My offline gaming has explained my online gaming history to me. I am a carebear because I want to play with friends, not against them.
I can be intensely competitive, but I prefer to be cooperative. We should all be on the same team. We should all win. I would rather have an allied victory than conquer the entire board; even in a single-player 4X, I have this unhealthy tendency to let others grow and prosper rather than playing aggressively. The joy of PvE is that we have two sides, and all the Ps are on one side of that divide.
I like my PvP in small, discrete units. FPS games are conveniently concentrated doses. Games with “duels,” “matches,” and “rounds” are usually the right length. If I need to log in more than once for a “campaign,” we are probably not in my element. I don’t want to spend days dwelling on how to destroy you. I can be rather good at that, but those are not thoughts I like having in my head.
I have had some pen-and-paper RPG time lately, and I like working as a party. If there is serious intra-party conflict, I would rather skip sessions when the problematic players/characters are present. I played Diplomacy this weekend, and eight hours of suspicion and war were incredibly engrossing but not how I want to regard my fellow players. I have also been playing some Eurogames lately, and I enjoy the common characteristic that, while the games are interactive and competitive, there are fewer instances of directly striking at each other, and players are rarely eliminated mid-game.
I do have a fondness for games where it is many-against-one, with a dungeonmaster-equivalent who is explicitly “against” and not a neutral arbiter. Those are hard to find and balance. Isn’t there a zombie FPS in which one player is effectively L4D’s Director, sort of a RTS versus his FPS friends? [Zombie Master -- thanks, Mikeful.] I like that notion more than survivors versus infected.