I finally found a program that will let me put in my equation so it makes visual sense.
lim 0 < θ < lim 1
Observe two changes: first, the α variable has been replaced with θ, also called theta. That is purely a symbol change. I wrote the equation in Desmos Graphing Calculator, and they didn’t have the variable symbol I was using before, so θ = α. They’re both the training method variable.
The second change, and the reason for this post, is the introduction of R. R is the symbol for Randomness. It’s been established in both Smash 4 and Ultimate that amiibo operate on some sort of underlying random number generation, as I outlined in Amiibo Science: RNG all the way back in August. We have to take randomness into account because it causes unplanned variations in the amiibo intelligence, as many experienced trainers have observed. One second, they’re racking up damage with moves you never taught them, and the next they’ve whiffed their KO move and are now offstage with no hope of recovering. Amiibo are random.
(The argument could be made that with precisely the same random seed, and with precisely the same circumstances in all other aspects, you could have deterministic amiibo matches. As far as we know that’s true, but amiibo and RNG are two very good examples of how Chaos theory plays out, and as such the argument is irrelevant unless we can simulate such circumstances and get the meta to agree on exactly what seed ought to be used.)
I applied the R variable solely to the combination of I^α and not C because it only applies to amiibo operation, not necessarily their character. However, it is important to note that I^α will be calculated first according to PEMDAS rules, and then randomness will be applied to it. It was ordered in this manner to outline an important point about amiibo training: you can start a solid amiibo and train it well, but even with all of your efforts it can still lose with plain ol’ bad luck.
Anyway, that’s the newest member of the Utility formula. I’m going to be throwing around revisions to the formula in the near future internally. We’re probably going to see this change yet again, as I don’t think this is the best that this model can get.
(Note: you could probably make the argument that the R variable can be applied to the C variable depending on the character. Mr. Game & Watch, Peach, Daisy and Luigi are all good examples of this due to the pieces of their moveset that contain random elements. Character-specific models are going to take a bit more time, as I’ve not got the Utility model to a place where I think it can accurately model the concepts important to amiibo training.)