Nintendude
Smash Hero
Let me start by saying I hate matchup "ratios." In a traditional sense, they are completely meaningless to me. Apparently one interpretation of them (the way Street Fighter does it apparently) is that, for example, a 7-3 matchup means that out of 10 matches one character wins 7 on average and the other wins 3 on average. Well, I think is fundamentally flawed. Taking 3 matches out of 10 is actually pretty decent, so how can this be considered a horrible matchup ratio? To use a more extreme example, why should 9-1 exist? If a character is losing so badly, why should he be able to win 1/10 games? It may be possible, but the ratio makes no attempt to illustrate it.
Now here's my approach to it:
Imagine an infinitely long stock match. Let's say that for a Fox vs. Falcon match, on average, for every 6 stocks Fox takes, Falcon takes 4 stocks.
This simply translates to 6-4 in ratio form. Simple.
Now let's consider Fox vs. Sheik. Let's say, for the purposes of this model, that for every 6 stocks Fox takes, Sheik takes 5. Let's scale this to a ratio in the 1-10 form:
6*10/(6+5) = 5.45
5*10/(6+5) = 4.55
This roughly translates to a 55-45 matchup (or 5.45-4.55), which can then be rounded to 5-5.
I'd say the biggest fundamental difference between this interpretation and the traditional, "Street Fighter," interpretation is you are ignoring the fact that matches are finite and end after 4 stocks. Well, to me this is actually a more meaningful approach. A ratio of 10-0 can either mean that a character is getting ***** repeatedly or the character is barely losing repeatedly. They are two completely different situations being represented by the same ratio, and it makes no sense. If you instead say that on average a certain character takes 4 stocks while losing 1, you can easily see that either character has a chance of winning, despite it being very uphill for the worse character.
Also, while I'm not knowledgeable about Street Fighter, I'm pretty sure the game is a traditional fighter in the sense that there aren't stocks - just individual games/matches. Basically, it seems to me like Street Fighter's games are sort of an equivalent to Smash's stocks. So, why are we carrying over Street Fighter's ratio interpretation to games rather than stocks?
If you wish you may skip this next section, cause it is really complex and strays from the overall point I'm trying to make.
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Now, how do we deal with counterpicks? This is really difficult but I have an idea. Let's examine Pikachu vs. Fox in SSB64 (cause SSB64 is simpler and doesn't use Dave's Stupid Rule).
For this example I'll say that the first stage in a best of 3 is random between Hyrule and Dreamland. Let's say that we determine the stock ratios to be 5-5 on Hyrule and 7-4 (Pika takes 7 for every 4 Fox takes) on Dreamland.
In a best of 3 set, Hyrule and Dreamland have equal chances of being chosen for the first match. If Hyrule gets chosen, both characters have an equal chance of winning. If Fox wins, Pika will counterpick Dreamland (or another stage that Fox sucks about as much on), and if Pika wins, Fox will pick Hyrule again. Then it happens again the third match.
Sample space:
H = Hyrule, D = Dreamland
(first stage, second stage, third stage)
HDD
HDH
DHD
DHH
HH
DD
Compute probabilities of each element in the sample space (based on chances of the characters winning on each particular stage). The probability of HH, for example, is .5*.5 = .25. I derived those numbers from H having a 50% chance of showing up in the first match and then a 50% chance of Fox losing, resulting in H being picked again.
With the probabilities of each one, you can then do a weighted average with the stock ratios determined up above to yield a composite stock ratio. Then use the scaling formula to convert it to a 1-10 ratio, round, and you got your matchup result.
As an example, consider this:
There are 2 stages, and on stage 1 the stock ratio is 6-4 and on stage 2 the stock ratio is 2-1. Stage 1 has a 70% chance of being played while stage 2 has a 30% chance.
First weight the stock ratios:
.7*6 + .3*2 = 4.8
.7*4 + .3*1 = 3.1
This means the average stock ratio is 4.8 to 3.1. Now scale it as a 1-10 ratio:
4.8*10/(4.8+3.1) = 6.08
3.1*10/(4.8+3.1) = 3.92
Round it and you get that this is a 6-4 matchup.
There are obviously issues with this approach, which are mainly due to preferences that players have for certain stages. For example, Peach in Melee has many viable counterpicking options, and it depends on the player's choice as well as what the opponent strikes. This makes it nearly impossible to calculate stage probabilities absolutely, but you can still approximate it (and also approximate certain stages as equal).
Also, this problem explodes enormously the more factors you consider. Here I only considered 2 stages and it is already very complicated. It's a start at quantifying counterpicks though.
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Start reading again if you skipped the counterpicks part.
One thing that this mathematical model makes possible is using actual data to determine matchup ratios. If you want to figure out Peach vs. Jiggly, take a look at matches between Mango, Armada, PC (when he went Jiggly vs. Mango), etc. Count total stocks that Jiggly lost in these sets and total stocks Peach lost in these sets. Then use the formula up above and get a ratio. If you are ambitious, you can break it up by stage and also compute the fractions of stage usage and use the formulas up above.
The biggest question in terms of using data is how do you know what valid data is? Like, I'm sure m2k would beat nearly anyone on any stage regardless of characters. Also, how old can data be while still being valid data? Yes, this is a serious issue to work out but that is not the focus of what I'm presenting here.
There are obviously other issues with this approach, but I believe it offers something much more concrete than has ever been used in the past. It allows for actual data input into equations to spit out a matchup ratio, rather than trying to arbitrarily weigh pros and cons of a matchup. Even without inputting data, it is much easier to conceptualize a character averaging 7 stocks to an opponent's 3 rather than trying to say a character can win 2 out of 10 matches.
Now here's my approach to it:
Imagine an infinitely long stock match. Let's say that for a Fox vs. Falcon match, on average, for every 6 stocks Fox takes, Falcon takes 4 stocks.
This simply translates to 6-4 in ratio form. Simple.
Now let's consider Fox vs. Sheik. Let's say, for the purposes of this model, that for every 6 stocks Fox takes, Sheik takes 5. Let's scale this to a ratio in the 1-10 form:
6*10/(6+5) = 5.45
5*10/(6+5) = 4.55
This roughly translates to a 55-45 matchup (or 5.45-4.55), which can then be rounded to 5-5.
I'd say the biggest fundamental difference between this interpretation and the traditional, "Street Fighter," interpretation is you are ignoring the fact that matches are finite and end after 4 stocks. Well, to me this is actually a more meaningful approach. A ratio of 10-0 can either mean that a character is getting ***** repeatedly or the character is barely losing repeatedly. They are two completely different situations being represented by the same ratio, and it makes no sense. If you instead say that on average a certain character takes 4 stocks while losing 1, you can easily see that either character has a chance of winning, despite it being very uphill for the worse character.
Also, while I'm not knowledgeable about Street Fighter, I'm pretty sure the game is a traditional fighter in the sense that there aren't stocks - just individual games/matches. Basically, it seems to me like Street Fighter's games are sort of an equivalent to Smash's stocks. So, why are we carrying over Street Fighter's ratio interpretation to games rather than stocks?
If you wish you may skip this next section, cause it is really complex and strays from the overall point I'm trying to make.
------------------------------------------------------------------------------------------------
Now, how do we deal with counterpicks? This is really difficult but I have an idea. Let's examine Pikachu vs. Fox in SSB64 (cause SSB64 is simpler and doesn't use Dave's Stupid Rule).
For this example I'll say that the first stage in a best of 3 is random between Hyrule and Dreamland. Let's say that we determine the stock ratios to be 5-5 on Hyrule and 7-4 (Pika takes 7 for every 4 Fox takes) on Dreamland.
In a best of 3 set, Hyrule and Dreamland have equal chances of being chosen for the first match. If Hyrule gets chosen, both characters have an equal chance of winning. If Fox wins, Pika will counterpick Dreamland (or another stage that Fox sucks about as much on), and if Pika wins, Fox will pick Hyrule again. Then it happens again the third match.
Sample space:
H = Hyrule, D = Dreamland
(first stage, second stage, third stage)
HDD
HDH
DHD
DHH
HH
DD
Compute probabilities of each element in the sample space (based on chances of the characters winning on each particular stage). The probability of HH, for example, is .5*.5 = .25. I derived those numbers from H having a 50% chance of showing up in the first match and then a 50% chance of Fox losing, resulting in H being picked again.
With the probabilities of each one, you can then do a weighted average with the stock ratios determined up above to yield a composite stock ratio. Then use the scaling formula to convert it to a 1-10 ratio, round, and you got your matchup result.
As an example, consider this:
There are 2 stages, and on stage 1 the stock ratio is 6-4 and on stage 2 the stock ratio is 2-1. Stage 1 has a 70% chance of being played while stage 2 has a 30% chance.
First weight the stock ratios:
.7*6 + .3*2 = 4.8
.7*4 + .3*1 = 3.1
This means the average stock ratio is 4.8 to 3.1. Now scale it as a 1-10 ratio:
4.8*10/(4.8+3.1) = 6.08
3.1*10/(4.8+3.1) = 3.92
Round it and you get that this is a 6-4 matchup.
There are obviously issues with this approach, which are mainly due to preferences that players have for certain stages. For example, Peach in Melee has many viable counterpicking options, and it depends on the player's choice as well as what the opponent strikes. This makes it nearly impossible to calculate stage probabilities absolutely, but you can still approximate it (and also approximate certain stages as equal).
Also, this problem explodes enormously the more factors you consider. Here I only considered 2 stages and it is already very complicated. It's a start at quantifying counterpicks though.
----------------------------------------------------------------------------------------------
Start reading again if you skipped the counterpicks part.
One thing that this mathematical model makes possible is using actual data to determine matchup ratios. If you want to figure out Peach vs. Jiggly, take a look at matches between Mango, Armada, PC (when he went Jiggly vs. Mango), etc. Count total stocks that Jiggly lost in these sets and total stocks Peach lost in these sets. Then use the formula up above and get a ratio. If you are ambitious, you can break it up by stage and also compute the fractions of stage usage and use the formulas up above.
The biggest question in terms of using data is how do you know what valid data is? Like, I'm sure m2k would beat nearly anyone on any stage regardless of characters. Also, how old can data be while still being valid data? Yes, this is a serious issue to work out but that is not the focus of what I'm presenting here.
There are obviously other issues with this approach, but I believe it offers something much more concrete than has ever been used in the past. It allows for actual data input into equations to spit out a matchup ratio, rather than trying to arbitrarily weigh pros and cons of a matchup. Even without inputting data, it is much easier to conceptualize a character averaging 7 stocks to an opponent's 3 rather than trying to say a character can win 2 out of 10 matches.
