Just duel it
Some games, of course, do offer a form of PvP to resolve this sort of conflict. This works, in the same sense that setting your house on fire to kill a fly will work. Sometimes it does, sometimes it doesn't, and in every event you're solving a problem with the wrong tools.
The central problem is that mechanics and characters do not line up conveniently. Even if you can just drop a duel in the middle of a crowded street, it's very possible for Character A to have far better equipment than Character B, or a better PvP setup, or a better class all around for PvP. And the situation isn't exactly conducive to a fair fight anyhow -- if both characters start off within arm's reach, then the ranged caster is going to have a hard time beating the heavily armored melee character, even if in-character the latter is meant to be more inexperienced and less capable in active combat.
Plus, well, there's always the simple problem that someone can be a great roleplayer without being much of a player. This isn't as common as you might think -- most of the people who spend a lot of time in the game will wind up getting up to reasonable levels of skill, by osmosis if nothing else -- but it really isn't cool to watch drama get deflated because one of the people involved just isn't very good at fighting back.
And all of that is without getting into games that require you to be in a specific spot to PvP or don't have open PvP in the first place. Try to pull an actual PvP duel off in Final Fantasy XI and it winds up just being silly.
Going through the emotions
The obvious workaround for these instances, and the one that a lot of people use, is the simple trick of using emotes to handle a duel. Write out an emote, give the other person reasonable amounts of time to respond, and onward you go. Assuming a reasonable level of maturity for both participants, this one is ironclad, right? It might not be as visually satisfying, but you can work around that.
The problem here is a more fundamental one, though: Combat in MMOs doesn't work.
Let's make something clear: I am not a martial arts master, or a martial arts novice, or even much of a martial arts aficionado. But I am still pretty certain that the best way to fight something is never to stand in place and swing wildly for several minutes. I'm also pretty sure that most human-sized animals do not require a few dozen stabs with a sword before they die, and that conversely a headbutt from a goat will not just knock you back a bit and deplete your health. Heck, I'm pretty sure that human beings don't have actually hit points to start with.
This doesn't bother me in the slightest, really. Of course, MMO combat is to real combat as an eight-foot-tall man with pointy ears and glowing eyes is to an actual human being. It's the same reason why save points, universal currency, and chat channels don't bother me. This is a game, not the Matrix. (And definitely not The Matrix Online at this point.) But the problem is that if you're dueling in emotes, you have to answer the question of how much actual combat in this world resembles actual combat in the real world.
Sound like a dumb question? It really isn't. Logically speaking, in real combat, levels and all of that are totally irrelevant. If your level stupid character is fighting a level awesome character, a smart strategy and some cleverness should still allow you to win. On the flip side, if you think that levels do matter, you have to essentially build in blind spots and dumb moves for your character to take, else the person who's best at describing a fight is going to win rather than the character who actually ought to win.
This does circumvent the problem of the previous example, but it creates an entirely new set of problems because you're essentially trying to create a fantasy of consensual combat without any actual rules to govern it. And for better or worse, everyone is going to have a slightly different idea of what each character is capable of.
Exit, pursued by a knife fight
Shakespeare's plays frequently had an easy out for violence that either couldn't be faked convincingly or had a foregone conclusion -- the characters in question would leave the stage, and then someone would come on in a bit to announce who won.
This functioned well enough for what it was, but it's kind of a cop-out. (The big climactic battle in Macbeth, for instance, happens entirely offscreen. For all we know, Macbeth threw some dirt and then ran off to France; we're just told he's dead.) But it produces another issue for roleplaying, the one that any comic book fan is familiar with. You say, "Well, clearly, character A would win this fight," and you're met with a resounding "nuh-UH!"
You can present evidence, but you're arguing opinions. There's never going to be a right answer.
And that's pretty much where you wind up with roleplaying combat in a whole. You have options, but none of those options is particularly good ones. I've seen all three in execution, and all three of them have some serious issues that don't seem to go away no matter how hard you squint. Unlike a lot of the issues my articles cover, this one doesn't present me with a really useful suggestion to fix the issue, but at the very least, understanding its existence helps.
If you do have a great suggestion, though, you can feel free to leave it in the comments or mail it along to firstname.lastname@example.org. And you can do the same for regular comments, too; there are more options here than just writing out suggestions. Next week, I'm going talk about something that might seem really innocent, and it is, right until it's relevant: time.
Every Friday, Eliot Lefebvre fills a column up with excellent advice on investing money, writing award-winning novels, and being elected to public office. Then he removes all of that, and you're left with Storyboard, which focuses on roleplaying in MMOs. It won't help you get elected, but it will help you pretend you did.