Just a little thought on sums of a specific form

I just had a nice thought about simple sums of the form .

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An interesting stochastic game (Part I)

We consider the following game played by two players (P1, P2). Both players choose a bitstring of length n (e.g. n=3, then available bitstrings would be 010, 110, 001, 111, and so on). Now a 0/1-coin (i.e. a coin with one side 0 and the other side 1, both occur equally likely) is thrown and the string is recorded. The game ends as soon as the last n cointosses form a string belonging to one of the two players. The player whose string is hit, wins.