AllegSkill - Player's ranking

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What follows is the method for updating a two-team game with an arbitrary number of players.

Getting ready

From a population of n players <math>\{1,...,n\}</math> let <math>k</math> teams compete in a match. Teams are defined by <math>k</math> non-overlapping subsets, <math>T_{j}\subset \{1,...,n\}</math>, of the player population, <math>T_{i}\cap T_{j}=0</math> if <math>i\ne j</math>.

In layman's terms, this means that no player can appear on more than one team at the same time.

Each player, <math>n</math>, has three variables associated with them:

• <math>\mu _n</math>, their average skill;

• <math>\sigma _n</math>, their uncertainty about <math>\mu _{n}</math>;

• <math>f_{n}</math>, the fraction of the total game played for their team.

We also have three very important constants to recall:

• <math>\beta</math>, standard variance around performance;
• <math>\gamma</math>, the dynamics variable that keeps sigma from reaching zero;
• <math>\epsilon</math>, the draw factor.

Packing teams up

Each team, <math>j</math>, has a mu and sigma derived from the ratings of its players according to the formulas below. We will use capitalization to help distinguish between a player's mu (<math>\mu</math>) and sigma (<math>\sigma</math>) and a team's mu (<math>\Mu</math>) and sigma (<math>\Sigma</math>).

<math>\Mu _j=\sum\limits_{n \in T_j}{\mu_n f_n}</math>

<math>\Sigma _{j}=\sqrt{\left( \sum\limits_{n \in T_j}{\left( \sigma_n^2 f_n +\beta ^{2}+\gamma ^{2} \right)} \right) -\beta ^{2} -\gamma ^{2}}</math>

Back to the 1 vs 1 scenario

For the next step, each team is designated according to whether they won or lost the game. We now calculate a new mu and sigma for our teams using the standard Trueskill update formulae. Definitions of <math>W_{win}</math> etc can be found in the Commander's ranking section.

<math>\Mu '_{w}=\Mu _{w}+\frac{\Sigma _{w}^{2}}{c}\cdot V_{win}\left( \frac{\Mu _{w}-\Mu _{l}}{c},\frac{\varepsilon }{c} \right)</math>

<math>\Sigma '_{w}=\sqrt{\Sigma _{w}^{2}\left( 1-\frac{\Sigma _{w}^{2}}{c^{2}}\cdot W_{win}\left( \frac{\Mu _{w}-\Mu _{l}}{c},\frac{\varepsilon }{c} \right) \right)+\gamma ^{2}}</math>

<math>\Mu '_{l}=\mu _{l}-\frac{\Sigma _{l}^{2}}{c}\cdot V_{win}\left( \frac{\Mu _{w}-\Mu _{l}}{c},\frac{\varepsilon }{c} \right)</math>

<math>\Sigma '_{l}=\sqrt{\Sigma _{l}^{2}\left( 1-\frac{\Sigma _{l}^{2}}{c^{2}}\cdot W_{win}\left( \frac{\Mu _{w}-\Mu _{l}}{c},\frac{\varepsilon }{c} \right) \right)+\gamma ^{2}}</math>

Note. The acute reader will notice that this is the point that keeps the current incarnation of AllegSkill from supporting multi team games.

To each one's own

Now we calculate the total variance:

<math>\beta _{total}=\sum\limits_{j=1}^{k}{\left( \sum\limits_{n \in T_{j}}{\left( \sigma _{n}^{2}f_{n}+\beta ^{2}+\gamma ^{2} \right)} \right)}</math>

Each team has a <math>V</math> and <math>W</math> factor:

<math>V_{j}=\tfrac{\sqrt{\beta _{total}}}{\sum\limits_{n \in T_{j}}{\left( \sigma _{n}^{2}f_{n}+\beta ^{2}+\gamma ^{2} \right)}-\beta ^{2}} \left( \Mu '_{j}-\Mu _{j} \right)</math>

<math>W_{j}=\tfrac{\beta _{total}} {\sum\limits_{n \in T_{j}}{\left( \sigma _{n}^{2} f_{n} + \beta^2 + \gamma^2 \right)} - \beta^2 }\left( 1 - \frac{\Sigma_{j}'^{2}}{\Sigma _{j}^{2}} \right)</math>

Then each player, <math>n</math>, is updated using the values calculated for their team, <math>j</math>:

<math>\mu '_{n}=\mu _{n}+\tfrac{\sigma _{n}^{2}+\gamma ^{2}}{\sqrt{\beta _{total}}}f_{n}V_{j}</math>

<math>\sigma '_{n}=\sigma _{n}+f_{n}\left( \sigma _{n}\sqrt{1-W_{j}\tfrac{\sigma _{n}^{2}+\gamma ^{2}}{\beta _{total}}}-\sigma _{n} \right)</math>

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