The idea is to increase the effective mmr of people who are in a premade to account for the fact that they will be coordinated and therefor getting more value from their abilities (which is effectively indistinguishable from having a higher mmr). This realization is what has driven the following adjustment.
Premade group member’s mmr= solo queue mmr + ß_N(group size dependant) + ß_M(exists only if there are other premades in the group like 2+2+1 or 2+3)
where ß= C(N^2)/5 +1 and N= the number of people in the premade and C is an arbitrary constant.
These are the values of ß for each premade size:
ß_2 = C(1,8)
ß_3 = C(2,8)
ß_4 = C(4,2)
ß_5 = C(6)
This means that ß grows quasi-lineally with premade size:
ß_3/ß_2 = 1,5555
ß_4/ß_3 = 1,5
ß_5/ß_4 = 1,42
These are the values of the 5 man premade compared to the others:
ß_5/ß_2 = 3,33333
ß_5/ß_3 = 2,14
ß_5/ß_4 = 1,42
In the case of mixed premades such as 2+2+1 and 3+2, where M= the size of the other premade in the group, the values for groups with more than one premade are like this:
2+2+1= ß_2 + ß_2 = C(3,6)
3+2= ß_3 + ß_2 = C(4,6)
The value of the 5man premade compared to the values of the combined premades:
ß_5/(2+2+1) = 1,66666
ß_5/(3+2) = 1,3
The value of a 4 man premade compared with combined premades:
ß_4/(2+2+1) = 1,1666666
ß_4/(3+2) = 0,913 => (3+2)/ß_4 = 1,095
The value of a 3 man premade compared with combined premades:
ß_3/(2+2+1) = 0,77777 => (2+2+1)/ß_3 = 1,28
ß_3/(3+2) = 0,6 => (3+2)/ß_3 = 1,666666
The value of a 2 man premade compared with combined premades:
ß_2/(2+2+1) = 0,5 => (2+2+1)/ß_2 = 2
ß_2/(3+2) = 0,39 => (3+2)/ß_2 = 2,56
The comparison between the values of both combined premades:
(3+2)/(2+2+1) = 1,277777
Commentary:
This is just a first approximation to get the perfect equation. To advance this further i would need to get the winrates of all premade sizes vs all premade sizes and so i would adjust the ratios between the values–if needed–and i would be able to get the value of the constant C for the game of HOTS. This is to say that this formula can be used to adjust the mmr inside of any 5v5 game and C is specific to each game and, probably, depends on time in some measure (under the assumption that the playerbase’s average experience vs premades increases with time).
The constant C must be finely tuned with the help of said statistics so that the win rates for all matchups (i.e. 5man vs full solo queue or 3+2 vs duo queue or any other matchup) are as close to 50% as possible.
After adjusting the players’ mmr using the here described method, one is to apply to it the common Glicko RD (rating deviation) treatment and you have then the tools to make balanced matches which result in less player frustration even with premades.
All the ratios between all values have been calculated based only on player perception of what is stronger than what and has no statistical foundation whatsoever. It would be fantastic to get the statistics to be able to make this equation exactly but, even with the inexactitude caused by the lack thereof, the incorporation of this adjustment to the current Glicko mmr system represents an upgrade in match making exactitude for any 5v5 game.
