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With the Big Leagues perk, you can gain a suite of melee bonuses, like increased damage, a chance to disarm or cripple, and the ability to hit every enemy in front of you.”— Fallout 4 loading screen hints
Big Leagues is a 5-ranked perk. With the first rank, attacks with melee weapons do 20% more damage. The second rank increases the damage bonus to 40%, and allows melee attacks to sometimes disarm your opponent. The third rank increases the bonus damage to 60%, and increases the chance to disarm your opponent. The fourth rank increases the bonus damage to 80% and allows you to hit all targets in front of you. The fifth and final rank increases unarmed attacks to do 100% more damage, and gain a chance to cripple opponents, or grand slam their head clean off!