EPTIf you have X% chance to win a best of one match and we assume (the assumption, referred to later) that you have that same X% to win each game in a best of three, you will have X^2 + (X^2 * (1-X)) *2 percent chance to win (50% would be written as 0.5 in formulas).
If we use the same assumptions for a best of five, the formula comes out to:
X^3 + (X^3 *(1-X))*3 + (X^3 *(1-X)^2) * 6
And for a best of seven, the formula is:
X^4 + (X^4 * (1-X)) * 4 + (X^4 * (1-X)^2) * 10 + (X^4 * (1-X)^3) * 20
Here's the interesting part. Let's say you have a 55% chance to beat an opponent in a Bo1 (pretty typical among pro players). In a Bo3, your win% goes up to 57.5%, in a Bo5 your win% goes up to 59.3% and in a Bo7 your win% goes up to 60.8%. Doesn't that seem a bit low? Like, even in a best of 7... typically considered the pinnacle of determining skill in sports, the 55:45 better player will only win about 6/10 times. A Bo7 only tacks on an extra 5.8% chance of winning to the better player. Even worse, a Bo3 provides almost zero benefit over a Bo1 when it comes to determining skill if we keep our assumption.
Of course, for someone with a bigger advantage, the winning chances do gain more from longer series. For someone with a 70% chance to beat an opponent in a Bo1 (70% win% is almost unheard of in professional gaming), his chances go to 78.4% in a Bo3, 83.7% in a Bo5, and 87.4% in a Bo7. In this case, a Bo7 tacks on an extra 17.4% chance to win. This seems better, but still far from the ultimate determination of skill. I mean, even if you were a dominant 70% even against other top players, making you the best player by far, you're still about a 1 in 8 chance of losing in a best of 7.
With these numbers in hand, what I'm getting at is that some of this talk about slumps and "best player" is completely overblown. Was Nestea ever really in a slump? Or did he just roll a 1-4 on a 10 sided dice in a few series? Was Jjakji ever really that good and is now slumping or did he just hit a very lucky streak and is now in an unlucky streak?
On the one hand, these numbers make repeat winners all the more impressive. On the other hand, it does force us to really question past winners. It's a situation where the odds of winning are terrible for everybody, but someone has to win and maybe sometimes those winners weren't really the best players, even at that time.
These numbers also bring up an interesting point. Are best of 3/5/7s really necessary? Percentage-wise, they barely tip the scales in most cases. People whining about losing an open bracket Bo1 really have nothing to whine about. Their odds to win would be practically no different in a Bo3. If we hold to our assumption, it seems like the real benefit is to fill a time-slot or create a story of comebacks or domination.
So the big question becomes, does the assumption hold? There are three scenarios that really effect the assumption, in my opinion.
The first scenario is where a losing player gets to pick the next map in a Bo3. If we make a new assumption that certain maps will favor one player over the other, then the player to win the first match has a bigger advantage than normal in the final two matches because he only needs to win one of them and he is guaranteed to get to play his favorite map of choice for the win (or win 2-0). In this case, I would suspect that the final win% in a Bo3 is even closer to the Bo1 win% than the formulas would conclude. Luckily, I do believe there is a solution to this. If you have a map pool of 7 maps, I think the 3 possible maps for a Bo3 should be determined via a veto system. So each player would veto 2 maps, and then the loser of round 1 could only pick from the remaining two maps, rather than his most imbalanced map in a 7 map pool.
The other scenario I want to discuss is a scenario where one player can only effectively use one strategy (whether that be cheese or just a single build that can be abused). Usually that build will be some sort of timing attack, but it could also be some triple expansion build that is extremely vulnerable to early timing attacks, but will beat other macro oriented play. In a case where a player only has one build, an opposing player may lose to it the first time, but prepare for it the next two and take two solid wins. In this case, a Bo3/5/7 certainly shines over a Bo1. Unfortunately, it's impossible to know exactly how much this effects a Bo3/5/7 (you won't necessarily know your opponent only has one build). But at least we can say it's something to defend the Bo3 that is so commonly played in tournaments these days.
The third scenario is about the issues of confidence/overconfidence/depression after winning/losing the first game of a set which may affect the assumption. However, those things are extremely player dependant, so I don't think I could adequately address them here.
Final Note: I arrived at my formulas by brute forcing the win/loss options and then counting the number of times each scenario came up, which helped me create the formula. Does someone have a better method for coming to those formulas so that a Bo9 or Bo11 or Bo21 would be easier to calculate? I am not a math expert by any means.
