Does Anyone Actually Pick AP Mages Because they are Strong?

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That's always been why you pick a Mage. You force itemization so that your ADC can perform better late game. A team of full armor tanks will steam roll an adc and an AD team. By using mages, you allow burst potential to knock out enemy carries as well as force MR, which is typically one to two less armor items built which greatly changes the ADC's DPS.

But that shouldn't be the reason you pick a mage...you should pick a mage because they are a strong champion that can carry a game with proper positioning and protection similar to an ADC. At this point, most of the high AP items are just so expensive that the lethality and crit/as AD's are much more gold efficient.

Rephrase: You are correct that you force the enemy team into building MR which helps your adc but that should NOT be the reason to pick an AP champion.

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The only superior thing AD can do is tank shred because of

can easily out damage all AD and AD carries on their team as long as the other team isn't 3-4 tanks stacking adaptive helms and spirit visages.

Yes, but the champions who go full AP(not lyandries, rylais, etc.) are not very Gold Efficient. Also, unlike many ADC's, most mages are immobile and have no self-peel which reduces their damage output dramatically as they have to play extremely safe or look at a gray screen.