Game Night

There is a grid of N rows and M columns, and it is filled with red and green tiles. The tiles are labeled such that the top left tile is $(1, 1)$ and the bottom right tile is $(N, M)$. On this grid, each tile has a score. This score is equal to the minimum Manhattan distance of the tile to any red tile on the grid. In particular, the score of a red tile is always $0$. Alice and Bob are playing