9/19/2017 – With a staggering share of the world’s elite playing in the World Cup, vying for a spot in the forthcoming Candidates in Berlin, it was a surprise to see so many of the biggest names falling like flies by the midway point, such as Carlsen, Kramnik, Caruana and more. However, the truth is Magnus Carlsen only had a 36% chance to even make the final! Here are some eye-opening stats and numbers. | Photo: ChessBase India YouTube

As the biggest names came up with the short end of the stick in match after match, it seemed like we would see an almost (though not quite) random pair of players by the final. One of the first surprise casualties was five-time world champion Vishy Anand, who in his heyday had made mincemeat of this exact format on several occasions. Then followed Vladimir Kramnik, Fabiano Caruana, Hikaru Nakamura, and of course the biggest name of all: Magnus Carlsen, the reigning World Champion, all eliminated before half the tournament was even done!

Some players opined somewhat sourly that this just proved the elite was overrated, and facing the wider gamut of players, had faced this hard reality. While this is certainly a possibility, the fact is that according to the laws of probability, their fall was actually mostly according to expectation.

The laws of probability are merciless and with the help of my colleague and friend, David Fadul (the same author of the Memory Technique series), we have computed the chances for a player to make it through the vicious event with his proverbial head still attached to his neck, and reach the final.

The first thing to realize is that the Elo-system in itself does not answer this question out of hand. Elo doesn’t measure a player’s chances of scoring a point, it measures how many points he scored against another player, and consequently what one can expect him to score based on his rating.

In order to be able to calculate the players’ winning chances in each and every match of two games, a draw rate needs to be established. There are no absolute answers to this, so we consulted the Mega 2017 and looked at the statistics of all the games between players rated 2600 or more over the last three years. The result was a very round 50%. The database was then consulted for all rapid games between players rated 2600 or more, and the result was only 40% of the games ended in draws. Blitz games ended peacefully only 30% of the time.

The draw rates used in these statistics are based on the result by top players over the past three years in Mega 2017

To understand the results that follow, it is also important to understand a few basic tenets of the World Cup and the KO system it adopts. To reach the final, a player must navigate through six matches of two games, and win them. Let’s stop and think about that for a second. If a player managed this feat in just classical games, never needing a tiebreak, he will have scored at least 75% of the points with never more than one draw in a row. After all, to win a match he needs to score either 1.5/2 or 2.0/2. That would mean a hefty 193 Elo over the average of the opposition. All it takes is one slip in six matches, and the player is out. By comparison, in a normal 12-round tournament, a single loss does not mean an end to dreams of gold. However, then there are the tiebreaks.

Each tiebreak is also a mini-match of two games, with ever faster time controls, until one of the players cracks. Aside from the Armageddon, each round means up to four matches, for a possible total of 24 matches of two games — just to reach the final. The variance of such encounters is huge, which means that while a higher rated player is a favourite to win his match each and every time, the chances of a slip increase the more matches he plays. How big a chance?

The odds of making the final of the World Cup are not quite a roll of the dice, but chance plays a big role all the same.

Let’s suppose you are rated 100 Elo more than each of your opponents. Of course, you are the favourite to win each encounter, but the question isn’t whether you will win most of your matches, the question is what the chance of losing just one is. After all, in this format, if you lose just one in six you are out.

Imagine you are rolling a die against someone. They can only beat you if they roll a 6. Anything else you win. You are a huge favorite, right? But what if you give them six tries? Even though they still need a 6 to beat you, the odds of rolling it at least once increase dramatically. That is the World Cup in a nutshell.

If one were to exclude all drawn matches, meaning draws go to the favourite, the chance of the +100 Elo player of losing just one match is 54.2%. In other words, he is a favorite to lose one. If you add the tiebreaks however, the chances worsen considerably, and now the +100 Elo player has a 73.7% chance of failing to make the final.

How big an Elo advantage would a player need to have to have just a 50% chance of making the final? An astronomic 139 Elo!

Ok, but what about Magnus in all this? As the highest rated player, his chance to make the final is the highest, whatever the number. Everyone else will consequently be worse.

Just as an exercise in probability, all the pairings that actually took place were considered, as well as his rapid and blitz ratings. The first round isn’t even taken into account since he enjoyed a near 600 Elo advantage. Even Dreev was a monster underdog, though not just because of his nominal Elo. His 200 Elo deficit is considerable, but it is worsened enormously due to Carlsen’s even larger Rapid- and Blitz-rating edge. Nevertheless, the chance of his wading through Balogun, Dreev, Bu, Svidler, Vachier-Lagrave, and Aronian without a blemish was a mere 36%.

The brutal crosstable of just the first few rounds

So wait, if the highest rated player in the field had a mere 36% chance of making it to the final, then what about those who actually make it? Of course their chances are smaller still, but remember that even in a lottery, someone will win, in spite of the odds. 36% is quite a lot, all things considered, and can hardly be compared to a lottery, but it does illustrate just how volatile this format is. While this means enormous stress on the players themselves, the upside is that fans and spectators are guaranteed a thrilling day of chess, every day, all the way until the end.

Final moments of Magnus Carlsen's 2017 World Cup

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As its so important to win first and the play for a draw, I wonder how its decided the colour for the first game?

I think Carlsen could draw at will against most all players including Bu ;) But Carlsen would beat most all players in tiebreaks except players like Aronian and MVL or Karjakin. But being Carlsen is different. He wants to win all games. This is why we like him. He is a fighter.

Albert. Thanks for your answer, again. Clarification request. Maybe I am difficult. Sorry. Carlsen's chances to be a finalist before the beginning of the World Cup were 36%. But then, the tournament began and Carlsen got out of the first two rounds, which then become known events, the probability of which not having to be considered anymore.

On these considerations I understood that you meant: “If Carlsen has a chance of 89.58% of beating a player with Bu’s rating in one match, he has only a 53% chance of beating such a player in 6 consecutive matches”. If I understand correctly.

However, Bu was beaten in the 3rd round – if you have to win 6 consecutive matches to go in the finals, then there would have been only 4 rounds left to reach the final before playing that 3rd round. Should we not then consider the probability that Carlsen beat a player with Bu’s rating in 4 consecutive matches rather than 6?

Well, this is complicated by the fact that there is also a probability that in the 4 following matches, Carlsen faces players rated higher than Bu but… one thing at a time – but I will not bother you on that last one :0).

On these considerations I understood that you meant: “If Carlsen has a chance of 89.58% of beating a player with Bu’s rating in one match, he has only a 53% chance of beating such a player in 6 consecutive matches”. If I understand correctly.

However, Bu was beaten in the 3rd round – if you have to win 6 consecutive matches to go in the finals, then there would have been only 4 rounds left to reach the final before playing that 3rd round. Should we not then consider the probability that Carlsen beat a player with Bu’s rating in 4 consecutive matches rather than 6?

Well, this is complicated by the fact that there is also a probability that in the 4 following matches, Carlsen faces players rated higher than Bu but… one thing at a time – but I will not bother you on that last one :0).

Before the start, if Carlsen had 36% chances of making the finals, it also means that there were 64% chances that the two finalists be SEC1 (someone else than Carlsen) and SEC2.

@spacerobe...

I think Carlsen having a 36% chance of reaching the final means the *combined* chances of *everyone else* reaching the final is 164%, ie, the sum of all players chances =200% (since we're talking 2 qualifiers for the final).

I think Carlsen having a 36% chance of reaching the final means the *combined* chances of *everyone else* reaching the final is 164%, ie, the sum of all players chances =200% (since we're talking 2 qualifiers for the final).

Impressive odds calculation for the World cup 2017, but this calculation shows only one side, the losing side, but what about the winning side ? If M.Carlsen having the highest ELO rating, had only 36% chance reaching the final game, someone must have had 64% chance reaching the final game. Anyway this is a clear signal that ELO rating isn't reliable at all, it's time to get rid of this rude, corruptive mechanism to define a players strenght.

@yesenadam - For the number for Carlsen, a universal 50% draw rate was not used, and indeed a different one was used for each of his opponents, based precisely on the average draw rate of players within that Elo difference (and over 2700). You point is quite valid, but rest assured it was factored in. For example, for MVL and Aronian, the draw rate used was 70% not 50%.

An analogy - the "Miracle on Ice" - USSR against USA. Maybe in an 8-match series, USSR-USA, it would have been 7-1. But there was only one game, and it was the one.

Thank you for the answer Albert!

"the truth is Magnus Carlsen only had a 36% chance to even make the final!" - well no, that's not "the truth", is it. I guess it would be very hard to present in a headline an accurate version of what the study revealed. Or any study.. That was my only problem with this. Interesting study, well explained - nice work.

You talk of the inevitable arbitrary decisions you made, like which games to include to fix a draw rate. How about using the draw rate of games between players rated, say, +/-50 of the actual players. No less arbitrary, but closer to uh - *tries to avoid saying "truth"* - it might be a better model. I was surprised in learning basic statistics how arbitrary elements like this are present even in the basic formulae of statistics. And there are fashions in exactly which formula to use etc. I had thought it was more, uh, scientific, I guess. But the truth is, it's not, 89.4% of the time.

You talk of the inevitable arbitrary decisions you made, like which games to include to fix a draw rate. How about using the draw rate of games between players rated, say, +/-50 of the actual players. No less arbitrary, but closer to uh - *tries to avoid saying "truth"* - it might be a better model. I was surprised in learning basic statistics how arbitrary elements like this are present even in the basic formulae of statistics. And there are fashions in exactly which formula to use etc. I had thought it was more, uh, scientific, I guess. But the truth is, it's not, 89.4% of the time.

If you're looking for other interesting statistics, count the number of matches where the player who scored first didn't win the match. It's remarkably low!

@Raymond - As the highest rated player, Magnus is a favorite in all matches, and individually he is even as high as a 89.58% against Bu Xiangzhi, tiebreaks (and TB ratings) included. However, his chance to beat Bu 6 straight times to reach the final, would drop to a mere 53%. Remember that it doesn't really matter what his chance to beat him once is, or three times. The World Cup is 6 rounds to reach the final, and it is the accumulated 6 times that pretty much kills everyone's chance of even having a 50-50 shot at making it.

Among the 4 remaining players we have from 128 at start, on classical live ratings as of September 19: world no 2 against world no 3, and world number 7 against world number 11. Maybe not statistically significant but does not seem shockingly against what could be expected from past history from Elo rating.

Of course, any rating system can only account from past history and project from it.

Actual level over the board in a given game can be different... Full suspense.

Of course, any rating system can only account from past history and project from it.

Actual level over the board in a given game can be different... Full suspense.

It would be interesting to know the probability of what actually happened to Magnus. The article indicates that had 36% chances of going to the final (round 7). But what was the probability of Magnus falling in round 3, as what happened for real?

They should at least "reseed" after each round like is done in Swiss tournaments and tennis. Otherwise, there is a reason this format was more or less abandoned in the 1850's...

It is interesting that the picture you chose shows two illegal dice!

The World Cup format is not suitable to classify for candidates, is just too random.

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