Getting the Drops Right

Today’s topic is arranging the drops in a double-elimination tournament. By drops, I mean the guides that show where the loser of a winner’s bracket match should reappear in the losers bracket. The goal is to avoid, as much as possible, repeating a pairing that happened earlier in the winners bracket.

First, let me show you why the drops matter, using as an example the simple 16-team double elimination format I introduced in an earlier post. This table shows the percentage of repeated pairings for nine of the matches in the losers bracket (in the other 21 matches, including, of course, all of the ones in the winners bracket, there were no repeats). The match numbers can be found on the new sample brackets page: 16upper, and 16lower, and the correct drops are shown on the latter. The figures are from my improved tournament simulator, and represent the results of running one million trials for each of the three drop patterns.

match correct
drops
awful
error
subtle
error
F1
0%
47%
0%
F2
0
47
0
F3
0
47
0
F4
0
47
0
H1
12
38
25
H2
12
38
25
J1
17
0
10
K1
33
34
34
L1
66
66
66
total
140
364
160

The “awful error” shows the result of carelessly simply listing the drops in order. The round B drops are simply dropped 1-2-3-4 instead of the proper 3-4-1-2. The subtle error comes from misplacing the C drops, 2-1 instead of the correct 1-2.

I’ll go into the details of how to determine what’s right tomorrow, but for now, it’s enough to note that there is a high price to pay for doing it wrong. With correct drops, there will be an average of 1.4 repeated pairings for the tournament. With the worst drops, this goes to 3.64. But even this understates the comparison.

With any drops, there is a two-thirds chance of a rematch in the championship game (L1), and another one-third chance of a rematch in the last game of the losers bracket (K1). But everyone expects such things to happen. In fact if the competition had no chance element, the championship game would always be a rematch. So, not counting those two matches, there will be an average of 0.4 rematches in the proper bracket, but 2.64 rematches in the bad bracket – more than six times as many.

Many of these rematches will be particularly bothersome because they occur in an early round – the F round, which (with the C round in the winners) is only the third round played. Thus, it’s quite likely that some poor player who wins one match and loses two will be out of the tournament having faced only two different opponents, and reasonably likely that they would have lost both matches to the same opponent. When this happens, the victim has good reason to complain about the way the bracket was constructed.

Fortunately, drops as bad as these are rare. But less egregious mistakes are quite common. The error shown above as a “subtle” error is made in the bracket available from printyourbrackets.com. And this is in a pretty small bracket that should be easy to analyze correctly.

My earliest investigations into tournament design, and also some of my most recent, have been concerned with getting the drops right. This is not because it’s a difficult issue, but because it’s one of the most common mistakes I see, and one where a proper cure doesn’t compromise any of the other goals of tournament design.

Improper drops have a modest effect on the tournament’s fairness. As noted before, the coefficient of fairness for the standard bracket is 10.86. With the subtle error, it is 10.84, and with the awful drops it is 10.81. This difference is small enough that it’s probably not, itself, a good reason to worry about getting the drops right – the good reason is to avoid repeat matches.

Tomorrow I’ll step through the process of determining the correct drops for a slightly more challenging tournament: a 32-team double elimination.