Guess 2/3 of the average

The logic behind it

In theory, if everyone can pick a number between 0 and 100, the average can also be anywhere between 0 and 100. If we all pick completely at random, the expected average is 50, and 2/3 of that is 33.3.

If we expect everyone to follow this logic, we expect everyone to pick 33.3 instead. The problem is that 2/3 of that is 22.2. This in turn leads to 14.8 – 9.9 – 6.6 – 4.4 – 2.9 and so on, until both the average and 2/3 of the average are 0.

Why doesn't everyone pick 0?

We humans are fairly logical, but not entirely. And if you think completely logically, you know that not everyone else does. These two things mean that you either expect others to think fewer steps ahead and act accordingly, or that you actually do so yourself. Either way, we typically end up with a number above 0.

In a contest in Denmark with 19,000 participants, 2/3 of the average came to 22, not 0.

What does it mean?

John Maynard Keynes, one of the most famous economists of all time, addressed this in 1936. He wrote about a competition in British newspapers in which readers had to pick the six prettiest faces from a hundred photographs, and whoever matched the majority's choice won. To win, you did not pick the faces you found prettiest, but the faces you thought others found prettiest, or one step further: the faces you thought others thought others found prettiest. Keynes believed the stock market works the same way. The price of a share depends not only on what the company is worth, or what the best analysts think it should be worth, but on what everyone thinks it is worth, or what everyone thinks everyone else thinks it is worth. This is called level-k thinking.

• Level 0 picks at random. The average then comes to around 50.
• Level 1 thinks everyone else is at level 0, and picks 2/3 of 50: around 33.
• Level 2 thinks everyone else is at level 1, and picks 2/3 of 33: around 22.
• Level 3 picks around 15, and so on towards 0.

The winning number in Denmark was around 22. Those who reasoned all the way through and picked 0 lost. Those who thought two steps ahead, or expected everyone else to, won.

That is the real lesson: in contests like this, it pays to be the smartest and most logical, but not without a sense of how far everyone else is thinking.

But it changes. In the economist Rosemarie Nagel's experiments from 1995, the same participants played several rounds. With each round, the numbers fell towards 0. When people see how others think, they learn, and they move towards the logical solution after all.

So: what would you guess now, knowing that everyone else has read this too?