In the last post, I compared the double-elimination portion of the Ottawa Men’s Bonspiel curling bracket against a slightly revised version of itself. But the more important comparison to make is with the a more conventional version of a double-elimination for 91 teams. And, as I’ll show, comparing the Ottawa bracket to a more conventional approach unexpectedly seems to open new avenues of inquiry for bye management.
As discussed in BBBR: 128s, there are a number of suitably-sized brackets from which to choose a comparator. In deciding which to use, I came upon another surprising virtue of the Ottawa-adapted format: it runs in just ten rounds, and there are almost no long waits except those caused by byes. The quickest of the 128s is the “128supershift“, which takes 11 rounds. So, to capture as much of the virtue of the Ottawa adaptation as possible in a (not very) conventional format, I chose the supershift.
Another convenient feature of the super shift is that it can be made to pay five places in the 28/18/18/18/18 payout pattern chosen for the Ottawa design. As before, I set luck at three.
The difference in fairness (C) is dramatic: 97.69 for Ottawa, and 86.05 for the supershift. This is only what we expected – it’s been clear from the beginning that the Ottawa pattern is not what you should choose to maximize fairness (C).
But there’s a surprising result with respect to fairness (B). The supershift comes in at 14.84, while the Ottawa design gets 6.97, or 6.12 in its tweaked version! What’s going on?
The key is to understand what causes the fairness (B) deficit. Both approaches are balanced brackets, so that there should be no fairness (B) problem if they were run with a full complement of 128 teams. Essentially all of the fairness (B) problems are coming from the byes.
What’s happening here is that the Ottawa design is automatically compensating for some of the unfairness inherent in the large number of byes. Teams that get byes are one round closer to most of the prizes, but they are actually ineligible for the Billings trophy because they never drop into that bracket. And the Billings trophy is the easiest one to win.
Note that this effect depends on the fact that the number of teams in Ottawa is close to one of the problematical numbers half way between powers of two. 91 teams is nearly 96 teams, and 96 is the maximally-awkward number of entries between a 64 bracket and a 128 bracket. If there were no byes necessary, there would be no advantage in terms of fairness (B).
This Ottawa experiment would seem to suggest that there might be other ways to ameliorate the unfairness caused by byes. We know that where there are byes, teams that don’t get them are at a disadvantage. Perhaps we should structure competition so as to mitigate this disadvantage, especially for tourneys near the difficult sizes: 12, 24, 48, 96, and so forth. I’ll explore a few of these in a subsequent post.
All in all, there is much to be said in favor of the Ottawa design. It runs in fewer rounds than any other known bracket. It sacrifices some measure of fairness (C), but then fairness (C) is among the most recondite qualities of a tournament. I can’t look at the design without seeing the aspects that compromise fairness (C), but it may well be that the average player, who has spent little or no time ruminating about the fairness of various bracket designs, will find it just as satisfactory as any other. And even I look at these brackets and see a tourney that I think would be fun to play.