More Better Bad Byes

In More Bad Byes, I looked at the consequences of grouping byes together in a bracket rather than spreading them evenly according to the conventional seeding pattern. Today I’ll follow up on a couple of suggestions that were made in the comments to that post.

Kevin H. suggested a format with the byes grouped just enough to allow the second round to start immediately. His bracket would equalize in the third round, rather than the second as happens with fully spread byes. As might be expected, his bracket does better than one with fully-grouped byes, but not as well as with a full spread.

Sean Garber, the author of the grouped-bye design that was the centerpiece of the analysis, mentioned that he had revised the drops for that design, and sent the new ones. His new drops made a modest improvement in fairness (C), but actually hurt fairness (B) a little, and increased the number of repeated pairings. I, guided by the pattern Sean established, uses the technique described in Getting the Drops Right, Part II to derive another new set of drops, which improved fairness (C) and reduced the number of repeats.

Here’s a summary of the results from this post and the last:

The analyzed brackets for Kevin’s suggested format are 24upperss and 24lowerss. There’s still a price to pay for getting the early start on the second round, but it’s considerably fairer, in all respects, to any version of the grouped-byes bracket. It seems reasonable to consider this a compromise between fairness and efficiency. It’s not clear how much time it will save, but it does have the important feature of getting everyone started at the beginning, and some directors have told me they consider that a highly desirable quality.

The new drops didn’t help the grouped-byes bracket very much, except that the TG drops helped avoid repeat pairings. The average repeats column in the table applies, as before, only to repeats that occur before that last two matches. Avoiding repeats was apparently not a high priority except for TG, but as there don’t seem to be any other big improvements to be had, one might as well do what can be done on that score. Here are the analyzed brackets that relate to the third and fourth lines of the table: 24groupuppersgrev; 24grouplowersgrev; 24uppertgrev; and 24grouplowertgrev.

One of the strengths of the CD shift is that it’s possible, with proper drops, to have repeats in only the last five matches in the lower bracket if the bracket is reasonably balanced. Grouped byes make this impossible – the quadrant that takes the D2 drop has to risk repeats because D2 represents two-thirds of the entire field, including all of the A drops.  But it’s still possible to limit the damage to that one quadrant.

Perhaps one of the hazards of innovations like grouped byes is that they give the bracket designer so many other things to think about that proper drops are likely to get overlooked. The bracket with nine possible repeaters has a particularly troubling possible repeat in the first quadrant of the second round of the lower bracket. It repeats nearly 12% of the time, and in some of those repeats, a player will be out of the tournament having won one match, and lost two to the same opponent.

Some other issues were also raised with respect to this experiment. Sean very sensibly observed that the simulations use a luck factor considerably lower than what would be needed to simulate the actual distribution of results for backgammon, the game for which he designed the bracket. But, as Sean was the first to observe, increasing the luck factor would only exacerbate the fairness problem because it would give players less opportunity of overcome the structural disadvantages imposed upon them by the bracket. So I resisted the temptation to fuss with the luck factor – as in my scientific work, I try to alter one variable at a time so that the influence of that variable shows itself more clearly.

Another trenchant observation is that these results don’t necessarily generalize to brackets with more bye, or with fewer. I chose 24 as a bracket to study because it’s right in the middle between 16 and 32, and thus might be maximally influenced by byes. But in other respects, it’s a very tractable number because the bracket as a whole can be brought back into balance after only two rounds. with any odd number of entries, perfect balance can never be restored. It may well be that there’s something to learn from taking Sean’s suggestion and studying the options for a 21-player bracket.