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.