Fairness and the Lopsided Bracket

What of the complaint that a bracket with shifted drops is lopsided? What sort of complaint is it, anyway, fairness (A), fairness (B), or fairness (C)? Potentially, at least, all three kinds of fairness might be of concern.

Here, again, is what a shifted bracket looks like. We’ll show only the lower bracket of a 16DE to make it easy to see.

That’s not the way a standard 16DE lower bracket looks:

In the shifted bracket, the top half receives drops in the third round, and the bottom half get a drop in the fourth round.

It’s a natural reaction to say that there must be something amiss here. Some parts of the bracket are being offered a different path to the championship than other parts, and there’s bound to be something fishy about that.

Is this example too subtle? Let me throw in a couple of other unbalanced brackets (see if you recognize them, or guess what sports they’re used with, before you continue):

The top two lines in the first are clearly better than the bottom two. And in the second bracket the higher lines are even more dramatically desirable. I doubt that I need to run simulations on either of these patterns to show that they’re grossly unfair.

Or are they? As it happens, each makes perfect sense in the context in which it’s used. The first is called a Page playoff, and is used in cricket, curling, and a few other sports. The second is the format for many televised bowling championships. It’s important to note the each is used only in the last stages of a larger tournament, and that the higher lines are earned by superior results in the earlier part.

In general, the common intuition about lopsided brackets is correct – if the bracket is out of balance, there will almost always be some compromise that benefits some lines, and disadvantages others. Thus, as a general rule:

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.

(Perhaps this will be the first of my maxims of tournament design, which I’ll collect and organize at some point.)

This rule is, I think, fairly intuitive. For this reason, any unbalanced design is susceptible to a fairness (A) complaint. While a tournament organizer who wants to do things better than they’re been done in the past has to be ready to defend useful innovations, it’s important to choose battles wisely. So it’s not enough that an unbalanced bracket is, in fact, fair in other respects. It also has to have enough positive virtue to make it worth defending against the intuitive fairness (A) complaint.

So, confronted with an unbalanced bracket, the question is whether there a reason for the design that outweighs the inequity.

How should we resolve that in the question of the shifted double-elimination lower brackets? Well, you can’t shift the drops in a bracket this way without unbalancing it a bit, but as we’ve already discussed, there are some very substantial benefits to making the shift. The tournament will need one less round to run. There will be fewer repeated pairings. And it will be a bit fairer (in the fairness (C) sense of making it more likely that the winner will be the best player).

And the inequities? Are there any?

Yes, there are. But they’re exceedingly small. In the initial run, when I was relying on the fairness (C) statistic to show these differences, they were so small that I ran 100 million (rather than my usual 10 million) iterations of the tournament simulator to be sure they were really there.

Take a look on the analyzed bracket (16-upper-shift-analyzed) at the lines in the second round of the upper bracket. The winning percentages for matches B1 and B3 are: 11.665%, 11.656%, 11.662%, and 11.667%. Now look at the winning percentages for the other lines, in matches B2 and B4: 11.659%, 11.654%, 11.660%, and 11.653%. It looks like there a tiny benefit to being in B1 or B3 rather than B2 or B4. But it’s a benefit of only a few thousandths of one percent in success rate.

As discussed in yesterday’s post, there’s reason to doubt that the winning percentage for such deeply buried lines is a reasonable indicator of equity. I reran the bracket with a revised simulator that also calculates the mean number of individual match wins by line. Here are the corresponding numbers of additional match wins to be expected from the lines in the B round. B1 and B3: 1.811, 1.812, 1.812, and 1.813; B2 and B4: 1.803, 1.803, 1.804, and 1.803. The effect has been considerably magnified, but it’s still small – slightly less than 0.01 additional win. Here are the new analyzed brackets: 16-upper-shift-analyzed-2 and 16-lower-shift-analyzed-2

(A note about the analyzed brackets: I’ve intentionally kept more digits for some of the statistics than are really significant. For that reason, lines that ought to have exactly the same win percentages or error rates show slight differences. By seeing how much variation there is between these equivalent lines, you can get a feel for how much random error there may be for the statistics in general/)

Why to the B-odd lines do a little better? It’s because the losers of B2 and B4 drop into the half of the lower bracket that’s going to receive the C drops in the next round, and the C drops will, on average, be slightly better than the players that the B1 and B3 losers might encounter in the same round.B2 and B4 are also in line to face a superior opponent that drops in from D1, but that’s a round further away, and so a little less significant.

It’s also worth noting that the inequity that shows up in the B round cannot be seen in the A round. That’s because of the artful placement of drops that we made to avoid, as much as possible, repeat pairings. A3 and A4, which send their winners to the slightly disadvantaged B2, send their losers into part of the lower bracket that enjoys the slight advantage. It might be argued that the only fairness (B) inequities that really matter are the ones that appear in the first round, as entrants might find themselves on either side. In response, one might argue that certain first-round lines might be unfair to better players, who are more likely to follow the path that starts with an initial wins. At this point, however, the fairness argument is very attenuated. I think we can safely judge the  fairness (B) inequity of the shifted 16DE to be negligible.

The striking thing about this out-of-balance shifted brackets is not that it has worrisome fairness (B) inequities, but that those inequities are so small.

Some would say that if you’re worrying about fairness (B) problems like this you just don’t have enough to worry about. I’m not sure I want to say that – I approve of a certain fastidiousness that wants to root out any imperfection in a tournament design, no matter how small. But here these imperfections are so incredibly small that it would be foolish not to embrace the real benefits of shifting the drops.