# Glicko rating system

The Glicko rating system and Glicko-2 rating system are methods for assessing a player's strength in games of skill, such as chess and go. It was invented by Mark Glickman as an improvement of the Elo rating system, and initially intended for the primary use as a chess rating system. Glickman's principal contribution to measurement is "ratings reliability", called RD, for ratings deviation.

Both Glicko and Glicko-2 rating systems are under public domain and found implemented on game servers online (like Lichess, Free Internet Chess Server, Chess.com, Counter Strike: Global Offensive, Guild Wars 2).[1] The formulas used for the systems can be found on the Glicko website.

The RD measures the accuracy of a player's rating, with one RD being equal to one standard deviation. For example, a player with a rating of 1500 and an RD of 50 has a real strength between 1400 and 1600 (two standard deviations from 1500) with 95% confidence. Twice the RD is added and subtracted from their rating to calculate this range. After a game, the amount the rating changes depends on the RD: the change is smaller when the player's RD is low (since their rating is already considered accurate), and also when their opponent's RD is high (since the opponent's true rating is not well known, so little information is being gained). The RD itself decreases after playing a game, but it will increase slowly over time of inactivity.

The Glicko-2 rating system improves upon the Glicko rating system and further introduces the rating volatility σ. A very slightly modified version of the Glicko-2 rating system is implemented by the Australian Chess Federation.

## Determination

These steps only apply to the original Glicko system, and not its successor, Glicko-2.

If the player is unrated, the rating is usually set to 1500 and the RD to 350.

### Step 1: Determine RD

The new Ratings Deviation (${\displaystyle RD}$) is found using the old Ratings Deviation (${\displaystyle RD_{0}}$):

${\displaystyle RD=\min \left({\sqrt {{RD_{0}}^{2}+c^{2}t}},350\right)}$

Where ${\displaystyle t}$ is the amount of time (rating periods) since the last competition and '350' is assumed to be the RD of an unrated player. If several games have occurred within one rating period, the method treats them as having happened simultaneously. The rating period may be as long as several months or as short as a few minutes, according to how frequently games are arranged. The constant ${\displaystyle c}$ is based on the uncertainty of a player's skill over a certain amount of time. It can be derived from a thorough data analysis, or estimated by considering the length of time that would have to pass before a player's rating deviation would grow to that of an unrated player. If it assumed that it would take 100 rating periods for a player's rating deviation to return to an initial uncertainty of 350, and a typical player has a rating deviation of 50 then the constant can be found by solving ${\displaystyle 350={\sqrt {50^{2}+100c^{2}}}}$ for ${\displaystyle c}$.[2]

Or

${\displaystyle {\sqrt {(350^{2}-50^{2})/100}}=c}$

### Step 2: Determine New Rating

The new ratings, after a series of m games, are determined by the following equation:

${\displaystyle r=r_{0}+{\frac {q}{{\frac {1}{RD^{2}}}+{\frac {1}{d^{2}}}}}\sum _{i=1}^{m}{g(RD_{i})(s_{i}-E(s|r,r_{i},RD_{i}))}}$

Where the following is defined:

${\displaystyle g(RD_{i})={\frac {1}{\sqrt {1+{\frac {3q^{2}(RD_{i}^{2})}{\pi ^{2}}}}}}}$

${\displaystyle E(s|r,r_{i},RD_{i})={\frac {1}{1+10^{\left({\frac {g(RD_{i})(r-r_{i})}{-400}}\right)}}}}$

${\displaystyle q={\frac {ln(10)}{400}}=0.00575646273}$

${\displaystyle d^{2}={\frac {1}{q^{2}\sum _{i=1}^{m}{(g(RD_{i}))^{2}E(s|r,r_{i},RD_{i})(1-E(s|r,r_{i},RD_{i}))}}}}$

${\displaystyle r_{i}}$ represents the ratings of the individual opponents.

${\displaystyle s_{i}}$ represents the outcome of the individual games. A win is 1, a draw is ${\displaystyle {\frac {1}{2}}}$, and a loss is 0.

### Step 3: Determine New Ratings Deviation

The function of the prior RD calculation was to increase the RD appropriately to account for the increasing uncertainty in a player's skill level during a period of non-observation by the model. Now, the RD is updated (decreased) after the series of games:

${\displaystyle RD'={\sqrt {\left({\frac {1}{RD^{2}}}+{\frac {1}{d^{2}}}\right)^{-1}}}}$