Damage Resistance

Damage Resistance (DR) is a derived statistic in the SPECIAL character system.

Any damage taken is reduced by this percentage (that is to say, a piece of armor with 10 DR will reduce damage taken by 10%). Damage Resistance can be increased by wearing armor, or by taking certain perks.

Fallout, Fallout 2, Fallout Tactics
In the original Fallout games, DR (Damage Resistance) is one of three stats by which a character can reduce or avoid damage. The other stats are AC (Armor Class) and DT (Damage Threshold). DR occupies the "final step" for combat and simulates the effect of how armor can help diffuse the energy of a bullet and reduce its lethality (like real-life Kevlar armor).

More specifically, after AC is checked for a successful hit and after DT is checked to reduce the damage, if there is any incoming damage, DR applies:
 * $$Final = \text{max}\Bigg(1, Adjusted \times \frac{100-\text{min}(DR,\ 90)}{100}\Bigg)$$

While DT can completely negate damage done, DR cannot. As the equation suggests, DR is capped at 90%, and any damage that goes past DT has a minimum of 1.

Initial level: 0
 * The Damage Resistance Modifier of ammo can pre-emptively reduce the DR of a target.
 * The Toughness perk increases normal DR by 10% per rank.
 * The Dermal Impact Armor implants increase normal and explosive DR by 5%.
 * The Dermal Impact Assault Enhancement implants increase normal and explosive DR by 10%.
 * The Phoenix Armor Implants increase laser, fire, and plasma DR by 5%.
 * The Phoenix Assault Enhancement implants increase laser, fire, and plasma DR by 10%.

Fallout 3
The maximum Damage Resistance is 85%, regardless whether this value is obtained by armor, perks, drugs or a combination of these. This makes the effects of Nerd Rage! and Med-X somewhat less useful, as even without chems, the Lone Wanderer can permanently gain up to 94% Damage Resistance through equipment and perks (with The Pitt additional content). The T-51b power armor and the Enclave Hellfire power armor plus their respective helmets have a max DR of 60. Combined with the five permanent perks (34%) gives the maximum DR without drugs or having less than 20 percent of your hitpoints. Without the additional content you would still be able to obtain a maximum DR of 91% with power armor. Note that these numbers assume a fully repaired suit of armor, which, when the T-51b armor is actually being worn, is only possible if the player boosts Crazy Wolfgang's repair skill to 100 by reverse pickpocketing workman's coveralls from Point Lookout onto him because there is only one copy of this armor and it can only be repaired by non-player characters. Enclave Hellfire armor can be found, but it is rare (only obtainable with Broken Steel loaded). For players who prefer non-power armor, you can still obtain a permanent DR of 82% (79% without The Pitt add-on) with ranger battle armor, ghoul mask, the ranger battle helmet and the perks listed below.

All add-on Combinations

 * Permanent DR of 73% (temporary of 83%), +5 melee weapons and 148 Action Points with the permanent perks below and Superior Defender, Action Boy/Action Girl, tribal power armor, Ledoux's hockey mask, Almost Perfect and waiting to collect all S.P.E.C.I.A.L. Vault Boy bobbleheads until level 30 has been achieved. This produces perfect SPECIAL stats with the exception of Agility which will be 9 due to the power armor effect. The tribal armor can be repaired with common T-45d power armor while the hockey mask is immune to item damage and can be repaired (if found damaged) with regular hockey masks.

The Pitt Combinations

 * Permanent DR of 73%, +1 Strength, +1 Luck, -1 Agility, +5 melee weapons and +65 AP with the permanent perks below and Action Boy/Action Girl, tribal power armor and Ledoux's hockey mask. The tribal armor can be repaired with common T-45d power armor while the hockey mask is immune to item damage and can be repaired (if found damaged) with regular hockey masks.
 * Replace the hockey mask above with Poplar's hood which has 2 lower DR than the hockey mask but can be worn with the ghoul mask producing 5 DR rather than 4. This combination repairs the Agility damage (+2 small guns, +2 sneak, +2 Action Points), adds +10 sneak, permanent 74% DR, +40 Action Points and feral ghouls will not become hostile. It is a possibility for those who have not yet found the hockey mask, or missed acquiring it altogether. This option, however, excludes obtaining the Barkskin perk which offers 5 DR itself.

Permanently increasing Damage Resistance

 * The Cyborg perk gives +10% damage resistance.
 * The Toughness perk gives +10% damage resistance.
 * The Barkskin perk gives +5% damage resistance.
 * The Survival Expert perk gives up to +6% damage resistance.
 * The Superior Defender perk gives +10% damage resistance when standing still.
 * The Pitt Fighter perk gives +3% damage resistance.

Temporarily increasing Damage Resistance

 * The Nerd Rage! perk gives +50% Damage resistance, but only if your Health is below 20%.
 * The Med-X drug gives +25% Damage resistance, and it is possible to double up the effect by wearing the prototype medic power armor and also manually administering the drug.

Fallout: New Vegas
In Fallout: New Vegas, DR is mostly replaced by DT (Damage Threshold). Med-X and the new items Slasher, battle brew, and Rebreather still provide bonuses to DR, which functions identically to the DR from Fallout 3. If all 3 items are taken together, they provide the maximum 85% damage resistance allowed by the game engine.

Damage Resistance is applied before damage threshold, contrary to the original Fallout games. So, for a character with 30 DR and 20 DT (i.e. a NCR Veteran Ranger), an attack that deals 80 damage is first reduced by 30% (leaving 56 damage), then the damage threshold is subtracted from that number, leaving a final damage of 36. As a result of this, a high damage resistance has a very large effect on a character's ability to withstand damage. For full damage formula see Fallout: New Vegas combat.

The only pieces of equipment in the game that raise DR instead of DT are: In addition, DR can be conferred via the console by entering player.forceAV DamageResist xx where xx is between 0 and 85. Values beyond 85 are ignored by the game; 85 is the maximum DR, a likely holdover in the game engine from Fallout 3.
 * Rebreather from the quest Volare!
 * Vault 11 jumpsuits
 * the normally non-playable trenchcoat found on the investigator during the quest Beyond the Beef.
 * Scientist outfits
 * Regulator duster

A rare few non-player character characters have perks which grant them DR in addition to the DT from their armor. These include:
 * All companions if using Ferocious Loyalty.
 * Sunny Smiles (10% DR)
 * Ranger Stella (10% DR)
 * Legion vexillarius (10% DR)
 * Praetorian guards (10% DR)
 * Lucius (10% DR)
 * Certain Legionary assassins (10% DR)
 * Ulysses (10% DR)
 * Blister (10% DR)
 * Beast (10% DR)
 * Bonesaw (10% DR)
 * Blade (10% DR)
 * Colonel Royez (10% DR)
 * Gaius Magnus (10% DR)
 * Mean Sonofabitch (20% DR)
 * Motor-Runner (20% DR)
 * Nightkin (30% DR)
 * Dog (but not God) (30% DR)
 * Certain NCR Rangers (30% DR)
 * NCR Veteran Rangers (30% DR)

Fallout 4
Damage resistance returns in Fallout 4. However, the stat is no longer a 1-1 relationship with actual % damage reduced, as its value can be in excess of 1000 on certain models of power armor.

Instead, the amount of damage reduction the player character gets out of the damage resistance is actually based on how much potential weapon damage is coming in; unlike previous games, net damage reduction is based on a ratio between the potential weapon damage done and the damage resistance, with diminishing returns for high damage resistance.

The amount of damage reduced by damage resistance climbs very quickly until damage resistance is about half of the potential weapon damage, after which diminishing returns means you get less and less damage reduction per point of damage resistance. Notably, if damage resistance is exactly equal to the potential weapon damage done, then exactly half of the damage is negated. In other words, if you have 50 ballistic damage resistance and are shot by an enemy doing 50 ballistic damage, then you'll receive 25 damage.

Note that the most important factor is the ratio between damage resistance and potential weapon damage.
 * 50 potential weapon damage against 0 damage resistance (no ratio): ~50 damage is done (0% damage reduction).
 * 60 potential weapon damage against 10 damage resistance (1/6 ratio): ~57 damage is done (~5% damage reduction).
 * 20 potential weapon damage against 10 damage resistance (1/2 ratio): ~13 damage is done (~35% damage reduction).
 * 20 potential weapon damage against 20 damage resistance (1/1 ratio): 10 damage is done (50% damage reduction).
 * 30 potential weapon damage against 60 damage resistance (2/1 ratio): ~10 damage is done (~61% damage reduction).
 * 25 potential weapon damage against 500 damage resistance (20/1 ratio, easily possible with power armor):~4 damage is done (~83% damage reduction).

The upshot is that unlike previous games, it is very hard to get to the point where e.g. enemy damage is reduced to the range ~20%. In addition, because the ratio between damage resistance and weapon damage is so important and because damage reduction grows very quickly at first for low values of the ratio, a higher damage resistance pays off mostly because it gives you better damage reduction against more-damaging attacks, not necessarily because it gives you better damage reduction against the same, weaker attack as before.

In other words, going from 20 to 40 damage resistance will only further reduce the damage done from a 10-damage shot by about ~12% (~61% to ~69% damage reduction), but it will dramatically boost your damage reduction against a 100-damage grenade by about 23% (~10% to ~30% damage reduction). However, significant mitigation against such high damage is harder to achieve, since it is harder to get enough damage resistance to generate an appreciably high resistance-to-damage ratio. In otherwords, getting a 5-1 ratio is easy against a 20 damage shot, but very difficult (essentially limited to power armor) against a 100 damage grenade.

Damage Reduction formula
All final values below are adjusted by difficulty (for a player attacking an enemy): Note that this is applied *after* the ratio and exponent are calculated, so damage resistance is applied on the base potential weapon damage, before difficulty-based multipliers are included in. This means that damage resistance is the same level of effectiveness (or ineffectiveness) on every difficulty level.
 * x2.0 Very easy
 * x1.5 Easy
 * x1.0 Normal
 * x0.75 Hard
 * x0.5 Very hard
 * x0.5 Survival

The basic damage reduction formula is close to (99.95% accurate):
 * $${\it DamageCoeff} = \text{Min}\left(0.99, \left[ \frac{\it Damage}{\it DamageResist}\right]^{0.366} \times 0.5\right)$$


 * Note that this is a coefficient (multiplier) for the net damage done not the actual damage reduction; so lower numbers are better (more damage is negated).
 * If you want the actual % damage reduction simply do $${\it DamageReduction} = 1 - {\it DamageCoeff}$$

There are variations on this DamageCoeff calculation based on weapon used:
 * Projectile and close combat weapons use PaperDamage (essentially, the number you see in the PIP Boy), RangeMultiplier, and PowerAttackMultiplier for the damage part.
 * Energy weapons use WeaponBaseDamage and RangeMultiplier for the damage part (essentially, they are more vulnerable to their respective damage resistance).


 * $${\it PaperDamage} = {\it WeaponBaseDamage} \times {\it Perk}_1 \times {\it Perk}_2 \times \dots \times {\it RangeMulti}$$


 * WeaponBaseDamage is affected by weapon mods and is e.g. doubled with a 2x charged Laser musket
 * RangeMulti is 1.0 at 100% WeaponRange or closer, 0.5 at 200% of WeaponRange or farther, and scales from 1.0 to 0.5 as range increase from 100% to 200% of WeaponRange.

For Projectile Damage

 * $${\it DamageCoeff} = \text{Min}\left(0.99, \left[ \frac{\it PaperDamage}{\it DamageResist\times RiflemanMulti}\right]^{0.366} \times 0.5\right)$$


 * A Hunting Rifle with 38 BaseDamage and 45.6 PaperDamage (PipBoy Value, from 1 rank of Rifleman) does 22.04 damage vs 50 DamageResist at point blank range.
 * $$\left(\frac{45.6}{50}\right)^{0.366} \times \frac{45.6}{2}$$
 * $$\frac{{({1.2} \times {38})}^{1.366}}{{2} \times {50}^{0.366}}$$
 * The same Hunting Rifle with 5 ranks of Rifleman, which reduces target damage resistance to 70%, has PaperDamage 76 and does 50.47 damage at the same range against the same target.
 * $$\frac{{({2} \times {38})}^{1.366}}{{2} \times ({{50} \times {0.7}})^{0.366}}$$

For Energy Damage

 * $${\it DamageCoeff} = \text{Min}\left(0.99, \left[ \frac{\it WeaponBaseDamage\times RangeMulti}{\it DamageResist\times RiflemanMulti}\right]^{0.366} \times 0.5\right)$$


 * in version 1.1.30 RiflemanMulti not used in energy resistance reduction (bugreport filed)
 * A Laser Rifle with 38 BaseDamage and 45.6 PaperDamage from 1 rank of Rifleman does 20.62 damage vs 50 EnergyResist at point blank range.
 * (38/50)0.366 × 0.5 × 45.6
 * $$\frac{{1.2} \times {38}^{1.366}}{{2} \times {50}^{0.366}}$$
 * The same Laser Rifle with 5 ranks of Rifleman, which reduces target damage resistance to 70%, has PaperDamage 76 and does 39.16 damage at the same range against the same target.
 * $$\frac{{2} \times {38}^{1.366}}{{2} \times ({{50} \times {0.7}})^{0.366}}$$

For Close Combat Damage

 * $${\it DamageCoeff} = \text{Min}\left(0.99, \left[ \frac{\it PaperDamage \times PowerAttackMulti}{\it DamageResist}\right]^{0.366} \times 0.5\right)$$


 * PowerAttackMulti = 1.5

Final Damage

 * $$\it FinalDamage = PaperDamage \times DamageCoeff \times HeadshotMulti \times SneakAttackMulti$$

Damage Reduction formula for VATS Critical Attack
CriticalDamage = [ PaperDamage x CloseCombatMulti ] + [ WeaponBaseDamage x CriticalMulti ]
 * CloseCombatMulti = 1.5 for most melee/unarmed weapons
 * CriticalMulti = 1 + BobbleheadBonus(0.25) + MagazineBonus(0.05 to 0.5) + WeaponModBonus(2.0)

Projectile / CloseCombat

 * $$FinalDamage = \text{Min}\Bigg(0.99, \Bigg[ \frac{CriticalDamage}{DamageResist}\Bigg]^{0.366} \times 0.5\Bigg) \times CriticalDamage \times HeadShotMulti \times SneakMulti$$

Energy

 * $$FinalDamage = \text{Min}\Bigg(0.99, \Bigg[ \frac{WeaponBaseDamage}{DamageResist}\Bigg]^{0.366} \times 0.5\Bigg) \times CriticalDamage \times HeadShotMulti \times SneakMulti$$

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