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I want to preface this by saying that I am in favor of having each series in a tournament be a separate event. I do not prefer the extended series method. However, after the last SOTG podcast where Day9 started to use math to describe the differences, but then stopped before taking it to the logical conclusion I was curious what the difference was.
Obviously the pros of the extended series BO7 include that you cannot be eliminated by someone who you have a winning or even record against. However, this does not stop problems of non-transitivity of superiority, a term that I believe I have just coined. By this I mean what Greg mentioned in SOTG where Player A may have an advantage over Player B, and Player B over Player C, but Player C has one over player A. In this case the extended series does not help us to determine who the best amongst the three is.
I believe this is a vital difference between SC2 and Halo. There are different races and play styles in SC2. This contrasts somewhat with halo where, while there are definitely different styles, most games come down to better execution. In Halo we are much less likely to run into non-transitivity of superiority.
Anyway, with that being said, there are a few things I wanted to figure out. Mostly the difference of the odds of a worse player winning a BO3 versus a BO7, but also the odds of the better player coming back when he happens to be down.
Here are the results:
![[image loading]](http://imgur.com/DyrBf.jpg)
I would tend to assume that in a high level tournament such as MLG Dallas the edges are between 50-50 and 80-20, and that therefore the extended series would allow the player with the advantage to advance roughly one time more per fifteen rematches that occur in the losers bracket. That is, it is roughly 6-7% more likely for the player with an advantage to win a BO7 than a BO3, although that number isn't fixed; it varies by the degree of skill difference and is maximized around 70:30.
Interesting things to note:
-The player starting down 1-2 is still expected to win the Bo7 series about a third of the time even if they are equally matched.
-The player starting down 1-2 is favored to win if he expects to win 65% of matches versus the given opponent.
-The player starting down 0-2 is favored to win if he expects to win 70% of matches versus the given opponent.
Please note, there are many confounding factors in analysis as it pertains to a tournament setting.
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Thanks for taking the time to make a chart and write it up. Interesting to see it all laid out.
The result with 70/30 "benefiting" most is cool. I wonder if you extended it to higher boX, what would that curve look like? I would expect it to be quite different than what you found, that is, a very small edge prefers the longer series the most. If you play a best of million and one, the better player should win the series 99.99% of the time no matter how much better they are, so 52-48 goes to the 52 way more than a bo3, and is preferred about 47% of the time, according to your chart.
Incidentally, a metagaming excellent player would prefer shorter series because it increases the chances that a weak player gets through on a fluke, who they can beat more easily than a strong player, n'est-ce pas?
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I'm not sure how practical this is to everything because it is essentially arbitrary percentages multiplied against one another, but it is interesting because at its most basic level there is some truth to this math. Theoretically there is a probability of each player beating their opponent even though there is literally no way to ever determine this.
In other words, this is probably the most concrete proof of how effective the series is, but in the grand scheme of things I don't think it means much :/
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On November 11 2010 14:00 EatThePath wrote:
Incidentally, a metagaming excellent player would prefer shorter series because it increases the chances that a weak player gets through on a fluke, who they can beat more easily than a strong player, n'est-ce pas?
I believe the best player in a tournament will always prefer the series be as long as possible.
I would guess that in a tournament with a couple dominant players it is the tier directly below them, but still well ahead of most of the rest who would benefit most from a shorter series. This player would enjoy both a decent shot of the couple of players above him being knocked out in a fluke and would still have a great advantage over the others. He would place worse, on average, but win the thing outright more often.
Of course that is a very specific example for a very general statement.
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On November 11 2010 14:08 imPERSONater wrote:
I'm not sure how practical this is to everything because it is essentially arbitrary percentages multiplied against one another, but it is interesting because at its most basic level there is some truth to this math. Theoretically there is a probability of each player beating their opponent even though there is literally no way to ever determine this.
In other words, this is probably the most concrete proof of how effective the series is, but in the grand scheme of things I don't think it means much :/
Well many teammates have a good idea of their odds of winning versus each other. This is because over many games the standard deviation gets smaller and smaller. You can never find one number, but you can make the range get tighter and tighter.
Of course this goes back to my caveat at the end of the original post; players may not play in tournaments as they do in practice, or there may be mind games that go on that favor one player in the event of teammates facing off.
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Canada8035 Posts
One thing to keep in mind is that with a longer BoX series, fatigue starts coming into play. It's entirely possible that the "better" player will tire faster than his opponent and make more mistakes because of that. It's pretty much impossible to gauge how much stamina a player has though.
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If the Bo3s are split, whoever won the second Bo3 advances (the one in the losers bracket). Is that taken into account?
Edit: Yes it is. The only Bo3 that matters is the second one. As for the Bo7 odds, a split Bo7 has the same odds as a non-split one.
So basically the math is all accurate, all that's left to prove is that the chances of winning a Bo7 are always higher than the chances of winning a Bo3.
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On November 11 2010 14:29 Spazer wrote:
One thing to keep in mind is that with a longer BoX series, fatigue starts coming into play. It's entirely possible that the "better" player will tire faster than his opponent and make more mistakes because of that. It's pretty much impossible to gauge how much stamina a player has though.
Yes, that is a confounding factor that is not possible to account for in generalized analysis.
On November 11 2010 14:34 jalstar wrote:
If the Bo3s are split, whoever won the second Bo3 advances (the one in the losers bracket). Is that taken into account?
That fact is irrelevant to the impact of a second Bo3 versus a continuation.
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On November 11 2010 14:09 MannerMan wrote:Show nested quote +On November 11 2010 14:00 EatThePath wrote:+ Show Spoiler +
Incidentally, a metagaming excellent player would prefer shorter series because it increases the chances that a weak player gets through on a fluke, who they can beat more easily than a strong player, n'est-ce pas?
I believe the best player in a tournament will always prefer the series be as long as possible. I would guess that in a tournament with a couple dominant players it is the tier directly below them, but still well ahead of most of the rest who would benefit most from a shorter series. This player would enjoy both a decent shot of the couple of players above him being knocked out in a fluke and would still have a great advantage over the others. He would place worse, on average, but win the thing outright more often. Of course that is a very specific example for a very general statement.
Yes I suppose it's pointless to talk about this without being more specific, but fun to think about the different situations. I like your take on it.
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http://www.wolframalpha.com/input/?i=x^2+2x^2*(1-x), x^4 + 4x^4*(1-x) + 10x^4*(1-x)^2 + 20x^4*(1-x)^3
The red line is the chance of winning a Bo7, the blue line is the chance of winning a Bo3. Hopefully it's clear that the red line is always above the blue line from x = 0.5 to x = 1 (x being the better player's chance to win). That's a graphical proof that a Bo7 is always better for the better player, an algebraic proof would be quite messy (take the equation for p(win a bo7) and subtract p(win a bo3), it should come out as > 0 any time 0.5 < x < 1)
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EPT
