If you read my previous blog post you saw I looked at how advantages effect 2 out of 3 sets as well as 3 out 5 since these were related to tournament success; however, I quickly found the math behind the problem grew too large too fast. So in order to actually get a result I wrote a computer code to simulate multiple tournaments and collect statistics. The math behind it is a little bulky, but the general idea is straight forward. Set up a bracket (size chosen by user), run each match, and record the results. Just like a real tournament except this one is controlled through probability. As I also stated in the last article, I don't want to get too technical because this is meant for a general audience. More technical explanations can be given upon demand (It includes words like Monte Carlo). I am also planning on releasing the code at a later date, but as of right now it is not the most user friendly.
With this code we can simulate a wide variety of tournaments by adjusting the individual match ups. By averaging over different tournament brackets it is possible to obtain the probability of winning for a scenario.
Introductory Data
So one of the first cases we can study is a very simple example. Let's say we have a tournament where every player has the same chance of beating any other player with one exception: You are the one player whose has different odds of beating the other players. This is actually a rather simple set up, but it gives a great deal of information.
For this data, I took player 1's match up probabilities against everyone else in the tournament and scaled them up incrementally from 5% up to 95%. Obvious at 0% the result is easy to predict since if you have no chance of winning a match you have no chance of winning a tournament and likewise at 100% if you always win every match up you will never lose a tournament.
I also changed the number of players to look at the effect of the bracket size. The numbers you see are in powers of 2 since new player slots are introduced when the bracket doubles in size. The size of a typical major is above 128 while local tournaments are around 8-32.
The contour plot shows the lowest win tournament percentages in blue while the highest are displayed in yellow. The x axis is the number of players in the bracket and the y axis is the match upwin percentage.
One of the first ways we can check the data for accuracy is to look for values which we can easily calculate own our own. The easiest case is when the match win rate is set to 50%. With the way the bracket was designed, if player 1 has a 50% win rate then every player has the same chance of winning. Thus the expected win rate is equal to 1/(# of players). If you calculate this for each bracket you see the simulation data matches this. So the simulation matches perfectly with the expected value for the easily calculable case.
The first trend of interest is that for any given win rate, as the bracket increases in size the win rate also declines. This is actually to be expected. If you have more players you would expect there to be more chances to for you to lose. While this is obvious, one of the things that was not entirely obvious was large this effect was. For values less than 50% the result does not change much (Obviously since you lose most of your matches), but for values above 50% there is a very interesting trend. Smaller win rates drop off much much faster than larger win rates!
If you notice at 60% for a small local you have a very reasonable chance of winning, but by the time you scale it up to the size of a major a 60% match win rate you will win roughly 1 out of every 20. This means that even if you are the best player if you are only slightly better you will still lose more majors than you win. A 60% advantage is not nearly enough to be a consistent winner because odds are you will get randomed out in a bracket that size. In fact only match win rates of 85% or higher actually translate to winning over half of your tournaments consistently. It actually speaks to the skill level of many of the top players when they are able to consistently win or place at majors. They aren't just slightly better, they have an enormous edge.
I will be going into much more detail in my upcoming posts. I will be going over probability of placing in the top 8, how first round byes effect your chances of winning, what happens when a top player is in your bracket, and other related topics. Hope to see you guys again.
Q&A
Q: But what if (Insert Player Here) has a bad day and gets knocked out early?
A: This method actually takes that into account. When we talk about win rates that includes fluke losses. I mean if a player like Daigo had a 96% win rate against other players that 4% are the rounds where Daigo had an off game and there is a small chance that he could have two bad games in a set resulting in getting knocked into losers early. This is all accounted for by the numbers.
Q: Well in a real bracket I'm going to have a lot of varying win rates. So why is this any good?
A: Yes in a real bracket you do, but the thing is the tournament win rate is still related to what you see. For instance if half your match win rates are 60% and the other half are 50% then your tournament rate is in between 50% and 60%, but you would expect a higher deviation. In other words it becomes bracket dependent. You could have all your advantaged match ups on your side of the bracket and like wise you can have all your even match ups. But at worst you will have the 50% probability and at best you will have the 60%.
More detailed situational data will be presented in the later additions. Right now this is still very general.
Q: But what about character advantages? How does that factor into this
A: That is also taken care of by the numbers. If I am equal skill to my opponent and we are playing a match up that is bad for his character it will be reflected by my win rate vs his win rate.
