Martingales and Infanticide

Assume that you live in a culture that, for some reason or another, values having a male heir in the family. Families adopt the strategy of having children until the first male child is born. If the natural sex ratio is 1:1, what is the expected sex ratio in this culture?

It remains at 1:1. Here’s why. Each family has 1 boy. As for females, there is a 50% chance of having no girls, 25% chance of having 1 girl, 12.5% chance of having 2 girls, and so forth. So the average number of girls is (n-1)/2^n summed from n = 1 to infinity.

Therefore just following the strategy above does not change the sex ratio. In fact, it is possible to prove that no stopping strategy would change the sex ratio. Here’s how:

After having m children, a couple has to decide whether to have the (m+1)st child, based on those m children. Since the (m+1)st child’s sex is completely independent of the already born m children, it does not on the average change the existing number of boys – girls. This means that the average with or without the stopping will remain the same, namely at 1:1.

How is one to get a skewed sex ratio then? Wikipedia (HT: Leeping):

However, other Asian regions also have higher than average ratios, including Taiwan (110:100), and South Korea (108:100), which do not have a family planning policy.[70] Many studies have explored the reason for the gender-based birthrate disparity in China as well as other countries. A study in 1990 attributed the high preponderance of reported male births in mainland China to four main causes: diseases which affect females more severely than males; the result of widespread under-reporting of female births;[71] the illegal practice of sex-selective abortion made possible by the widespread availability of ultrasound; and finally, acts of child abandonment and infanticide.

This is actually a special case from the very powerful and counter-intuitive Optional Stopping Theorem.

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