Defining and explaining the Utility formula

This article is part of a series on the Utility formula. In this series, we will attempt to model out what makes amiibo better or worse in competitive settings.


Utility = (I^α)(C^α)

lim 0 < α < lim 1

I’ve made some alterations to the formula introduced in I C what I couldn’t C before… in order to better explain the alpha variable. It’s now describing utility , which equals I to the power of alpha times C to the power of alpha, with alpha being between 0 and 1. This is going to be a very heady post no matter what, but let me try to take some of the weight off and explain what’s going on.

First, the formula above is not a mathematical equation, it’s a model. I will use the words formula and model interchangeably, but technically speaking this is a mathematical model. We will not be attempting to mathematically quantify specific amiibo. These are concepts that are being explained through the use of modeling functions, in the same manner as real-world disciplines such as economics and statistics.

Let’s start by defining the concepts behind each of the variables in the equations above.

Utility is the competitive value of the amiibo. Individual amiibo with more utility do better generally, while individual amiibo with less utility do worse. If you have two amiibo in a tournament setting, then the one with more Utility will win. (We’ll go over the exceptions to that rule in future posts.)

I: Intelligence. The I variable is somewhat tricky to pin down conceptually, so I’ll do my best to explain it.

Suppose you have two people: one is very smart, and the other is very dull. If you weren’t able to provide some sort of IQ test, how would you determine the intelligence of each person?

Maybe you could interact with them and get an idea of how smart each person is. After all, you know when somebody is smart. It’s not very hard to tell. But what are you looking for when trying to determine if someone is smart? Maybe you’re asking how well they adapt. You could be looking for their articulation skills, or the flexibility of their understanding of a complicated concept.

Determining the intelligence of an amiibo is the same way. If you want to know if a specific amiibo is intelligent and to what degree, you would ask yourself questions like these:

  • How flexible is the amiibo?
  • How well does it adapt to its opponent?
  • Does it use the proper move at the proper time?
  • Does it parry often?
  • Does it have bad habits that it keeps returning to?
  • Is the base AI attached to its character a generally useful AI?

This is clearly not an exhaustive list of criteria to determine the intelligence of an amiibo. In fact, given enough time you could probably come up with a thousand questions to find its intelligence, but you get the idea. The intelligence of an amiibo doesn’t just come down to “does it KO more often than it gets KOed”, as intelligence is not the only factor at play.

It is also important to note that Intelligence is character-specific. Mario and Greninja have different base intelligences than each other, and such have a different I variable.

C: Character. We aren’t training disembodied intelligences to fight. Every amiibo has an attached character: some of them have King K. Rool, while others have Mario. The idea of Character is pretty straightforward as well: it’s just their moveset and attributes that the I is controlling.

The more useful their moveset and attributes in amiibo-style combat, (I refer specifically to ESI moves, which we’ll get to) the higher the C variable will be. Characters that have no useful moves, like Sheik, would have a low C variable while characters with many useful moves such as Bowser would have a very high C variable. The C variable can only be changed when the character is patched, when the moveset is changed in the case of the Mii Fighters, and in the case of Pokemon Trainer when the Pokemon is switched out.

ESI moves: while technically not being a variable, ESI moves play the largest part in determining the size of the C variable. As I explained in I C what I couldn’t C before…, which I recommend you go back and read to get a more thorough understanding of them, ESI moves are moves that basically work best with the nature of amiibo AI. Because amiibo can’t combo, meaning they can’t think ahead as to which move should be used next, they tend to rely on single-input moves that they can use repeatedly in order to win. That’s where the abbreviation ESI comes from: Effective Single Input. They’re moves that take one input, and are also generally effective due to their high firepower.

Having said that, ESI moves are not the complete definer of the composition of the C variable. Technically speaking, the entire moveset of the character comprises the C variable. ESI moves comprise most of the C variable because historically, ESI moves have been a uniting factor in nearly every top-tier amiibo across all metagames, and thus play the largest part in determining the size and effectiveness of the C variable. There are some notable exceptions: Mii Gunner in Ultimate vanilla has no ESI moves (her most useful move is her side special missile, which doesn’t have KO power until high percentages) but still remains a top tier amiibo. We’ll explain situations like that in future posts.

Alpha variable (α): the alpha variable is the input that you, the trainer, have on your amiibo. The more effective your training, the smarter your amiibo will be (a higher I variable) and they will use their more effective moves (a higher C variable). Think of alpha variable as training. You can train a Link to do poorly, resulting in unwanted behaviors like jumping around and dodging, and only using arrows, or you can train a Link to perform well, resulting in proper movement, desirable behaviors and using the right moves. The Link’s I and C is the same: the base Link AI and Link moveset hasn’t changed, the only thing that has changed is their training.

The alpha variable is between 0 and 1 because there are limits to amiibo training. You can’t possibly train an amiibo to be a perfect replica of the best Smash players in the world, but you also can’t train a completely worthless amiibo. Thus, we have the variable as approaching the limit of 0 and the limit of 1 because you can never technically reach it. If you aren’t familiar with that concept, that’s okay, it’s a bit high-level and is typically only taught in calculus courses.

(Again, go back and read I C what I couldn’t C before… before continuing past this point. It’s vital that you have a full understanding of all of these concepts, lest you misunderstand the next articles in this series.)

Having covered all of the variables, let’s pretend for a moment that we were to conceptually “solve” this equation using arbitrary numbers and determine the Utility of your specific amiibo.

Assume that we have numerically quantified the I and C variables of Ganondorf. We’ll say that his I variable is 30 because he’s not very smart, and that his C variable is 90 because his moves are useful. (I am only bringing in actual numbers to demonstrate how the model works. As I said earlier, we are not mathematically quantifying specific amiibo. We’re not driving anywhere, I’m just showing how the engine runs, so to speak.) Now suppose that you did a good job training him, and your alpha variable is 0.9. Remember that alpha variable is between 0 and 1.

So this would look like Utility = (30^0.9) x (90^0.9) = 1225. Now let’s say that you did just a little bit better job of training him, and your new alpha variable is 0.91. The new equation would come out to: Utility = (30^0.91) x (90^0.91) = 1325.

Looking at that equation, we can get a real feel for just how important proper training is. The difference between training them well enough at a 0.9 level versus a 0.91 level is a difference of 100 units of Utility. I believe this equation properly emphasizes the importance of good training, and the above example displays that.





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