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?