blog - steve t. piantadosi

I played around with a fun problem this weekend with Celeste. We started to talk about descendants and ancestors. What are the odds that someone descended from me will be alive T years in the future? What are the odds I will have no descendants in 10000 years? What is the relationship between the number of kids I have and the odds my descendants will all die out?

There are some funny phenomena relevant to this. Mitochondrial Eve is one. The number of people descended from Genghis Kahn is another. In Knuth’s Volume 1, he talks about a “proof” by H. W. Watson that under certain assumptions, infinitely many people will be born in the future, but each family line will die with probability 1 (pg 383). This is of course logically inconsistent, and Knuth proves so: that in an infinitely tall tree, you can find an infinite path of descent. If people live forever, someone’s family will live forever. Which should be obvious.

Here’s a simple model for thinking about infinite family trees. Suppose that there is some bounded population P. To create people for the next generation, the following is repeated: two people are chosen at random, bred, and their (one) child inhabits the next generation. That child is a descendant of their two parents. This is repeated until the next generation is filled up with people. And so on. This neglects a lot of important things–like gender, natural selection, sexual selection, and the grossness of incest. But maybe its not such a bad start…

First, two observations. If the population size is bounded, then it is eventually going to be the case that everyone is a descendant of me, or nobody is. This is because everyone being my descendant, and nobody being my descendant are fixed points—once either is true, it is true for all of time. And each generation there is some probability of either of these happening, so given enough time, one must come true. The second thing to notice is that the expected number of descendants of me at the next time step is 1-(1-p)^2 = p(2-p), where p is the proportion of people who are my descendants at the current time step. This is because the probability of getting someone in the next generation who is not a descendant of me is the probability that neither parent was a descendant of me, or (1-p)(1-p). (This also shows that p=0 and p=1 are the only fixed points). Thus, at each generation the number of my descendants should grow by a factor of (2-p). Things are looking good.

But of course there is a lot of variability, especially when you only have a few kids initially. So how many kids should someone have in order to be pretty sure your genes will take over? Since I know more about perl than stochastic processes, I wrote a little script to figure this out. If anyone wants to solve this analytically, I’d like to know, because I can’t run the perl script very quickly on a population of size 6 billion.

Here are some results, showing the probability that everyone is eventually a descendant of me for various population sizes and number of kids I have, averaged over 100 runs:

So, you don’t need many kids to eventually take have a good odds of taking over. At least when the population is relatively small. But in the larger population, probably most people’s kids reproduce themselves more readily than in this model, so you don’t need many kids to eventually take over. I wonder if anyone has attempted to model the distribution of family names along these lines…