Adding the Third Bracket

Adding a third bracket to a double elimination or a consolation tourney is uncommon, but not unknown. In this post, I’ll walk through the process of creating three third brackets.

One will be constructed on entirely conventional principles as a triple elimination 32 bracket. I’ll do only enough of this process to show why you probably don’t want to run such a tourney. The other two will be “last chance” brackets – third brackets added to a consolation tourney which are never reconciled to either of the other brackets. I’ll show the differences involved in doing this for brackets of 32 and 48.

In the next post, I’ll return to the post-mortem analysis of the Big, Peculiar Bracket to consider its third tier, which presented unusually difficult challenges.

Triple Elimination 32

Drawing a third bracket to make an unshifted 32 into a full triple elimination is a relatively straightforward process.

The second bracket has absorbed five rounds of drops from the top bracket in this pattern: A.B.|.C.|. D.|.E. The third bracket will need to accommodate drops from all eight of these rounds. Because the second bracket is unshifted, the number of players in each set of drops is an even power of two: 8,8,4,4,2,2,1,1. That makes it easy give each round a number of matches that’s also a power of two.

For fairness (b) sake, one tries to put all of each set of drops into the same round, and to avoid any rounds that take more than one set of drops.

The main difficulty accommodating the drops is that there are two rounds of eight at the beginning. There’s little choice by to drop them together into a single round of 16. This is unfortunate, as it yokes eight teams with no wins with another eight teams with one win. But after that there need be no round overlaps. The pattern is FG.|.H.J.|.K.L.|.M.N. Here’s a schematic that shows the lines, rounds, and sets of drops: triple elimination lines.

On the schematic I’ve shown which round takes each set of drops, but I haven’t bothered to figure out the individual drops. That’s because I think this format is entirely unworkable. The blamed thing has 15 rounds. It’s also got four possible recharge rounds, marked on the diagram with an “r”. A minimum of one, and a maximum of three of these recharge rounds will need to be played, making the format 16 to 18 rounds in all. And notice that, after winning five rounds in the top bracket, the E1 winner will have to wait either nine of ten rounds for the other two brackets to resolve themselves to produce a single challenger.

 

Last Chance 32

I doubt that the full triple-elimination 32 will ever be attempted, but there is a three-bracket 32 that is being played at a backgammon tournament in Wisconsin as I write this. Here’s the way that bracket was constructed.

The first important feature is that the three brackets aren’t reconciled with each other, so that the competition has a consolation and a last chance. It also saves rounds by not dropping the loser of the finals in the top two brackets, and another round with a shift in the second bracket.

The second bracket patters is A.B.C.D.|.|, the “CD shift”. The later “ED shift” is preferred in most contexts, but there is no E drop from the upper bracket because that player is paid. That leaves five sets of drops from which the third bracket needs to be built. But, as the second bracket is shifted, the drops do not all come in powers of two – the third set of drops is from a round for which the shift has transformed one round of four and another round of two, and that upsets the usual pattern. That means that there are going to have to be some unbalanced drops.

The pattern I settled on was EF.FG.H.J.|.|. The eight F drops are divided between the first two rounds of the last chance bracket, creating a significant imbalance in each. There will be some players with no wins and others with one win in the first round of the last chance, and some players with one win and some with two in the second round. But at least that are no rounds that take three different sets of drops, and the imbalance is gone by the third round. Here’s the last chance bracket: 32 Last Chance.

As always, the upper bracket resolves in five rounds. The second bracket takes six, but waits on the first round above it, so it resolves in round seven. The last chance bracket resolves in six rounds also, but because its first round takes drops from two different rounds, it waits on two rounds from the second bracket, which means that the tourney as a whole plays out in nine rounds, which is only half the maximum number of rounds needed for the full triple elimination of the same size.

Getting the drops right for a third tier bracket is more difficult than it is for a second tier because the drops represent a greater number of possible competitors. But the process is essentially the same. It was possible to avoid repeat pairings in only the first two rounds of the third bracket.

Last Chance 48

Against the chance that the Wisconsin tourney would attract more than 32 players in some division, I also prepared a 48 bracket with a consolation and a last chance.

A standard 48 bracket is really just a 64 bracket that’s been pruned of 16 lines, effectively hiding the first 16 byes. There are two schools of thought about whether those 16 lines should all come from one half of the draw or should be spread evenly – this is the issue discussed in great detail in the series of posts starting with Bad Byes. I won’t revisit that analysis here, but chose to spread the byes evenly.

The format for the middle bracket, 2436lower48, was AB.|.C.D.E.|.|. The six rounds of drops were 16,8,8,6,4,2. Here only one round had to take more than one set of drops, in the patters FG.|.H.J.K.L.|.|, 48 last chance. The tourney as a whole plays out in ten rounds.

As with the 32 last chance, it was possible to avoid repeat pairings all together for only the first two rounds of the 48 last chance.