How the ratings are calculated
This is a brief summary of the ratings calculation method.
A full explanation of the method has been published in the
Journal of Applied Statistics, 30, 361-372 (2003).
The actual functions used are listed
here in pdf format.
Players may be loosely classified according to rating by the following scheme:
over 2300 - grand master
over 2100 - senior master
over 1900 - master
over 1700 - expert
over 1500 - apprentice
less than 1500 - novice
The program is designed to give a rating for each player that
takes into account performance and the strength (or otherwise) of partners
and opponents.
The rating is calculated on a tournament-by-tournament basis, and it should be
appreciated that the program is a RATINGS system, and not a RANKING system.
Earlier algorithms (dating back to the mid-1980's) were fairly successful,
but anomalous results would arise in certain cases.
The ideal program needs to be flexible enough to cope sensibly with tournaments
that have:
many players or very few players
many games or very few games
people playing in their first ever event
players returning to the game after a long absence
singles games or pairs games (sometimes both!)
fixed partnerships (and sometimes fixed opponents) or varying partnerships
It would also be nice if the algorithm had a reasonably firm mathematical
basis that avoided making too many arbitrary judgements over the parameters
that influence ratings.
The key parameters calculated each tournament for each player are:
the player's rating
the player's Rating Reliability Factor (RRF); this provides a
measure of the confidence that the program has in that player's rating.
In the new algorithm, an RRF of 100 is equivalent to an estimated uncertainty
in the rating of +-70 points, while an RRF of 0 is equivalent to an
uncertainty of +-350 rating points. Players who play in most tournaments
will eventually gain an RRF of 100, while those who only play a few games
a year will have low RRF values.
The adjustment to a player's rating each tournament takes into account:
the points per game (p.p.g.) achieved.
the number of games played in the tournament.
the strength of the partner(s) and opponent(s).
the initial uncertainty in the player's rating. There will be smaller
adjustments for players with low uncertainties compared to those with
high uncertainties
the uncertainties in the ratings of partner(s) and opponent(s); for
instance, there will only be small adjustments for players partnering
complete beginners (as the beginner has a high uncertainty in rating).
Similarly only small adjustments will be made for players partnering
someone returning to the game after a long absence.
Prediction of scores
The predicted score in a game of pairs is given by:
PREDICTEDSCORE = 3.5 + 3.55 erf((RATING + PARTNERRATING - OPP1RATING - OPP2RATING)/1600)
(subject to the predicted score not being outside the range 0-7). Here
"erf" denotes the error function.
Similarly for a game of singles:
PREDICTEDSCORE = 3.5 + 3.55 erf((RATING - OPPRATING)/800)
This function is shown in the graph below:
It can be seen that an average ratings points difference between pairs of:
100 points predicts a 4-3 win
205 points predicts a 4.5-2.5 win
315 points predicts a 5-2 win
440 points predicts a 5.5-1.5 win
590 points predicts a 6-1 win
805 points predicts a 6.5-0.5 win
(Incidentally, the ratings may be used to calculate
handicaps
on the basis of this function).
Calculation of Ratings
The first step in the program is to calculate a tournament rating for each
player in the tournament. This is the rating that the player would need to
have had in order to be predicted the same number of total points as was
actually achieved.
The second step in the program is to calculate the estimated uncertainty
in the tournament rating. This has two contributions. One is due to
the finite number of games played - effectively each game can be treated
as being an independent "measurement" of tournament rating. The second
contribution is due to the uncertainties in rating of partner and opponents.
These factors can be estimated by appropriate statistical analysis.
At this stage OLDRATING and OLDERROR are known, and the
TMTRATING and TMTERROR are known. The new rating is calculated as the
best average of two Normally distributed observations with these
properties. This is the
average of the OLDRATING and TMTRATING weighted by the reciprocal of the
square of the uncertainty in these values. The
uncertainty (NEWERROR) in the rating is calculated in a similar manner,
but the minimum uncertainty is set at 70 rating points.
Tournament newcomers
Tournament newcomers are assigned an initial rating of about 1500, and
an uncertainty of 350 rating points. The actual value used depends on
whether the rating is being calculated for the newcomer, or whether a
tournament rating is being calculated for a player who partnered or
opposed the newcomer.
If a calculated rating becomes less than that of a "nominal beginner"
(1500 rating points), then a small adjustment to it is made to ensure that
the rating never becomes significantly less than a beginner.
A lower rating limit of 1320 is chosen. There are no players worse
than 1320.
An additional small increase in the
uncertainty of a player's rating is also made for these players, to
allow for the fact that experience shows that people of this
quality can show rapid sudden improvements (e.g. if they start practising).
Inactive players
The uncertainty in rating of players who haven't played for a while needs
to be increased.
No adjustment is made if the player's last tournament was within
the previous 4 months. After the 4 month period, a player's
uncertainty is increased by 5 SQRT(M-4), where M is the number of months since
the last tournament
played, whenever ratings are calculated for an open tournament; no
uncertainty increase is made for "closed" events such as World Singles and
University matches.
Any player not competing in a tournament for over a year is
removed from the published ratings. When they
next play in a tournament, they will
re-enter the ratings at a starting point adjusted slightly from their last
rating. Players lapsing with ratings higher than 1500 have their rating
adjusted towards 1500 by 15% of the difference (to allow for being "out of
practice"), while those with ratings less than 1500 have their ratings
adjusted upwards towards 1500 by a factor of 50%.