"George John" writes:
The USCF system uses a Boltzmann distribution and FIDE a Gaussian.
And, the two are very close to each other. Are they both wrong?

They are close to each other when the ratings are close together and
the parameters are chosen appropriately. I don't think what the USCF
uses is the same as the Boltzmann distribution, but it's pretty close
too. The big difference between the USCF formula and Elo's original
one is that the USCF uses a much larger standard deviation.

I know how to compute We (score expectancy). I do NOT know how to
compute the win expectancy or draw expectancy. We know that Sloan will
tend to score about .35 points for every game played or 35 points in
ever 100 games. What we don't have a clue about is what the
distribution of draws and wins will be that add up to 35. BTW, this is
something that I have long been interested in knowing more about in
general.

The draw probability is to some extent up to the players. For
example, some openings are more likely to lead to draws than others.
So if one player really really wants a draw, he can choose drawish
openings and so forth. Assuming constant draw probability is a
somewhat bogus approximation since the draw probability depends (among
other things) on the match equity at any round: if Sloan wins game 1,
he might then choose play drawishly in all three remaining rounds.
But if he draws games 1 and 2 and loses game 3, he has to go all-out
for a win in game 4, draws be damned. Similar considerations hold for
Brock.