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Did anybody ever look into the difference between SOS in Swiss and McMahon tournaments? Isn't McMahon SOS (neglecting bars and with 1 rank = 1 point difference) similar to SOR + Swiss SOS?
McMahon SOS has a much wider range (depending on the maximum rank difference in the tournament) in comparison to Swiss SOS. Shouldn't then a single point of McMahon SOS be much less significant (because of the added randomness of the rank data) than a single point of Swiss SOS?
I don't think that the rank data adds randomness. In fact, it should improve SOS. In discussion of SOS, it is often remarked that an unlucky first or second round pairing can be disastrous for your SOS. Since McMahon simulates the early round results through the initial MMS, the data added to your SOS should be far more reliable than that in normal Swiss.
This also bears out if you consider SOS as a measurement for "Strength of Opposition". Suppose that the lower bar is at 10k, so 5k start with 5 points, 1d with 10 points. A 5k having a SOS of 35 after 5 rounds has, without a shadow of a doubt, faced far easier opponents than a 1d having SOS 60, even though the difference of 25 SOS is entirely the result of their 5 point initial MMS difference working over 5 rounds.
I don't doubt that 25 points SOS shows the difference in opponents. But the 5k with five wins and the 1d with five wins are not tied anyway in a McMahon system, so it is misleading to compare a tie breaker here.
Also it seems obvious to me that the SOR consists of information (about strength) + disinformation (about strength) + randomness due to rounding. While in general this may add information it may be the case that the added randomness unduely affects the last digit.
These two players might in fact be tied, if the 5k won all and the 10k lost all. The 5k might have his SOS as 5+6+7+8+9=35, the 1d as 13+13+12+11+11=60
But regardless, that was not the point I was trying to make. The 25 point difference example is extreme, but shows that SOS measures something (the question is then whether a 1 point SOS difference is still significant, though).
Yes, the point you were trying to make is conceded (and was never denied) :) I justed wanted to hint at the possibility that SOS works slightly differently in Swiss and McMahon systems (and here probably in the band system and in the 1 rank = 1 MMS system)
McMahon SOS has a much wider range (depending on the maximum rank difference in the tournament) in comparison to Swiss SOS. Shouldn't then a single point of McMahon SOS be much less significant (because of the added randomness of the rank data) than a single point of Swiss SOS?
I do not see any randomness coming from the rank data. If anything, the initial MMS removes noise from unlucky early pairings, as Herman said earlier. I don't think you meant to say that an apple will get smaller just because it came from a wagonful of apples, so maybe there is just some kind of misunderstanding going on here?
I justed wanted to hint at the possibility that SOS works slightly differently in Swiss and McMahon systems
Hinting at possibilities is of course all well and good when it is done in a constructive spirit. Are you still seeing some differences, and if you are, how do you think they should be taken into account when choosing tournament systems and/or tie breakers?
Cheers,
-Bass
If you take one apple out of a dozen you can be pretty sure to have an apple if you take something which has the same weight as an apple from a whole wagonload of apples there is the possibility to end up with a stone or a dead bird instead. The possibility for this is much higher with industrial agriculture (with high numbers), ain't it?
Even if totally unbiased (no sandbaggers, no overrated players, highly unlikely) rank data inevitably comes with randomness due to rounding. This doesn't mean it shouldn't be used or something (because the positive effects may be larger), but it should at least be recognized that introducing such a randomness multiplied by the number of rounds may have some effects on the reliability of the last digit which is what counts for a tie breaker.
I do not think rounding adds randomness, on the contrary. Even if some randomness was created, why would you want to multiply by the number of rounds?
Do you mean to say that the opponents' MacMahonScores become _less_ reliable every round the tournament is continued?
EDIT: ah, scratch that, I think I got your meaning now: every round you face a new opponent, so any initial randomness would indeed get accumulated, if it weren't for the fact that every round of MacMahon actually fixes any possible initial biases a little.
Working with the idea that each rank contains randomness, would the randomness be evenly distributed above and below the "actual rating" ? If it is evenly distributed, either exactly or significantly even, then each additional round should decrease the effect of the randomness.
To use the apple analogy: not sure that "is an apple" vs "is a stone or dead bird" is the right analogy. Perhaps we might talk of the weight of an apple from the dozen or from the wagonload. The more apples we weigh, the better our estimates regarding both the range of the weight of apples as well as the weight of "an apple".
Each round of play would reduce randomness due to the random effects offsetting to some extent and out reduce unusual results due to the repeated sampling of performance using a variety of opponents.
Sorry for the bad analogy. And you're indeed right with randomness overall (relative to the total amount of SOS), I am not sure regarding the single or second SOS point breaking the ties. (In one round the maximal difference for one player is 0.5 = half a rank, in three rounds even though much less likely you may have up to three times 0.5 making, as you compare two players the maximum error is twice that amount. The likelyhood for the different rounding errors is flat for single rounding error and increasingly resembles a normal distribution with increasing number of rounds.) I have to take a second look when less tired or bored during my holidays, but I believe it is an idea worth playing around with. Assuming evenly distributed rounding errors for strength, how likely is it that a 1 point or even 2 point difference in SOS between two players result entirely from the rounding error in the strength of their opponents where a finer grained strength measure would have resulted in no SOS difference?