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| RenSC2 United States. May 26 2012 05:06. Posts 265 | Profile Blog # |
Have you ever wondered how much the percentages shift when a game goes from a best of 1 to a best of 3/5/7? How much difference does it really make? I was curious and so I did a little math.
If 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.
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| | Playing better than standard requires deviation. This divergence usually results in sub-standard play. |
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| McFortran United States. May 26 2012 07:01. Posts 79 | Profile # |
On May 26 2012 05:06 RenSC2 wrote:
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.
Suppose you require W wins and L losses. The sequences of wins/losses must always end with a win, so you need the number of possible combinations resulting from combining (W-1)+L objects, with (W-1) and L of them indistinguishable from each other. This is given by (W-1+L)!/[(W-1)!L!].
For example, in a Bo7, the number of ways of winning 4-3 is 6!/(3!3!)=20. The number of ways of winning 7-3 is 9!/(6!3!)=84, etc.
You could actually test if independence is a valid assumption from TLPD. Obviously if we knew the win rates of players in a matchup, it would be easy to determine any covariance between matches in a BoX. Without it, however, the sequence of wins and losses in a best of 5 or 7 works as well. Given all of the Bo7's that ended 4-3, for instance, it could be tested to see how often the sequence LLLWWWW occurs compared to how often it should under independence. I'm not sure if there's enough data to do a reliable inference test, however.
On May 26 2012 05:06 RenSC2 wrote:
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.
Yeah, people place way too much emphasis on winning tournaments. Consider a 64 player single elimination Bo3 tournament where every player is of equal skill with independence between games. The winner of a Bo3 will win, on average, 83% of their games in that Bo3 (given that they win, they have an equal chance of going either 2-0 or 2-1). The winner will have won all of their Bo3's, giving them a win rate throughout the tournament of 83% on average, yet they have the exact same skill as the other players. |
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| Iranon United States. May 26 2012 12:02. Posts 963 | Profile Blog # |
Your example of a player favored 55-45 isn't the best way to look at things. If two players are that evenly matched (yes, 44-55 is a very even match, contrary to the large-scale balance stats we're used to looking at), they SHOULD still both have a pretty good shot at taking the other one out, even in something crazy like a BO9.
If you look at a more reasonable example, say one player has a 66% chance to win, things look a little different. I'm pulling the 66% number not quite out of thin air -- in chess, a difference of 100 Elo is significant, but not crushing. A 1400 player should lose to a 1500 player about 2/3 of the time, and a 2300 player should beat a 2200 player about 2/3 of the time. (This compounds exponentially in theory, but of course a 1000 player would never actually beat a 2000 player, even if they played way more than the expected 3^10 games.)
In a BO1, the 1500 player has a 67% chance to win. (against a 1400 opponent)
In a BO3, the 1500 player has a 75% chance to win.
In a BO5, the 1500 player has a 80% chance to win.
In a BO7, the 1500 player has an 83% chance to win.
That's exactly what I'd want to see. If there's a noticeable difference in player skill, then any series at all should be much better than a BO1, and really long series (7+) should be all but a forgone conclusion.
Edit: I'm an idiot, I somehow didn't see the paragraph where you included 70%. Whoops. I'll leave the post up anyway. You ask "isn't it crazy that even if you're clearly the better player (70% win chance), your opponent still comes out on top 1 time in 8?". Well no, not really. A 30% loss rate sounds low when we're used to almost all pros being on roughly even turf, but 30% is not all that much of an unlikely event. If you roll two dice and get a 12, you don't flip a shit because you hit an outcome that has probability 3%. You should be an order of magnitude less surprised when the 70% win chance player drops a game. It happens. Last edit: 2012-05-26 12:10:43 |
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