Just currious about the scaling of the Crown of the Tyrant

Does it count it`s sockets or does it count every socket?
Does it multiply in a party where multiple people wear it?
Does the added damage and the -10% res to enemyes multiply if multiple people wear it?

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Does it count it`s sockets or does it count every socket?

Only its socket.

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Does it multiply in a party where multiple people wear it?

You cannot stack the same damage type among party members, but each member can be affected by 1 instance of each damage type if other players are doing different damage types. For example, my Tyrant gives added fire and yours gives added cold. We will both get added fire and added cold in a party. But if we both use a Tyrant with added fire, only 1 will be applied.

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Does the added damage and the -10% res to enemyes multiply if used with Mirror Arrow or Blink Arrow?

The added damage works the same on minions as it does on players. Your minions will have the same added damage mods you have, just like any aura (assuming they're in range of course).

The reduced resists never stacks, from players or minions.

Aggromagnet is correct. Basically you can ever only be affected by any Aura or AoE mechanic once unless specified otherwise. When applicable, the strongest available version. This goes for enemies and minions as well. Socket modifiers are always local except for the few cases that specify additional gear.