Game Of Dots

Alice and Bob are playing a game on a grid consisting of $n\times m$ dots. The players take turns adding connecting adjacent dots, i.e. adding one edge at a time to each $1\times1$ square and for each $1\times1$, the player adding the final edge to it gets $1$ point and an additional chance. The player will get an additional chance until the edge added by him/her doesn't complete a $1\times1$ squa