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On July 23 2013 00:17 Quoonit wrote:
Correct me if I'm wrong (I may be missing some complexity or nuance here), but the following is true, and is independent of whether or not the series is continuous or broken into a Bo3 and extended into a Bo7.
If player A is better than B, say having a 51% chance of winning any single game vs. player B, then the more games they play, the law of large numbers will push the winrate of each player towards the projected underlying probability. Which is to say that smaller samples are noisier than large ones.
So, using a simple binomial distribution with the set of binomial parameters:
% chance of success (probability that the better player will win):
# successes needed (required # of wins in a series):
# trials (# of games in a series)
The probability of the better player winning always goes up with a larger set of games.
For example, with the initial binomial parameters (for a Bo3 series with 2 extremely evenly matched players):
% chance of success (probability that the better player will win): 51%
# successes needed (required # of wins in a series): 2
# trials (# of games in a series): 3
The better player wins that series 51.499% of the time.
If the series is extended to 7 games however:
% chance of success (probability that the better player will win): 51%
# successes needed (required # of wins in a series): 4
# trials (# of games in a series): 7
The better player wins 52.19% of the time.
For larger discrepancies between skill level (70% to 30% chance of winning between players), the dichotomy between short and long series increases even further:
% chance of success (probability that the better player will win): 70%
# successes needed (required # of wins in a series): 3 (short), 7 (long)
# trials (# of games in a series): 2 (short), 4 (long)
The better player wins the short series 78.4% of the time, and the long series 87.4% of the time.
This trend is also independent of how the series is broken up. For example, using our 70/30 split again. Here are the chances of certain outcomes in a Bo3 series:
Better player wins Bo3: 78.4%
Better player wins 2-0: 49%---------------------------------->Better player goes on to win Bo7 extended series 96.9% of the time
Better player wins 2-1: 29.4%-------------------------------->Better player goes on to win Bo7 extended series 91.6% of the time
Worse player wins Bo3: 21.6%
Worse player wins 2-0: 9%------------------------------------>Better player goes on to win Bo7 extended series 52.8% of the time
Worse player wins 2-1: 12.6%------------------------------->Better player goes on to win Bo7 extended series 65.2% of the time
Using the % chance of each Bo3 result to weight the Bo7 results:
87.4% chance of the better player winning the Bo7 extended series overall, which is identical to the result for continuous Bo7 series
Given a simple statistical model, this is the best answer.
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Katowice25012 Posts
On July 23 2013 02:05 Theberlinwall wrote:
well said.
on a different note i never actually thought that extended series was used to resolve potential map score issues. The rules makes a little more sense now. However, I still dislike it from a spectator point of view.
This is what motbob's anecdote from Lee gets at but is a bit muddled, the basic idea here is that a player cannot be knocked out while still having a positive record against another player due to the format being double elim. It's a fine enough ideal and if you talk to most players they love it, but it tends to make spectating kind of lame which is why people on TL rally against it.
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On July 23 2013 02:32 Heyoka wrote:Show nested quote +On July 23 2013 02:05 Theberlinwall wrote:
well said.
on a different note i never actually thought that extended series was used to resolve potential map score issues. The rules makes a little more sense now. However, I still dislike it from a spectator point of view. This is what motbob's anecdote from Lee gets at but is a bit muddled, the basic idea here is that a player cannot be knocked out while still having a positive record against another player due to the format being double elim. It's a fine enough ideal and if you talk to most players they love it, but it tends to make spectating kind of lame which is why people on TL rally against it.
Weird, I thought most players hated it with spectators being more split, at least in polls. The anti-extended-series people are much more vocal though.
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Right at least I'm polls, you are much more likely to get self-selection bias with strong negative opinions being overrepresented to some degree.
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A few random notes from calculating probabilities:
If Player A and Player B were on equal ground (50/50 chance of winning a game)...
There is a 50% chance for A to eliminate B in two unweighted series.
There is a 50% chance for A to eliminate B under extended series.
There is a 6.25% chance that, without extended series, A can be eliminated by B in two series but have a better overall map score between them.
There is a 75% chance for A to win in an extended series given a win previously.
There is a 25% chance for A to win in an extended series given a loss previously.
If Player A were slightly better than Player B (51/49)...
There is a 51.5% chance for A to eliminate B in two unweighted series.
There is a 52.19% chance for A to eliminate B under extended series.
There is a 6.37% chance that, without extended series, A can be eliminated by B in two series but have a better overall map score between them.
There is a 76.42% chance for A to win in an extended series given a win previously.
There is a 26.46% chance for A to win in an extended series given a loss previously.
If Player A were significantly better than Player B (70/30)...
There is a 78.4% chance for A to eliminate B in two unweighted series.
There is a 87.4% chance for A to eliminate B under extended series.
There is a 6.17% chance that, without extended series, A can be eliminated by B in two series but have a better overall map score between them.
There is a 94.94% chance for A to win in an extended series given a win previously.
There is a 60.03% chance for A to win in an extended series given a loss previously.
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motbob
United States12546 Posts
Upon further review, I reported the results of the second type of test I ran correctly, but it's actually a useless test in regards to whether extended series are good or bad. Can you spot why?
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On July 23 2013 02:55 motbob wrote:
Upon further review, I reported the results of the second type of test I ran correctly, but it's actually a useless test in regards to whether extended series are good or bad. Can you spot why?
Because extended series are bad! No math required.
I think we should tell the NFL about the extended series. So regular season results carry over into the Superbowl! A terrible comparison? Certainly.
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United States47024 Posts
On July 23 2013 01:15 KissMeRed wrote:
Scenario 1:
Player A wins 2-0 in winner's bracket. Player B wins 2-1 in loser's bracket rematch. Player A leads 3-2 overall in maps, but Player B advances. I consider this unfair.
Scenario 2;
Player A wins 2-0 in winner's bracket. Player B wins 2-0 in loser's bracket rematch. The series is a tie 2-2 overall in maps, but Player B advances solely based on the order of the matches. A won earlier and B won later, but B advances. Again, I consider this unfair.
If tournaments were simply about number of games won/head-to-head, then we would always run round robin tournaments with no bracket stage and decide the winner by the player with the most wins/best head-to-head in ties.
The nature of a bracket tournament is that it ascribes particular importance to specific games/series'. In this case, a match deeper in a tournament is considered more important. Losses in the loser's finals are considered more severe than losses in an earlier stage, perhaps because the match carries more importance and you are expected to bring more of your skill to bear.
You could actually have a similar discrepancy in head-to-head results vs. match winner in a Group Stage->Single Elim tournament, but we don't consider this a problem because bracket stage games are considered more important than Group Stage games and losing them is supposed to be more consequential.
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Technically, if we're talking about loser's bracket rematch, MLG Lee is right, it does benefit the more skilled player. It does so at the expense of other players at the same level of the bracket, though, and is primarily based on the luck of the bracket placement.
The problem isn't extended series, it's the circumstances under which it comes into play.
I do think that extended series is a good alternative to traditional double elimination for the Winner bracket vs Loser bracket match, because it gives a slight advantage to the underdog without being unfair to the winner of the first series, doesn't give an unfair advantage to one part of the bracket over another, ensures that all games actually matter, and is more interesting/easy to follow for the spectator.
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On July 23 2013 00:17 Quoonit wrote:
Correct me if I'm wrong (I may be missing some complexity or nuance here), but the following is true, and is independent of whether or not the series is continuous or broken into a Bo3 and extended into a Bo7.
If player A is better than B, say having a 51% chance of winning any single game vs. player B, then the more games they play, the law of large numbers will push the winrate of each player towards the projected underlying probability. Which is to say that smaller samples are noisier than large ones.
So, using a simple binomial distribution with the set of binomial parameters:
% chance of success (probability that the better player will win):
# successes needed (required # of wins in a series):
# trials (# of games in a series)
The probability of the better player winning always goes up with a larger set of games.
For example, with the initial binomial parameters (for a Bo3 series with 2 extremely evenly matched players):
% chance of success (probability that the better player will win): 51%
# successes needed (required # of wins in a series): 2
# trials (# of games in a series): 3
The better player wins that series 51.499% of the time.
If the series is extended to 7 games however:
% chance of success (probability that the better player will win): 51%
# successes needed (required # of wins in a series): 4
# trials (# of games in a series): 7
The better player wins 52.19% of the time.
For larger discrepancies between skill level (70% to 30% chance of winning between players), the dichotomy between short and long series increases even further:
% chance of success (probability that the better player will win): 70%
# successes needed (required # of wins in a series): 3 (short), 7 (long)
# trials (# of games in a series): 2 (short), 4 (long)
The better player wins the short series 78.4% of the time, and the long series 87.4% of the time.
This trend is also independent of how the series is broken up. For example, using our 70/30 split again. Here are the chances of certain outcomes in a Bo3 series:
Better player wins Bo3: 78.4%
Better player wins 2-0: 49%---------------------------------->Better player goes on to win Bo7 extended series 96.9% of the time
Better player wins 2-1: 29.4%-------------------------------->Better player goes on to win Bo7 extended series 91.6% of the time
Worse player wins Bo3: 21.6%
Worse player wins 2-0: 9%------------------------------------>Better player goes on to win Bo7 extended series 52.8% of the time
Worse player wins 2-1: 12.6%------------------------------->Better player goes on to win Bo7 extended series 65.2% of the time
Using the % chance of each Bo3 result to weight the Bo7 results:
87.4% chance of the better player winning the Bo7 extended series overall, which is identical to the result for continuous Bo7 series
This is correct. Then extend the logic thusly:
The odds of winning a Bo3 from your binomial math listed above, using 51% as the chance of the "better" player winning to demonstrate his edge, (51.499%) and the odds of him winning an extended Bo7 (52.19%).
Now, consider a player that must win either both Bo3 matches, or the second Bo3. That leaves us with:
•AA -win
•BA -win
•AB -loss
•BB -loss
write probability of A winning as p(A):
=> 2C2*p(A)^2 + 2C1*p(A)*(1-p(A))/2
=>1*p(A)^2 + 2/2*p(A)*(1-p(A))
=>p(A) * (p(A) + 1 - p(A))
=>p(A)
=> .51499
51.499% is the chance of a better player winning two Bo3 in double elim format (where second winner wins).
52.19% is the chance of that same player winning a Bo7.
We can do this again to the 70/30 split then too and say:
Chance of winning Bo7 (as above): 87.4%
Chance of winning the correct two Bo3's is the same as winning one of them: 78.4%
The better player always has the advantage, but his advantage is greater in a Bo7 than two Bo3 series. Flukes or not, one can fluke any game in any situation and these must be considered independent events.
Therefore Bo7 is the better format (in purely mathematical theory, excluding maps and map choice).
QED unless I'm mistaken. Feel free to point it out if so 
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First of all, excuse me, but I always thought that winning games is what determines the better player, not the other way around.
The fact that player agreed to participate in a tournament under specific conditions means he has to reconcile himself with a possibility of paradoxes like those mentioned in this thread. I never liked or supported the whole 'Loser/Winner brackets' format, but I can see the point behind their existence: to give defeated players/teams bigger margin of error and extend their tournament experience by granting them chances for a comeback.
But 'extended series' format - apart from OP's point of not really letting better player through - is absurd itself. Imagine Greece playing Euro 2004 soccer championship final from the 2-1 goal advantage their earned against Portugal in a group stage game - total nonsense. In Starcraft under those rules, there will be evantually tournaments with one matchup of players who didn't met before, and other match, with players who did met. Two matches at the same stage of tournament played under different rules - only because of random luck. Something is really wrong.
For years, instead of going for such complications, events had 1-game advantage for winners' bracket comptetitor, winner picks maps/ map order or some other handicaps (Day[9] once even mentioned 1 Bo5 advantage for winners' bracket player, but to me it's way too extreme). Everyone seemed happy with those simple conditions, so why MLG bothers to confuse them?
(Yet still, as I said, you participate ---> you can't complain about the format of shit you got into -- so it all comes down to us, fans complaining about the fact that math sort of disproves the logic background of this format.)
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TL;DR: Setting up a test where it's already known who the better player is is flawed since the premise of the competition itself is to determine who the better player is. The better player is defined as the player who wins the most number of games in an odd number of sets/games; the better player ALWAYS advances in ES because the better player is defined by who wins the most games in a Best of X series.
Assuming the data is correct, I find the arguments on either side of the aisle regarding "does the better player have a better shot of advancing" flawed in the definition of "better player".
I thought the point of having competition is that we don't know who the "better player" between a set of two players is. The "better player" isn't something that's a definitive, ELO-like thing; such an objective way of telling us who is a "better player" does not exist and may even come to be rejected if it did. We have tournaments and leagues so when someone says "you're considered a better player than that other guy...so prove it".
I think the premise behind the old MLG reasoning regarding the top team and up-and-comer team works more like this:
"Well, that team won 3-0 and the other won 3-2. They've each won a Bo5, so we actually don't know who the better team is. In fact, we have the tricky situation where, cumulatively, the team that's set for elimination actually won more maps than the other team, but the other team can argue that they won a series and better teams are determined by defined sets of games. To see who's better we need to have another set to see who's better (like how tennis or volleyball operate) or we can instead just make the one set longer."
If, in the MLG example given in the OP, MLG thought that the lesser team advanced then they're wrong.They'd also be wrong if they thought the better team did advance. The fact of the matter is that we may think we know who the better team/player going into the tournament is, but the point of having the competition is to actually find out since we don't know for certain.
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On July 23 2013 02:32 Heyoka wrote:Show nested quote +On July 23 2013 02:05 Theberlinwall wrote:
well said.
on a different note i never actually thought that extended series was used to resolve potential map score issues. The rules makes a little more sense now. However, I still dislike it from a spectator point of view. This is what motbob's anecdote from Lee gets at but is a bit muddled, the basic idea here is that a player cannot be knocked out while still having a positive record against another player due to the format being double elim. It's a fine enough ideal and if you talk to most players they love it, but it tends to make spectating kind of lame which is why people on TL rally against it.
What?????? Who
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On July 23 2013 04:39 wingpawn wrote:
But 'extended series' format - apart from OP's point of not really letting better player through - is absurd itself. Imagine Greece playing Euro 2004 soccer championship final from the 2-1 goal advantage their earned against Portugal in a group stage game - total nonsense. In Starcraft under those rules, there will be evantually tournaments with one matchup of players who didn't met before, and other match, with players who did met. Two matches at the same stage of tournament played under different rules - only because of random luck. Something is really wrong.
For years, instead of going for such complications, events had 1-game advantage for winners' bracket comptetitor, winner picks maps/ map order or some other handicaps (Day[9] once even mentioned 1 Bo5 advantage for winners' bracket player, but to me it's way too extreme). Everyone seemed happy with those simple conditions, so why MLG bothers to confuse them?
These are two separate issues.
I don't think there can be a very convincing argument for extended series within the loser's bracket of a tournament, because both players should go into that series as equals and it introduces an extra luck factor into the bracket for the players.
It's pretty standard, however, for double elimination formats to force the loser's bracket to beat the winner's bracket player in two matches before the winner's bracket player wins one. In that specific case, extended series has an effect that isn't innately unfair, just different. I don't think it's any more complicated than the alternative either, since you can display the score in X-Y and get a clear picture of what each player must do to win. In double elimination, you have to know which series is which to understand what's going on.
Suppose the bracket is Bo3 throughout. Player X beat Player Y 2-0 and put him in the loser's bracket and they now meet in the Grand Finals.
Player X wins game 1, but loses game 2 and 3. In traditional double elimination, Player Y won the first series and forced a second Bo3. In extended series, the score is now 3-2 in favor of Player X.
Player Y wins game 4 and Player X wins game 5.
In traditional double elimination, we have a 1-1 going into a deciding game. In extended series, Player X wins the tournament after game 5.
Now consider the same situation after a 2-1 first meeting in both cases. In standard double elimination, the result is the same. In extended series, Player Y wins the tournament after game 4.
You can look at this a few ways, 1) you have more games so it's more fair and there's more content in the tournament 2) each game really mattered in this case, so you have to be at your best throughout the whole tournament.
I don't think either are clearly better and I don't think extended series is inherently worse, provided it's only in the above situation.
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On July 23 2013 05:27 TrippSC2 wrote:+ Show Spoiler +On July 23 2013 04:39 wingpawn wrote:
But 'extended series' format - apart from OP's point of not really letting better player through - is absurd itself. Imagine Greece playing Euro 2004 soccer championship final from the 2-1 goal advantage their earned against Portugal in a group stage game - total nonsense. In Starcraft under those rules, there will be evantually tournaments with one matchup of players who didn't met before, and other match, with players who did met. Two matches at the same stage of tournament played under different rules - only because of random luck. Something is really wrong.
For years, instead of going for such complications, events had 1-game advantage for winners' bracket comptetitor, winner picks maps/ map order or some other handicaps (Day[9] once even mentioned 1 Bo5 advantage for winners' bracket player, but to me it's way too extreme). Everyone seemed happy with those simple conditions, so why MLG bothers to confuse them?
These are two separate issues. I don't think there can be a very convincing argument for extended series within the loser's bracket of a tournament, because both players should go into that series as equals and it introduces an extra luck factor into the bracket for the players. It's pretty standard, however, for double elimination formats to force the loser's bracket to beat the winner's bracket player in two matches before the winner's bracket player wins one. In that specific case, extended series has an effect that isn't innately unfair, just different. I don't think it's any more complicated than the alternative either, since you can display the score in X-Y and get a clear picture of what each player must do to win. In double elimination, you have to know which series is which to understand what's going on. Suppose the bracket is Bo3 throughout. Player X beat Player Y 2-0 and put him in the loser's bracket and they now meet in the Grand Finals. Player X wins game 1, but loses game 2 and 3. In traditional double elimination, Player Y won the first series and forced a second Bo3. In extended series, the score is now 3-2 in favor of Player X. Player Y wins game 4 and Player X wins game 5. In traditional double elimination, we have a 1-1 going into a deciding game. In extended series, Player X wins the tournament after game 5. Now consider the same situation after a 2-1 first meeting in both cases. In standard double elimination, the result is the same. In extended series, Player Y wins the tournament after game 4. You can look at this a few ways, 1) you have more games so it's more fair and there's more content in the tournament 2) each game really mattered in this case, so you have to be at your best throughout the whole tournament. I don't think either are clearly better and I don't think extended series is inherently worse, provided it's only in the above situation.
What I feel after thinking those rules through:
Okay, so long story short, instead of no odds/1 game odds in Bo3 or Bo5, winner bracket guy gets 1 game/2 game odds in Bo7, depending on his previous performance against that rival, right? Strange, but might be acceptable for many, I guess.
It's largely a matter of taste, but I always felt that 'the underdog' shouldn't be punished with any point disadvantage at all. After all, he is so often punished for 1-2 defeat in super-close series that could've gone either way. Maybe the losers' brackets should merge with winners' at earlier stage of the tournament? Or, perhaps, to compensate for having 'weaker' players in their bracket, losers should make some sort of group stage between each other to increase the number of games and difficulty of getting through the bracket, so it could match the difficulty of winners?
Nauseating issue. Just copy/paste old BW OSL format people and live happily ever after 
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On July 23 2013 02:38 jalstar wrote:Show nested quote +On July 23 2013 02:32 Heyoka wrote:On July 23 2013 02:05 Theberlinwall wrote:
well said.
on a different note i never actually thought that extended series was used to resolve potential map score issues. The rules makes a little more sense now. However, I still dislike it from a spectator point of view. This is what motbob's anecdote from Lee gets at but is a bit muddled, the basic idea here is that a player cannot be knocked out while still having a positive record against another player due to the format being double elim. It's a fine enough ideal and if you talk to most players they love it, but it tends to make spectating kind of lame which is why people on TL rally against it. Weird, I thought most players hated it with spectators being more split, at least in polls. The anti-extended-series people are much more vocal though.
That has proven to be false on these forums. I already posted the old data where it was 2/3 opposed and I wonder which players Heyoka is talking about because I've seen quite a number opposed to it. Maybe he's talking about TL players like Tyler & possibly Jos. Keep the math blogs coming on extended series. Weeeeeeeeeeeee.
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The result in the original post might surprise people, but it makes sense when you consider the implications of player skill in a broader bracket. Consider the following:
Suppose we have two players that I shall name GOOD and BAD. Further suppose that I am omnipotent and can determine with full accuracy that GOOD beats BAD exactly 70% of the time. We place these two players into a double elimination tournament with extended series, each round being best of 3.
Scenario 1: GOOD beats BAD in the winner's bracket (expected result). BAD drops to lowers. Because BAD is unfavored against GOOD, it is extremely likely that BAD is also unfavored against other players as well. Because GOOD is favored against BAD, it is extremely likely that GOOD is also favored against other players as well. As a result, BAD has a high probability to be knocked out of the tournament in lowers long before he ever meets GOOD again. The simple conclusion is thus: is GOOD beats BAD in the winner's bracket, there is a low probability that an extended series will even happen. For the sake of argument, lets say there is a 5% chance that GOOD meets BAD again. When this does occur, GOOD has a very high chance to beat BAD as a result of the extended series setup (GOOD begins with a lead).
Scenario 2: BAD beats GOOD in the winner's bracket (unexpected result / the "fluke"). GOOD drops to lowers. As we said before, because BAD is unfavored against GOOD, it is extremely likely that BAD is also unfavored against other players in the tournament. Consequently, BAD has a high probability of falling to lowers sooner rather than later. Similarly, since GOOD is favored against BAD, GOOD has a high probability of advancing through lowers. Therefore, there is a much higher probability that GOOD will meet BAD an extended series will happen. Lets suppose there is a 20% that GOOD meets BAD again. When this does occur, BAD has quite an edge due to the extended series setup (BAD begins with a lead). So, although GOOD is favored in an individual match against BAD, BAD still has a higher probability of winning in an extended series.
Based upon these (somewhat winged) numbers, we see that, when an extended series DOES occur, MUCH more often it is a bad player starting with a lead against a good player. So, "worse players" will win more often in an extended series double elimination bracket.
To clarify, we are not saying that worse players have a statistical advantage overall. Rather, in a double elimination bracket, most of the extended series will be a worse player beginning with a lead over a better player.
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On July 23 2013 02:32 Heyoka wrote:Show nested quote +On July 23 2013 02:05 Theberlinwall wrote:
well said.
on a different note i never actually thought that extended series was used to resolve potential map score issues. The rules makes a little more sense now. However, I still dislike it from a spectator point of view. This is what motbob's anecdote from Lee gets at but is a bit muddled, the basic idea here is that a player cannot be knocked out while still having a positive record against another player due to the format being double elim. It's a fine enough ideal and if you talk to most players they love it, but it tends to make spectating kind of lame which is why people on TL rally against it.
it would be a fine enough ideal if MLGs format (the way brackets feed in to one another) didnt cause players to meet each other more often than you would expect through random chance. because at most mlg events to date the way brackets had been preformed rather than having randomized group selections it was specifically designed to make people play the same people over and over. this is bad from a purely "we want our bracket to be good sense" but also undermines any argument that they dont want players to go out with positive win rates vs people. because the easiest step to stopping that happening is to stop players meeting twice before the final.
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It's bad either way. Tournaments are not meant to determine (let alone aid) better players, they're meant to determine winners. The format doesn't matter in that context.
Extended series is just a tunnel vision solution to the inherent ugliness of double elimination systems when it comes higher-lower bracket interaction. There's really no better alternative to extended series - they're all terrible, because the underlying format (double elimination) is terrible.
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EPT
