I'd like to play a game with you. Send me, privately, a real number between 0 and 100, inclusive. (No funny business. If you say "my age", I'm going to throw it out.) The winner of this game is the person who, after a week, guesses the number closest to 2/3 of the average guess. I will reveal the average guess, and will confirm the winner's claims to have won, but I will reveal no specific guesses.
Suppose that you're a rational person. You also know that everyone else who plays this game is rational, you know that they know that, you know that they know that, and so on. Therefore, you conclude that the best guess is P. Since P is the rational guess to make, everyone will guess P, and so the best guess to make is P*2/3. This gives an equation that we can solve to get P = 0.
I propose that this game be used as a sort of test to see how well Aumann's agreement theorem applies to a group of people. The key assumption the theorem makes--which, as taw points out, is often overlooked--is that the group members are all rational and honest and also have common knowledge of this. This same assumption implies that the average guess will be 0. The farther from the truth this assumption is, the farther the average guess is going to be from 0, and the farther Aumann's agreement theorem is from applying to the group.
Update (June 20): The game is finished; sorry for the delay in getting the results. The average guess was about 13.235418197890148 (a number which probably contains as much entropy as its length), meaning that the winning guess is the one closest to 8.823612131926765. This number appears to be significantly below the number typical for groups of ordinary people, but not dramatically so. 63% of guesses were too low, indicating that people were overall slightly optimistic about the outcome (if you interpret lower as better). Anyway, I will notify the winner ahora mismo.
The purpose of this game, admittedly, is to test just how complacent / obedient the Overcoming Bias / Less Wrong community has become.
Think about your assumptions:
First you've got "common rationality". But that's really a smokescreen to hide the fact that you're using a utility function and simply, dearly, hoping that everybody else is using the same one as you!
Your second assumption is that "you gain nothing by defecting alone".
There's no meaningful sense in which you're "winning" if everybody guesses zero and you do too. The only purpose of it, the only reward you receive for guessing 0 and 'winning', is the satisfaction that you dutifully followed instructions and submitted the 'correct' answer according to game theory and the arguments put forth by upper echelons of the Less Wrong community.
In fact, there is much to gain by guessing a non-zero number. First of all, it costs nothing to play. Right away, all of your game theory and rationalization is tossed right out the window. It is of no cost to submit an answer of 100, or even to submit several answers of 100. Your theory of games can't account for this - if people get multiple guesses, submitted from different accounts, you'll be pretty silly with your submission of 0 as an answer.
"But that would be cheating." Well, no. See, the game is a cheat. It's to test "Aumann's agreement theorem" among this community here. It's to test whether or not you will follow instructions and run with the herd, buying into garbage about a 'common rationality' and 'unique solutions', 'utility functions' and such.
You see, for me at least, there's great value in defecting. You of course will try to scare people into believing they're defecting alone, but here you're presupposing the results of the experiment - that everybody else is dutifully following instructions. So anyway, I would be greatly pleased if the result turned out to be a non-zero number. It would restore my faith in this community, actually. And to that end, I would submit a high number...
If I were to play.
I will be very surprised if more than half of the answers are 0.