Suppose there is a function ƒ of (possibly infinitely) many variables that characterizes a team's strength relative to all other teams. Then any rating system is an approximation to ƒ using a subset of the variables that contribute to ƒ. We define strength of schedule as
| Eq. 1 |
| Eq. 2 |
SOS is a function of the same variables as ƒ, and in particular most systems' approximation of ƒ use (opponents') winning percentage to approximate SOS.
Let F and G be rating systems that approximate ƒ and SOSF and SOSG be the corresponding versions of SOS as defined within those systems.
The Performance Against algorithm is the equivalent of an integral with respect to winning percentage for a discrete system, so we can write
We find that all functions such pairs of approximations to ƒ tested so far result in nearly the same ordinal ranking after their values are subjected to the PA transformation.
By equation 2,
Having found that PA(F) and PA(G) result in nearly the same rankings without regard to the rankings by F and G for all of the systems that have been tested, we conjecture that all approximations to the unknown function ƒ have the property that
| Eq. 3 |
In terms of the PA algorithm, this implies that the derived SOS (dSOS) results in the same ordinal ranking for all rating systems. (The dSOS values will of course be different because the systems are in different "units", and even the ordinal ranking will be different to the degree that a system defines SOS in terms other than winning percentages - typically adjustments for home field advantage, margin of victory, etc. The same is true for a rating system's equivalent of Winning Percentage.)
What the PA algorithm does is very much the same as histogram-style reports that show records versus teams ranked in various "tiers" (top 25, top 50, etc.) by some rating system, except it defines a unique "tier" for each team based on its performance against teams rated by the same system. Once that value is assigned to every team, the teams' opponents' values can be can be calculated based upon the games they won against their opponents.
Basically what the PA does is use all head-to-head-to-head results to provide a correction to the arbitrary rating system used to originally rank the teams. So if team A is ranked lower than team B by rating system R, but A beats team B, the PA algorithm gives more weight to team A, taking into account the strength of other opponents that were defeated by both A and B and either A or B. Regardless of whether rating system F had B rated higher than A and G either had a different order or a different margin of difference for the teams, PA adjusts both based upon how all teams performed relative to the rating system.
So the PA transformation provides a correction to a ranking F and a ranking based on dSOSF is for all practical purposes a universal ranking of schedule's strengths.