[wgardner]

And a follow up:

Take the above derivation, but let's pretend that each match score is only the result of 1 team's efforts, not 3. So in this case, each row of A would only have a single 1 in it, not 3.

In this pretend case, the OPR IS exactly just computing the average of that team's match scores(!). A' A is diagonal and the diagonal elements are the number of matches that a team has played, so Inv (A' A) is diagonal with diagonal elements that are 1/ the number of matches that a team has played.

Then the i,jth elements of Inv (A' A) A' are just 1/the number of matches a team has played if team i played in match j or 0 otherwise.

The variance of the Oest values in this pretend case is the variance of the prediction residual / number of matches that a team has played, and thus the standard error of the Oest value is the standard error of the match predictions divided by the square root of the number of matches that a team has played.

So this connects Oblarg's statements to the derivation. If match results were solely the result of one team's efforts, then the standard error of the OPR would just be the standard error of the match prediction / sqrt(n), where n is the number of matches that a team has played. But match results aren't solely the result of one team's efforts, so the previous derivation holds in the more complicated, real case.