Swiss and math

[Victor Plotkin]

As some of you know, Hou Yifan intentionally lost her last game in Gibraltar as a protest against artificial pairings. She had to play 7 women in 10 rounds. Her logic was fairly simple: it's just almost impossible to have 7 female players in 10 rounds, because there were just 37 women players among 255 participants.

I decided to check, how low the probability of having 7 or more women is.

a) A basic calculation. We don't know how works the Swiss System, everyone has an equal chance to play Hou Yifan. Hou Yifan can't play with herself, so we have 254 potential opponents and 36 women. The probability of 7 out of 10 women is:

P(7)= (36*35*34*33*32*31*30*218*217*216) / (254*253*252*251*250*249*248*247*246*245) * 120.

120 is the number of different combinations 7 out of 10. 10!/ (7!*3!) = 10*9*8/6 = 120.

P(7)= 0.000055 = 0.0055%.

That means, these strange pairings should happen every 18.000 events.

Theoretically, we should add the probability of having 8, 9, or even 10 female opponents. Sure, P(8) is much lower than P(7). I don't want to enter additional calculations, so we can estimate:

P(7-10) = 0.00006 = 0.006%. This is the probability of playing at least 7 women players in 10 rounds.

b) Now we can take into account some rules of swiss pairings and make 2 adjustments:

1. Hou Yifan was at 22nd ranking position and played close to her rating the whole tournament. In this case, it's very unlikely for her to face somebody from the lower half of the tournament, except for the 1st round.

2. In a big swiss event, one usually doesn't play against the opponent with a very close rating (or ranking). A "dead zone" is around 60 points in this kind of event. In other words, Hou Yifan could not play against opponents with 2591-2711 rating.

Indeed, she played against a "bottom half" opponent only in 1st round and never played against 2591-2711 players. Taking 1 and 2 into account, we reduce the number of potential opponents to 1-10 and 36-125 ranking. So, we have exactly 100 potential opponents and 25 women among this group.

Now, the probability of playing against women is much higher. 25/100 = 25%. In previous example (basic calculation) we had 36/254=14.2%.

Using the similar formula, we get

PP(7)= (25*24*23*22*21*20*19*75*74*73) / (100*99*98*97*96*95*94*93*92*91) * 120 = 0.0019 = 0.19%.

By adding to this number PP(8), PP(9) and PP(10) we should get about

PP (7-10) = 0.22%

So, these 2 adjustments changed the probability from P(7-10)= 0.006% to PP(7-10) = 0.22%, that means almost 40 times higher.

0.22% means that such a strange pairing could happen every 450 cases. Taking into account the total number of players in Gibraltar, this situation should happen every second tournament with one of the players.

[Mathieu Cloutier]

Re: Swiss and math

That means, one needs to play about 18.000 events to have such a strange pairings.

I don't disagree with your approach and I didn't calculate the odds myself, but your numbers look reasonable.

However, I slightly disagree with the wording of your conclusion. The point is that we already had thousands of women playing in thousands of events over the course of history. Even if Hou's pairing are highly unlikely, it was bound to happen at some point.

[Victor Plotkin]

Re: Swiss and math

I don't disagree with your approach and I didn't calculate the odds myself, but your numbers look reasonable.

However, I slightly disagree with the wording of your conclusion. The point is that we already had thousands of women playing in thousands of events over the course of history. Even if Hou's pairing are highly unlikely, it was bound to happen at some point.

I have a problem to enter a big post on this forum. To save my time, I usually enter the first part and after that "edit" the initial post. Probably, your reply was done before I finished entering the whole post.

[Mathieu Cloutier]

Re: Swiss and math

I see your updates now. And yeah, that's about it. Unlikely events happen all the time, it doesn't mean there's something fishy just because of that.

However, do we have any official statement from the Gibraltar organizers on the matter? If they didn't manipulate the pairings and they can prove it, Hou Yifan's 'move' looks pretty dumb.

[Egidijus Zeromskis]

Re: Swiss and math

However, do we have any official statement from the Gibraltar organizers on the matter?

Here it goes from the press release http://www.gibraltarchesscongress.com :

"The Organisers of the Tradewise Gibraltar Chess Festival firmly reject the suggestion of any "fixing" of pairings. That the Women's World Champion chose to react in the way she did is a matter of considerable regret, both for us and also, we feel, for the global chess community"

and this was from a Youtube interview:
"Although sympathetic about her reason for giving her game away, Mr Callaghan (a tournament organiser) was quite clear that he did not believe the pairings had been wrong or that the move by Hou Yifan had damaged the tournament. But he did refer to the world champio n having had a “bad day at the office”.

[Wayne Komer]

Re: Swiss and math

Chess and math

February 10, 2017

There has been an extensive discussion by chess.com chat room readers on the topic of the probability of Hou Yifan meeting 7 women as opponents in 10 rounds.

https://www.chess.com/news/view/naka...age=8#comments

Some of the more interesting observations:

- Hou Yifan's opponents were mainly from the range Elo 2447-2587, with only Adams higher-rated and her first-round opponent (by coincidence also a woman) lower-rated. The probability to face several woman already becomes higher than by looking at the entire field. Why did Hou Yifan face more women than other female players? Within a given scoring group, she is more likely to be in the upper Elo half paired against the lower Elo half. Other women often end up in the lower half, to be paired against Hou Yifan (obviously only once) or against men.

Hou Yifan's pairings were still "anomalous", but calculations by others underestimate the probability of her facing many women. If we compare her with Iturrizaga, her neighbor in the seeding list then having a similar tournament (but for the fact that Iturrizaga won, and Hou Yifan de facto forfeited in the last round): In round 1, both had an opponent rated about 2300 - Hou Yifan's happened to be female, Iturrizaga's male. Then Hou Yifan had eight games against the Elo zone with a rather high women density (only Adams higher-rated) and played six women. Iturrizaga had seven games against the "women Elo zone" (Adams and Nakamura higher-rated) and played four women. 4/7 (57%) is less than 6/8 (75%) but also appears anomalously high. Iturrizaga didn't complain, maybe he actually enjoyed it.

- Some other pairings were also (arguably) unlikely or 'improbable': There were three Chinese in the field - Ju Wenjun got both others in rounds 8 and 10. There were three Bulgarians in the field - Cheparinov got Topalov and Stefanova in consecutive rounds.

Random picks for other players: Sam Shankland played three Germans in rounds 6-8. While in total 26 Germans participated (including FMs and untitled), only five fell into the Elo range (about 2440-2600) that Shankland faced after round 1. Jan-Christian Schroeder faced four (out of fifteen) players from India.

Two compatriots out of two possible for Ju Wenjun and Cheparinov. Three Germans (in consecutive rounds) out of five possible for Shankland. Four Indians out of fifteen possible for Schroeder. Seven women out of twenty-eight possible (excluding those rated below 2300) for Hou Yifan. All unlikely based on probability theory, but not impossible. Players not named Hou Yifan may not even have noticed - if they did they may not have bothered. In any case they didn't complain, and possible complaints wouldn't be taken seriously.

- "In an interview with the Chess News in May, Hou spoke out against the World Chess Federation’s different rules in women and men’s tournaments, and she chose not to defend her women’s world championship title this month in Iran. She has also pledged to attend only open (mixed-sex) events in the future."

http://www.csmonitor.com/World/Europ...tar-tournament

- Its a bit puzzling to me, how most here are wasting time, discussing probabilities, instead of focusing on the real point that Hou is trying to make, with her protest. The same point, Ms Polgar has been trying to make for years, (both woman world champions) regarding the focus that chess makes to male dominance. Equality, fairness, I'm sure , are the only things they are seeking. Narrow minded people, of course, will focus solely , on the shell of the problem, and disregard the essence of the problem. "She should apologise, and so on and so forth.. Typical behaviour, that aims to silence the voice of the oppressed. But as history as shown several times, one single voice, can surpass the critiques of a thousand, as the group mentality, usually, follows not, what is correct , moral, and just.

(Hikaru Nakamura) - I normally don't comment, but I am going to add a quick comment. It has to be noted that I found it very strange how I got two whites (rounds 5,6) and then two blacks in a row (rounds 7,8) or that I thought the pairings in round 10 were weird with Anton being due black and getting white or how Ju Wenjun didn't drop as the lowest in the score group. However, all that being said, I inquired and the arbiters were helpful and even double checked to make sure. I have absolute confidence in Laurent Freyd and all the other arbiters who did everything 100% correctly and by the book.

- From my own experience as an arbiter in Gibraltar I know that specific complaints are investigated in detail to explain pairings to the player. This is time-consuming but a natural service to the players. However, general remarks get a standard answer 'we use this particular software and don't change pairings manually'. "

Also I can (largely) explain "weird" last-round pairings. Rather obvious why Cheparinov rather than Ju Wenjun was floated down - Ju Wenjun was already floated down just before in round 9. Color assignments are slightly trickier, but there's also an obvious reason for it: In round 9, Anton won with black to become sole leader with 7.5/9. By color alternation he was due white in the final round, even if he was due black by color balance - double white twice earlier in the event thus so far 5 whites and 4 blacks. ALL players with 7/9 but Yu Yangyi had white in round 9: MVL (against Yu Yangyi) and Nakamura (floated down against Caruana) drew with white, the five others won with white - relatively normal given that three of them were also clear Elo favorites (95 points or more).

Thus some players had to end up with six whites and four blacks, indeed it was also the case for Gelfand who did/had to get double white in the final rounds. Edouard had five whites, four blacks and a bye. Anton had to be paired against Adams - highest-rated player from the tie for second he hadn't yet faced (he already played Nakamura and MVL). Switching their colors would mean that Adams ends up with a +2 color balance (six whites, four blacks) - in Gibraltar, giving such an advantage to an Englishman rather than a Spaniard would _appear_ just as biased. However, bias in favor of some players was unavoidable and software-generated - the software considers nothing but previous score, color history and starting rank. Any manual adjustments would have led to manual bias rather than automatic bias - the software already optimized pairings, the result was somewhat "weird" but understandable.

GMHikaru could have figured out all this himself (it took me about ten minutes) but had the right to ask the arbiters. A software error couldn't be all ruled out: Similar situations (most players in contention for top prize money being due black in the final round) probably occurred before, while this one is pretty Gibraltar-specific: most if not all other major Swiss opens have an odd number of rounds thus color imbalance +1 or -1 for everyone. Gibraltar has ten rounds thus perfect color balance for most players, but +2 or -2 for a few. If such color imbalances are considered "unacceptable" (FIDE Swiss pairing rules only prohibit +3 or more, -3 or less or triple consecutive white/black), the alternative was to cancel the final round? Which obviously wasn't an option .... .

Once again, Hou Yifan's protest is simply unfounded: the software doesn't even consider names, sex, nationality, age, .... when it generates pairings.

- The pairing process is random in the sense that the output can appear random. Indeed, Hou Yifan getting seven women in nine rounds is a random result, consistent with pairing rules that were defined before the tournament - when nobody knew which pairings would arise. The pairing process is well-defined in the sense that there are clear rules. The FIDE Handbook has details, the basic rules are actually fairly simple:

- Players with the same number of points are generally paired against each other. Only if the number of players in a scoring group is uneven, or if a player has already faced all opponents from his/her scoring group someone will be floated up or down.

- Consider this: If you use the law of probability to calculate the probability of being paired against 7 women in 10 games, then you are assuming that the pairings are random. But they are not random at all, they are made according to the Swiss system.

There are known statistical anomalies that can happen from Swiss pairings. For example, a player in the middle of the rating field will sometimes lose all their white games and win all their black games. I've seen this happen more than once. It happens because the Swiss is geared to give you White and Black alternately, and to give you a stronger opponent when you win and a weaker opponent when you lose. Sometimes these effects synchronize, and you get alternately Black against a weak opponent and White against a strong opponent (or vice versa), producing results that at first glance seem extremely improbable.

See:
http://forum.chesstalk.com/showthrea...tar-2017/page3
_______

After his fourth-place finish in the 1962 Candidates tournament in Curacao, Bobby Fischer complained that the Russians had fixed the competition against him with draws amongst themselves. When no one did anything, he published The Russians Have Fixed World Chess in Sports Illustrated and that caused quite a stir and he was criticized. Hou Yifan felt that she was treated unfairly and took action to publicize it and must take the consequences as Bobby did. I think she was brave.

[Mathieu Cloutier]

Re: Swiss and math

Hou Yifan felt that she was treated unfairly and took action to publicize it and must take the consequences as Bobby did. I think she was brave.

Well, it appears she was mostly careless and impulsive. Not brave.

[Brad Thomson]

Re: Swiss and math

If no gender distinctions were made in chess none of this would even have been possible.

[Brad Thomson]

Re: Swiss and math

Here is the game:

1.g4 d5 2.f3 (Babu stated later "I was scared when she played 1.g4 and I suspected preparation! But when I saw 2.f3, I figured out that something was wrong.") ...d5 3.d3 (Also playable is Nh3.) ...Qa4+ 4.Kd2 h5?? (Disgraceful. The only move is ...Qe1+ hoping to illicit 5.Ke3 d4+ 6.Ke4 Nf6+ 7.Ke5 Nc6+ 8.Kf4 Bd6+ 9.Kg5 h6#.) 5.h3 hg and White resigns.

[Neil Frarey]

Re: Swiss and math

Here is the game:

1.g4 d5 2.f3 (Babu stated later "I was scared when she played 1.g4 and I suspected preparation! But when I saw 2.f3, I figured out that something was wrong.") ...d5 3.d3 (Also playable is Nh3.) ...Qa4+ 4.Kd2 h5?? (Disgraceful. The only move is ...Qe1+ hoping to illicit 5.Ke3 d4+ 6.Ke4 Nf6+ 7.Ke5 Nc6+ 8.Kf4 Bd6+ 9.Kg5 h6#.) 5.h3 hg and White resigns.

Moron Opening?

Or do we play Nh3 earlier on? Followed by f3, Nf2?

It's been awhile...

[Brad Thomson]

Re: Swiss and math

It's been awhile...

Yes, but I believe that you and I did indeed pioneer the opening Yifan played and we deserve full credit for it. Perhaps Robert Hamilton could help us?