Two players, Alice and Bob take turns in filling in the cells of a $19 \times 19$ grid (which is initially empty). Alice writes a $1$ in her chosen cell, and Bob writes a $0$ in his chosen cell. When all cells are filled in, they calculate the sum of each row, and the sum of each column. Let the largest row sum be $A$, and let the largest column sum be $B$. If $A > B$ then Alice wins the game. Bob wins if $B > A$, and it is a draw if $A = B$. Does either player have a winning strategy?
Points I have gotten:
Alice can force a draw as the diagonal has an odd number of cells, so it's Alice who can avoid ever making an off-diagonal move what's not a response to the opponent.
Alice can simply copy cat off Bobs move, mirroring his moves along a diagonal, and choosing a random choice in the diagonal when he chooses a square on the diagonal.