<< [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.]
According to the rules of the game, no communication is possible. This could be ensured by simply not revealing to the players the plays of any of the other players, or even the extent to which each of them won -- until the game is over. (Also, just as a motivation to have an opportunity to spend your winnings -- if any -- the game could be in principle played at the moments 1 - 1/n hours, n = 1,2,3,.... Then after one hour the game is over.) The question asks for the optimal strategy. I think if there is any optimal strategy, it must be 1,1,1,1,1,1,.... FOR: If you play any other strategy at all, you risk the possibility that exactly one opponent is playing the 1,1,1,1,... strategy -- guaranteeing that you will end up losing money. Whereas if you play 1,1,1,1,... you are guaranteed to do no worse than break even. --Dan