Pity timers and guaranteed pulls
If you keep paying for random pulls, the game eventually promises you the prize.
A guarantee after enough tries sounds kind, but it sets a price on the item and keeps you pulling until you reach it.
What it does to the fun
The guarantee feels like a safety net, but it changes how stopping feels. Once a counter shows you're most of the way to a sure thing, walking away means wasting everything already spent, so the closer you get, the harder it is to leave. Like a stamp card with two spaces left, quitting feels silly. And when the prize finally lands, the counter goes back to zero and the pull starts again.
What you're actually buying
More pulls. The pity counter turns a random chance into a known price: keep spending and you will get the result.
How it works
After Japan banned the most extreme version of gacha in 2012, what survived came back with disclosed odds and pity timers. A pity timer guarantees a top-rated result after a set number of unsuccessful pulls. That is better than no guarantee, but it also tells you how much you need to spend, and the closer the counter gets, the harder it is to stop. A related variant used in some Japanese games lets you spend a separate currency to choose a character after a fixed number of pulls.
When it's fine
As a safety net it is better than none, if the pity point is reasonable, clearly shown, and spending limits are on.
Where it came from
Japan's Consumer Affairs Agency banned kompu gacha in May 2012. What survived mutated into disclosed drop rates and pity timers, and the mechanic travelled west as loot boxes.
Rules and regulation
In May 2012 Japan's Consumer Affairs Agency banned kompu gacha. Disclosed odds and pity timers are what survived.
Historical prices and revenues are in US dollars at the time unless marked A$.
By the numbers
Japan's Consumer Affairs Agency ruled kompu gacha illegal in May 2012; what survived mutated into disclosed drop rates and pity timers.
Games that use it

Used to, but dropped it

Related badges
Covered in Video Games & Money: Ep 30 · Gacha.