A roleplaying game is a game in which:
A. One participant takes the role of a referee or "game master"
B. The game has no built-in end state. In other words, the rules don't explain how the game ends.
I think B is the key point, since it folds aspects of competition (or the lack thereof) neatly into a broader, much more easily tested case. It is also, I believe, the critical point that makes an RPG different from everything else.