Win probability?

[llimllib]

I'm trying to figure out how to calculate the win probability from two TrueSkill ratings. You have draw probability, as match_quality, but not win probability.

Any idea how to calculate it? I've been reading lots of stuff on TrueSkil, but it's a bit over my head.

[sublee]

You can calculate with β constant. β means the skill class width which is the difference between two ratings with 80% of win probability. You can find more information about β in 14p of The Math Behind TrueSkill.

[llimllib]

That paper tells me that player A, of rating R, will beat player B of rating S 80% of the time if R-S=β, but it doesn't give me a function from (R,S,β) to win percentage.

[sublee]

According to the paper, the ratio of A's winning is 4^((R-S)/β):1.

1. 0 -> 1:1 -> 50%
2. β -> 4:1 -> 80%
3. 2β -> 4²:1 -> 94%

So we can make a naive win_probability function such as:

def win_probability(greater, lesser):
    exp = (greater.mu - lesser.mu) / BETA
    n = 4. ** exp
    return n / (n + 1)

But I think that this function isn't trustworthy because it ignores the sigma values.

[llimllib]

I don't see that equation in the paper? Could you point me to it?

I feel like an idiot because I'm reading the paper over and over and not finding it.

[llimllib]

Furthermore, what I'd really like to determine is Pwin(β,µ1,σ1,µ2,σ2), but the paper only gives Pdraw, and I deeply do not understand the derivation for it. I assume you don't know of any way to derive a function Pwin?

[sublee]

Here are 3 players:

1. A player (best) has Rating(μ+β, σ₁)
2. B player (normal) has Rating(μ, σ₂)
3. C player (worst) has Rating(μ-β, σ₃)

You know that A player beats B player 80% of time and B player beats C player 80% also. Then what of time A player beats C player? See the below proportional expressions. The win probability can be represented as ratio expression.

1. A : B = 4 : 1 = 4² : 4
2. B : C = 4 : 1
3. A : B : C = 4² : 4 : 1

We can also generalize about nβ. The nβ of difference follows 4ⁿ : 1 of win probability.

You're right. I don't know how calculate win probability with sigma values. TrueSkill system doesn't offer Pwin function. So the result of my function is just expected value not correct value.

[llimllib]

right... I should have just thought a bit more. Thanks for the explanation, and concurring with me that there's no Pwin.

[warner121]

This gives the probability of a win, taking the current sigma values into account (from Herbrich and Graepel's equation 1.1):

from trueskill import Rating
from trueskill.mathematics import cdf

def Pwin(rA=Rating(), rB=Rating()):
    deltaMu = rA.mu - rB.mu
    rsss = sqrt(rA.sigma**2 + rB.sigma**2)
    return cdf(deltaMu/rsss)

There's no accounting for the draw here, though. Perhaps you could subtract 'half a draw' from the result?

[sublee]

@warner121 Thanks to tell the code! It seems to be pretty nice. I also want to see the original documentation of Herbrich and Graepel's equation 1.1. Can you give me that?

[warner121]

@sublee No problem, thank you for sharing your code. I found the paper at http://research.microsoft.com/pubs/74419/TR-2006-80.pdf.

[nbonfire]

This win probability is good for 1v1, but what would need to change to give the win probability of an n v n match?

[azurespace]

@nickdanger3d In the formula, we obtained a normal difference distribution of two ratings and calculate its CDF. I think we can sum players' ratings in a team into one normal distribution before that. (See: http://mathworld.wolfram.com/NormalSumDistribution.html )

However, what should I do if a player partial played? Do you people have any idea?

[nbonfire]

@azurespace Just to be clear, you're saying the win probability for a match of n v n assuming no one partially plays would be something like this:

def Pwin(rAlist=[Rating()],  rBlist=[Rating()]):
    deltaMu = sum( [x.mu for x in rAlist])  - sum( [x.mu for x in  rBlist])
    rsss = sqrt(sum( [x.sigma**2 for x in  rAlist]) + sum( [x.sigma**2 for x in rBlist]) )
    return cdf(deltaMu/rsss)

[azurespace]

@nickdanger3d Exactly. I think it makes sense when the skill in target game is additive. If it isn't, we must think the again with the paper ...