The Swiss Tournament

by Dr. John P. Pratt

11 March 2006

©2006 by John P. Pratt. All rights Reserved.

The Swiss system tournament, popular in chess, could be fun for sports also.

A tournament which has been very popular with chess players for the last century is called the Swiss style tournament.[1] It has many features which make it attractive for a variety of team sports, such as basketball. Those features include:

Here's how the new tournament works. The entries all play the same number of matches, in "rounds." If there is an odd number of entries, then another imaginary entry named "Bye" is entered as the lowest ranked, which loses every game.[2] Three rounds are needed for five or six entries, four rounds for up to 12, five for up to 24, with an additional round needed for each doubling of the number of entries.[3]

All of the teams are ranked according to some sort of preliminary ranking, which does not have to be very accurate. If no such initial ranking is available, the tournament works well enough with random seeding, and always correctly determines the winner. The top half of the entries are paired with the bottom half, such that the top entry plays the highest in the bottom half, the second in the top plays the second in the bottom and so on. After the first round, each of the winners gets one point and the losers get zero. In the case of tied games, each team gets half a point.

Let's look at an example of ten teams, each named for an animal, with the first letter of the name representing how good the team really is. For example, the Apes are the best team (starting with "A") and the Jackals (starting with "J", the 10th letter) are the worst team. Suppose that the preliminary ranking was good enough to get all of the teams to with three places of where they really should be. For example, the Apes should be ranked first, but they were entered as fourth. Here is how the first round might look.

Team Team Name Round 1
   vs.   Wins
1 Bears 6 1
2 Dogs 7 1
3 Foxes 8 1
4 Apes 9 1
5 Hares 10 1
6 Colts 1 0
7 Eels 2 0
8 Jackals 3 0
9 Gators 4 0
10 Ibises 5 0
After Round 1

The first round is supposed to be a sort of "warm-up" round where all of the top half teams are expected to win. In this case, that happened, but just barely. The Bears had to play the Colts, the Dogs played the Eels, and the Hares played the Ibises, which are all really next to each other in ability. Thus, while the tournament tries to have widely separated teams play in the first round, it won't happen if they are not ranked well. If there is an upset, the tournament handles it well as we will see in the next rounds.

In the second round, the attempt is made to pair all of those with the same score in the same manner. That is, the top half of all the entries with one point are paired against the bottom half that have one point.[4] If there is an odd number of teams, then the bottom team with 1 point plays the highest team with a half point (if any), or zero points. The process is repeated for the lower groups. The resultant pairings and scores are shown for the example for Round 2.

Team Team Name Round 1
   vs.   Wins		 Round 2
   vs.   Wins
1 Bears 6 1 3 2
2 Dogs 7 1 4 1
3 Foxes 8 1 1 1
4 Apes 9 1 2 2
5 Hares 10 1 6 1
6 Colts 1 0 5 1
7 Eels 2 0 9 1
8 Jackals 3 0 10 0
9 Gators 4 0 7 0
10 Ibises 5 0 8 1
After Round 2

First look at the pairing. Teams 5 and 6 were paired first because there was an odd number with 1 win, so the lowest team with a win (the Hares) were paired with the highest team with no wins (the Colts). Then the top half of the teams with one win were paired against the bottom half with one win, and similarly for zero wins. The final column shows the total number of wins after Round 2 (remember that the names of the teams tell who will win). After repeating the process for Round 3, the score sheet looks like this:

Team Team Name Round 1
   vs.   Wins		 Round 2
   vs.   Wins		 Round 3
   vs.   Wins
1 Bears 6 1 3 2 4 2
2 Dogs 7 1 4 1 6 1
3 Foxes 8 1 1 1 10 2
4 Apes 9 1 2 2 1 3
5 Hares 10 1 6 1 7 1
6 Colts 1 0 5 1 2 2
7 Eels 2 0 9 1 5 2
8 Jackals 3 0 10 0 9 0
9 Gators 4 0 7 0 8 1
10 Ibises 5 0 8 1 3 1
After Round 3

Again, look at the pairing, which now becomes more tricky. We have to repeat the process and also make sure that no two teams have played each other before. We begin at the top and work down. There are only two teams with two points, so they are paired (the Bears and the Apes, the top two teams). There are six teams with one point, and ideally the top three are to be paired against the bottom three. That works for the first pair, namely the Dogs and the Colts. At first it looks like it will work for the second pair (the Foxes and Eels), but that would leave the third pair to be the Hares and the Ibises who already played each other in the first round. Thus, the Foxes are instead paired with the next team down, being the Ibises. That leaves the Hares to be paired with the Eels. Finally the two teams with no wins are paired. If this all sounds complicated then you see why computer programs often do the matching.

Now look at the scores after Round 3. The Apes are already the only team with 3 wins and the Jackals the only team with 0 wins, so the top and bottom teams are pretty much known already. The fourth round will help confirm those positions, as well as help to break the four-way tie for second place. Here's how it looks after Round 4.

Team Team Name Round 1
   vs.   Wins		 Round 2
   vs.   Wins		 Round 3
   vs.   Wins		 Round 4
   vs.   Wins
1 Bears 6 1 3 2 4 2 7 3
2 Dogs 7 1 4 1 6 1 10 2
3 Foxes 8 1 1 1 10 2 4 2
4 Apes 9 1 2 2 1 3 3 4
5 Hares 10 1 6 1 7 1 8 2
6 Colts 1 0 5 1 2 2 9 3
7 Eels 2 0 9 1 5 2 1 2
8 Jackals 3 0 10 0 9 0 5 0
9 Gators 4 0 7 0 8 1 6 1
10 Ibises 5 0 8 1 3 1 2 1
After Round 4

Again look at the pairings. Begin with the Apes, only team with 3 points. They cannot be paired with the highest team with 2 points (the Bears) because they've already played, so they are paired with the next team with 2, the Foxes. Then the Bears cannot be paired with the next team with 2 points (the Colts) because they played in the first round, so the Bears are paired with the only other team with 2 points (the Eels). Then the remaining team with 2 points (the Colts) are paired with the highest team possible with 1 point, being the Gators (they've already played both the higher teams, the Dogs and the Hares). There are three teams left with one point. Before pairing them, we note that the Jackals, with 0 have already played two of them (the Gators and the Ibises). Thus, the Jackals must play the Hares, which leaves the Dogs to play the Ibises. If those last two had already played, we'd have to go back to the top and work down again through a different route, such as the Apes playing the next team down, the Colts.

Tie Breaking

After the end of Round 4, the Apes are still the clear winner, but there is a tie for second place. There is also a four-way tie for fourth place. If the Apes had lost, there would also have been a tie even for first place. Because there are always many ties in the Swiss system, several ways to break ties have been used.[5] We will use the simplest and one of the most effective. One reason the ties were caused came by simply counting wins, without regard to how good the opponent was. That can be compensated for by adding up all the wins of each opponent, on the assumption that the ones who played the tougher opponents are better. Thus two more columns are added to the score sheet. First a column for total wins by opponents, and then for the final ranking of the team after this tournament. In the case of a tie even in opponents points, then the team that was initially ranked higher wins. Opponent points are only totalled for teams with a tied score. So here is the final score sheet.

Team Team Name Round 1
   vs.   Wins		 Round 2
   vs.   Wins		 Round 3
   vs.   Wins		 Round 4
   vs.   Wins		 Opp.
Wins		 Rank
1 Bears 6 1 3 2 4 2 7 3 11 2
2 Dogs 7 1 4 1 6 1 10 2 10 4
3 Foxes 8 1 1 1 10 2 4 2 8 5
4 Apes 9 1 2 2 1 3 3 4 - 1
5 Hares 10 1 6 1 7 1 8 2 6 7
6 Colts 1 0 5 1 2 2 9 3 8 3
7 Eels 2 0 9 1 5 2 1 2 8 6
8 Jackals 3 0 10 0 9 0 5 0 - 10
9 Gators 4 0 7 0 8 1 6 1 9 8
10 Ibises 5 0 8 1 3 1 2 1 6 9
Final Ranking

Analysis

First, let's note that the final ranking came out excellent in this case. If we list the teams from the last column in order, they are Apes, Bears, Colts, Dogs, Foxes, Eels, Hares, Gators, Ibises, Jackals. So only the Eels and Foxes are backwards and the Gators and Hares. It must be stated that such is an unusally good turn out, in fact, it was the best of all the examples I tried. In other cases the Eels tied for second, and other teams were also three places from where they should have been. But this example was better to show how it should work.

Another problem I see with this tournament, is that it doesn't really save all the close games for the last round, as was hoped. For example, the top two seeded teams played numbers 7 and 10 in the last round, and indeed none of the teams played any close competitors, except the Apes and the Foxes, who were seeded third and fourth. But in reality, even the Apes and Foxes were not close in ability.

To me the real problem with this tournament is that the entire basis is not correct, namely that the teams should be ranked by the number of wins, ignoring who was played (except as an afterthought in the case of a tie). So I have proposed another tournament, called the Rapid Ranker which has all of the features of the Swiss system, but has nearly every match in the final two rounds be between competitors who are ranked next to each other. It is described in detail in another article.[6] When this same example is played in the Rapid Ranker, only two teams are out of order by one place (Foxes and Gators). While I believe the Rapid Ranker usually ranks teams better than the Swiss system, it will probably take a detailed statistical study to prove that.

Notes

  1. The best description of the Swiss System known to me is from Scholastic Chess of Indiana, found at scichess.org/faq/swiss.html. Note that in chess they also try to alternate which color pieces a player uses. That is like offense or defense and is simlar to who gets the ball first in a ball game. In most sports directional advantages are decided by the toss of a coin rather than having been included in the tournament pairings, so I ignore that feature in this article.
  2. This is a minor departure from the official Swiss system, which pairs the bye with the lowest ranked entry. That practice gives the lowest player a win in the first round which is contrary to the more standard practice of treating the "bye" exactly like the entry who failed to show up to the tournament because they knew they would lose all matches. After much testing, it became clear to me that this is a flaw in the standard system which scrambles results unnecessarily. The bye should be treated exactly as all other entries rather than having a special rule for it.
  3. To correctly determine the winner, there must be n rounds for up to 2n players. That means that 4 rounds will determine the winner correctly for up to 16 players. But the problem is that with more than 12 entries in 4 rounds, there is usually a 4-way tie for second place between competitors that never played each other. If the goal is to rank more players correctly than just the winner, then 12 is the limit for four rounds. There are tie breaking methods, but they are already pushed to the extreme just to handle up to 12 players.
  4. Note that the teams are not reordered after each round. In this case it would make no difference because all of the top entries won. But if there had been an upset, then reordering would cause the winner to be listed as the new #5, and the loser to be the new #6. This would lead to better subsequent pairings. After spending much analysis on this possible improvement to the Swiss system, it became clear that it was too complicated to relist and renumber the teams after each round. Even though the order in the official Swiss system might rely too heavily on the initial ranking, it can be compensated for with the standard tie-breaking method which is discussed in this article. Thus, the only change I propose to the Swiss system is the treatment of "byes" discussed in footnote 2.
  5. For a summary of several methods, see "Tie Breaking Methods" at scichess.org/faq/tiebreak.html.
  6. The Rapid Ranker is described at www.johnpratt.com/items/docs/ranker/ranker.html.