Lethal Company v81時点のQuotaの計算に関する仕様の解析メモです。
- Build ID:
22825947 - Manifest ID:
6423525044216269478
Quotaの増加量のランダム性
Quotaの増加量のランダム性は、TimeOfDay.quotaVariables.randomizerCurveで定義される、右肩上がりのランダマイザ曲線によってもたらされます。
このランダマイザ曲線は、以下の制御点を持つエルミート曲線です。
x |
y |
in_tangent |
out_tangent |
|---|---|---|---|
| 0 | -0.50302887 | 7.455404 | 7.455404 |
| 0.11723505 | -0.13017726 | 0.5548811 | 0.5548811 |
| 0.88036245 | 0.15344214 | 0.5221589 | 0.5221589 |
| 1 | 0.5030365 | 7.0514693 | 7.0514693 |
AssetRipperで抽出したUnityアセットから表の値を取得するには、対象のSampleSceneRelay.unityを指定して以下を実行します。
asset='ExportedProject/Assets/Scenes/SampleSceneRelay.unity'
yq -r '.. | select(tag == "!!map" and has("quotaVariables")) | .quotaVariables.randomizerCurve.m_Curve[] | [.time, .value, .inSlope, .outSlope] | @tsv' "$asset"
曲線計算には、Unity 2022.3.62f1のAnimationCurveが使われています。
preWrapModeおよびpostWrapModeは、ClampForeverに設定されています。
これにより、[0, 1]の外側の評価値は、端点の値になります。
青点はAnimationCurveのキーフレーム（制御点）です。
赤点は、Luck補正で入力分布の境界になる点（x=0.01800, x=0.68875）と、曲線がy=0を横切る点を示しています。
ランダマイザ曲線の描画スクリプト（Python）
v72のグラフでは、サンプリング点が[0, 1]の範囲を超えるケースがあったため、グラフの右端点で外挿していましたが、v81ではサンプリング前にサンプリング点が[0, 1]の範囲でクランプされるようになったため、それにならってグラフの端点も[0, 1]の範囲に変更しました。
Details
実行例
uv run --script --locked quota_randomizer_curve_v81.py
Pythonスクリプト quota_randomizer_curve_v81.py
# /// script
# requires-python = ">=3.14"
# dependencies = [
# "matplotlib>=3,<4",
# "numpy>=2,<3",
# ]
# [tool.uv]
# exclude-newer = "P7D"
# ///
from dataclasses import dataclass
from typing import List
import bisect
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.ticker import MultipleLocator
# Unity AnimationCurve keyframes
# preWrapMode/postWrapMode: ClampForever
KEYFRAMES_RAW = [
{"time": 0.0, "value": -0.50302887, "inTangent": 7.455404, "outTangent": 7.455404},
{"time": 0.11723505, "value": -0.13017726, "inTangent": 0.5548811, "outTangent": 0.5548811},
{"time": 0.88036245, "value": 0.15344214, "inTangent": 0.5221589, "outTangent": 0.5221589},
{"time": 1.0, "value": 0.5030365, "inTangent": 7.0514693, "outTangent": 7.0514693},
]
@dataclass
class Key:
time: float
value: float
inTangent: float
outTangent: float
# Unity-style Hermite evaluator (non-weighted)
class Curve:
def __init__(self, keys: List[Key]):
self.keys = sorted(keys, key=lambda k: k.time)
def evaluate(self, t: float) -> float:
# ClampForever: range outside [0, 1] returns endpoint values.
if t <= self.keys[0].time:
return self.keys[0].value
if t >= self.keys[-1].time:
return self.keys[-1].value
times = [k.time for k in self.keys]
i = bisect.bisect_right(times, t) - 1
k0 = self.keys[i]
k1 = self.keys[i + 1]
dx = k1.time - k0.time
u = (t - k0.time) / dx
h00 = 2 * u**3 - 3 * u**2 + 1
h10 = u**3 - 2 * u**2 + u
h01 = -2 * u**3 + 3 * u**2
h11 = u**3 - u**2
return h00 * k0.value + h10 * dx * k0.outTangent + h01 * k1.value + h11 * dx * k1.inTangent
def main():
curve = Curve([Key(**k) for k in KEYFRAMES_RAW])
# In v81, both boundary points and highlighted ranges stay within [0, 1], so no extrapolation is needed.
ts = np.linspace(0.0, 1.0, 500)
ys = np.array([curve.evaluate(t) for t in ts])
plt.figure(figsize=(16, 9), dpi=300, facecolor="white")
plt.plot(ts, ys, label="randomizerCurve", color="#0072B2", linewidth=2.4)
plt.scatter(
[k["time"] for k in KEYFRAMES_RAW],
[k["value"] for k in KEYFRAMES_RAW],
label="Keyframes",
color="#0072B2",
zorder=3,
)
for k in KEYFRAMES_RAW:
plt.annotate(
f"({k['time']:.4f}, {k['value']:.4f})",
xy=(k["time"], k["value"]),
xytext=(4, -7),
textcoords="offset points",
fontsize=9,
color="#0072B2",
ha="left",
va="top",
bbox={"boxstyle": "round,pad=0.18", "fc": "white", "ec": "none", "alpha": 0.85},
)
# Representative v81 luck values: -0.012, 0, 0.2075.
range_specs = [
(0.0, "Default Luck (0.0)", "#56B4E9"),
(-0.012, "Lowest Luck (-0.012)", "#E69F00"),
(0.2075, "Highest Luck (0.2075)", "#009E73"),
]
y_min, y_max = plt.gca().get_ylim()
y_span = y_max - y_min
arrow_base = y_max - y_span * 0.10
arrow_step = y_span * 0.08
for idx, (luck, label, color) in enumerate(range_specs):
x_min = max(0.0, min(1.0, 0.0 - luck * 1.5))
x_max = max(0.0, min(1.0, 1.0 - luck * 1.5))
y_arrow = arrow_base - idx * arrow_step
plt.axvline(x_min, color=color, linewidth=1.8, alpha=0.95, zorder=1)
plt.axvline(x_max, color=color, linewidth=1.8, alpha=0.95, zorder=1)
plt.annotate(
"",
xy=(x_max, y_arrow),
xytext=(x_min, y_arrow),
arrowprops={"arrowstyle": "<->", "color": color, "lw": 2.2},
zorder=2,
)
plt.annotate(
f"{label}: [{x_min:.5f}, {x_max:.5f}]",
xy=((x_min + x_max) / 2, y_arrow),
xytext=(0, 8),
textcoords="offset points",
color=color,
fontsize=10,
ha="center",
va="bottom",
bbox={"boxstyle": "round,pad=0.18", "fc": "white", "ec": "none", "alpha": 0.9},
)
# Mark boundary points where luck adjustment changes the effective input distribution.
special_xs = [0.018, 0.68875]
for x in special_xs:
y = curve.evaluate(x)
plt.plot(x, y, "ro", zorder=4)
plt.annotate(
f"({x:.5f}, {y:.4f})",
xy=(x, y),
xytext=(-4, 6),
textcoords="offset points",
color="red",
fontsize=10,
ha="right",
va="bottom",
bbox={"boxstyle": "round,pad=0.15", "fc": "white", "ec": "none", "alpha": 0.75},
)
# Also mark where the curve crosses y=0 as a reference point.
zero_crossings: list[float] = []
for i in range(len(ts) - 1):
if ys[i] * ys[i + 1] < 0:
t0, t1 = ts[i], ts[i + 1]
y0, y1 = ys[i], ys[i + 1]
t_zero = t0 - y0 * (t1 - t0) / (y1 - y0)
zero_crossings.append(t_zero)
for x in zero_crossings:
plt.plot(x, 0.0, "ro", zorder=4)
plt.annotate(
f"({x:.5f}, 0.0000)",
xy=(x, 0.0),
xytext=(-4, 6),
textcoords="offset points",
color="red",
fontsize=10,
ha="right",
va="bottom",
bbox={"boxstyle": "round,pad=0.15", "fc": "white", "ec": "none", "alpha": 0.75},
)
plt.xlabel("Input")
plt.ylabel("Output")
plt.legend(loc="lower right")
plt.title("Quota Randomizer Curve v81 (Manifest ID: 6423525044216269478)")
plt.grid(True, which="both", axis="both")
plt.gca().xaxis.set_major_locator(MultipleLocator(0.05))
plt.gca().yaxis.set_major_locator(MultipleLocator(0.1))
plt.savefig("quota_randomizer_curve_v81.png")
if __name__ == "__main__":
main()
lockファイルquota_randomizer_curve_v81.py.lock
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revision = 3
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exclude-newer-span = "P7D"
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LuckとQuotaの関係
Luckを上げるほどQuotaは上がりにくく（下振れしやすく）なり、Luckを下げるほどQuotaは上がりやすく（上振れしやすく）なります。
Luckの合計値はTimeOfDay.CalculateLuckValue()関数で計算されます。この関数は、Quotaを設定するTimeOfDay.SetNewProfitQuota()関数で使用されています。
初めてQuotaを満たしたときは、Luck = 0が使用されます。2回目以降にQuotaを満たしたときは、前回Quotaを満たしたときに設置されていた家具によるLuckの合計値が使用されます。
例えば、Quota 1の達成時に設置されていた家具のLuckの合計値が、Quota 3の上振れ・下振れに影響します。そして、Quota 2の達成時に設置されていた家具のLuckの合計値が、Quota 4の上振れ・下振れに影響します。Quota 2の上振れ・下振れは、常にLuck = 0で計算されます。
v81では、ランダマイザ曲線の入力値は以下で計算されます。
$$ x = \operatorname{clamp}(u - 1.5L, 0, 1) $$
ここで、(u)はRandom.Range(0f, 1f)、(L)はLuck値です。
Luck = 0の場合、入力値は[0, 1]、出力値は[-0.50302887, 0.5030365]、期待値は0.012733341965388603になります。
Luckの最小値-0.012の場合、入力値は[0.018, 1]です。u >= 0.982の範囲ではClampによりx = 1になります。出力値は[-0.3845052318032696, 0.5030365]、期待値は0.02972983489597595になります。
Luckの最大値0.2075の場合、入力値は[0, 0.68875]です。u <= 0.31125の範囲ではClampによりx = 0になります。出力値は[-0.50302887, 0.07265585366115436]、期待値は-0.1964269360444696になります。
Quotaの計算式
\(t\) 回目のQuotaの増加量 \(\Delta q_t\) は、以下の式で表されます。\(t\) はQuotaを満たした回数、\(r\) はランダマイザ曲線の出力です。
$$ \Delta q_t = 100 (1 + \frac{t^2}{16}) (1 + r) $$
\(t\) 回目のQuota値 \(q_t\) は、以下の式で表されます。増加量はintキャストで小数点以下が切り捨てられます。
$$ q_0 = 130 $$
$$ q_t = q_{t-1} + \lfloor \Delta q_t \rfloor \quad (t \geq 1, t \in \mathbb{Z}) $$
最終Quotaの最小値は、Quota 1の達成時にLuckがマイナスの家具をすべて設置していて、その後のQuota達成時にも設置し続けていた場合に達成されます。
最終Quotaの最大値は、Quota 1の達成時にLuckがプラスの家具をすべて設置していて、その後のQuota達成時にも設置し続けていた場合に達成されます。 ただし、運の問題を除いても、これは通常のゲームプレイでは達成不可能です。1 Quotaで購入可能な家具は3種類のみ（常設の備品を除く）であること、家具のラインナップはQuota達成時に更新されること、すべての家具の購入に必要なクレジットをQuota 1で得ることは非常に困難または不可能であること、が理由です。 そのため最大値は、Luckシステムの効果を説明するための、理論上の値として扱ってください。
Luckが初期値の場合のQuota値
Luckが初期値（0）の場合のQuota値は、以下の通りです。
| 回数 t | 最小値 (r=-0.50302887) | 最大値 (r=0.5030365) | 期待値 (r=0.012733341965388603) |
|---|---|---|---|
| 0 | 130 | 130 | 130 |
| 1 | 182 | 289 | 237 |
| 2 | 244 | 476 | 363 |
| 3 | 321 | 710 | 521 |
| 4 | 420 | 1010 | 723 |
| 5 | 547 | 1395 | 982 |
| 6 | 708 | 1883 | 1311 |
| 7 | 909 | 2493 | 1722 |
| 8 | 1157 | 3244 | 2228 |
| 9 | 1458 | 4155 | 2841 |
| 10 | 1818 | 5244 | 3575 |
| 11 | 2243 | 6530 | 4442 |
| 12 | 2739 | 8033 | 5454 |
| 13 | 3313 | 9770 | 6624 |
| 14 | 3971 | 11761 | 7965 |
| 15 | 4719 | 14024 | 9490 |
| 16 | 5563 | 16579 | 11211 |
| 17 | 6510 | 19444 | 13141 |
| 18 | 7566 | 22637 | 15293 |
| 19 | 8737 | 26178 | 17679 |
| 20 | 10029 | 30085 | 20312 |
Luckが最小値の場合のQuota値
Luckが最小値（-0.012）の場合のQuota値は、以下の通りです。
t = 1については、Luck = 0として計算しています。t >= 2では、Luck = 最小値（-0.012）として計算しています。
| 回数 t | 最小値 (r=-0.3845052318032696) | 最大値 (r=0.5030365) | 期待値 (r=0.02972983489597595) |
|---|---|---|---|
| 0 | 130 | 130 | 130 |
| 1 | 182 | 289 | 237 |
| 2 | 258 | 476 | 365 |
| 3 | 354 | 710 | 525 |
| 4 | 477 | 1010 | 730 |
| 5 | 634 | 1395 | 993 |
| 6 | 834 | 1883 | 1327 |
| 7 | 1084 | 2493 | 1745 |
| 8 | 1391 | 3244 | 2259 |
| 9 | 1764 | 4155 | 2883 |
| 10 | 2210 | 5244 | 3629 |
| 11 | 2737 | 6530 | 4510 |
| 12 | 3352 | 8033 | 5539 |
| 13 | 4063 | 9770 | 6729 |
| 14 | 4878 | 11761 | 8093 |
| 15 | 5805 | 14024 | 9644 |
| 16 | 6851 | 16579 | 11394 |
| 17 | 8024 | 19444 | 13356 |
| 18 | 9331 | 22637 | 15544 |
| 19 | 10781 | 26178 | 17970 |
| 20 | 12381 | 30085 | 20647 |
Luckが最大値の場合のQuota値
Luckが最大値（0.2075）の場合のQuota値は、以下の通りです。
t = 1については、Luck = 0として計算しています。t >= 2では、Luck = 最大値（0.2075）として計算しています。
| 回数 t | 最小値 (r=-0.50302887) | 最大値 (r=0.07265585366115436) | 期待値 (r=-0.1964269360444696) |
|---|---|---|---|
| 0 | 130 | 130 | 130 |
| 1 | 182 | 289 | 237 |
| 2 | 244 | 423 | 337 |
| 3 | 321 | 590 | 462 |
| 4 | 420 | 804 | 622 |
| 5 | 547 | 1078 | 827 |
| 6 | 708 | 1426 | 1088 |
| 7 | 909 | 1861 | 1414 |
| 8 | 1157 | 2397 | 1815 |
| 9 | 1458 | 3047 | 2302 |
| 10 | 1818 | 3824 | 2884 |
| 11 | 2243 | 4742 | 3572 |
| 12 | 2739 | 5814 | 4375 |
| 13 | 3313 | 7054 | 5304 |
| 14 | 3971 | 8475 | 6368 |
| 15 | 4719 | 10090 | 7578 |
| 16 | 5563 | 11913 | 8944 |
| 17 | 6510 | 13957 | 10475 |
| 18 | 7566 | 16236 | 12182 |
| 19 | 8736 | 18763 | 14075 |
| 20 | 10028 | 21551 | 16164 |
Quota値の計算スクリプト（Python）
Details
実行例
uv run --script --locked quota_table_v81.py
Pythonコード quota_table_v81.py
# /// script
# requires-python = ">=3.14"
# dependencies = []
# [tool.uv]
# exclude-newer = "P7D"
# ///
import bisect
import math
KEYS = [
(0.0, -0.50302887, 7.455404, 7.455404),
(0.11723505, -0.13017726, 0.5548811, 0.5548811),
(0.88036245, 0.15344214, 0.5221589, 0.5221589),
(1.0, 0.5030365, 7.0514693, 7.0514693),
]
def curve(t: float) -> float:
if t <= KEYS[0][0]:
return KEYS[0][1]
if t >= KEYS[-1][0]:
return KEYS[-1][1]
times = [k[0] for k in KEYS]
i = bisect.bisect_right(times, t) - 1
t0, y0, _, out0 = KEYS[i]
t1, y1, in1, _ = KEYS[i + 1]
dx = t1 - t0
u = (t - t0) / dx
h00 = 2 * u**3 - 3 * u**2 + 1
h10 = u**3 - 2 * u**2 + u
h01 = -2 * u**3 + 3 * u**2
h11 = u**3 - u**2
return h00 * y0 + h10 * dx * out0 + h01 * y1 + h11 * dx * in1
def simpson(f, a: float, b: float, n: int = 200000) -> float:
if a == b:
return 0.0
if n % 2:
n += 1
h = (b - a) / n
s = f(a) + f(b)
for i in range(1, n):
s += (4 if i % 2 else 2) * f(a + i * h)
return s * h / 3
def randomizer_stats(luck: float) -> tuple[float, float, float]:
shift = 1.5 * luck
if shift >= 1:
r = curve(0)
return r, r, r
if shift <= -1:
r = curve(1)
return r, r, r
if shift >= 0:
hi = 1 - shift
expected = shift * curve(0) + simpson(curve, 0, hi)
return curve(0), max(curve(0), curve(hi)), expected
lo = -shift
expected = simpson(curve, lo, 1) + lo * curve(1)
return min(curve(lo), curve(1)), curve(1), expected
def print_table(luck: float, count: int = 20) -> None:
stats_zero = randomizer_stats(0)
stats = randomizer_stats(luck)
q_min = q_max = q_exp = 130
print(f"| 回数 t | 最小値 (r={stats[0]:.4f}) | 最大値 (r={stats[1]:.4f}) | 期待値 (r={stats[2]:.4f}) |")
print("|--------|-------------------:|-------------------:|-------------------:|")
print(f"| 0 | {q_min} | {q_max} | {q_exp} |")
for t in range(1, count + 1):
r_min, r_max, r_exp = stats if t > 1 else stats_zero
base = 100 * (1 + t**2 / 16)
q_min += math.floor(base * (1 + r_min))
q_max += math.floor(base * (1 + r_max))
q_exp += math.floor(base * (1 + r_exp))
print(f"| {t} | {q_min} | {q_max} | {q_exp} |")
def main():
for luck in [0, -0.012, 0.2075]:
print(f"Luck = {luck}")
print_table(luck)
print()
if __name__ == "__main__":
main()
lockファイル quota_table_v81.py.lock
version = 1
revision = 3
requires-python = ">=3.14"
[options]
exclude-newer = "0001-01-01T00:00:00Z" # This has no effect and is included for backwards compatibility when using relative exclude-newer values.
exclude-newer-span = "P7D"
v72からv81への主な差分
前回の記事 の読者、またはすでにv72のQuotaシステムに関する知識のある読者向けに、v72からv81への主な差分を説明します。
Luckの反映方法の変更
Luckをランダマイザ曲線の入力値へ反映する計算式が変わりました。
v72:
$$ x = u |L - 1| $$
v81:
$$ x = \operatorname{clamp}(u - 1.5L, 0, 1) $$
v72では、Luckはランダマイザ曲線のサンプリング点の乗数として機能していました。 v81では、ランダマイザ曲線のサンプリング点を左右にずらす量として機能するようになっています。
どちらもLuckを上げるほどQuotaは上がりにくくなり、Luckを下げるほどQuotaは上がりやすくなる点は同じですが、v81の方がLuckの値あたりの効果が大きくなっています。
一部家具のLuckの変更
v81では、家具のLuckが一部変更されています。Luckの反映方法は変更されましたが、家具のLuckの値自体はv72から一部家具を除いて変わっていません。
変更された家具のLuckは以下の通りです。
| 家具名 | v72 | v81 |
|---|---|---|
| Electric chair | 0.018 | 0.012 |
これに伴い、Luckの最大値は0.2135から0.2075に変わりました。
Quotaの期待値の変化
代表的なLuck条件ごとのrの期待値は以下の通りです。
| Luck条件 | v72 | v81 |
|---|---|---|
| 初期値 | 0.012733341965388603 |
0.012733341965388603 |
| 最小値 | 0.018547213404534198 |
0.02972983489597595 |
| 最大値 | -0.03943367667392469 |
-0.1964269360444696 |
また、Quota 1の達成時にLuck条件を満たし、その後も満たし続けた場合のQuota 20の期待値は、以下の通りです。
| Luck条件 | v72 | v81 |
|---|---|---|
| 初期値 | 20312 |
20312 |
| 最小値 | 20428 |
20647 |
| 最大値 | 19279 |
16164 |