Approximating Formulas for the US Chess Rating …

1 Approximating Formulas for theUS Chess Rating SystemApril, 2017 This document provides approximation Formulas forplayers to compute and update their US Chess rat-ings. The actual Rating system is quite complex,and performing the computations by hand is virtu-ally impossible. It should be noted that, on occasion,the Formulas presented below produce ratings con-siderably different from ratings calculated under theactual algorithm, so that the Formulas below shouldserve only as a rough guide. A detailed description ofthe new system s Formulas is posted at the US Chessweb site ( ), and is available on re-quest from US and established ratings:Players ratings are considered provisional if theyhave played 25 or fewer games (rather than 20 un-der the old system), and established if having playedmore than 25.

2 There are two different Formulas tocompute ratings. The criterion for using the differ-ent Formulas depends on whether the player has com-pleted 8 tournament games. The formula for ratingsbased on 8 or fewer games is called the special rat-ing formula , and the other is called the standard formula . A provisional Rating is updated using thespecial formula if the number of completed games is8 or fewer, and the standard formula if the number isgreater than 8. Established ratings are based on thestandard Rating formula :If a player has a Rating based on 8 or fewer games, oris unrated, then the new Rating can be approximatedby the old provisional Rating formula , that isRpost=N Rpre+mRavg+ (W L)400N+mwhereRpreis the player s pre-tournament Rating ,Nis the number of games upon whichRpreis based,mis the number of games the player completes in thetournament,Ravgis the average of the opponents ratings,Wis the number of wins, andLis the num-ber of losses.

3 If the player is unrated, setN= 0 andRpre= 0. The final result is rounded to an : Suppose a player rated 1500 based on 6games competes against players rated 1400, 1550 and1650, winning the first, losing the second and draw-ing the third. In this case,Ravg= (1400 + 1550 +1650)/3 = ,m= 3,N= 6,W= 1,L= 1,andRpre= 1500. Then, from the special Rating for-mula,Rpost=6(1500) + 3( ) + (1 1)4006 + 3= 1511 The final result has been rounded from should be noted that, from the Approximating for-mula, a player could gain Rating points by losing to ahigh rated player, or lose Rating points with a win overa low rated player. The actual Rating procedure cor-rects for these possibilities. Furthermore, the actualformulas first calculate ratings for unrated opponents,thereby making use of all game Rating formula :To approximate one s Rating using the standard for-mulas, a player needs to know (or approximate) thenumber of games played in tournaments, only if lessthan 50.

4 LetNbe the number of previous games,but setNto 50 if the number of games is 50 or , if the player has a pre-tournament Rating lessthan then 2355, the player computesNr= 50/ + (2569 Rpre)2If the player s Rating is 2355 or greater, then setNr= 50. Finally, letNebe the smaller ofNandNr. This number, the effective number of gamesupon which a Rating is based, can be calculated be-fore entering a : Suppose a player s pre-tournament ratingisRpre= 1700, based onN= 30 games. Then, ac-cording to the formula above,Nr= 50/ + (2569 1700)2= is smaller than 30,Ne= is theeffective number of games for this next step in the calculation is to determine thevalue ofK, the value that governs the magnitude ofrating changes.

5 Lettingmbe the number of gamesthe players completes in the tournament,K= 800/(Ne+m).It is worth noting that for high-rated players compet-ing in time controls quicker than G/60 the value ofKhas a different formula ; see the main Rating systemdocumentation for details. Finally, onceKhas beencomputed, the formula for updating a player s ratingis given byRpost=Rpre+K(S E) +BwhereRpreis the pre-tournament Rating ,Kis thevalue just computed,Sis the total score in the tour-nament (counting 1 for each win, for each drawand 0 for each loss),Eis the sum of winning ex-pectancies (described below), andBis a possiblebonus amount (described below). The final value isrounded to the nearest integer. In the actual formu-las, the Rating is roundedawayfrom the calculateE, the winning expectancy for each op-ponent must be calculated and then summed.

6 Theformula for the winning expectancy between a playerwith ratingRpreand an opponent with ratingRoppis given byWe(Rpre, Ropp) =110 (Rpre Ropp)/400+ is computed for each opponent, and the resultsare totaled to produce a value bonus,B, is automatically 0 if the player hascompleted fewer than 3 games, or played more thantwice against any opponent. If the player has com-pleted 3 or more games, and no more than twiceagainst each opponent, then a comparison is madebetween the valueK(S E) andT m , wherem isthe larger ofmand 4 (in other words, 3-round eventsare treated as 4-round events when computing thebonus amount). As of May 1, 2017, the value ofTis 14, though this is subject to change values ofTresult in greater difficulty in earn-ing bonus points.

7 IfT m is larger or equal, thenthe bonus is 0. But ifK(S E) is larger, then thebonus is the differenceB=K(S E) T m .This is the extra amount added to one s Rating forhaving an unusually strong : Suppose a player is rated 1300 based on45 games, and competes in a full-Kevent against 4distinct opponents rated 1250, 1400, 1500, and 1550winning three and drawing one. With these results,S= 1 + 1 + 1 + = , we compute the value ofNrasNr= 50/ + (2569 1300)2= that the lower of and 45 isNe= value ofKfor this player in this tournament isgiven byK= 800/(Ne+m) = 800/( + 4) = winning expectancy against the opponent rated1250 is110 (1300 1250)/400+ 1= 1/( + 1) = , the winning expectancies against the otherthree opponents are computed as , Adding these four values results inE= , becauseK(S E) = ( ) = larger thanT m= 12 4 = 24,the bonus is 24 = The final approximatedrating is thereforeRpost= 1300 + ( ) + = is then rounded to floors: Rating floors exist at 100, 1200, 1300.

8 , 2100. Noplayer s Rating can drop below 100. However, an in-dividual s personal absolute Rating floor is normallyhigher than 100, and depends on the number of ratedgames won and drawn, and the number of eventsin which the individual has completed at least threegames. Details are provided in the full Rating player s Rating floor is calculated by subtracting200 points from the highest attained established rat-ing, and then using the floor just below. For example,if a player s highest Rating was 1941, then subtract-ing 200 yields 1741, and the floor just below is the player s Rating cannot go below 1700. If aplayer s highest Rating was 1388, then subtracting 200yields 1188, and the next lowest floor is the personalabsolute floor, which is this player s floor.