CFB Rankings Methodology

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Methodology

The inspiration for this ranking model came from a personal desire to objectively rank college football teams solely based on game outcomes.

Year after year, I find that college football discourse is filled with subjective opinions about which teams are better or worse, often influenced by biases, media narratives, and historical prestige. I am tired of hearing debates about hypothetical matchups when we are lucky enough as fans to witness so many real games each season.

Thus, I was inspired to spend time developing a model that ranked teams solely based on game outcomes, without the influence of subjective factors such as recruiting rankings, preseason expectations, or traditional power ratings. I chose to implement this using a constraint optimization approach, specifically a convex quadratic programming model, because I had not seen many existing models that took this approach in a transparent and mathematically rigorous way. At the core, this model is very simple, it tries to assign ratings to teams such that:

  • Teams that win games have higher ratings than the teams they beat relative to a factor of the margin of victory, adjusted for home/away/neutral site.
  • Teams that lose to FCS opponents are penalized severely.
  • In order to avoid arbitrary early-season rankings, teams' ratings are tied more closely to their prior-season performance early in the season, until enough games are played to connect enough teams through common opponents. Imagine a network graph where teams are nodes and games are edges; as more edges are added, the relative positions of nodes become clearer.
  • It minimizes total ranking inconsistency subject to logical constraints about game results.

Additionally, this model has been a great exercise for me to apply my optimization and mathematical modeling knowledge from my undergraduate background in industrial engineering and operations research, my cloud computing and automation experience from working professionally as a data engineer, and my passion for college football.

Unlike some computer rankings, my code and methodology are fully open and transparent for anyone to review, critique, and improve upon, so please feel free to reach out to me if you have any suggestions, thoughts, or questions. I plan on continuing to refine and improve this model over time and adding more features to this app as well. I hope you find this model as interesting and useful as I have building it.

Decision Variables & Parameters

Sets

  • \(\mathcal{T}\): set of all FBS teams
  • \(\mathcal{G}\): set of all games \((i,j,k)\) where team \(i \in \mathcal{T}\) played team \(j \in \mathcal{T}\) in their \(k\)th matchup
  • \(\mathcal{F}\): set of FBS teams that lost to FCS opponents
  • \(\mathcal{M}\): set of margins of victory \(\forall (i,j,k) \in \mathcal{G}\)

Variables

  • \(r_i \in \mathbb{R}_+\): rating for team \(i\), \(\forall i \in \mathcal{T}\)
  • \(r_{\text{fcs}} \in \mathbb{R}_+\): rating for the dummy FCS team

Parameters

  • \(\lambda(w)\): Prior regularization weight where

\[ \lambda(w) = \begin{cases} \frac{7-w}{700} & \text{if } w < 7 \\ 0 & \text{if } w \geq 7 \end{cases} \]

where \(w\) is the week number.

  • \(\gamma_{\text{margin}} = 0.05\): Small margin penalty coefficient
  • \(\gamma_{\text{loss}} = 0.5\): FBS Loss penalty coefficient
  • \(\gamma_{\text{fcs}} = 5.0\): FCS Loss penalty coefficient
  • \(r_{\text{min}} = 0.01\): Lower bound for FBS team rating
  • \(r_{\text{max}} = 100\): Upper bound for FBS team rating
  • \(r_{\text{fcs-min}} = 0.01\): Lower bound for FCS team rating
  • \(r_{\text{fcs-max}} = 15\): Upper bound for FCS team rating
  • \(\text{prior}_i\): prior rating for team \(i\) from previous season's final rankings (default \(1\) for new FBS teams), model for prior is same as this model, but run on prior season's data

Margin Scaling Parameters

  • \(\alpha\): Scaling factor for home/away/neutral site adjustments where

\[ \alpha = \begin{cases} 1.0 & \text{neutral site} \\ 0.8 & \text{home team win} \\ 1.2 & \text{away team win} \end{cases} \]

  • \(g_{\text{median}} = 7.0\): Rating points that the median margin should represent
  • \(g_{\text{max}} = 20.0\): Maximum rating gap any margin can demand
  • \(h_{\text{margin}} = \frac{g_{\text{median}}}{\sqrt{\text{median}(M)}}\): Scaling factor for margins
  • \(m_{i,j,k} = \alpha_{i,j,k} \cdot \min(h_{\text{margin}} \cdot \sqrt{\text{M}_{i,j,k}}, g_{\text{max}})\): Adjusted margin of victory for game \((i,j,k)\)

Objective Function

\[ \begin{aligned} \text{minimize} \quad & \sum_{(i,j,k) \in \mathcal{G}} \gamma_{\text{margin}} \cdot [\max(0, r_{\text{loser}} + m_{i,j,k} - r_{\text{winner}})]^2 & [\text{Margin Penalty}] \\ & + \sum_{(i,j,k) \in \mathcal{F}} \gamma_{\text{fcs}} \cdot [\max(0, r_{\text{i}} + m_{i,j,k} - r_{\text{fcs}})]^2 & [\text{FCS Margin Penalty}] \\ & + \sum_{i \in \mathcal{T}} \gamma_{\text{loss}} \cdot (r_i \cdot \frac{\text{losses}_i}{\text{wins}_i + \text{losses}_i})^2 & [\text{Loss Rate Penalty}] \\ & + \sum_{i \in \mathcal{T}} \lambda \cdot (r_i - \text{prior}_i)^2 & [\text{Prior Regularization}] \end{aligned} \]

  • Margin penalties give a quadratic penalty for any game where the rating difference does not sufficiently explain the margin of victory. The margin is:
    • Scaled by the square root to reduce the impact of extreme margins
    • Multiplied by \(h_{\text{margin}}\) to normalize around the median margin
    • Capped at \(g_{\text{max}}\) to prevent blowouts from dominating
    • Finally adjusted by \(\alpha_{i,j,k}\) for home/away/neutral site effects
  • Loss rate penalties discourage teams with poor win-loss records from having high ratings.
  • Prior regularization ties team ratings to their prior season's ratings early in the season when data is sparse.

Constraints

FBS rating bounds:

\[ r_{\text{min}} \leq r_i \leq r_{\text{max}} \quad \forall i \in \mathcal{T} \]

FCS rating bounds:

\[ r_{\text{fcs-min}} \leq r_{\text{fcs}} \leq r_{\text{fcs-max}} \]

Implementation Notes

  • Prior ratings are carried over from the previous season's final model output. New FBS teams receive a default prior rating of \(15\).
  • The \(\lambda\) parameter enforces prior-season influence, linearly dropping to \(0\) from Week \(1\) to Week \(7\), eventually to rely entirely on current season results.
  • FCS losses are handled via a dummy FCS team whose rating is constrained between \(r_{\text{fcs-min}}\) and \(r_{\text{fcs-max}}\). This allows the model to penalize FBS teams losing to FCS opponents without needing to explicitly rate every FCS team.
  • Margin penalties encourage appropriate rating separation based on game margins.
  • Loss rate penalties discourage high ratings for teams with poor win-loss records.
  • Prior regularization prevents extreme ratings early in the season when data is sparse.
  • Home/away adjustments use \(\alpha\) multipliers to account for game location advantage.
  • The optimization is solved using CVXPY with default convex solvers, returning team ratings sorted in descending order.