Preston’s Paradox | Hacker News Reader

Preston’s Paradox

allendowney.com

67 pointssebg52 comments

There is an extremely important difference between "every woman has fewer children than her mother" and "on average a woman has fewer children than her mother".

You can see that in the first scenario the human race dies out in about 12 generations as the maximum possible number of children any woman can have decreases by 1 each generation.

The second scenario is easily compatible with sustained long-term population growth. For example each woman initially has four kids, two of whom are childless and two of whom have four kids, etc. This doubles the population each generation but on average a woman has half as many children as her mother (two fewer).

It doubled in the first generation (one woman and presumably one man produced four kids)

In the following generation, the population didn't increase. Four people and their four partners collectively produced eight kids.

Ah, great catch. I keep forgetting about sexual reproduction. To correct the point while keeping the math easy, make it 8 kids each, half of whom have 8 kids and half none, etc. Then each couple contributes on average to two couples in the next generation, etc.

> I keep forgetting about sexual reproduction

a problem all too common in these fields

So the average shifts slightly up in the short term because of whales, but the population still will trend to zero in the long term. Not really a paradox, just a slightly unexpected short term statistical anomaly.

I don't know if "whales" is the right metaphor in this situation.

I think it's actually pretty well known that with a stable population there are always a lot more people with fewer kids than their parents have than there are people with more kids. After all, a decent fraction of people never have any kids at all, but everyone's parents had to have at least one.

> Starting from this distribution, what would happen if every woman had the same number of children as her mother? A woman with 1 child would have only one grandchild; a woman with 2 children would have 4 grandchildren; a woman with 3 children would have 9 grandchildren, and so on. In the next generation, there would be more big families and fewer small families.

I don’t think this is correct, because not all children are female. Under this scenario a woman with one child would have either one or zero grandchildren, a women with 2 children would have 0, 2 or 4 grandchildren, etc.

I guess the point the article is making (that the average number of kids needs to decrease to keep the population from exploding) still stands.

Side note is that if the rule is to have exactly the same number of kids as your mom, then the average asymptotically approaches the woman of the first generation who has the most kids.

So the king did better than nothing.

> Preston concludes, “Those who exhibit the most traditional behavior with respect to marriage and women’s roles will always be overrepresented as parents of the next generation, and a perpetual disaffiliation from their model by offspring is required in order to avert an increase in traditionalism for the population as a whole.”

This has huge political implications.

This is why most religions are actually fertility cults of one type or another. Disobey the fertility cult and you get the Onan treatment.

Also this explains why many religions staying power. "Have lots of kids and teach them your religion".

I wonder if there is a more important paradox "wealthy people trying to increase their wealth will influence government laws so extremely selfishly that the world will suffer a catastrophic event that drastically lowers human population."

I missed the part where the paradox was explained.

If every woman goes from having 3 children to 2, why did the rate go up?

Suppose there's 1 woman who had 10 kids, and everyone else had 3. Everyone then has max kids according to the "1 fewer than mother" rule. After 5 generations that 1 woman has 10 * 9 * 8 * 7 * 6 * 5 = 151200 descendants that will each have 4 kids. Everyone else went extinct several decades ago. So the average number of children per woman increased from 3 to 4.

> Everyone then has max kids according to the "1 fewer than mother" rule.

1 fewer is not the rule.

The rule is "every woman has to have fewer children than her mother"

There will be some whose mother had 10 then the daughter had 1. This narrative implies that every woman is having as many kids as possible all the time, which is both unrealistic and misleading by statistics. With a narrative sample size of one, the story is indicating that the number of people inevitably increased because of the rule rather than explaining this is an edge case paradox.

Yeah, but TFA then shows a graph of 1-fewer-than-mother.

That's the point. It's making an implicit assumption by graphing the maxima rather than demonstrating the range, making the paradox seem more likely than it is.

But after 10 generations they all go extinct.

Because of the shape of the distribution.

Those who had >= 4 were sufficiently numerous for the second generation going to 3.3.

P.S. also, further iteration is not discussed. Possibly the average for that particular curve would come down on the second or third generation.

Because every woman didn't go from 3 to 2. The average was 3. Some families had 10 and went to 9.

okay, I think this one made it make sense for me.

Also the "Eventually it goes to zero, but at first it increases".

Maybe that's what the graphs were saying, and usually I skim text and read graphs, but the graphs were confusing

Imagine a population of 2 women, one who has 2 children and one who has 100. Average is 51. Next generation 2 women have 1 child each and 100 women have 99 each, so the average is about 98. Eventually it will go down to zero, but initially it goes up.

You're supposing they bear females only ?

Another way to look at it is to focus on the children. 99 children have a mother who had lots of children, and 2 children whose mother didn't. Basically, if you pick a child at random, they're more likely to come from a family with a lot of children.

A similar paradox is the friendship paradox. On average, your friends have more friends than you do.

Ah, if a^i_t is the number of children in the t'th generation in the i'th 'matrilineal line':

    a^i_t = 0.5 * k^i_t * a^i_{t-1} * (k^i_t - 1)

Fertility rate has a quadratic relationship with number of kids per woman, since half the kids will be women capable of kids themselves. Whatever, my superscripts are fucked, but the idea is that the relationship being quadratic means high-birth women are going to be much more influential than low-birth women and so linear measures of central tendency don't capture that.

Cool.

Interesting and new to me, at least.

There is no place I can see to give feedback. Perhaps the author reads HN?

> how many children they have ever born

Surely "borne"?

In regards to children, as in giving birth, "born" is correct.

>Both born and borne are forms of bear. Born is commonly used with the sense of bear meaning "to give birth." Borne is used in reference to carrying something (physically or figuratively) [...] [1]

https://www.merriam-webster.com/words-at-play/borne-vs-born-...

'and, occasionally, in the "give birth to" sense.' Odd choice of words to elide.

GP has a valid point. "Borne" is the customary construction for the active past participle. Look at the very last example in your own link. It's a subtle point, really just amounts to the fact that with "have" and "had," borne is the typical collocation, as contrasted with "have been" and "had been," where born is normally used.

“Birthed” might be the correct version.

idk if its 'more correct' but it seems to read more clearly

In American English "born" means given birth to, while "borne" means physically carry something (possibly a child, but not necessarily).

The "paradox" is the incredible, insane 20-th century population explosion.

Paradox, or just pointing out human inability to multiply?

Not really a paradox, but also not intuitive.

If the average number of offspring were 3, and also every woman had exactly 3 children, then mandating that every woman has one fewer children than her mother would stabilize the population in one generation, and wipe it out shortly after.

The "paradox" happens because of the skewed distribution of family sizes.

To the young people here. Ignore the last line in the article.

You should never ever allow statistics at this level impact your decision to start a family. If you are intelligent enough to be reading an article about statistics on the Internet, please reproduce.

I don't have any good answers here, but it would seem that a negative consequence of welfare for lower income classes would have them disproportionately incentivized to have children (as their welfare would represent a much higher % of their total income). With this being the case I have to imagine that's a large contributing factor to associating large families with lower income and thus it loses the former advantage of large families being a high status indicator and thus higher income (and presumably higher intelligence/ better adjusted) families were responsible for a larger percentage of the population.

I have no idea what a humane incentive would look like, but I do wonder if we could find one if giving higher income families more incentive to reproduce relative to lower income families (or more directly I'd want to incentivize higher intelligence and better adjusted families, I just don't know how to measure that so I use income class to differentiate).

Presuming in all of this, that it isn't controversial to want intelligent and adjusted people to be responsible for the lions share of genetic makeup of future generations.

Well, I guess eugenics is back... It was a nice 100 years.

Scary "Idiocracy" figures aside, this discussion is incomplete without the incel statistics. The data is relatively conclusive that anyone below a certain threshold has no chance of getting a date. I think you will find that all the eugenics you could possibly want is already happening, the "program" is being administered by everybody, and the social norms that permit it are unimpeachable.

It has little to nothing to do with genes. You want a productive working class that produces more than it consumes to support the people who - for whatever reason - do not.

You could provide incentives to those who would rather not produce to straighten up and fly rite.

I'm sure you mean to sound scary with the word "eugenics" but this word is also the root cause of outlawing incest. Do you think we should legalize incest? I'd assume not, and that's quite literally a eugenics law, so no it was not 100 years ago that society agreed that eugenics in some forms was a good idea. It's very much alive today and the vast majority of people would argue that's a good thing.

Of course, since I feel you've already tried to demonize my position I guess I'll go ahead and restate again that I don't know of a humane way to implement eugenics such that we end up with more children from wealthy families. So if you want to take a stab at understanding me in a more charitable light, you can take that to me all the scary ways you can think of incentivizing such behavior are things I would be against.

Or, you know, reduce welfare and help people get off welfare.

Ideally there would be a solution to bridge the huge class gap and better distribute resources such that middle class would move to the upper class and leave a vacuum pulling poor people up, but I do find it considerably disturbing to see how many people downvote ideas on social media. We're not yet a society that's mature enough to entertain ideas we disagree with, it seems.

https://pubmed.ncbi.nlm.nih.gov/12321292/

>A 100% increase in the per child welfare benefit resulted in a 2% increase in the number of children. The policy implication is that a considerable increase in welfare benefits will have only trivial behavioral impacts for the poor on family size decisions.

At face value this, to me, only suggests that welfare is already so high that anything further is suffering from diminishing returns. To me, that wouldn't imply there are no policies (social or legal) that would encourage wealthy families to have more children.

> You should never ever allow statistics at this level impact your decision to start a family. If you are intelligent enough to be reading an article about statistics on the Internet, please reproduce.

I think that reading articles on advanced topics on the internet at best demonstrates one's self-assessed, not actual, intelligence. There are more than enough people in the world who are sure that they're the smart ones.

Dunning-Kruger is the relevant theory here: Maybe we have too many self-assessed smart people, and too few self-doubting ones.

This. We're already facing large reductions in population world-wide due to low fertility. Sure, the overall turning point is a few decades away, but it's certain, and in many countries it's in the past.

Aren't you just citing different statistics?

when we say the overall turning point is a few decades away, under most assumptions that don't involve massive war or pestilence or ecological collapse, even with a substantial drop in fertility, my child(ren), by the time they are old, will still be living in a world with higher total population than the current one.

we are basically in no risk of running down the population within any of our lifetime...

... on the proviso that everything doesn't go pear- shaped.

And with a recognition that those risks of things that might go pear-shaped are probably highly correlated with the carrying capacity of the planet with regards to human population numbers.

Planetary "carrying capacity" is not at all well defined. It depends on too many things like natural developments (think year 536), how the economy is organized, and powered, where and how many nukes have exploded, the quality of life, unforeseeable technological and scientific developments, dynamical ecosystems, ... So I'm curious what you mean by it? And how can you declare there's "basically no risk" when we still have superpowers with atom bombs going to war?

Maybe the king should have said "1 less factorial number than the mother".

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