π-Ratings

π-ratings (Pi-ratings) are nonlinear power ratings based on changes in play-by-play win probabilities throughout a game, naturally deemphasizing non-competitive game situations without applying arbitrary cutoffs. π-ratings are adjusted for strength of schedule (SOS) and represent the probability of beating an average FBS opponent (i.e., a team with a 50% π-rating) on a neutral field. π-ratings strike a natural balance between Elo-style ranking systems that only account for wins and losses, good for explaining past results, and pure points-based systems that are designed for predicting future outcomes.

λ-ratings (Lambda-ratings) are linear power ratings that reflect the expected margin of victory or defeat against an average FBS opponent on a neutral field (i.e., a team with a λ-rating = 0.0). These are classical power ratings in that they measure how many points better or worse a team is than average in a linear fashion. To obtain a predicted net score between two teams at a neutral site, simply take the difference in the two teams’ λ-ratings.

Examples

In the 2022 preseason, Alabama has an initial π-rating = 99% and a λ-rating = +28.3. The most average FBS team is Kansas, with a π-rating = 50% and a λ-rating = +0.0.

Thus, at a neutral site (no home-field advantage for either team), Alabama is estimated to have a 99-percent chance of beating Kansas, and the Crimson Tide would be predicted to beat the Jayhawks by 28.3 points, on average, if the game were replayed hundreds or thousands of times.

Quotients

The Quotient in the rightmost column is the basic, SOS-adjusted statistic that forms the basis of both π-ratings and λ-ratings. Derived from statistical predictive models covering 15 years of FBS football games, the Quotient is equal to λ / 246.4. Knowing the Quotient is useful for computing more advanced win probabilities and net score predictions between any two FBS teams at any stadium (home, away, neutral, etc.). Even partial home-field advantage can be assigned, which can be useful for “neutral” games that presumably favor one team over the other, like Georgia playing Oregon at Mercedes-Benz Stadium in Atlanta in Week 1 of the 2022 college football season. The Quotient-based formulas are given at the bottom of the page, below the power ratings lookup table below.

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Pi & Lambda Ratings Lookup Table

Quotient Formulas

To compute the win probability of Team A vs Team B, look up their corresponding Quotients.

We’ll label these Q(A) and Q(B). All the computations below will be relative to Team A; i.e., the win probability and net points predictions will be for Team A.

Next, determine the home-field advantage (HFA) index:

• HFA = -0.50 → Team B’s home stadium
• HFA = 0.00 → Purely neutral site
• HFA = 0.50 → Team A’s home stadium

Alternative HFA indexes can also be used, which requires some judgment on behalf of the user. For example, in the aforementioned Oregon v. Georgia game in Atlanta, the College Football Atlas prediction models would assume an HFA = -0.25 for Oregon, since they are traveling across the country to the home state of the Bulldogs and will likely play in front of a hostile crowd. However, Georgia is not ascribed a full home-field advantage since they, too, must travel at least some distance and play away from the familiar confines of Sanford Stadium in Athens.

Note that there is no perfect answer for the HFA index. Use your best judgment. Choosing something reasonable within the [-0.50,+0.50] range will yield good results.

Once you have acquired team Quotients and set the HFA index, you can calculate win probability and predicted net score using the following formulas:

• First, compute Q = Q(A) – Q(B)

Net Score Prediction

• Net Score (A) = 5.34*HFA + 246.4*Q

Estimated Win Probability

• Win Probability (A) = 100 * { 1 / [ 1 + exp(-0.764*HFA – 37.8*Q) ] }

In the Win Probability formulas, exp refers to the natural exponent function. Calculators and spreadsheets, like Excel, will make this function readily available. The factor 100 converts the initial win proportion into the more familiar win probability.