# A Notation for Bracket Structure

Tourneygeek has been quiet for a while, but that is not (or at least not mostly) because I’ve been idle. The new simulator is nearly ready to run, and there should soon be plenty of results to report. I hope these will include substantial progress on such matters as a coefficient of competitiveness, fairness (D), and new measures of efficiency based on modeling playing time.

In the meantime, I’ve been coding several basic tournament designs for the new simulator format. In the course of this, I’ve developed a shorthand way of describing lower brackets that is handy for showing how they are (or are not) shifted.

The idea is to show how the drops from the rounds in the bracket above are distributed among the rounds in the bracket receiving the drops.

The unshifted 16-team double elimination bracket receives its drops in a pattern that can be notated thus:

A . B . | . C . | . D . X . R

Here’s what this means. The A drops go into the first round of the lower bracket (which I usually call the E round, as A-D are all that are needed to describe the four rounds in the upper bracket). B’s drop into round F. The vertical bar indicates that round G is a consolidation round, which receives no new drops. C drops go into the H round, and then the I round is another consolidation. The single D drop goes into the J round. The X indicates that the next round unifies the upper and lower bracket, and the R shows the recharge round – the possible repeat of the K round where the tourney is a strict double elimination in which the survivor of the lower bracket has to defeat the upper bracket champion twice to win the tournament.

Here’s the same notation for an unshifted 32-team double elimination tourney:

A . B . | . C . | . D . | . E . X . R

For 32 teams, you need two extra rounds – one for the E drop, and another consolidation round. Note that the length of the string is a handy way to count the number of rounds in the tournament (though you have to add one to account for the first round in the upper bracket that’s needed to fill the first matches). Thus, the 16-team tourney runs nine rounds, and the 32-team tourney runs 11.

Larger tourneys follow the same pattern. Here are the strings for 64- and 128-team tourneys with no shifts:

A . B . | . C . | . D . | . E . | . F . X . R

A . B . | . C . | . D . | . E . | . F . | . G . X . R

Where elements of the full double-elimination format are omitted (usually to make the tournament run faster), the string is shortened appropriately. For example, if a 32-team tourney is run not as a full double elimination, but as a single-elimination with consolation, and the upper bracket pays the top two places (so that the E round does not drop), the string is this:

A . B . | . C . | . D

This shows the appeal of the format – almost as many individual games are played, but the tourney runs in seven rounds rather than eleven.

The notation provides a good way to show bracket shifts. Here, for example, is the shifted 16-team bracket:

A . B . C. D . | . X . R

This saves a round (and except for events with a high degree of skill progression) enhances fairness.

For the 32-team bracket, there are two possible shifts, which I have called the “CD” shift, and the “ED” shift, respectively. Here’s I’ll show them without the final recharge round:

A . B . C . D . | . | . E . X

A . B . | . C . D . E . | . X

The pattern is to shift two adjacent rounds to the left, eliminating the two consolidation rounds that separate them in the unshifted bracket, and adding one consolidation round to the right of the shift, for a net reduction of one round.

Here’s a 64-team pattern, which permits a double shift:

A . B . C . D . | . E . F . | . X

The double shift, together with eliminating the recharge round, has reduced a 13-round tourney to ten rounds.

One final bit of notation, which will explain why there are dots separating the rounds. Sometimes a single round in the lower bracket receives drops from more than one round above. This happens naturally for a 24-team tournament:

AB . C . D . | . | . X

This is simply a 32-team format in which the A’s and B’s drop into the same round because all of the A’s get a bye in what would otherwise be the first round of the lower bracket.

Other formats that require such combinations are likely to be ungraceful accommodations to very difficult situations. Here are two patterns in which some rounds receive drops from more than one round, and some rounds drop into more than one round:

AB . BC . DE . F . G . | . | . | . |

A . ABCD . E . F . G . | . | . | . |

These pattern was actually under consideration for the hideously difficult last chance that’s part of the Big, Peculiar Bracket. Needless to say, there are some substantial fairness objections to brackets that look like this. The extreme shifts, together with the large number of successive consolidation rounds at the end, suggest a perceived need to squeeze as many rounds as possible out of the tourney.

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