Infinite Grid Game

After a long period of relaxation Alice and Bob decided to play a game. This time of course not a number game. The rules of the game are as follows: There is a vehicle situated at the point (m, n) of a rectangular grid. Only one corner of the rectangular grid is defined, i.e. the left-top point (0, 0), which is also known as the origin. The grid extends infinitely towards east and infinitely towa