The Elo system was invented by
Arpad Elo and is the most common rating system. It is used by
FIDE, other organizations and some Chess websites such as
Internet Chess Club and
chess24.com. Elo once stated that the process of rating players was in any case rather approximate; he compared it to "the measurement of the position of a cork bobbing up and down on the surface of agitated water with a yard stick tied to a rope and which is swaying in the wind". Any attempt to consolidate all aspects of a player's strength into a single number inevitably misses some of the picture. FIDE divides all its normal tournaments into categories by a narrower average rating of the players. Each category is 25 rating points wide. Category 1 is for an average rating of 2251 to 2275, category 2 is 2276 to 2300, etc. Women's tournaments currently commence 200 points lower, including its Category 1. The USCF uses the USCF system, a modification of the Elo system, in which the K factor varies and it gives bonus points for superior performance in a tournament. USCF ratings are generally 50 to 100 points higher than the FIDE equivalents.
Example Elo gives an example of amending the rating of
Lajos Portisch, a 2635-rated player before his tournament, who scores 10½ points of a possible 16 winning points (as this is against 16 players). First, the difference in rating is recorded for each other player he faced. Then the expected score, against each, is determined from a table, which publishes this for every band of rating difference. For instance, one opponent was
Vlastimil Hort, who was rated at 2600. The rating difference of 35 gave Portisch an expected score of "0.55". This is an impossible score as not 0, or 1 but as this is higher than 0.5 even a draw will slightly damage Portisch's rating and slightly improve Hort's rating, so (ignoring their other results in the tournament) moving their ratings slightly closer together. Portisch's expected score is summed for each of his matches, which gave him a total expected score of 9.66 for the tournament. Then the formula is: : new rating = old rating + (K × (W−We)) K is 10; W is the actual match/tournament score; We is the expected score. Portisch's new rating is 2635 + 10 × (10.5 − 9.66) = 2643.4.
Linear approximation Elo devised a linear approximation to his full system, negating the need for look-up tables of expected score. With that method, a player's new rating is R_{ new } = R_{ old } + K \left( \frac{W - L} {2} \right) + \left( \frac{K}{4C} \right) \sum_i D_i where
Rnew and
Rold are the player's new and old ratings respectively,
Di is the opponent's rating minus the player's rating,
W is the number of wins,
L is the number of losses,
C = 200 and
K = 32. The term
(W-L) / 2 is the score above or below 0.
ΣD / 4C is the expected score according to: 4C rating points equals 100%. The USCF used a modification of this system to calculate ratings after individual games of
correspondence chess, with a
K = 32 and
C = 200. ==Glicko rating system==