The transversal game

Two players take turns claiming empty cells from an $n\times n$ grid. The first player (if any) to occupy a transversal (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 optimal play? Our aim in this paper is to show that for $n\ge 4$ the first player has a winning strategy. This answers a question of Erickson.