Math in Video Games: XP Curves, Loot Odds, and Power-Ups

The spreadsheet I did not expect to need

My son hit level 15 and announced that he was nearly finished. He had been playing for weeks. So I did the arithmetic on the back of an envelope, and found that the five levels from 11 to 15 had taken him longer than the first ten levels put together. He was not behind schedule. He was not even unlucky. He was on a curve, and the curve had quietly taken over.

That is the whole subject of this article. A game is a machine built out of three numbers, and once you can see the three numbers you can predict the machine. They are the cost of a level, the chance of a drop, and the rate a power-up works at. Designers spend months tuning them. Every one of them is a piece of school arithmetic wearing a costume.

A curve is a brake, not a reward

Most games make each level cost more experience than the last. Suppose level 1 takes 100 XP, level 2 takes 120, level 3 takes 144. Each one is twenty percent dearer. Add it up and the first ten levels need about 2,600 XP altogether, while the first fifteen need about 7,200. Levels 11 to 15 alone account for roughly 4,600.

Here is the part that surprises people. The cost of a level is rising quickly, but the reward for reaching one usually stays flat. A new level gives you the same single skill point it gave you at level three. So the ratio between what you gain and what you spend is quietly getting worse.

That ratio is what a game designer is actually managing. A rising cost with a flat reward is a negative feedback loop: the more skill you bring, the harder the game pushes back. Without it, an experienced player would reach the end in an afternoon and there would be no game left. Our why each step multiplies page covers the growth itself, but the design point is the opposite of the maths point. The growth is not there to reward you faster. It is there to slow you down.

Rarity is a stack of fractions

A drop table is usually shown as percentages: 60% common, 30% rare, 9% epic, 1% legendary. It reads like four separate facts. They are really one nested stack. Your epic is also drawn from the remaining tenth after commons and rares are gone, and your legendary is one in a hundred of what is left. Read as nested fractions, a 1% legendary rate sits inside the 10% that is not common.

This is why designers do not just print the headline number. A 1% drop rate sounds generous right up until you remember it is also 1 in 100 pulls, not 1 in 100 boxes, and not a guarantee in any hundred. Knowing how to read a small percentage as a fraction is the whole skill, and our percentages guide covers the conversion in full.

The loneliest number in the game

Here is the number nobody prints. If a drop has a 1% chance each pull, the chance of getting nothing at all across 100 pulls is not 1% either. Multiply 0.99 by itself a hundred times and you get about 0.366. So the chance of nothing is roughly 37%, and the chance of at least one drop is about 63%.

That 37% is the honest face of a 1% rate, and it is why 100 pulls does not feel like 100 chances. It is closer to a coin flip than to a certainty. This is the same weighted-average machinery behind expected value, but the point here is different: the average drop rate says nothing about how long you will actually wait.

And this is exactly why pity timers exist. When a game guarantees a drop after so many attempts, it is not fixing the odds. It is changing the rules of the game in writing. The 1% was always 1%; the pity counter is a promise about the hundredth pull that the odds never made.

Damage per second, and why buffs run out

Power-ups are a rate problem wearing a costume. If an ability deals 100 damage every 10 seconds, you get six uses a minute and 600 damage a minute. Any comparison between two abilities is really a comparison of two of those numbers, and if a skill costs a cooldown as well as mana, the cooldown is the number doing the deciding. Our damage per second article is about exactly this habit of reducing a messy choice to one per-one number.

Buffs are where it gets more interesting, because most games will not let you stack the same bonus forever. The first might give 100% more damage, the second 80%, the third 60%, and the tenth nothing at all. Each extra copy is worth less than the last. Designers call this diminishing returns, and the reason is the same negative feedback seen in the XP curve: without it, a player who farms one easy encounter forever would break the game and there would be nothing left to play.

So the same mechanism that makes a game slow down a good player also stops a lazy strategy from winning. Both are the designer's decision, written once in a spreadsheet, and both are just a fraction getting smaller on purpose.

What to try this week

Pick a game you have played more than twenty hours. Find its level requirements and check whether each one is roughly ten, twenty or fifty percent higher than the last. Then work out how many levels that single rule added to the end of your playthrough. If the answer is larger than you expected, that is the curve doing its job, and it has been doing it to every player who ever wondered why they stopped levelling.

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