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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 21:07:54 -0500

At 10:33 PM 12/6/2003 +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.

For Player 1:

initial mwc = 50%, final mwc = 100%
netSkill = -50% (Player 1 gave up 50% mwc, Player 2 gave up 0% mwc in this hypothetical example)
net luck = +100%
result = +50%

Btw I checked this by playing a 1pt match against gnubg with manual dice,
where I tried to make all possible errors but gave gnubg such poor dice that
it lost anyway; An analysis gives me a total error rate of -128% and a luck
adjusted result of -137%.

(Note: the total error rate = luck adjusted error rate if gnubg's match analysis is perfect; since gnubg's match analysis is not perfect, the above result seems reasonable.)

Suppose that Player A (playing perfectly) plays Player B (an extremely weak player, e.g. a computer making all checker and cube decisions randomly). In general Player A is > 99.9999% favorite (let's round this to 100%). However, if these matches are analyzed, sometimes Player A will get very good dice and will quickly win the match. In these matches, player B won't have many opportunities to make errors and may only make, say, 25% mwc worth of errors, resulting in a luck adjusted result of -25% mwc. In other matches though, Player B will get very good dice which will prolong the match, giving him more opportunities to make errors. In these matches he might make, say, 75% mwc worth of errors, resulting in a luck adjusted result of -75% mwc.

On average player B will make 50% mwc errors per match. In a given match he may make less than or more than 50% mwc worth of errors. Player A's expected result is +50% mwc (i.e. he expects to win 100% of the matches).

I think that those numbers are not right. If I would do a prediction of the
mwc of player 1 against player 2 based only on the match above, I would say
that player 1 apparently gives away half his mwc twice during a match,
regardless of what the situation or matchscore is at those times. So his mwc
will be on average 50% * .5 *.5 = 12.5%.

I disagree. In the extreme example that I gave, I'd be annoyed if gnubg didn't report Player A's luck adjusted result as > +50% mwc in some of its matches. Gnubg is just reporting what happened in the individual match (I like how it does it).

It's important to analyze a large number of matches. What matters is the average luck adjusted result. In extreme examples, as I gave above, the length of the match (which is positively correlated with Player B's net luck) significantly affects the luck adjusted result of an individual match.


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