Performance rating (chess)

A player's performance rating in a series of games is the Elo rating a player would need to have to expect to get their actual total score against the opponents they faced in those games. A practical way to understand the performance rating centers around the fact that a player's actual rating changes after each game played. By the definition, the only way a player's actual rating would not change after this series of games is if their rating at the start of these games was already their performance rating over the series. With this definition, individual game results do not directly factor into the calculation. Unlike the linear and FIDE methods, however, the ratings of individual opponents do affect the calculation.[1][2]

FIDE calculates a player's performance rating ${\displaystyle r_{p}}$ as ${\displaystyle r_{p}=r_{c}+d_{p}}$, where ${\displaystyle r_{c}}$ is the average rating of the opponents and ${\displaystyle d_{p}}$ is an additional rating difference based on the player's total score divided by the number of rounds played. That fractional score is called ${\displaystyle p}$. There is no analytic expression for ${\displaystyle d_{p}}$. Instead, FIDE provides a lookup table for the values of ${\displaystyle d_{p}}$ based on the values of ${\displaystyle p}$ rounded to the nearest hundredth. The values of ${\displaystyle d_{p}}$ for common lengths of tournaments (eight to eleven rounds) are listed below under rating difference examples.[3][4]

Like the true definition, the FIDE method also does not depend on individual game results. Unlike the true definition, the FIDE method does not depend on individual opponent ratings.[3]

Rating difference examples

Note: Zero scores have ${\displaystyle d_{p}=-800}$, even scores have ${\displaystyle d_{p}=0}$, and perfect scores have ${\displaystyle d_{p}=800}$.

Eight rounds

Negative scores
Score	½	1	1½	2	2½	3	3½
${\displaystyle p}$	0.06	0.13	0.19	0.25	0.31	0.37	0.44
${\displaystyle d_{p}}$	-444	-322	-251	-193	-141	-95	-43
Positive scores
Score	4½	5	5½	6	6½	7	7½
${\displaystyle p}$	0.56	0.63	0.69	0.75	0.81	0.87	0.94
${\displaystyle d_{p}}$	+43	+95	+141	+193	+251	+322	+444

Nine rounds

Negative scores
Score	½	1	1½	2	2½	3	3½	4
${\displaystyle p}$	0.06	0.11	0.17	0.22	0.28	0.33	0.39	0.44
${\displaystyle d_{p}}$	-444	-351	-273	-220	-166	-125	-80	-43
Positive scores
Score	5	5½	6	6½	7	7½	8	8½
${\displaystyle p}$	0.56	0.61	0.67	0.72	0.78	0.83	0.89	0.94
${\displaystyle d_{p}}$	+43	+80	+125	+166	+220	+273	+351	+444

Ten rounds

Negative scores
Score	½	1	1½	2	2½	3	3½	4	4½
${\displaystyle p}$	0.05	0.10	0.15	0.20	0.25	0.30	0.35	0.40	0.45
${\displaystyle d_{p}}$	-470	-366	-296	-240	-193	-149	-110	-72	-36
Positive scores
Score	5½	6	6½	7	7½	8	8½	9	9½
${\displaystyle p}$	0.55	0.60	0.65	0.70	0.75	0.80	0.85	0.90	0.95
${\displaystyle d_{p}}$	+36	+72	+110	+149	+193	+240	+296	+366	+470

Eleven rounds

Negative scores
Score	½	1	1½	2	2½	3	3½	4	4½	5
${\displaystyle p}$	0.05	0.09	0.14	0.18	0.23	0.27	0.32	0.36	0.41	0.45
${\displaystyle d_{p}}$	-470	-383	-309	-262	-211	-175	-133	-102	-65	-36
Positive scores
Score	6	6½	7	7½	8	8½	9	9½	10	10½
${\displaystyle p}$	0.55	0.59	0.64	0.68	0.73	0.77	0.82	0.86	0.91	0.95
${\displaystyle d_{p}}$	+36	+65	+102	+133	+175	+211	+262	+309	+383	+470

Because of the need to have a lookup table to calculate the rating difference ${\displaystyle d_{p}}$ in FIDE performance ratings, another simpler method instead calculates the rating difference as ${\displaystyle d_{p}=8p-400}$, where ${\displaystyle p}$ is the percentage score in this case. The overall performance rating is then calculated as ${\displaystyle r_{p}=r_{c}+d_{p}}$, the same as the FIDE method. This method is sometimes called the linear method due to the linear dependence on the percentage score ${\displaystyle p}$. Like the true definition, the linear method also does not depend on individual game results. Unlike the true definition, the linear method does not depend on individual opponent ratings.[1]

Different methods for calculating the performance rating generally give similar results. The only score in which all methods give exactly the same result is an even score against opponents with no skew away from their average rating, in which case the performance rating is the average of the opponents' ratings. There are larger discrepancies closer to zero scores or perfect scores, or with a large skew away from the average of the opponents' ratings (in which case the individual ratings have a larger effect). The true definition of the performance rating gives -∞ for a zero score and ∞ for a perfect score, whereas the other methods yield finite values.[1]

As a specific example, if a player scores 2½/3 against three opponents rated 2400, 2500, and 2600, their performance ratings with the different methods are 2785 (true definition), 2773 (FIDE), and 2767 (linear).[1]

FIDE Handbook: Rating System