Re: [Bug-gnubg] Ratings as random walks

[Douglas Zare]

Quoting Joern Thyssen <address@hidden>:

> On Sun, Jun 15, 2003 at 06:54:09AM -0700, Rod Roark wrote
> > Joern, I'm realizing that the standard deviation of a FIBS
> > rating increases with the number of matches played, so it's
> > not going to be meaningful to publish it in isolation.
> > 
> > For some theory behind this, consider that the rating is a
> > type of one-dimensional random walk[1], since each new rating
> > must be close to the previous one.  There's some academic
> > discussion at http://www.chm.uri.edu/urichm/chm531/walk/walk.html,
> > and in particular the statement:
> > 
> >   Equation (35) implies that the standard deviation of
> >   the random walk probability distribution function grows
> >   as the square root of the number of steps.
> 
> Yes, but to get the 95% confidence interval you should devide by the
> square root of the number of steps (following the discussion last week).

You are comparing apples with oranges. 

Of course the ratings after subsequent matches are correlated, hence the 
observed standard deviation is not accurate. This is also a random walk with a 
restoring force, not an unbiased random walk.

See the discussion of Kevin Bastian's statistics in the appendix of 
http://www.gammonvillage.com/backgammon/news/article_display.cfm?
resourceid=764 . That article also has a description of the stable 
distribution, assuming that one's opponents are correctly rated. The actual 
standard deviation should be higher. Since what you have observed is much 
lower, I don't think you have a representative sample.

Douglas Zare