Many thanks to Paul Almond for developing the initial form of this argument.

My previous post was somewhat confusing and potentially misleading (and the idea hadn't fully gelled in my mind). But here is a much easier way of seeing what the SIA doomsday really is.

Imagine if your parents had rolled a dice to decide how many children to have. Knowing only this, SIA implies that the dice was more likely to have been a "6" that a "1" (because there is a higher chance of you existing in that case). But, now following the family tradition, you decide to roll a dice for your children. SIA now has no impact: the dice is equally likely to be any number. So SIA predicts high numbers in the past, and no preferences for the future.

This can be generalised into an SIA "doomsday":

  • Everything else being equal, SIA implies that the population growth rate in your past is likely to be higher than the rate in the future; i.e. it predicts an observed decline, not in population, but in population growth rates.
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If all parents of every generation rolled a d6 and had that many children then population growth stays constant - at a rate of 1.75x per generation, ignoring for the sake of argument the effect of people failing to live up to their dice roll for various reasons. In this concrete scenario, 6/21 children in the general population were born to parents who rolled a 6, 1/21 to parents who rolled a 1, etc. So yes, you would be are six times as likely to be born to parents who rolled a 6, but population growth is constant. If an argument says that population growth in falling in this scenario it's doing something wrong.

This argument seems to be using the fact that you are more likely to have parents who rolled a 6, and somehow taking it to mean that more 6s were rolled for the previous generation in general. This is simply not correct.

I'm honestly not sure whether this is a flaw in your argument, or a flaw in another argument that you're trying to highlight, so sorry if I'm just stating the obvious here.

There are extra subtleties, I've realised. I'm working on them now...

If I understand correctly, it predicts lower birth rates, but says nothing about death rates, is that right? "Population growth rate" usually means the difference.

Clearly this "prediction" is wrong. Just model the population growth based on dice throw. Each pair of parents has on average 3.5 children, so the population grows exponentially. The real question is, where did the logic go so wrong?

First, what are the aposteriori probabilities of each outcome of the dice roll by your parents, according to the SIA? How do you calculate that? Forget 6-sided die, let's try something simpler: a random choice of 0 or 1 child (furthermore, only one parent is required for procreation). SIA, SSA or any other model implies apriory probabilities of 0.5 for each outcome but aposteriori probabilities 0 and 1. Thus the "prediction" is that the growth rate was zero in the past, but will be negative in the future.

Now the issue is much clearer: if you trace your reasoning into the past, you see that only those parents who had children got accounted for. Those who had zero children dropped out from the calculations, since there is no one to remember them. And the number of people who had exactly one child (culminating with you) has always been constant (namely 1).

Extrapolating this to your case, your "prediction" under-samples those parents who had fewer children.

Imagine, that you have a 3 sided dice. This way the population is stable.

Yet, your parents have probably more children than you will have. You will have 1, 2 or 3 (p=1/3 for each case) - 2 on average.

But since you are alive, it is only 1/6 that your parents have 1 child, 1/3 that they have 2 and 1/2 that they have 3. It looks like you will have smaller number of children than your parents. Most probably.

A small modification of the initial post and there is no population growth or decline.

Yes, this was an example I considered, too, but it does not seem to highlight the problem with under-sampling of the low-sibling families as much.

A side note. In the real world, on average, one HAS less children than his parents. Parents can't have zero children, a child can.

In the real world, on average, one HAS less children than his parents. Parents can't have zero children, a child can.

Both are true, but the former doesn't follow from the latter. In particular, I suspect the former was false a few decades ago.

Indeed, this self-selection is the reason that the SIA Doomsday is a bad prediction.

I find this asymmetry between the past and the future in the circumstances of the dice rolling parents a bit shocking. As something was indeed wrong with the SIA. A very interesting gedanken experiment!

Sounds like quantum immortality run in reverse.