Equipment Maths -- Version 2.0

[Malex]

** WORK IN PROGRESS**

Starting over due to a better mathematical model. It is not accurate at very high values or at zero, but as I get better data on the extremes, it should fix. (I could fix the zero issue by replacing the (+1.0421 with +1, but then it becomes more inaccurate at the end points, so I'm leaving it as is and data can fix it later.)

Also, I believe some of my base assumptions in my original experiments may have been wrong, so I have to do some additional tests to see.

New Mathematical Model:

Let x = Stat / 100
-0.1027x³ + 0.1539x² + 0.5608x + 1.0421 = Proportional Effect


Zero Stat Adjusted Model:

Let x = Stat / 100
-0.1027x³ + 0.1539x² + 0.5608x + 1 = Proportional Effect

Spoiler: Test 1

Spoiler: Example: Falcon Punch with +151 Attack

Base = 25
Stat = 151
x = 1.51

Expectation:
-0.1027(1.51)³ + 0.1539(1.51)² + 0.5608(1.51) + 1.0421 = 1.8862243223

25 * 1.8862243223 = 47.1556

1 Falcon Punch = 47%
4 Falcon Punch = 189%

Actual:

1 Falcon Punch = 47%
4 Falcon Punch = 186%


98.41% accurate. Despite not having accurate data because of the rounding and truncating of the real numbers in calculations to integers in the game display.

Spoiler: Example: Falcon Punch with -111 Attack

Base = 25
Stat = 151
x = 1.51

Expectation:
-0.1027(-1.11)³ + 0.1539(-1.11)² + 0.5608(-1.11) + 1.0421 = 0.7496878937

25 * 0.7496878937 = 18.7421973425


1 Falcon Punch = 19%
2 Falcon Punch = 37%
3 Falcon Punch = 56%
4 Falcon Punch = 75%

Actual:

1 Falcon Punch = 18%
2 Falcon Punch = 36%
3 Falcon Punch = 54%
4 Falcon Punch = 71%'

94.67% accurate

Spoiler: Summary

This formula is approximately 95% accurate at areas where it is of low accuracy. This could be caused by the issue of Real numbers being displayed as integers.

An interesting note: If I force-fix the formula to "+1" as the final term. The accuracy of 0, 111, and 151 all increase from about 95% to 99%. Will update with a "Fixed Formula"

Work in progress... I have to leave for now. Will try to work on this later tonight.

 

[Indigo Jeans]

Amazing work man. How did you even manage to begin figuring out an equation to model the damage growth?

 

[Jaxas]

So, after 1.5 days of testing and trying my best to calculate some type of rule for how equipment works. Here's what I have come up with. I haven't done REAL math in a long time, so I've hit a wall.


Basically, I was messing around in custom mode. As you'll all find out tomorrow, there is a practice area in the custom set up so you can test your build on a sand bag. This is how I got all my numbers quickly, but I won't bore you with all that. Not only because it's just a data set, but because I believe there are data errors. (The game only displays integers, and I don't think the game calculates with them.)

Questions:

• Is there a maximum for each stat?
  • Data suggests YES. However, proof required.
• Is there a minimum for each stat?
  • Unknown.
• Do all characters benefit from stats in the same way?
  • Example: Does Attack on a power character do more than an Attack on a speed character?
    • Data suggests that all characters are affected proportionally from stats, and all interact with it the same way.
• Do all qualities of characters benefit from stats in the same way?
  • Example: Do certain moves benefit from stats more than others?
    • Data suggests that character attributes are treated equally. All attacks will receive the same proportionally increase/decrease.

Things to know about equipment:

• There appears to be a +200 Maximum
  • I believe this because of the custom equipment interface has a hard cap there. I was able to meet 200 with my equipment, but I was unable to surpass. (I haven't been farming equipment on my japanese version.)
  • I achieved +200 with a +80 and two +60s, but they were for speed and had special effects. I will note that Villager's speed and jump felt like they were doubled.
• It is possible to go below -100
• Stats are random
• There are character / franchise specific equipment
• I don't know any of the special effects of equipment, so all testing was done with "clean" equipment

I loaded up Bowser, and just hit with uncharged F-smashes and recorded the data. Eh, I'll include the numbers after all.

Spoiler: Bowser F-smash

Format = Stat - Damage
-88 -- 18
-85 -- 18
-80 -- 18
-74 -- 18
-71 -- 19
-66 -- 19
-60 -- 19
-55 -- 20
-50 -- 20
-42 -- 20
-36 -- 21
-31 -- 21
-25 -- 21
-20 -- 22
-14 -- 23
0 -- 24
21 -- 28
30 -- 29
35 -- 31
36 -- 31
37 -- 31
39 -- 31
41 -- 32
51 -- 33
56 -- 34
65 -- 35
71 -- 36
76 -- 36
80 -- 38
86 -- 38
90 -- 38
95 -- 39
102 -- 40
113 -- 40
117 -- 41

Now, to the untrained eye, it might just look like nothing special, but I noticed two things. First, is that the negative stat modification is half the strength of the positive stat modification. Secondly, there appears to be a weak diminishing returns on the stats.

I wanted to achieve the highest and lowest stats I could. I was able to meet 200, but what about the lowest possible?? I was able to get -101 Speed on Villager. (Maybe the game only works with -100 though, it was too close to tell.) I noticed that he "felt" like he was "about half" his normal speed. I referenced my above data, and sure enough, it looked like -100 was tending to half.

I achieved +200 on a character and did 200% damage.
Villager had -101 speed and felt half as fast.

I referenced the data above and sure enough, the "negative" stat was half the strength.


As far as diminishing returns go, that's kind of easy. When I was looking at my data, I noticed that for each +20 stat you get approximately +4 damage on Bowser's f-smash. So, each +5 = +1 damage for the move. However, that falls apart at about the 50s. Based on the data close to zero, it should be +50 = +10, but here we see +51 = +9. It gets worse as you move on. You see a difference of +10 between 102 and 113, but no increase in damage. So the rate of change lowers over time.
For those who had trouble understanding that bit, here it is in simpler terms. "The higher the stat, the less effective more of it will be."




I tried my best to come up with a model to describe the way equipment behaves, but I haven't done real math in so long. Here is what I came up with, and I feel that it is pretty accurate, and maybe others of you with better skills can come up with a more accurate model.

Let x = stat/100

+% variance = sin(xπ/4)
-% variance = sin(xπ/4) /2


There's my model. Let's try it with a character and stat adjustment I haven't tried yet.

Character chosen at random: Diddy Kong
Stat value chosen at random: 62
Fully charged F-Smash: 7 + 16 = 23
Fully charged D-Smash: 17
Fully charged U-smash: 4 + 4 + 9 = 16 (here's where real numbers vs integers affect the data. The % total went 4, 7,16. Which means "4" is probably something more like 3.5)

Prediction of stat value +62 (Achieved with 3 clean equipment of +35, +41, -14)

sin( (0.62 * π ) / 4) = ~.4678 or +46.78% the base damage. Multiply base by 1.4678
So, I expect to see the new values to be:

Expected Values of Full charged smashes:
F-smash:10.2755 + 23.4869 = 33.7624 = 34
D-smash: 24.9548 = 25
U-smash IF 4: 5.8717 + 5.8717 + 13.2114 = 24.9548 = 25
U-smash IF 3.5: 5.1378 + 5.1378 + 13.2114 = 23.487 = 23


Actual Values of full charged smashes:
F-smash: 11 + 23 = 33
D-smash: 25
U-smash: 6 + 6 + 13 = 23

Based on the results. You can see that my prediction of U-smash being 23 was correct, despite the floating damage integers adding to 25. I suspect that the variable is added to the floating number separately than the actual damage calculation. I'll do one more for a negative value.

Value at -77 attack. (remember to half it, since it is negative)

sin( (-0.77 * π ) / 4) = -0.5686 / 2 = -0.2843 or -28.43%. Multiply base by 0.7157

Expected:
F-smash: 5.0099 + 11.4512 = 16.4611 = 16
D-smash: 12.1669 = 12
U-smash(4): 2.8628 + 2.8628 + 6.4413 = 12.1669
U-smash(3.5): 2.50495 + 2.50495 + 6.4413 = 11.4512

Actual:

F-smash: 6 + 12 = 18
D-smash: 14
U-smash: 3 + 3 + 7 = 12


Unfortunately, it seems that my model has broken down somewhat. However, I think it could just be an error in the data. I'll come back to it when the chance, but I can't right now.





What does it all mean?

It means that equipment CAN be managed for competitive play. I'm not really an advocate for equipment in competitive play. However, with a hard cap per stat and diminishing returns, it looks like it is possible, or at least more possible than same detractors are making it out to be.

I don't think +200 defense Shulks will be fun, though, and because of extreme cases like it, they won't make it, competitively.

Just as a heads up (I haven't read the full thing yet, though I intend to) this is how percentages work:

• Damage percentages are tracked along with a decimal value
• During normal matches, the decimal is always not displayed (rounded down)
• During the Wifi waiting room, the decimal is rounded to the nearest integer (always rounded)

But I have no idea as to training, since I have the demo only (for another... 9 hours!)

 

[ScubaF_ingSteve]

As much as I think its cool, I'm not sure if they'll have a place in the tournament scene, but I LOVE the data. Data makes me happy.

 

[Conda]

Equipment is intended to overpower your character to humorous effect. Why some feel like we have to try and make it legal in competitive tournaments confuses me to no end. We don't have to enable every single feature for competitive play

 

[Requiem]

Whether it actually is tourney material or not, and regardless of my opinion on it, amazing job man. That's some dedication right there!

 

[Malex]

Equipment is intended to overpower your character to humorous effect. Why some feel like we have to try and make it legal in competitive tournaments confuses me to no end. We don't have to enable every single feature for competitive play

If you had read the concluding thought, you'd know I am not an advocate of equipment in tournaments. This is just an effort to learn their mechanics better because a lot of people have similar feelings. However, these feeling are based on nothing. We don't have a reasonable understanding of the mechanics.

I've seen people talking about immortal shulks and OHKO Link arrows. And I am sorry, but it is just not the case. I don't like the idea of requiring players to farm equipment to make their best builds. I don't like the idea that equipment is random. However, I think it's something we can break down and assess and make a determination.

 

[Conda]

If you had read the concluding thought, you'd know I am not an advocate of equipment in tournaments. This is just an effort to learn their mechanics better because a lot of people have similar feelings. However, these feeling are based on nothing. We don't have a reasonable understanding of the mechanics.

I've seen people talking about immortal shulks and OHKO Link arrows. And I am sorry, but it is just not the case. I don't like the idea of requiring players to farm equipment to make their best builds. I don't like the idea that equipment is random. However, I think it's something we can break down and assess and make a determination.

My apologies I was mostly being humourous. I'm definitely going to have fun with equipment too

 

[Count Bleck]

This is all extremely useful information. I will be pinning this at the top of my thread for easy access. Thanks for your contribution to the Equipment metagame.

 

[Loki]

How much of an effect do values around 5 or 10 have? are they neglectable or have enough impact to be considered relevant?

 

[popsofctown]

This kind of stuff lays the groundwork for a "balanced brawl" "mod" that could be sidetourney worthy three years down the road.

 

[Malex]

How much of an effect do values around 5 or 10 have? are they neglectable or have enough impact to be considered relevant?

EDIT: I originally did this calculation incorrectly. (Radians vs Degrees, % vs decimal) These are numbers are now corrected and are more accurate.

+% variance = sin(xπ/4)

Stat Values:

+5 Attack

Expectation: +3.926%

+10 Attack

Expectation: +7.846%

Test: Falcon Punch

Base: 25
Expected
+5 Attack: 25.9815%
+10 Attack: 26.96%

Actual:
+5 Falcon Punch: 26%
+10 Falcon Punch: 28%


Result: Model was able to predict the new values of Falcon Punch. While it is apparent that these are not 100% accurate, it is because of the inaccurate data we can observe (Real numbers being displayed as integers, causing loss of information and accuracy) and my inability to come up with a more precise model. 2x Falcon Punch yielded 55%, so it is less than 28, but more than 27. However, for now, it appears to be a workable predictive model for positive stat adjustments.

+5 stat = +3.926%
+10 stat = +7.846%


I'm assuming this question was asked in regards to the suggestions going around of wearing equipment with equipment with special affects but equalizing the stats to near zero, so we can play with bonus effects. (Easy perfect shield, no lag aerials, etc.)
Unfortunately, the lower the stat value, the more adjustment you get per stat. In simpler terms, the bonus you get from 0 to 1 is bigger than the bonus you get from 199 to 200.

 

[Davis-Lightheart]

Can you say exactly what a 0-1 bonus is like? If that doesn't provide the lowest results, what number would?

 

[Malex]

Can you say exactly what a 0-1 bonus is like? If that doesn't provide the lowest results, what number would?

No. I mean the bonus granted from the distance from 0 to 1 is a bigger bonus than the distance from 199 to 200.

They both are a distance of 1 (+1 stat) but you get a bigger effect from 0 to 1 than you do from 199 to 200. Does that make better sense?

 

[Davis-Lightheart]

You're losing me here. Could you provide an example?

 

[Malex]

You're losing me here. Could you provide an example?

It's called diminishing returns. The more you have of something, the less valuable it is.

Example:
(Note: These are arbitrary numbers and do not reflect data)

Let's say you have 10 stat and get +10% effect. You might suspect that it's 1 stat = 1% effect. So you go to 20 stat and have +19% effect. Then you go to 30 stat and have +27% effect. and so on.. if we make it look cleaner...

+Stat = +% Effect
0 = 0
10 = 10
20 = 19
30 = 27
40 = 34
50 = 40

As you can see, each distance of 10 stat you cover grants you a smaller bonus than the distance before it. 0-10 gives you +10, but 40-50 gives you +6.

 

[Davis-Lightheart]

So what would 0-1 give you? I get that the bonus shrinks over time, but would a 1 really be the equivalent of 20 increase over what a 10 would give?

 

[Loki]

Oh my, then it looks that to some degree even such low values have an impact, even if barelly noticeable.

Have you tested this only for % output?, what about KB?, Do you feel it scales the same way or similar? What about defensive stats?

 

[Malex]

So what would 0-1 give you? I get that the bonus shrinks over time, but would a 1 really be the equivalent of 20 increase over what a 10 would give?

sorry it took so long to get back, make sure you have your calculator in radians! I'll go back and fix the math I did wrong when I get the chance.

According to what I came up with:

x = stat / 100
+% variance = sin(xπ/4)

Bonus from 0 - 1 = 0.7854%
Bonus from 9 - 10 = 0.7832%

So, the 1st stat bonus is 0.0028% better than the tenth.

Bonus from 198 - 199 = 0.000092525%
Bonus from 199 - 200 = 0.000030842%

So, the 199th stat is pretty much 3x better than the 200th stat and the 1st stat is 25,465x better than the 200th stat.

Oh my, then it looks that to some degree even such low values have an impact, even if barelly noticeable.

Have you tested this only for % output?, what about KB?, Do you feel it scales the same way or similar? What about defensive stats?

I have only tested % output because that is the only "hard data" that I can collect. Perhaps I can do some speed tests using frames. But I expect that all stats will follow the same rules.

 

[Davis-Lightheart]

So my thought is though, by the time you reached the hundreds wouldn't you have all the former stats though? Just because the bonus diminishes over time doesn't mean the final result will be better with a 1 than a 200, right?

 

[Malex]

So my thought is though, by the time you reached the hundreds wouldn't you have all the former stats though? Just because the bonus diminishes over time doesn't mean the final result will be better with a 1 than a 200, right?

if your question is "Does a +200 stat give a better bonus than a +1 stat?" the answer is yes, obviously.

 

[Davis-Lightheart]

if your question is "Does a +200 stat give a better bonus than a +1 stat?" the answer is yes, obviously.

This was my question from the beginning:

How close is a 1 to the stat of 0? Does it have an impact on results in a meaningful way? If not, at what point do these stats start to matter?

 

[Malex]

This was my question from the beginning:

How close is a 1 to the stat of 0? Does it have an impact on results in a meaningful way? If not, at what point do these stats start to matter?

depends on who you ask. What % increase would be meaningful? less than 1%? less than 10%?

+2 to a stat is already over +1%, so I mean, I would say that any stat change is meaningful.

 

[Davis-Lightheart]

Something around or below a 3% bonus is negligible. What stat changes would produce around that range?

 

[Malex]

Something around or below a 3% bonus is negligible. What stat changes would produce around that range?

Based on the assumption that a 3% shift is negligible (which I would disagree since we have no data to support this), you couldn't have stats higher than 4. (4 is just over 3%) However, my negative model is inaccurate, I still have to figure something out for that, but you could probably have up to -8 or so.

So, my answer right now, until I come up with something better for negatives, would be -8 to + 4.

 

[Malex]

Updated OP with a better model.

 

[Signia]

Why is this in 3DS competitive discussion?

 

[Doval]

I tested Mario's fully charged up and forward smashes with different Attack amounts in the testing room:

F-Smash:
-100: 18
-50: 20
0: 24
25: 29
50: 34
75: 37
100: 41
150: 45
200: 48

Up Smash:
-100: 15
-50: 17
0: 20
25: 24
50: 27
75: 31
100: 34
150: 37
200: 40

I'm way more inclined to think they implemented a piecewise linear function than some complicated polynomial. If you round up, those numbers suggest:

-100 to 0 Attack: x0.75 to x1
0 to 100 Attack: x1 to x1.7
100 to 200 Attack: x1.7 to x2

 

[Doval]

Went back and got more precise numbers. This time I went to Training Mode and used Mario's down tilt 40 times on Bowser at Final Destination with each Attack amount. I waited until Bowser was standing before hitting him again; for some reason, hitting him while airborne would randomly increase the damage slightly and he'd end up with around 2% more total damage after 20-40 hits.

https://docs.google.com/spreadsheets/d/1-GtanSjAp-xLeqepoGA_SmH0YYeFzDJnmOZ5IE9b8p8/edit?usp=sharing

The Damage column is the damage per hit, calculated by dividing the 40 hits column by 40. It should be accurate down to the first decimal place, and reasonably close in the second decimal place. It's clear now that it isn't linear, though I don't know what formula would give those numbers.

EDIT: The forum destroyed the formatting so I added a link to the spreadsheet and a screenshot for convenience.

 

[Balgorxz]

I'm glad stats barely change low % attacks, atk also affects knockback right?
there a way we can figure how much does attack affects knockback?

anyways thanks a lot for this, this can help people that might host equiments tournament in the future

 

[Leebee]

yeah, I'm most interested in seeing how (negative) defense stats affect knockback/hitstun!

thanks for all the dedication!

 

[Doval]

I'm glad stats barely change low % attacks, atk also affects knockback right?
there a way we can figure how much does attack affects knockback?

Low % attacks are affected just as much as high % attacks. Double damage is double damage. Knockback is affected by damage so by increasing the damage of an attack, you're indirectly increasing its knockback too.