Many games involve the concept of prestige. That being said some require it in order to continue progressing, while for others its a question of whether it is worth losing my current progress in order to progress faster from a game's starting point. In order to better understand the latter case, consider the following simplified example in which x denotes progress in a game and f(x) denotes the multiple of speed to be gained by a prestige relative to a game's starting speed (which for simplicity's sake I will set equal to 1). Also, set x = 10 to be a game's completion point and f(x) = x as a generic "speed multiplier function". Then minimizing the time required to complete this game (with a single prestige) requires minimizing the function g(x) = x + 10/x. Setting the first derivative of g(x) equal to 0 and solving yields x = sqrt(10). So in a very general sense, I should optionally prestige when I can increase my speed of progression by a factor of sqrt(10). Obviously, things can get a lot more complicated than this (e.g. multiple prestiges realistically required, multiple layers of prestige, far more complicated speed increase functions etc.). Although the unique circumstances of each game must always be considered, this serves as a good rule of thumb for optional speed increasing prestiges.
In a the more general case in which x = Infinity (or something close to it) is a game's completion point, you should prestige every time you can multiply your speed by Euler's constant e, which is approximately 2.71828. Just remember prestige ends in e. Sorry about the lack of generality on my initial post; I was trying to start a conversation about using mathematics to determine when to sacrifice one's progress for an optional prestige.
