Game of Cycle

Karan and Arjun got bored of finding who is more lucky based on coin tosses. So they reached Avi, to seek for a better game. Avi made them sit on two isolated screens where they can't see each other's screen and told them a number $N$. Now both of them get a few pairs of numbers on their screen and they just have to choose any random out of them (Just like how you choose head or tail without any o