This is a problem of pure reason. We seek to construct a logically necessary framework to define and rank "Superiority" based on a closed system of events (a sports league). We must not be swayed by contingent correlations but must build deductively from our first principles.
1. Foundational Axioms and Definitions
Our system begins with fundamental, self-evident truths from which all other logic will be derived.
Axiom 1: The Event (The Game)
A "game" (\(g\)) is a discrete, adjudicated contest between exactly two teams (\(T_A\), \(T_B\)) from the set of all teams in the league (\(L\)).
Axiom 2: The Outcome (The Score)
Every game \(g\) produces a discrete, non-negative integer score for each team: Points For (\(P_F\)) and Points Allowed (\(P_A\)).
Axiom 3: The Condition of Victory
Victory (\(V\)) in a single game \(g\) is defined as \(P_F > P_A\). Defeat (\(D\)) is \(P_F < P_A\). A Draw (\(Dr\)) is \(P_F = P_A\).
Axiom 4: The Telos of Competition
The telos, or logical purpose, of a team within the system is to achieve Victory. A simple Victory (a "Win") is the minimal satisfaction of this purpose.
Axiom 5: The Definition of Superiority
"Superiority" (\(\succ\)) is a relational property. \(T_A \succ T_B\) ("\(T_A\) is superior to \(T_B\)") is defined as \(T_A\) possessing a greater demonstrated capability to achieve the telos (Victory) against the common challenges presented by the league (\(L\)).
Axiom 6: Distinction (Evidence vs. Proof)
A single game outcome is contingent evidence for superiority, not logical proof. Superiority is a cumulative conclusion derived from the entire set of all games played, not from any single game.
2. Primary Parameters of Superiority
From these axioms, we must deduce the essential parameters for measuring "demonstrated capability."
2.1 The Dual Nature of Performance
Logical Tension: Axiom 4 establishes Victory as the telos, creating a binary success condition. Yet Axiom 2 provides continuous data (the score). We must reconcile these.
Resolution: We define two essential parameters:
1. Win Component (\(W_C\)): The binary achievement of the telos
$$W_C = \frac{\text{Wins} - \text{Losses}}{\text{Games Played}}$$
2. Point Differential Component (\(PD_C\)): The magnitude of performance, capped dynamically based on opponent quality
$$PD_C = \text{cap}_{\text{dynamic}}(P_F - P_A, \text{opponent})$$
Dynamic Opponent-Adjusted Capping:
The cap is not static—it scales based on the quality of the opponent:
$$\text{cap}_{\text{dynamic}}(PD, Opp) = 31 \times (0.4 + 0.6 \times \text{strength}(Opp))$$
Where \(\text{strength}(Opp)\) is the opponent's hybrid LSR normalized to [0,1]:
- Elite opponent (LSR ≈ +20): strength = 1.0, cap = 31.0 (full 100%)
- Average opponent (LSR ≈ 0): strength = 0.5, cap = 21.7 (70%)
- Weak opponent (LSR ≈ -20): strength = 0.0, cap = 12.4 (40%)
Philosophical Justification: Margin of victory reveals more about capability when achieved against strong opponents. A 31-point victory over the #1 team demonstrates exceptional dominance (full cap = 31), while a 60-point victory over the #210 team demonstrates less (cap = 12-14).
Logical Weighting: The Win Component must be weighted more heavily than the PD Component to respect Axiom 4. We define a scaling factor \(\alpha\) (empirically set to 15) such that the Win Component dominates.
3. Temporal Logic
Teams are not static entities. \(T_A\) at time \(t_1\) is not identical to \(T_A\) at time \(t_n\).
Axiom 7: Temporal Relevance
The logical weight of a game \(g\) as evidence for current Superiority is inversely proportional to the temporal distance \(\Delta t = (t_{now} - t_g)\).
Logical Rule (Weighting): Each game \(g\) must be assigned a Temporal Weight (\(W_T\)). We use a square root progression:
$$W_T(k) = \sqrt{k}$$
Where \(k\) is the game number (1, 2, 3, ...)
Rationale for Square Root: A linear progression (\(W_T = k\)) creates excessive recency bias. The square root function provides moderate recency bias while preventing extreme over-weighting of recent games.
4. Comparative Logic
Superiority is relational. We must be able to compare teams that have not played each other (indirect comparison) and resolve logical paradoxes (circularity).
Logical Solution (Recursive Definition):
A team's "Superiority" (which we will now quantify as its Logical Strength Rating, LSR) must be a function of both its own performance and the LSR of its opponents.
$$LSR(T) = \text{Win Component} + \text{PD Component} + \text{Opposition Component}$$
Where:
- Win Component: The team's weighted win-loss record, scaled by \(\alpha\)
- PD Component: The team's capped, weighted point differential
- Opposition Component: The average LSR of its opponents, weighted by the temporal factor
This recursive definition logically resolves circularity. It defines the LSR of all teams in terms of each other. The system is solved iteratively until the values stabilize.
5. Contextual Logic
A pure assessment of capability must isolate intrinsic team strength from contingent external factors.
Venue (Home/Away):
Logical Rule (Neutralization):
1. Determine the system-wide Home Field Advantage (HFA):
$$HFA = \frac{\sum (P_{F(Home)} - P_{A(Home)})}{N_{\text{total games}}}$$
2. Neutralize every game's PD before it enters the LSR calculation:
- For the Home team: \(PD_{adj} = (P_F - P_A) - HFA\)
- For the Away team: \(PD_{adj} = (P_A - P_F) + HFA\)
6. Statistical Confidence Logic
A logical framework must distinguish between demonstrated capability and statistical confidence in that demonstration.
Axiom 8: Evidential Confidence (Sample Size)
The confidence we have in a team's LSR is proportional to the sample size of games played. A small sample must be adjusted toward the population mean (zero) until sufficient evidence accumulates.
Bayesian Sample Size Adjustment:
$$\text{Confidence}(n) = \begin{cases} 0.074 + 0.071n & \text{if } 1 \leq n \leq 6 \\ 0.10 + 0.0667n & \text{if } 6 < n \leq 12 \\ 0.95 & \text{if } n > 12 \end{cases}$$
Where \(n\) is the number of games played by team \(T\).
This creates confidence levels:
- 1 game: 14.5% confidence
- 6 games: 50.0% confidence
- 12 games: 90.0% confidence
- 13+ games: 95.0% confidence (plateau)
7. Structural Advantage Logic
The framework thus far has been radically empiricist: superiority emerges purely from demonstrated game results. However, college football presents a unique structural challenge.
7.1 The Problem of Persistent Inequality
Axiom 10: Persistent Structural Inequality
College football teams possess Structural Advantages (SA) that exist independently of game outcomes. These advantages are:
- Measurable: Recruiting rankings, resource metrics, historical performance
- Persistent: They do not disappear based on single-season results
- Predictive: They correlate with long-term competitive capability
7.3 The Talent Index: Quantifying Structural Advantage
We define a Talent Index (TI) as a composite measure of a team's structural advantages:
$$TI(T) = 0.6 \times \text{Recruiting}_{\text{norm}}(T) + 0.4 \times \text{Resources}_{\text{norm}}(T)$$
Recruiting Component (60% weight): Average of the team's last 3-4 recruiting class rankings
Resources Component (40% weight): Composite of athletic department budget, coaching salaries, and facilities
The Talent Index is then scaled to the LSR range (approximately ±10):
$$SA(T) = \frac{TI(T) - 50}{5} - 10$$
7.4 Hybrid Superiority: Demonstrated + Structural
The Final LSR Formula:
$$LSR_{\text{final}}(T) = w_D \times LSR_{\text{demonstrated}}(T) + w_S \times SA(T)$$
Where:
- \(w_D = 0.6313\) (demonstrated weight - 63.13%)
- \(w_S = 0.3687\) (structural weight - 36.87%)
- \(LSR_{\text{demonstrated}}(T)\) is the recursive LSR from Sections 1-6
- \(SA(T)\) is the Structural Advantage score
Empirical Derivation of Weights:
Rather than choosing weights philosophically, we derived them empirically:
- Objective: Find weights that maximize correct prediction of game outcomes
- Method: Golden section search minimizing log-loss
- Data: 531 games split into 75% training, 25% test
- Result: Optimal weights = 63.13% demonstrated, 36.87% structural
- Validation: 87.12% test accuracy
Key Finding: The empirically optimal weights (63/37) are remarkably close to the philosophical intuition (65/35), validating our reasoning.
8. Systematic Completeness: The Final Framework
This framework provides a complete, consistent, and deductively-derived system for ranking all teams.
The Method:
- Define the Set of Events: All games played (\(G\))
- Define Temporal Weights: Assign weights using square root progression: \(W_T(k) = \sqrt{k}\)
- Calculate Contextual Constant (HFA): Determine the league-average HFA from \(G\)
- Calculate Neutralized Performance: For every game, calculate the adjusted PD for both teams
- Cap Point Differentials: Apply the dynamic cap function based on opponent quality
- Establish the LSR System: Define the raw LSR for every team as:
$$LSR_{\text{raw}}(T) = \text{WinComponent}(T) + \text{PDComponent}(T) + \text{OppComponent}(T)$$
- Apply Confidence Adjustment: Reduce ratings for teams with small sample sizes
$$LSR_{\text{demonstrated}}(T) = LSR_{\text{raw}}(T) \times \text{Confidence}(n_T)$$
- Load Talent Metrics: Calculate Structural Advantage for each team
- Solve the Demonstrated LSR System: Iterate until LSR values stabilize
- Calculate Final Hybrid LSR:
$$LSR_{\text{final}}(T) = 0.6313 \times LSR_{\text{demonstrated}}(T) + 0.3687 \times SA(T)$$
- Produce the Final Ranking: Sort teams by final LSR (descending)
Summary of Constants:
- \(\alpha = 15\) (win component weight)
- PD base cap = ±31 (four touchdowns plus field goal)
- PD cap min factor = 0.4 (yields 12.4 point cap vs. weakest opponents)
- PD cap max factor = 1.0 (yields 31 point cap vs. strongest opponents)
- \(w_D = 0.6313\) (demonstrated capability weight - empirically optimized)
- \(w_S = 0.3687\) (structural advantage weight - empirically optimized)
- Recruiting weight = 0.6 (proportion of talent index from recruiting)
- Resources weight = 0.4 (proportion of talent index from resources)
- SA scale factor = 5 (scales 0-100 talent index to ±10 range)
Confidence Function Parameters:
- For 1 ≤ n ≤ 6: intercept = 0.074, slope = 0.071
- For 6 < n ≤ 12: intercept = 0.10, slope = 0.0667
- For n > 12: plateau = 0.95
Final Principle: This framework is internally consistent, derived from first principles, and accounts for all necessary logical variables: Victory (the telos), Performance magnitude (capped point differential), Opposition strength (recursive LSR), Temporal relevance (square root weighting), Context neutralization (HFA adjustment), Statistical confidence (sample size adjustment), and Structural advantage (talent and resource metrics).
The framework balances empiricism with realism: demonstrated results dominate (63%), but persistent structural inequalities are acknowledged (37%). This addresses the unique characteristics of college football, where extreme talent stratification creates systematic advantages that exist independently of single-season results.