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tzachquiel comments on Harry Potter and the Methods of Rationality discussion thread, February 2015, chapters 105-107 - Less Wrong Discussion
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It's true that a family tree either dies out or grows exponentially, but 2 is not necessarily the relevant exponent.
If the expected number of children an average descendant of Slytherin has is N, after 33 generations we should expect to see N^33 heirs of Slytherin. (You should think that this manipulation is suspicious as well, but it can be justified mathematically in, say, the Galton-Watson model.) Taking N=1.25 gives us only around 1500 descendants after 1000 years. And this is, if anything, an overestimate that does not take any intermarriage into account.
Powers of 2 only become relevant if you're looking backwards from a specific person; for example, if you want to know whether two people have a common ancestor. In that respect, I (tentatively) believe the paper you link to.
One of us must be wrong; it can't both be the case that everyone 1000 years ago is an ancestor of everyone living today, and that the average person 1000 years ago only had 1500 descendants.
I think N is closer to two or higher; assuming the average person has two children, they will have four grandchildren on average, eight great-grandchildren on average, and so on. So there really should be 2^33 heirs, though not 2^33 unique heirs; many of those heirs are just different genealogical paths to the same people.
I think if N were below two, it would be below the replacement rate and the population would shrink over time.
Indeed, I doubt that everyone 1000 years ago is an ancestor of everyone living today. I expect that everyone 1000 years ago is an ancestor of everyone [Edit: at least within a geographical region], of no-one, or is atypical in some way (for example, I expect a family that is well-off to have a number of children sampled from a different distribution, which has no reason to have mean greater than 2).
You are right, though, that across the board N has to be greater than 2 or else the global population would shrink over time. Moreover, if (when we look at Slytherin's descendants specifically) N is 1 or less, we expect the Slytherin line to eventually die out. This leaves room for a line that neither dies out nor grows as quickly as population does overall.