I heard this puzzle from Bob Koca.  Suppose we play misere tic-tac-toe (a.k.a. noughts and crosses) where both players are X.  Who wins?

That particular puzzle is easy to solve, but more generally, has $n \times n$ impartial tic tac toe, in both normal and misere forms, been studied before?

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EDIT: Thane Plambeck's paper, mentioned at the end of his answer below, coined the term <b>Notakto</b> for this game. That name seems to have caught on; for example, there is now a <a href="https://en.wikipedia.org/wiki/Notakto">Wikipedia article on Notakto</a>.