The Big, Peculiar Bracket, a Postmortem, Part III

In two earlier posts, I examined a set of brackets for a 96-team double-elimination tourney with a third consolation bracket. This format was originally discussed in a post called A Big, Peculiar Bracket (BPB).

These analyses identified a number of apparently sub-optimal features of the bracket. In this post, I’ll report the results of series of simulations of the BPB, and compare them with an alternative bracket that maintains the same overall structure, but addresses several of the identified issues. The result is that the BPB can be somewhat improved by fixing them.

A subsequent post will address the more fundamental issue of whether the unusual structure of the BPB makes it perform better than a more conventional bracket.

The distinctive feature of the BPB is that it consists of a set of six 16-team brackets, upper and lower, together with some extra brackets, some of them contingent, to make the 16 small brackets into a single double elimination, together with a third bracket that serves as a consolation. Here are the three parts of the BPB: MichUpper, MichMiddle, and MichLower.

As noted in the previous analysis, the apparent 6 x 16 structure of the upper and middle brackets is functionally identical to a 3 x 32 pattern, where some of the lines have been hidden away in mini-brackets on the upper portion of the page containing the middle bracket. For the simulation, I’ve maintained the odd labeling system this implies, with round “Z” replacing round “E”, which then becomes the label of the first round in the middle tier. But I’ve otherwise treated the BPB as a 3 x 32.

To see the effect of some of the anomalies noted previously, I’ve also simulated a similar tourney, in which the individual 32s, upper and middle, have been drawn according to tourneygeek’s best practices, most notably substituting a A.B.|.C.D.E.| structure for the A.B.C.D.|.|.E structure of the losers brackets.

Tourneygeek did not have a recommended third bracket for a 96-team tourney, so I drew one using the method discussed here. The groups of drops that needed to be accommodated were: 24, 24,12, 12, 9, 6, 3, which, in order to form a bracket that plays in the same number of rounds as the bottom bracket of the BPB where arranged thus: FG.GH.JK.L.M.|.|.|. Here is that complete lower bracket, with drops: M3x32consol.

Both the BPB and the alternative bracket were simulated three times. They were run with a full 96-teams, and also with 85 teams and 76 teams. These latter numbers were chosen because they were the actual number of entrants for the two events that were run using the BPB at the Michigan Summer Backgammon Championships.

As the BPB was intended for use in a backgammon tourney, I’ve used the luck factor previously developed for backgammon. Both tourneys were run as elite tourneys – that’s arguably a bad choice for the 76-player tourney, which was the intermediate-level event in Michigan, but I didn’t want to introduce another extraneous factor that would hinder the comparison of the 76 with the 85 and the 96. All six of the simulations consisted of 200K trials .

The tourneygeek simulator cannot, as it currently exists, deal with the contingent bracket that resolves the last six players in the double elimination. So I truncated the analysis at the X round, awarding the players on those six lines their expectations based on the assumption that the remainder of the tourney would be resolved by coin flips. These payouts (and expected payouts) were based on the actual payouts of the open division in Michigan.

Here are the results of the simulation, showing the fairness (C) and fairness (b) coefficients, in their new versions (for which a lower number indicates more fairness):

 BPB96 BPB85 BPB76 alt96 alt85 alt76 f(C ) 73.18 71.55 69.93 f(C ) 72.24 70.66 68.83 upper bracket f(b:A) 1.77 15.90 19.45 f(b:A) 1.69 16.18 18.99 f(b:B) 1.99 4.80 5.33 f(b:B) 1.84 5.02 5.99 f(b:C) 2.22 4.45 4.75 f(b:C) 1.85 2.94 2.41 f(b:D) 2.40 2.97 4.10 f(b:D) 2.17 2.18 2.21 f(b:Z) 2.67 2.61 2.63 f(b:E) 2.54 2.54 2.50 middle bracket f(b:E) 5.68 11.09 13.74 f(b:F) 3.65 16.56 13.17 f(b:F) 1.47 7.96 13.56 f(b:G) 4.42 7.98 5.72 f(b:G) 1.33 3.41 5.48 f(b:H) 0.94 2.04 2.78 f(b:H) 1.16 1.53 2.61 f(b:J) 5.34 5.01 4.53 f(b:J) 1.09 0.94 1.14 f(b:K) 1.97 1.84 1.60 f(b:K) 0.98 1.05 0.85 f(b:L) 0.95 0.86 0.73 f(b:L) 3.40 2.89 2.99 f(b:M) 0.24 0.21 0.18 lower bracket f(b:M) 0.45 1.80 1.55 f(b:N) 9.42 30.96 20.55 f(b:N) 5.95 12.80 32.46 f(b:P) 4.68 5.88 7.04 f(b:P) 5.80 6.13 6.52 f(b:Q) 5.78 5.95 6.99 f(b:Q) 7.50 7.96 7.73 f(b:R) 4.19 3.90 4.63 f(b:R) 6.44 6.21 6.30 f(b:S) 3.01 2.74 3.02 f(b:S) 3.45 3.44 4.13 f(b:T) 1.92 1.90 2.09 f(b:T) 1.94 2.27 2.57 f(b:U) 0.41 0.48 0.39 f(b:U) 0.37 0.45 0.71 f(b:V) 0.21 0.23 0.20

There’s a wealth of information in this table. But here are a few observations.

First, in every case, the alternate structure is better than the BPB with respect to fairness (C). It’s likely that one of the principal reasons for this is the use of the A.B.|.C.D.E.| shift pattern rather than the A.B.C.D.|.|.E shift used by the BPB. This shows up in the figures for the last round in the middle bracket, f(b:L) for the BPB, and f(b:M) for the alternate. The fairness numbers, 3.40, 2.89, and 2.99 as against 0.24, 0.21, and 0.18 may seem small, but this is the round that decides whether the player is in the big money in the double elimination or drops fairly deep into the consolation.

Comparing the full 96 team numbers with the 85 and 76 numbers shows that byes are the principal source of f(b) problems. In every case, the chief shock to the f(b)’s happens in the first round of each bracket, where there are byes. (It looks like the M round in the BPB is an exception to this, but it really isn’t – in the BPB, 11 or 20 byes is enough to wash out the M round entirely.

In every case, the f(b) problems eventually wash out of the middle and lower brackets, yielding f(b) figures near one. This doesn’t happen in the upper bracket because of the imbalance introduced by the contingent match that’s used to squeeze the three top brackets into a power of two playoff. The bottom third of the upper bracket draw are all playing for a line that’s worth more than \$200 less than the lines the other two thirds of the draw are playing for. The equalization in the lower bracket is slow in the BPB because of the failure to interleave the various drops, which tends to push the problem forward into later rounds.

The single worst round is the experiment is N in the BPB76. N is bad because it is the only round that receives three different sets of drops, and it’s particularly toxic because the large number of byes in the 76 gives several E drops byes into that round, which effectively gives that round a fourth set of drops to deal with.

The BPB85 outperforms the Alt85 with respect to fairness (B) (which is equal to f(b:A)). It’s not easy to see why this should be true. The 11 byes needed by the BPB85 cancels the M round nearly exactly, so that the 12.80 suffered in the N round is relatively benign compared to the 30.96 suffered by the Alt85. Perhaps the 11-bye model hits a sweet spot in the BPB that’s strong enough to outweigh the BPB’s various infelicities. But the fact that Alt85 still beats BPB85 still suggests that the Alternate structure is superior, even with this number of byes.

There are other aspects of the BPB that might be examined. One of its less endearing quirks is that it has some flow problems. There are a few matches for which, on average, one player will need to wait about three hours – I personally was unlucky enough to get the long wait between my first and second matches. As noted in part II of this postmortem, the drops to the lower bracket are not optimized to avoid rematches, though the drops to the middle bracket are fine.

All in all, however, it would appear that the Alt bracket is significantly better than the BPB. But this shouldn’t be a surprise. The Alt bracket sweated a welter of details that the BPB did not. And, as a practical matter, directors shouldn’t have to worry about those things. It’s enough that tourneygeek has worked through them, so that future directors can be guided simply by finding the 96Alt rather than the BPB on the printable brackets page.

You might note, however, that the 96Alt format does not appear, at least yet, on the printable brackets page. It’s still a pretty warty bracket. Those warts may be smaller than the warts of the BPB, but that doesn’t mean that running any tourney as a 3 x 32 rather than a conventional 128 is a good idea. We need to find a way to weigh the unavoidable problems of the 3 x 32 approach against the gains realized by avoiding byes. If the 96 does well, perhaps it can be admitted to the pantheon of the printable bracket page, but it’s not there yet. That is a topic for another post.