What "game"is Buck playing and what does he hoppe get out of it?

What point in the book or chapter are you referring to?

Refer to chapter 4 of the book

Buck is playing something like the Game of Thrones, except for dogs. Francois discovers that Spitz is missing and praises Buck. As the dogs are about to be harnessed, Buck trots up to the spot that Spitz occupied. Francois brings Sol-leks to the position of leader instead. Buck is furious and springs upon him. He will not let Francois harness the team. Francois retaliates by bringing in a heavy club. Bitterly snarling, but remembering his lesson at the hands of the man in the red sweater, Buck stays just out of reach. Both men try to catch him, but Buck will not relent. He wants the leadership. AFter an hour, Francois finally throws down the club, and makes a place for Buck in front.