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

The game is played on the board. The board has some (finite) number of lines drawn on it. The 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 the player who begins has a winning strategy when the game is played on n*m rectangular grid?