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Assume we have $n$ players (each knows the number of competitors). Each has to chose a natural number and the player that has selected the smallest number, that appears uniquely, is going to win (if such a player exists). Hence, for example, if one player selects number 3 and number 1 and 2 has been selected two or more times and no one else has selected number 3, that player is going to win.

I am wondering whether with game theory one can figure out an optimal strategy (like selecting a probability distribution for each natural number to select it).

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