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.

Overcoming Bias: No Evolutions for Corporations or Nanodevices

I have made a serious conceptual mistake, adopting darwinian evolution as the default explanation for almost any generational process for the past ten years. The requirements for darwinian evolution to be effective, as enumerated by Eliezer:

• Entities that replicate
• Substantial variation in their characteristics
• Substantial variation in their reproduction
• Persistent correlation between the characteristics and reproduction
• High-fidelity long-range heritability in characteristics
• Frequent birth of a significant fraction of the breeding population
• And all this remains true through many iterations

are not that easily satisfied.

It is easy to argue that objects fall at the same speed in vacuum, regardless of mass. Just imagine two identical uniform cubic bricks dropping from a height. Since the two bricks are identical, they drop at the same speed. Now imagine placing the starting bricks closer and closer to each other, until eventually they join up. The resulting brick has twice the mass of the originals, and assuming there is not discontinuity in behaviour, it should also fall at the same rate.

Similarly, imagine an economy with four people in it - a baker, a shoemaker, a farmer and a blacksmith. The four people sell each other goods at prices that result in no savings or loans. Now imagine another world, except it only has 3 people, and the shoemaker and the blacksmith are now the same person - i.e. the shoemaker/blacksmith makes in total the same things as the two people in the first world, and consumes in total the same amount as the two people in the first world. This second situation does not change the standard of living of the baker and the farmer at all, but assuming equal incomes in the first world, income inequality is higher in the second world.

If I am the farmer, should I care that I live in the second world instead of the first?

wikipedia