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# Rating Calculations

The rating scheme used by PlayChess is based upon the rating schemes used by many other organisations (FIDE, ICCF, IECG).

The application - and even more the deduction - of the rating scheme requires a lot of mathematics. But don't worry: as a player you don't need to understand this scheme - PlayChess does all the math automatically for you.

The following "mathematical treatise" is given here just to

• demonstrate that there is no hidden magic in the rating calculation
• enable the mathematicians to check our numbers
• prove how extremely clever we are ... :-))

## Rating Scheme for Dummies

If you are just a humble human being, not a god of math, here are the essential ideas of the rating system:

• A rating number measures the strength of a player.
The higher the number the stronger the player.
• If two players of same strength (say rating 1000) play against each other,
their rating performance will be as follows:
• The game is drawn (½-½)
The rating performance of both players is 1000,
because they played like expected.
• Player A wins (1-0)
The rating performance of player A is 1400 (+400 points).
The rating performance of player B is  600 (-400 points).
• The future ratings of both players will be calculated from the rating performance of the single game played AND from the rating they had prior to the game.
In the example above (1-0), player A had played 4 games before, and player b had played 1 game before. The new ratings of the players are calculated as follows:
Player A: New rating R = (4 × 1000 + 1400) / 5 = 1080
Player B: New rating R = (1 × 1000 +  600) / 2 = 800
• If you win or draw against a stronger opponent (higher rating), your own rating will rise even stronger.
• If you lose or draw against a weaker opponent (lower rating), your own rating will drop even stronger.

The ratings of experienced players (more than 10 games) are calculated by a more complicated formula. There, the simple "arithmetic mean" formula above is not sufficient, because the rating of these players would change VERY slowly (imagine players with more than 100 games!).

# The Complete Rating Scheme

Principle. The rating scheme is a numerical scheme, in which percentage results can be exchanged into rating differences and rating differences into percentage performance probabilities. The basis of the scheme is the normal probability distribution.

Provisional and Established Ratings. A player has an established rating, if he/she has finished at least 10 games, otherwise his/her rating is called provisional.

Cut Off. To keep the influence of largely different ratings small for the preliminary (and established) ratings, a rating cut-off is used. Anytime the difference is larger than 400 points, the opponents rating is treated as being 400 points higher/lower than the players rating.

The performance probability is calculated by the formula

P(D) = 1/(1+10^(-D/400))        (1)
P(D) is the performance probability
D is the difference of the ratings of the two players.

The expected rating changes based on the percentage result is given by

D(p) = -400 * log10((1-p)/p)        (2)
D(p) is the expected rating change
p is the percentage result of the player
and D(0) = -800 and D(1)= 800
The percentage result is calculated by
p = (2*W+D)/(2*N)        (3)
p is the percentage result
W is the number of wins
D is the number of draws
N is the number of finished games

For the calculation of the ratings, the opponents "Tournament Entry Ratings", which are valid on the day of the rating run, are used.

## Calculation of Established Ratings

For each finished game the rating change is calculated by

dR = k*(W-We)        (4)
dR Rating change
W True game result (win 1, draw 1/2)
We Expected result
k Development coefficient

The expected result We is calculated using formula (1) with the rating difference of the two opponents. The development coefficient is a stabilisation factor and is given by

k = r*p        (5)
 r = 10 if Ro >= 2400 r = 70-Ro/40 if 2000 < Ro < 2400 r = 20 if Ro<= 2000 p = 1 if Pn >=80 P = 1.4-Pn/200 if 30 < Pn < 80 P = 1.25 if 15 < Pn <= 30 P = 1.5 if Pn <= 15
 Ro is the Rating from the previous rating list Pn is the number of all rated games upto the previous rating list.

The next rating is calculated by

Rn = Ro + SUM(dR)        (6)
Rn new rating of the player
Ro old rating of the player
SUM(dR) Sum of all rating changes as calculated in (4) for each game

## Calculation of Provisional Ratings

The Provisional Rating is calculated by

Rp = Rc + D(p)*F        (7)
Rp Provisional Rating
Rc Average of the opponents tournament entry ratings
D(p) Expected rating change based on (2)
F A Correction factor given by F = -2*p*p + 2*p + 0.5