Damage modifier

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Damage modifiers refer to the buffs or debuffs applied to a unit that causes it to deal or receive increased or decreased damage. These are percentage values. They don't add extra flat damage but modify all damage normally done.

These buffs or debuffs are received by champion abilities, neutral buffs, on-hit effects from items and runes.

Damage dealt modifier

These are buffs that allow the champion that has them to deal increased or decreased magic or physical damage. The raw value from the attack or ability is increased or decreased by the modifier and then applied to the target. All damage modifiers stack multiplicatively.

Increasing damage dealt

Champion abilities

Runes

Flat bonus damage

While technically not damage modifiers, these abilities are included for completeness.

Champion abilities

Items

Doran's Shield (Unlike most sources of bonus damage, this is added after damage modifier calculations)
Doran's Ring (Unlike most sources of bonus damage, this is added after damage modifier calculations)

Decreasing damage dealt

Champion abilities

Summoner Spell

Exhaust

Damage received modifier

These are buffs or debuffs that allow the champion that have them to receive increased or decreased magic or physical damage from autoattacks, abilities and item's actives. The final value from the attack or ability is increased or decreased by the modifier and then directly applied to the target's health.

Armor and magic resistance damage reduction percentage count as one source of damage received modifier against physical and magical damage respectively. For example, a champion with 300 armor has 75% damage reduction from physical damage.

Damage reduction from armor and magic resistance and from any other sources stack multiplicatively.

Increasing damage received

Decreasing damage received

Champion abilities

Summoner Spells

Challenging Smite

Items

Ninja Tabi

Damage immunity

There are some instances in the game where a champion is immune to receiving any kind of damage (including true damage) for a few seconds.

Champions

Items

Flat Damage Reduction

Several champions have abilities that reduce damage by a flat amount. Some flat damage reductions are factored in after armor or magic resistance. This makes it significantly better the more resistances you have. In addition, periodic damage effects will have each tick of damage reduced. For example, Malzahar's Malefic Visions ticks 8 times, so 1 flat damage reduction would reduce its total damage by 8.

Flat damage reduction does not work against true damage.

Champions

Items

Guardian's Horn