Neither my SOWP nor my WtdW report constitute a "rating system" but they can be useful for analyzing the behavior of "advanced" systems because those depend upon the directed games graph. Here I've used A→B to mean only that A won over B, but those systems can be modeled by adjusting the length and/or thickness of the arrow based upon, say, game location and margin of victory.but one could use the graph itself as a "skeleton" for a rating system. The "weighted wins" function and "second order winning percentage" function each have some desirable properties that anything based upon either inherits.
The SOWP is based upon simple counting statistics. SOWP is just:
|SOWP =||WW + TT × ½|
|WW + LL + TT + UU|
One must be careful using property 2 - until all teams with at least one win are connected to all teams with at least one loss there's a great deal of imprecision in the SOWP's implied SOS, and therefore the SOWP rank. The UU term in the calculation provides a handy measurement of that. Just define the precision of the SOWP as:
|#teams − 1|
Early in the season, the precision will be less than a tenth, indicating a 90% uncertainty in most teams' rankings.
Let RT be the ordinal ranking of team T (1 being best, N being worsst where N is the number of ranked teams). If we replace the terms in the WtdW sum above with FA,B and FB,A for simplicity, the WtdWRating is:
|∑wins( FA,x × (N + 1 - Rx) ) − ∑losses( Fx,A × Rx )||†|
Let the ranking be that as defined in the Maj column in the Majority Consensus Summary (for the games through 15 Sep.)
There's a bit of a problem with the ranking. We have
|86||New Mexico St|
|New Mexico State→UTEP→New Mexico|
|UTEP→New Mexico→New Mexico St|
When we calculate WtdWBMaj we get a new ranking:
|67||New Mexico St|
So, according to the way the teams' wins are embedded in the directed games graph, we can conclude that the win that violates the ranking is UTEP's over New Mexico.
And to resolve the question about Alabama's rating, we notice:
†— The formula is a little more complicated than shown. When both F(A,x) and F(x,A) are defined, then for wins the multiplier is
F(A,x) − F(x,A)and for losses F(x,A) − F(A,x)This accounts for cases where an intermediate win is by a team that itself has first or second -order losses to other teams in the chain.
This is necessary in sports (such as baseball) where it is common for teams to play more than once in a season.