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John_Maxwell_IV comments on The Aumann's agreement theorem game (guess 2/3 of the average) - Less Wrong
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I've played this game, with an actual small prize to give some incentive in favor of cooperating with the experiment. I was surprised at the number of intelligent-seeming people who did not understand that 0 was the "rational" solution. I was unsurprised at the number of people who understood, and submitted answers they knew were irrational just for fun.
This is a bad test of an agreement theorem. There's no reason to believe that participants are motivated to agree, or that their expression of guess is the same as their belief in the "correct" guess.
Right. I'm going to guess the smallest number I think no one else will guess to maximize my chances of being the only winner. I don't place any value in winning along with a lot of other people.
<strike>Also, the OP is wrong that the iterated 2/3 process will eventually produce 0. If everyone plays 1, then 2/3 of 1 will round back up to 1.</strike> Edit: Sorry, in a version of this game I once played you were restricted to guessing integers.
2/3 is a valid guess.