Swiss system ARMATI
by Vincent AUGER1
The basic principles of the Swiss Tournament system
Competitions are played in “rounds” during which all players play one game versus another player as determined by the pairing for that round. The pairings are generated by the umpire, often with the aid of a computer.
At the end of each round, each player receives points as determined by the result of his game and he adds them to the points obtained in the previous rounds.
For the next round, each player is paired against another player with a similar total number of points. Players NEVER play each other more than once in a competition.
The result of this system is that the best players accumulate more points by scoring victories and are finally pitted against each other in the last rounds since they have similar total values. That property is also true for middle level or weaker players whose results also lead to similar totals. The overall effect is therefore that players are pitted against players of more or less equivalent value as the tournament progresses, hence their games become more balanced and more challenging. The principal benefits of this system are that all players get to play the same number of games (unlike a knockout tournament) and the system is very efficient at finding the strongest player in the competition with relatively few games or rounds (unlike a round robin tournament).
The value awarded to games (Scoring systems)
The most basic system, and easiest to compute, is a simple Win / Draw / Loss score (typically 2 / 1 / 0). The drawback of that system is that it does not discriminate very well, hence that several players may end up with the same total at the end of a competition.
Several solutions exist to avoid this issue :
1) Adopt secondary scores which are used for tie breaking
2) Adopt a different scoring system, with a wider range of values, which will minimize the probability of ties.
If the tie concerns only 2 players and if they have played each other, then their game result (if not a draw) can be used to untie them. This is by far the best tie-break mechanism, unfortunately it is not always available as the players may not have met during the competition.
Buchholtz (also known as SOS = Sum of Opponent's total Scores), M-Buchholtz (similar but not taking into account the best and the worst totals), Progress (sum of the player's total scores at the end of each round), SODOS (Sum Of Defeated Opponent's total Scores: similar to Buchholtz but counting only the scores of the players against whom you have won your game) are typical tie breakers. They can be used in series (first use the total score, if equal use the M-Buchholtz score, if equal again, use the Buchholtz score, if equal again, use the SODOS). Note that all these tie-breaking systems (except Progress) rely only on your opponents' results (i.e. on the toughness of the opposition you had to fight against), not on the quality of your games.
Goal averages (e.g. the sum of the percentages of enemy key units killed, compared to the enemy breakpoint) are sometimes also used as ties breakers.
The BHGS in UK has adopted the different score system solution. Unfortunately, the 32-0 system they have adopted has several drawbacks, some of which are fatal in my point of view. I devised for DBM players in France a system where scores range from 5 to 35, but I do not think it should be used for ARMATI, especially since the number of draw results is extremely low in ARMATI competitions. Note also that even with a wide ranging scoring system, ties do sometimes occur and that some tie-breaking system is always necessary. On the other hand, any tie breaking system used is bound to involve some bit of luck. Hence some competition organisers will limit their use only to the first places, e.g. to identify precisely the podium players and distribute the corresponding awards.
Should you wish to experiment with other score systems, I think it is imperative that the following rules are respected to reward good strong competitive behaviour:
* A win is always better paid than a draw which is itself always better paid than a loss
* the difference between the best loss and the worst win is at least 4 times the difference between the best win and the worst win (6 times if you play 6 round competitions) : this ensures that if only one player wins all his game, he is guaranteed to finish the competition in first place.
* the best draw is equal or more than the average of the best win and the best loss. This means that a player with 2 favourable draws will be ranked ahead of a player with a win and a loss.
Please note that it is feasible to use a given score system to arrange the pairings and another to determine the final rankings of all players. This must be done with care and the two systems used must have very similar proprieties (e.g. 2 Wins and a loss are always less than 3 wins in both score systems).
For pairing purpose, I strongly recommend the use of the simple 2 / 1/ 0 score. A wider score system would just artificially separate game results which are essentially equivalent (a win is a win, whether it was obtained with no losses or many. This affirmation is highly debatable but otherwise, it reduces the number of "playable" armies).
Pairing the first round
For the first round, you can pair the players randomly. Another way, more complicated but leading to better results, is to first rank all players present according to their estimated strength (if available, ELO or GLYCKO ranking are extremely valuable for this). The best pairing is then to pair the first half against the second half (i.e. the first player of the first half plays against the first player of the second half, the second of the first against the second of the second, and so on). Another common method for the first round is to pair local players with ones who have travelled to the competition or to disallow any pairings between players from the same club in the first round. This is to increase opponent variety for players.
In small arenas, the first half against second half system may results in the initial players pairing being often the same from one competition to the next.
Nonetheless, the main advantage of that pairing technique is that since all initial games pit players whose strength are widely separated, the games often result in clear victories. Hence, the group of winners is very numerous in the second round and the best players have sufficient opposition not to be pitted against each other. This ensures them 2 clear victories to start the competition and means that they have a high total when they play each other in the later rounds. The eventual winners will therefore have high Buchholtz or SODOS secondary scores.
See also the comment on accelerated pairing at the end of the discussion on the number of rounds.
Pairing later rounds
After the first round, you must first group the players in "classes" or score-groups of players having the same number of points. These classes can be large if the scoring system does not have many values (e.g. 2 / 1 / 0) but will be very small (often reduced to one player) if you use a scoring system with a large range. The main and most important principle is that players of a given class MUST be pitted against other players of their own class. Within a class, you can pair the players randomly or again use the first half against the second half technique.
Several problems can arise:
1) The number of players in a class is odd: the solution is then to move one player from one class to the next one (either under or above) to ensure an even number player in each class. Such moved players are called floaters.
2) All players in a class have already fought each other (this typically happens when only 2 players had a draw result in the first round, they then find themselves in the middle class but cannot play each other again): the solution is then to regroup the players in that class with the next one and retry pairing.
Chess players have created very elaborate and complex rules to decide which player should be chosen as a floater. In practice I pair the class in descending order (i.e. I pair the players with the highest number of points first) and I use the top player from the next class as a floater when I need one. A general principle they use is that a given player should have as close as possible to zero floats during an event (counting floats upwards as positive and downward as negative). This is because a player who receives a long sequence of downward floats will have an easier competition than their score deserves.
When using a wide ranging scoring system, the single player classes lead you to pair the players in descending order (first against second, third against fourth).
If using the first half against second half technique, the order within a class can either be determined by the tie-breaking mechanism chosen or by the initial estimated strength (or even both, but preferably in that order).
The final ranking
My preferred ranking system uses the following score system in the following order
1) Total score in 2 / 1 / 0 game results
5) Any goal average system
6) Pick at random
This system places emphasis on the number of victories, then on the quality of the opposition (if 2 players achieved 2 wins and 2 defeats, then the one who fought against high ranked players will be given advantage, typically WLWL is often better rewarded than LWLW). The SODOS is very good to eliminate draw seekers. The M-Buchholtz score reduces part of the luck factor included in having been pitted against the best / worst player of the competition.
The 6th step has been included as a last resort system, but it is very unlikely that you will ever have to use it. Should you find yourself in that case, a worthy alternative is to give the title to the youngest player.
The number of rounds
The basic rule for the number of rounds to be played is that the system is likely to pick the strongest player as the winner in N rounds for 2 to the power N players i.e. 3 rounds for 8 players, 4 rounds for 16, 5 for 32, etc..
The total number of players should not be too high when compared to the number of rounds (i.e. of games played by each player). In particular, a 4 round competition should not have more than 24 players, preferably 16 or less. On the other hand, pairing can become a nightmare if the number of rounds is too high with respect to the number of players. If there are N-1 rounds available for N players then you can instead play a round-robin competition, where each player plays all other players present.
e.g. Britcon 2002 and 2003 pairings were fortunately done by computer. With 6 rounds and 10 players, the impossibilities due to players having already met each other becomes very numerous in the late rounds.
If not enough rounds are played for the number of players present, the risk of having unresolved ties on the podium becomes quite high. This risk is reduced by the several tie-breaking mechanisms provided, but a (un)fair amount of luck will often be found in the final rankings.
One solution to ameliorate the issue of not having enough rounds compared to a high number of players is to use accelerated Swiss pairing. It is meant to work with seeded players (i.e. players ranked according to their estimated strength), 2-1-0 score and the first half against second half technique. The accelerated system works in the following way:
Give a virtual victory to all players of the first half, thereby creating artificially 2 classes of players before the first round. Pair players according to the normal rules (i.e. first quarter against second quarter, then third quarter against fourth quarter). Keep the virtual victory in and pair players for the second round still using the normal rules. At the end of the second round, suppress the virtual victory given to all players. Pair normally your players in all subsequent rounds. More detailed info on accelerated pairing can be found on Rob Brennan's site http://iworg.com/rob/.
(un)Fortunately, that issue does not currently plague ARMATI competitions.
I have developed an Excel spreadsheet which implements the preferred choices listed above and some others. It requires an Excel 2000 or higher.
Unfortunately, the help page of that spreadsheet is quite reduced and currently written only in French.
1 The author would like to thank Rob Brennan with whom he had long discussions about the subjects contained in this paper and who provided many helpful comments on the early draft of this document. Many of the idea exposed here have been refined and ameliorated due to its beneficial influence. More info on these subjects can be found on his site : http://iworg.com/rob/
Armati Second Edition copyright 2003 'The Strength Trainer'
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