TITLE: A Transversal Achievement Game on a Grid
QUESTION [7 upvotes]: Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an n x n grid.
The first player (if any) to occupy some transversal (i.e., a set of n cells having no two cells in the same row or column) is the winner.  What is 
the outcome of the game given best possible play by both players?

REPLY [3 votes]: As Michael Albert mentioned, an interesting variant is when X wins by building a permutation, while O wins only by preventing X from building a permutation.  (The standard term for such a game is maker-breaker.)
According to my coding (which should not be regarded as infallible), X wins this game on a $4\times 4$ grid both when X goes first and when O goes first.  By a straight-forward inductive argument, this implies that X wins the maker-breaker game on an $n\times n$ grid for all $n\ge 4$.
The question remains of whether this has anything to do with the version of the game posed by Martin, where both X and O are trying to build permutations.