The advantages/disadvantages listed are somewhat biased to my experience as a herald and a fencer, but I've tried to consider all the effects they may have.
Single Kill, Double Elimination
This one can be considered the baseline tourney format, and is certainly the most common in Lochac.Each round, fighters are paired up with someone they haven't fought yet (if possible), and if there's an odd number, one fighter is given the bye (which may be fought against a designated person not otherwise participating, but need not be). Each bout goes to a single kill. If a fighter is defeated twice, they're eliminated from the tourney, so the field starts to narrow sharply after the second round.
A common variation on this is to alter the number of losses required for elimination, or to fight the final pairing as a Best-of-Three (or five, &c.). Double kills can either be taken as a loss for both, or refought.
This format requires a lists officer to be in attendance and monitoring the progress, recording losses and determining the next pairings. Often, the herald will assist by recording each bout's results.
A double elim tourney has the drawbacks of requiring an attentive lists officer (who needs to be both recording results and working out the next round at once), one per field if multiple fields are running, and can rather limit the amount of fights that the participants get each. Advantages can be the speed of the tourney, as rounds get successively shorter, and the definitive result at the end (thus its use for many Crown Tourneys).
The single kill, double elim tourney tends to have the highest formality, with every bout announced and salutes made. This doesn't mean that others can't have this level of formality also, but they tend not to as often.
Atlantian Speed Tourney
The speed tourney, which apparently came out of the Kingdom of Atlantia, can be seen as a variation of the single kill, double elim tourney, designed to increase the speed and decrease the lists work.The fighters line up, then the line splits in half and the fighters pair up with someone from the other side of the line. The fights are done with multiple going at a time, if space allows (so, it requires a larger field than a plain single kill, double elim tourney). If the first bout is won, that fighter goes to the winner's circle, and if lost, to the loser's circle. The second round is fought in among each of these circles. The winner of a fight in the winner's circle stays in that circle, while the loser moves to the loser's circle. The winner of a fight in the loser's circle stays in that circle, while the loser is eliminated from the tourney.
In terms of a double elim tourney, the winner's circle can be seen as "two lives remaining", and the loser's circle as "one life remaining".
The speed tourney requires almost no lists or herald intervention, other than to start it, though the multiple fights mean that more marshals are required, and there can be some confusion among the fighters if they're unfamiliar with the format. It's also very quick, as the name implies, and has almost constant action, so is better for spectators, though can be somewhat unsatisfying to those who want to fight more, as with a double elim.
Round Robin
In a round robin tourney, each round the fighters are paired up with someone they haven't fought yet. There are no eliminations, and a round robin of N fighters will have N-1 rounds, as each fighter faces each other fighter once. The winner is the one with the highest number of victories, or may be decided in a final bout after the main rounds.A round robin has the drawbacks of the amount of time required, as each round stays full, but is easier on the lists officer, as they can prepare the full list of rounds in advance, and then just need to record scores. It also allows more bouts for each fighter, as they face every other fighter once.
Meatgrinder
The meatgrinder format has been gaining possibilities, and is a sure way to make sure fighters get plenty of action. An example of a meatgrinder for five is thus: The fighters are numbered, 1, 2, 3, 4, 5. In round 1, fighter 1 takes the field, and faces fighters 2, 3, 4, and 5. In round 2, fighter 2 takes the field, and faces fighters 3, 4, 5, and 1. And so on for rounds 3, 4, and 5.Essentially, the fighters form a queue, and then fight the rest of the queue in order, then join the back of the queue (which gives them a break between their round on the field, and their next fight).
Meatgrinders are long, but take almost no lists intervention past counting victories (a common variation is whether to count only victories during that fighter's round holding the field, or all victories). The herald's interaction is also greatly reduced, just making sure the fighters know who's up next.
Every fighter will face each other fighter twice, once each as the attacker and defender. My first tourney took this format, and with 10 fencers, that made for 90 bouts (10 rounds, each of 9 bouts). About three hours long. While it's hard to go unsatisfied after a meatgrinder, be careful about setting them during summer, when fighters may overheat.
Valhalla
The Valhalla tourney format is a semi-melee, even more so than the Atlantian speed tourney. It requires no lists officer, and likely no herald.All fighters take to the field, and fight a series of individual bouts among themselves as they choose (or it may be treated as a full melee, depending on the preferences of the participants). If defeated, the fighter leaves the field. However, when a fighter is defeated, all those that they had sent off the field themselves are brought back to life. The winner is the last fighter standing - and thus, the person who necessarily defeated all of the others themselves, without dying in the interim.
Because of this, the winner of a Valhalla tourney is undoubtedly the true victor, even if by endurance rather than sheer prowess. However, because of the constant resurrections, the Valhalla tourney can take several hours (or ten minutes, it all depends), and grows with numbers much more fiercely than other formats. It's possible to get only a single fight in a Valhalla tourney, if you're the first victim of the victor, and they're on particularly fine form, but it's unlikely.
I'm aware there are a great many other formats, but as I've not participated in them (or perhaps they were unmemorable to me), I shan't list them at this point.
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