EPTLet x be: points held + Bonus Pool - division offset
Let u be: average points of everyone in Diamond League + average unused bonus pool - average division offset
Let n be: the number of players currently in Diamond League
Your skill relative to the average diamond player, assuming you are in diamond = (x-u)/[(Sum of all x-u)^2/n)^(1/2)]
I would like to seek the opinions of the community to see if my thinking is correct. I realized that this is an imperfect measure but I believe it is much better than just stating how many points you have. If my thinking is correct, I was wondering if TL could create an automated algorithm that can allow us to calculate this on the spot by putting in our battle.net profile URL.
PS: This is obviously for Diamond players only.
Intro
The purpose of this post is to give a metric that can allow players to understand truly how they rank compared to other players within their league, across divisions and regardless of bonus pool. The reason I'm making this post is because a lot of people say things like "I'm 2100 points, but I play people between 2800 and 3200 all the time, so I must be closer to 3000, really" or "I have 1800 points but over 1000 in my bonus pool, so I should therefore be around 2800 in reality". I want to create a standard phrase that people can say and instantly have people understand where their skill level is without any kind of second guessing. Remember that I am trying to create a relative metric. I am NOT trying to derive the actual MMR number.
So, as we all know, Ladder Points do not adequately show what your actual rating is within the matchmaking system (MMR). This has been done on purpose by Blizzard and it is essentially designed to make you want to continue playing and not destroy your ego. There's already been a lot of analysis and explanation done on this and I would first like to draw the reader to these amazing posts by Excalibur_Z that essentially explain how the system works. This isn't mandatory reading to understand my post, but it's pretty insightful stuff.
SC2 Ladder Analysis Part 1, an overview of the system: http://www.teamliquid.net/forum/viewmessage.php?topic_id=118212
SC2 Ladder Analysis Part 2, greater depth about matchmaking and promotion: http://www.teamliquid.net/forum/viewmessage.php?topic_id=142211
SC2 Ladder Misconceptions FAQ:
http://www.teamliquid.net/forum/viewmessage.php?topic_id=150367
Most importantly, I think everyone needs to read this post (also by Excalibur_Z):
http://www.teamliquid.net/forum/viewmessage.php?topic_id=142211
This last post talks a lot about Division Offset, which basically used information from Blizzard's top 200 rankings compared to the top 200 rankings from sc2ranks.com (which uses pure points to rank players) to calculate a modifier that, when subtracted from your current points, can give you an idea of what your rating is regardless of your division.
What we Know
So let's take stock of what we know so far:
1) Points are not equal amongst divisions, but if we use the Division Offset as calculated by Excalibur_Z, we can come close to understanding our "divisionless" rank according to points.
2) As time goes on, everyone gets a Bonus Pool. This Bonus Pool essentially gives you double points for winning until it runs out. Everyone gets Bonus Pool at the same rate, regardless of whether you play or not.
3) Because of Bonus Pool, the average points across diamond players will constantly rise as time goes on. For example, a few months ago I had 1200 points and Gretorp (who is also in my division) had about 2200. I now have 2200 points and Gretorp has like 3200. Does that mean I am now as good as Gretorp was a few months ago? Hell no. What's important is our relative ranking.
4) As you play more games, the system gets better at matching you against players with the same skill as you - that is, your MMR Confidence Interval shrinks. As this interval gets small enough, you will eventually converge to a 50% win rate, provided your skill does not change faster than the average skill increase of all players in Diamond.
5) From SC2ranks.com, we know how many points every diamond player has.
6) Though not currently displayed on sc2ranks.com, your bonus pool is presumably fetchable from your battle.net profile.
Far's Ranking Metric
So from these 5 things, I suggest that we start using the following metric to measure skill.
Let x be: points held + Bonus Pool - division offset
Let u be: average points of everyone in Diamond League + average unused bonus pool - average division offset
Let n be: the number of players currently in Diamond League
Your skill relative to the average diamond player = (x-u)/[(Sum of all [x-u]^2)/n)]^(1/2)
So for example, if you get a number like 1.5, that means you are 1.5 standard deviations above average (quite a bit above average). If you get like -0.5, that means you are half a deviation below average. If we wanted to get numbers closer to what we're used to with ELO, we could just multiply this by 1000.
Explaining the Metric
Points held + Bonus Pool - division offset
This part essentially gives us your unmodified points ranking. THIS IS NOT YOUR MMR.
Why do we add Bonus Pool?
From points 2, 3 and 4 under "what we know", provided your confidence interval is sufficiently small and your skill is rising at roughly the same rate as the average skill in diamond league, you should have close to a 50% win rate. If that's the case, you would lose exactly as many points as you win, except that when you win you get double points until your bonus pool runs out. If we assume that the system is right about your skill and that your skill has not rusted while not playing and that you've not magically gotten better without laddering, we should add Bonus Pool.
[i]Why do we subtract division offset?[i]
Because not all divisions are equal (what we know #1). So in order to compare across divisions, we have to make them equal.
Average points of everyone in Diamond League + average unused bonus pool - average division offset
This tells us the current unmodified point value of the average player. When we subtract the two, the difference tells us how many more (or less) points you have than the average diamond player.
[(Sum of all [x-u]^2)/n)]^(1/2)
This is basically the unmodified standard deviation of points in Diamond league. Here's why this is important:
Consider two players: A and B. 2 months into the season, A has 100 points more than B. 4 months later, A has 500 points more than B. Does this mean that the skill gap between A and B is now 5x greater than it was before? No! Everyone got more points. What's important is their relative distance in skill from average player. That is what we are trying to measure when we divide your point difference from the average player by this number.
And for those of you not familiar with basic statistics, we need to square all x-u, because if we don't, we will get both positive and negative values, which, when added together, would give us 0 (by definition). So we square them to make the negative values positive and then that the root at the end to "undo" the squaring.
Why this is still an imperfect measure and why there are still some challenges in measuring this
1) Division offset is imperfect and we need more data to make it perfect
2) There are many divisions that we don't have offset values before, simply because nobody from those divisions has ever appeared in the top 200
3) New divisions are being created every day
4) In order for this to be accurate as intended, we need to assume some sort of symmetric distribution. This is most certainly not the case as we probably have a fat lower tail (low diamond players are closer in skill to each other than high diamond players).
5) It is possible to have 0 points in diamond league, lose a game and not get negative points.
6) There are many platinum players with MMRs sufficiently high enough to get into Diamond that simply haven't been promoted yet. There are also some Diamond players with MMRs low enough to get demoted but haven't yet.
7) If the MMR threshold for Diamond is rising (a fair assumption if we conclude that the population is growing and that most new entrants are not diamond players), there should be inactive diamond players that should not be in diamond league, but still are because you can't get demoted without logging on (not sure if this is correct).
This has just been a random thought experiment while I avoid sleeping in order to get to boxing day shopping early enough! I appreciate your comments and feedback. I'm far from certain that everything I typed here is correct.
-Far.257
