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Theorem 2: Foul Up 3

Claim

When leading by 3 with fewer than 12 seconds left, intentionally fouling is better against average-to-good 3PT shooters and worse against poor shooters.

Results

Foul Up 3 Heatmap

Green cells show situations where fouling improved the historical win rate; red cells show where normal defense was better. Each cell value is the win-percentage gain (in percentage points) from fouling versus playing normal defense, based on NBA play-by-play data (2019--2024).

Conclusion

Foul the opponent when they are a competent 3PT shooting team (≥ 40%). Against poor 3PT teams, normal defense remains the safer choice.