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Defined as the Sum of Defeated opponent's McMahon Scores, SoDoS has the following property:
SoDoS differences are NOT always invariant under change of origin of the McMahon Pairing scale.
The origin of the McMahon Pairing scale does vary: it is zero at 20 kyu in many European tournaments, it is zero at shodan in the UK. To convert from the UK to the European value we need to apply the transformation:
SoDoS(EU) = SoDoS(UK) + 20*D, where D is the number of defeated opponents.
This transformation arises from the fact that any player's European McMahon Pairing score is 20 more than the UK score i.e.
MMS(EU) = MMS(UK) + 20
Now when you sum the EU MMS scores over just D players, you get the sum of the UK MMS scores + 20*D.
The implication of this is that two UK players may end up with the "same" SoDoS, but have done so by beating "different" numbers of opponents. In this case their position in the European version of the rank list may change, as the European SoDoS's will be different.
Harleqin: I do not see any application of a comparison between SoDOS scores on different tournaments. After all, they are on different tournaments, aren't they?
Geoff: No, we are comparing SoDoS for the same tournament, published under two different systems. I have to make this transformation whenever I send UK results to the EGF rating system.
mgoetze: I fail to see how SODOS is relevant to anyone's rating.
So does Geoff iirc, but it is still used as a tiebreaker anyway.
Geoff:
Here is an example of a 3 round tournament illustrating my concerns about using SoDoS as a tie breaker.
We have a shodan Alan winning 2 games playing:
Bob(1k)-, Cath(1d)+ Dave(1k)+
We have a 1kyu Juliet winning 3 games playing:
Karen(1d)+ Lionel(1k)+ Martin(1k)+
It is not necessary to show the entire tournament results table. Here is a summary of the key information for Alan's and Juliet's opponents' scores in UK and EU styles.
Alan's opponents:
UK MMS Euro MMS Name Wins initial final initial final Bob(1k) 2 -1 1 19 21 Cath(1d) 1 0 1 20 21 Dave(1k) 2 -1 1 19 21
Juliet's opponents:
UK MMS Euro MMS Name Wins initial final initial final Karen(1d) 0 0 0 20 20 Lionel(1k) 2 -1 1 19 21 Martin(1k) 2 -1 1 19 21
Suppose we use SoDoS as the one and only tie breaker. Then we can construct the portion of the final ranklist showing both Alan's and Juliet's position. The column MMSi is the initial McMahon Pairing score, and MMSf is the final McMahon Pairing score.
Then the final position of Alan and Juliet in the UK scale is:
Wins MMSi MMSf SoDoS WHO CONTRIBUTES TO SODOS Alan(1d) 2 0 2 1+1=2 Cath+Dave Juliet(1k) 3 -1 2 0+1+1=2 Karen+Lionel+Martin
Alan and Juliet are ranked equal, and they split a box of chocolate.
However in the EU scale:
Wins MMSi MMSf SoDoS WHO CONTRIBUTES TO SODOS Juliet(1k) 3 19 22 20+21+21=62 Karen+Lionel+Martin Alan(1d) 2 20 22 21+21=42 Cath+Dave
Now Juliet is ahead of Alan and gets all the chocolate!
I am not worrying here about which result is better!. All I care about is that they are different.
Note that if you used SoS (Sum of all opponents McMahon scores) as the one and only tie breaker, then this effect does not happen because you are summing over all games, not just a selection.
SODOS is Sum Of Defeated Opponents Scores.
This is sometimes used as a secondary tie breaker after SOS. It assumes that your strength is defined by the strength of the people that you beat.
Of course some would say that the strength of the people you lose to is equally important. :)
SODOS (also sometimes abbreviated as SDOS) can be used in round robin tournaments, where SOS is of no use.
Matti
I have moved the example of the SODOS tiebreaker conversion from UK to EU scales posted by Steve Bailey to the Discussion page.
Geoff Kaniuk
(wms: In general, this example seems incredibly unclear to me. It is comparing SODOS under apparently UK standard McMahon Pairing and Euro standard McMahon Pairing, but yet it doesn't take all the differences and group them together in any one place; you have to pick apart the numbers to realize what starting score a shodan gets in England. And what the heck does a statement like "for rating = dan-1 or 0-kyu" mean? Then there is all this X and Y business, and X1 and Y2 and...well, I hope you see my point. Please, can somebody rewrite it so that it can be understood without spending 1/2 hour studying it? The point is quite simple, it should be possible to make the explanation simple also. Further, it shows that SODOS alone is a bad tiebreaker, but it isn't usually used that way, usually it is a secondary tiebreaker after SOS; when used this way, does the same problem exist?)
Christoph Gerlach: Using SODOS as a tie breaker after SOS is also questionable. If SODOS will discriminate between two players, these two players have the same SOS. Let's say Player 1 has a higher SODOS, this means that he has won to stronger opponents compared with Player 2. But he also lost to weaker opponents compared with Player 2 (we know this because both have the same SOS). Would anyone really think it is more significant to win against stronger opponents than losing to weaker opponents?
Anonymous: So clearly then, never use a palindrome when an acronym will do.
Jens Baaran: Hi Christoph, I would say, if a player has more wins than losses, it's better to look at the wins than looking at the losses, simply because there is more data available to base a judgement upon. So when it comes to deciding the top places in a tournament using SODOS as a secondary tiebreaker seems reasonable to me (I haven't studied the above mentioned example though).
Christoph Gerlach: Since SODOS is a tie breaker usually used after the number of wins we can assume that everyone has the same number of wins if SODOS is used to break ties.
Jens Baaran: Hm, I don't quite see your point. Perhaps I haven't made my point clear enough. I'm not saying, that one should look at the number of wins/losses for a secondary tie-breaker. Of course this doesn't make sense. I am saying, that using SODOS (i.e. a number based on the defeated opponents' score) for a secondary tiebreaker as it is used today is better than using a similar number based on the scores of the opponents the players lost to, when one wants to decide the top places of the tournament.
Geoff Kaniuk:
Hi wms. I have moved the example to the discussion page as the SODOS page seemed to be getting unreadable. Please let me know if any further clarification is needed.
As to the question: does it happen when SODOS is used as a second tie-breaker? The answer is Yes in principle, players get different SODOS increments when converting between different McMahon Pairing origins. The difference may or may not be enough to change their actual rank position.
In my experience seeing an actual difference in this case is rare. The problem is that it can happen, and so probably does.
Jeff Boscole:
Readers are asked to review a discussion on rec.games.go which presents statistical analysis from tournament simulations. Minimal tournament entropy is obtained by summing SOS and SODOS for tiebreakers, even for end-of-round pairing methods.
- jb
RobertJasiek: What you have analysed empirically is in the context of defining "fairness of a pairing method" as "minimal entropy". It does not conclude that SODOS is useful per se but that it has a useful meaning for this definition in the type of tournaments (Swiss?) for that you have done analysis. Your analysis does not clarify why your definition should be chosen or how and in which sense it might compare to other definitions. Broader definitions would be needed to enable us to also consider all earlier criticism of SODOS again (why won games are / should ever be emphasized more than lost games; the flaw of SODOS in McMahon) and compare both that criticism and your concept of minimal entropy to other tiebreakers (used for pairings and / or / either or used for ordering the final resuls) for a given tournament system.
Jeff Boscole:
My tournament simulation runs "adjacency pairing" with no pairing avoidance for multiple matches. Justification for the "fairness" definition is not required because neither Robert nor anyone else has proposed any alternatives. Minimal entropy just means minimum complaints and minimum arguments about the tournament. Because SOS = SODOS + SOLOS there is a simple choice which to use in combination with SOS (but not both). I invite Mr. Jasiek to propose any tiebreaking methods he wishes to test and we will test them. A number of them have been tested already.
- jb
RobertJasiek: I hope to find time some time to offer other definitions of fairness so that they are useful for tiebreaker study. - What you might study: Evaluating usage of not using any tiebreaker at all. Explain the nature of additional or lost information gained from using (particular) tiebreakers. How is that information related to the particular player's playing skill and responsibility and why is it or is it not just playing with numbers for its own sake or information on that the player does not have influence? If part of the information is asssociated with either type, then which percentage, what does that mean for the player's skill, and how should it be interpreted?
Jeff Boscole:
In each case additional rounds are usually the better method of obtaining skill results. Tiebreakers are topical only for the context of limited tournament time, or for awarding prizes. Direct skill is not being measured: only the likelihood of having more skill. Robert is correct that initial round pairing has some influence, yet if he posits no knowledge of player skill from the outset then we have no idea whether initial rounds matter. The "information difference" among choice of tiebreakers is rather small even as tiebreakers may be compared. Along with tiebreakers used for pairing midway through a tournament we need also to consider various pairing strategies.
- jb
RobertJasiek: Thanks. - If you should have time, I would also like to see the following being tested: SOS-1. SOS-2. Direct comparison provided a) it is applied iteratively (if meaningfully possible) and b) it is applied only if the tied players have played an implicit round-robin (or multiple thereof) within the Swiss tournament's pairings, i.e., each tied player has played equally many games against each other of the tied players.
Jeff Boscole:
I am guessing that SOS-1, SOS-2, refers to SOSOS & SOSOSOS. These were investigated and appear to drift away from being tied more directly to the informative player statistics. Robert's request to examine tiebreak first from SOS and then SODOS turns out to be inconclusive (see 15 Mar '07 post to rec.games.go). For later Rounds, and in smaller tournaments, successive tiebreaks may offer some slight advantage yet for tournaments of 48 or more players a simple sum of SOS+SODOS appears to be less entropic. Tiebreaking first by SOS and then SODOS seems to work well for tournaments having player numbers some power of 2, yet does not otherwise perform as well as SOS+SODOS.
- jb
RobertJasiek: SOS-x means to modify SOS: for every player and for x rounds, throw away his worst (smallest) values as noise.
Jeff Boscole:
You have described several ways to interpret SOS-1 (SOS-2) so I am first trying the approach of applying a rule to ignore one (or two) minimum opponent McMahon Score(s) from each player's SOS (but not SODOS). Using that idea, my tournament simulator does not detect any appreciable improvement for this tiebreaking context: MMS, then SOS, then SODOS. Since sidebar data is slightly useful, any impairment to collecting SOS (and SODOS) data results in marginally deleterious effects upon entropy minimization. The arguments which stem from examination of "defeat chains" among near-equally ranked players are not particularly strong because there is also remains the marginal likelihood of some "unrepresentative" game result. On rec.games.go I also detailed a means whereby the tournament end-result can be evaluated for fairness. Because initial entropy for an n-player tournament (by the way I am measuring it) also corresponds to tetrahedral numbers (n-1)*n*(n+1)/6 we may obtain a precise estimate for expected initial entropy - `ent' - and then invoke the level of signicance p>=0,05 for identifying a tournament entropy value which must be less than "ent^((1-p)/(1+p))" for a fair tournament result (where pairing methods were being uniformly applied).
- jb
SODOS differences are NOT always invariant under change of origin of the McMahon Pairing scale.
The origin of the McMahon Pairing scale does vary: it is zero at 20 kyu in many European tournaments, it is zero at shodan in the UK.
Here is an example of a 3-round tournament illustrating concerns about using SoDoS as a tie breaker.
We have a shodan Alan winning 2 games playing:
Bob(1k)-, Cath(1d)+ Dave(1k)+
We have a 1kyu Juliet winning 3 games playing:
Karen(1d)+ Lionel(1k)+ Martin(1k)+
It is not necessary to show the entire tournament results table. Here is a summary of the key information for Alan's and Juliet's opponents' scores in UK and EU styles.
Alan's opponents:
UK MMS Euro MMS Name Wins initial final initial final Bob(1k) 2 -1 1 19 21 Cath(1d) 1 0 1 20 21 Dave(1k) 2 -1 1 19 21
Juliet's opponents:
UK MMS Euro MMS Name Wins initial final initial final Karen(1d) 0 0 0 20 20 Lionel(1k) 2 -1 1 19 21 Martin(1k) 2 -1 1 19 21
If we use SoDoS as the one and only tie breaker, then we can construct the portion of the final ranklist showing both Alan's and Juliet's position. The column MMSi is the initial McMahon Pairing score, and MMSf is the final McMahon Pairing score.
Using the UK scale, we have Alan and Juliet ranked equal and sharing any prize:
Wins MMSi MMSf SoDoS WHO CONTRIBUTES TO SODOS Juliet(1k) 3 -1 2 0+1+1=2 Karen+Lionel+Martin Alan(1d) 2 0 2 1+1=2 Cath+Dave
Using the EU scale Juliet is ahead of Alan, Juliet gets the prize:
Wins MMSi MMSf SoDoS WHO CONTRIBUTES TO SODOS Juliet(1k) 3 19 22 20+21+21=62 Karen+Lionel+Martin Alan(1d) 2 20 22 21+21=42 Cath+Dave
It does not matter here which result is better. It demonstrates that SODOS is flawed and should not be used.
Note that if you use SOS (Sum of all opponents McMahon scores) as the tie breaker, then this effect does not happen because you are summing over all games, not just a selection.
Read the example above again comparing either scheme with one where a shodan scores -100. In that case a player with no wins scores 0 whilst one with several wins will be far worse off with a score of minus several hundred!
wms: I find this argument to be totally bogus. If I understand correctly, the argument is: SODOS changes based on your origin, therefor it is a lousy tiebreaker in MM tournaments. My answer: Each tournament has only one origin, so within the context of the tournament the SODOS is reliable, therefore the "problem" doesn't exist.
Harleqin: For SODOS, losing a game is worth the same as winning against a player with 0 MacMahon points. Obviously, below the origin, you will get behaviour inverse to that over the origin. This means that for getting the desired direction of your tiebreaker, you have to make sure the origin is lower than the last player. Also, the effect of losing a game is directly proportional to the distance from the origin. This means that also with swiss pairing, the top players (whose opponents have points close to the number of rounds) will see a big effect of winning or losing on their SODOS, while the last players (whose opponents have points close to 0) will not see any big effect.
wms: Yes, that is what I alluded to in the next paragraph. For high ranked players, when the weakest players started with 0 MM points, SODOS will place players with more wins above those with fewer wins, and when both players have the same number of wins, it will behave the same as Swiss-style SODOS. I find this behavior good, so therefore, I would call SODOS a good tiebreaker (if you set up your tournament properly).
Further, I find SODOS to be a better first tiebreaker than SOS if the bars and MM points are set up so that people only have positive initial scores. My reasoning is that in a tournament, the only thing a player has control over is when they win and when they lose. If the stronger players have positive starting scores, then the SODOS will prioritize the player with more wins over the player with fewer wins, so a player starting with 19 points who gets 3 wins will usually have a higher SODOS than a player starting with 20 points who gets only 2 wins. Meanwhile, using SOS as the first tiebreaker will usually give proritize the player who started with 20 points and won fewer games, because they started at a higher point, and thus probably played opponents with more points.
On KGS, originally I did use SOS then SODOS as the MM tiebreakers (same as used in Swiss tournaments), but found that the "tiebreaker" would always just hand the win to the player who started with more points, instead of the player who actually won more games. This struck me as a terrible tiebreaker since it would use something utterly beyond the control of the players to break the ties, instead of using actual tournament performance. I switched to SODOS first, and since then I can't find a single tournament where comparing wins vs. tiebreaker results "feels" wrong.
Chris Hayashida: Having just run the Cotsen Go Tournament, I find myself thrust in the middle of all of these tie-breaker problems. I think example above uses SODOS incorrectly. If I understand SODOS correctly, it should only be applied after the win-loss record is taking into account.
The example above is flawed in that Juliet, with 3 wins, is being compared against 2 wins. Juliet should always win because of a better record, regardless of SODOS. SODOS only comes into play when comparing two players with an equal number of wins.
wms: Chris, I think that usually in MM tournaments, the winner is not the player with the most wins. The winner is the player with the most MM points. Since you can gain (at most) one point on the current leader per round, that means that if you start more than 5 points below the top player in a 5 round tournaments, you have zero chance of being the overall winner. This sounds unfair, but the reason for it is that MM is a system that lets you have one tournament with people of all different ranks in it, and have the games be "close." In a Swiss tournament, the weaker players don't play each other, so you get a lot of badly lopsided games. But if MM gave the win to the player with the most wins, then the tournament wouldn't be won by the best player, it would instead be won by the biggest sandbagger, which wouldn't seem very satisfying either.
So MM gives the win to the most points at the end, and the best overall player wins, and the weaker players enter the tournament for the fun of it and have good games.
There's nothing stopping a TD from handing out prizes for "most wins", etc., which can give weaker players something to try for, but the tournament winner should be the best player in the tournament, which (if the tournament is set up properly) will be the player with the highest MM score at the end.
I thought that Cotsen was a Swiss tournament. Is it MM?
Chris Hayashida: The Cotsen Go Tournament is run as a Swiss-McMahon tournament. I am not clear on what the difference is. We do tiebreakers based on win-loss record, then use SOS and SODOS to break ties. Because each rating is more or less a separate band, it seems that the player with the stronger entry rating (and loses in later rounds) seem to have better tie break scores.
Bass: (2008-01-20) I strongly disagree with wms. Use of SODOS as described in the above comments is fundamentally wrong. The lengthy comment below tries to explain why, so please bear with me.
MacMahon is a system that is to be used in a tournament whose purpose is to find an order for all players, and to give the better score to the stronger player, giving this tournament's success an emphasis in evaluation of strength. If you have any other purpose for the tournament, or any other definition for "better", using MacMahon will very likely be counterproductive.
Now, if the MacMahon system fails to produce a difference in the score, you may want to use a tiebreaker to find out which one of the players was better. In this case you must remember that "better" is already defined, and it would be silly to change it at this point. So if you break ties, you should do in a way that is consistent with your definition of "better". If you use SOS, you will compare the strength of the opposition, as found out using your definition of "better". (It does not matter what your definition of "better" was, this is a property of the SOS tiebreaker.) So SOS is indeed a good way to break ties, when tiebreaking is necessary. Or if you really want to ignore the losses in the tiebreaker, you could imaginably average over the defeated opponent's scores. But if you use SODOS with all players having positive scores, you are actually rewarding the player that is worse by your definition of better. (As elaborated above by wms, when the players started from a different band, SODOS-cum-positive-mms produces results that are opposite to those given by SOS.)
If you do not like the definition of "better" used by the MacMahon system, you should use another tournament system. You should not kludge a buggy tiebreaker to produce results you happen to like, because if that tiebreaker is not consistent with the rest of the tournament system, then your tournament works the way you want only in the case of ties.
All in all, please do not combine SODOS with MacMahon. Thank you.
wms: Bass, you make some claims that I don't see. Please explain, how does SODOS reward the player that is worse by my definition of better? An example would be helpful I think.
I think what Bass is getting at is that if you define player A as better than player B, by giving player A more McMahon points at the start of the tournament, and they end up on the same McMahon score (so B has more points than A), then SODOS pretty much guarantees that B will be ranked higher. --Herman Hiddema
Bass: Thanks Herman, that was my point exactly. If you decide that "more MMS is better", then (if you really want to break ties) the logical follow-up is "getting the same MMS against better players is better". In other words, SOS is always consistent with the bigger plan. Therefore, an argument against SOS is usually an argument against "more MMS is better", which usually means that there is something wrong in the tournament system itself, and another one might be better.
Bass: Also, if you cannot avoid MacMahon for some reason and still really really want to award any MMS ties to the player with more wins, you should make that your primary tie breaker instead of cleverly exploiting a SODOS bug to provide the same effect. This allows you to choose a primary tie breaker other than SODOS for players that indeed have the same MMS and the same number of wins.
wms: But if starting points means a better player, then why hold the tournament at all? Starting points is the estimate of a player's strength, used to guarantee they play others close in strength. It's nothing more. By your reasoning, if A starts out with a MMS of 10, and B starts out with 9, then B wins two more games then A and thus beats A in tournament standing, the "wrong" player is winning the tournament. That's ridiculous. Tournaments should award winners to the player who had the best performance in the tournament, giving that precedence over the player who was seeded the highest. Using SODOS does exactly that. If two players end at the same MMS, then the one who won the most games (which will be the one seeded lower, since he had to win more to get to where he is) is considered the better player. I see that as rewarding exactly the correct player. (If they have the same number of wins, then again SODOS works great, it awards the better position to the player who had to beat tougher opponents).
Herman Hiddema: Well, an example of something that would certainly feel wrong to me would be:
To me, the above results show that Alice is about 1k, scoring 50-50 against 1k players, but also just 50-50 against 2k players. Bruce shows himself to be a solid 1d, scoring 50-50 against other 1d and also 50-50 against 2d players. I would, based on these results, think that Bruce is the better player. But SODOS will award that distinction to Alice. Wrongly, IMO.
Now the other way around:
To me, the above results show that Alice seems stronger than 1k, scoring 100% against 1k and 67% against 1d. Bruce shows himself to be a very weak 1d, scoring 0% against other 1d and just 33% against 1k players. I would, based on these results, think that Alice is the better player.
In both of the above cases, SOS will (generally) agree with me. SODOS will do one case wrong. So I think this shows SOS is the better tiebreaker.
tderz:^{[1]}
pwaldron: It is important to distinguish between using SODOS in a European McMahon tournament vs. a North American event. From the examples above it appears that Europeans are trying to break ties regardless of initial score, which leads to trouble.
In North America, players are initially grouped into divisions and prizes are awarded by division. In such a case SODOS is only used to break ties within a division (and usually only after SOS fails to break the tie first). Used in this manner SODOS makes sense because in order to be tied players must have won the same number of games and the origin of the McMahon zero doesn't matter.
Herman: In Europe, tie breakers are really only used to determine the winner of a tournament. There are usually no divisions, so there is no need to break ties within them. Instead, players are awarded prizes based on their performance, which usually means that any player who scores 4/5 or better gets a prize, regardless of their starting rank or SOS/SODOS/etc.
You can only use SODOS as a tie breaker if your tournament is so small that you only have one McMahon group. (So that you are basically just doing swiss pairing) Even then, you should consider using SOS first.
Some acceptable reasons for considering SODOS are
However, you are probably better off using another tie breaking method, as SODOS tells nothing about the strength of the opponents the player lost to, and can therefore sometimes be considered unfair. (or more fair, depending on your point of view, of course)
[1]tderz: Has the following already been addressed?
Above we can read that
with the underlying assumption that "if you have defeated a stronger set of opponents than another player on the same final score, then you are slightly stronger/better than they are."
Now let's imagine a new type of score called
This weakness indicator SOL(T)OS should give us the same information of strength, be it with another polarity.
If SODOS as tie-breaker is senseless in Swiss and round-robin tournaments,
it doesn't make directly sense in McMahon where several diverging examples can cloud the proof.
Perhaps you guess it at this point, I find SODOS as tie-breaker for McMahon tournaments quite senseless. McMahon tournaments are used for enabling many people to play interesting even games in a big tournament comprising players from a broad range of strengths. If the downside is that the tournament produces several winners, so be it.
If there is still need to end up with only one winner (e.g. un-dividable ticket or representation rights), then use again real Go strength as criterium (e.g. best-of-3 (rapid?) or a last decisive game) as opposed to chart reading.
A McMahon with more planned rounds, if needed with shorter time, e.g. 45 minutes - all of this can really help to produce clear winners.
If TDs (and the participants) do not have time for these measures, perform a lotterie (!) - which is as fair to all (except for those who are very fast at calculating which 'SOxyz' indicator would serve them best this time). Of course these rules for deciding on winnars should be made know to everyone beforehand.