Tournament rankings: some statistical questions

[Wadih Khoury]

Thread only for those who like to see things through a mathematical lens

In a Swiss tournament, if I recall correctly you need

• Roundup(Ln2(X)) round to determine the winner with X players
• So for 50 players, you would need (Ln2(50), rounded up which is 6 rounds.

If you want to get the top Y players, you need to add

• Roundup(Ln2(Y)) rounds
• So if you want the top 4, you add an extra Ln2(4)=2 rounds.

First question then: Do we need

• Roundup(Ln2(x)+Ln2(Y)) rounds or
• Roundup(Ln2(X)) + Roundup(Ln2(Y) )

Now once this is sorted, looking at some recent tournament structures like the 2019 UKCC or the 2020 Online British, is it fairest (mathematically) to have one big tournament to determine the top Y players for qualification purposes, or to have n tournaments and qualify Y/n in each of them?

For me, without mathematical proof, it would appear that splitting qualifiers in n tournaments makes qualification more variable and uncertain, and therefore less fair: even for stronger players, it is difficult to constantly hit 90+% (which is usually needed if Y/n is between 1-3), while they are often almost certain to be part of Y qualifiers in a tournament that has enough rounds.

If we take the recent British which had 3 qualifiers for a total of 12 places (3 of which were on cumulative score, which I'll ignore here for simplicity), which is best. Some qualifiers had 80-90 participants. For simplicity, let's assume 100 unique players

1. 3 qualifiers of 7 rounds as per the tournament
2. 3 qualifiers of Ln2(100)+Ln2(4)= 9 rounds
3. a single qualifier of Ln2(100)+Ln(12)= 11 rounds

I think we can say that option 1 is not enough mathematically, but what about 2 vs 3?

[Matthew Turner]

Wadih,
There are some interesting ideas here and you might well be able to apply this to tennis, but I think there are some significant problems when applying it to chess

The most obvious problem is the impact of draws, but having a qualifier and championships is also hugely beneficial for online events. Banning a player after the qualifying rounds is (potentially) much less disruptive than allowing them to compete all the way through to the finish.

[Wadih Khoury]

The most obvious problem is the impact of draws, but having a qualifier and championships is also hugely beneficial for online events. Banning a player after the qualifying rounds is (potentially) much less disruptive than allowing them to compete all the way through to the finish.

I didn't make a point against qualifiers, rather wondering which type of qualifiers is mathematically the most correct (a single one with a lot of rounds? Multiple ones with less rounds? Or put in another way is it better to take the top 5 in 4 different qualifiers or the top 20 in a single qualifier?)
Then if 2 models are equally valid, there may be a subjective flavour (e.g: do you want something that gives more cup or league type results? Do you want more or less variance and unusual results? etc...)

[Matthew Turner]

Wadih,
As an organiser you really decide how many rounds you are going to have, so having a lot more extra rounds is not normally an option.
If you have a lot of players, should you have one big qualifier or split it up into smaller qualifying groups? This is really a question about what the players will perceive as being fairest.
If you have one big section then you will need to tiebreak and potentially a significant number of players on the qualifying mark will miss out and feel aggrieved
If you split sections then it will be very hard to make all sections equally strong, so a number of players will feel hard done to by being placed into (what they perceive as) a particularly hard section.

I don't think you can really attempt to answer this in the way you are trying to.