Something very key to consider here is activities like football, checkers and hangman, are competitive. RPGs can be, but they aren't really about the players competing against each other (in fact competitive campaigns are often frowned upon---especially if the competition involves PCs fighting one another). There is a style of play where they are competing against the GM but adversarial GMing is also generally frowned upon and I think most groups take a more cooperative approach. Even me, I run games with the kind of GM authority we are talking about but I have no issues asking players for input, being transparent about my rulings, etc. For me the aim isn't to beat the players.