2/2/2018 – The Gibraltar Masters wrapped up Thursday, with Levon Aronian in first place. This year round ten passed without incident, in contrast to 2017 when, on February 2nd, the story of the day was a rare scandal involving women's World Champion Hou Yifan deliberately losing a game in protest of the high number of women she was paired against. She was further confounded when a similarly unlikely string of pairings happened in October at the Isle of Man Open. Johannes Meijer looks at the odds in detail. Hou did not return to Gibralter in 2018, but instead competed in the Tata Steel Chess Masters. | Photo: Alina l'Ami

Imagine, you are at a tournament with 255 players of which 43 are female. You are to play ten rounds. How many female opponents would you expect to face? Three? Five? I am pretty sure you wouldn't say seven. Yet, this was exactly the number of female players Hou Yifan faced at the Gibraltar Open 2017 when, a year ago today, she threw her last game in protest of these seemingly odd pairings.

How probable is such a pairing? Could it have happened by chance at all?

Let's expose this strange occurrence to statistics. In this article, we will do it for you — and uncover the truth about Hou Yifan's pairings in Gibraltar. In the article Investigating Hou’s pairings, Grandmaster John Nunn told Albert Silver that precisely calculating the odds would be an exercise in futility. This seems indeed to be the case, but it is however possible to estimate these odds with the aid of a simple probability model.

One of the goals of the organizers of the Tradewise Gibraltar Masters tournaments is to attract as many as possible world class chess players at the end of the month of January. They do this by offering very attractive awards for all and special awards for women. Many women attend and compete in Gibraltar head-to-head with strong and very strong male grandmasters.

Obviously that was the intention of world champion Hou Yifan when she travelled in January, 2017, to Gibraltar. She had decided to focus on playing the strongest competition she could, rather pursue women-only honours, but in Gibraltar she had to play seven women and three men. That wasn’t exactly according to plan. Beforehand she must have expected that she had to face no women, like in 2015, or just two, as in 2012. Hou Yifan was clearly affected by her pairings and threw her last game in protest.

*[The contemporaneous interview with Hou is no longer available on the Gibraltar Chess YouTube channel, but can still be found on YouTube, as well as the live commentary surrounding the incident.]*

Hou Yifan explaining her protest in 2017

Let’s have a look at the probability of her unusual pairings. How do we calculate the probability that a participant of a given Gibraltar Open, with a total of N* *participants of which *K* are women, that lasts *n* rounds, has to face *k* women, with *k* a number between 0 and *n*? This looks very much like an experiment where we have a population of (*N-K*) black and *K* white balls and draw randomly *n* times a ball, without replacement, and count the number of *k** *successes, i.e. white balls — which in this case shall mark the female players. Without replacement implies that after drawing a ball we don’t put it back. In this way we avoid drawing the same ball twice or more. Said in a different way: a chess player doesn’t have to play the same opponent twice or more. This problem has been solved a long time ago and leads to the hypergeometric distribution.

A question that has to be answered is whether the hypergeometric probability model adequately describes the reality of an open chess tournament. This is not obvious at all. Everybody who has played in an open tournament knows that at the end of the tournament he or she will tend to finish at about the same place as at the beginning of the tournament. His or her gain or loss of rating points will be close to zero. This is true for most of the participants but not for the youngsters who hope to gain rating points and for the veterans who hope to limit the loss of rating points. While playing in an open tournament, after a few rounds you start to recognize the faces around you. The group of players with the same score becomes smaller after each round and the pairings seem to become less random. So preserving some doubt that our model describes adequately the reality of an open tournament seems more than justified.

In order to test the hypergeometric model, I had a close look at the last six Gibraltar Open tournaments. Specifically, I counted the number of participants that faced *k* women during the *n* rounds of the tournament, with *k* a number between 0 and *n*. For the results of these investigations see table 1. The critical reader will notice that in this table a number of games are missing. This is due to the fact that there were women who took a day off during one or more rounds and because I overlooked every year one or two games that were played by women.

Table 1. The observed number of players facing k women during n=10 rounds.

Looking at this table we notice that in 2012, one participant, Soumya Swaminathan, faced six women and that in 2017 something even more remarkable occurred when two participants, Ketevan Arakhamia-Grant and Bodda Pratyusha, faced six women and one participant, Hou Yifan, faced seven women. It is astounding that all four players are women. One might think of a conspiracy if it wasn’t for the fact that a computer took care of the pairings.

Information about the participants that faced five, six and seven female players during these six tournaments can be found in table 2. The name of Andy Baert from Belgium appears twice in this table. In 2014 and 2016 he attracted a combined total of ten female chess players.

Table 2. The players that faced k=5, 6 or 7 women during the Gibraltar Opens of 2012-2017.

The next step is to compare the observed numbers of players that faced *k* women with the numbers of players that are predicted by the hypergeometric model. The results of my calculations for the year 2013 can be found in table 3. The total number of participants of the Tradewise Gibraltar Masters 2013 was N=247, there were K=34 women and the tournament lasted n=10 rounds. Three of the 340 games played by the female participants are missing.

Table 3. The Expected and Observed number of players facing k women during n=10 rounds.

For the calculation of the probabilities p(k) the online calculator of René Vápeník can be used. The observed number of players that faced *k* women are given in the Obs(k) column, while in the Exp(k) column the expected number of players can be found. In order to check the "goodness of fit" of the hypergeometric distribution we used Pearson’s chi-squared test. With the online calculator of Kristopher J. Preacher it could be determined that *p(χ^2=4.16)* = 0.53. The probability* p(χ^2)* = 0.53 is much higher than p=0.05, a conventional criterion for statistical significance. If the *p-value* lies above p=0.05 then we can accept our model, otherwise we have to reject it. We conclude that for the Gibraltar Open 2013 our model is certainly acceptable.

Table 4. The χ^2 and p(χ^2) values of the Gibraltar Opens 2012-2017.

The results of chi-squared "goodness of fit" tests for the years 2012-2017 can be found in table 4 above. According to these results, the expected values agree well with the observed values in all six cases. We observe that for the hypergeometric model five of the six values of *p(χ^2)* lie far above the critical value of p=0.05. Only the value of *p(χ^2)* for 2017 lies rather close to the critical value of p=0.05. These results confirm that we can use the hypergeometric model to estimate the probability that a participant faces *k* women during the *n* rounds of a Gibraltar Open tournament. The unexpected result suggests that the pairing process in Gibraltar was quite random.

For the Tradewise Gibraltar Masters 2017 tournament, with N=255, K=43 and n=10, I found that the odds that a participant had to face k=7 female opponents were 1 in 5311. So we could expect that, after playing roughly 21 tournaments with the same characteristics, one of the participants would eventually face seven female chess players. So far, in Gibraltar 15 tournaments have been played, most however with slightly less female participants. Anyway, the goddess of chance, Tyche (which means luck in Greek) decided that the honor of playing seven female opponents would be bestowed during the Tradewise Gibraltar Masters 2017 tournament on Hou Yifan, the world’s strongest female player. An honor that left her utterly confused.

Rd. | Bo. | SNo | Name | Rtg | FED | Pts. | Res. | w-we | |

1 | 22 | 143 | WGM | Pourkashiyan Atousa |
2303 | IRI | 5,0 | s 1 | 0,11 |

2 | 19 | 85 | GM | Zhukova Natalia |
2447 | UKR | 5,5 | w 1 | 0,24 |

3 | 11 | 47 | GM | Muzychuk Anna |
2558 | UKR | 6,5 | s ½ | -0,13 |

4 | 16 | 51 | GM | Muzychuk Mariya |
2546 | UKR | 6,0 | w 1 | 0,36 |

5 | 4 | 5 | GM | Adams Michael | 2751 | ENG | 7,5 | s 0 | -0,36 |

6 | 19 | 81 | GM | Cramling Pia |
2454 | SWE | 5,5 | w ½ | -0,25 |

7 | 20 | 78 | IM | Ider Borya | 2463 | FRA | 5,5 | s 1 | 0,26 |

8 | 15 | 38 | GM | Ju Wenjun |
2583 | CHN | 7,0 | w 0 | -0,59 |

9 | 23 | 66 | IM | Batsiashvili Nino |
2492 | GEO | 6,0 | s 1 | 0,29 |

10 | 17 | 37 | GM | Lalith Babu M R | 2587 | IND | 7,0 | w 0 | -0,59 |

Two other female players Ketevan Arakhamia-Grant and Bodda Pratyusha, who each played six female opponents, kept her company. Together, these three events made it a unique tournament. In retrospect, I believe that in Gibraltar, Tyche wanted to honor all female chess players in a highly original way.

After Gibraltar, Hou Yifan played in mixed tournaments in Sharjah, Shen Zhen, Karlsruhe & Baden-Baden, Moscow, Geneva, Biel and Tbilisi. Well, slightly mixed, because she was the only woman among forty-six men. She ended this string of tournaments on a high note winning the Biel tournament in August ahead of grandmasters like Etienne Bacrot and Pentala Harikrishna.

Hou Yifan in Biel, 2017 ǀ Photo: Pascal Simon

Hou Yifan’s next stop was the Isle of Man in the Irish Sea where she played in the Chess.com Isle of Man Open Masters 2017. This tournament was her first really mixed tournament after the Tradewise Gibraltar Masters 2017. The big surprise that awaited Hou Yifan on the Isle of Man was that in the first, second, third and fourth round her opponents were all women. It must have spooked her. What was Tyche trying to tell Yifan?

Data for the expected and observed number of participants of the Isle of Man Open 2017 that played *k* women during the n=9 rounds of this tournament can be found in table 5. The six games that are missing weren’t played. The differences between the values in the Exp(k) and Obs(K) are all very small and consequently the value of *p(χ^2)* is close to one — so almost a 100% match between expected and observed pairings. In this particular case, this agreement between prediction and reality is almost too good to be true. One of the predictions of the hypergeometric probability model, with N=159, K=22 and n=9, was that approximately three participants would face k=4 female players. These players turned out to be Andrew J. Ledger, Michael Babar and Hou Yifan. Completely explained by statistics — nothing strange happened at all!

Table 5. The Expected and Observed number of players facing k women during n=9 rounds.

During the Isle of Man Open, a rather improbable event occurred that went *unnoticed*. As reported by Sagar Shah in Indians at the Isle of Man (October 15th, 2017), there were K=30 players from India among the N=159 participants. It turned out that the number of players that would face *k* Indian players during the n=9 rounds of the tournament could be predicted rather well with the hypergeometric model. According to this model, we could expect that 1.76 participants would face k=5 Indian players and that 0.23 participants would face k=6 Indian players. The two participants that faced five Indians were Boris Gelfand and Varuzhan Akobian, while Adhiban Baskaran must have been quite amazed that he had to face six players, five of them in a row, from his native country. The wheel of fortune clearly turns in mysterious ways and not only for Hou Yifan!

Rd. | Bo. | SNo | Name | Rtg | FED | Club/City | Pts. | Res. | we | w-we | K | rtg+/- | |

1 | 6 | 7 | GM | Gelfand Boris | 2737 | ISR | 5,0 | w ½ | 0,41 | 0,09 | 10 | 0,90 | |

2 | 36 | 86 | Raja Harshit |
2423 | IND |
4,0 | s ½ | 0,81 | -0,31 | 10 | -3,10 | ||

3 | 30 | 91 | IM | Degtiarev Evgeny | 2412 | GER | 4,0 | w 1 | 0,82 | 0,18 | 10 | 1,80 | |

4 | 21 | 63 | IM | Nihal Sarin |
2483 | IND |
5,0 | s 1 | 0,74 | 0,26 | 10 | 2,60 | |

5 | 15 | 97 | IM | Harsha Bharathakoti |
2394 | IND |
5,0 | w 0 | 0,83 | -0,83 | 10 | -8,30 | |

6 | 23 | 45 | GM | Sunilduth Lyna Narayanan |
2568 | IND |
6,0 | s 0 | 0,64 | -0,64 | 10 | -6,40 | |

7 | 33 | 84 | GM | Sundararajan Kidambi |
2426 | IND |
4,0 | w 1 | 0,80 | 0,20 | 10 | 2,00 | |

8 | 24 | 78 | IM | Swayams Mishra |
2444 | IND |
5,0 | s ½ | 0,79 | -0,29 | 10 | -2,90 | |

9 | 23 | 68 | IM | Batsiashvili Nino | 2472 | GEO | 4,5 | w 1 | 0,76 | 0,24 | 10 | 2,40 |

In his article Investigating Hou’s pairings, Albert Silver wrote that the Gibraltar pairings and the Isle of Man pairings have been thoroughly checked. They were found to be beyond any suspicion of wrongdoing. Silver:

"These findings have been passed on to Yifan, who was forced to wonder what bizarre twist of fate had made her victim to such an incredible series of coincidences. Nevertheless, she was reassured, as were we all, that everything was in order."

It is nice that our hypergeometric probability model provides extra evidence for these findings.

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