The Swiss tournament system is popular in a wide array of sports and games. It provides a good balance between keeping the tournament length reasonably short and reaching an outcome that approximately reflects the relative strength of the participants.
While variations exist, the simple explanation for the Swiss system is that for a set number of rounds each participant plays on other participant, the participants are matched against opponents that have won approximately as many games as themselves. Thus after a few rounds participants will play opponents of approximately the same strength, and in the last rounds the top players will clash, fighting for the win.
One downside of the Swiss system is the strict requirement for playing rounds in order. All games of the preceding round must be finished before even the planning of the next round can begin. In many tournament settings this is not a big issue, but any tournament that requires each match to be scheduled individually will require a significant amount of time per round to allow competitors headroom for scheduling issues.
But is scheduling only one match per round for each competitor really necessary? The answer I have found is: No. While using past results for scheduling future rounds is necessary for the system to work, relaxing the system a little does no major harm. The Swiss system has self-correcting nature, a good player that lose in an early round will get easier matches for a while, thus tending to catch up, while the opposite will happen to a lesser player with a good streak in the first rounds. Thus it is the final rounds where accurate results are most important.
Based on these observations I have devised the follow Swiss variant, which I call the Nohat Tournament System:
For the first round 2, 4, 6 or 8 matches are played by each competitor, the pairing can be either random or, based on a previously established rating, pairing competitors so that they each meet a mix of high and low skilled opponents.
For the following rounds each competitor play two matches with pairings generated based on tournament performance like in standard Swiss.
In the final round each competitor plays only one match, this reduces the likelihood of big swings in the final round that cannot be corrected because there are no more rounds.
In order to determine round count I have set up the following formula to determine how many players a particular round combination is suitable for: Add 1 to the number of matches in each round, multiply the results together, lift the result to a power of 0.8. This gives the maximum number of players that that combination is suitable for. For instance a tournament with 6 matches in the initial round, three more rounds of 2 matches each, and a final round of 1 match would give (7*3*3*3*2)^0.8 ≈ 115.34, thus allowing up to 115 players. This limit makes a situation where two or more players both have perfect records at the end of the tournament improbable, though this can also be achieved at higher limits by pairing specifically to avoid this issue.
In order to quantify the differences between Nohat and regular Swiss I have set up a simulation. Download program and run result here, you can simply look at the run results, or you can run the program with different parameters using node.js. Note that the simulation include a number of setups where clearly too few rounds are played, for 64 players the above formula suggests playing at least 8 rounds of Swiss, or a 4-2-2-2-1 format Nohat tournament.
The first 3 results given for each simulation tell us how well the tournament result reflect the skill of the players. These numbers give the average number of positions that tournament results deviate from rank according to rating for certain groups of players. Mostly these numbers seem to depend on the number of matches played with Swiss doing slightly better per match, but much worse per round.
The next number is the fraction of the 10 possible top 5 matches that has been played. If this number is too low the tournament might feel unfinished to the top players, with many potentially exciting matchups not having happened. Above 90% one might fairly call the tournament too long, with too few top matches during the final rounds.
Next is the probability that multiple players scored a perfect record. If this happens it is a clear sign that the tournament was too short, and whoever places second will probably feel that they got robbed. For Swiss this is impossible when 2^rounds is greater than or equal to the number of players. For a Nohat tournament this property is more fluid, but the probability does seem negligible at the recommended number of rounds. Note that in order to speed up the process the simulation use a very simple matching algorithm, with proper matching this stat should improve somewhat.
Finally there is the average difference in skill rank between players who are matched against each other. With random matchups or round robin this number would be 21.67, a difference of over 400 rating points. The Swiss and Nohat systems make this number somewhat lower, reaching 13 to 16 in the most relevant configurations for this player count. For amateur competitions more equal matches would generally be desirable, but without introducing McMahon points or similar there is little that can be done to even matches when the field is this wide. Overall Swiss does better by approximately 0 to 3 ranks depending on the exact choice of tournament models. This stat in particular does better when keeping the initial round smaller, so maybe save the big 6 and 8 match initial rounds for tournaments where the field is expected to be tight.
A simple pairing mechanism for the Nohat system is to do a normal single round pairing multiple times, this method however fails to take full advantage of the system. A proper Nohat pairing should always schedule either two games per player, or the entire round in one go. This way no byes are needed, except for the last round, it is also in some cases possible to avoid some matchups between players with equal points, if for instance 5 players are in the lead with the same amount of points they may simple play 5 games between them with no games against lower scored players.
For Tak tournaments we have used a simulated annealing solver, optimizing on score difference squared. Some pairing systems prioritise balancing the top matches. Usually this makes no difference as the top matches typically end up as balanced as they can be using an unbiased solver anyway.
Dos and don'ts
Do not make rounds bigger than 2 matches after the initial round, as these could cause a case of elevator ranking, where some players alternate being too high and too low in the table.
Do not attempt to pair players against those equally skilled in the first round, as this would result in a very mixed table with uneven matches simply delayed until the following rounds.
Do combine the Nohat and McMahon systems if appropriate. If done, the initial big round must be dropped, the McMahon points will act in its place.
In a scenario where the top is expected to be tight with a lot of equally skilled competitors it may be okay to play 2 matches in the final round. If in doubt, always play 1 match in the final round.
One may end a tournament with multiple 1 match rounds. I believe this to be unnecessary, but it does not introduce any flaws to the system.
Do not use the Nohat system for fixed schedule tournaments. A 2 match round cannot necessarily be played within two time slots. Furthermore the primary Nohat advantage of fewer rounds is moot as in this case it is the match count that determines tournament length. Play regular Swiss instead.
Champions League Swiss
While the systems are designed independently, there are similarities between Nohat and the system proposed for the 2024 Champions League, which in the press has been referred to as a Swiss system. The Champions League system is basically like the first round of a Nohat tournament, with 10 matches, and no further rounds played. This system seemingly defies some of the principles set up in this post, namely it does nothing to make the top teams play one another, and as a result it is quite possible that multiple teams will get a perfect score. The reason that this system still works is that it is only a qualification for the next stage, the top 8 are all equally winners, and any team with a perfect or near perfect result should easily be in that group.
I wouldn't have called this system "Swiss" as it is in most respects much more closely related to the group stage that it replaces. But I find it an interesting development none the less.