Dictionary of chess terms. Creating a tournament schedule Berger table for a mixed system of 27 teams

What is this thing, when and where is it used. Today, under our close gaze, the Berger coefficient is, in its way, the “half-brother” of Buchholz.

What it is?

The Berger coefficient is an additional numerical indicator and is used to rank the participants in the standings. It is taken into account only in case of equal points of participants.

The author of the idea is the Czech Oscar Gelbfus, who proposed a similar method of ranking in 1873. The Berger coefficient entered the tournament practice starting from the tournament in Liverpool in 1882 thanks to the efforts of William Sonneborn and Johann Berger.

As you can see, the history of the distribution of seats with the help of "Berger" has passed more than a solid test of time.

Berger's coefficient applied in round robin tournaments . When all participants play among themselves in turn.

How to count?

I hasten to reassure you, there is no higher mathematics here. If you wish, you can calculate everything in your mind.

The formula for calculating the Berger coefficient is as follows:

KB = SumB + ½ SumH, where

AmountB- The sum of points of opponents from which the participant won

Amount- The sum of points of opponents with whom the participant tied.

The points of opponents to whom the participant lost are not taken into account. Rather, the sum is considered equal to zero.

For example:

In the table above, Sidorov and Kuznetsov scored 4 points each. To rank in the final table of the tournament, we will calculate the "Berger" of these participants:

Sidorov: 1 + ½* (5 +4.5 +4 +2.5) = 9

Kuznetsov: (2.5 +1) + ½* (4.5 +4) = 7.75

Thus, Sidorov is ahead of Kuznetsov in the standings with equal points in an additional indicator - the Berger coefficient.

Berger's logic

Any additional indicator that affects the final distribution of places in the table must have a certain logic. How to carry the "grain of justice" in yourself.

Berger's logic is determined by the formula for calculating the odds: the player who scores more points against stronger opponents has an advantage.

I will not say that such logic is unconditionally fair and cannot raise questions.

Perhaps that is why, in recent years, in order to determine prizes, often instead of additional indicators, additional games with a shortened control are practiced. Whatever you say, the result at the board is always a priority.

However, it is hardly possible to do without additional indicators, especially when distributing non-prize places. In almost a century and a half of the history of chess, no one has come up with anything more adequate than the CB.

The Berger coefficient is still alive and well just as it was in 1882. in Liverpool.

Simplified count

Since about the eighties, simplified counting has also come into practice.

It's even easier: The points of defeated opponents are added, the points of those who lost are minus (taken with a minus sign). The sum is considered simple arithmetic addition.

This method simplifies the calculations.

Common Mistake

For a tournament fight, the following situation is usual: before the last round, the participants estimate the coefficients. In order to choose tactics for the last game. For example, the chess player Petrov thinks:

“It’s enough for me to make a draw, because if Ivanov beats Pupkin and catches up with me on points, Berger is better for me”

And Petrov agrees to a draw in a position with excellent chances of winning, anticipating the rewarding procedure.

However, when calculating the coefficients, it suddenly turns out that his Berger is worse than Ivanov's!

The secret is simple. In the last round, games were played and points were awarded. Petrov, in his estimates, was guided by the “spectacle weight”, which was relevant until the last round.

Well, when you play in a team, there is a coach or another person who "counts" all these nuances. Often online during the last round. It is also not difficult to make some kind of calculator.

However, being distracted by such things during the game is very risky. I guess it's overkill to explain that the best math is winning over the board.

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Yesterday in the Premier League held a meeting of sports directors of clubs, which discussed the calendar of the second stage of the championship. "Soviet sport" knows some details.

WHY DID YOU REJECT THE MANUAL OPTION?

As Sovetsky Sport previously reported, after the blind draw procedure was abandoned (due to the inability to take into account extremely important factors: the climate, the participation of teams in European competitions), it was decided to work out the option of manually compiling the calendar. With this option, the above factors could be taken into account, but in reality it turned out that it would be difficult to “include” all the wishes of the clubs in the schedule.

In general, early planning of the calendar, especially in a situation where the first eight have decided with a high degree of probability, is an absolutely normal phenomenon. There were questions on the ethical side - they say, how can you take into account the matches with the participation of Anji, if Krasnodar did not lose the chance to break into the top eight? But, firstly, they worked on the calendar, when the gap between 8th place and 9th was about 10 points, and secondly, it would be somehow strange if the league took care of the schedule the morning after the 30th round.

Yesterday, RFPL President Sergei Pryadkin told a Sovetsky Sport correspondent Sergei EGOROV the following: "The calendar will be drawn up according to the sports principle."

What does it mean? According to our data, we are talking on the scheduling of the tournament according to the so-called Berger tables.

WHAT IS THE BERGER TABLE?

The table, named after the famous Austrian chess player and chess theorist Johann Nepomuk Berger, is a way of making a calendar.

Clubs are assigned a number corresponding to their place in the Russian Championship. Each club, except for the one that received the first number, consecutively plays with rivals in ascending numbers. That is, the team that takes eighth place plays in the first round with the first, in the second round - with the second, in the third round - with the third and so on until the seventh round. The eighth round of the opponents coincides with the second, the ninth with the third, and so on. The last round will repeat the first, only the opponents change fields.

If the position of the teams after the 30th round remains the same as after the 28th, the top eight clubs will receive the following numbers: 1. Zenit, 2. CSKA, 3. Lokomotiv, 4. Dynamo, 5. Spartak, 6. Rubin, 7. Kuban, 8. Anji.

With this system, the sports principle is preserved - the strongest team starts the tournament with the weakest. And so that our readers can follow a possible calendar online, we publish both the Berger table and an approximate calendar as of the 28th round. And you can update the schedule after each of the two remaining tours in the first stage.

The final version (unless, of course, it is decided to abandon the Berger table) we will find out on the evening of November 6, when the last match of the 30th round ends.

WHAT WILL HAPPEN TO THE SECOND EIGHT?

The calendar of the second G-8 will be determined by a blind draw on November 7 during a solemn ceremony. As Sergey Pryadkin already said in an interview with Sovetsky Sport, the matches of the teams of the first and second eights will be held on the same dates, the days of the tours will coincide.

Sonneborn-Berger system- a method for determining the best result (coefficient) if several participants in the tournament scored the same number of points. The coefficient of the participants is equal to the sum of the points of the opponents they won against and half the points of the opponents with whom they drew.

In fact, the Sonneborn-Berger odds system gives an advantage to a player who won against strong players and lost against weak ones over a “normal” player who lost against strong ones and won against weak ones. The Sonneborn-Berger odds are widely used, especially in round robin tournaments.

The Sonneborn-Berger system is not objective, therefore, in important cases (definition of a champion, admission to the next stage of a major competition), it is customary to hold an additional competition. The mixed method is also used (in case of equality of points in the additional competition, the Sonneborn-Berger coefficient decides).

Along with the Sonneborn-Berger coefficient system, other methods are used to identify advantages in case of equality of points: by the number of wins, by the result of a meeting between themselves, etc.

Not so long ago, the first World Cup in Russia ended. The fanfare rang out, the champions, winners and losers went home. Someone earlier, someone later, someone with regret, someone is happy, and someone did not leave anywhere 🙂 The last championship gave a lot of emotions, many bright matches, a great final and something else. Namely, a situation that is unique in my opinion, when one of the teams left the group due to ... fewer yellow cards! Note about this situation.

So, we will talk about group H, although there was a moment when a similar situation was in group B, where Spain and Portugal could really even reach the draw! First, a few words about why this is possible at all.

The round robin system, for all its merits, is not without drawbacks, the main of which is the issue of distribution of places in case of equality of points. Smart people have come up with a lot of all sorts of additional coefficients, some of which are de facto and de jure standards. For football, coefficients are not used (it is not very clear why), instead they are considered (at least for the 2018 World Cup):

• the difference between goals scored and goals conceded. The logic is simple - who scores more and concedes less, the higher. Let us now leave the discussion on the adequacy of this approach, we will only accept the fact that it is used as the first additional indicator.
• the number of goals scored. The logic is the same - who scores more, that is, who is more aggressive, more interesting, more militant. Again, there is no purpose to criticize the system. This is the second additional indicator.
• difference of yellow cards. This is what I think is bullshit.

Let's be clear! As a chess player, it will be convenient for me to compare with chess. The difference between goals scored and goals conceded is about the same as counting the number of moves in games. Roughly speaking, Vasya Pupkin beats Kesha Popkin in 20 moves, and Fedya Ruchkin beats the same Kesha Popkin in the longest endgame of 140 moves. Between themselves they played a draw, even in 10 moves, even in 150 - it doesn't matter. Who is stronger - Vasya Pupkin or Fedya Ruchkin? According to the first additional coefficient - Vasya, because he beat Kesha faster. Rave? Rave. Maybe Kesha just didn't get enough sleep, messed up something in the opening, blundered etc. Again, against Fedya Ruchkin, Kesha fought like a hero, but still lost. Why is Vasya stronger? Maybe, on the contrary, he is weaker, because he easily broke Kesha's resistance, while Fedya rolled out the hardest ending and was eventually rewarded. Maybe the one who put in more effort is stronger? Also nonsense. And who is really stronger? The correct answer is nobody.

A football example: let the conditional Russian team defeat the conditional Chinese team with a score of 5:0. Eagles! And the conditional French team defeated the same Chinese team with a score of 2:0. Between themselves, Russia and France played boring 0-0. Under the current system, Russia is higher, as the difference between goals scored and conceded is higher. The disadvantage of the system is that it does not take into account the fact that all teams have different styles, and whipping boys are not always whipping boys. And in general, there are a lot of accidents that can change not only the score, but the entire course of the match.

The same goes for goal difference. The system is imperfect, not always fair (really!), but it exists and everyone is used to it. Let! But yellow cards… It's not even nonsense, it's indescribable. It is clear that FIFA thus fights for the purity of the game, for the notorious Fair Play, but is it really that important?? I will express my personal opinion - yellow cards in football are the same element of strategy as everything else. How many we have seen at this championship, at others, but anywhere tactical violations? A lot of! Not always it was yellow cards, but nonetheless. Again, there are teams that are rougher, there are less. You don't have to treat everyone with the same brush. It is clear that rudeness, real rudeness, should be punished on the field, but such purely tactical violations as disrupting an attack are quite possible. And this is the same element of the game as a corner kick! Weeding out teams according to this principle is like weeding out chess players by the number of moves in the game that he made on the first line of the conditional Stockfish ...

That critic is bad who does not offer an alternative, but only babbles. I will be a good critic. Let's look at methods that could more honestly (in my opinion) resolve such controversial situations and let's find out who is still worthy of reaching the 1/8 finals - Japan or Senegal.

This is how the group H table looks like. It and the pictures of the flags are taken from the Eurosport website.

As you can see, Japan and Senegal have completely the same indicators. 4 points each, goal difference 4-4. In a personal meeting, it will also not be possible to select - a draw 2:2. With fewer yellow cards, Japan came out in 1/8. It's funny that the Japanese coach admitted that with the score 0-1 in last game against Poland, his team defended and did not go forward. Cynically? Why not a remedy?

In chess, situations where the number of points is the same are extremely frequent. Since there are no goals scored and other things, you have to invent all sorts of coefficients and systems. We will start with them.

Berger coefficient was invented by the Czech master Oscar Gelbfus (suddenly, right?) a long time ago and has been used by chess players for more than a hundred years. Agree, time. Exhaustive quote from Wikipedia:

The Berger coefficient of a certain participant is the sum of all the points of opponents against whom this participant has won, plus half the sum of the points of opponents with whom this participant has drawn. The idea on which the coefficient is based: of two participants equal in number of points, the one who won against stronger opponents, that is, those who scored more points, is stronger. Therefore, a participant with a higher Berger coefficient is awarded a higher final place in the tournament.

We count! Japan won against Colombia (the score is not important), which ended up with 6 points, tied with Senegal, which had 4 points, and lost to Poland (in general, it does not matter how many points Poland has in this case). Therefore, Japan's Berger coefficient is 6 (100% of Colombia) + 2 (50% of Senegal) + 0 (0% of Poland) = 8. Senegal won against Poland, who ended up with 3 points, tied with Japan ( 4 points) and lost to Colombia (again, no matter how many points). Senegal's Berger coefficient is 3 (100% of Poland) + 2 (50% of Japan) + 0 (0% of Colombia) = 5.

Sonneborn-Berger coefficient

Sonneborn-Berger coefficient - the same Berger. It is slightly adapted for chess so as not to count halves, but for football there is no such problem. Just to show it, Japan has 12 (200% of Colombia) + 4 (100% of Senegal) + 0 (0% of Poland) = 16, Senegal has 6 (200% of Poland) + 4 ( 100% from Japan) + 0 (0% from Colombia) = 10.

Koya system

Koya system - another method based on the same idea - the stronger the opponent that was defeated, the stronger you are! Let's go back to Wikipedia:

The Koya system takes into account the number of points scored against all opponents who achieved a score of 50% or more (i.e. scored more than 50% of the maximum possible points).

Fun fact, but with this scoring system (3 for a win, 1 for a draw), 50% is 4 points (1 win, 1 draw and 1 loss = 4).

We count! Dropping Poland (3 points< 50%) и считаем сколько очков набрали Япония и Сенегал в матчах с Колумбией и между собой. Япония - 3 (победили Колумбию) + 1 (ничья с Сенегалом) = 4. Сенегал - 1 (ничья с Японией) + 0 (поражение от Колумбии) = 1. При прочих равных, система Койя делает то же, что и Бергер, но наглядней.

These three systems are logical and seem to me exhaustively show why Japan's advance to the 1/8 finals is more fair than Senegal's (to be honest, I was rooting for Senegal in this group!). Moreover, the Japanese proved that they did not get there by chance and not in vain. They were on the verge of a sensation...

I wish FIFA would turn to something more reasonable than the number of yellow cards and (oh my god!) the draw! All the same, not for the championship of the court tournament.

Laplacian potential

If you have read up to this point, but were waiting for pythonic revelations, then I also have them. A little hard math never hurts! 🙂

To begin with, I strongly recommend reading and understanding (as far as possible). Let's agree that the balance equation is what we need (we can consider the group as a balanced system). Participants play with each other and "self-evaluate" each other. Since the result (crosstable) of the group is an adjacency matrix, we can easily construct a Kirchhoff matrix. Like this one:

Such a matrix is ​​built in an elementary way, by adding a sign (-) to the points obtained. We need to add such values ​​​​to the main diagonal so that the sum of the column is equal to 0. In order to obtain the values ​​​​of potentials and fluxes from our Kirchhoff matrix (aka Laplacian), we must find additional minors of the matrix (they will be potentials) and multiply them by the corresponding value of the main diagonal. It seems that everything is complicated, but let's look at the code:

The minor is calculated as follows: one column and one row are deleted from the original matrix and the determinant is considered.

Since we have assumed that in our Laplacians the sum of each column is zero, then the value of the potential is determined only by the crossed out column - the row can be any. It is convenient to strike out the same line as the column - then you do not need to think about the sign of the determinant.

That is why we cross out the same row as the column. The results obtained are the potentials, that is, the weights of the participants. If the potential is multiplied by the value of the main diagonal of the original matrix (that is, by its degree), we get the value of the flow (the table is sortable).

Let's look at the results. The team's potential (the absolute values ​​of the numbers are not important, only relative ones) is the "weight" of the team, it can be conditionally said that this is the strength of the team in this tournament. That is, it is already enough for us to calculate the potential in order to understand who is stronger. It is clear from the table that Japan is stronger again. The situation with streams is more interesting. Since the higher the potential of the team, the more valuable the points received from it by others, Japan, which defeated the Colombians, received even more flow than Colombia itself. A similar story is with Poland, which defeated a strong (in terms of potential) Japan.

Of course, the calculation using Laplacians and the balance equation is much more complicated than the Koya system or the Berger coefficient, in addition, there is one more question:

What exactly should serve as the basis for ranking - potentials or flows - requires separate consideration in each task, since it is determined by the applied aspect.

And yet, in my opinion, the proposed methods make it possible to unequivocally identify the strongest team in case of equality of points (the values ​​of the flows and potentials of Colombia and Poland can be omitted), because, I repeat, the tournament is not for the championship of the yard.

There is no need for revolutions, there is no need to remove the goal difference, everyone is used to it, but that's why it is impossible to use the Koya system (or the Berger coefficient) instead of absurd cards (and / or, as an option, instead of the number of goals scored), and even more so, if suddenly all the indicators will turn out to be equal (no matter how good Koya and Berger are, this is possible) not to arrange a random draw, but to uncover the Laplacians. The calculations are not that complicated. Recent times FIFA introduces technologies - spray, video replays ... Why not make the rules of the group stage of the World Cup more balanced?

If the world has survived on balance, then such a system has a chance.

What will the experts say?

Berger coefficient- a way to determine places in competitions among participants who have scored an equal number of points. The method of determining the place by the Berger coefficient was originally developed for round-robin (everyone plays with everyone) chess tournaments. Later, this method was applied to other competitions, such as shogi and go.

Order of Evaluation

In round-robin tournaments, where a certain constant number of points is awarded for a victory, a draw and a defeat (for example, in chess 1 point is given for a victory, 0.5 points for a draw, 0 points for a defeat; less often - 3 - for a victory and 1 for draw, for example, in London Chess Classic 2010), it often happens that two or more participants score the same number of points. To determine which of these participants ranked higher, the Berger coefficients of the participants are calculated.

The Berger coefficient of a certain participant is the sum of all the points of opponents against whom this participant has won, plus half the sum of the points of opponents with whom this participant has drawn. The idea on which the coefficient is based: of two participants equal in number of points, the one who won against stronger opponents, that is, those who scored more points, is stronger. Therefore, a participant with a higher Berger coefficient is awarded a higher final place in the tournament.

The Berger coefficient was invented for round-robin tournaments, but can, if necessary, be used in other drawing schemes, where the players whose places need to be distributed play an equal number of games. It can also be used in tournaments according to the Swiss system, although the Buchholz coefficient is traditionally used there. In round-robin tournaments since 1985, the “simplified Berger” (proposed by M. Dvoretsky) has also been used: the points of all opponents against whom the chess player won are taken with a plus sign, and all those to whom he lost - with a minus sign, by the sum and is considered the best result. This allows you to reduce the calculations and not pre-divide in half most of the results.

Example

The final table of a hypothetical round-robin tournament:

Designations: 1 - victory, ½ - draw, 0 - defeat, KB - Berger coefficient.

Participants Sidorov and Kuznetsov scored the same number of points, 4 points each. Which of them will take third place is decided by the Berger coefficient.

Sidorov's Berger coefficient is: 2.5 (half of Ivanov's points) + 2.25 (half of Petrov's points) + 2 (half of Kuznetsov's points) + 1.25 (half of Smirnov's points) + 1 (all of Vasiliev's points) + 0 ( all points of Nikolaev) = 9.

Kuznetsov's Berger coefficient is as follows: 0 (for the loss to Ivanov) + 2.25 (half of Petrov's points) + 2 (half of Sidorov's points) + 2.5 (all Smirnov's points) + 1 (all Vasiliev's points) + 0 (all points Nikolaev) = 7.75.

Thus, participant Sidorov has a higher Berger coefficient than participant Kuznetsov (9 versus 7.75), so the third place is awarded to Sidorov. The Berger coefficient is higher for someone who wins or ties with stronger players (players who score more points). In the example above, winning against a participant with zero points does not contribute to the Berger coefficient.