mathematics – What mathematical function best models player improvement over time?

Basically, I’m looking for the formula of the learning curve, so I can better predict how players are doing and encourage them to improve. I’m ideally looking for a few choices, as I’m sure not every learner improves along the same function, but if there’s one standard choice that’s reliable, that’s great too.

Not much in the way of details I can add here to fill up this box :]

Ideas: