should have been obvious that I was not even a little bit oblivious to either of them, although Matthews makes it perfectly clear here that not only was she unfamiliar with the Wright-Fisher model, she had never heard of Moran, let alone Charlesworth, Hill, or Feselstein.
Meanwhile, there are 21 references to Wright-Fisher alone in Probability Zero.
What Hancock clearly didn’t understand is that neither of the models he cites are capable of addressing the MITTENS-related problem of calculating the number of effective generations available over a specified period of time. I thought it was surprising that he didn’t mention the Charlesworth model or the Hill ratio, both of which are more relevant to d than either Wright-Fisher or Moran. I suspect the reason Hancock did not do so is because he had no idea why it was necessary for me to develop d in the first place nor does he comprehend how it is utilized. He just reacted in a Pavlovian manner to seeing “overlapping generations” and didn’t stop to think about why overlapping generations would be of any interest to me.
It’s actually a very strange criticism when you stop and think about it. “This guy is so stupid and ignorant that he derived from scratch the same mathematical tools that the legendary scientists whose work established our field created! He’s just reinventing a very important wheel on his own! What a dummy!”
That’s close to the nicest criticism I’ve ever received. And, of course, I didn’t do that anyhow. I utilized those tools to develop a new tool that does something different, as well as something considerably more important than its original purpose.
The reason it is obvious that I knew about all of these various models is due to a) the references to them in PZ and in b) the paper in which d is derived, The Selective Turnover Coefficient: Distinguishing Selection from Drift in Age-Structured Populations, which has been in the repository since December 24, 2025 and was even included as Appendix A in the first edition of Probability Zero. Below is a partial selection from the References section of the STC paper:
- Charlesworth B (1974) Selection in populations with overlapping generations. VI. Rates of change of gene frequency and population growth rate. Theoretical Population Biology 6:108–133.
- Charlesworth B (1994) Evolution in Age-Structured Populations, 2nd ed. Cambridge University Press.
- Coale AJ, Demeny P, Vaughan B (1983) Regional Model Life Tables and Stable Populations, 2nd ed. Academic Press.
- Felsenstein J (1971) Inbreeding and variance effective numbers in populations with overlapping generations. Genetics 68:581–597.
- Fisher RA (1930) The Genetical Theory of Natural Selection. Clarendon Press, Oxford.
- Hill WG (1972) Effective size of populations with overlapping generations. Theoretical Population Biology 3:278–289.
- Hill WG (1979) A note on effective population size with overlapping generations. Genetics 92:317–322.
- Wright S (1931) Evolution in Mendelian populations. Genetics 16:97–159.
At first glance, it might look as if Wright-Fisher and Moran are doing the same thing as my Selective Turnover Coefficient because they have an application to overlapping generations. And yet, they’re not. The application is different. What the Wright-Fisher and Moran models do with overlapping generations is adjust the drift dynamics. When generations overlap, drift slows down because the gene pool isn’t being completely reshuffled every generation. Both models address this slowdown, Wright-Fisher by scaling N down to an effective population size N_e, Moran by building overlap into its architecture, and both provide correct predictions for how fast alleles drift to fixation or loss under a neutral regime.
In other words, they’re both answering this question: how fast does genetic drift move neutral alleles through the population?
What d does is very different: The Selective Turnover Coefficient addresses both drift and selection because, as the name suggests, it concerns selective constraint, which determines how much of the gene pool is replaced with every generation. It quantifies the fraction of the gene pool that is effectively flushed out of the population each generation, computed forward from real demographic schedules: age-specific mortality μ(x), survival l(x), and reproductive value v(x).
d = T · [ ∫ μ(x) · l(x) · v(x) dx / ∫ l(x) · v(x) dx ]
So d is designed to answer an entirely different question: how fast does the entire gene pool turn over?
The reason the distinction matters is that the Wright-Fisher model is functionally equivalent to d = 1, which is complete and instantaneous generational replacement. Every individual reproduces, every individual dies, the slate is wiped clean. The Moran model relaxes this for drift purposes, but neither model is at all capable of tracking what incomplete turnover does to the rate at which selection can fix new mutations throughout the population, or cleanse the genome of deleterious mutations. For example, if you look at the demographic statistics, then you will see that a human generation presently represents only about 24 percent of the total living population, not 100 percent as both the Wright-Fisher and the Moran models assume.
That’s why d is necessary. For wild chimpanzees, d ≈ 0.86, not too far from Wright-Fisher’s d = 1. For humans in the neolithic, d ≈ 0.45. It is now around 0.015. For the full human lineage since the CHLCA, the time-weighted average over 6.3 million years comes to 0.80. These aren’t drift corrections, they are selective throughput corrections. They tell you that in a species where grandparents coexist with their grandchildren and can occupy reproductive slots across decades, selection cannot flush the gene pool as fast as either the Wright-Fisher or the Moran model indicates.
Here’s an easy way to explain why d is necessary. There have been 252,000 generations since the chimp-human divergence. Applying either Wright-Fisher or Moran to that number changes nothing. You’ve still got 252,000 generations with either model. But applying time-weighted d over the entire period corrects for the available number of effective generations. It shows that you’ve actually got 201,600 generations with which to work, 50,400 generations less than you would naively assume. It doesn’t really matter much for MITTENS, which holds up fine with or without incorporating d, but it does provide a more accurate result. Which, of course, is why I developed it in the first place.
But, as we subsequently learned, and as those who have read any of the Biostellar novels know, the most important thing d does is something neither the Wright-Fisher nor the Moran model is able to do: it determines whether natural selection is at work on a given population, and if not, helps determine how fast that organism’s genome is degrading. Both classic models adjust for overlapping generations by scaling the effective population size, which corrects the drift dynamics but leaves the generation count untouched. The Selective Turnover Coefficient reduces the raw number of generations to the effective generations that actually represent gene-pool replacement, then enters the picture again as a differential through the per-generation rate of selective frequency change, because selection can only operate on the fraction of the population that turns over. This d² scaling is what connects demography to the selection-cost ceiling: it tells you, for any species with known mortality and reproductive-value schedules, whether the selection budget can keep pace with the mutational load on the genome or not.
And when it cannot, that means the genome is not being kept healthy by purifying selection. It is degrading over time due to deleterious mutations, and d quantifies the rate. Which is why the Selective Turnover Coefficient may be the most important thing to come out of Probability Zero since it may explain the observed decline in global human fertility and genetic fitness. And let’s face it, anyone who watches the Evo-Avengers’ videos has been provided with more than seven hours of visual evidence for not only the existence of d, but the degradation of the human genome.