A splashfight. Did a lot as a kid and never lost.
Well, 20 feet will almost certainly mean mutual destruction with such pistols, so I’m going to specify that the combatants turn back to back and face either other, draw out their swords and shoot one another. Hopefully no deaf policemen will hear the noise, and come and kill the two dead boys…
First thing that comes to mind is, I could lose this contest, quite possibly to the death. That thought eliminates consideration of some weapons, like baseball bats. The hell if I want to die by being bludgeoned to death with a baseball bat.
The rules here are unclear on whether this can be turned into a contest that’s strictly one of intellect. If it can be, I’m not gonna go with chess, despite a decent facility for the game. There are still plenty of people in this world who can beat me with ease.
The game I’d choose is the 2-person version of Blokus. I’m pretty good at it, and it’s pretty obscure. I know some Dopers have played it, but I’d bet most haven’t, and I’d feel pretty comfortable that the vast majority of people who would challenge other people to a duel would have little if any experience with the game.
Since the terms of the duel are to the blood, to the wound, or to the death, as the other person chooses, the blood/wound/death would be administered to the loser by the officiants of the duel in as quick, painless, and impartial a manner possible in all cases. (E.g. death by guillotine if ‘to the death’ is chosen.)
If I can’t make the contest one of intellect, then I’m gonna go with blondebear’s razor-edged Frisbees, with the following additional details. Each duellist will start off with one Frisbee, and will be equipped with metal forearm protectors sufficient to deflect an incoming Frisbee. (Other than those protectors, the duellists will be wearing matching lightweight tunics that the Frisbees will easily slice through, and nothing more.)
The walls of the enclosure shall be padded with foam to minimize deflections, and there shall be a line diagonally across the 20-foot square that neither contestant may cross, on pain of losing the duel. There shall be a very thin foot-high barrier defining this line, which shall ensure that when a Frisbee comes to rest, it’s on one side of the line, or the other; the contestants will be unable to fight for possession of a thrown Frisbee that is on the ground, but whoever’s side it lands on, can pick it up (carefully - you don’t get gloves!) and throw it at the other duellist.
Actually, my original choice was going to be the Holy Hand Grenade of Antioch, but the footnote excluded it on multiple grounds.
HK MP5, fixed stock, dialed to 3-round bursts.
If no handguns are allowed, an 8-foot spear.
Balloons and blunderbusses. Just leave the pickelhaube behind.
So, the minimum damage is to blood?
Diabetes lancets I guess. Dialed up to 4.
I wouldn’t want to kill anyone, and I don’t think those are capable of killing anyone.
Ikea to the death!
Both parties receive an unassembled Ikea chair in an unopened box, and the instructions are in Swedish. You must assemble the chair correctly with no parts left over before wielding it against your opponent.
Thumbtacks, to blood. Similar idea to Ashtura but more readily available and probably easier to grip. Nobody’s going to die, or even be significantly injured, it’ll be over quickly, and I’ll just hold my arm out with an expression on my face that expresses just how stupid this whole duel thing is.
ETA: I just noticed/realized that the opponent gets to choose whether it’s to the death or not, which my mind rejected probably because it just means I’m going to die (I refuse to fight). I’ll need to think on the best weapon to allow for a quick kill but enable a painless blooding, in the non-existent chance this asshole doesn’t want to kill me.
Blood poising, perhaps, or maybe your opponent is a hemophiliac. Either way…YOU MONSTER!!
Foam pool noodles. We’re going to be here a while…
I’d like to note that these restrictions are really cramping my style in picking a weapon that can kill me instantly and also allow me to live in the 0% chance this guy doesn’t pick death. (If you’re wondering why I’m assuming that, it’s because I’m both clearly not a threat, and that there’s no prestige to be gained from beating my soft flabby self in combat. Nobody will fight me for status. They will “duel” as a way to murder.)
Given my druthers I’d really like to give them a bazooka with a thumbtack taped to the barrel (to cover the 0% likely option), but I suspect that’s banned. 
Atom Bombs, at ten paces.
Or, sullen looks at 3/4 of a mile.
I would bring my kindness.
And I would kill you with my kindness.
I’m going with a variation on the challenge between Bilbo and Gollum:
First person to correctly identify the item in my pocket wins. My opponent can even go first.
M1 Abrams.
Yes, I was going to say,
“Black Queen’s rook to Chronos’ windpipe. Checkmate!”
I also thought of Wiffle bats, but then I considered being bludgeoned to death with whiffle bats and reconsidered. Your best bet is to go with something you might be better at or more comfortable with than the other guy. And I think choosing a game or such id out, unless you want to be hit upside the head with a "Candyland’ box.
Just because your ‘weapon’ is a chessboard or such doesn’t mean the other person can’t use their hands and feet.
In MY hypothetical reality, I am exempted from all rules and conditions that I find inconvenient to me.
A cast iron skillet