Chapter 21: Key Takeaways - Win Probability Models

Quick Reference Summary

This chapter covered building, calibrating, and applying win probability models for college football.

Core Concepts

Game State Features

Key Metrics

Essential Formulas

Logistic Win Probability

WP = 1 / (1 + exp(-z))

Where: z = β₀ + β₁·score_diff + β₂·time + β₃·field_pos + ...

Win Probability Added

WPA = WP_after - WP_before

Total game WPA = Final WP - Initial WP

Expected Calibration Error

ECE = Σ (n_bin / n_total) × |predicted_avg - actual_avg|

Pregame WP from Spread

WP = 1 / (1 + 10^(-spread / 10))

Example: 7-point favorite
WP = 1 / (1 + 10^(-7/10)) = 0.72

Code Patterns

Basic Win Probability Model

from sklearn.linear_model import LogisticRegression

def train_wp_model(plays: pd.DataFrame) -> LogisticRegression:
    """Train basic WP model."""
    features = ['score_diff', 'time_remaining', 'field_position',
                'down', 'distance', 'possession']

    X = plays[features].values
    y = plays['home_win'].values

    model = LogisticRegression()
    model.fit(X, y)
    return model

WPA Calculation

def calculate_wpa(plays: pd.DataFrame, wp_model) -> pd.DataFrame:
    """Calculate WPA for each play."""
    df = plays.copy()

    # Get WP before each play
    df['wp_before'] = wp_model.predict_proba(df[features])[:, 1]

    # WP after is next play's WP before
    df['wp_after'] = df.groupby('game_id')['wp_before'].shift(-1)

    # Handle game-ending plays
    df['wp_after'] = df['wp_after'].fillna(df['home_win'])

    # Calculate WPA
    df['wpa'] = df['wp_after'] - df['wp_before']

    return df

Calibration Check

def check_calibration(predictions: np.ndarray,
                     outcomes: np.ndarray,
                     n_bins: int = 10) -> dict:
    """Check model calibration."""
    bins = np.linspace(0, 1, n_bins + 1)
    calibration = []

    for i in range(n_bins):
        mask = (predictions >= bins[i]) & (predictions < bins[i+1])
        if mask.sum() > 0:
            pred_mean = predictions[mask].mean()
            actual_mean = outcomes[mask].mean()
            calibration.append({
                'bin': f'{bins[i]:.1f}-{bins[i+1]:.1f}',
                'predicted': pred_mean,
                'actual': actual_mean,
                'error': abs(pred_mean - actual_mean)
            })

    df = pd.DataFrame(calibration)
    ece = df['error'].mean()

    return {'calibration': df, 'ece': ece}

Model Comparison

Logistic Regression

Gradient Boosting

Neural Network

Fourth Down Decision Framework

Decision Options

1. Go for it: Convert or turnover on downs
2. Punt: Better field position for opponent
3. Field Goal: If in range

Expected WP Calculation

E[WP_go] = P(convert) × WP_convert + (1-P(convert)) × WP_fail

E[WP_fg] = P(make) × WP_make + (1-P(make)) × WP_miss

E[WP_punt] = WP_after_punt

Break-Even Analysis

Go for it when: E[WP_go] > E[WP_punt]

Break-even rate: P* where E[WP_go] = E[WP_punt]

Calibration Guidelines

Interpreting Calibration Curves

Calibration Methods

1. Platt Scaling: Logistic regression on predictions
2. Isotonic Regression: Non-parametric mapping
3. Temperature Scaling: Divide logits by constant

Common Pitfalls

1. Ignoring Time-Score Interaction

Wrong:

features = ['score_diff', 'time_remaining']  # No interaction

Right:

df['score_time'] = df['score_diff'] * df['time_remaining']
features = ['score_diff', 'time_remaining', 'score_time']

2. Poor End-of-Game Handling

Wrong:

# Missing data for final plays
df['wp_after'] = df['wp_before'].shift(-1)  # NaN at end

Right:

df['wp_after'] = df['wp_before'].shift(-1)
df.loc[df['wp_after'].isna(), 'wp_after'] = df['home_win']

3. Ignoring Pregame Probability

Wrong:

# Start both teams at 50%
initial_wp = 0.5

Right:

# Incorporate team strength
initial_wp = calculate_pregame_wp(home_team, away_team)

Evaluation Checklist

Before Training

• [ ] Clean play-by-play data
• [ ] Define game state features
• [ ] Determine outcome variable
• [ ] Split train/validation/test

Model Training

• [ ] Feature engineering complete
• [ ] Cross-validation performed
• [ ] Baseline comparison done
• [ ] Hyperparameters tuned

Calibration

• [ ] Calibration curve plotted
• [ ] ECE calculated
• [ ] Brier score measured
• [ ] Calibration applied if needed

Validation

• [ ] Out-of-sample testing
• [ ] Extreme case analysis
• [ ] Visual inspection of predictions
• [ ] Domain expert review

Quick Reference Tables

Approximate WP by Situation

Typical WPA Values

Next Steps

After mastering win probability models, proceed to: - Chapter 22: Machine Learning Applications - Chapter 23: Network Analysis in Football - Chapter 27: Building a Complete Analytics System