Brawl Mechanics: Damage & Knockback Formulae

ColinJF

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This research was completed by Amazing Ampharos and I (Colin) over the last few days.

1.How Damage is Stored
2. A Note on Methods
3.The Damage Formula
4. A Note on Training Mode
5.The Knockback Formula
6. Further Research

Abstract

In this post, we present an accurate damage formula and an accurate knockback formula. In particular, we develop new results in the areas of how damage is stored, stale move decay, Lucario's aura power, pokemon fatigue, charging smashes, and more. We develop exact models for many of these things that predict the precise damage, including decimal places. We also present a correct formula for knockback.

1. How Damage is Stored

At least three decimal places of precision are required to explain the exact results we obtained using our damage formula. Whether damage is actually a decimal number is difficult to determine, but whatever it is, it is precise enough to store the equivalent of three decimal places. Contrary to some previous reports, the maximum damage is in fact 999%. Damage over 999% is discarded.The game truncates damage for the purposes of display. For example, 125.651% is displayed as 125%.

2. A Note on Methods

To determine the actual damage a move was dealing, we used it repeatedly and recorded the sequence of numbers after each use. Using a program I wrote, we were able to determine the actual damage the move was dealing, to full precision. At first, we had to work within training mode to avoid stale moves, but once we cracked stale moves precisely, we were able to account for these in the sequence and determine the exact damage of moves directly from sequences of numbers in versus mode.

The source for this program is provided here in case other people want to do further research involving decimal damage:

http://colin.shoddybattle.com:81/BrawlDamage.txt

3. The Damage Formula

This formula gives you the exact damage that a move will do in any mode of the game. This formula works for both training mode and versus mode.

damage = base damage × stale multiplier × charge multiplier × aura multiplier × fatigue multiplier

We explain the value of each of these multipliers in the following sections.

3.1 Base Damage

Every move has some integer damage associated with it. This is its base damage. However, most of the time, the actual damage dealt by a move is not an integer, but this is because of multiplicative factors in the damage formula.

3.2 Stale Move Multiplier

The formula for the stale move multiplier is



This is explained below, including the definition of s.

In training mode, stale moves are not in play, so the value of this multiplier is exactly1. If the move is not affected by stale move decay, then the value of this multiplier is 1 even in versus mode. The following moves are not affected by stale move decay: Luigi's down taunt; DK's cargo throws; tether attacks; Zero Suit Samus's neutral air.

In versus mode, for moves affected by stale move decay, this factor is never 1. If a move is not in the stale move queue before being used, the value of this multiplier is exactly 1.05.

In versus mode, for moves affected by stale move decay, a move is placed in the stale move queue after it is used. The stale move queue contains nine entries, and each entry has a numerical weighting as depicted in the following diagram.



The left of the diagram represents the most recent move to be added to the queue, and the far right represents the oldest move to the added to the queue. Then we have

stale multiplier = 1 – s

where s is the sum of the values of the positions in the queue occupied by the move being used. For example, if the move being used occupies the .1 position, the .06 position, and the .03 position, then the stale multiplier is 1 - (0.1 + 0.06 + 0.03) = 0.81.

3.3 Charge Multiplier

This is the formula for the charge multiplier



For partially charged smash attacks, the value will be somewhere between 1 and 1.4, but we were unable to determine the exact distribution experimentally; it is too difficult to test. However, we speculate that it follows the same distribution as the aura damage multiplier. We speculate that if you charge the smash 1/n of the way, the multiplier is the same as the aura damage multiplier for (170-75)/n+75 %.

3.4 Aura Multiplier

This is the formula for the aura multiplier:



where



In coin mode, each coin counts as a stock for the purpose of the stock multiplier.

The damage multiplier is in the interval [0.7, 1.4]. It is 0.7 for all damages <= 20% and 1.4 for all damages >= 170%. It is exactly 1 only when the damage is 75%. A very good approximation for the damage multiplier can be found by reading the value from this chart:



For example, the damage multiplier for 95% is 1.09.

This chart was created by using quintic interpolation between some exact values that we worked out.

3.5 Fatigue Multiplier

Not everything is known about this multiplier. However, it is 1 if the character is not a pokemon trainer pokemon. Otherwise, it is in the interval [0.7, 1] where 1 is fully unfatigued and 0.7 is fully fatigued. There is in fact a spectrum of fatigue levels and the multiplier can take on various values in between 0.7 and 1. The multiplier cannot take on every value in this interval, however. When the pokemon shows no signs of fatigue, the multiplier is 1.

4. A Note on Training Mode

Our damage formula explains one aspect of training mode that was previously a mystery. In training mode, the stale move multiplier is 1, since stale move decay is not in play. As a result, most moves do their integer base damage always. In versus mode, however, the stale move multiplier is never 1 for moves that respect stale move decay. As a result, moves in versus mode do not usually do integer damage. This is the cause of the inconsistencies noticed between training and versus mode.

5. Knockback

[NOTE: This knockback section has been superseded by a new article. Click here to read it.]


The starting place for our research on Knockback was MrSilver's “Character weight list, fall speed list and random things” thread. MrSilver's work is a good starting place, but is incorrect in several ways.

Throughout this section, the words “launch speed” and “knockback” are used interchangeably because the quantity that corresponds to knockback on the results screen is the “max launch(er) speed”.

5.1 Constants Associated with Characters

Firstly, for the purpose of calculation of knockback, every character has two associated constants, namely weight1 (w1) and weight2 (w2). MrSilver determined both of these constants for every character, but mysteriously referred to weight2 as “fall speed”. In fact, weight2 has nothing to do with fall speed. Here is the list of weight1 and weight2 for each character. (These values are copied from MrSilver's document and included for ease of reference with the following formula.) Weight1 is the first value and weight2 is the second value

Bowser: 1.111, 0.91
Donkey Kong: 1.091, 0.20
Snake: 1.076, 0.16
King Dedede: 1.071, 0.29
Charizard: 1.061, 0.33
Ganondorf 1.056, 0.91
Samus: 1.051, -0.54
Yoshi: 1.045, -0.05
Wario: 1.045, 0.30
R.O.B.: 1.040, -0.32
Ike: 1.035, 0.28
Captain Falcon: 1.030, 0.92
Link: 1.030, 0.46
Wolf: 1.020, 1.84
Lucario: 1.010, -0.40
Ivysaur: 1.010, -0.16
Mario: 1, 0.00
Luigi: 0.995, -0.32
Sonic: 0.985, 0.33
Ness: 0.980, -0.12
Pit: 0.980, 0.00
Lucas: 0.980, 0.49
Diddy Kong: 0.975, 0.55
Toon Link: 0.970, -0.16
Ice Climbers: 0.970, 0.07
Peach: 0.960, -0.37
Marth: 0.944, -0.25
Sheik: 0.935, 2.21
Zelda: 0.935, -0.26
Falco: 0.919, 1.23
Olimar: 0.919, -0.47
Zero Suit Samus: 0.914, 0.44
Fox: 0.910, 3.41
Pikachu: 0.904, 0.40
Meta Knight: 0.904, 0.68
Kirby: 0.899, -0.44
Squirtle: 0.884, 1.05
Mr. Game & Watch: 0.884, 0.01
Jigglypuff: 0.849, -0.78

5.2 Constants Associated with Moves

Every move has a base knockback constant (b) and most moves have a knockback growth constant (g). For moves with fixed knockback, the base knockback constant is defined as the knockback taken by Mario. For moves with variable knockback, the knockback growth constant is defined as

g = (f - m)/e

where m is the knockback taken by Mario at 0% before the hit, e is any damage you like (say, 100), and f is the knockback taken by Mario at e% before the hit.

Then the base knockback constant is defined as

b = m – dg

where m is the knockback taken by Mario at 0% before the hit, d is the damage the move deals, and g is the knockback growth constant.

Moves with fixed knockback have have two weight variability constants (s1, s2), which describe how much the knockback of the move varies with weight1.

All sufficiently strong moves have a weight2 scale constant (c) which describes how important weight2 is to the move. c can take on a variety of values. For some moves, such as Ness's up throw, c is over 160.

5.3 Knockback Equation

We determined that the knockback equation has two cases depending on the value of a constant k.

To calculate the knockback, first work out k.



where b is the base knockback constant, d is the damage the move would deal, x is the damage of the target before the hit, g is the knockback growth constant, w1 is the weight1 of the target, and s1, s2 are the weight variability constants.

Then we have the equation for knockback:



where w2 is the weight2 of the target, and c is the weight2 scale constant.

5.4 Additional Multipliers for Knockback

We found a few things that cause simple multiplications of the final knockback value. A move being “super effective” is a multiplier of 1.1, and a move being “not very effective” is a multiplier of 0.9. Obviously, this only affects the Pokemon Trainer. In fast mode special brawl, all knockback is multiplied by 1.5, and in slow mode special brawl, all knockback is multiplied by 0.5. Heavy mode special brawl does not affect knockback. As was previously known, attacking a target who is charging a smash other than Ness's up and down smashes is a multiplier of 1.2.

6. Further Research

It is not known how stale move decay affects knockback precisely. Another area for further research is how charging smash attacks affects knockback.


Colin and Amazing Ampharos
 

Amazing Ampharos

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The main purpose of this post is to confirm my collaboration on this project. If anyone wants the raw data I happen to have saved (it's really boring), feel free to PM me.

For the sake of this post not being totally boring, here's a list of curiosities that came up during this research.

Falcon Punch actually has two hitboxes. The tip of the range is slightly weaker than the rest of it, and yes that applies to reverse Falcon Punch as well.

Doing the same attack multiple times in a row if it's "too fast" will prevent it from entering the stale move queue. For instance, doing Ness's up tilt on some target (including damagable stage features) at max speed won't cause every hit to enter the queue.

Samus's Charge Shot doesn't power up continuously. It can do 3%, 5%, 8%, 11%, 14%, 18%, 21%, or 25% base damage depending on how much it was charged. Lucario and R.O.B.'s similarly charged moves are weird by the way; don't ask me about them.

Mr. Game & Watch's down throw has a negative "c" value, and it does more knockback to characters with more negative weight2 values. Granted, the knockback on this move is almost always irrelevant, but it's interesting regardless.

Snake's down throw actually inflicts no knockback at all. If it's the only move you do in a match, the game will report the victim as having received no knockback during the match.

Mr. Game & Watch and Peach's jab1 attacks have zero knockback growth. King Dedede's down throw is not alone in this lovely attribute (and there are more than just these three, but I'll leave it as an exercise to the reader to find them).

Three smashes are not powered up by the smash charge multiplier. These are the up and down smashes of Ness and the up smash of Snake.

Fox's laser does more damage at close range than at far range.
 

Mr. Escalator

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Colin would mention little tidbits of this research as it progress, and to be honest, the way he explained it over mIRC was much clearer than it is here xD

The knockback is what I find most interesting. Weight2 is p. cool stuff

<@Colin> interesting fact: jigglypuff has the best weight2
<@Colin> (fox has the worst weight2)
<@Colin> this is why jigglypuff takes less knockback than bowser from game & watch's up smash at 0%

G&W has positive2, but only slightly, so he's okay!
 

PK-ow!

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The knocback section seems more interesting. Let me try and attach some meaningful descriptions.

EDIT: Quotes removed at behest of author.


So, how to make sense of weight2? It features in one knockback equation. It is interesting because it features in a term with c as a coefficient, and both these values can be positive or negative. It is also interesting because it is 0 for Mario. I don't think you can explain this as just an artificial zero point; Mario is the unique character who is unaffected by the weight2 scale constant.

If it is an artificial zero, I'd like to see if there is a simpler 0 point. As in, is there a way to rewrite this so that w2, or c, is always positive or always negative?

In absence of that, you can't say w2 makes your launch speed higher or lower, nor anything about c. All you can say is that w/c > 0 means you'll go farther and w/c < 0 means you go a little slower (both given that k exceeds a certain value).

However, it appears GnW's dthrow negative c is exceptional, so you can generalize: w2 larger positive means that you are knocked farther by sufficiently large k moves, and larger negative means you are knocked less.

5.2 Constants Associated with Moves

g = (m – f)/(d – e)
EDIT: Quote simplified.

You should probably rewrite that as (f - m)/(e - d), so that it's a quotient of positives rather than negatives.
Then you can say it's the ratio of the difference of knockback at e% and at 0%, over the difference of e% and the damage percent.

. . . what if you pick e less than the damage? Then g is negative, since f is for sure still greater than m.
. . . and then g is not a constant.

:confused:

. . . this sort of stopped me cold. What's going on?
 

ColinJF

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Please don't quote large parts of the post so that I can change if it mistakes are noticed.

And yes, I did make a mistake there.

It's supposed to be

g = (f - m)/e

Sorry about that minor typo.
 

Big O

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I remeber testing a lot of this stuff out and got really strange results. The biggest one I can remember being that there seemed to be a different w1 and w2 values for each move. DK's 9 punch and fully charged punch do about 29.38%, but when calculating the initial knockback using MrSilvers weight 1 values and finding out the weight 2 #'s for both punches the calculated values were different from the in-game numbers. When I calculated the 9 punch's knockback the results were spot on. When I tried the same for fully charged punches the numbers were always somewhat off. In his test he used Ike's fsmash which is pretty close in power to a 9 punch. The weaker fully charged punch was always off because I assumed MrSilver's weight list to be 100% accurate. When I tested the compared the growth of incremental knockback for 9 punches and fully charged ones they were slightly different. The 9 punch's values were extremely close to MrSilver's while the fully charged ones were somewhat off. This might only apply to specials or whatever though.

Some of the details are in here http://www.smashboards.com/showthread.php?t=176033&page=4. Hopefully you can verify this or see if I was careless in something.
 

ColinJF

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Also if you wouldn't mind editing the quote in your post to avoid confusion, that would be good.
 

ColinJF

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I remeber testing a lot of this stuff out and got really strange results. The biggest one I can remember being that there seemed to be a different w1 and w2 values for each move. DK's 9 punch and fully charged punch do about 29.38%, but when calculating the initial knockback using MrSilvers weight 1 values and finding out the weight 2 #'s for both punches the calculated values were different from the in-game numbers. When I calculated the 9 punch's knockback the results were spot on. When I tried the same for fully charged punches the numbers were always somewhat off. In his test he used Ike's fsmash which is pretty close in power to a 9 punch. The weaker fully charged punch was always off because I assumed MrSilver's weight list to be 100% accurate. When I tested the compared the growth of incremental knockback for 9 punches and fully charged ones they were slightly different. The 9 punch's values were extremely close to MrSilver's while the fully charged ones were somewhat off. This might only apply to specials or whatever though.

Some of the details are in here http://www.smashboards.com/showthread.php?t=176033&page=4. Hopefully you can verify this or see if I was careless in something.
Can this not be explained by the different levels of charge of the punch having different c values? Anyway, we calculated a few w2 values ourself using various moves and they were the same as his values, so it could be unique to certain specials as you said.
 

Amazing Ampharos

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I think the giant punch actually transforms into a different move when it is fully charged so a different c value for just the fully charged version (as compared to all other charges) is to be expected. Likewise, I'm pretty sure that Ike's Eruption and the Wario Waft transform into different moves at full charge (and in those cases as well, they are actually more powerful shortly before they are fully charged).
 

Big O

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I did calculate different c values for differnt levels of charge but the problem was the fact that the relative incremental knockback values, w1, were not the same for both. Another problem was that the w2 values were also different for each move which made the c values kinda worthless.
 

ColinJF

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The game rounds the "launch speed" figure in an as yet unknown way, by the way. This is particularly obvious when experimenting with fast mode because the multiplier is clearly x1.5 in the code, but it doesn't always work out to exactly 1.5 when you check it yourself (varying around 1.51 for example). When you consider that the w1, w2 values are themselves derived from these rounded values, and then rounded themselves, and you are comparing to other rounded values, there is plenty of error coming in. It's possible this is all we are seeing here.

The damage section of the document on the other hand is focused on exact results since they are possible to get. (We can't get more decimals for launch speed since the game doesn't tell you them.)
 

Crystanium

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Is there an easier way to explaining this stuff? Math has never been my forté. Some of us who are mathematically illiterate don't get any benefits from this.
 

Big O

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The game rounds the "launch speed" figure in an as yet unknown way, by the way. This is particularly obvious when experimenting with fast mode because the multiplier is clearly x1.5 in the code, but it doesn't always work out to exactly 1.5 when you check it yourself (varying around 1.51 for example). When you consider that the w1, w2 values are themselves derived from these rounded values, and then rounded themselves, and you are comparing to other rounded values, there is plenty of error coming in. It's possible this is all we are seeing here.

The damage section of the document on the other hand is focused on exact results since they are possible to get. (We can't get more decimals for launch speed since the game doesn't tell you them.)
It would probably be easier to see what I mean with actual numbers. When actually comparing the numbers given in-game when punched at zero and 300% for each character, you subtract the knockback at 300% from the knockback at 0% and divide the result by 300 to get the incremental knockback. You then compare the incremental knockback of the move done on Mario to the same move done on say Fox. When you do this with the 9 punch, Mario's growth/Fox's growth is a teeny bit less than .910. For the fully charged though you get the same character weight ratio as a bit less than .908. Since the knockback values in mph are pretty high this difference is probably too high to be affected by rounding error. Also some things like Bowser having a drastically different (negative) w2 value for fully charged punches also probably have little to do with rounding error.
 

ColinJF

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Noted. It sounds like Giant Punch deserves some specific research when we do further research on knockback, but I don't have any hypotheses to offer now.
 

Zhamy

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Doing the same attack multiple times in a row if it's "too fast" will prevent it from entering the stale move queue. For instance, doing Ness's up tilt on some target (including damagable stage features) at max speed won't cause every hit to enter the queue.
Any particular reasoning for this, or is it just faulty mechanics?

Is there an easier way to explaining this stuff? Math has never been my forté. Some of us who are mathematically illiterate don't get any benefits from this.
1) All damage, for the purposes of their testing, is assumed to have 3 decimal places trailing it for sake of accuracy.

2) We wrote a program in Java and basically did the same attacks over and over and over and over and had the program record and interpret the results for us. That's where our data comes from.

3) DAMAGE
3.1) Every attack has a "base damage," that is, the level of power at which the other factors are applied.
3.2) Stale moves regenerate at a specific rate, where the closer you get to replenishing the move, the stronger it becomes. A completely fresh, never before used move has a power multiplier of 105% (Certain moves always remain at 100% multipler despite stale moves)
3.3) A charged smash attack will have a power multiplier of 140%, to find the multiplier for partly charged smashes, they estimate it's the same formula for Lucario's Aura.
3.4) Lucario's Aura: Stock and damage both affect it.
3.5) Fatigue: If you're not PT, don't worry about it. If you are, the worse off your pokemon will be is with a 70% multiplier.

4) Training Mode: The reason that training mode damage =/= versus mode damage is because in training mode, stale move decay is always a 100% multipler, thus the discrepancies.

5) Knockback
5.1)The original list of weights (there are two for each character!) referred to one as fall speed - this is incorrect, but the numbers are right.
5.2)There is Base Knockback and Knockback Growth - rather self explanatory, but you can run the numbers yourself.
5.3) Knockback Equation: Unless you really want to do the numbers, it's nothing new.
5.4) Simple enough to quote

We found a few things that cause simple multiplications of the final knockback value. A move being “super effective” is a multiplier of 1.1, and a move being “not very effective” is a multiplier of 0.9. Obviously, this only affects the Pokemon Trainer. In fast mode special brawl, all knockback is multiplied by 1.5, and in slow mode special brawl, all knockback is multiplied by 0.5. Heavy mode special brawl does not affect knockback. As was previously known, attacking a target who is charging a smash other than Ness's up and down smashes is a multiplier of 1.2.
6)
It is not known how stale move decay affects knockback precisely. Another area for further research is how charging smash attacks affects knockback.
We still don't know how DDD's chaingrab works/fails, basically.
 

Zhamy

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Just FYI, my "summary" up there is a gross oversimplification, but it's a very short summary of what they did.
 

Amazing Ampharos

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Any particular reasoning for this, or is it just faulty mechanics?
I suspect it is caused by a move "comboing" into itself only counting as one move. Presumably the reason it was programmed like this was to stop something like a rapid jab from very quickly refreshing stale moves, and it was probably made general to account for things like pummels or very fast tilts (Ness/Lucas, etc.). You can actually get a feel for just how fast the attacks have to be by looking at pummels; I have some old research on the matter.

http://www.smashboards.com/showthread.php?p=5408607

I assume the "unsure" group is actually just rounding error and should fit in with the "Does not force ground break" group.

This does have the unusual implication that the game considers you to be doing a "combo" on a stage feature, but remember that the programmers were just a tiny bit crazy and it should be alright.
 

ColinJF

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By the way, one consequence of these results that may not be immediately obvious is that if you graph knockback versus damage for a move that initially gives knockbacks below 2550, you notice a "jump" in the graph just before the knockback is about to equal 2550. Whether it jumps up or down depends on whether w2 * c is positive or negative.
 

leafgreen386

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Interesting. Nice research you've got there. But I have one little nitpick. When you said:
In fast mode special brawl, all knockback is multiplied by 1.5, and in slow mode special brawl, all knockback is multiplied by 0.5. Heavy mode special brawl does not affect knockback.
I noticed the slow/fast thing, too, when doing research into how brawl handles momentum (which is still incomplete). Although I think that saying the differences in launch speed are a direct modifier on "knockback" is incorrect. Launch speed isn't actually knockback. It's just what the game uses to ultimately derive knockback values (and as an added note, to also calculate hitstun). Since slow brawl has characters flying half the distance per frame than they normally would, it would only make sense that launch speed is appropriately halved. But there is a key distinction here, in that the actual knockback the move has remains (mostly?) unchanged; you're still flying the same distance. I think the most appropriate way you could define launch speed is as the number of units the game sends you on the first frame of knockback (which may or may not be completely true, but it sure seems to be the most likely candidate until things are looked at a lot more closely), which then becomes smaller and smaller as the character eventually comes to a stop.

That said, launch speed is still a very good (indirect) measure of how much knockback a move deals. I just don't think it's accurate to use the two terms interchangeably.
 

Villi

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By the way, one consequence of these results that may not be immediately obvious is that if you graph knockback versus damage for a move that initially gives knockbacks below 2550, you notice a "jump" in the graph just before the knockback is about to equal 2550. Whether it jumps up or down depends on whether w2 * c is positive or negative.
Meaning at certain percents, if w2*c is negative, some moves might cause more knockback than if the percents were slightly higher?

It doesn't seem like w2*c is a very significant number for knockback, anyway... or is it? Though it probably explains some small vertical kill discrepancies I've noticed...
 

Steeler

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colin you are a huge nerd

and i love you

3.5 Fatigue Multiplier

Not everything is known about this multiplier. However, it is 1 if the character is not a pokemon trainer pokemon. Otherwise, it is in the interval [0.7, 1] where 1 is fully unfatigued and 0.7 is fully fatigued. There is in fact a spectrum of fatigue levels and the multiplier can take on various values in between 0.7 and 1. The multiplier cannot take on every value in this interval, however. When the pokemon shows no signs of fatigue, the multiplier is 1.

very interesting stuff. thanks for this!
 

Adapt

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Meaning at certain percents, if w2*c is negative, some moves might cause more knockback than if the percents were slightly higher?
naw... the w2*c is for differences between characters. If you want a good example look at this:



See how the characters no longer follow their weight trend exactly when you get to 2550
Which coincidentally is the point where DDD can no longer chaingrab the character.

DDD's Dthrow has no knockback growth btw.

It doesn't seem like w2*c is a very significant number for knockback, anyway... or is it? Though it probably explains some small vertical kill discrepancies I've noticed...
As you can see above... for a select few characters (with high w2) it can be quite a difference. Fox is a great example of this.
 

ColinJF

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Villi is correct. For sufficiently weak moves being used on characters with a negative weight2, there exists a percent X such that the character takes more knockback at X - δ than they do at X + δ where δ > 0 is a small amount of damage. The stronger the move is though, the smaller X is, and for sufficiently strong moves, there is no such percent.

Adapt's chart is a good example of how big the difference is, but since the move he used has fixed knockback, you don't notice this phenomenon.
 

CR4SH

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****ing EPIC. The best post I've ever seen on these boards, and I love it. I mother****ing love it. I wish there were more offensive swear words so I could accurately articulate the depth of my feelings. I love data, I love this. I love this more than I love my dog.

You win.
 

Adler

Smash Apprentice
Joined
Nov 20, 2008
Messages
128
Location
My teacher says &quot;in my own world&quot;.
I'll try not to post every Question which came to mind as I read that (My Brain hurts).

How does charging other Attacks besides Smash Attacks work?
I mean like Aura Sphere, Charge Shot, Link/Toon Link's Up-B, Peanut Popgun, etc.

Also how does Aura affect the size of an Aura Sphere?

How long does it take for PKMTrainer's Stamina to deplete (is it affected by the Moves you use and/or does it constantly run out)?

How is the Power of Wario Waft calculated?
 

Ralk

Smash Rookie
Joined
Dec 11, 2007
Messages
17
Location
Pennsylvania
Just wanted to commend the OP on some great work. You've just written the first journal paper on smash.
 
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