1. The input to the algorithm should be only number of wins and losses
   The reason we only want to use wins and losses is because we want the rating system to have the same properties as winning percentage, and that’s all that is used to calculate it.

   WP = W ( W + L )

2. A winless team should have a zero value
   This follows from the fact that until a team beats another team, there is no evidence that it should be ranked higher than any team
3. An undefeated team should have a value proportional to the “quality” of its opponents
   This is why there are so many ranking systems - there are an infinite number of ways to characterize the “quality” of teams against which their Winning Percentage was achieved

The RPI is usually defined as the sum of 25 percent of a team's winning percentage (WP), 50 percent of the team's opponents' winning percentage (OWP), and 25 percent of the team's Opponents' Opponents' Winning Percentage (OOWP). Sometimes it is described as 25 percent winning percentage and 75 percent strength of schedule, with SOS defined as 2/3 × OWP + 1/3 × OOWP.

There are several problems with this definition as an alternative to Winning Percentage for ranking teams. The most obvious is that it fails criterion 2, since 75 percent of the formula is based upon how well other teams have performed and criterion 3 because the "quality" of an undefeated opponents is not based upon how well its opponents have performed. That eliminates it as a useful measure of probability that a team would win against a lower-ranked team, but even if that were corrected there still would be problems because of the way OWP and OOWP are defined.

The RPI definition of OWP is a weighted average of the opponents' winning percentages (where opponents' winning percentages do not include games between the team and the individual opponent). The weight is the number of games played agaist the opponent, so just playing a game adds to the team's OWP by the RPI definition.

This would be equivalent to defining a batting average in baseball by the average of the BA for each game played. A 0 for 5 day followed by a 3 for 4 day would give (.000 + .750) = .375 instead of 3 for 9 = .333. In basketball, a player in a 3-game tournament who hits 2 of 10 shots, then 3 of 6, then 4 of 10 would have a shooting percentage of (.200 + .500 + .400)/3 = .433, when in fact for the tournament she was 9 for 25 = .360.

There's no other formula in all of sports statistics that makes this mistake.

"True OWP" is just the total of opponents' wins minus the team's losses divided by the total of opponents' games played minus the team's games played - in other words, a true percentage. The errors are not very large in the RPI's OWP component, but they become significant when used in the RPI definition of OOWP.

The RPI's definition of OOWP is the average opponents' OWP's. This doesn't take into account that the team's OOWP includes the records of the team itself and the records of teams that are also included in the OWP.

Most of the teams that have a high RPI ranking have a high RPI because their WP+OOWP is high, and their OOWP is only high because it includes the team's WP in its OOWP.

The RPI fails to pass criteria 2 and 3 just becaue a team's ranking depends more upon how their opponents have done against other teams. It turns out that the RPI also doesn't conform to criterion 1, since the RPI values depend more on the schedules than the wins and losses.

For any column vector V that assigns value V_i to each team i, let square matrices W and G be defined by W_i,j = wins by team i over team j, and G_i,j = games played by team i against team j and define [ result ]^* to mean result_i is divided by the number of games played by team i.

Then

PA = [ W × [ W × V ]* ]*

defines the PA algorithm.

SOSV = [ G × V ]*

is equivalent to team i's strength of schedule with respect to rating V.

[ W × V ]^* is just the value of a win over team i according to the characterization by V of team i's opponents, and PA characterizes a team by its performance against that characterization of its opponents.

dSOS = [ G × [ W × V ]* ]*

With that definition of V, the Performance Against SOS equations can be written as:

aSOS  =  [ W × OWP ]* dSOS  =  [ G × aSOS ]* PASOS  =  [ W × aSOS ]*

The intermediate term [ W × V ]^* is given a unique value for reporting purposes. It is the value of a win over team i in the PASOS measurement, and that would be a useful input to a team's scheduling process.

Both aSOS and dSOS consist of elements that are a "percentage of a percentage", and PASOS is a percentage of a percentage of a percentage, so to translate the values of PASOS into the same range as Winning Percentage we define adjusted Winning Percentage as:

aWPi = 3√PASOSi

and

dSOSPi = √dSOSi

Only the dSOS value applies to the team corresponding to the winning percentage; the aSOS is used to calculate the teams' opponents' PASOS (aWP) values. Plotting dSOS and WP against PASOS rank so that teams with the same winning percentage are separated:

The PASOS uses only wins and losses (criterion 1); a winless team has a zero value since SOS is weighted only by wins (criterian 2); and an undefeated team has just PASOS = DSOS (criterion 3).

It is worth noting that the PASOS' definition of schedule strength is not very different from the RPI's. Although there is no fixed ratio that applies to every team, overall the contributions to DSOS are roughly 2/3 opponents' winning percentage and 1/3 OOWP (plus a smaller fraction of OOOWP). The most important difference is that there are no duplications in the PASOS definitions of WP, OWP, and OOWP.

The effect of a team's winning percentage on its own SOS in the RPI definition is clearly visible when compared to the PASOS definition with no duplication. The difference between the trend lines indicates the degree to which a team "played itself", and at the lower values lost to itself!

The algorithm was validated by correlating the expected winning percentage to actual winning percentage and by applying it to a different sport (basketball).

For a game between team i and team j let

P = aWPi   ;   Q = aWPj

Here we've assigned a value to a win based upon the opponent's rank: 1 for a win against a top 25 team, 1/2 for a top 50 opponent that's not a top 25, 1/4 for a top 100 team opponent that's not in the top 50, 1/8 for a win vs a team in the 101-200 range, and zero for a win over a team in the 200+ range. The difference between this report and the PA is that we also count losses: 1/8th for each loss to a top-25 team, 1/4th for a loss to a top 50 team, 1/2 for a loss to a top 100 team, and 1 for each loss to a +100 team. All teams can be ordered by this metric.

A more precise ordering can be obtained by setting V to be the rating used to produce such a report and applying the Performance Against algorithm. Instead of a range of opponents' values such as 1-25, we just use the exact value for each opponent. For the RPI:

Applied to the PASOS itself, the result is:

• The PASOS combines WP, OWP, and OOWP more accurately than the RPI.
  Because only wins can contribute to a team’s rating, each game can contribute only once. The reason the RPI appears to be more accurate in basketball is that the errors in OWP and OOWP tend to be in the same direction.
• The aWP is an efficient adjustment to WP based upon schedule strength.
• The PASOS has additional benefits that aren’t provided by the RPI.
  An opponent's aSOS explicitly defines the value of a win over that team. This means that the effect of any particular win is easily visible, quantifying the concept of a "quality win".
• The Performance Against algorithm can be used to define a method-independent schedule strength measurement.
  The PA is a generalization of the "gory details" reports used within rating systems to qualify a team's ranking. Instead of "top 25" that varies among systems, it uses the actual values within the system for each opponent.

The last property of the generalized Performance Against method provides a mechanism for comparing systems that incorporate factors other than wins and losses. Typically these involve game location (as the ISR does) and/or margin of victory.

References and Credits