About Championship Tournament Systems

In this text I will attempt to define and discuss about rigorous championship systems. Detailed calculations about various systems are skipped.

Definitions
Qualification is an optional stage before the championship, in which some or all championship participants have played. It may consist of sub stages. It is not the interest of this text.

Play-off or tie break is an optional stage after the main championship stage. In a tie break shorter time limits might be used.
All the championship participants play in the main stage.

Goals
In this text, the most important goal of the tournament is to
determine the unique champion.

We can have an additional goal, which is to:
determine lower places, possibly shared

Basic Conditions
Here are conditions I select for the tournament system to fulfill:

A. “The champion should have more wins, McMahon points or fewer losses than the second.”
B. “The tournament must terminate.”

For a player to be the champion we expect him having beaten his strongest rivals. In my opinion, if players have tied on number of wins or McMahon points, it is unsatisfactory to determine the champion by tie breakers like SOS, SODOS, SOSOS etc. Some people suggest using players’ direct encounter as tie breaker. However, it does not bring any new information about players’ performance, it does not cover all cases and it is statistically meaningless. I would use it only as a last expedient. Due to other conditions we may have to restrict condition A to the last stage of the tournament. In this case I would add:”The champion may not score less wins or McMahon points and more losses totally in the last stage and the stage prior the last.”

If we want to be sure that the winner is the strongest, we might try to use a condition: “The champions must win at least two more games than the second”. However, if the top two players beat each other alternately, the tournament does not finish. We want to be able to schedule the tournament.
The following properties for the system are also desired:

C. “The players should not meet more than once.”
D. “The championship should finish in fixed number of rounds.”

However these properties may conflict with property A or with each other. In those cases I choose the property A to override C or D.

Implications
The only tournament system that perfectly satisfies both criteria A and D is knock out. In some tournaments the A criterion is relaxed. In this case only the last stage of the tournament is knock out, but there may be earlier one which is not. The knock out may be as short as one round.

Unless the system is knock out, the criteria C creates a problem, if attempted to use together with A criterion. In case of tie for the first place, it would prevent a play off. Therefore the C criterion should not be used in a play off or tie break stage.

Round Robin
In round robin each player plays against each other player. All places get resolved, although ties are possible. It is a good idea to resolve tie for the first place with additional game(s). The number of rounds is rather high compared to number of players. Double round robin is even more accurate, but it is also longer and rarely used. Tie break may be needed.

Swiss, McMahon and related systems
So far we know knock out is a system to determine the champion quickly. It does not allow a player to lose a game and win the championship. Adding a round or more we can get Swiss system or McMahon, but then we may have a tie for the first place. To avoid the question of tie breakers, we could define a system:
”A top player with one loss, should be allowed to win the championship, while a top player with two losses should not”.

The top player means here a player in the highest starting McMahon group or any player in the Swiss system. For time being we discuss about Swiss system.
We want to predict the number of round after which, we have a single winner. Allowing no losses, and 2^n players we have a winner after n rounds. If we have p players, 2^n < p < 2^(n+1), then we need n or n+1 rounds. So the number of rounds may vary by one. If we allow one loss, the number of rounds may vary by two or with unsuitable pairing by three. When in the tournament we achieve the moment where we have one player left with no losses and one with one loss, we don’t know if we need one or two games to finish.

Numbers of rounds needed to select a winner with at most one loss with Swiss system:


With numbers in between, the number of rounds to determine the single champion may vary by two. For example with 24 players we need 7 – 9 rounds. Generally adding more rounds to a tournament will improve the accuracy of placement of the players, but it does not necessarily help sorting the top. A three way tie at the top can easily persist when the tied players play another round against new opponents.

If McMahon is used, one can simulate a couple of rounds with Swiss system and find out the sizes of the two highest groups. One round difference can be handled in the tournament schedule either by adding an optional playoff round or by letting the unique winner out from the tournament before the last round, whichever is preferable. Note, that players who are still in completion for championship are allowed to rematch, if no other pairing in their McMahon groups is available.

(*) With 32 players, if we want to finish with exactly 8 rounds, we might define the round 8 as final, and the winner of the final is the champion, regardless the number two having one or two losses.
Double knock out is similar to Swiss, but the common format requires more rounds.

Allowing two losses for the champion
Suppose we define: “A player may become the champion if he has lost two games, but he can't become the champion if he has lost three games”. A best of five match fits to this definition. If we want to have more players the system becomes complex and it is hard to define a rigorous system. We might have a two stage tournament were in the first part a player may lose one game and the second part is a best of three match. However, if a player has won all his games in the first stage, he will miss the championship title after losing two games in the second. I will not attempt to discuss further about this kind of systems in this text.

Conclusions
If we have a tight schedule, then knock out is the solution. If we have for a day or a half or time for (optional) separate play-off, we can use round Robin or Swiss system. I recommend McMahon only if there is no prior qualification stage.

Edit: corrected a typo
edit2: removed text about copyright, although not giving it up

Just edit it and remove everything.

Some general remarks (some are quite critical)

- I like the formula's (if only to see how you calculated your numbers)
- Number of rounds needs to be exact (if nescesary adjust the number of players)

- leave McMahon and Double Round Robin out of the discussion (Swiss in itself is allready difficult enough)

- it is a bit inconsistent at places

If the goal is to determine the champion then a tie break (in whatever form) is part of the tournament (otherwise the goal of a single champion cannot always be reached)

If For a player to be the champion we expect him having beaten his strongest rivals. then SODOS is a satisfactory tie breaker. (because the strongest rivals have an higer score)
(But SODOS does favour wins against stronger players higher than wins against weaker players)


- What do you mean with "All places get resolved" in:
In round robin each player plays against each other player. All places get resolved, although ties are possible.

Bug:

2^n < p < 2^n is a bug, I think you mean 2^n < p < 2^(n+1)

If you are really worried about copyricht send it to me in a PM

I will not distribute it
Scouts Honour

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Promotor and Librarian of Sensei's Library

Put it on your website and link to it.

Or better yet, just stop worrying about something so silly and put it on here, copyright or no.

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Occupy Babel!

A nice start for a slightly refreshed view on the topic! What follows is my criticism only:

* Order of further places needs to be studied in greater details.

* Sharing the title (or other places) is a principle option that you do not discuss. Just declare your assumption that by all means you want to avoid it. Otherwise study about it would be needed, too.

* I disagree about Direct Comparison (but do not discuss it now).

The singular of "criteria" is "criterion".

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A good system naturally covers all corner cases without further effort.

I edited the text as a reply to Robert's and Harleqin's comments.

If the readers find it convenient enough I will put it on my web page instead of having it here. You can say it by a private message.

I don't understand the reason to claim copyright in the first place.
(copyright only protects text not the ideas in the text) are you planning to publish a book about it?

I would prefer the discussion paper with the discussion. (It makes copying and pasting easier)

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Promotor and Librarian of Sensei's Library

The condition: “The champion should have more wins, McMahon points or fewer losses than the second.” means that if any tie for the first palce remains, at least one more game is to be played.
I was trying to say that all places are resolved with roughly equal accuracy.

This is equivalent as saying that Play Off is the prescribed tiebreaker



Here you get back to the same problem.

For tounamentsystem there is no fixed terminology where everybody on agrees.
So all terms need to be explained.

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Promotor and Librarian of Sensei's Library