three or more players ranks probability

[dotcom]

I think, if I want to calculate the win percentage for two players, I should calculate the difference distribution.
X - Y ~ N(μ1 - μ2, σ12 + σ22)
I think that it is sufficient to definitely integrate N in the interval of 0 or more.

But, if game is contained three or more players, how should I predict the ranking probability?

a = Rating(mu= m_a, sigma= s_a)
b = Rating(mu= m_b, sigma= s_b)
c = Rating(mu= m_c, sigma= s_c)

# I want to calculate the probability that rank c > b > a.

[sublee]

Simply isn't it P(c>b) * P(b>a)?

[dotcom]

In general,
P(c>b>a) = P(b>a|c>b)P(c>b) ≠ P(c>b)P(b>a)
Because "simply P(b>a)" differs from "P(b>a) with b that satisfies c>b" .

P(c>b>a) can be evaluated using the multivariate normal cumulative distribution function
X = C - B
Y = B - A
We want P(X>0, Y>0).
However, computing multivariate normal cumulative distribution function is so difficult.

4 variables example:
P(d>b>c>a) = P(d>b|b>c,c>a)P(b>c|c>a)P(c>a)

ok, I will make an effort to consider whether it can be implemented and send a pull request, if you hope.

>>> a = Rating()
>>> b = Rating()
>>> c = Rating()
>>> ranks_probability([(a,)(b,),(c,)], ranks=[0,1,2])
0.1210593088888

Probrem

• At this time, how to evaluate the team rate is a problem.
• and, How should we evaluate the rating of an asymmetric team?
• If the number of variables is too large, there is a possibility that the calculation amount will be too high.

[sublee]

Oh, awesome! Thank you for letting me know my mistake.

The function ranks_probability looks good but I don't want to adopt it as a part of this library. There are 2 reasons:

• This library is just a transparent implementation of the TrueSkill paper. In my perspective, it should not provide any additional functionality.
• As you mentioned, the function is hard to apply every valid conditions in TrueSkill such as 3v2v1 or 1v1v1v1v...v1.

Instead, would you paste the function as code at this issue?

[dotcom]

ok. I agree with the implementation philosophy that this library is a transparent implementation of Trueskill.
I will only discuss (or paste function) the implementation here on this matter.

[sublee]

Thanks for your kindness!

[lukebyrne]

@dotcom did you ever get around to a ranks_probability method?