In Bad Byes, I discussed the problems with grouping byes together in a bracket for a single-elimination tournament. Today, with the help of the newly-developed fairness (B) metric, I’ll extend that analysis with an examination of a double-elimination format.
As before, I’ll analyze a tournament with 24 entries. There are some alternatives I’ll look at some day, but the usual way to handle this situation is to use a 32 bracket with eight byes. The question is whether it’s better to concentrate those eight byes in one half of the bracket, or spread evenly through the bracket, guided perhaps by the seeding lines.
As with so many other tournament design issues, the decision pits one goal against another. If the byes are grouped, the players drawing the bye can begin play immediately – their second-round matches are against other players who drew byes, and so are ready to go.
How much time this saves is an open question. Starting a round earlier may help move things along if it happens that the slow matches that would otherwise hold up the tournament happen to be in the half of the bracket that starts early. It’s not as helpful as a bracket shift, which actually reduces the number of rounds that need to be played, and it may do no good at all if the slow matches that hold up the tournament happen to be in the part of the bracket that doesn’t start early.
Against this possible efficiency gain, one has to weight the fairness loss resulting from the severely un-balanced bracket. In a single elimination format, grouping the byes caused the fairness (C) statistic to drop from 2.167 to 2.128. And the fairness (C) metric is not particularly good at reflecting fairness (B) problems in the early rounds of a tournament. Applying the new fairness (B) measure shows the problem even more clearly. For the single-elimination, fairness (B) drops from 2.924 all the way to 0.971.
But perhaps things aren’t so bad for the grouped byes in a double-elimination format. The lower bracket offers players another path to victory, and perhaps that other path is less compromised by grouping the byes, so that the loss of fairness is mitigated.
Now, when testing an idea with my simulator, I have generally drawn up what I think is the best possible bracket that embodies the idea. I want optimum drops to reduce the number of repeat pairings, for example, and as much bracket balance as the idea allows. But frankly I don’t know how to draw a lower bracket to go with and upper where the byes are grouped. I know that it’s going to be harder to avoid repeat pairings, but I don’t know just how much I want to tinker with the overall balance to reduce the repeats. I can’t really take fairness as my guide because I’ve already decided, in the interest of efficiency, to accept a degree of imbalance that I’d otherwise avoid.
So I’ve decided to run this test not on one of my own brackets, but on one created by a friend. My friend is an experienced tournament director, and a very creative drawer of brackets. I don’t always (or even usually) agree with him about every detail, but I almost always learn something when I look at one of his designs. He is, for example, the person most responsible for introducing me to the shifted bracket, and I’ve become a big fan of shifted brackets.
Here are analyzed versions of the brackets I’m testing: 24groupcdupper, and 24groupcdlower. For comparison, I’ll use a design of my own that spreads the byes in the way I recommend: 24uppercd and 24lowercd.
There are a few things to note about my friend’s bracket. It’s basically a CD shift, and so I’ve chosen a CD shift to compare it against. In general, I prefer the ED shift, but CD makes sense in this case because my friend’s bracket was designed not as a full double elimination, but rather as a championship with consolation in which the loser of E1 doesn’t drop – that team is simply awarded second place in the championship. Since E1 is not dropping, the ED shift is unavailable, and choosing the CD makes perfect sense.
Another anomaly is in the details of the drops. I initially approached the problem of modeling this format as a matter of running a 32 bracket with eight byes. But this way the drops are done in this bracket can’t be accommodated by a 32 bracket – I would have to have used a 64 bracket with 40 byes, and my simulator wouldn’t do that without some signification changes. For that reason, I made a separate version of my simulator specifically for running this format. In my main simulator, I handle byes by putting in entrants with skill levels so low that they’ll never win a match. But in the bespoke version, I created only the lines I needed. As a result, the tally for the mean number of wins by skill rank does not compare directly, because it included byes as wins in the comparison version, but not in the other one.
The results? There are a number of interesting things to note on the analyzed brackets, but I’ll summarize the highlights:
- The grouped byes bracket was a bit lower on fairness (C): 2.880 as opposed to 2.910. I’m still getting used to the revised measure, but I have the sense that this is a fairly substantial difference;
- The grouped byes were awful on fairness (B): 1.963 as opposed to 4.928. On fairness (B), the grouped byes were even worse than a single elimination run on the spread byes, which came in at 2.924. When you contrive to make a double elimination less fair than a single elimination, you must be doing something wrong;
- The grouped bye bracket did lead to more repeat pairings. Eight of the grouped-bracket matches could be rematches, while that was true of only five on the comparison bracket. discounting the last two matches, which is only fair because they wouldn’t be played at all if the text bracket was used, as intended, for a consolation rather than a full double elimination, the grouped bracket averaged 0.639 repeats for the tournament, as opposed to 0.377 for the comparison bracket;
- The new fairness (B) statistic punishes only inequities occurring on starting lines, but in the test bracket there are some eye-popping differences elsewhere in the bracket, also. Look, for example, at the first round of the lower bracket, where all four quadrants are all different from each other, and the third quadrant is very different indeed.
All in all, the experiment shows again that grouping the byes, even in a double elimination, impairs the fairness of the bracket substantially. To my mind, you’d have to really, really, want to start the second round early to justify treating your players so unfairly.