2 +-------------------------------------------------+
3 | Vek-splanation of the Glicko Ratings System |
4 +-------------------------------------------------+
6 As you may have noticed, each FICS player now has a rating and an RD.
8 RD stands for "ratings deviation".
13 The new system with the RD improves upon the binary categorization that was
14 used before on fics and elsewhere, where players with fewer than 20 games were
15 labeled"provisional" and others were labeled "established". Instead of two
16 separate ratings formulas for the two categories, there is now a single
17 formula incorporating the two ratings and the two RD's to find the ratings
18 changes for you and your opponent after a game.
23 The Ratings Deviation is used to measure how much a player's current rating
24 should be trusted. A high RD indicates that the player may not be competing
25 frequently or that the player has not played very many games yet at the
26 current rating level. A low RD indicates that the player's rating is fairly
27 well established. This is described in more detail below under "RD
30 How RD Affects Ratings Changes
31 ------------------------------
33 In general, if your RD is high, then your rating will change a lot each time
34 you play. As it gets smaller, the ratings change per game will go down.
35 However, your opponent's RD will have the opposite effect, to a smaller
36 extent: if his RD is high, then your ratings change will be somewhat smaller
37 than it would be otherwise.
39 A further use of RD's:
40 ----------------------
42 Vek asked Mark Glickman the following:
44 > Given player one with rating r1, error s1,
45 > and player two with r2 and s2, do you have a formula for the probability
46 > that player 1's "true" rating is greater than player 2's ?
52 1/(1 + 10^(-(r1-r2)f(sqrt(s1^2 + s2^2))/400) )
54 where f(s) is [the function applied to RD in Step 2 below].
59 In this system, the RD will decrease somewhat each time you play a game,
60 because when you play more games there is a stronger basis for concluding what
61 your rating should be. However, if you go for a long time without playing any
62 games, your RD will increase to reflect the increased uncertainty in your
63 rating due to the passage of time. Also, your RD will decrease more if your
64 opponent's rating is similar to yours, and decrease less your opponent's
65 rating is much different.
67 Why Ratings Changes Aren't Balanced
68 -----------------------------------
70 In the other system, except for provisional games, the ratings changes for the
71 two players in a game would balance each other out - if A wins 16 points, B
72 loses 16 points. That is not the case with this system. Here is the
73 explanation I received from Mark Glickman:
75 The system does not conserve rating points - and with good
76 reason! Suppose two players both have ratings of 1700,
77 except one has not played in awhile and the other playing
78 constantly. In the former case, the player's rating is not
79 a reliable measure while in the latter case the rating is a fairly
80 reliable measure. Let's say the player with the uncertain rating
81 defeats the player with the precisely measured rating.
82 Then I would claim that the player with the imprecisely
83 measured rating should have his rating increase a fair
84 amount (because we have learned something informative from
85 defeating a player with a precisely measured ability) and
86 the player with the precise rating should have his rating
87 decrease by a very small amount (because losing to a player
88 with an imprecise rating contains little information).
89 That's the intuitive gist of my extension to the Elo system.
91 On average, the system will stay roughly constant (by the
92 law of large numbers). In other words, the above scenario
93 in the long run should occur just as often with the
94 imprecisely rated player losing.
96 Mathematical Interpretation of RD
97 ---------------------------------
99 Direct from Mark Glickman:
101 Each player can be characterized as having a true (but unknown) rating that
102 may be thought of as the player's average ability. We never get to know that
103 value, partly because we only observe a finite number of games, but also
104 because that true rating changes over time as a player's ability changes. But
105 we can *estimate* the unknown rating. Rather than restrict oneself to a
106 single estimate of the true rating, we can describe our estimate as an
107 *interval* of plausible values. The interval is wider if we are less sure
108 about the player's unknown true rating, and the interval is narrower if we are
109 more sure about the unknown rating. The RD quantifies the uncertainty in
110 terms of probability:
112 The interval formed by Current rating +/- RD contains your true rating with
113 probability of about 0.67.
115 The interval formed by Current rating +/- 2 * RD contains your true rating
116 with probability of about 0.95.
118 The interval formed by Current rating +/- 3 * RD contains your true rating
119 with probability of about 0.997.
121 For those of you who know something about statistics, these are not confidence
122 intervals, but are called "central posterior intervals" because the derivation
123 came from a "Bayesian" analysis of the problem.
125 These numbers are found from the cumulative distribution function of the
126 normal distribution with mean = current rating, and standard deviation = RD.
127 For example, CDF[ N[1600,50], 1550 ] = .159 approximately (that's shorthand
128 Mathematica notation.)
133 Algorithm to calculate ratings change for a game against a given opponent:
135 Step 1. Before a game, calculate initial rating and RD for each player.
137 a) If no games yet, initial rating assumed to be 1720.
138 Otherwise, use existing rating.
139 (The 1720 is not printed out, however.)
141 b) If no RD yet, initial RD assumed to be 350 if you have no games,
142 or 70 if your rating is carried over from ICC.
143 Otherwise, calculate new RD, based on the RD that was obtained
144 after the most recent game played, and on the amount of time (t) that
145 has passed since that game, as follows:
147 RD' = Sqrt(RD^2 + c log(1+t))
149 where c is a numerical constant chosen so that predictions made
150 according to the ratings from this system will be approximately
153 Step 2. Calculate the "attenuating factor" due to your OPPONENT's RD,
154 for use in later steps.
156 f = 1/Sqrt(1 + p RD^2)
158 Here p is the mathematical constant 3 (ln 10)^2
162 Note that this is between 0 and 1 - if RD is very big,
163 then f will be closer to 0.
165 Step 3. r1 <- your rating,
166 r2 <- opponent's rating,
169 E <- ----------------------
170 -(r1-r2)*f/400 <- it has f(RD) in it!
173 This quantity E seems to be treated kind of like a probability.
176 --------------------------------------
177 1/(RD)^2 + q^2 * f^2 * E * (1-E)
179 where q is a mathematical constant: q = (ln 10)/400.
181 Step 5. This is the K factor for the game, so
183 Your new rating = (pregame rating) + K * (w - E)
185 where w is 1 for a win, .5 for a draw, and 0 for a loss.
187 Step 6. Your new RD is calculated as
190 -------------------------------------------------
191 Sqrt( 1/(RD)^2 + q^2 * f^2 * E * (1-E) ) .
193 The same steps are done for your opponent.
198 A PostScript file containing Mark Glickman's paper discussing this ratings
199 system may be obtained via ftp. The ftp site is hustat.harvard.edu, the
200 directory is /pub/glickman, and the file is called "glicko.ps". It is
201 available at http://hustat.harvard.edu/pub/glickman/glicko.ps.
206 The Glicko Ratings System was invented by Mark Glickman, Ph.D. who is
207 currently at the Harvard Statistics Department, and who is bound for Boston
210 Vek and Hawk programmed and debugged the new ratings calculations (we may
211 still be debugging it). Helpful assistance was given by Surf, and Shane fixed
212 a heinous bug that Vek invented.
214 Vek wrote this helpfile and Mark Glickman made some essential
215 corrections and additions.
217 Last major update: April 19, 1995.
218 Minor revisions: August 28, 1995 by Friar.