This is a very good question, but it is quite complicated and long to explain in details, so I will try to do my best.

If you can model how the weather behaves in mathematical formulas, you can model anything, especially games, especially because using formulas for the progressions of a game is much more powerful and easier than doing it with a data table or with an infinite if.

Nowadays, there are many tools that allow you to do what you are looking for, it is quite common to use Python and libraries like Pandas, it is a very powerful tool that allows you to do everything, play with statistical distributions, progressions, etc. but they are a more advanced level and you would have to invest a good amount of time in learning them.

 But any spreadsheet, such as excel or libre office calc, google sheet, etc, should be enough to do the work you need.

As far as I know, there is no area of mathematics focused exclusively on what you are looking for. It is rather a set of different disciplines, perhaps the closest would be series and progressions (arithmetic and geometry).
But if you allow me the comparison, it's like wanting to learn Russian just to translate the buttons that say "new game" and "exit".

If you are interested, perhaps the easiest thing to do is to look at some exponential progressions and perhaps logarithms. The rest will be a bit of experience and asking if you are looking for something specific, for example in RPG it is very common to use formulas with S-shaped curve or sigmoid curve.