The Big, Peculiar Bracket, a Postmortem, Part I

In an earlier post, I discussed a large bracket that was being prepared for a major backgammon tournament the summer in Michigan. The tourney was to be a full double-elimination format, with a consolation as well, capable of accommodating 96 entries.

For reasons I don’t fully understand (and would probably be impolitic to discuss if I did), the director chose not to use the bracket I prepared. The tournament was played using another very interesting and elaborate format that presents a number of bracket design issues.

In this post, I’ll discuss the structure of the format that was played. In subsequent posts, I’ll report some simulation results, and compare the format used to the unused one I suggested, and to a more conventional bracket.

Here are my re-drawings of three bracket sheets for the Michigan tourney: MichUpper, MichMiddle, and MichLower.  

The upper bracket is a set of six nearly conventional 16 brackets. Apart from the fact that there are six of them, the only notable peculiarity is that the seeding lines have been manipulated so that if there are fewer than 96 entries some of the byes will be grouped, allowing a few second-round matches to begin immediately.

The middle bracket has a number of noteworthy features. The lower part of the sheet is occupied by a set of six 16-player losers brackets. The upper part of the sheet is a series on mini-brackets that take the six winners from the first sheet and the six winners from the losers brackets for a playoff to determine the overall winner.

The first three of these playoff mini-brackets, the ones with matches K1, K2, K3, L1, L2, L3, Z4, Z5, and Z6 are, in essence, the missing bits and snippets necessary to transform the six 16 brackets on each sheet into three 32 brackets. This presentation is, to be kind, eccentric – it would be much clearer, and functionally identical, to draw the three 32 brackets instead.

The shifted structure for a 16 bracket is A.B.C.D.|. For some bazaar reason, only two of the six losers brackets drawn this way. Of the other two, two are A.B.C.|.|, with a double-helping of C drops, and two are A.B.|.D.|, with a double helping of D’s. This can only exacerbate the imbalance of the losers bracket.

But a larger problem is that the shift is done on the approximate 16-shift pattern at all. Since the six 16 brackets are, for all intents and purposes, really three 32 brackets, they ought to be using the best 32 bracket shift for high-luck games like backgammon, which is A.B.|.C.D.E.|. Instead, by drawing the brackets as if they were 16, the brackets have to use the inferior A.B.C.D.|.|.E pattern. There aren’t any drops from an E round because they’ve been hidden away in the K round in one of the mini-brackets, but it comes to the same thing. So, not only is the superficial 6 x 16 organization of the bracket harder to understand, it requires the use of an inferior shift.

One other stylistic oddity makes the middle bracket hard to understand. Almost always, the match identifiers are assigned systematically. Thus, for example, you should be able to find the matches in a straightforward order, reading A1, A2, … , A48; B1, B2, … , B16, etc. from top to bottom on the sheet. For no apparent reason, the middle bracket breaks this pattern. The beginning of first losers bracket round is labeled E1, E2, E9, E10, E17, E18. Rounds F, G, and H are similarly out of order, before sanity is restored with the J round. It’s hard to imagine any reason at all, let alone a good one, for doing this.

The most innovative thing about the tourney format is the way that the six survivors – three undefeated players, and three one-loss players coming up through the middle bracket, are managed so as to yield a single winner, runner up, and third place, all while respecting the principle of double elimination. That logic is embodied in the last of the three-match mini-brackets and the two contingent brackets at the top of the sheet. Here are those brackets:

The X4, X5, and X6 matches are always played. Z1, Z2, and Z3 are the three winners of the 32 winners brackets that emerge from the other mini-brackets. Z10, Z11, and Z12 are, similarly, the winners of the three emergent 32 losers brackets.

X5 is the key match because it pits an undefeated player against one with a loss. If the Z10 player wins, there will, at the end of the X round, be one undefeated player left, and four with one loss. The undefeated player goes straight to the final, awaiting the winner of a playoff among the other four.

But if Z3 wins match S5, there will be four players left, two undefeated, and two with one loss. Then, the bracket is resolved in the way a four-player page playoff is resolved, with the loser of the battle of the undefeated getting another chance to make the playoffs by playing the winner of the two one-loss players.

This is, in a way, ingenious. Hey presto!, the 32 byes needed to play a 96-team tournament on a conventional 128 bracket simply disappear. And getting rid of 32 byes is bound to improve the f(B) statistic (which was formerly known as fairness (B)).

There is, however, a price to pay. It’s always a red flag when you introduce something that’s not in perfect balance, and it’s particularly worrisome when you do it late in the bracket, especially so late as to affect the money places. Here, the unbalanced item is match X5, and the victim is the Z3 player.

Here’s the problem. When Z3 loses match X5, the winner of X4 gets to go directly to the final. But when Z3 wins, Z3 always goes instead to a semi-final. Earning line Z3 by winning the virtual third 32 bracket is less valuable than winning either of the other two 32’s because it never gets to win by the shorter path. That means that all of the lines that feed up to Z3 have a lower expectation than the apparently equivalent lines that feed Z1 and Z2. This will necessarily hurt the f(B) fairness of the entire bracket.

This may be a price worth paying. The many byes that will be needed to fill out the standard 128 bracket are going to exact a toll measured by f(B), too. So perhaps this is the best pattern. But in order to make the choice intelligently, we should test the two ideas to see how the two sources of unfairness compare.

And, if one knows that the last third of the bracket is getting shafted by the X5 problem, it would seem that you should at least try to compensate a little by directing whatever byes are necessary into the unfairly disadvantaged bottom third. There doesn’t seem to have been any effort to do this.

Such are the peculiarities that appear from a surface inspection of the upper and middle brackets used in Michigan. There peculiarities of the bottom, or “Consolation” bracket are also numerous, and will be analyzed in a subsequent post. And yet another post will show the result of running the big, peculiar bracket through simulation.