In
the Glicko system each rated player has two calculated
rating statistics.
The first, and most important,
is the rating level.
This is the central estimate of the player’s DBM
ability. In other words it is the statistically best,
unbiased estimate that can be made from the information
available. A competitive player should be aiming to
maximize their rating level.
This is set at 2000 for new players.
The second statistic is their
rating Reliability
Measure or “RM”
from here onwards (this is called the ratings deviate or
RD in the chess Glicko system, a term I dislike). The RM
is like a standard deviation of the rating level or, in
laymen terms, an indicator of how much faith can be placed
in their rating level. A high RM means that the level is
less reliable while a low RM indicates that the level is a
more likely to be an accurate reflection of the player’s
true ability.
Please note that although a
particular player’s rating level may be considered less
reliable, it is not a biased value. Based on the evidence
available, there is as much chance of it being in an
underestimate of the player’s ability as there is of it
being an overestimate.
A player would normally hope
that their RM is low but a high RM, that is one indicating
less reliability, does not mean less DBM playing ability.
The RM is a function of the number of games played not
won, the RM of the opponents in those games and the length
of time since the player’s last rated competition game.
A player’s rating level is a
function of his or her results in DBM games against rated
opponents. It is not based at all on tournament placings.
The difference between two
players’ rating levels is an indicator of the expected
probability of one player defeating the other. The player
with the higher rating level always has a higher expected
probability of winning. The exact expected probability of
this player winning is a function of the difference in
ratings and the RM of both players. If two players have an
equal rating level then each is expected to have a 50%
probability of winning a game between them, regardless of
their RMs.
One other concept should be
mentioned. A player’s RM increases with time since his
last rated game. As time passes with the player not taking
part in any rated games, the player’s rating level does
not itself change since no firm data has been received
indicating that he is a better or worse player. However,
the faith that can be placed in his old rating level is
reduced by increasing the RM. The rate of increase in the
RM over time is only moderate, designed so that players
who are only able to take part in two tournaments a year
are not severely disadvantaged.
For example: If a player has an
RM of 87.5 (on the verge of moving from reliability C to
B) and plays no more rated games it will take about 22
months for his RM to naturally degrade to 122.5 (the
transition from C to D).
However after 13 months he will be dropped from the
Top Players list.
For any series of competition
games by a particular player against a certain set of
opponents, it is possible to calculate an expected DBM
score from each of those games. (see example below).
This is calculated based on the
rating level and RM of each opponent and the player’s
own pre-competition rating level. If the player’s actual
DBM score exceeds his expected DBM score then the
player’s rating level will increase. If the player
scores less than expected than his rating level will
decrease. The amount of increase or decrease in the rating
level depends upon the extent of the difference between
the actual and expected score and the players RM and
opponent’s RM.
Over a competition it is quite
likely that a player will score more than expected in some
games and less than expected in others. If all of his
opponents had the same RM, then if the player’s total
actual DBM score for the tournament exceeds his total
expected DBM score for the tournament, his rating level
will increase. Also, since the player has now played more
rated games, his RM will decrease, indicating greater
reliability, from what it was immediately before the
competition commenced.
Example
A
quick example for one player at a four round competition
may help to illustrate the workings. The figures below
utilize the formulae and parameters in Mathematical
formulae and statistical basis (see below).
At the commencement of a
competition player A has a rating level of 2,100 and an RM
of 80 (a reliability rating of B) ((an RM consistent with
him having played approximately 12 games, say 3
competitions, per year in the last few years). He plays 4
games ( 3 wins, 1 loss) against the following players with
the results shown in the table below (each player’s
rating is as at the start of the game):
Based on his own and his
opponent’s ratings a knowledgeable observer might have
expected Player A to lose his first game to B and his
fourth game heavily to E. He might have been expected to
have won his second game against C and for his third game
to be a narrow victory over D (or to win slightly more
often than he loses). One of the actual results is much
better than expected (against B), the other three similar
to expectations. In total, A’s actual score has exceeded
his expected score by 8.8. He has had a good tournament.
Based on these results, at the
end of the competition Player A’s rating would increase
from 2,100 to 2,137. His RM would decrease from 80 (B) to
75.0 (B). However if he does not play another competition
for 4 complete months his RM will return to 77.6 (B).
If at the start of the
competition, player A’s rating level had been 2,300 and
RM 80 (B) then his expected total score would have been
25.6, quite similar to but lower than his actual result.
Then at the end of his tournament his rating level would
have increased to 2,306 and his RM to 74.8
(B).
Similarly if his rating level
had been 1,800 and RM 80 (B) his expected total score
would have been only 8.9 and his post competition rating
level increased to 1,860.
If his rating level had been
2,100 the same as in the table but his pre-competition RM
equal to 150 (D, consistent with his having probably
played only two or three competitions prior to this) his
expected score would have been 19.8 (his early good
results increased his mid-tournament rating more quickly
than if his reliability rating had been 80 (B)). After his
good result his rating would have changed to 2,181.
Because his pre-competition rating level in this case was
less reliable (had a higher RM) than the example in the
table, his good result here has had a greater effect. The
less reliable the original rating level (ie A’s track
record is shorter), the greater its responsiveness to new
information. In this case the post competition RM would
have dropped to 123.2 (D).
The formulae following have
been adapted from those developed by Dr Mark E. Glickman
for use on the Free Internet Chess Server. At this stage
only a few adjustments have been made to the original
formulae to make them suitable for use when rating DBM
players on the basis of competition results.
As can be seen there are a
number of complex steps involved in calculating each
player’s rating after a competition. However all
calculations are performed by a computer program. Once
data is loaded, it only takes a split second on a modern
PC to calculate new ratings for all players.
In the formulae and steps that
follow, a DBM player A has played n
rated games in a particular competition.
A’s actual DBM scores out of
10 in each of the n
games are S1, S2,
S3,
…, Sn.
A’s opponents in his n games had rating levels of L1,
L2, …, Ln and reliability
measures of RM1,
RM2, …, RMn at the start of
their game with A.
A’s rating level at the end
of his last tournament was L.
If A is a new player set L
equal to 2,000. At the end of each of
A’s n games, his rating level (calculated below) is
.
is equal to L.
A’s reliability measure at
the end of his last tournament was RM.
If A is a new player set RM
equal to 400. At the end of each of A’s n games, his
reliability measure (calculated below) is
.
is equal to
from step 1
below.
It has been t
complete months
from A’s last tournament to this. If A is a new player
set t equal to 0.
Step 1.
Calculate
A’s time-adjusted pre-competition reliability measure at
the start of this competition, RMt,
according to:
where:
Step 2.
Calculate
the auxiliary constant p
according to:
Step 3.
Calculate
the auxiliary constant q
according to:
Steps 4 to 10 below are
repeated for each of A’s n games.
Step 4.
For
each rated game, calculate the “attenuating factor” fi
for use in later steps according to:
Step 5.
For
each rated game, calculate A’s expected DBM score Xi
as a proportion of the maximum score available (ie 0.00 to
1.00) according to:
Step 6.
For
each rated game, calculate the weighting value Ki
according to:
Step 7.
For
each rated game, calculate the change to A’s rating
level Di
according to:
Step 8.
Calculate
the A’s new rating level according to:
Step 9.
For
each rated game, calculate the proportional change to
A’s reliability measure Ei
according to:
Step 10.
Calculate
A’s new reliability measure according to:
Step 11.
Set A’s new post-competition
rating level equal to
and
post-competition reliability measure equal to
.