For any finite matrix, the first player wins

http://dev.whydomath.org/Reading_Room_Material/ian_stewart/ChoosilyChompingChocolate.pdf

The chomp game discussed in the link is the game that you have proposed. However, for an infinite matrix, the first player does not necessarily win. Furthermore, this is the type of game in which one player is guaranteed to win, but the mechanism by which it occurs is uncertain.