RPR Derivation

A team's RPR, or Relative Power Rating (denoted ) is an attempt to quantify team's power relative to other teams. For a game between teams and , we approximate the probability of team winning as .

We focus on computing point outcomes simply because game/set/half outcomes are dependent on the order of points, whether the game was run with sets or stones, the length of each set, etc., all of which are hard to control for and can add significant noise.

Our data comes from a set of teams who played matches. Let represent the number of points won by team in game .
For each row, we observe an estimate of the probability that team wins. We can set this equal to the differences in team and 's RPR:

It is of note that this is equivalent to the other way around - we can swap and and it's the same equation.
If we fill this out for every game, we get a large system of equations. We can write this in matrix form:

with containing the coefficients in the appropriate locations, being the vector of all teams's RPRs, and containing the score of the team with the positive coefficient.

This matrix is most certainly singular, since we have likely played many more games than we have teams. Thus we compute the least squares solution for , giving the best estimate assuming normally distributed noise.

We compute this using a sparse Cholesky decomposition of . We do this because we know that only two entries of each row can be nonzero, and thus is very sparse.

There is one issue remaining, however: because RPR is inherently relative, the absolute zero point is a free variable. For the purposes of numerical stability, therefore, we add an additional row of ones to the bottom of and a single zero to the end of , which ensures that the sum of all RPRs is zero. This just shifts the values to be centered around zero; it does not change anything else.

Obviously this is an approximation, and a very crude one at that. It is only one dimension and thus can only measure one dimension of a team's performance; specialties which may lead to drastically different results than RPR predicts will get averaged out. RPR also claims some matchups should have a higher than 100% win rate, which is obviously false. I am not a statistician and therefore do not care; RPR is meant to be a simple way to find vaguely correct power ratings given large amounts of unstructured data.