Theory Fighter meets Theory: How do matchup numbers affect sets and tournaments?

 Preface:

This article can get a little lengthy and also makes some math references, but in general I tried to keep the math language down to make it more readable for a general audience.  If you are the kind of guy who likes math then I can write a more detailed description how I got these results. 

I will also note this is my first time trying to type on this website so some of the formatting may be screwed up because it didn't copy perfectly from MS Word.

P.S.  If you are lazy and don't care about how I got this, skip to the results

Introduction:

Tier lists and match up charts…We all know of them and we all know how much trouble they can cause. They are often our way of ranking against all the other characters in the game. In an ideal world there would be no problem with match up charts, but of course we know quite well that our world is far from ideal. A single trip to any major message board will show you this fact.  You end up with arguments that are a combination of egos, inexperienced players, over-exaggerations, under-exaggerations, trolls, random posters who add nothing to the debate, and then the occasional knowledgeable poster who finally adds to the discussion. So I decided to do a mathematical analysis that would (hopefully) be much less controversial and also provide some useful information at the same time. Now before we get to the meat of this I just wanted to state my credentials.  I graduated from Washington State University (GO COUGS)  with a degree in both Chemistry and Mathematics.  I am currently doing my PHD in Chemistry at Louisiana State University (GO TIGERS!) specializing in Physical Chemistry which is a probability heavy field.

So one of the common tools we use in rating characters are match ups.   We usually define this as the chance of winning a match up given two players of equal skill.  This is usually a measure of how balanced or unbalanced a matchup is.   Although everyone should know, we say a 5-5 is a balanced match up or that it is a coin flip who will win.   A match up that is say 8-2 is a match up where one character’s tools overpowers the other’s to the point that the matchup is very one sided.   Now I am going to keep this general to keep the arguments to a minimal so I won’t use any game specific examples. 

You might say this next part is obvious for 95% of the population of the world.  If I wanted to know my chance of winning a 6-4 match up in my favor it is simply I have a 60% chance. That is rather obvious, but just saying the chance of winning a tournament match of this type is simply 60% is terribly wrong!  In actual application we never play just a single game, we always play sets!  In a set we play till one person hits the critical number.  In a 2 out of 3 set we stop if a player wins 2 games because the third one is irrelevent.   

So the question becomes: "What is the probability of winning a set?" The approach we can take is to treat this in the same manner in the same way someone would treat a simple coin flip. Fortunately this math is well explored.  Let’s say we have a fair coin where the chance of getting heads or tails is equal, basically a 50/50 chance.  We want to know the chance of getting 2 heads out of 3 flips (which translates to winning a best 2 out of 3 set).  According to a normal coin flip there are 8 possible outcomes which you can write out to prove it to yourself.  Out of those 8 possibilities here are the orders that would qualify as getting two heads

HHT, HTH, and THH

So there are 3 possibilities and so the chance of getting two heads is 37.5% or 3/8.  Now at the same time we would also say that getting 3 heads is just as good because in a tournament if I win the first two matches we don't even play the third one. So we add the 1/8 chance of getting 3 heads and we find our chance is 50% which is what we would expect. Both players have an equal chance of winning the set given an equal match up.  Now you might ask,  what happens if the chances of getting heads and tails are not equal (For example: Chance of getting heads is 60% and Tails is 40%).   Fortunately for us this math has already been worked out and is ready for us to use. The name in math for this distribution is called the binomial distribution. If you really want to look this up go for it, it is very easy to find and use. If you wish to replicate my results for a set of any size all you need to do is add up the terms that are equal to or greater than the number of wins required to take the set.  So for a FT5 (or best 5 out of 9) all you need to do is add up the probability you will win 5, 6, 7, 8, or 9 times.

The results for individual matches:

So what does this break down to?  Well for starters let’s just examine how good/bad match ups can affect a simple tournament match.

   This table contains the calculated results of common sets that occur within a standard tournament.  The way to read this should be straight forward,  just find what the match up of interest is (For example if your character has a 4-6 match up against your opponents) and this will tell you the chance of winning each set type.  

As you can see the probability of winning a 2 out of 3 set quickly differs from the individual chances of winning a single game.  Each time you add an additional round the chance of overcoming a bad match up goes down and the chance of winning a good match up goes up. You can also see that in grand finals the match is inherently slanted toward the winner side because the loser must win twice while the winner only needs a single win.  

So this gives us the chance for each individual match inside of a tournament so our next question is how can we use this information to calculate the chance of winning a tournament which is a combination of several of these matches? In order to extrapolate this into a full sized tournament there are a couple issues to deal with. The tournament size will determine the probability because the more matches you play the more chances for something to go wrong.  For simplicity I am going to use a 4 man double elimination bracket as an example since it is the simplest case and actually shows some rather interesting information.  All the probabilities will drop as the tournament size increases so these in a sense are the "best case" odds.  

The second issue is that every tournament is a different combination of match ups and the probability will reflect that. Sometimes you might get unlucky and get a bad match up out of the gate or you might never run into one. So to get around this complication I suggest using an average match up instead of an absolute match ups. There are two ways to do this.   

The most accurate way is to average the match ups for a specific tournament bracket.  For instance let’s say my characters match ups with the other 3 characters are 3-7, 5-5, and 5-5. So my average match up for the tournament will be about 4.3-5.7 or slightly better than 4-6. You can also find the error associated this rather easily. For this example the standard deviation is about 1. So we would expect this characters chance for any given tournament set up is between the chance of 5-5 or 3-7 which.  The problem with this method is if you extend it to large tournaments it becomes a nightmare to calculate. 

Now the second way is to create an average match up for my character’s match up as a whole.  Or in other words average your match up chart and figure out your character’s average match up.   The problem with this approach is the fact that some characters are more likely to be in a tournament than others.You would have to work out some system to make sure it is weight properly, but I am not going to get into that right now. For my small example the first method will work just fine.  

When all is said and done: 

When you finally finish with the math (It took me two days to finally work it all out), YOU FINALLY get to the Results!  Here they are boys. 

 You can also display this information as a graph for easy visualization:

These results shows that based on shear probability that the chance of winning even with a small tournament bracket rapidly changes with the average matchup.  Just having a slight edge can almost double your chances to win while having a slight disadvantage can drop you by a large amount.   The 5-5 probability makes sense since if everyone has the same chance out of 4 players then each player has a ¼ chance. For my example of a character who’s average match up in the tournament is 4-6 with an error of 1. The most likely value is around 8-9%, but depending on luck you can be between 1-25% if you get a favorable bracket.  If I get screwed and have to play the 3-7 more than once or early on then my chances go down, while if I never play the 3-7 match up then my chances go up to 25%. 

What to take home from this:

So you might be asking the question, ok why should I care? What these results tell us is that bad and good match ups have a massive effect on the ability of a character to make it through a tournament and win it. Now you might be thinking “Oh so that means I have to play top tier in order to win.”, but there is another side of this.   

Let’s say you have a match up that is really a problem for you even if it isn’t supposed to be bad. Having this one match up that gives you difficulty can provide your own personal barrier to winning and drop your chances. It is an encouragement to get better in those match ups because as you see above just increasing your odds of winning a single match up greatly increases your odds of winning the whole thing! Just giving yourself even a slight edge over everyone else massively improves your odds of winning. It isn’t about necessarily what we traditionally think about match up numbers, but rather you can also view this as your own personal match up chart and how it effects you!

I may release a more detailed version if people wish to see it, but for now that is all.

-LoyalSol