Unfair Game

As Dhruvi and Aashka don't like dhaval's lecture, they are playing a game. The game starts with set $A$ of $N$ non-negative integers. Aashka takes first turn and after that two players move in alternating turns and both play optimally. Now, during each move a player can select any number $X$ from the set $A$ and convert $X_{decimal}$ into $X_{binary}$. Then, she can select any $i^{th}$ bit o