# Slow, Shifty Brackets

In Getting the Most from an Afternoon, I found that shifting the lower bracket resulted in a considerably slower event. This result was explained, in Making a Double Elimination Run Faster, to be caused by the fact that the shift caused the lower bracket matches to have to wait on the results of the upper bracket matches, which were played to a larger number of points.

This is not a general result – playing longer matches in the upper bracket is common in backgammon tourneys, but rare in most other events. But a further experiment shows that, for backgammon, it’s somewhat robust.

In the initial experiment, with 16 brackets, the choice was between saving a round with a bracket shift and saving a round by not dropping the D round, instead just paying second-place to the lower of the upper bracket final. With a 16 bracket, you can’t do both.

I reran the experiment on 32 brackets, where you can do both. Upper bracket matches were played to seven points, and lower bracket matches to five points. Here, the unshifted bracket with no E drop (A.B.|.C.|.D.|) runs eight rounds, while the shifted bracket, also omitting the E drop (A.B.C.D.|.|), runs seven rounds. Nonetheless, the unshifted bracket runs faster. Here are the mean start times by round in the lower bracket:

A.B.|.C.|.D.|:     12:59; 1:52; 2:33; 3:09; 3:49; 4:21; 5:02
A.B.C.D.|.|:          1:02; 1:59; 3:03; 4:07; 4:46; 5:30

(Here, for the sake of simplicity, I’ve used the later of the two start times for the shifted bracket. In rounds showing starts of 3:03 and 4:07 there are some matches that don’t receive drops, starting at 2:47 and 3:49, respectively.)

The unshifted bracket makes up for the extra round, and then some, by not having to wait for slower matches.

Here’s another example, comparing times for a 64 double elimination, with and without a shift. Note that the shift saves two rounds, not just one, but still manages to run more slowly.

A.B.|.C.|.D.|.E.|.F.X:   12:59; 1:52; 2:34; 3:10; 3:53; 4:26; 5:08; 5:40; 6:22; 6:53; 7:24
A.B.C.D.|.E.F.|.X:           1:04; 2:02; 3:09; 4:12; 4:53; 5:42; 6:37; 7:15; 7:49

Before discarding shifts all together, however, it would be well to bear in mind a few other considerations.

First, this effect is limited to situations where the matches in the upper bracket take longer than the ones in the lower bracket. That’s usually the case for backgammon tourneys, but it’s uncommon elsewhere.

Second, A.B.C.D.|.| is the recommended shift for a 32 bracket only where the E round does not drop. Where the E round drops, previous simulations have shown that A.B.C.D.|.|.E is inferior to A.B.|.C.D.E.|. Breaking up the initial A.B.C.D run of consecutive drops with an earlier consolidation round breaks the logjam, so that the shifted bracket runs nearly as quickly:

A.B.|.C.|.D.|.EX:    1:00; 1:55; 2:37; 3:14; 3:56; 4:31; 5:14; 5:48; 6:21
A.B.|.C.D.E.|.X:      1:02; 1:59; 2:43; 3:02; 4:15; 5:21; 5:56; 6:30

Finally, there’s probably a price to be paid for the extra round that the simulator doesn’t catch. The simulator operates on the unrealistic assumption that each round starts immediately when the two players clear their earlier matches. In practice, an extra round break usually means extra time for the players to find each other, go to the bathroom, order fresh drinks, and so forth.