It’s time to address the thorny issue of byes in elimination tournaments.

So far, we’ve been considering only tournaments that conveniently happen to have a number of entries that’s an even power of two: 16 or 32. If you’re running an elite tournament, with people clamoring to get it, you can, if you like, decide to accept only such an convenient number of entries. But many tournaments are not at all elite, and gratefully take more of less everyone who shows up. And if the number of people who show up is not a power of two, you generally award the number of first-round byes necessary to bring the number of entries up to a power of two for the second round.

In this post, I’ll begin to discuss the effect this has on the tournament. In later posts, I’ll look at a number of other issues, but here I’ll just offer one analyzed bracket so that you can begin to see the effect byes have on a tournament.

The tournament I’ll look at is blind draw, double elimination, and has 24 entrants. It’s a non-elite event, so we won’t truncate the distribution of the entries.

The usual way of accommodating this number is to use at 32 bracket, padded with eight byes. (There are some exotic alternative approaches that will be explored later.) Here’s a suitable bracket:32upper. Let’s use an unshifted lower bracket: 32lowerus.

This upper bracket has seeding lines that are useful not only for seeding, but also for guiding an optimal distribution of the byes. In this case, the 24 names are entered onto the lines numbered one through 24, and the other lines become byes. Instead of 16 games in the first round, there will be only eight, and the first round of the lower bracket will be eliminated entirely as all of its eight games will become byes.

Here are the analyzed brackets: 24upperns and 24lowerus.

The chief thing that stands out is that there’s a distinct fairness (B) problem. We haven’t seen anything like it so far – there was a hint of a problem with the shifted bracket we looked at in fairness and the lopsided bracket. But there, the problem was barely detectable. And it didn’t show itself until the second round, which suggests that whatever small inequity there was in the second round was offset by a compensating difference in the way that drops were treated in the first round.

But here, there’s no avoiding the conclusion that there’s a fairness (B) problem from the first round. The players who get byes win about 4.47% of the time, while the ones who have to play the first round win only about 4.02% of the time. A little less than half a percentage point may not sound like a lot, but bear in mind that the numbers are small to begin with, so that the bye teams’ winning chances are more than 10% higher than the non-bye teams.

But, as I’ll show soon, it could be worse. The way that the byes were distributed evenly through the bracket has a way of evening things out. Note that the people who get byes play their first rounds against players who won in the first round, and who are thus, on average, considerable better than they are. Since there is no elite threshold, the expected skill lever (essentially a Z score) is zero, but by winning one round their next opponents will be from a pool that have Z scores at very nearly 0.4.

The bracket is also kind to those who don’t get a bye in the first round and lose. They get an immediate bye of their own.

The group with the most to complain about are the folks who win one match, and then lose the second – they drop into the same round as players who lost in round one. But even they have had a pretty good chance of getting to the third round with a win over a player who is, on balance, not as good.

The disadvantage of not getting a bye is real, but it is dissipated by the third round. And there’s really no obvious alternative – it’s not good if you don’t get a bye, but the inequity has been minimized. As we’ll see soon, byes are not always so benign.

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