riddle of the day

This game looks psychological: guessing what other people will think. But if everyone reasons perfectly, the crowd collapses toward a single number.

The 2/3 Contest

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Master plays

Riddle statement

A group of people takes part in a contest.

Each person must choose a number between 0 and 100.

The average of all chosen numbers is calculated.

The winner is whoever chose the number closest to:

\frac{2}{3}

of that average.

If all players are perfectly rational, all want to win, and all know that everyone else reasons this way too, what number should they choose?

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Solution

The ideally rational answer is:

$ 0 $

Here is why.

No one should choose more than:

$ \frac{2}{3}\cdot 100 = 66.66\dots $

because the average can never exceed 100, so two thirds of the average can never exceed 66.66.

But if everyone knows this, then no rational player will choose more than 66.66.

Therefore, the average also cannot exceed 66.66.

Then the winning number cannot exceed:

$ \frac{2}{3}\cdot 66.66\dots = 44.44\dots $

But if everyone knows this, no one should choose more than 44.44.

Apply the same reasoning again:

$ \frac{2}{3}\cdot 44.44\dots = 29.62\dots $

And again.

And again.

Each round of reasoning lowers the maximum rational possible choice.

The only number that survives this argument repeated indefinitely is:

$ 0 $

So, under perfect rationality and common knowledge of that rationality, everyone should choose 0.

Answer: they should choose 0.