EDIT2: Oh I see the confusion now. It's from the previous competition where I questioned choice of game going to 3rd place winner rather than THE winner, I still think it's a mistake because people might be willing to throw the 2nd place for 3rd just to choose the game. Choice of game used to be nice reward and the winner could always withdraw itself if they think they will win (after all it's friendly competition,right?), that was practiced for a little bit in the beginning.
Yep, that's it.
As for the winner picking the next game, take a look at this, especially the last column.

18 games played, 15 games won by two people. I don't want to pick a game that I like only to withdraw from the competition because I probably will win it. So I would let someone else choose the game, maybe the runner-up. Then I might win the game again, let the runner-up choose the next one ... That's basically what we're doing now, without losing time by going back and forth about who will pick the next game.
Choice of game used to be nice reward
If you win one game, yes. If you win ten, no.
edit: I sent a pm to Mr. Peabody, let's see if he wants to do the next game.