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Re: [Bug-gnubg] real match winning chances

From: Douglas Zare
Subject: Re: [Bug-gnubg] real match winning chances
Date: Sun, 7 Dec 2003 09:47:37 -0500
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Quoting Joern Thyssen <address@hidden>:

> On Sat, Dec 06, 2003 at 09:16:50PM -0500, Christopher D. Yep wrote
> > 
> > >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 
> Yes :-) Small typo...
> > (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).
> That's what I expected, but I can't find a convincing argument. 

You can have a net skill of -50% on average if you resign everything.
There is a positive probability of winning a game, indeed, winning a backgammon,
even if you are trying to lose (without resigning). Imagine both players get
5-5 for a while, then one escapes, bears in, crashes to the deuce point, and
bears off while the other gets stuck on the ace point by rolling doubles of a
blocked number.

The average outcome equals 50% plus the average net skill. Since the average
outcome is positive, the average net skill is greater than -50%.

It does not make sense for someone to be a 130%:-30% favorite. However, there
will be some matches in which one player gives up 80% mwc more than the
opponent. I think of it like the speed of a falling object measured in
miles/year. Although an object may be moving at a rate of 10^5 miles/year, this
does not mean that left alone for a year, it would actually fall 10^5 miles.
It's not constant or sustainable, though it does tell you something accurate at
the moment, say about how much energy is required to catch the falling object.

Since you can't win 130% of the matches in which your opponent gives up 80% more
mwc than you do, you also don't necessarily win 60% of the matches in which
your opponent gives up 10% mwc more than you do. I think Hank Youngerman has
some interesting statistics about how frequently the "stronger" player wins
when the favorite % is about 60% in a database of actual matches. I wouldn't be
shocked if in some populations, the side rated to be a 60% favorite loses most
of the time, since it may be tougher to play properly while leading. My guess
is that he also finds that individuals tend to play worse when they win, or are
winning in the match.

It may be possible to change units to something not resembling miles fallen per
year. You could talk about mwc/game given up, or EMG/game. You could also try
to take a different average, the geometric mean of something, or a
rootmeansquare, or some more complicated f^-1(avg(f(*))). However, I don't see
a good way to preserve the meaning that the simple arithmetic average of the
favorite % should converge to the percentage of matches won. 

Douglas Zare

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