Game of Thrones is Παιχνίδι των Θρόνων, right?
Yes, although the official translation (that you will find in books etc) of it is "Παιχνίδι του Στέμματος" = Game of Crown.
So can παιχνίδι be used eg as in "game of cricket"?
Hmmm, that's better translated as αγώνας κρίκετ = cricket match. Sports use αγώνας.
He has a game