I've never been a PvP fan. I find it creates too many personal issues and arguments. But for me it boils down to how one person's game affects others. I choose to play PvE, but I really enjoy playing WITH others in a cooperative manner. So I do like multiplayer online games, but I prefer to play alongside colleagues to achieve a common in-game goal. Again, my choice, and why I joined Mobius' group.
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THIS
Same for me. I like COOP games a lot more than PVP games. Playing with friends or other people and having a social experience is a lot more fun to me than shooting someone (virtually) to pieces.
Most FPS are PVP, which is no real problem. I played CS for years, also competitively, but I always put fun over the competitive aspect (can't deny when you win the one comes with the other...). But it's the social aspects that make completely repetitive games like CS playable for such a long time, at least for me.
I remember one ESL clanwar, when we all started to do and talk utter nonsense while defending bombspots. Hitting "Use Button" (sounded like *moeb* and others could hear that) on the PCs ingame, acting like we were playing CS in CS, yelling "you cheater" in the TS. Until the two guys defending the other bombspot rushed in. Of couse they were confused and one of them asked "What's going on HERE?!". Next thing was someone saying "Hey, who's defending the other bombspot now?" followed by "Bomb has been planted"-message. We laughed so hard for a few seconds, went to the other spot and somehow still managed to win the war.

THIS is what remains in memory. It's the social interactions that sometimes lead to weird storys. I don't even remember what the name of the other clan was, because it doesn't matter... shooting someone is not a social interaction.
League of legends, Dota, Starcraft, Call of Duty, all got big from their single player campaigns right?
Nope! The pvp.
I have to disagree, too. I can talk for CoD and SC. They would never have gotten very famous without their stunning SP campaigns. The SC SP is still one of the best storys and campains game-wise and I know a lot of people (me included) who only played SC because of singleplayer. Same goes for CoD.
Same thing for Diablo 2, which was mainly a COOP game for me with a stunning story.
Counter-Question: What makes games like Left 4 Dead or Borderlands so famous?... it's because they are actually meant for coop play.
Other example which may suit you more: TeamFortress2! It has no SP, but what makes it famous is the Team aspcet. Teamplay is key, which keeps it interesting over a long time.
I don't say "take PvP out of the game" because it really belongs there. It just needs to be limited somehow. Best thing to do this is gameplay wise. Through authority NPCs rushing in to help (when the bounty is very high even the military could take notice and send squad) in the faction controlled systems, through fees and bounty, which of course need to be balanced. Also a designated "pirate" just isn't allowed to land on factional stations when there's a bounty on his head and thus he can't buy fuel and equipment there. In a "social" game, ED is one of them (social, factional, economic simulation), there have to be downsides when you are an outlaw. I trust Frontier in this; they'll find a way to do it. Because farming noobs in the starting systems (which are faction controlled) should not happen.
As many tried to explain before. It really is a big issue that when a PvP player attacks a PvE player, he puts the PvE player where he doesn't want to be and spoils his fun for the game by doing so (depends on the PvE player of course how much, but it essentially is like that). Additionally the PvE player will lose the fight in 95% of cases, because the PvP player picked a victim which is no match for him and his equipment.