After looking a quite a number of analyzed brackets, I’m finally beginning to see some patterns that should have been obvious long ago. I now begin to think that I can explain why the bracket shift works for some events, and not for others.
I’ve been distracted by the question of bracket balance. Balance is, to be sure, important, and I’m not going to lighten up on my crusade against bad drops and grouped byes. But two other features are more important: the number of rounds in the path to success, and the progression of skill levels.
The most striking thing, I think, about the analyzed bracket posted yesterday is how very flat it is. Look at the rounds receiving the drops in the lower bracket – here it is again: OdniLower. In the second round of the consolation, the B drops arrive in the bracket with only a 0.002 skill advantage over the survivors among the A drops. The C drops are only 0.003 better than their opponents. And even the D drops are only 0.005 better.
That means, for example, that the team that begins (win, loss) is only a tiny bit better than the team that begins (loss, win). It stands to reason that there should be some difference. The (win, loss) player beat a slightly better player on average, than the (loss, win) team, and lost to a slightly better player also. This is because the teams get better as one progresses in either bracket.
But with luck set at 75% (and, in effect, higher still because the truncated distribution of skill makes it a somewhat smaller factor), the skill progression from round to round is small. Through the five rounds of the upper bracket it progresses thus:
0.798 –> 0.865 –> 0.938 –> 1.018 –> 1.104 –> 1.197
The reason that the drops don’t arrive with a much higher skill differential is that the round skill progression just isn’t big enough to make (win, lose) much harder than (lose, win). Compare that progression with the same bracket after the luck and skill are equalized:
0.798 –> 0.970 –> 1.170 –> 1.390 –> 1.619 –> 1.848
Now consider the progression in the consolation. Odin:
0.731 –> 0.790 –> 0.855 –> 0.926 –> 1.005 –> 1.087 — 1.176
with the progression of the other:
0.625 –> 0.745 –> 0.902 –> 1.065 –> 1.273 –> 1.469 –> 1.649
In both cases, the figures for the lower bracket are given for the teams progressing up through the bracket, not for the drops. But, as we’ve seen, in the backgammon bracket, this matters very little, because the skill advantage for the B, C, and D drops is:
0.002, 0.003, and 0.005
But for the other, the difference is more significant:
0.025, 0.049, and 0.095
In a conventional, unshifted bracket, the A’s and B’s drop into adjacent rounds, but for every round after that, the is compensated for having had to face tougher opposition in the upper bracket by being allowed to skip one round in the lower bracket.
In the earliest analysis on tourneygeek, the striking finding was that this was simply too much compensation. The reason that the shifted bracket was fairer was that, by squeezing a round out of the tourney as a whole, it removes one of these round bonuses. But I later discovered a case in which the skill progression was so severe in the upper bracket that the conventional bracket, with its one-round bonus was fairer than the shifted bracket. That happened when the strength of the upper bracket increased because the teams were seeded.
In the comments to Bad Byes, Sean Garber was on track when he observed that the really important thing was to equalize the number of rounds. He’s exactly right if the game in question is backgammon. Grouping byes is still, I think, a bad idea, but it’s not as bad in backgammon, where the skill progression is so small. And I think it may be no accident that I never encountered shifted brackets until I saw them in use in backgammon tourneys, where they’re so desirable.