The bracket-shifting technique illustrated in the case of a 16 DE tournament can be generalized to larger brackets, but there are some additional considerations.
The bracket shift works, essentially, by taking two adjacent rounds of drops, which would ordinarily be separated in the lower bracket by a non-drop consolidating round, and pushing them into earlier rounds of the lower bracket, causing there to be one fewer round in the tournament as a whole.
In a 32 bracket, there are two ways to do this. You can either shift the C and D drops earlier, leaving the E drop in its usual relative position, or shift D and E drops earlier, leaving the C drops in place:
In both cases, the design saves a round by eliminating the consolidation round before each of two adjacent sets of drops, adding a new consolidation round after them. In a 64 bracket, it would be possible to save two rounds, shifting C, D, E, and F drops all back. But in a 32 bracket, the D drops will shift, and you have to decide whether it is the C drops or the E drops that move with them.
Is one of these shifts better than the other?
The CD shift is better at avoiding repeat pairings in the early rounds. Proper drops work very compactly, so that it’s possible to avoid repeats entirely for all but the last five matches, two fewer than either the ED shift or the standard. The total mean number of rematches is not much less than it is for the ED drops – 1.328 for CD, 1.343 for ED, but more of those rematches tend to come later, when everyone knows that rematches are likely. Both shifts do better than the standard bracket, which averages 1.438 rematches.
But the ED shift beats the CD shift in the fairness statistic: 11.03 for ED, 10.85 for CD. Again, both are better than the standard bracket at 10.82. To see why this is the case, recall the earlier analysis of the shifted 16. The largest factor that compromised the fairness of the standard 16 was the over-generous treatment of the last drop. In the standard 16 bracket, the last drop was 39% worse than the opponent it faced. While it may seem odd it’s not really a disadvantage to drop into a tougher match, it simply shows that the team has been promoted beyond its merit. The shift in the 16 bracket made this better, but only by a considerable overcorrection: instead of being 39% worse than its foe, it was 19% better.
In the standard 32 bracket, the E drop gets an even bigger boost – it’s 42% worse than its opponent. The CD shift only exacerbates the problem. By being a little bit fairer in the earlier rounds of the lower bracket, it produces slightly better opposition for the E drop, so that the E drop is now 44% worse. But the ED shift very nearly restores the D drop to its proper level – it’s now only 3% better than its opponent.
Here’s an operational consideration to keep in mind while deciding which shift to use. The CD shift will eliminate some players from the tournament sooner than the ED shift will. Maybe this means they’ll go home early, or maybe they’ll hang around and play in more side events. In either case, that’s probably a good thing, at least as far as they’re concerned.
Now that I’ve seen it in the data, the injustice of the E drop sticks out like a sore thumb for me, and so I’m inclined to favor the ED shift. But that may not be the most rational choice. The vast majority of participants in your tournaments will never have read tourneygeek.com, and thus be blissfully unaware of how awful the E drop can be. But they will notice the earlier repeated pairings. And, whether they know why it happened or not, some of them will enjoy a bit more free time because they’ve dropped out of the tournament a little earlier. So maybe you should choose the CD shift.
What’s clear, I hope, is that either one of the shifted brackets is better than the standard bracket in all respects, excepting with respect to the reflexive fairness (A) complaint that can attend any innovation. Once people come to expect shifted brackets, the tournament world will be a slightly better place.
To help that happen, I think I need to stop talking about the conventional bracket architecture as standard. Standard tends to imply that it’s the right choice, or at least that it’s a safe choice. Words matter. And, to that end, I hereby rename what I’ve been calling the standard bracket. Henceforth, it will be known on tourneygeek as the shiftless bracket.