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Re: [Bug-gnubg] real match winning chances

From: Christopher D. Yep
Subject: Re: [Bug-gnubg] real match winning chances
Date: Sat, 06 Dec 2003 21:16:50 -0500

At 10:03 PM 12/6/2003 +0000, Joern Thyssen wrote:
On Sat, Dec 06, 2003 at 10:33:27PM +0100, Misja Alma wrote
> Sorry, I think I didn't express very clearly what I meant, the example I
> gave could be more complete. Here is my explanation again:
> So 2 players start a match, and at that point both have 50% chance.
> When player 1 makes a mistake which costs him half his mwc, this costs him
> .5 * 50% = 25% mwc. Gnubg will find this also.
> But now let's assume that the player gets lucky and comes back to 50% mwc
> again. At that point he makes again a mistake which costs him half his mwc;
> so 25%. What gnubg does is it addes up error1 and error2, for a total of
> 50%. If these were all the errors both players made this match, it will then
> estimate player 1 to have 0% mwc in an analysis.
> The luck rate will do something similar: After his first error player 1 must
> have had luck worth for 25% mwc. Suppose that player1 won the match anyway
> after his second mistake, he must have had another portion of luck worth for
> 75% mwc this time. Both are added up for a total of 100%. final - initial =
> netLuck + netSkill; Filling the numbers in gives that netSkill must be zero.

Don't you mean -50%? final-initial=50% and netluck=100%.

Anyway, I'm starting to understand you :-)

netskill = -50% => 0% MWC derives from assuming that player 1 starts out
at 0% instead of 50% since his net skill is -50%. As you correctly point
out, the changes in MWC occur in discrete steps, so there is a non-zero
chance that player 1 might get so lucky that he actually wins despite
his skill being -50%.

The netskill for an individual match can vary widely, even if Player A is a perfect player and Player B plays extremely poorly (so that Player A wins close to 100% of its matches against Player B). (See the previous note I sent.) For more realistic cases, even if Player A wins 60% of the time against Player B, sometimes Player A's luck adjusted result will be +20%, sometimes +0% (so that on average his luck adjusted result will be +10%), even when analyzed by a perfect bot.

I'm not sure how to model this mathematically?!  I'm also wondering if
it's possible to have a net skill of zero?

Presumably you mean is it possible to have net skill of -50% mwc (luck adjustments done by gnubg, in an unbiased manner). For an individual match, it's definitely possible (even if gnubg actually did perfect luck adjustments). It's not possible to have net skill of -50% mwc on average (i.e. for an infinite set of matches).


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