# A Balanced Bracket

On his website tournamentdesign.org, Joe Czapski has some ingenious tournament ideas.

One of his chief thoughts is to try to equalize the treatment of the upper and lower brackets in double- (and even triple- and quadruple-) elimination tournaments. He has a suite of Excel-based brackets for download. None of the ones I looked at looks like anything I’ve ever seen.

I’ve analyzed the “balanced” bracket for a 16-team double elimination so that I can relate it to the more conventional brackets already discussed.  Analyzed brackets for these designs are available here: balanced-bracket-bd-analyzed and  balanced-bracket-seeded-analyzed. These are large, single sheets, as the structure doesn’t lend itself to a clean division between upper and lower bracket. I’ve revised the bracket in non-substantive ways, using the the (A1, A2, …, B1, …, etc.) labels I’ve used other brackets, and moving the lines around a bit so that all the matches in a round are in a column. I’ve revised the drops a little, but only to make then a little more orderly – one of the interesting aspects of the bracket is that there’s no wrong way to arrange the drops. Duplicate pairings are an issue, but none of the duplication comes from bad drops. And to keep it comparable with the previous analysis, I’ve eliminated the possible rematch if the winners bracket champion loses a first round of the final.

bracket rounds fairness duplicates
16 DE, balanced, blind draw
7.5
10.76
1.63
16 DE, balanced, seeded
7.5
11.43
1.67
16 DE, shifted, blind draw
7
10.97
1.20
16 DE, shifted, seeded
7
12.28
1.23
16 DE, unshifted, blind draw
8
10.86
1.41
16 DE, unshifted, seeded
8
12.53
1.35

The balanced bracket is listed as taking 7.5 rounds because one round consists entirely of a match that is not played a little more than half the time. In this table, unlike the one in the pervious post, I’ve counted all duplicates, not just duplicates occurring before the last two rounds because in the balanced bracket one of the last two rounds doesn’t always happen.

A close inspection of the analyzed brackets suggests some reasons that the balanced design falls a bit short in fairness of the standard set by the other two. The distinctive structural feature is that each of the first three sets to drop = the A’s, B’s, and C’s – forms a separate mini-bracket that is resolved before it encounters a team dropping from any other round. This makes the path much easier for some teams than others.

Recall that the reason shifted brackets did poorly in seeded tournaments is that there was a mismatch in skill levels. The early rounds of the lower bracket were deprived of most of their better teams because those teams had been protected in the upper bracket by the seeding, and the shift sent the later drops too deep into the lower bracket. The balanced bracket has this problem for a somewhat different reason – it creates regions that have quite different skill levels because they’re populated by different drops.

Compare, for example, the generous treatment afforded the A drops by the balanced bracket. A drops are, on average, less skillful than the other teams – after all, they lost in the first round. In the balanced bracket, where teams need six wins to win the tourney, an A drop can get half-way to the goal exclusively by playing other A drops. Thus, while the future of an A drop is not bright in any format, it’s better in the balanced bracket. Although the skill level is the same for an A drop in any format, the balanced blind draw bracket gives the A drops collectively a 14.8% chance to win the tournament. In an unshifted blind draw, the A’s get only 6.7% of the wins, and in a shifted blind draw, they get 7.5% of the wins.

Here’s another way to see the inequity based on internal evidence from the analyzed balanced blind draw bracket. Compare the fate of the player who loses the first round played (thus becoming an A drop) and then wins a round with a player who wins the first round and then loses the second (thus becoming a B drop). The average W-L B dropper has an error rate of about 42, which is about four points better than the average L-W A dropper, whose error rate is about 46. But the A dropper, who gets to look forward to two more rounds paired against other A drops, has a 3.70% chance of winning the tournament, a full point better than the 2.67% chance afforded the B drop, who has to play better opponents in the next few rounds.

As expected, all of the formats gain something in fairness with seeding, as the whole point of seeding is to manipulate the pairings to the advantage of the more skillful players. The interesting finding of a couple of days ago was that conventional, unshifted brackets reap greater rewards than shifted ones. And the current figures show that this is even more true in balanced brackets. In a blind draw, there are likely to be some pretty good players among the A drops who just happened to have been drawn against other good players. If the bracket is seeded, there will be fewer of these, and the bracket full of A drops will be an even more appealing place to get a couple of cheap wins.

Joe Czapski’s Balanced 16 DE doesn’t shine by any of the metrics- it doesn’t excel at saving rounds, fairness, or avoiding duplicates. But it’s worth considering on other grounds. His main goal, as I understand it, was to create a format that keeps people playing, and he’s done that – no one sits out for more than one round in his format, and you can’t say that of either of the alternative 16DE’s I’ve discussed. And even if you don’t find that compelling, you’re likely to find something else that’s interesting and useful to you at tournamentdesign.org.