Chapter 25: Case Study 1 - Building an Elo Rating System for NBA Prediction

Introduction

Elo ratings, originally developed for chess, have become one of the most popular and effective approaches for rating teams and predicting game outcomes across sports. This case study walks through the complete development of an NBA Elo rating system, from initial implementation through calibration and validation.

Part 1: Elo Rating Fundamentals

The Core Algorithm

The Elo system is based on a simple principle: when two opponents compete, the outcome updates both of their ratings based on the difference between the expected and actual results.

Key Components:

1. Expected Score Calculation:

E_A = 1 / (1 + 10^((R_B - R_A) / 400))

Where E_A is Team A's expected win probability and R_A, R_B are the ratings.

2. Rating Update:

R_A_new = R_A + K × (S_A - E_A)

Where S_A is the actual result (1 for win, 0 for loss), and K is the learning rate.

3. Home Court Adjustment: Add a fixed amount (typically 100 points) to the home team's rating for prediction purposes.

Implementation

class NBAEloSystem:
    def __init__(self, k_factor=20, home_advantage=100, initial_rating=1500):
        self.k_factor = k_factor
        self.home_advantage = home_advantage
        self.initial_rating = initial_rating
        self.ratings = {}

    def get_rating(self, team):
        return self.ratings.get(team, self.initial_rating)

    def expected_score(self, rating_a, rating_b, home_advantage=0):
        """Calculate expected win probability for team A."""
        return 1 / (1 + 10 ** ((rating_b - rating_a - home_advantage) / 400))

    def update_ratings(self, home_team, away_team, home_score, away_score):
        """Update ratings after a game."""
        # Get current ratings
        home_rating = self.get_rating(home_team)
        away_rating = self.get_rating(away_team)

        # Calculate expected scores
        home_expected = self.expected_score(home_rating, away_rating, self.home_advantage)

        # Actual result
        home_actual = 1 if home_score > away_score else 0

        # Update ratings
        self.ratings[home_team] = home_rating + self.k_factor * (home_actual - home_expected)
        self.ratings[away_team] = away_rating + self.k_factor * (home_actual - 1 + home_expected)

Part 2: Calibrating the Parameters

K-Factor Optimization

The K-factor determines how quickly ratings change. Too high, and ratings are noisy; too low, and they adapt slowly to real changes.

Testing Process: 1. Split data into training (seasons 1-5) and validation (season 6) 2. Test K values from 10 to 40 3. Measure prediction accuracy on validation set

Results:

Optimal K-Factor: 20 (balances responsiveness and stability)

Home Court Advantage

Testing different home court adjustments:

Optimal HCA: 100 Elo points (approximately 3.6-point spread advantage)

Season Carryover

Between seasons, ratings should regress toward the mean to account for roster turnover:

Formula:

R_new_season = R_old_season × carryover + mean_rating × (1 - carryover)

Testing carryover rates:

Optimal Carryover: 75% (moderate regression to mean)

Part 3: Enhancements

Margin of Victory Adjustment

Basic Elo only considers wins/losses. We can incorporate margin of victory:

MOV Multiplier:

mov_mult = log(abs(margin) + 1) × (2.2 / ((rating_diff × 0.001) + 2.2))

This multiplier: - Increases for larger margins - Decreases for expected blowouts (to prevent rating inflation)

Impact: - Accuracy: 65.8% → 66.4% - Brier Score: 0.219 → 0.214 - Spread RMSE: 11.8 → 11.2 points

Recency Weighting

Recent games matter more than early-season games. We can apply a recency adjustment:

def recency_weight(games_ago, decay=0.995):
    return decay ** games_ago

This slightly improves late-season predictions.

Part 4: Validation

Accuracy Over Time

Tracking season-long accuracy:

Comparison to Benchmarks

The enhanced Elo system approaches but doesn't exceed market accuracy.

Against the Spread

Testing the model against point spreads (converting Elo difference to spread):

Conversion:

spread = (elo_diff + hca) / 28  # ~28 Elo points per point of spread

Results (2000 games): - ATS Accuracy: 51.8% - Not statistically significant (p = 0.21) - Conclusion: No edge against the market

Calibration Check

The model is well-calibrated across probability ranges.

Part 5: Practical Applications

Pre-Game Predictions

For a game between Team A (Elo: 1620) vs Team B (Elo: 1480) with Team A at home:

Elo difference: 1620 - 1480 + 100 = 240
Expected win prob (A): 1 / (1 + 10^(-240/400)) = 80.4%
Predicted spread: 240 / 28 = 8.6 points

Power Rankings

Current Elo ratings translate to power rankings:

Playoff Predictions

Using Elo for playoff series simulation:

1. Calculate single-game win probability
2. Simulate 7-game series (thousands of times)
3. Report series win probability and expected games

Example: - Team A Elo: 1650, Team B Elo: 1550 - Home court to Team A - Game-by-game win probs: 75%, 75%, 68%, 68%, 75%, 68%, 75% - Series win prob for A: 87% - Expected games: 5.2

Part 6: Lessons Learned

What Works Well

1. Simplicity: Basic Elo captures most predictable variance
2. Adaptability: Ratings update automatically with each game
3. Interpretability: Easy to explain and understand
4. Calibration: Produces well-calibrated probabilities

Limitations

1. No roster information: Injuries not incorporated
2. No matchup specifics: Style matchups ignored
3. Slow to adapt: Takes 15-20 games to converge
4. Market benchmark: Doesn't beat betting markets

Recommendations

1. Use Elo as a baseline model
2. Combine with other approaches (efficiency models, injury adjustments)
3. Don't expect to beat the market consistently
4. Update parameters annually

Exercises

Exercise 1

Implement the basic Elo system and run it on one NBA season. Compare your ratings to final standings.

Exercise 2

Test different K-factors on your implementation. Create a graph showing accuracy vs. K-factor.

Exercise 3

Add margin-of-victory adjustment to your system. Measure the improvement in prediction accuracy.

Exercise 4

Use your Elo system to simulate the NBA playoffs. Compare to actual results.

Conclusion

An Elo rating system provides a robust, interpretable foundation for NBA game prediction. While it doesn't beat betting markets, it achieves approximately 66% accuracy with well-calibrated probabilities. The system serves as an excellent baseline for more sophisticated models and provides intuitive power rankings that capture team strength throughout the season.