Yesterday began the discussion of using byes to fill out a bracket when you don’t happen to have a number of entries that is a power of two. I illustrated how the seeding lines could be used to ensure an even spread of the byes through the bracket, and showed how this played out in a sample 24DE tournament.

Not all directors, however, use the seeding lines to distribute byes. There are some who like to group the byes together so that the second round can begin immediately. This is usually a bad idea.

Here’s an illustration of the two approaches:

In the bracket on the left, the byes are distributed according to the seeding lines, but on the right they’re been collected together in the lower part of the bracket.

This potentially violates the maxim from the lopsided brackets post:

Unbalanced brackets are likely to be inequitable, and so are to be avoided unless there is some good reason that the bracket needs to be unbalanced.

The bracket on the right is obviously lopsided. Is there a good reason for drawing it that way?

The best explanation I’ve heard for the appeal of the bracket is that it helps things flow at the start of the tournament. By grouping the byes, the part of the bracket inhabited by the byes can start the second round at the same time as the first. There is no reduction in the number of rounds, so grouping the byes won’t really save much time. But if the advantage is not large, at least there is one. And if the potential fairness (B) problems are an tiny as the ones we found when considering the bracket shift, it makes sense to group the byes.

So, how big a fairness problem do we have with grouped byes?

It’s pretty darned big.

I’ll show this a couple of ways. In this post, I’ll show how grouping the byes works for a 24 single elimination tournament. In the next post, we’ll see if we can also make a sensible comparison for a double elimination tourney of the same size.

Here are two 24SE analyzed brackets, one with spread byes, and one with grouped byes: 24sespread and 24segrouped. Let’s look first at the spread byes.

As we might expect, the players who draw a bye get a substantial advantage. And that advantage is somewhat larger in this bracket than it was in the 24DE we looked at last time – the double elimination format gives the disadvantaged non-byes another route to victory, and this dilutes their disadvantage. Here, with no such alternate path, the winning chances of the byes, at about 4.64%, are almost 18% higher than those of the non-byes, at 3.93%. Another way to think about this is that in a little more than 3% of the tournaments using this bracket, the byes have shifted the win away from a non-bye to a bye. That’s not a trivial fairness (B) problem, but unless we can find a better way to accommodate 24 players, it’s one that we’ll just have to live with.

Now look at the second analyzed bracket. The discrepancy has grown. The non-byes now have winning only 3.52% of the time, while the byes are getting 5.47%! in percentage terms, the byes’ advantage has almost tripled, and now almost 10% of the time the bracket takes a victory away from a non-bye and awards it to a bye.

For reasons explained in the earlier lopsided bracket post, this kind of discrepancy is predominantly a fairness (B) concern rather than a matter of fairness (C). But here the imbalance is so large that it begins to show up in the fairness (C) statistic: the bracket with the grouped byes gets 2.128, while the spread byes get 2.167. That seems like a small difference, and perhaps it is. But we have very few other brackets for which the new fairness (C) measure has been calculated, and until we do it’s hard know just how significant that extra 0.039 is.

Why is there such a large imbalance? In the earlier analysis of one of Joe Czapski’s balanced brackets, we saw that one of the problems was the way he handled the first-round losers, or A drops. They dropped into a separate, sheltered part of the bracket, where they could get as many as three wins playing only against other A drops.

In the bracket with the grouped byes, the entire lower section of the draw is such a sheltered sub-bracket! The bye there is a gift that keeps on giving. This is in stark contrast to the fate of a player with a bye in the spread bracket, where the player with a bye is, just like any other winner, promoted into a tougher part of the bracket.

For my money, it’s hard to see how the limited advantage gained by starting a few round two games early is worth the price you pay in terms of fairness (B). Perhaps the most charitable explanation for the continued use of bye-grouped brackets is that people just don’t realize how unfair they are. But now you do. If you can think of a another reason justifying the use of grouped byes, please, Please, share it in the comments.

Oh well, you might say, things won’t be so bad if we use the grouped byes in a double elimination formats – we saw a few paragraphs back that that seems to attenuate the unfairness somewhat. Well, maybe. But, as I’ll show in the next post, grouped byes introduce another difficulty in the context of double eliminations.