Here is a simple game I've invented (if the idea is not fresh, then please let me know):

The game is played on a board.
The board has some (finite) number of lines drawn on it.
A pawn is placed on each intersection point of (two or more) lines.
Two players take alternate turns removing pawns.
On each turn, a player removes one or more pawns.
All pawns removed in a single turn have to be taken from the same line.
The player who cannot make a move loses (alternatively: the player who takes the last pawn wins).

Here is my question:
For what values of m and n does the player who begin have a winning strategy
when the game is played on an $n\times m$ rectangular grid?