Guess the number

I haven’t posted a puzzle in a while.  The following problem has the usual nice characteristics; it works on a cocktail napkin or as a programming problem, via exact solution or simulation, etc.

I am thinking of a randomly selected integer between 1 and m=10 (inclusive).  You are the first of n=3 players who will each, in turn, get a single guess at the selected number.  The player whose guess is closest wins $300, with ties splitting the winnings evenly.

Here is the catch: each player may not guess a number that has already been guessed by a previous player.  As the first player, what should your strategy be?  Which player, if any, has the advantage?  And what happens if we vary m and n?

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