Getting the Drops Right, Part II

So, having made a case for the importance of putting the drops in the right places, let’s turn to the issue of how to get them there.

As a practical matter, the easiest thing to do is to download or copy from some source that can be relied upon to get it right. Unfortunately, there’s no site, other than this one, that I think can recommend on this point. I hope, in the fullness of time, to build out the collection of sample documents I make available here to the point where a tournament organizer can find a useful sample for any but the most exotic tournament. But don’t hold your breath. In the meantime, let me explain a general approach that I’ve found useful.

First, take a blank bracket sheet, and label the bracket from which the drops come (here, 32upper) to show which of the tournament’s entries might make it to any particular line. You don’t need to separate the lines that are paired against each other in the first round because those pairs will forever have the same possible conflicts. I use letters for small tournaments, but where there are more than 26 first-round matches it makes sense to use numbers instead. Here’s a part of such a sheet for a 32-team tournament:

I start with the lines for the second round, with match A1 getting A, A2 getting B, and so forth. For the third round, the labels start representing a range rather than an individual letter: labels like A-B and C-D in the third round, A-D in the forth, and so on.  It’s important to do this in the most straightforward and boring way – the drops themselves will be out of the natural order, but don’t start trying to distribute them yet.

I then print slips of paper representing every match that needs a drop. This may seem needlessly fussy, and it isn’t strictly necessary for smaller tournaments, especially if you’ve done it a time or two. But I find it indispensable for larger, complex tournaments.

The first label is A1_A, the next A2_B, and so forth. I color code the labels, usually by round, with black labels for the A round drops, blue for the second, etc. Another good use of color is to color code regions of the bracket. It’s often helpful to rotate the sequence of drops from one quadrant to another, so in the example above I might just as well have made the slips for the first quadrant, A-D, black, the second E-H quadrant blue, and so forth.

Then, with a copy of the bracket you’re dropping to (here 32lowerstandard) as a guide, arrange the slips so that no letter repeats within the branching for as many rounds as possible.

Here’s a picture of the arrangement of slips I settle on for the 32-team tournament with the most ordinary arrangement of lines:

In general, you want to keep the drops together so that they can combine into the same compact combinations that you have represented in the drops from later rounds. You’ll notice that I don’t even bother to cut apart pairs of drops from the A round – I know from experience that I’ll want to keep them together. And look at the top quadrant – I have A-D for the four A round drops, and E-H for the two B round drops. That makes it immediately clear that I can drop either C3_I-L, or C4_M-P into that quadrant, but not C1 or C2.

If the B drops had fit with the A drops less compactly, it might have been impossible to find good spots for the C drops. I sometimes find it useful to make a guess about which is the last set of drops I can accommodate with no potential repeats and then work backwards, to the left. I try for an orderly rotation so that the solution for one part of the bracket will automatically reproduce itself elsewhere. Here, the B drops were shifted down one quadrant, and the C drops down two quadrants.

The first priority is to make as many entirely conflict-free early rounds as possible. It’s early repeat pairings that people find most annoying.

When you get to the point at which new drops cannot be made conflict free, try to make them so that the conflicts that can occur have as many intervening rounds as possible. Here, it would be a mistake to drop D2_I-P in on top of C3_I-L and C4_M-P. (That is essentially what happened with the “subtle error” discussed yesterday.) Better to drop the D’s as shown, where the possible conflicts need to survive more rounds before they meet.

It might be a good idea to make your pairings on a copy of the bracket itself rather than, as in the picture, on the table top. I usually have the bracket sheet off to the side where I can see it, but can still slide the slips around easily. But if you don’t pair directly on the bracket, you have to keep in mind when there are rounds that don’t receive any drops that still may have conflicts. In the example, the first conflicts will be in the round between the C drops and the D drops (matches 56 and 57 in the table below). In the picture, I’ve left a gap to remind myself to consider that round, though I ultimately conclude that I can’t do anything about it without causing larger problems elsewhere.

Assessing the relative likelihood of late-round conflicts can be exceedingly difficult in large tournaments with complicated structures. If you make as many conflict-free rounds as possible, and then make plausible guesses about the late drops, I think you’re entitled to declare victory. Now that I have the tournament simulator, I’m looking forward to testing my intuition about these late drops.

So how does this bracket work in simulation? Here’s what I get from 10 million trials. Only seven of the 62 matches are possible repeat pairings:

This amounts to an average of 1.439 repeats per tournament, or only 0.485 repeats if you don’t count the last two games. This is comparable to the results we got for the best of the 16-team tournaments yesterday. There are twice as many matches in a 32-team tournament, but there is also a larger bracket to spread them in.

One other interesting result of the simulation: the fairness coefficient is 10.82, very close to the 10.86 we got for the analogous 16-team bracket. I hope that the fairness coefficient will be similarly well behaved in other applications.