Key Takeaways: Game Simulation

One-Page Reference

Monte Carlo Fundamentals

Core Idea: - Generate many random scenarios - Aggregate results to estimate probabilities - Explore full distribution of outcomes

How Many Simulations? | N | Standard Error | |---|----------------| | 1,000 | ±1.6% | | 10,000 | ±0.5% | | 100,000 | ±0.16% |

Score-Based Simulation

Simple Model:

Home ~ N(μ_home + HFA, σ)
Away ~ N(μ_away, σ)

Parameters: - σ (score std): ~10 points - HFA: ~2.5 points - Score correlation: 0.10-0.15

Output: Distribution of final scores and margins

Drive-By-Drive Simulation

Drive Outcomes (League Average): | Outcome | Probability | |---------|-------------| | Touchdown | 22% | | Field Goal | 15% | | Punt | 38% | | Turnover | 12% | | Other | 13% |

Process: 1. Simulate drive outcome 2. Update score 3. Switch possession 4. Repeat until game ends

What Simulation Provides

Beyond Point Estimates: - Full probability distributions - Confidence intervals - Scenario probabilities - Path-dependent outcomes

Key Questions Answered: - P(home wins by 10+)? - P(total > 45)? - P(overtime)? - Expected score distribution

Season Simulation

Uses: - Playoff probability - Division title odds - Draft position estimates - Schedule strength impact

Process: 1. Simulate each game 2. Calculate standings 3. Determine playoffs 4. Repeat many times 5. Aggregate probabilities

Live Win Probability

Factors: - Current score - Time remaining - Possession - Field position

Method: 1. Set current game state 2. Simulate game to completion 3. Count home wins 4. Repeat many times

Validation Checklist

• [ ] Score distributions match historical
• [ ] Margin distributions match historical
• [ ] Close game frequency realistic
• [ ] Blowout frequency realistic
• [ ] Win probabilities calibrated
• [ ] Overtime frequency correct

Common Score Patterns

Most Common NFL Scores: 17, 24, 20, 27, 14, 10, 21, 13

Score Correlation: - High-scoring games: both teams elevated - Low-scoring games: both teams depressed - Typical correlation: ~0.1-0.15

Simulation Limitations

Does NOT Provide: - Better point estimates than models - Certainty about outcomes - Immunity to bad assumptions

Watch For: - False precision (more sims ≠ better model) - Assumption sensitivity - Over-interpretation

Practical Applications

Quick Implementation

# Basic game simulation
def simulate_game(home_mean, away_mean, n_sims=10000):
    home_scores = np.random.normal(home_mean, 10, n_sims)
    away_scores = np.random.normal(away_mean, 10, n_sims)

    margins = home_scores - away_scores
    home_win_prob = np.mean(margins > 0)

    return {
        'home_win_prob': home_win_prob,
        'mean_margin': np.mean(margins),
        'std_margin': np.std(margins)
    }

Key Insight

Simulation reveals uncertainty. The goal isn't predicting exactly what will happen—it's mapping the landscape of what might happen and how likely each scenario is. Every probability comes with a distribution.