Balance

The math, step by step

Everything needed to repeat this test by hand or in code. The numbers of each weapon, ammunition, armor piece and enemy are the game's own and are on their pages in this wiki; below are the rules and the numbers this version used.

1. Who fights

Each tier is tested on its own: the gear of the tier against the enemies of the same tier. Faction weapons and the starting gear fight in tier 1, which is their level.

The character has the same mastery level L in every weapon and armor class, and Vitality at the same level: health = 60 + 10 × (L − 1).

The specialisation is not chosen: it is the highest level whose XP requirement fits in 25% of the XP that mastery level L requires, which is the share of mastery XP that also goes to the specialisation.

No medications on the belt, no buffs and no accessories.

2. One attack

T = a + b + A − H − R
D = 6 + M
Critical:  a = b = 10  or  T − D ≥ 10
Hit:  Critical  or  T > D

a, b

two ten-sided dice (1 to 10), rolled again for every shot

A

accuracy of the attacker: weapon + ammunition + mastery bonus (“+N hit chance”); for an enemy, the accuracy of its attack minus its hit penalty

H

hit penalty of the armor the attacker wears: the sum of the three pieces (enemies: 0)

R

range penalty: 0 from the minimum to the optimal range of the weapon, 4 beyond the optimal, 6 closer than the minimum; beyond the maximum range the weapon does not fire

M

defense modifier of the target: the sum of its three armor pieces plus the set bonus (enemies: 0)

Each of the 100 results of the two dice is equally likely, so every chance is a count. Example: with accuracy 0 against defense 6, 85 of the 100 results hit and 15 of those are critical.

Hit chance →

3. The damage of a hit

Damage = max(1, ⌊x × (1 + B) × K⌋ − AR)

x

an integer drawn with equal chance from min to max: the damage range of the weapon (melee) or of the ammunition (firearms), each end plus the mastery bonus (“+N min dmg”, “+N max dmg”)

B

damage bonus or penalty of the terrain for the weapon class (players only) plus the perks that give a percentage of damage; they add up

K

1.5 on a critical hit, otherwise 1

AR

armor rating of the target against the damage type of the attack; a perk can make a hit ignore it

⌊ ⌋

rounded down; a hit always takes at least 1

4. Armor

AR(t) = AR₁(t) + AR₂(t) + AR₃(t) + S(t) + G
M = M₁ + M₂ + M₃ + S(M)
H = H₁ + H₂ + H₃

ARᵢ(t)

what piece i resists of damage type t: its prime armor rating if t is its prime type, its weak rating if t is its weak type, its normal rating otherwise

S

set bonus, only when helmet, chest and legs are of the same armor class: the armor mastery adds, once, its “+N armor rating” to every damage type, its “+N prime resist armor rating” to the prime types of the pieces and its “+N defense modifier” to M

G

cover armor of the terrain, against every damage type (players only)

Mᵢ, Hᵢ

defense modifier and hit penalty of piece i

Armor perks come from the specialisation of the class worn on most pieces; with three different classes, the helmet's. Enemies use the prime, weak and normal armor ratings of their own page.

5. One fight

1. The fighters start 4 steps apart (the distance goes from 0 to 5). A coin decides who acts first in the first turn; from then on, who acts first alternates every turn.
2. At the start of each turn, whoever is bleeding loses health. Then each fighter takes one action.
3. The action, in this order: stunned, it is lost; with an empty clip, reload, which takes the whole action; farther than the optimal range of the weapon, one step closer; otherwise, attack with every shot the weapon fires per turn, one roll per shot, until the clip is empty or the target falls.
4. The fight ends when the health of one of them reaches 0. After 200 turns without that it is a draw, counted as not won.
5. Effects (stun, bleeding, fear) start on the turn after they are applied. A stun costs one action; bleeding from a perk takes 5% of the maximum health per turn until the end of the fight, since there is no bandage; fear lowers accuracy.
6. Perks do what their game text says, with the percentages read from the text. Where the text leaves room: “each turn” is each attack; bonuses in % of damage add to each other; a count of consecutive hits taken restarts when the attacker misses; a count of consecutive turns firing restarts when the fighter moves or reloads; damage that a perk or an enemy ability adds as a fixed amount enters x, before the multiplications.
7. Enemy abilities: Deep Cut and Carrion Shriek use the two numbers of the game file as amount and turns and come back after their cooldown; Metamorph copies your damage type and range; Paradox adds your armor to its damage (our reading of the game description).
8. Guesses where the game does not say: fear from perks takes 2 off accuracy for one turn; enemies have no natural defense.

6. Ammunition and terrain

Each weapon uses, against each enemy, the ammunition with the highest expected damage per turn among those it accepts: same caliber, of the tier being tested or below, or faction ammunition (tier 2 rounds only fit tier 2 and 3 weapons, tier 3 rounds only tier 3 weapons).

E = n × (P₁ × μ₁ + P₂ × μ₂)

n

shots the weapon fires per turn

P₁, P₂

chance of a normal hit and of a critical hit, with A = accuracy of the weapon plus the ammunition, no penalties, and D = 6

μ₁, μ₂

average of max(1, x − AR) and of max(1, 1.5 × x − AR) over the damage range of the ammunition, AR being the enemy's armor rating against that damage type

An enemy that appears in more than one terrain is fought in each of them, and the results are weighted by the number of map hexes of each terrain where it appears:

P = Σ (hₜ ÷ h) × Pₜ

hₜ, h

hexes of terrain t where the enemy appears on the official map, and all the hexes where it appears

Pₜ

chance of winning with the damage bonus and the cover of terrain t; enemies that are not on the map fight with no terrain effect

7. How many fights, and the margin of error

Two armor combinations are the same fight against an enemy when they give the same hit penalty, the same defense modifier, the same armor rating against that enemy's damage type and the same perks. Each distinct fight is simulated once and its result serves every combination that shares it: nothing is sampled or left out.

First every distinct fight is repeated 30 times. Then, for each weapon and enemy, the 5 best armor profiles and every full set of a single armor class are repeated 200 times, and the new result replaces the first.

p = v ÷ n
σ = √(p × (1 − p) ÷ n)

v, n

fights won and fights simulated; p is the chance of winning

σ

standard error of p: at most 9.1 points with 30 fights and 3.5 points with 200

Taking the best of several estimates leans slightly upward; that is one reason for the tolerance of 5 points used below.

Random numbers: Python's random.Random (Mersenne Twister), seeded for each distinct fight from the text “balance|weapon|helmet/chest/legs|enemy|damage bonus|cover|s” (“f” in the second round), with the names as in the addresses of this wiki, by h ← (h × 131 + character code) mod 2,147,483,647, starting at 0. The same data gives the same numbers again; another program should agree within the margin of error, not digit by digit.

8. From the fights to the tables

m(w, e) = max { p(w, c, e) : c }
t(e) = max { m(w, e) : w }

p(w, c, e)

chance of winning with weapon w and armor combination c against enemy e

m(w, e)

the weapon with its best armor: the highest p among the combinations of the second round

t(e)

the best any weapon does against enemy e

• Enemy with a real choice: t(e) above 5% and at least one weapon with m(w, e) more than 5 points below t(e). Only these enemies count for “among the best” and “far behind”.
• Among the best: m(w, e) ≥ t(e) − 5 points. Far behind: m(w, e) ≤ t(e) − 20 points. The only best: the weapon reaches t(e) and every other one is at least 5 points below.
• Wins on average, for a weapon: the average of m(w, e) over every enemy of the tier, with or without a choice.
• Armor class: for each weapon and enemy, the highest p among the combinations with three pieces of that class. “All weapons averaged” is the average of that over the weapons and then over the enemies; “with the best weapon” takes the best weapon against each enemy; “among the best” and “far behind” compare that best with t(e).
• Each piece on its own: the average of p in the first round over every weapon, every combination that has the piece and every enemy. “In a best loadout” counts the enemies that some weapon beats and against which a loadout within 5 points of t(e) uses the piece.
• Mixing classes beats the full set when the best combination with pieces of more than one class is more than 5 points above the best one with a single class.
• Summary: “among the best against at least n of N” lists what is among the best against 80% or more of the enemies with a choice; “almost never” means among the best against 10% or fewer and, for weapons, far behind against half or more.