‘Skill’ is not a single number!
While we represent skill as a single number (i.e., your Rating), it is, in fact, a range. Specifically, skill is a range of probabilities around an average or mean. The probability is greatest that a player will perform very close to their mean and becomes less likely as you move further away from that point.
If you are playing tennis against World No. 1, Novak Djokovic, you are expected to lose. But, if you play him a million times, you might win one or more times (he might be ill or have an injury). As the best in the world, Djokovic’s mean is higher than everyone else’s, but the probability that he will lose to you is not zero, it’s just very low.
So, a player’s skill can be accurately represented by a bell-curve (specifically, a Gaussian Distribution), as below:
The y-axis of this graph indicates the probability that a player will perform at a given level. Their average or ‘mean’ skill value is denoted by the Greek letter ‘mu’ (µ). This represents the level of skill (shown on the x-axis) at which the player is most likely to perform.
But the OpenSkill algorithm uses two key factors to reach the single number that we use to represent players’ skill levels. The second factor, represented by the Greek letter ‘sigma’ (σ), is the player’s Standard Deviation1. This reflects the degree of uncertainty of a player’s performance and determines how wide or tight the curve is. When the algorithm has lots of information about how a player has performed, the curve will be quite tight – reflecting a higher degree of confidence as to their skill level. If the algorithm does not have much information (or the player plays very inconsistently), the curve will be quite wide.
In summary, σ represents the uncertainty surrounding a player’s expected performance, while µ represents the player’s average (and therefore most expected) performance.
Our algorithm starts with no information about how skilled a player is, so all players start with a µ of 1200, exactly halfway between 0 and 2400 – i.e. dead average.
With no information, there is total uncertainty (i.e., no degree of confidence) around this estimate, and so all players start with a σ value of 400, which is extremely high. This sigma value decreases as the player plays more matches and the algorithm obtains more information about their skill.
Mu and sigma form the basis of how we calculate changes in players’ Ratings. After every game, these two values change. We then use a simple formula to convert them into the single number that denotes your Rating.