Lean C1: A Regularized Team-Strength Rating for College Football

1. Inputs and rating

Lean C1 produces one scalar strength per team, ordered from strongest to weakest. It combines opponent-adjusted game-margin and drive summaries, pairwise-result estimates, existing internal component blends, joint team-strength summaries and observed personnel descriptors. Personnel inputs describe passing, rushing, route-running, blocking and defensive contributors: current and prior grades, returning contribution, transfers, exposure and missingness. These facets measure observed performance rather than innate talent or verified depth-chart position groups. Related input families overlap; an input slot is not an independent source of evidence. Bookmaker information and incumbent ratings are excluded from the learned features.

For game g, let x_h and x_a denote the two teams’ feature vectors, y_g the home-win indicator, and v_g the nonneutral-site indicator. The training logit is η_g = βᵀ(x_h − x_a) + γv_g. The minimized objective is:

Weights are two when the unchanged AVR control’s absolute standardized strength gap is below 0.25, and one otherwise. Published strengths use βᵀx without γ, then map supported utilities to the supported control teams’ mean and standard deviation. Unsupported teams retain their exact control value; inadequate annual fitting support triggers full control fallback. Ranking strengths are not calibrated win probabilities. Pairwise logistic models have established precedent [1]; no novelty is claimed for this general model class.

2. Lean encoding

Five report-matching-share fields are globally zero in the qualified source banks. Two groups each contain five identical missingness fields. Removing the five zero fields and replacing each duplicate group with one representative leaves 247 inputs from 260, a 5% reduction.

Each representative is multiplied by √5. If the five original coefficients equal b, its consolidated coefficient is √5b. The product and squared coefficient norm remain unchanged: (√5x)(√5b) = 5xb and (√5b)² = 5b². Simply deleting four duplicate columns would change effective regularization. Consolidation preserves the profiled ridge objective, not the norm of arbitrary unequal duplicate coefficients. Nonconstant signals remain represented even when earlier fitted coefficients were zero. Future source changes must satisfy the same zero and equality invariants.

Independent checks covered 62 saved solutions, 166 boards and every evaluated game. Saved-head conversion changed gaps by at most 1.29 × 10⁻¹⁴. A fresh lean refit preserved all annual selections, fallbacks and 7,717 game-ordering signs. Numerical differences moved four team-origin ranks by one place across two complete boards; exact full-list equality is not claimed.

3. Annual learning

For season Y, both choices use all admitted outcome-training rows through Y−2, validation in Y−1, and refitting through Y−1. Recent-three-season and expanding describe feature/source histories, not outcome-year trims. The grid crosses those histories with λ ∈ {0.1, 1, 10}. Minimum support is 100 training/refit pairs, 100 total validation pairs and 25 close-strength validation pairs.

Eligible choices must have prior-year full ordering error no more than one percentage point above the control. Selection minimizes close-strength error, then full error; ties prefer stronger regularization and expanding history. Historical source and outcome admission applies a 24-hour completion buffer. Weekly origins govern the reported evaluation. Annual fitting and feature construction use earlier information, but this reconstruction cannot certify every original publication clock.

4. Historical evaluation

The target contains 7,717 FBS-versus-FBS games from 2016 through September 20, 2026. A stronger-rated team losing counts as one ordering error; exact rating ties count as one-half. Scoreboard ties are excluded. The 1,937-game close band is fixed from the control rather than redefined by the candidate.

Lean C1’s overall difference from A0 is −0.350 percentage points. Its individual 95% interval is [−0.935, +0.228] under 20,000 paired UTC calendar-week resamples and [−1.117, +0.331] under 20,000 football-season/week resamples. Close-band differences are −1.187 points, with intervals [−3.286, +0.886] and [−3.802, +1.339]. All include zero. Lean C1 exactly inherits C1’s historical error vectors; it has one fewer overall error and three more close-band errors than P1.

Native editions require qualified clocks and at most ten days’ information age. FPI is a predictive strength measure [2]. MAS is distinct from Massey Power [3]; Sagarin Rating is distinct from Predictor. No all-four superiority or one-point noninferiority survives both declared simultaneous bounds. Different maximal cohorts cannot be ranked directly against one another.

5. Interpretation and limitations

All historical seasons are development; close weighting followed earlier error analysis. The seven-procedure close/talent program tests 54 own-system, 84 common-cohort and 84 maximal-cohort contrasts. Six separate contrasts compare lean C1 with C1 and P1. Simultaneous bounds are conditional on each finite program and exclude earlier adaptive search. Favorable counts do not establish improvement. PFF publication precision, source-age differences and unknown-source exclusions remain limitations. Computed 269-team boards provide no FCS outcome validation.

Historical boards regenerated with this version are reconstructed ratings, not original contemporaneous releases. Operational adoption is a product choice; it does not convert development evidence into future validation. A genuinely future evaluation is required for broader performance claims. This paper concerns team strength and ordering only; résumé rankings and the game simulator are outside its scope.

Disclosure

Andy Volpe develops AVR. This working paper reports owner-directed research. Independent AI-agent checks and an AI expert-panel synthesis support verification; they are not external peer review. Published October 13, 2026.

References and evidence

1. Turner, H., and Firth, D. (2012). Bradley–Terry Models in R: The BradleyTerry2 Package. Journal of Statistical Software, 48(9), 1–21. doi:10.18637/jss.v048.i09.
2. ESPN (2017). Football Power Index terminology.
3. Massey, K. (n.d.). Rating-system description. NCAA.
4. Volpe, A. (2026). Lean C1 aggregate evidence supplement. Method definitions, matched counts, uncertainty and qualification.
5. Volpe, A. (2026). The AVR Rating System. Earlier control-system evaluation; its reported counts concern A0 rather than lean C1.