TrueSkill [1] is a widely accepted algorithm for matchmaking, in which player skills are initialized with a random variable following Gaussian distribution and then get updated through team-based match outcomes. Eventually, with sufficient amount of matches, player skill random variables will converge to some means with very small variance.
There are multiple open sourced libraries implementing the algorithm. For example, one in Python [2] and another in C# [3]. However in these open sourced libraries I don’t see how to predict new matches given learned player skill mean and variance.
Today I checked again the paper and other materials to assure that to predict a new match where draw is not allowed, whichever team has higher sum of means should be predicted to win. Why? Let’s use the example from the original TrueSkill paper.
In the factor graph above, we want to compare $latex t1$ vs $latex t2$ and $latex t2$ vs $latex t3$. Let’s take the comparison between t1 and t2 for example. We are essentially comparing $latex p1$ vs $latex p2+p3$. $latex s_1 \sim \mathcal{N}(\mu_1, \sigma_1^2)$ and $latex p_1 \sim \mathcal{N}(s_1, \beta^2)$. To get the distribution of $latex p_1$, we have:
$latex \int_{-\infty}^{\infty} \frac{1}{\sigma_1 \sqrt{2\pi}} e^{-\frac{(s_1 – \mu_1)}{2 \sigma_1^2}} \frac{1}{\beta \sqrt{2\pi}} e^{-\frac{(p_1 – s_1)}{2 \beta^2}} ds= \frac{1}{\sqrt{\sigma_1^2 + \beta^2} \sqrt{2\pi}} e^{- \frac{(p_1 – \mu_1)^2}{2(\sigma_1^2+\beta^2)}} &s=2$
Therefore $latex p_1$ is another random variable following normal distribution. $latex p_2$ and $latex p_3$ can be derived to be random variables in a similar fashion. Since we know that add/substract multiple normal random variables will result in a random variable with the mean equal to add/substract of the means of participating variables, we conclude that the predicted outcome between two teams relies on whether the sum of means of one team is greater than that of the other team. In our example, $latex d_1 \sim \mathcal{N}(\mu_1 – \mu_2 – \mu_3, \sigma’)$ for some $latex \sigma’$ not important to know for prediction.
if you want to know the winning probability of a player over another, checkout [5] and [6]
[1] http://research.microsoft.com/en-us/projects/trueskill/
[2] http://trueskill.org/
[3] http://www.moserware.com/2010/03/computing-your-skill.html
[4] http://www.tina-vision.net/docs/memos/2003-003.pdf
[5] http://stackoverflow.com/questions/28031698/with-the-trueskill-algorithm-given-two-players-ratings-how-can-i-calculate-th
[6] https://github.com/sublee/trueskill/issues/1#issuecomment-10491635