Game on a Grid

Alice and Bob are playing a game. A single pawn is placed on a grid. The grid consists of cells (X, Y) for non-negative integers X and Y. Some cells of the grid have been marked as impassable. A move consists of choosing some positive integer D and moving the pawn from (X, Y) either to (X − D, Y) or to (X, Y − D), of course, if this cell belongs to the grid. In other words, we move the pawn a posi