For those who aren't aware, Richard Bartle is one of the creators of the first MUD, and thus one of the pioneers of online gaming. He researched the personality traits of online gamers and divided them into four main groups: Explorers, Achievers, Socializers, and Killers. The test I linked to tries to gauge which of those groups you best fit in, and how strong your leanings are toward each. I test as an EASK, with Explorer way ahead and Killer as a distant fourth.
The most obviously applicable trait to PvPing is Killer. That's what it means after all, your interest in fighting and defeating other players. It's just not that strong in me, all told. That's not to say I don't get competitive; I absolutely do. But I need more of a goal than the defeat of another player to keep interest for very long.
The fact that Explorer and Achiever are my number one and two traits is part of it too. If you think about it, those two are pretty much the single-player traits; the ones that pertain to personal growth. You may group with others to explore places you can't reach alone, or to fight monsters that can't be soloed, but ultimately the grouping is a means to those ends. I enjoy grouping with friends, and I'm definitely more social nowadays than I used to be, but ultimately I want to be able to go at my own pace and do my own thing. I'm more likely to hang out in voice chat and talk with people while I'm off soloing than to actively seek a group.
None of this is to say that I actively avoid group play; far from it. I raided actively for years in WoW, and I doubt I would have still been playing by Burning Crusade if not for having a guild of friends and my original raid to keep me coming back. And then during Burning Crusade Belghast lured my real-life friends and I into House Stalwart, and from then on it was those connections that kept me around.
So how about you? Does PvP get your blood pumping or leave you cold? Would you rather be playing in a group or running off on your own?