Basic Rating Formulae
by Mark Orr
created August 31, 2012; last updated Nov 28

This article describes the basic formulae used by the ICU for calculating ratings: Elo formulae for players with full ratings and averaged performance ratings for players with provisional ratings. There are fairly standard as rating systems go but the ICU system also has some less common components: see the articles on bonus points and provisional ratings.

## Elo Formulae

These formulae are used for players with full ratings (20 or more games).

R = Ro + K × (A − E)

where

• R is the new rating after the event
• Ro is the old rating before the event
• K is the K-factor which determines the maximum change per game
• A is the actual score (counting 1 for a win, ½ for a draw and 0 for a loss)
• E is the expected score based on the old rating

A player's expected score in a single game, e, depends on the difference, D, between her rating and her opponent's. If both players have the same rating then they each have an expected score of 0.5. As the rating difference increases, the expected score goes up for the higher-rated player and down for the lower-rated according to:

e = 1/(1 + 10D/400)

The rating difference is obtained by subtracting the opponent's rating from the player's rating. The player's expected score for the event is the sum of expected scores in each game:

E = e1 + e2 + …

### Example

A player whose old rating is 2000 and who has a K-factor of 40 plays two rated games: a win against a player of the same rating and a draw against a player rated 2200. With an actual score of 1.5 (1 + ½), the new rating is:

= 2000 + 40 × (1.5 - 1/(1 + 100/400) - 1/(1 + 10-200/400))
= 2000 + 40 × (1.5 - 1/(1 + 1) - 1/(1 + 3.162))
= 2000 + 40 × (1.5 - 0.5 - 0.240)
= 2000 + 40 × (1.5 - 0.740)
= 2000 + 30.4
= 2030

In the example the player gained because her actual score (1.5) was higher than her expected score (0.740). New ratings are always rounded to the nearest integer.

## Performance Ratings

These formulae are used for players with provisional ratings (less than 20 games).

First, the player's performance rating in the event, Re, is calculated by taking:

• the rating of each player beaten and adding 400
• the rating of each player lost to and subtracting 400
• the rating of each player drawn

then summing these figures and dividing by the number of games played. Finally, a weighted average is taken between the accumulated performance rating before the tournament and the performance rating during the tournament:

R = (Ro × Go + Re × Ge)/(Go + Ge)

where

• R is the new rating after the event
• Ro is the old rating before the event
• Go is the number of games in all previous events
• Re is the performance rating in the event
• Ge is the number of games in the event

If a player starts a tournament with a provisional rating, these calculations are used throughout, even if the player reaches 20 rated games before the last round.

### Example

A player whose old provisional rating is 1000 after 10 games plays 2 games in a new event: a win against a player of the same rating and a draw against a player rated 1200. The performance rating for the event (Re) is:

= ((1000 + 400) + (1200 + 0))/2
= (1400 + 1200)/2
= 1300

The player's new rating, R, is then:

= (Ro × Go + Re × Ge)/(Go + Ge)
= (1000 × 10 + 1300 × 2)/(10 + 2)
= (10000 + 2600)/12
= 1050

In this example the new rating is closer to the old rating than to the new performance rating because the 10 previous games gives the old rating a higher weighting than the performance rating based on only 2 games.