Author: Russell Reagan
Date: 18:51:23 12/14/03
Go up one level in this thread
On December 14, 2003 at 05:29:04, George Tsavdaris wrote:
>I don't understand. This means that for 20 participants the number of round
>must be Log2(20)= 4.32 ---> 5. Too little rounds i think.
Here is how I understand this.
This depends upon what the tournament is supposed to produce. Is the tournament
supposed to produce one clear winner (a "champion"), or do you care about the
accuracy of the rankings of lower finishers?
A rough formula for determining the number of top finishers that will be
accurate is:
(5 x number of rounds) - number of players
number of rounds = ------------------------------------------
7
So for the WCCC, there were 16 participants and 11 rounds. This means that ((5 x
11) - 16) / 7) = 5.57. So approximately the top 5 places will be accurate. This
means that you can't accurately say that the person who finished 8th is really
better than the person who finished 9th or 10th. You can only make statements
like that for the top 5 finishers in this case.
In an event where you only care about crowning one champion, as in a "world
champion," you only want the first place to be accurate ideally. When you add
more rounds beyond log2(participants), you get a more accurate 2nd place
finisher, and then 3rd place, and so on.
This would make you think that the more rounds, the better the results, but I
believe that the problem is that when you add more rounds beyond the "optimal"
number of rounds, then you create a log jam at the top. If you are just holding
a tournament for fun, that's fine. If you are holding one for the "world
championship," that isn't so good, because you really only care about who wins,
not who finishes 2nd or 3rd. A log jam for places 2-5 is fine as long as we have
one clear winner.
As for how the swiss system works and why 5 rounds is enough for 20
participants, it would probably be easier to understand if we used a game that
didn't allow for draws, because draws change the math a little bit. If you are
using the swiss system in a game that allows for draws, then you actually will
need less than log2(participants) rounds, so even 4 would be acceptable.
Let's assume for a minute that there would be no draws, just for the sake of
explaining why ths swiss system works. Let's say we have 16 participants. In a
swiss tournament, you play other participants with the same number of points as
you (if possible), and you don't play the same person twice. So each round every
undefeated participant will play another undefeated participant, and after the
game, we will have eliminated half of the undefeated participants.
Before round 1: 16 undefeated participants
After round 1: 8 undefeated participants
After round 2: 4 undefeated participants
After round 3: 2 undefeated participants
After round 4: 1 undefeated participant, the winner
This is intended to make the good participants play other good participants and
"weed out" people as quickly as possible. For only 16 participants, it would
probably make more sense to just have a round robin. The swiss system was
created for something like a national championship where there may be many
thousands of participants, and you want to have one clear winner (obviously a
round robin is not possible).
Now for an example of why playing more rounds than optimal can mess things up.
If there are 16 participants, then the optimal number of rounds is 4. Assuming
no draws, then after 4 rounds we have the following standings:
Rank Points
1. 4
2. 3
3. 3
4. 3
5. 3
6. 2
7. 2
8. 2
9. 2
10. 2
11. 2
12. 1
13. 1
14. 1
15. 1
16. 0
After this round, we have a clear winner. Notice that after 1st place there is a
log jam. If we play another round we could have three participants with 4
points, then we either have to have tie breaker rounds or use tie breakers. I'd
rather have 4 rounds and a clear winner than three people tied for first and the
"champion" be determined by a tie breaker. When our goal is to determine a world
champion, we are willing to accept a log jam between places 2-5 to get one clear
winner.
This is how I understand it anyway.
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