The USCF Rating System

Prof. Mark E. Glickman

Boston University

Thomas Doan

Estima

November 3, 2011

The following algorithm is the procedure to rate USCF events. The procedure applies to two

separate rating systems: the Quick Chess (QC) system, and the Regular system. The QC

system governs events with time controls of G/5 through G/60. Regular events have time

controls of G/30 or slower. The formulas apply to each system separately. Events having

time controls between G/30 and G/60 are rated in both systems (i.e., dual-rated).

1 The Rating Algorithm

Before an event, a player is either unrated, or has a rating based on having played N games. A

player's rating is termed \provisional" if it is based on 25 or fewer games, and is \established"

otherwise. Assume the player competes in m games during the event. Post-event ratings are

computed in a sequence of ve steps:

The rst step sets temporary initial ratings for unrated players.

The second step calculates an \e
ective" number of games played by each player.

The third step calculates temporary estimates of ratings for certain unrated players

only to be used when rating their opponents on the subsequent step.

The fourth step then calculates intermediate ratings for all players.

The fth step uses these intermediate ratings from the previous step as estimates of

opponents' strengths to calculate nal post-event ratings.

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The calculations are carried out in the following manner:

Step 1: Set initial ratings for unrated players.

Initial rating estimates are set for all unrated players in an event. The purpose of

setting initial rating estimates for unrated players is (1) to be able to incorporate

information about a game result against an unrated player, and (2) to choose among

equally plausible ratings during a rating calculation for an unrated player (see the

details of the \special" rating formulas in Section 2.2.

An initial rating for an unrated player is determined in the following order of prece-

dence.

If an unrated player has a FIDE rating, use a converted rating according to the

following formula:

USCF =

(

720 + 0:625 FIDE if FIDE< 2000

􀀀350 + 1:16 FIDE if FIDE 2000

If the FIDE rating is over 2150, then this converted rating is treated as based on

having played 10 games (N = 10). If the FIDE rating is 2150 or less, then this

converted rating is treated as based on having played 5 games (N = 5).

If an unrated player has a CFC rating over 1500, use a converted rating according

to the following formula:

USCF = 1:1 CFC 􀀀 240:

This converted rating is treated as based on having played 5 games (N = 5).

If an unrated player has a CFC rating of 1500 or less, use a converted rating

according to the following formula:

USCF = CFC 􀀀 90:

This converted rating is treated as based on having played 0 games (N = 0).

If a player has a foreign national rating, but no CFC, FIDE (or USCF) rating,

the USCF oce may at their discretion use a rating of their determination on a

case-by-case basis. In such a case, the rating is treated as based on having played

0 games (N = 0).

If the event is a regular event and the player has no regular rating, but has a

QC rating based on at least four games, then use the QC rating as the imputed

rating. This rating is treated as being based on 0 games (N = 0).

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Conversely, if the event is a QC event and the player has no QC rating, but has

a regular rating based on at least four games, then use the regular rating as the

imputed rating. The rating is assumed to be based on the lesser of 10 and the

number of games on which the regular rating itself is based (N = 10 or N = prior

number of regular games, whichever is smaller).

Otherwise, impute an age-based rating according to the following procedure. De-

ne a player's age (in years) to be

Age = (Tournament End Date 􀀀 Birth Date)=365:25:

The formula for an initial rating based on age is given by

USCF =

(

50 Age if 3 Age 26

1300 otherwise.

The rating is assumed to be based on 0 games (N = 0). If an unrated player

does not provide a birth date, but is inferred to be an adult (e.g., through an

appropriate USCF membership type), then the initial rating is set to be 1300

with N = 0, treating the player as a 26-year old in the Age-based calculation. As

a practical concern, if \Age" is calculated to be less than 3 years old, then it is

assumed that a miscoding of the player's birthday occurred, and such a player is

also treated as a 26-year old in the Age-based calculation.

If no international rating or birth information is supplied, and if the player does

not have a non-correspondence USCF rating, impute a rating of 750. This rating

is assumed to be based on 0 games (N = 0).

Step 2: Calculate the \e
ective" number of games played by each player.

This number, which is typically less than the actual number of games played, reects

the uncertainty in one's rating, and is substantially smaller especially when the player's

rating is low. This value is used in the \special" and \standard" rating calculations.

See Section 2.1 for the details of the computation.

Step 3: Calculate a rst rating estimate for each unrated player for whom Step 1 gives

N = 0. For these players, use the \special" rating formula (see Section 2.2), letting the

\prior" rating be R0. However, for only this step in the computation, set the number

of e
ective games for these players to 1 (this is done to properly \center" the ratings

when most or all of the players are previously unrated).

If an opponent of the unrated player has a pre-event rating, use this rating in the

rating formula.

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If an opponent of the unrated player is also unrated, then use the initial rating

imputed from Step 1.

If the resulting rating from Step 3 for the unrated player is less than 100, then change

the rating to 100.

Step 4: For every player, calculate an intermediate rating with the appropriate rating for-

mula.

If a player's rating R0 from Step 1 is based upon 8 or fewer games (N 8), or

if a player's game outcomes in all previous events have been either all wins or all

losses, then use the \special" rating formula, with \prior" rating R0.

If a player's rating R0 from Step 1 is based upon more than 8 games (N > 8),

and has not been either all wins or all losses, use the \standard" rating formula

(see Section 2.3). Note that the standard formula is used even if the \e
ective"

number of games from Step 2 is less than or equal to 8.

In the calculations, use the opponents' pre-event ratings in the computation (for players

with pre-event ratings). For unrated opponents who are assigned N = 0 in Step 1, use

the results of Step 3 for their ratings. For unrated opponents who are assigned N > 0

in Step 1, use their assigned rating from Step 1.

If the resulting rating from Step 4 is less than 100, then change the rating to 100.

Step 5: Repeat the calculations from Step 4 for every player, again using a player's pre-

event rating (or the assigned ratings from Step 1 for unrated players) to perform the

calculation, but using the results of Step 4 for the opponents' ratings. If the resulting

rating from Step 5 is less than 100, then change the rating to 100.

These ve steps result in the new set of post-event ratings for all players.

2 Details of the Rating Algorithm

This section describes the details of the rating algorithm, including the computation for

the \e
ective" number of games, the \special" rating formulas, and the \standard" rating

formulas.

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2.1 E
ective number of games

For each player, let N be the number of tournament games the player has competed, or, for

unrated players, the value assigned from Step 1 of the algorithm. Let R0 be the player's

pre-event rating, or, for unrated players, the imputed rating assigned from Step 1. Let

N =

(

50=

q

1 + (2200 􀀀 R0)2=100000 if R0 2200

50 if R0 > 2200

(1)

De ne the \e
ective" number of games, N0, to be the smaller of N and N. As a result of

the formula, N0 can be no larger than 50, and it will usually be less, especially for players

who have not competed in many tournament games. Note that N0 is a temporary variable

in the computation and is not saved after an event is rated.

Example: Suppose a player's pre-event rating is R0 = 1700 based on N = 30 games. Then

according to the formula above,

N = 50=

q

1 + (2200 􀀀 1700)2=100000 = 50=

p

3:5 = 26:7

Consequently, the value of N0 is the smaller of N = 30 and N = 26:7, which is therefore

N0 = 26:7. So the e
ective number of games for the player in this example is N0 = 26:7.

2.2 Special rating formula

This procedure is to be used for players with either N 8, or players who have had either

all wins or all losses in all previous rated games.

The algorithm described here extends the old provisional rating formula by ensuring that

a rating does not decrease from wins or increase from losses. In e
ect, the algorithm nds

the rating at which the attained score for the player equals the sum of expected scores,

with expected scores following the \provisional winning expectancy" formula below. For

most situations, the resulting rating will be identical to the old provisional rating formula.

Instances that will result in di
erent ratings are when certain opponents have ratings that

are far from the player's initial rating. The computation to determine the \special" rating

is iterative, and is implemented via a linear programming algorithm.

De ne the \provisional winning expectancy," PWe, between a player rated R and his/her

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i-th opponent rated Ri to be

PWe(R;Ri) =

8><</P>

>:

0 if R Ri 􀀀 400

0:5 + (R 􀀀 Ri)=800 if Ri 􀀀 400 < R < Ri + 400

1 if R Ri + 400

Let R0 be the \prior" rating of a player (either the pre-event rating for rated players, or the

Step 1 imputed rating for unrated players), and N0 be the e
ective number of games. Also

let m be the number of games in the current event, and let S be the total score out of the

m games (counting each win as 1, each loss as 0, and each draw as 0.5).

The variables R00

and S0, which are the adjusted initial rating and the adjusted score, re-

spectively, are used in the special rating procedure. If a player has competed previously, and

all the player's games were wins, then let

R0

0 = R0 􀀀 400

S0 = S + N0

If a player has competed previously, and all the player's games were losses, then let

R0

0 = R0 + 400

S0 = S

Otherwise, let

R0

0 = R0

S0 = S +

N0

2

The objective function

f(R) = N0 PWe(R;R0

0) +

 

mX

i=1

PWe(R;Ri)

!

􀀀 S0

which is the di
erence between the sum of provisional winning expectancies and the actual

attained score when a player is rated R, is equal to 0 at the appropriate rating. The goal,

then, is to determine the value of R such that f(R) = 0 within reasonable tolerance. The

procedure to nd R is iterative, and is described as follows.

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Let " = 10􀀀7 be a tolerance to detect values di
erent from zero. Also, let x0 = R00

􀀀 400,

y0 = R00

+ 400, and, for i = 1; : : : ;m, xi = Ri 􀀀 400, yi = Ri + 400. Denote the unique xi

and yi, i = 0; : : : ;m, as the collection

Sz = fz1; z2; : : : ; zQg

If there are no duplicates, then Q = 2m+2. These Q values are the \knots" of the function

f (essentially the value where the function \bends" abruptly).

1. Calculate

M =

N0R00

+

Pmi

=1 Ri + 400(2S 􀀀 m)

N0 + m

This is the rst estimate of the special rating (in the actual implemented rating pro-

gram, M is set to R00

, but the nal result will be the same { the current description

results in a slightly more ecient algorithm).

2. If f(M) > ", then

(a) Let za be the largest value in Sz for which M > za.

(b) If jf(M) 􀀀 f(za)j < , then set M za. Otherwise, calculate

M = M 􀀀 f(M)

 

M 􀀀 za

f(M) 􀀀 f(za)

!

If M < za, then set M za, and go back to 2.

If za M < M, then set M M, and go back to 2.

3. If f(M) < 􀀀", then

(a) Let zb be the smallest value in Sz for which M < zb.

(b) If jf(zb) 􀀀 f(M)j < , then set M zb. Otherwise, calculate

M = M 􀀀 f(M)

 

zb 􀀀M

f(zb) 􀀀 f(M)

!

If M > zb, then set M zb, and go back to 3.

If M < M zb, then set M M, and go back to 3.

4. If jf(M)j ", then let p be the number of i, i = 1; : : : ;m for which

jM 􀀀 Rij 400:

Additionally, if jM 􀀀 R00

j 400, set p p + 1.

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(a) If p > 0, then exit.

(b) If p = 0, then let za be the largest value in Sz and zb be the smallest value in Sz

for which za < M < zb. If

za R0 zb, then set M R0.

R0 < za, then set M za.

R0 > zb, then set M zb.

If the nal value of M is greater than 2700, the value is changed to 2700. The resulting value

of M is the rating produced by the \special" rating algorithm.

2.3 Standard rating formula

This algorithm is to be used for players with N > 8 who have not had either all wins or all

losses in every previous rated game.

De ne the \Standard winning expectancy," We, between a player rated R and his/her i-th

opponent rated Ri to be

We(R;Ri) =

1

1 + 10􀀀(R􀀀Ri)=400

The value of K, which used to take on the values 32, 24 or 16, depending only on a player's

pre-event rating, is now de ned as

K =

800

N0 + m

;

where N0 is the e
ective number of games, and m is the number of games the player completed

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in the event. The following are example values of K for particular values of N0 and m.

N0 m Value of K

6 4 80

6 6 66.67

6 10 50

20 4 33.33

20 6 30.77

20 10 26.67

50 4 14.81

50 6 14.29

50 10 13.33

If m < 3, or if the player competes against any opponent more than twice, the \standard"

rating formula that results in Rs is given by

Rs = R0 + K(S 􀀀 E)

where the player scores a total of S points (1 for each win, 0 for each loss, and 0.5 for each

draw), and where the total winning expectancy E =

Pmi

=1We(R0;Ri).

If both m 3 and the player competes against no player more than twice, then the \stan-

dard" rating formula that results in Rs is given by

Rs = R0 + K(S 􀀀 E) + max(0; K(S 􀀀 E) 􀀀 B

p

m0)

where m0 = max(m; 4) (3-round events are treated as 4-round events when computing this

extra term), and B is the bonus multiplier (currently B is set to 6 as of June 2008). The

quantity

max(0; K(S 􀀀 E) 􀀀 B

p

m0)

is, in e
ect, a bonus amount for a player who performs unusually better than expected.

The resulting value of Rs is the rating produced by the \standard" rating algorithm.

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2.4 Rating oors

The absolute rating oor for all ratings is 100. No rating can be lower than the absolute

rating oor. An individual's personal absolute rating oor is calculated as

AF = min(100 + 4NW + 2ND + NR; 150)

where AF is the player's absolute oor, NW is the number of rated games won, ND is the

number of rated games drawn, and NR is the number of events in which the player completed

three rated games. The formula above speci es that a player's absolute oor can never be

higher than 150. As an example, if a player has earned 3 wins, 1 draw, and has competed

in a total of 10 events of at least three ratable games, then the player's absolute oor is

AF = 100 + 4(3) + 2(1) + 10 = 124.

A player with an established rating has a rating oor possibly higher than the absolute

oor. Higher rating oors exist at 1200, 1300, 1400, : : :, 2100. A player's rating oor is

calculated by subtracting 200 points from the highest attained established rating, and then

using the oor just below. For example, if an established player's highest rating was 1941,

then subtracting 200 yields 1741, and the oor just below is 1700. Thus the player's rating

cannot go below 1700. If an established player's highest rating was 1388, then subtracting

200 yields 1188, and the next lowest oor is the player's absolute oor, which is this player's

current oor.

A player who earns the original Life Master (OLM) title, which occurs when a player keeps an

established rating above 2200 for 300 (not necessarily consecutive) rated games, will obtain

a rating oor of 2200. Achievement of other USCF titles do not result in rating oors.

A player's rating oor can also change if he or she wins a large cash prize. If a player wins

over $2,000 in an under-2000 context, the rating oor is set at the rst 100-point level (up to

2000) which would make the player no longer eligible for that section or prize. For example,

if a player wins $2,000 in an under-1800 section of a tournament, then the player's rating

oor would be 1800. Floors based on cash prizes can be at any 100-point level, not just the

ones above based on peak rating.

3 Updating USCF ratings from foreign FIDE events

The USCF regularly updates ratings based on performances in FIDE-rated non-USCF events

to obtain more accurate ratings for its players. The following describes the procedure used

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to update USCF ratings based on performance in FIDE events.

Each time a FIDE rating list is produced, the USCF oce identi es all players who

appear with a \USA ag" (usually US residents) and who have played at least one

FIDE-rated game in the set of events/tournaments that are included in producing the

rating list. This may include US residents who are not USCF-rated.

For each identi ed US player, all the player's opponents are identi ed along with their

FIDE ratings.

The opponents' FIDE ratings are converted to the USCF scale using the conversion in

Step 1 of the rating algorithm. If an event is known to be a youth event, such as the

World Youth Championships, then the following conversion is used for all opponents:

USCF = 210 + 0:93 FIDE:

The standard rating formula (with bonus) is then applied to update the player's USCF

rating based on the opponents' converted ratings. The standard formula is applied

only once, as opposed to twice in the usual algorithm.

4 Miscellaneous details

The following is a list of miscellaneous details of the rating system.

All games played in USCF-rated events are rated, including games decided by time-

forfeit, games decided when a player fails to appear for resumption after an adjourn-

ment, and games played by contestants who subsequently withdraw or are not allowed

to continue. Games in which one player makes no move are not rated.

The rating calculations apply separately to the QC and regular chess rating systems.

Other than the use of imputing initial ratings for unrated players, there is no formal

connection between these two systems.

After an event, each players' value of N is incremented by m, the number of games

the player competed in the event.

Individual matches are rated with the following restrictions:

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1. Both players involved must have an established published rating, with the di
er-

ence in ratings not to exceed 400 points.

2. The maximum rating change in a match is 50 points; the maximum net rating

change in 180 days due to match play is 100 points; and the maximum net rating

change in 3 years due to match play is 200 points.

3. The bonus formula does not apply to matches.

4. Rating oors are not automatically in e
ect in matches. Instead, if a player has a

match result that would lower the rating to below that player's oor, this will be

treated as a request to have that oor lowered by 100 points. If the USCF oce

grants this request, the rating will drop below the old oor and the new oor will

be 100 points below the old oor.

Ratings are stored as integers. During the rating of an event, intermediate computa-

tions are done using oating point arithmetic. When a post-event rating is less than

the pre-event rating, the rating is rounded down. Conversely, when a post-event rating

is greater than the pre-event rating, the rating is rounded up. The practical e
ect is

that a positive result earns at least one point per event, and a negative result loses at

least one point per event.

The USCF Executive Director may review the rating of any USCF member and make

the appropriate adjustments, including but not limited to imposition of a rating \ceil-

ing" (a level above which a player's rating may not rise), or to the creation of \money

oors" (rating oors that are a result of winning large cash prizes).

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