Gacha maths

Every pity system, in one table

31 games, no two promising the same thing: where each one's odds start climbing, where they stop, and the four that never owe you the unit.

By the GachaData team Updated 6 min read

“Pity” is one word doing at least four different jobs. In one game it is a ramp that quietly raises your odds from pull 74. In another it is a counter that hands you the unit outright at 200 and does nothing whatsoever before then. In a third it guarantees a rarity and then draws at random from twenty-six units, which is not the same promise at all. And in two of the games here it does not exist at all.

Every game explains its own system in its own words, on its own terms, and never next to anyone else’s. Here they are next to each other.

The table

GameSoft pity fromHard pitySparkGuaranteed byWhat the mechanic does
Star Savior——200200200-pull spark guarantee
Silver and Blood—60—not carriedHard pity at 60 · ~50% featured rate-up
Mecharashi3960—120Soft pity from 39, hard pity 60 · 60/40 featured · ramp as the game publishes it
Path to Nowhere5180—160Soft pity from 51, hard pity 80 · 50/50 featured
Mobile Suit Gundam U.C. ENGAGE——200200200-pull spark guarantee
Reverse: 19996170—140Soft pity from 61, hard pity 70 · 50/50 featured
Blue Protocol: Star Resonance6580—80Soft pity from 65, hard pity 80 · no featured split — the pool's SSR is the advertised reward
Limbus Company——200200200-pull spark guarantee
Punishing: Gray Raven—60—120Hard pity at 60 · 70/30 featured
Arknights5199—no pull guaranteeSoft pity from 51, hard pity 99 · ~50% featured rate-up · ramp as the game publishes it
Chaos Zero Nightmare5870—no pull guaranteeSoft pity from 58, hard pity 70 · ~67% featured rate-up
Tower of Fantasy——120120120-pull spark guarantee
Epic Seven——121121121-pull spark guarantee
Azur Lane——200200200-pull spark guarantee
Snowbreak: Containment Zone71100—100Soft pity from 71, hard pity 100 · no featured split — the pool's SSR is the advertised reward
Wuthering Waves7080—160Soft pity from 70, hard pity 80 · 50/50 featured
hololive Dreams——200200200-pull spark guarantee
Persona5: The Phantom X7080—160Soft pity from 70, hard pity 80 · 50/50 featured
Honkai: Star Rail7490—180Soft pity from 74, hard pity 90 · 56/44 featured
Genshin Impact7490—180Soft pity from 74, hard pity 90 · 55/45 featured
Duet Night Abyss7490—180Soft pity from 74, hard pity 90 · 50/50 featured
Zenless Zone Zero7490—180Soft pity from 74, hard pity 90 · 50/50 featured
Umamusume: Pretty Derby——200200200-pull spark guarantee
Blue Archive——200200200-pull spark guarantee
CookieRun: Crumble——180180180-pull spark guarantee
Fate/Grand Order——330330330-pull spark guarantee
DIGIMON UP——200200200-pull spark guarantee
Whiteout Survival———24010 shards recruit the hero · +1 chest shard every 24 Epic Recruitments
Dragon Ball Z Dokkan Battle———not carriedNo pity on the pull — every summon independent (the game's Summon Point exchange ceiling is not in our data yet)
Seven Knights Idle Adventure——1,0001,0001000-pull spark guarantee
Rise of Kingdoms———no pull guaranteeNo pity — every summon independent

Soft pity from is the pull where the per-pull rate starts climbing, listed only where the climb is separate from the cap. Hard pity is the pull that forces the top rarity. Spark is a separate counter that buys the unit outright regardless of what you have pulled. Guaranteed by is the pull at which the game finally owes you the specific unit you are chasing — which, as the column shows, is a much rarer promise than the other three. Rows are in the order of what a featured unit costs, and the table is regenerated from each game’s own Pull Planner model on every build.

Three things the column layout makes obvious

1. Hard pity and a guarantee are different promises

Look at Arknights and CookieRun: Kingdom. Both have a hard pity — 99 and 100 — and neither has a value in the Guaranteed by column. That is not an omission. Hitting hard pity in either game gives you a top-rarity unit, chosen from a pool. The unit you were actually pulling for is one member of that pool, and the game never owes it to you.

A game can have pity and still never guarantee you the unit. Two of the 31 here (Arknights and Chaos Zero Nightmare) have a pity mechanic that promises a rarity and nothing more specific — and two more have no pity at all, which lands them in the same column for a completely different reason.

The spark games make the opposite promise and make it plainly: Fate/Grand Order’s 330 and Blue Archive’s 200 are counters that end in the unit itself, whatever the pulls did on the way. A spark set high enough stops being a promise and becomes a formality — Seven Knights Idle Adventure’s sits at 1,000, past where almost every player has long since landed the unit by chance.

2. The advertised rate tells you almost nothing about the ramp

Several games here advertise the same 0.6% and they do not behave alike. Genshin Impact, Duet Night Abyss, Honkai: Star Rail and Zenless Zone Zero start climbing at 74 and cap at 90. Persona5: The Phantom X starts at 70 and caps at 80 — the same headline number, ten pulls cheaper at the ceiling. CookieRun: Crumble advertises 1.25% as well and never climbs at all, which is why the identical rate costs it 110 typical pulls against Star Rail’s 90. What you experience is set by the ramp, not the rate; soft pity explained walks through what the climb actually does to a pull.

3. “No pity” is a real design, not an oversight

Dragon Ball Z Dokkan Battle and Rise of Kingdoms have no pity on the pull. Every summon is independent, which means there is no number of pulls after which anything improves. That is exactly why they sit at the expensive end of what a featured unit costs, and why their unlucky cases run so much further than everyone else’s — see the unlucky tail. Rise of Kingdoms shows where that design ends up when the per-unit chance is tiny: 2,488 pulls on average, with nothing at any count to stop it.

What this table does not tell you

It compares mechanics, not value. A 200-pull spark is only generous relative to what a pull costs and how often the unit you want is on a banner — neither of which is in this table. It also describes the banner type each game leads with; several games run additional pools with their own rules. And a pity counter that carries between banners is worth considerably more than one that resets, a distinction the columns above deliberately do not try to compress into a number.