[Warning: Maths] On natures, IVs and base stats

[Glint]

(If anybody knows a better place to put this post, please move it there)

 

Hi all!

 

This is my first post over on the reborn side of things, although to be honest I didn't really know where to put this ... Anyway! Before we go any further, a warning:

 

This post contains maths. If you do not like maths, you will not like this post. But TL;DR, nature important. There's also a summary at the end, so maybe go read that too.

 

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Introduction

 

With that warning out of the way, let's get started. During a recent run of mine I caught this beautiful Zangoose called Disaster:

 

Spoiler

 

I thought she was pretty neat, 26 Atk IVs and +Spd nature, despite only having 12 Spd IVs. However, I showed Disaster to a couple of people and got mixed reactions, with some thinking that she was absolutely trash because of the 12 Spd IVs. It got me thinking "I'm pretty good at maths...let's figure out just how good she is". So that's what we're gonna do here, but a little bit more generally. I'll try and answer the question:

 

Just how good is a good nature?

 

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Calculating the IV-Nature equivalent: A tale of floor functions

 

To answer this question, we first need to know how a stat is calculated. Stats other than HP (which is irrelevant here, since we can't change it with nature) can be calculated using:

 

Stat = Floor[ Nature * (Floor[ (2 * Base + IV + Floor[ EV/4 ]) * Level/100) ] + 5) ]

 

Where Nature is 1 for a neutral nature, 0.9 for a hindering nature and 1.1 for a beneficial nature, whilst the floor function is defined such that it picks out the largest integer less than or equal to its argument. e.g: Floor[ 1.2 ] = 1, Floor [ -3.7 ] = -4.

 

Ok great. So what next? To try and quantify how good or bad a nature is, I'll try to convert it into and IV equivalent. So let's suppose something has a neutral nature (Nature = 1) and IV IVs. I'll answer the question "for a different nature, which we will call Nature', what IV value, which we will call IV' , would give the same stat total?".  Seems simple right? Let's get started.

 

Well, there are a few problems here. First off, the floor function has no inverse so we can't just "solve for IV' ". But what we can do is use the definition of the floor function to say that:

 

Stat <= Nature' * (Floor[ (2*Base + IV' + Floor[ EV/4 ]) * Level/100) ] + 5) < Stat + 1

 

Great! That's one of the floor functions gone at least. We can tidy up a little more and get the following inequality:

 

(Stat/Nature') - 5 <= Floor[ (2 * Base + IV' + Floor[ EV/4 ]) * Level/100) ] < ((Stat + 1)/Nature') - 5

 

Aaaand, we're stumped again. Getting rid of this next floor function isn't very easy (trust me, I've tried...a lot). If anybody knows how to get the most general solution here then please tell me! Since I haven't been able to figure it out though, we'll stick with a more specific case. If we're at Level = 100, we see that everything inside the floor function is an integer, so the floor function goes away nicely. Leaving us with:

 

(Stat/Nature') - 5 <= 2 * Base + IV' + Floor[ EV/4 ]  < ((Stat + 1)/Nature') - 5

 

Which we can solve quite easily for IV':

 

(Stat/Nature') - 5 - (2 * Base)  - Floor[ EV/4 ] <= IV' < ((Stat + 1)/Nature') - 5 - (2 * Base)  - Floor[ EV/4 ]

 

But we want our new IV stat, IV', to be an integer. So we're going to stick in some floor and ceiling (the opposite of the floor function...great names I know) functions of our own in. Giving:

 

Ceiling[ (Stat/Nature') - 5 - (2 * Base)  - Floor[ EV/4 ] ] <= IV' < Floor[ ((Stat + 1)/Nature') - 5 - (2 * Base)  - Floor[ EV/4 ] ]

 

For those of you still reading, congratulations! We've found the range of values of IV' that will give the same total stat with a nature Nature' as something with IV and Nature.

 

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A worked example

 

A worked example is always nice to put these things into perspective and give you an idea of exactly what I've been trying to do here. Let's suppose we're ready to take on the champion with our trusty Armageddon:

 

Spoiler

 

He's got perfect attack IVs and EVs, so we're set, right? Well, let's find out just how good Armageddon is using what we've learnt so far. With those stats and that nature, Armageddon has:

 

Stat = 219

Base = 60

EV = 252

 

Using what we derived so far, this means that if instead of a neutral nature Armageddon had a beneficial nature, Nature' = 1.1, he'd have the same stats with:

 

IV' ~ 12

 

Yep, that's right! Stated in words:

 

"Armageddon with 31 Atk IVs and a neutral nature is the same as a Torchic with 12 IVs and a beneficial nature."

 

Oops! Looks like Armageddon isn't so perfect at all... Better get back to breeding!

 

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Calculating the Base Stat-Nature equivalent: It's easier this time...because we've done it already

 

We could have asked the question above in a different way: "for a different nature, which we will call Nature', what base stat value, which we will call Base', would give the same stat total?". All we're doing here is saying that instead of treating the beneficial nature like having more IVs, we could also picture it like having a higher base stat to start with.

 

Given what we've done so far, this isn't too hard to calculate. We just rearrange what we have above to find:

 

Ceiling[ (1/2) * ((Stat/Nature') - 5 - IV  - Floor[ EV/4 ]) ] <= Base' < Floor[ (1/2) * (((Stat + 1)/Nature') - 5 - IV  - Floor[ EV/4 ]) ]

 

That wasn't so bad now, was it? So what does this mean for our trusty Armageddon? Well, just like before we can go ahead and plug in the numbers for Armageddon:

 

Stat = 219

IV = 31

EV = 252

 

And we find using the above:

 

Base' ~ 50

 

And once more in words, to make it clear:

 

"Armageddon with 31 Atk IVs and a neutral nature is the same as a Torchic with 50 base attack and a beneficial nature."

 

This, coincidentally, is exactly the attack stat of an Oddish. So maybe we should have said it a little differently:

 

"Armageddon with 31 Atk IVs and a neutral nature hits as hard as a very angry Oddish."

 

Wow, we really need to work on Armageddon...

 

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So what does this mean for Disaster, our beatiful Zangoose?

 

So just how good is Disaster? After all, that's why I started thinking about this in the first place. Let's calculate the Lv 100 IV and base stat equivalents, just like we did for Armageddon.

 

We'll have to work slightly differently here, since we're starting with a beneficial nature and trying to find the "bad nature" equivalent instead. Unfortunately the floor functions make this hard to do analytically, so I told my computer to do it for me instead.

 

For Disaster's speed stat, we have:

 

Nature = 1.1

Base = 90

IV = 12

EV = 252

 

So if instead she had a neutral speed nature, what IVs/base stat would she need to be as fast? Our computer tells us that:

 

IV' ~ 38

Base' ~ 103

 

So our fabulous Disaster (I hope somebody gets this reference) is as good as a Zangoose with neutral nature and 38 Spd IVs, or a base stat of 103. To put that into perspective, that's faster than both Garchomp and even Oddish.

 

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Summary

 

If you're reading this coming straight from the start, or you've struggled to follow along, I'll try to summarise nicely:

 

1. Nature is very important. Probably even more so than IVs and base stats, depending on what you're using to start with. I didn't realise quite how important it was until I started making this post, and I've pleasantly surprised myself even.
2. Adamant nature Oddish hits as hard as Serious Torchic.  Yep, it's true.
3. Disaster is beautiful and very fast. So stop being mean to her.

 

That is all.

 

Peace!

 

 

[Aphelli]

Wow, math.

Your post made me realize that indeed, natures had effects of considerable magnitude – I didn’t think that it could offset subpar IVs that well. Yes, Disaster is beautiful and deadly and very fast. 

Don’t be too upset about not resolving analytically the inequation, since you would still plug the solution in a computer to get the result. The solution I found out is the following: a <= floor(x) < b iff ceil(a) <=x < ceil(b). And then plug it in. (Where “ceil(x)” is the smallest integer no less than x). 
 

What would be interesting to see is how the effect scales for higher-BST Pokemon. I’m giving the figures at level 100, for fully EV-IV’d mons. 
 

A favorable nature is equivalent to adding 10% of the base stat plus 4-5 points (I didn’t check the rounding but it shouldn’t be too much of an issue – same below). So a +Speed Mew has the equivalent of a Serious mon with 114 base speed. 

It is also equivalent to adding 20% of the base stat, plus 9, as IVs. A +Speed Mew has the equivalent of 60 Speed IVs.

 

Then again, it’s fun to get worked over these numbers, but for example, the defence stat only is used as a factor in the formulas, so a *1.1 nature means (up to roundings) 1.1 times less damage, full stop. That’s useful, but is it really as game-changing as speed (which is why you chose your example very well)? 

[R.B]

Worth noting that you get the ability to change natures around the midpoint of the game (and early game battles generally don't tend to hinge on the specific stats of a given mon), whereas IVs can never be changed except by breeding or a mod, so even if nature is more impactful than IVs (which you've established pretty convincingly), it still makes sense to ignore it when evaluating a mon (at least if that mon is intended to be kept around long-term).

[stibarsen]

In math we have a super advanced technique called "ignoring all the floor and ceiling functions for the sake of simplicity." Doing that, and also ignoring the +5 because it's insignificant in most cases at high levels, we're just looking at how much we need to increase [IV] to multiply (2*[Base] + [IV] + [EV]/4) by 1.1. For example, if we want to ask the question, "at what value of the base stat does beneficial nature + 16 IVs become better than neutral nature and 31 IVs," assuming max EVs in that stat, you just do (2*[Base] + 16 + 63) * 1.1 = (2*[Base] + 31 + 63), which yields [Base] = 35.5. So, basically everything except certain stats on chanseys, blisseys, shuckles, and so forth.

 

Also yes what R.B said about nature being easily mutable in this game while IVs are by default fixed for life (though that'll probably change with the postgame).

[Glint]

1 hour ago, Mindlack said:

Wow, math.

Your post made me realize that indeed, natures had effects of considerable magnitude – I didn’t think that it could offset subpar IVs that well. Yes, Disaster is beautiful and deadly and very fast. 

Don’t be too upset about not resolving analytically the inequation, since you would still plug the solution in a computer to get the result. The solution I found out is the following: a <= floor(x) < b iff ceil(a) <=x < ceil(b). And then plug it in. (Where “ceil(x)” is the smallest integer no less than x). 
 

What would be interesting to see is how the effect scales for higher-BST Pokemon. I’m giving the figures at level 100, for fully EV-IV’d mons. 
 

A favorable nature is equivalent to adding 10% of the base stat plus 4-5 points (I didn’t check the rounding but it shouldn’t be too much of an issue – same below). So a +Speed Mew has the equivalent of a Serious mon with 114 base speed. 

It is also equivalent to adding 20% of the base stat, plus 9, as IVs. A +Speed Mew has the equivalent of 60 Speed IVs.

 

Then again, it’s fun to get worked over these numbers, but for example, the defence stat only is used as a factor in the formulas, so a *1.1 nature means (up to roundings) 1.1 times less damage, full stop. That’s useful, but is it really as game-changing as speed (which is why you chose your example very well)? 

 

It certainly surprised me as to how much of an effect they have. As good as +/- 10% sounds, I usually put natures on the backburner compared to IVs. But they can have an equal if not greater effect, mostly depending on EVs and base stats!

 

Your numbers on the Mew look about right, at least according to what I've looked at. I get about 114-115 Spd or 60-61 IVs for you example. It's a fun little puzzle! I think I first started thinking about this when I realised that I could make my Timid Gengar almost as fast as my Adamant Crobat, and at Lv 85 they were only ~10 speed apart, despite the 20 base speed difference.

 

1 hour ago, R.B said:

Worth noting that you get the ability to change natures around the midpoint of the game (and early game battles generally don't tend to hinge on the specific stats of a given mon), whereas IVs can never be changed except by breeding or a mod, so even if nature is more impactful than IVs (which you've established pretty convincingly), it still makes sense to ignore it when evaluating a mon (at least if that mon is intended to be kept around long-term).

 

This is a very good point! I'd not considered this in the context of Reborn/Rejuv/(Deso? Not sure if Deso has this feature), but it's definitely worth taking into consideration.

 

36 minutes ago, stibarsen said:

In math we have a super advanced technique called "ignoring all the floor and ceiling functions for the sake of simplicity." Doing that, and also ignoring the +5 because it's insignificant in most cases at high levels, we're just looking at how much we need to increase [IV] to multiply (2*[Base] + [IV] + [EV]/4) by 1.1. For example, if we want to ask the question, "at what value of the base stat does beneficial nature + 16 IVs become better than neutral nature and 31 IVs," assuming max EVs in that stat, you just do (2*[Base] + 16 + 63) * 1.1 = (2*[Base] + 31 + 63), which yields [Base] = 35.5. So, basically everything except certain stats on chanseys, blisseys, shuckles, and so forth.

 

Also yes what R.B said about nature being easily mutable in this game while IVs are by default fixed for life (though that'll probably change with the postgame).

 

This is a nice way of looking at the problem! Ignoring the floor/ceiling functions would definitely have made my life easier, I just wanted a nice analytic expression for the allowed values of IV/Base that would give the same stat...unfortunately I was only able to do it for Lv 100.

[shadow of mars]

This is a very cool post, and yes natures are very important

 

But the reason some people may look down on your beautiful Zangoose is because a single speed point makes all the difference in pokemon (either you go first or you don't). That's diffent from atk/def where super effective moves usually do more than enough to OHKO/2HKO without perfect IVs. 

Yes she may be faster than an adamant/serious/whatever garchomp, but no one uses that. Pokemon is dominated by  +speed natures and 252 evs on every pokemon that isn't a defensive wall, so in reality you are not outspeeding any garchomp and will get outspeed by things like +speed Rotom-A, Nidoking, Excadril etc. (and dying without accomplishing anything, instead of the other way around) with such a below average speed IV.

 

Anyway, she's still faster than +speed base 80 pokemon, so she isn't terrible indeed :)

[Mr. SU]

On 7/19/2021 at 2:33 PM, Glint said:

Nature is very important. Probably even more so than IVs and base stats, depending on what you're using to start with. I didn't realise quite how important it was until I started making this post, and I've pleasantly surprised myself even.

Not surprised. My skuntank in reborn's atk IV is literally 1 but he has a naughty nature so he's never really had damage problems. though I only really ever use him for toxic stall, (seriously this helped so much against shelly and taka & zel early on, he even stalled out the dittoceus by dodging 3 focus blasts in a row and the AI brainfarting and not realizing judgement turns psychic in this terrain) and taking out frail psychic types with sucker punch.

 

Also excellent taste, I love zangoose