The current puzzle at The Grey Labrynth is an amusing math puzzle.
http://www.greylabyrinth.com/puzzles/puzzle.php?puzzle_id=206
Hmm, if I were in this situation (ok, maybe I could not play infinite rounds), I would try to deterministically play 3,2,1, 3,2,1, 3... If player 2 and player 3 adapt to this (playing 2,1,3,2,1,3, and 1,3,2,1,3,2,..., resp.) then we all would get the maximum (symmetric) result: 0, at the same time avoiding any collision. At the beginning there will be probably a phase in which it is not clear who plays 1 and 2 when I play 3. This could be resolved by selecting at random before the "sync" has happened. At the moment I cannot see how the 2 other guys (and myself) could have a better strategy than this... Choosing random numbers all the time has two disadvantages: first, all 3 should have the same distribution (which is not possible, since the interval to choose from is not defined), second, you have collisions that reduce your expectation value... [On the other hand, all 3 could communicate by choosing a very obvious code, e.g. 1=a, 2=b,... (all 3 must speak the same language) and deal out how to play. Since you play afterwards infinite rounds this communication would not reduce your expectation value.] Christoph