Before reading the studies, we did this exercise in my Experimental Econ class a couple years ago. However, beforehand the teacher didn't let any of us know P=0 even though it should have been obvious.
We did the test 4 times in a row.
There were 12 students in my class (an upper division econ class at a private school)
Test 1: I guessed 20 (answer was 22, I was closest)
Test 2: I guessed 12 (got it exactly)
Test 3: I guessed 7 (split the reward with one other student)
Test 4: I guessed 3 and the answer was 2
If more tests were done I could only assume the whole class would have eventually gone to 0.
When reading the paper it amazed me how many people put 0 as the answer on single trials. Yes, P=0 but a lot of people don't know that (the study was done by advertising a monetary award in the newspaper) and even more may know that and still guess what others will put.
The logical way to look at the test is breaking it down into what level you think people will guess on.
Level 1: everyone guesses 100 so guess 66.66
Level 2: What idiot would guess 100, everyone guesses ~67 so guess 2/3*66.66 = 44.44
Level 3: But everyone will think ~44 so guess ~30
Level 4: Guess 30*2/3= ~20 and so on
There's no reason for the game to go all the way down to 0. If everyone is playing 1, that's an equilibrium because 2/3 of 1 is closer to 1 than 0.
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.