The NPR blog Planet Money is asking you to guess a number:

This is a guessing game. To play, pick a number between 0 and 100. The goal is to pick the number that’s closest to half the average of all guesses.

So, for example, if the average of all guesses were 80, the winning number would be 40.

The game will close at 11:59 p.m. Eastern time on Monday, October 10. We’ll announce the winner — and explain why we’re doing this — on Tuesday, October 11.

This is a famous game that has been used in numerous experiments investigating whether real people are as rational as game theory and economic theory assumes they are.  Powerful logic suggests that you should guess the number zero:

1. For sure the average will be no greater than 100 so half the average will be no greater than 50.
2. Anybody who is smart enough to figure this out will guess something no greater than 50 so the average will be no greater than 50 and half the average will be no greater than 25.
3. Anybody who is smart enough to figure this out will guess something no greater than 25, etc.

Of course time after time in experiments the actual guesses are very far from zero, demonstrating that people are in fact less rational than economic theory assumes.

Planet Money, however is an intelligent blog and when they analyze the results of their experiment, they won’t jump to that conclusion. They will be insightful enough to see past the straw man.

It all starts at point 2.  It is true that people who are smart enough to figure out point 1 will guess something no greater than 50, but almost all of those people are also smart enough to know that there is a sizeable proportion of people who are not that smart.  And thus these smart people, if they are rational, will not deduce in point 2 that the average will be no greater than 50.  The induction will not take them past point 2.

In fact, some of the smartest and most rational people in the world, professional chess players, guess numbers around 23 when they play these experiments. (To be precise, the chess players were playing a version of the Beauty Contest were you are supposed to guess 2/3 of the average. Their guesses would be somewhat lower in the Planet Money version, see below.) And that is because if someone is indeed as rational as game theory and economic theory assumes she is, and also she is smart enough to know that

1. Not everybody is that rational,
2. Most of the rational people know that not everybody is that rational,
3. Most of the rational people know that most (but not all) of the rational people know that not everybody is that rational

etc., then she will never choose anything close to zero.  Indeed, according to my calculations, the ultra-rational guess in the Planet Money Beauty Contest is about 16.  Here is how I came up with that number.

I think that

1. About 2/5 of the Planet Money readers will be confused by the rules of the game and guess 50.
2. Another 3/10 will be smart enough to know that the rational thing to do is to guess something less than 50, and reasoning as in the straw-man argument they will guess 25.
3. The remaining 3/10 of the population are the really smart ones.
What will the really smart ones guess?  If they agree with me about the remaining 7/10 of the population then for sure they will not guess anything less than
$\frac{1}{2}((\frac{2}{5}) 50 + (\frac{3}{10}) 25 ) = 13.75$
because half the average will not be less than that.  And if they agree with me that 3/10 of the population are really smart and won’t guess anything less than 8.75, then in fact they won’t guess anything less than
$\frac{1}{2}((\frac{2}{5}) 50+(\frac{3}{10}) 25 +(\frac{3}{10})13.75)$
which is around 15.  Notice that, unlike the strawman argument, the implications of rationality are now implying higher and higher guesses, not guesses converging to zero anymore.  And if we take this to its logical conclusion and assume that this 3/10 of the population are the hyper-rational decision-makers that economic theory assumes, then the winning guess in the Planet Money Beauty Contest will be the value $x$ that solves
$x = \frac{1}{2}((\frac{2}{5}) 50+(\frac{3}{10}) 25 +(\frac{3}{10})x)$
which is about 16.  And that is what I just guessed.