# Intro

I’m using this topic to continue a discussion started in this post, where I answered the question “What is the average TrueSkill rating of a carry, captain and jungle main?”:

I want to combine it with the results of a discussion on Discord about players’ main roles.

# What is a main, mathematically?

We are going to look at a sample of players from VainSocial who played at least 20 ranked matches.

## Facts

Here you can see how commonly a player picks a role. One would assume that there are as many captain mains as there are jungle or carry mains and that the majority of players plays all three roles about 33% of the time.

Nope.

On the right side are the most dedicated players. About 188 players play captain 90% of the time for example, which means that they play carry or jungle 10% of the time. Any player that is a captain main is not a carry or jungle main at the same time. Likewise for carry or jungle mains.

Oddly enough, the 3 curves do not perfectly fit onto each other. The captain peak is shifted to the left and the curve is higher on the right extreme.

That means that captain is less common to be played by someone who plays “flex”, and that there are twice as many hardcore captain mains as there are pure carry or jungle mains.

A possible explanation for that is that there are two stereotypes of players: “captain mains” and “damage mains”. A damage main switches between carry and jungle, but rarely plays captain. See below.

## Interpretation

Let’s divide players equally into two categories: flex pickers and mains.

A flex picker is in the middle of the 3 curves. A main occupies space in the right (main role) and the left (other roles) area.

## maths

I summed all 3 graphs, used relatives, and applied a regression.

I used x^4, it fits close enough. X is pick rate, Y is relative amount of players with that pick rate.

`f(x) = -1.73 x^4 + 5.11 x^3 - 5.01 x^2 + 1.60 x^1 + 0.04`

Now we want to divide the areas like this

- A1 below the graphs at 0% <= x <=
`x_a1`

belongs to the mains - A2 below the graphs at
`x_a1`

< x <=`x_a2`

belongs to the flexers - A3 below the graphs at
`x_a2`

< x <= 100% belongs to the mains

We want to divide 50/50 into mains/flexers so`A1 + A3 = A2`

.

The same player who mains a role does not pick two other roles, so`A3 = 2*A1`

.

wolframalpha helps:

```
F(x) = -0.346 x^5 + 1.278 x^4 - 1.67 x^3 + 0.8 x^2 + 0.04x + c
A1 = F(x_a1) - F(0) = F(x_a1) - c
A2 = F(x_a2) - F(x_a1)
A3 = F(1) - F(x_a2) = 0.102 + c - F(x_a2)
A1 + A3 = F(x_a1) + 0.102 - F(x_a2)
A1 + A3 = A2
A3 = 2 * A1
=> 3A1 = A2
*1:
A2 = A1 + A3
<=> F(x_a2) - F(x_a1) = F(x_a1) - F(x_a2) + 0.102
<=> 2 F(x_a2) = 2 F(x_a1) + 0.102
<=> F(x_a2) = F(x_a1) + 0.051
*2:
A1 + 2A1 = A2
=> F(x_a2) - F(x_a1) = 3 F(x_a1) - 3c
<=> F(x_a2) = 4 F(x_a1) - 3c
*1 and *2:
4 F(x_a1) - 3c = F(x_a1) + 0.051
<=> 3 F(x_a1) - 3c = 0.051
<=> F(x_a1) - c = 0.016999
=> x_a1 = 0.139045
*1 again, we need x_a2
F(x_a2) = F(x_a1) + 0.051
=> F(x_a2) - c = 0.016999 + 0.051
<=> F(x_a2) - c = 0.067999
0 <= x_a2 <= 1, so:
x_a2 = 0.410716
```

As demonstrated, mains’ two non main roles are between 0% and 14% pick rate on average, and their main roles are between 41% and 100% pick rate.

Flexers’ pick rates averages are between 14% and 41%.

Therefore, any player that picks a role 41% the time or more mains that role according to the given definition. Logically, a player can main two roles at the same time.

More in this post: Analysis: Role "mains"

## Distribution of mains by tiers

The relative majority of captain mains is between tier 5 and 7.

See also @TheCopyCat’s article: https://docs.google.com/document/d/1MryxSYDTNlr2HVjznpARhZcNyaYhDr0_s-YcL6wL-0Y

# Mains and their rankings

A month ago, we split TrueSkill - which originally was our match maker rating for both casual and ranked - into a ranked TrueSkill, a casual TrueSkill, Blitz TrueSkill, … and a TrueSkill shared between all modes (which is used internally for placement and analysis). Since what I wrote in the article linked in the introduction used the TrueSkill shared between all modes, I’ve done the same analysis again with our ranked TrueSkill and the average skill tier.

This time, we are going to use 41% pick rate to qualify for a “main” and look at players from tier 7 and above.

- carry: 1 391 players
- captain: 1 343 players
- jungler: 1 480 players

- carry: tier 8.94
- captain: tier 8.84
- jungler: tier 8.8

- carry: ranked TrueSkill 2 230
- jungler: ranked TrueSkill 2 117
- captain: ranked TrueSkill 1 966

TrueSkill is only based on win/loss, your own, ally and enemy ratings. It is not an algorithm we have developed, it is from Microsoft Research: Microsoft Research / TrueSkill Rating System

As before, carry mains’ ratings are the best using both rankings.

Surprisingly though, the average skill tier of a captain main is above a jungler main’s - while the TrueSkill is lower.

Is that correlation or causation (Wikipedia / Correlation does not imply causation)?

Are carries better or are better players just carries by chance?

I will stop the analysis here because I do not have any more answers or theories yet.

Edit 2018-02-08: Added link to “hardcore main” / “two role flex” analysis, corrected grammar