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Sounds like the big bad is a Complete Monster (YMMV) and the anti-hero is a proponent of Pay Evil unto Evil.
Anything as far as intensity and "worst atrocities", would just make it Pay Evil unto Evil as an Exaggerated Trope. See PlayingWith.Pay Evil Unto Evil and the examples under "exaggerated". Bob's actions are pretty harsh regardless of Emperor Evulz's original crime.
Since you are involving an audience reaction here, there is also a bit of Catharsis Factor (YMMV) if the audience strongly appreciates seeing the anti-hero really doing terrible things to the big bad.
Edited by KD
You nailed it. Thank you for the detailed explanation. I was surprised to find a lot of media using these same tropes recently. In a lot of these situations, Bob undergoes a training of sorts after having been subjected to Emperor Evulz's cruelty in order to acquire a certain set of skills. In all these stories, Bob's methods are extremely graphic and sadistic, but you are clearly supposed to empathize and root for Bob to be victorious and Emperor Evulz's to get his comeuppance in the most graphically cathartic way possible.
Gray-and-Black Morality or A Lighter Shade of Black. Also Deliberately Bad Example
Edited by molokai198
Is there a trope for this situation: when the big bad is SO bad, so irredeemable that no matter what extreme the hero - who is usually his main victim - goes to, you not only root for him but can't help but feel the atrocities he commits in retribution are justified?It Would be as if said big bad was tailor-made for the hero to be able to commit the worst atrocities and we would still call him a hero.