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Author Topic: Win Probability Calculator  (Read 18485 times)

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jonts26

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Win Probability Calculator
« on: December 03, 2012, 01:09:40 pm »
+4

Hey, I was bored and I'm a huge nerd so I made a series win probability calculator in excel. It's still mostly functional in google docs and open office if you don't have excel.

Just plug in each player's mean skill and who goes first. Also you can adjust the first turn advantage parameter.

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D Bo

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Re: Win Probability Calculator
« Reply #1 on: December 03, 2012, 01:10:33 pm »
0

Wow, that is super-nerdy. I like it!
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WanderingWinder

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Re: Win Probability Calculator
« Reply #2 on: December 03, 2012, 01:31:31 pm »
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You're telling me that you know the underlying TrueSkill WE algorithm, and it's just a logistic curve?

DStu

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Re: Win Probability Calculator
« Reply #3 on: December 03, 2012, 01:35:35 pm »
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You're telling me that you know the underlying TrueSkill WE algorithm, and it's just a logistic curve?

WE?
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WanderingWinder

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Re: Win Probability Calculator
« Reply #4 on: December 03, 2012, 01:37:48 pm »
+2

Win Expectancy. Apologies, should have written that out.


I am developing a rating system... now(?)... mostly stemming from my experience with Dominion, Chess, and a smattering of other games. Hopefully an improvement.

jonts26

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Re: Win Probability Calculator
« Reply #5 on: December 03, 2012, 01:39:28 pm »
0

Win Expectancy. Apologies, should have written that out.


I am developing a rating system... now(?)... mostly stemming from my experience with Dominion, Chess, and a smattering of other games. Hopefully an improvement.

Logistic curve as I discussed here: http://forum.dominionstrategy.com/index.php?topic=5193.msg127303#msg127303
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toaster

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Re: Win Probability Calculator
« Reply #6 on: December 03, 2012, 03:31:15 pm »
0

Nice work.  Next step: start running tournament simulations!   ;)
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zahlman

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Re: Win Probability Calculator
« Reply #7 on: December 03, 2012, 05:07:52 pm »
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Wait, is "logistic" a real name for a real class of functions? o_O

(I wish Noscript's XSS filtering didn't play havoc with Wolfram Alpha links...)
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DStu

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Re: Win Probability Calculator
« Reply #8 on: December 03, 2012, 05:11:12 pm »
0

http://en.wikipedia.org/wiki/Logistic_function

:e Actually, I thought TS was Gaussian?
« Last Edit: December 03, 2012, 05:12:43 pm by DStu »
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greatexpectations

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Re: Win Probability Calculator
« Reply #9 on: December 03, 2012, 05:23:43 pm »
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Actually, I thought TS was Gaussian?

yup.
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rrenaud

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Re: Win Probability Calculator
« Reply #10 on: December 03, 2012, 05:25:35 pm »
0

If it doesn't have a variance parameter in it, it's definitely not trueskill.

The logistic and cumultive normal functions are pretty similiar looking.



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DStu

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Re: Win Probability Calculator
« Reply #11 on: December 03, 2012, 05:28:04 pm »
0

The logistic and cumultive normal functions are pretty similiar looking.
Yeah. both increasing, and reaching from 0 to 1. :P
And symmetric, to add something nontautologic...
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zahlman

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Re: Win Probability Calculator
« Reply #12 on: December 03, 2012, 05:39:18 pm »
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Basically, they're both sigmoids (which is what I'd have called the logistic function if I looked at it and didn't know the exact formula).
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DStu

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Re: Win Probability Calculator
« Reply #13 on: December 03, 2012, 05:45:09 pm »
+1

but Gaussian is decreasing much faster with ~exp(-x^2), while logistic ~exp(-x)
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SirPeebles

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Re: Win Probability Calculator
« Reply #14 on: December 03, 2012, 05:48:29 pm »
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It's odd really.  I know logistic functions well from teaching so many semesters of differential equations, but I never used them elsewhere when I was a student, nor do I ever use them in my research.  I hope that some of you use this stuff, and that it's more than just a contrived plaything for my students.
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rrenaud

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Re: Win Probability Calculator
« Reply #15 on: December 03, 2012, 06:38:14 pm »
0

Maybe this picture is better?  It's inverse CDF of normal vs inverse of logistic, scaled at p = .5.



http://en.wikipedia.org/wiki/Logit#Comparison_with_probit
Quote
Closely related to the logit function (and logit model) are the probit function and probit model. The logit and probit are both sigmoid functions with a domain between 0 and 1, which makes them both quantile functions — i.e. inverses of the cumulative distribution function (CDF) of a probability distribution. In fact, the logit is the quantile function of the logistic distribution, while the probit is the quantile function of the normal distribution. The probit function is denoted , where  is the CDF of the normal distribution, as just mentioned:

As shown in the graph, the logit and probit functions are extremely similar, particularly when the probit function is scaled so that its slope at y=0 matches the slope of the logit. As a result, probit models are sometimes used in place of logit models because for certain applications (e.g. in Bayesian statistics) implementation of them is easier.
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jonts26

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Re: Win Probability Calculator
« Reply #16 on: December 03, 2012, 07:08:17 pm »
0

So I'm not intimately familiar with the trueskill workings.

Does anyone know how the beta parameter for the game (25 in this case) interacts with individual player variances? Certainly I could just use the cfd of the difference in the individual player normal curves, but I'm pretty sure the individual variances don't account for the inherent randomness of the game, only in the players actual skill value.

Anyway, that's why I left them out.

Also I used the logistic distribution instead of the normal because that one paper I linked in the other post said it did a better job of prediction, but that's as much research as I did.
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DStu

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Re: Win Probability Calculator
« Reply #17 on: December 03, 2012, 07:11:55 pm »
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Really don't like that statement of wikipedia. Of course in the middle, they can be fitted to be quite similar, but  they behave differently quite soon at the edges. Especially if you want to do some bayesian statistics, where some more "extreme" regions of the distribution can come into play quite easily, you better don't just change the distribution just because it's easier to compute.

Of course assumed that you have chosen the original distribution in your model for some better reason than being easily But this happens with Gaussian quite often...
« Last Edit: December 03, 2012, 07:14:10 pm by DStu »
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rrenaud

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Re: Win Probability Calculator
« Reply #18 on: December 03, 2012, 08:01:37 pm »
0

Kaggle ran a chess prediction competition, and the winner's entry was based on TrueSkill.

http://people.few.eur.nl/salimans/chess.html

But he won basically because he milked side information (the data contained #matches per participant.  High #matches came from advancing further in tournaments, doh!, see WW's post next).

It's not clear to me that the logistic or normal is distribution is more natural than another for player skills though.  Nor that either are a particularly good fit.

Many uses of the logistic I've seen are motivated because it has a super easy to compute derivative, which makes it easy to optimize with gradient descent.
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WanderingWinder

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Re: Win Probability Calculator
« Reply #19 on: December 03, 2012, 08:19:07 pm »
+1

Kaggle ran a chess prediction competition, and the winner's entry was based on TrueSkill.

http://people.few.eur.nl/salimans/chess.html

But he won basically because he milked side information (the data contained #matches per participant.  High #matches came from advancing further in tournaments, doh!).

It's not clear to me that the logistic or normal is distribution is more natural than another for player skills though.  Nor that either are a particularly good fit.

Many uses of the logistic I've seen are motivated because it has a super easy to compute derivative, which makes it easy to optimize with gradient descent.

I don't think you understood that correctly: it wasn't so much that people with more games went further in tournaments, but that people who played stronger opponents were likely to have done better - particularly if way outrated, it probably means that the event is a swiss that your player has done well in, hence overpredict him a bit. (Incidentally, if you know much about chess, there are very few knockout tournaments...). On the other hand, his closest competitors all did some version of the same thing, too.
The method he used was not deemed feasible as a practicable rating system for FIDE (the world chess federation) to use.
(I am already acquainted with this stuff from my knowledge of the chess world; for a more personable write-up, see say this article: http://www.chessbase.com/newsdetail.asp?newsid=7277 )

Logistic seems much more natural to me than normal - you can get logistic out of the Odds Ratio really simply, and this is not too hard to give some justification for. Normal... I don't see any particular reasoning for. Again, not that either are super hot in all contexts.

SirPeebles

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Re: Win Probability Calculator
« Reply #20 on: December 03, 2012, 08:33:46 pm »
+6

Really don't like that statement of wikipedia. Of course in the middle, they can be fitted to be quite similar, but  they behave differently quite soon at the edges.

Even the math discussions devolve into bickering about edge cases  ::)
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Kirian

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Re: Win Probability Calculator
« Reply #21 on: December 03, 2012, 09:16:54 pm »
0

Off-topic but possibly important for theory:  I can't see the attachment in Chrome.  I was able to download it in Firefox, however.
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shark_bait

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Re: Win Probability Calculator
« Reply #22 on: December 03, 2012, 09:20:45 pm »
0

Highlight the text and move your cursor to the right.
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Re: Win Probability Calculator
« Reply #23 on: December 03, 2012, 09:33:52 pm »
0

Highlight the text and move your cursor to the right.

Highlight which text?  (Not that it matters much, the bug is in SMF I'm sure, and not particular to this forum).

This calculator really highlights some cool things!

1. The necessity of a multiple-round matchup.  My advantage in my first round only moves from 76.6% to 74.5% when going from 50% (really, "none") first-person advantage to 70% first-person advantage.

2. The effects of loser-goes-first.  For fun, try changing the first person advantage to something really low, like 0.2, and then look at your graph.  It becomes horribly skewed, as the winner has the advantaged position each time!
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shark_bait

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Re: Win Probability Calculator
« Reply #24 on: December 03, 2012, 09:35:38 pm »
0

Highlight the OP text,

Quote
Hey, I was bored and I'm a huge nerd so I made a series win probability calculator in excel. It's still mostly functional in google docs and open office if you don't have excel.

Just plug in each player's mean skill and who goes first. Also you can adjust the first turn advantage parameter.

And then bring your cursor to the right.  You should get the attachment from chrome.
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