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Tiny documentation error with explanation of beta parameter #11

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rsimmons opened this Issue Dec 29, 2015 · 5 comments

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rsimmons commented Dec 29, 2015

The documentation describes beta as "the distance which guarantees about 75.6% chance of winning". I think the correct percentage should be 76.025% (rounded however you wish). While the difference is trivial, it might confuse other people.

I was curious where the 75.6 magic number came from so derived what it should be, using the formula for computing win probability (mentioned in another issue). If you consider a match of two players, with the player sigmas and draw margins being 0, and the difference in rating means equal to beta, the win probability simplifies to cdf(1/sqrt(2)), which is about 0.76025.

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sublee Dec 30, 2015

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Thank you for the report. I agree on your opinion.

Should we describe Beta as 76.025% (Φ(1/√2))? Or approximately 80%? Which do you prefer?

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sublee commented Dec 30, 2015

Thank you for the report. I agree on your opinion.

Should we describe Beta as 76.025% (Φ(1/√2))? Or approximately 80%? Which do you prefer?

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sublee Dec 30, 2015

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Which win probability formula you used? If you used the last one, should the simple formula be Φ(1/√(2β)) instead of Φ(1/√2)?

Φ(delta of μ÷√(players×β²+sum of σ))
→ Φ(β÷√(2β²+0))
→ Φ(1÷√(2β))
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sublee commented Dec 30, 2015

Which win probability formula you used? If you used the last one, should the simple formula be Φ(1/√(2β)) instead of Φ(1/√2)?

Φ(delta of μ÷√(players×β²+sum of σ))
→ Φ(β÷√(2β²+0))
→ Φ(1÷√(2β))
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rsimmons Dec 30, 2015

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I think there's a mistake in that simplification: Φ(β÷√(2β²+0)) has √(2β²) in the inside denominator, which simplifies to β√2, so we get Φ(β÷(β√2)) → Φ(1/√2).

I would probably describe beta as "the distance which guarantees about 76% (specifically Φ(1/√2)) chance of winning".

Also, thanks so much for your work on this project. When I discovered the TrueSkill site/papers I was excited to use it but the thought of implementing it from scratch made my heart sink a little :)

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rsimmons commented Dec 30, 2015

I think there's a mistake in that simplification: Φ(β÷√(2β²+0)) has √(2β²) in the inside denominator, which simplifies to β√2, so we get Φ(β÷(β√2)) → Φ(1/√2).

I would probably describe beta as "the distance which guarantees about 76% (specifically Φ(1/√2)) chance of winning".

Also, thanks so much for your work on this project. When I discovered the TrueSkill site/papers I was excited to use it but the thought of implementing it from scratch made my heart sink a little :)

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sublee Dec 30, 2015

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Oh, you're right. I had a mistake. Thank you for letting me know.

Would you give me a pull request to add you to the contributor list? If you don't care about the contributor list, I'll update the document myself.

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sublee commented Dec 30, 2015

Oh, you're right. I had a mistake. Thank you for letting me know.

Would you give me a pull request to add you to the contributor list? If you don't care about the contributor list, I'll update the document myself.

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sublee Dec 31, 2015

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I released TrueSkill-0.4.4 with your patch. http://trueskill.org/#version-0-4-4

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sublee commented Dec 31, 2015

I released TrueSkill-0.4.4 with your patch. http://trueskill.org/#version-0-4-4

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