KöMaL 2019 B5052

Two players, First and Second take turns in writing a number 0 or 1 in the fields of a $19\times 19$ table, initially all blank. When all fields 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 First wins the game. Second wins if $A<B$, and it is a draw if $A=B$. Does either of the players have a winning strategy?