WARNING: Math ContentThis post contains thorough explanations of the methodology and formulas involved in tabulating the rating points that will be used in the "Chase the Ace" Qualifiers to determine the Playoff roster and seeding. If you are not interested in the exact math, a short summary follows:
Alright, for those still with us, we have the full summary. The rating points calculated in this qualifier system are based on Arpad Elo's rating system for chess. Although this system was originally designed for rating chess players, it has been adapted to other sports (most notably Association Football) and non-sports games, including online games. Professional leagues and sanctioned tournaments of Trading Card Games also commonly use Elo ratings to rank players.The core concept of an Elo rating is that it represents a player's relative strength in comparison to other players, which allows you to calculate the expected result of a match between two rated players A and B. This expected score is derived for player A from the following formula:Player A's Expected Score = 1/1+10^(Player B's Rating - Player A's Rating)/400(So the expected score for player B is the reverse: Player A's Expected Score = 1/1+10^(Player A's Rating - Player B's Rating)/400)If you add the two expected scores, you'll always get 1, because this formula estimates the chances of Player A beating Player B, or vice versa.Let's say GM_Moshup and GM_Chaos both fight at the beginning of the tournament, and Chaos won with 3 lives remaining (that's embarassing). Both players would have a rating of 1500 since they just started, so they would both have an expected score of 0.5, representing a 50% chance (by the numbers) of each one winning.You can paste this formula into Microsoft Calculator to see the expected score of 0.5 for yourself: 1/(1+10y((1500-1500)/400))To calculate how many points Moshup is going to lose, we use this formula:Player A's New Rating = Player A's Current Rating + K(Player A's Actual Score - Player A's Expected Score)In our case, Moshup lost so his actual score is 0 (winning would be 1), and his expected score was 0.5, so we can plug it in:Moshup's New Rating = 1500 + K(0-0.5)You know that Moshup is losing points because a negative number [K(-0.5)] is being added to his current rating, but what's K?The K factor is a variable used in the Elo system to determine the maximum change in rating that can occur after each update to the rating. This is an important part of helping the ratings meaningfully reflect the actual skill of a player. A high K factor means your rating changes faster because each update is bigger, so similarly a low K factor indicates less change in rating. In chess, the K factor changes based on your rating; it starts off really high and gets really low, to reflect that when you're just starting out, not much is known about your skill level, so the goal is to quickly get you to the relative skill range that you should be in. Once you've established yourself and you're ostensibly only playing against other players of relative skill to you, you want the change to be small so that a single update doesn't put you ahead of a player who is actually better than you. FIDE (Fédération Internationale des Échecs, the world chess federation) uses a K factor of 25 for new players and 10 for pros. That's great since they're trying to identify Grandmaster chess players over years of play and hundreds of competitions, but we're trying to rank players in only 8 hours, so our system is going to move faster.In this tournament, the K factor will change based on how many lives the winner has remaining when the match ends by the following chart:1 Life Remaining: K=322 Lives Remaining: K=403 Lives Remaining: K=48So now that we know how the K factor changes based on the outcome of the match, the formula can be resolved to Moshup's New Rating of 1476.Since Moshup's rating decreased by 24 points, Chaos' rating must have increased by 24 points to 1524. You can try it for yourself: Chaos' New Rating = 1500+48*(1-0.5)From this, it's easy to see how the K factor changes your rating. Since both players had an expected score of 0.5, the change in rating is going to be half of K. If the difference in rating between two players is high enough, it's actually possible for no rating change to occur. Even if you blow out the noob with a 3-0 win, if your expected score is .990 you won't get any points from that victory, this will occur at a difference of around 800 points. It is unlikely in my opinion that this level will be reached by anyone in the tournament due to the short spread of games being played, but if someone *does* get to 2300, they won't get any points from fresh blood. I say this is unlikely because even at a max K factor of 48 (every game you play is a win and a blowout), most of your opponents will be fresh blood (1500 rating), therefore it should take over 300 games to reach the point where you no longer gain rating. At 8 hours to play, over 35 games an hour would be difficult to achieve.Now GM_Chaos is the king of the room, and GM_Takeda enters and beats him, but just barely with only one life remaining.GM_Chaos' Expected Score = 1/(1+10y((1500-1524)/400))GM_Takeda's Expected Score = 1/(1+10y((1524-1500)/400))GM_Chaos' Expected Score = 0.534 (round to 3 digits)GM_Takeda's Expected Score = 0.466GM_Chaos' New Rating = 1524+32*(0-0.534)GM_Takeda's New Rating = 1500+32*(1-0.466)GM_Chaos' New Rating = 1507GM_Takeda's New Rating = 1517Here you can see that even though Chaos lost to a weaker opponent (no offense Takeda), the fact that Takeda barely won makes the loss only 17 points; less than Chaos won from Moshup, so he's still ahead. Still, 17 points is more than the 16 (half of K=32) that would be lost if he faced an even opponent, so Chaos should get his act together if he wants to qualify.
Now GM_Takeda's the king, and GM_Moshup rejoins the room. Takeda's rating is 1517, Moshup's is 1476, run the numbers:
GM_Moshup's Expected Score = 1/(1+10y((1517-1476)/400))GM_Takeda's Expected Score = 1/(1+10y((1476-1517)/400))GM_Moshup's Expected Score = 0.441GM_Takeda's Expected Score = 0.559
Takeda and Moshup fight pretty frequently, I'd say they're pretty evenly matched. Their ratings disagree, but only slightly; they favor Takeda slightly to win because he has a better rating currently. Since they only have 7:30 to duke it out, they hit the time limit, causing the game to end in a tie. If you consider a tie to be half a win and half a loss, then your actual score when tying is 0.5. This works just fine in the Elo formula for ratings change:GM_Moshup's New Rating = 1476+32*(0.5-0.441)GM_Takeda's New Rating = 1517+32*(0.5-0.559)GM_Moshup's New Rating = 1478GM_Takeda's New Rating = 1515
So if you'll notice here, Takeda actually *lost* points (albeit only 2 points) for that tie. This is because Takeda was expected to perform better than Moshup; Takeda was expected to *win* the match. Since Takeda did not win the match, his rating will suffer for it.
In our tournament, although there will be multiple rooms, there is one master rating database. So no matter which room you play in, your cumulative rating from your prior record will be used to compare you to your opponent and the result will update to the master roster. We may not publish your ratings in realtime, but they are calculated in the order you fought your matches in, so they will always be accurate when they are calculated.Any questions?
From Ryugen
Wow...Im in high school and even this formula gave an epic headache Dx
Ok...Even though i may not understand more than half of this >_> Points are factored by
Lol...Also...By Opponent strenght, do you determine by Cash/GP Equipment or by PvP skills or by Victory %?
Is there anything else im missing?
ArkMaster364: Wow...Im in high school and even this formula gave an epic headache Dx
I may agree with you, but im in middle so yeah. Im not very well at caculating Q.Q
Hmmm I get it.
I wouldnt have been able to come up with that formula on my own, but I get what the formula is.
We need a epic youtube clip that says something like "WARNING, MATH APPROACHING" with the super smash bros "challlenger approaching" screen.
johniscool24: ArkMaster364: Wow...Im in high school and even this formula gave an epic headache Dx I may agree with you, but im in middle so yeah. Im not very well at caculating Q.Q
shadeofred: Hmmm I get it. I wouldnt have been able to come up with that formula on my own, but I get what the formula is. We need a epic youtube clip that says something like "WARNING, MATH APPROACHING" with the super smash bros "challlenger approaching" screen.
With a buffed up Pi and Omega symbol with HUGE arms and an OPed sword/Axe? xD
ArkMaster364: johniscool24: ArkMaster364: Wow...Im in high school and even this formula gave an epic headache Dx I may agree with you, but im in middle so yeah. Im not very well at caculating Q.Q Its kinda embarrasing D=...Im a Senior who has B to A grades in Math...and I tried to make sense of this but failed miserably...the way i failed on getting the Table of Elements on Quemistry Last year D=
ArkMaster364: failed on getting the Table of Elements on Quemistry Last year D=
failed on getting the Table of Elements on Quemistry Last year D=
I take it you didnt do too well on spelling either....
All thoughout school I was like 2-3 years ahead in math..... and I'm going into a CS major, so I better be able to do calculations!
Very interesting method. I like it.
shadeofred: ArkMaster364: failed on getting the Table of Elements on Quemistry Last year D= I take it you didnt do too well on spelling either.... All thoughout school I was like 2-3 years ahead in math..... and I'm going into a CS major, so I better be able to do calculations!
Then you shall be my math teacher :D
johniscool24: shadeofred: ArkMaster364: failed on getting the Table of Elements on Quemistry Last year D= I take it you didnt do too well on spelling either.... All thoughout school I was like 2-3 years ahead in math..... and I'm going into a CS major, so I better be able to do calculations! Then you shall be my math teacher :D
I meant that i couldnt Completely memorize the ToE...and in my last quiz of that...I got 70%...If i gotten one more wrong, it wouldve been a D >_>...and that would mean i would have to take the WHOLE grade again....My entire future depended on me memorizing a bunch of combinations and some letters reffering to an specific Element...Im not good at thinking on my feet D=
Then don't think on your feet..... by all means, sit down!
On topic: so was this formula already made into a program by you guys? You said the scores would be updated in real time, so does that mean you guys got a program or the GM's are just gonna keep track of all that math in their heads?
Zhon spotted a typo but who cares, as long as there aren't two many.
I've seen many variants of this system but never knew the actual name for it, and never knew it was used in Chess (my favorite non-digital game... lol).
So just curiously, was there a hunt for a program that would calculate and track this data, or did someone over at Ntreev write up an application? Or are you going PEN & PAPER STYLE? (I wouldn't do that even if I was paid!)
-edit- Shadeo beat me to it
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Excel I presume. I could do it on excel.
Wth? mods can edit posts in this section?
Ugh, that would have saved me a lot of emberassments in the past. >_<