Magic resistance

Magic Resistance (or MR) is a stat that all units have, including minions, monsters, and buildings. Increasing magic resistance reduces the magic damage the unit takes. Each champion begins with some magic resistance which may increase with level. You can gain additional magic resistance from abilities, items, and runes. Magic resistance stacks additively.

Damage reduction

 * Note: One can include the magic penetration in all the following ideas by enumerating it with a due amount of corresponding negative MR.

Magic resistance reduces the damage of incoming magic damage by a percentage. This percentage is determined by the formula: Damage Reductiontotal magic resistance(100total magic resistance). For example, a champion with 150 points of magic resistance would receive 60% reduced damage from magic damage. Incoming magic damage is multiplied by a factor based on the unit's magic resistance (same with armor):

$$\pagecolor{Black}\color{White}{\rm Damage\ multiplier}=\begin{cases} {100 \over 100+{\it MR}}, & {\rm if\ }{\it MR} \geq 0\\ 2 - {100 \over 100 - {\it MR}}, & {\rm otherwise} \end{cases}$$

Examples:
 * 25 magic resistance &rarr; magic damage (20% reduction).
 * 100 magic resistance &rarr; magic damage (50% reduction).
 * &minus;25 magic resistance &rarr; magic damage (20% increase).

Stacking magic resistance
Every point of magic resistance requires a unit to take 1% more of its maximum health in magic damage to be killed. This is called "effective health".


 * Example: A unit with 60 magic resistance has 160% of its maximum health in its effective health, so if the unit has 1000 maximum health, it will take 1600 magic damage to kill it.

What this means: by definition, magic resistance does not have diminishing returns, because each point increases the unit's effective health against magic damage by 1% of its current actual health value whether the unit has 10 magic resistance or 1000 magic resistance.

For a more detailed explanation, see this video.


 * Unlike health, increasing magic resistance makes healing and shielding more effective because it requires more raw damage from your enemies to remove the bonus health granted. This is called indirect scaling.

Magic resistance as scaling
These use the champion's magic resistance to increase the magnitude of the ability. It could involve total or bonus magic resistance. By building magic resistance items, you can receive more benefit and power from these abilities.

Champions

 * ,, and
 * ,, and

Ways to reduce magic resistance
See magic penetration. Note that magic penetration and magic resistance reduction are different.

List of champions' magic resistance

 * All melee champions (except for ) and, have a base magic resistance of.
 * All ranged champions (except ),, and most shapeshifter champions , have base magic resistance of.
 * is an exception with a magic resistance of . is also an exception with.

Optimal efficiency (theoretical)
''Note: Effective burst health, commonly referred to just as 'effective health', describes the amount of raw burst damage a champion can receive before dying in such a short time span that he remains unaffected by any form of health restoration (even if the actual considered damage is of sustained form). Unless champion's resists aren't reduced below zero, it will always be more than or equal to a champion's displayed HP in their health bar and it can be increased by buying items with extra health, armor and magic resistance. In this article, effective health will refer to the amount of raw 'magic damage' a champion can take.''

In almost all circumstances, champions will have a lot more HP than MR such that the following inequality will be true: Champion-HP > Champion-MR100.

If this inequality is true, a single point of MR will give more 'effective health' to that champion than a single point of HP.

If (HP < MR100), 1 point of HP will give more effective health than 1 of MR.

If (HPMR100), 1 point of HP will give exactly the same amount of effective health as 1 point of MR.

Because of this relationship, theoretically, the way to get the maximum amount of effective health from a finite combination of HP and MR would be to ensure that you have exactly 100 more HP than MR (this is true regardless of how much HP and MR you actually already have).


 * Example: Given a theoretical situation where you start off with 0 HP and 0 MR and are given an arbitrary sufficient number of stat points (x ≥ 100), each of which you can either use to increase your HP or MR by 1 point, the way to maximize your effective health is to add points to your HP until your HP(MR100)(0100)100, and then split the remaining stat points in half, spend half on your HP and half on your MR.

However, this is only theoretically true if we consider both HP and MR to be equally obtainable resources with simplified mechanism of skill point investment. In reality a player buys these stats for gold instead. As gold value of MR (derived from cost of basic magic resistance item) is currently (as of season six) times higher than gold value of HP (derived from cost of basic health item), we theoretically can maximize effective health represented by product of  HP(MR100) with gold as input variable by satisfying the following equation: HP(MR100). The graph and conclusions obtained by solving it are mentioned in the analogous section about armor.


 * Example: Given a theoretical situation where you start off with 0 HP and 0 MR and are given an arbitrary sufficient amount of gold (x ≥ ), which you can either use to increase your HP or MR, the way to maximize your effective health is to buy HP until your HP(MR100)(0100)675, and then split the remaining gold in half, spend half on your HP and half on your MR (as former is times cheaper than the latter, it would lead to buying  times more additional HP than MR and thus naturally reaching equality in the equation above).

Now we just formulated a simple rule of preserving equilibrium (or maximum effective health):

Once equilibrium state is reached, all we need to do to preserve it is to always distribute gold equally into all involved stats for the rest of the game.

... or in our case, always 50% gold into HP and 50% gold into MR.

Again this model is highly simplified and cannot be exactly applied in cases when we are buying any other item than, or  (for example if our decision-making process would involve  instead of , the above model would need to use equilibrium constant ). Even considering the purchase of different MR or HP items with differing gold efficiencies (quite natural expectation under real circumstances) makes use of single constant utterly impossible. Going even further, the continuous model simplifies a discrete character of real shopping, as you cannot really buy for, so with that much gold you opt to buy either a single  or a single , drastically changing the equilibrium constant to 6.

However, thankfully to almost linear item stats' gold efficiency a player can use weakened base equilibrium condition in a form: HP ≈ (MR100) safely enough to speed up decision-making. The important thing to remember is that there is no reason to hold to it too strictly.

Note: In case of armor only the basic constant  is slightly changed to .

This information is strongly theoretical and due to game limitations from champions' base stats, innate abilities and non-linearity of gold value of item stats (gold value of stats differs for different items or is even impossible to be objectively evaluated due to interference of unique item abilities), the real equilibrium function is too complicated to be any useful.

The complexity of this problem provides space for players' intuition to develop and demonstrate their itemization skills. If given sufficient amount of time, each player could perfectly analyze situation at any given moment when he exited the shop and tell what should he buy at that moment for available gold to maximize own effective health. The sheer impossibility of doing such thing in real time creates opportunity to develop the skill. Not only that but often choosing to maximize current effective health leads to suboptimal decision branches in the future. The summary on end game screen about type of fatal damage taken is a key part of this decision process as well.

Instead, broadly speaking, items which provide both HP and MR give a very high amount of effective health against magic damage compared to items which only provide HP or only provide MR. These items should be purchased when a player is seeking efficient ways to reduce the magic damage they take by a large amount. Furthermore, these items are among all available items the best ones to distribute their gold value equally among both HP and MR, thus working perfectly for rule of preserving equilibrium.

Trivia
Last updated: March 12, 2018, patch V8.5 


 * One of the biggest amount of magic resistance any champion other than a ​ with ​ Gathering Storm​, can obtain is  (which reduces magic damage by %), being a level 18.


 * MR
 * 1
 * 1 near 3 enemy champions
 * 1 fully stacked
 * A combination of 3 of the following:
 * 1
 * 1 near 3 enemy champions
 * 1 fully stacked
 * A combination of 3 of the following:


 * bonus MR:
 * Items = +  +  +  = 415 MR
 * Runes = +  = 128 MR
 * MR = 415 + 128 = 543 bonus MR
 * bonus = + 543 =  bonus MR


 * MR:
 * Base stats: MR
 * Items = +  +  +  = 415 MR
 * Runes = +  = 128 MR
 * Buffs = +  =  MR
 * MR Amplification = 1 +  +   =
 * MR = ( + 415 + 128 + ) + =  MR


 * Prior to Patch 3.10, a level 18 with 1, 5 , 3 points in , 3 points in , , , , , an allied  aura, a full page of Scaling Magic Resist runes, and  active gave a total of approximately 1004 magic resistance. Switching  for  and having an enemy  with the same setup use  on  and the allied  use  on the enemy , yielded a total of approximately 1297 magic resistance. This is the highest possible amount of magic resistance, which is  reduction.
 * If disconnected from the fountain and had the same setup, he will have 2108 magic resistance. This is a  reduction.
 * NOTE: This calculation does not include the mastery.

Magic resistance Magieresistenz Resistencia mágica Résistance magique Odporność na magię Магическое сопротивление 魔法抗性