Damage Resistance

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

Any damage taken is reduced by a percentage based either completely or in part in this number, depending on the game. Damage Resistance can be increased by wearing armor, or by taking certain chems or perks, depending on the game.

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%.
 * The Prizefighter reputation title will increase normal DR by 5%.

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 damage against 0 damage resistance (no ratio): ~50 damage is done (~0% damage reduction).
 * 50 damage against 10 damage resistance (1/5 ratio): ~45 damage is done (~10% damage reduction).
 * 50 damage against 25 damage resistance (1/2 ratio): ~32 damage is done (~35% damage reduction).
 * 50 damage against 50 damage resistance (1/1 ratio): 25 damage is done (50% damage reduction).
 * 50 damage against 100 damage resistance (2/1 ratio): ~19 damage is done (~61% damage reduction).
 * 50 damage against 250 damage resistance (5/1 ratio): ~14 damage is done (~72% damage reduction).
 * 50 damage against 1000 damage resistance (20/1 ratio): ~8 damage is done (~83% damage reduction).

The upshot is that unlike previous games, it is very hard to get to the point where enemy damage is reduced to the range of ~20% or less. 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: 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.
 * Very easy: x2.0 for the player, x.5 from enemies
 * Easy: x1.5 for the player, x.75 from enemies
 * Normal: x1.0 for the player and from enemies
 * Hard: x0.75 for the player, x1.5 from enemies
 * Very hard: x0.5 for the player, x2.0 from enemies
 * Survival: x.75 for the player (boosted higher depending on adrenaline level), x4.0 from enemies as of 1.5 update (x0.5/2.0 for Survival difficulty that shipped with the game)

Note also that companions are unmodified. On all difficulty levels they do 1x damage and take 1x damage from enemies. This means that companions generally become a bit more effective at higher difficulty levels since they will generally be taking less damage and doing more damage than the player.

Note that regardless of where you hit an enemy/are hit by an enemy, the total damage resistance of all currently equipped pieces of armor is used for the formula.

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 Poison Damage
Poison deals damage each second, as a distinctly inflicted hit, typically for 10 seconds; for example, if the player has poison resistance 5 and suffers a poison 3 hit, then once per second for 10 seconds, they will suffer a 3 damage hit, resisted with resistance 5, which otherwise follows the rules for projectile damage.

For Bleeding Damage
Bleeding works like poison, except that there is no such thing as bleeding resistance.

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 Radiation Damage
Radiation damage has two subtypes. Almost all of it in the game is poisoning; each point of radiation poisoning from a radiation "hit" is resisted by radiation resistance (as per the energy damage resistance formula), and resolves as a .1% loss of maximum hp for each point that gets through. The only perk that raises radiation poisoning damage inflicted by a weapon is Nuclear Physicist, and radiation immunity makes a target ignore radiation poisoning entirely.

Non-poisoning radiation damage follows the normal rules for energy damage, including modification by perks; it ignores radiation immunity, and creatures with radiation immunity typically have no radiation resistance.

For Explosive Damage
Explosive modifies other damage types - for example, something can inflict explosive energy damage. Explosive damage uses the same formula as the underlying type, but is affected by additional perks and effects that interact with explosives. How the padded and dense armor mods work is currently unknown, but they would check to see if damage is explosive before applying or not.

Combined Damage Types
When something has multiple resistible damage types (ballistic, energy, radiation, and poison are the currently resistible types, with any of them capable of being explosive or not, and radiation can be poisoning or not - almost all radiation sources are poisoning), one "hit" resolves as a hit for each damage type applied. Likewise, when something deals both direct and area of effect damage, a target is hit by each one individually. For example, if one removes the Lorenzo's Artifact gun mod from the original gun and attaches it to an Irradiated Gamma Gun with an Electric signal carrier antennae, a charged shot from the weapon will inflict 5 distinct "hits": radiation poisoning, radiation damage, explosive radiation poisoning, explosive radiation damage, and explosive projectile damage.

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

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

Armor Piercing

 * $${\it Damage Increased With AP} = \frac

{\left[ \frac{\it Damage}{\it DamageResist \times 0.6}\right]^{0.366} \times 0.5} {\left[ \frac{\it Damage}{\it DamageResist}\right]^{0.366} \times 0.5} = 1.2056$$

Due to the way the damage formula works, the 40% armor piercing of some weapons mods actually result in a flat +20% effective damage done, on any foe, regardless of the weapon base damage or the DR of the target.

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