I think in the ideal MMO a limited amount of players should be able to become "heroes", seeking to keep their status (becomes it has advantages), and others should try to acheive it. It's what EvE online does extremely well : players are all racing against each other to become the king of the hill, the bandit warlord, the corporation manager, the head of the security forces, and so on. What is especially interesting is that players are very far from just fighting each other over the actual ingame combat. They start social influence games inside corporations, try to take over political structures etc. That's the kind of (slightly emergent) competition I want to see in a truly pvp MMO.
Edit : let's expand a bit :
First, the gameplay itself should require more player-player interactions, and not necessarily competitive or cooperative. The classical problem with this is that it's very difficult to have meaningful interactions with people you don't already know outside the game, without devoting waaaay too much time to the game. Even a casual player without friends playing the same game should be able to play with others, and that I think is a real challenge in game design.
Secondly, in most classical MMO's, when you have seen/defeated everything that the game had to offer, that's pretty much it. You can still play for pixel crack, sure. But that's empty. The end game shouldn't be a competition over nothing like a CS game, but it shouldn't be a quest for super rare pixel crack either. I think end game players should, depending on what path they chose, be more like guild masters, organising their alliance, training other players, etc. and generally monopolizing the benefits of your guild. But also fight against the inside threats, especially people from the same organisation as you trying to take command.
