A friend asked me for a specialized bracket for a backgammon tourney he’s running soon. He wants a 128 bracket that’s a full double elimination with a progressive consolation as a third bracket. There’s an unexpectedly nice one available.
Most backgammon tourneys do not attempt double elimination – it simply takes too long to play. But a few, especially among those played on longer, holiday weekends do.
I recalled from earlier work that there’s a remarkable “super shift” that squeezes an extra round out of the second bracket in a 128DE, and also scores well on fairness. But what happens when you need a third bracket? In every other case, adding a third bracket to a shifted bracket necessitated some pretty ugly compromises.
In this particular case, however, there’s something astonishingly nice: 128SSthirdA. Unlike any other third bracket I’ve ever drawn, it is a strictly progressive consolation, with no split or overlapping round drops. What’s more, it avoids rematches entirely for the first three rounds of the consolation, and they’re infrequent after that.
tourneygeek.com 
As I’ve said before, I’m interested in seeing how a 256 double-elimination could work.
The main advantage 256 has over 128 is that 256 is a power of 16, so no consolidation rounds are necessary until after the winners’ bracket final has been played. The sole H drop goes into losers’ quarterfinals, joining the seven entrants already there.
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Hmmm. A.B.C.D.E.F.G.H.|.|.|.X. That is a remarkably clean design. A 256 in only 13 rounds (unless you insist on a recharge, which I expect will be shown to be counterproductive), saving a full 3 rounds over an entirely unshifted 256DE: A.B.|.C.|.D.|.E.|.F.|.G.|.H.X.
I’ve never been asked for a bracket that big, but it’s worth looking into. I’ll test it as soon as I get done with my current project, which is trying to optimize a 96DE with consolation.
I’m guessing that it will work best for relatively high-luck games. If there were a high level of skill progression, I expect that lonely H drop will feel a bit ill-used to drop all the way to the loser’s quarterfinals. It might be interesting to determine the luck factor at which the triple shift loses its fairness (C) advantage over the unshifted version.
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Yeah, when you have a 3-Round Bye, you might as well give a prize to the guy who wins the Winner’s Bracket, (I’m talking Trophy and non-monitaries, not just a bit of the pot,) like Hogsback for the Ottawa Bonspiel, and a similar (perhaps reduced value) prize for winning the Loser’s Bracket, with the Championship just awarding a sizable infusion of cash/gift card. A 3-round rest has the possibility of the rest generating “rust”; i.e. taking the WB out of “game mode” sufficiently for the LB to have an “edge” even though they played every round and likely the fatigue from doing so.
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