10.0. The working of the FIDE Rating System

The FIDE Rating system is a numerical system in which percentage scores are converted to rating differences and vice versa. Its function is to produce scientific measurement information of the best statistical quality.
10.1	The rating scale is an arbitrary one with a class interval set at 200 points. The tables that follow show the conversion of percentage score `p` into rating difference `dp`. For a zero or 100% score dp is necessarily indeterminate. The second table shows conversion of difference in rating `D` into scoring probability `PD` for the higher `H` and the lower `L` rated player respectively. Thus the two tables are effectively mirror-images. The table of conversion from percentage score, p, into rating differences, dp p dp p dp p dp p dp p dp p dp 1.0   .83 273 .66 117 .49 -7 .32 -133 .15 -296 .99 677 .82 262 .65 110 .48 -14 .31 -141 .14 -309 .98 589 .81 251 .64 102 .47 -21 .30 -149 .13 -322 .97 538 .80 240 .63 95 .46 -29 .29 -158 .12 -336 .96 501 .79 230 .62 87 .45 -36 .28 -166 .11 -351 .95 470 .78 220 .61 80 .44 -43 .27 -175 .10 -366 .94 444 .77 211 .60 72 .43 -50 .26 -184 .09 -383 .93 422 .76 202 .59 65 .42 -57 .25 -193 .08 -401 .92 401 .75 193 .58 57 .41 -65 .24 -202 .07 -422 .91 383 .74 184 .57 50 .40 -72 .23 -211 .06 -444 .90 366 .73 175 .56 43 .39 -80 .22 -220 .05 -470 .89 351 .72 166 .55 36 .38 -87 .21 -230 .04 -501 .88 336 .71 158 .54 29 .37 -95 .20 -240 .03 -538 .87 322 .70 149 .53 21 .36 -102 .19 -251 .02 -589 .86 309 .69 141 .52 14 .35 -110 .18 -262 .01 -677 .85 296 .68 133 .51 7 .34 -117 .17 -273 .00   .84 284 .67 125 .50 0 .33 -125 .16 -284     Table of conversion of difference in rating, D, into scoring probability PD, for the higher, H, and the lower, L, rated player respectively. D PD   D PD   D PD   D PD   Rtg Dif H L Rtg Dif H L Rtg Dif H L Rtg Dif H L 0-3 .50 .50 92-98 .63 .37 198-206 .76 .24 345-357 .89 .11 4-10 .51 .49 99-106 .64 .36 207-215 .77 .23 358-374 .90 .10 11-17 .52 .48 107-113 .65 .35 216-225 .78 .22 375-391 .91 .09 18-25 .53 .47 114-121 .66 .34 226-235 .79 .21 392-411 .92 .08 26-32 .54 .46 122-129 .67 .33 236-245 .80 .20 412-432 .93 .07 33-39 .55 .45 130-137 .68 .32 246-256 .81 .19 433-456 .94 .06 40-46 .56 .44 138-145 .69 .31 257-267 .82 .18 457-484 .95 .05 47-53 .57 .43 146-153 .70 .30 268-278 .83 .17 485-517 .96 .04 54-61 .58 .42 154-162 .71 .29 279-290 .84 .16 518-559 .97 .03 62-68 .59 .41 163-170 .72 .28 291-302 .85 .15 560-619 .98 .02 69-76 .60 .40 171-179 .73 .27 303-315 .86 .14 620-735 .99 .01 77-83 .61 .39 180-188 .74 .26 316-328 .87 .13 over 735 1.0 .00 84-91 .62 .38 189-197 .75 .25 329-344 .88 .12
10.2	Determining the Rating `Ru` in a given event of a previously unrated player.
	10.21	First determine the average rating of his competition `Rc`. (GA `94) In a Swiss or Team tournament: this is simply the average rating of his opponents. The results of both rated and unrated players in a round robin tournament are taken into account. For unrated players, the average rating of the competition `Rc` is alo the tournament average `Ra` determined as follows: (GA `94) Determine the average rating of the rated players `Rar`. Determine p for each of the rated players against all their opponents. Then determine dp for each of these players. Then determine the average of these dp = `dpa`. `n` is the number of opponents. Ra = Rar - dpa x n/(n+1)
	10.22	If he scores 50%, then Ru = Rc. (GA `94)
	10.23	If he scores more than 50%, then Ru = Rc + 12.5 for each half point scored over 50%. (GA `94)
	10.24	If he scores less than 50% in a Swiss or team tournament (GA `94): Ru = Rc + dp.
	10.25	If he scores less than 50% in a round-robin (GA `94): R(u) = R(c) + d(p) x n/(n+1).
10.3	The Rating Rn which is to be published for a previously unrated player is then determined by taking the         weighted average of all his Ru results. e.g. A player has Ru results of 2280 over 5 games, 2400 over 10 games and   2000 over 5 games: Rn = [ 2280 x 5 + 2400 x 10 + 2000 x 5 ] / 20 = 2270.
	10.31	Where a player`s first performance(s) is less than 2005, the result(s) is ignored.
	10.32	Rn for the FIDE Rating list (FRL) is rounded off to the nearest 5 or zero. (GA `93) Only Rn>= 2005 are considered.
10.4	Any unrated player who plays in 50% or more of the rounds in the Olympiad and who scores 50% or better will be entered on the FRL at 2200 (2050 for players in the Women`s Olympiad) unless that player`s performance qualifies for a higher rating. (GA `94)
	10.41	A player who has qualified under this rule for a rating of 2200 (or 2050) before the last round of the Olympiad can ignore all his subsequent games. In such a case, the Chief Arbiter shall certify that the player`s results are valid for the indicated rating.
10.5	If an unrated player receives a published rating before a particular tournament in which he has played is rated, then he is rated as a rated player with his current rating, but in the rating of his opponents he is counted as an unrated player.
10.6	Determining the rating change for a rated player:
	10.61	Determine Rc For a Swiss, team or individual match this is the average rating of the player`s opponent(s). A difference in rating of more than 350 points will be counted for rating purposes as though it were a difference of 350 points (cf 10.67). For a round-robin (GA `94): Rc = Ra (n+1) - R ---------------- n where R  = rating of the player n  = number of opponents Ra  = average rating of the whole tournament as calculated in 10.21b when unrated players are competing.
	10.62	See B.01./9.4. for an Honorary Rating which regulation also applies for 10.21b.
	10.63	R is the published rating of the rated player R-Ra = D. Use 10.1b. to determine PD.
	10.64	W is the score achieved and We is the expected score. (GA `94) We = PD x n for a Swiss or individual match. DR = K (W - We). DR is the rating change. K is the development coefficient. K = 25 for a player new to the rating list until he has completed events with a total of at least 30 games. K = 15 as long as a player`s rating remains under 2400. K = 10 once a player`s published rating has reached 2400, and he has also completed events with a total of at least 30 games. Thereafter it remains permanently at 10.
	10.65	To determine Rn for a rated player, determine DR for each tournament in which he has played. Ro is the old rating. SDR is the summation of DR for each event in which the player has participated. Rn = Ro + = SDR
	10.66	Rn is rounded off to the nearest 1 or 0.
	10.67	IScheveningen tournaments/matches are not rated if unrated players participate.
	10.68	Determining  the Ratings in a round-robin tournament. The example has been created to demonstrate the methodology and also to show how the invitees should be chosen carefully. What follows differs from the Handbook. The workings are shown more clearly and the arithmetic has been corrected. player rating score p dp Rc Ru Rc Ru D PD We DR               (new) (new)         A 2600 8 .89 351 2320   2373   227 .79 7.11 +8.9 B 2500 7 .78 220 2331   2350   150 .7 6.3 +7 C u 7     2348 2411 2355 2418         D 2400 6 .67 125 2342   2348   52 .57 5.13 +8.7 E u 6     2348 2386 2352 2390         F 2150 4 .44 -43 2370   2359   -209 .23 2.07 +28.95 G 2300 3 .33 -125 2353   2353   -53 .43 3.87 -13.05 H u 2     2348 2128 2332 2112         I u 1     2348 1997 2286 1935         J 2300 1 .11 -351 2353   2353   -53 .43 3.87 -43.05 Rar  = 2600 + 2500 + 2400 + 2150 + 2300 + 2300 divided by 6 Rar  = 2375 dpa  = 351 + 220 +125 -43 -125 -351 divided by 6 dpa  = 29.5 Ra  = 2375 -29.5 x 9/10 Ra  = 2348 For Player C Ru = 2348 + 5 x 12.5  = 2411 For Player E Ru = 2348 + 3 x 12.5  = 2386 For Player H Ru = 2348 - 220  = 2128 For Player I Ru = 2348 - 351  = 1997 However Player1 is more than 350 points below players A, B, C, D, E. Player H is more than 350 points below A. Player A                                                                                                          Rc(new)            =2366 Player B, I                               counts as 2150                                                            Rc(new)            =2344 Player C, I                               counts as 2061. 2061-2032=29.29/9=3                        Rc(new            )            =2351 Player D, I                               counts as 2050                                                            Rc(new)            =2344 Player E, I                                counts as 2036                                                            Rc(new)            =2348 Player F, I                                counts as 2500                                                            Rc(new)            =2359 Player G,                                 all players are within 350 points                           Rc(new)            =2353   Player H, A                              counts as 2500                                                            Rc(new)            =2337   Player I, A, B, C, D, E,                        counts as 2382                                                            Rc(new)            =2305   Player J                                   all players are within 350 points                           Rc(new)            =2353   To determine the rating changes for the rated players: An example: Player G. D = 2300-2353=-53. P(D)=0.43. We = 0.43 x 9 = 3.87                      DR = (3-3.87) x 15 = -13.05 for G F was a poor choice of player for the tournament. He dragged down the average rating too much. If a player rated 2380 or higher had replaced him, C would achieve a better rating even with one point less. This is because, for unrated players with plus scores the average rating of the field is extremely important. Had I`s expected score been so poor, he should not have been chosen, everybody suffered.