The Extended Euclidean Algorithm and Its Application to NFL Preseason Betting Predictions Using AI and Machine Learning

1. Introduction

The NFL preseason is a notoriously difficult betting landscape. Starters play sparingly, depth charts are fluid, and coaching priorities often have little to do with winning games outright. For sports bettors, the preseason is a game of uncertainty — predicting which team will care more, which third-string QB will overperform, and whether a coach values evaluation over victory.

In such an unpredictable environment, mathematical precision and data-driven balancing become essential. That’s where the Extended Euclidean Algorithm (EEA) — a centuries-old mathematical tool — can play an unexpected role when paired with AI-driven sports betting models.

While the EEA originated in number theory for computing greatest common divisors (GCD), its principles translate surprisingly well into reconciling model outputs, optimizing ensemble predictions, and finding balanced weights between competing AI-driven signals — all crucial in the preseason’s messy data landscape.

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2. The Extended Euclidean Algorithm: A Brief Overview

2.1. Core Purpose

The Euclidean Algorithm is used to find the greatest common divisor (GCD) of two integers — the largest number that divides both without a remainder.

The Extended Euclidean Algorithm takes it a step further.
It not only finds the GCD but also solves for integers $x$ and $y$ in the equation:

$$a \cdot x + b \cdot y = \gcd(a, b)$$

These coefficients $x$ and $y$ are known as Bézout coefficients.

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2.2. Why This Matters for AI Betting Models

In the context of AI:

• Think of $a$ and $b$ as two conflicting predictions from different models (e.g., Model A predicts a team will win 27–10, Model B predicts they will lose 17–14).

• The EEA provides a systematic way to combine these predictions by finding optimal integer or rational weights that reconcile differences while keeping prediction integrity.

• This can be extended to multi-model ensembles, where balancing weights can be computed iteratively for more than two models.

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3. NFL Preseason Betting: A Unique Challenge

3.1. Data Chaos

NFL preseason betting predictions are difficult because:

1. Roster Rotation – Starters may play only a few snaps; backups dominate most of the game.

2. Unclear Motivation – Coaches sometimes prioritize playbook experimentation over scoring.

3. Limited Historical Data – Preseason performances are poor predictors of regular season form.

4. Volatile Player Usage – Third-string QBs can suddenly play entire halves.

This volatility means traditional betting models (based on regular season data) often underperform. Instead, AI-driven models must rely heavily on depth charts, training camp reports, preseason-specific trends, and simulated matchups.

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4. Integrating the Extended Euclidean Algorithm into Preseason AI Models

4.1. The AI Prediction Pipeline

A robust preseason betting AI system might include:

1. Data Gathering:

   • Historical preseason results

   • Depth chart and roster data

   • Training camp beat reports

   • Player efficiency metrics from prior seasons

2. Model Training:

   • Logistic regression for win probabilities

   • Gradient boosting for scoring predictions

   • Neural networks for player-level simulation

3. Multiple Prediction Outputs:

   • Model A: Score-based simulation

   • Model B: Player-prop-based win probability

   • Model C: Historical preseason trend model

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4.2. Where the Extended Euclidean Algorithm Fits

Let’s say:

• Model A predicts a spread margin of +6 for Team X

• Model B predicts a spread margin of -2 for Team X

We want a balanced prediction that accounts for both models’ tendencies while weighting them in a mathematically principled way.

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Example Calculation

1. Step 1: Represent Predictions as Integers

   • Convert predictions into scaled integers:

     • $a = 60$ (represents +6.0 points from Model A)

     • $b = -20$ (represents -2.0 points from Model B)

2. Step 2: Apply EEA

   • We find:

     $$\gcd(60, -20) = 20$$

     And EEA tells us:

     $$60 \cdot (1) + (-20) \cdot (-2) = 20$$

     So $x = 1$, $y = -2$.

3. Step 3: Normalize to Prediction Weights

   • Weight Model A at +1 part

   • Weight Model B at -2 parts

   • Final reconciled prediction:

     $$(1 \cdot 6.0) + (-2 \cdot -2.0) = 6.0 + 4.0 = +10.0 points$$

   • This balanced output considers both optimism (Model A) and pessimism (Model B), while avoiding simple averaging that might dilute strong signals.

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4.3. Why This Helps in Preseason

The EEA-based weighting:

• Preserves important model differences instead of washing them out via averaging

• Dynamically adapts when one model is historically more reliable in certain preseason scenarios (e.g., when depth players decide the game)

• Helps identify when models are fundamentally disagreeing — a sign of an unpredictable matchup where bankroll exposure should be reduced

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5. Practical Use Case: Betting NFL Preseason Games

Imagine Week 2 of the NFL preseason:

• Model A uses player depth data and predicts Team A wins by 7.

• Model B uses historical preseason trends and predicts Team B wins by 3.

• Model C uses coach motivation indicators (e.g., tendency to try in preseason) and predicts Team A wins by 1.

Using EEA iteratively:

1. Combine A & B into a reconciled prediction

2. Combine that with C’s output for a final number

3. Compare to market line and determine if there’s an edge

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6. Betting Strategy Implications

Using EEA in AI-driven preseason betting:

• Identify False Confidence – When two strong models heavily disagree, the EEA reveals underlying volatility.

• Optimize Bankroll Allocation – Confidence can be tied to GCD-derived weight stability.

• Spot Market Inefficiencies – If reconciled predictions deviate sharply from market consensus, you may have a true edge.

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7. Limitations and Enhancements

Limitations:

• Relies on having multiple quality predictive models — bad inputs = bad outputs.

• Works best when predictions are quantitatively measurable (spreads, totals).

• EEA is not inherently probabilistic — needs to be paired with probability-based betting frameworks like Kelly Criterion.

Enhancements:

• Apply EEA to probability distributions instead of point estimates.

• Use Bayesian weighting on top of EEA coefficients for stability.

• Combine with Alpha-Max + Beta-Min methods to merge optimism-pessimism balancing with GCD-driven weighting.

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8. Conclusion

The Extended Euclidean Algorithm may have been born in number theory centuries ago, but its application to AI-driven NFL preseason betting is both innovative and practical. In an environment plagued by uncertainty and unpredictable player usage, balancing and reconciling multiple model predictions is essential.

By using the EEA, bettors can:

• Integrate conflicting AI predictions mathematically

• Avoid naive averaging

• Gain a sharper, balanced edge in volatile preseason markets

In short:
The NFL preseason might be chaotic, but the Extended Euclidean Algorithm offers a way to bring order to the chaos — and in sports betting, that’s a mathematical advantage worth having.