Step 1: Determine ratings deviation

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 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 is 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 ${\displaystyle c}$.^[10]

Or

${\displaystyle c={\sqrt {(350^{2}-50^{2})/100}}\approx 34.6}$

Step 2: Determine new rating

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

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

where:

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

${\displaystyle E(s|r_{0},r_{i},RD_{i})={\frac {1}{1+10^{\left({\frac {g(RD_{i})(r_{0}-r_{i})}{-400}}\right)}}}}$

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

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

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

${\displaystyle RD_{i}}$ represents the rating deviations 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:

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

Step 1: Compute ancillary quantities

Across one rating period, a player with a current rating ${\displaystyle \mu }$ and ratings deviation ${\displaystyle \phi }$ plays against ${\displaystyle m}$ opponents, with ratings ${\displaystyle \mu _{1},...,\mu _{m}}$ and RDs ${\displaystyle \phi _{1},...,\phi _{m}}$, resulting in scores ${\displaystyle s_{1},...,s_{m}}$. We first need to compute the ancillary quantities ${\displaystyle v}$ and ${\displaystyle \Delta }$:

${\displaystyle v=\left[\sum _{j=1}^{m}g(\phi _{j})^{2}E(\mu ,\mu _{j},\phi _{j})\{1-E(\mu ,\mu _{j},\phi _{j})\}\right]^{-1}}$

${\displaystyle \Delta =v\sum _{j=1}^{m}g(\phi _{j})\{s_{j}-E(\mu ,\mu _{j},\phi _{j})\}}$

where

${\displaystyle g(\phi _{j})={\frac {1}{\sqrt {1+3\phi _{j}^{2}/\pi ^{2}}}},}$

${\displaystyle E(\mu ,\mu _{j},\phi _{j})={\frac {1}{1+\exp\{-g(\phi _{j})(\mu -\mu _{j})\}}}.}$

Step 2: Determine new rating volatility

We then need to choose a small constant ${\displaystyle \tau }$ which constrains the volatility over time, for instance ${\displaystyle \tau =0.2}$ (smaller values of ${\displaystyle \tau }$ prevent dramatic rating changes after upset results). Then, for

${\displaystyle f(x)={\frac {1}{2}}{\frac {e^{x}(\Delta ^{2}-\phi ^{2}-v-e^{x})}{(\phi ^{2}+v+e^{x})^{2}}}-{\frac {x-\ln({\sigma ^{2}})}{\tau ^{2}}},}$

we need to find the value ${\displaystyle A}$ which satisfies ${\displaystyle f(A)=0}$. An efficient way of solving this would be to use the Illinois algorithm, a modified version of the regula falsi procedure (see Regula falsi § The Illinois algorithm for details on how this would be done). Once this iterative procedure is complete, we set the new rating volatility ${\displaystyle \sigma '}$ as

${\displaystyle \sigma '=\exp\{A/2\}.}$

Step 3: Determine new ratings deviation and rating

We then get the new RD

${\displaystyle \phi '=1{\Big /}{\sqrt {{\frac {1}{\phi ^{2}+\sigma '^{2}}}+{\frac {1}{v}}}},}$

and new rating

${\displaystyle \mu '=\mu +\phi '^{2}\sum _{j=1}^{m}g(\phi _{j})\{s_{j}-E(\mu ,\mu _{j},\phi _{j})\}.}$

These ratings and RDs are on a different scale than in the original Glicko algorithm, and would need to be converted to properly compare the two.^[8]