Guess the smallest number Three people play a game where each of them writes down a positive integer at the same time. The one who writes a unique and smallest number wins one dollar from every other person. This means if two people happen to write down the same number, then the third one gets one dollar from each of them regardless. Now what number would you put if you were to play this game?

Empirically, if two of the three individuals each writes down a positive integer $N_i$ independently with probability $$P(N_i=n)=2^{-n}$$ then whatever the third writes, the expected outcome for each of the three is $0$ and so the third may as well follow this strategy too (in which case there would then be a tie with probability $\frac17$).  
It also seems that the first two follow a different iid strategy then the strategy above will produce a positive expected outcome for the third.  So it looks like the optimal strategy in the absence of collusion.  
The first two can develop a collusive strategy: suppose the first always writes $1$ and the second $2$, with them sharing the winnings after the game.  The third player would then always lose.  
