German rating number

The German rating number ( DWZ for short ) is a rating number in chess to compare the skill level of individual players.

General

The introduction of the DWZ was decided by the German Chess Federation (DSB) after reunification. By January 1, 1993 , the DWZ was introduced across the board and replaced the Ingo system of the German Chess Federation of the FRG and the NWZ system of the German Chess Federation of the GDR. The DWZ is comparable to the FIDE Elo rating , but has been continuously developed over the years. Experience from the Ingo system and the NWZ system were taken into account during development. The player strengths range from around 500 (beginners) to over 2800 (world champions), but theoretically it is open upwards and downwards. The evaluation of the tournaments is carried out by DWZ speakers, who forward the recorded tournaments to the evaluation center of the DSB, where the central evaluation database (ZDB) is kept. A chronological recalculation will then be carried out there according to the end date of a tournament.

In contrast to the Ingo system, a higher DWZ means a higher skill level. The DWZ is made up of the rating as a measure of skill level and the index, which is separated from it by a "-". For players who do not have a DWZ, but have an Elo rating from FIDE, the latter is given the index 6 and adopted as a DWZ. For players who have neither a DWZ nor a FIDE Elo rating, but have a national rating (NWZ), this is used and possibly first converted into a DWZ. In this case the index is set to zero.

Calculation of the DWZ

The basis for the calculation is a normal distribution , the density of which is known as the Gaussian bell curve. An integral is used for the calculation to determine the profit expectation. Only results achieved on the chessboard against opponents with a DWZ are taken into account.

Basic formula

The DWZ is calculated as follows:

${\displaystyle Z_{n}=Z_{0}+{\frac {800}{E+n}}(W-W_{e})}$

Z ₀ : previous DWZ

Z _n : new DWZ

W : Points scored

W _e : expected points

n : number of games played

E : development coefficient

Expected and scored points

The points scored are the sum of the game results, with a win counting as 1 point, a draw as 0.5 points and a loss as 0 points. ${\displaystyle W}$

${\displaystyle W={\frac {1}{n}}\sum _{i}^{n}s_{i}}$

The expected points are accordingly the sum of the expected game results. Thanks to the appropriate point scale, the expected result of a game is simply the probability of winning , which only depends on the DWZ difference between the rating of the player in question and that of his opponents . ${\displaystyle W_{e}}$ ${\displaystyle P(D)}$ ${\displaystyle D}$ ${\displaystyle Z}$ ${\displaystyle Z_{i}}$

${\displaystyle W_{e}={\frac {1}{n}}\sum _{i}^{n}P(Z-Z_{i})\ }$

A normal distribution is assumed for the probability of winning . ${\displaystyle {\mathcal {N}}(\mu =0;\sigma ^{2}=80000)}$

The distribution function of the normal distribution is through

${\displaystyle F(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}e^{-{\frac {1}{2}}\left({\frac {t-\mu }{\sigma }}\right)^{2}}\mathrm {d} t}$

given. With the corresponding values ​​for and the following function of the winning probability results: ${\displaystyle \mu }$ ${\displaystyle \sigma ^{2}}$

${\displaystyle P(D)={\frac {1}{400{\sqrt {\pi }}}}\int _{-\infty }^{D}e^{-\left({\frac {t}{400}}\right)^{2}}\mathrm {d} t}$

Since the formula is somewhat unwieldy, the following approximate formula is used to estimate the results, which corresponds to a centered logistic distribution with a resulting standard deviation of approx. 315. It is also used when calculating the Elo rating. ${\displaystyle (\alpha =0;\beta ={\tfrac {400}{\ln 10}})}$

${\displaystyle P(D)\approx {\frac {1}{1+e^{-{\tfrac {\ln 10}{400}}\cdot D}}}={\frac {1}{1+10^{-{\tfrac {1}{400}}\cdot D}}}}$

For the values ​​of the expected points there is a table in the scoring rules of the German Chess Federation. However, this only contains corresponding entries for the amounts of the rating number differences.

The development coefficient

Another component is the expansion coefficient E . It is made up of the basic value E ₀ , the acceleration factor a and the braking allowance B :

${\displaystyle E=a\cdot E_{0}+B}$

With

${\displaystyle E_{0}=\left({\frac {Z_{A}}{1000}}\right)^{4}+J}$

The component J depends on the age of the player. For teenagers up to 20 years: J = 5, for juniors (21-25 years): J = 10, for players over 25 years: J = 15.

${\displaystyle a={\frac {Z_{A}}{2000}}}$

The following applies to the acceleration factor a : This must not be greater than 1 and not less than 0.5. In addition, it will only be calculated if the player is under 20 years old and more points have been scored than expected. If this is not the case, the acceleration factor is 1. The acceleration factor helps younger players improve their DWZ faster.

${\displaystyle B=e^{\frac {1300-Z_{A}}{150}}-1}$

The brake surcharge B is only calculated for players with a DWZ below 1300 and only if the points scored are less than or equal to the expectation, otherwise the brake surcharge = 0. The brake surcharge is there to ensure that the DWZ of poorer players does not decrease so quickly.

E also depends on the number of tournaments that have been rated so far. The index i of the first DWZ is 1 and is increased by 1 after each tournament evaluation.

The following also applies:

${\displaystyle 5\leq E\leq {\begin{cases}\min(30;5i)\quad {\text{für }}B=0\\150\qquad \qquad {\text{ für }}B>0\end{cases}}}$

This formula is only "evaluated" if there is already a DWZ, that is, i is greater than zero. This usually increases E to 5 if the coefficient is below 5, or decreases to 30 if E is greater than 30. If there are brake supplements ( B > 0), E can also go beyond this.

In the last step, E is rounded to the nearest whole number.

Calculation of an initial DWZ

The basic formula cannot be used to calculate a “first DWZ”. Based on the point yield in percent, a DWZ difference is determined with the help of the probability table. To this value you add the DWZ average of the opponents and get a “first DWZ”. The prerequisite for this is five or more games played and a point yield of at least half a point and less than 100% of the possible points. At least one draw must have been achieved or given up.

Exemplary representation of the meaning of the DWZ

German rating number	Approximate meaning
<1000	Beginner
1000-1300	More advanced
1300-1600	Normal club player
1600-1900	Above average club player
1900-2100	Outstanding club player
2100-2300	League player
2300-2500	Bundesliga player
2500-2700	Grandmaster
> 2700	World class player

Web links

• Evaluation regulations (regulations for the DWZ skill level evaluation of chess players in Germany) - German Chess Federation
• DWZ database - German Chess Federation
• An unofficial DWZ calculator according to the current ranking
• An unofficial DWZ computer with a storage facility

Individual evidence

1. ↑ Probability table of the scoring system of the German Chess Federation