World Cup: the odds of making the final

Easier said than done

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.

Always bet on 'the field'

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!

Mister 36%

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