Building a Better Bracket Redux

One of the earliest important results in tourneygeek was that the shifted bracket was, for many purposes, not just a clever way to save a round in a two-bracket tourney, but a way to enhance fairness (C) at the same time. See Building a Better Bracket and subsequent posts.

Recently, however, Slow, Shifty Brackets showed that shifted brackets were, for one particular type of tourney, slower to run than unshifted brackets. This may be added to the finding in Seedy and Shiftless that in some cases unshifted brackets outperform shifted ones with respect to fairness (C).

The initial posts on the virtues of shifted brackets are so old that they preceded the availability of the current simulator, and used a version of the fairness (C) measure that no longer seems adequate. So it’s high time to take another look at the bracket shift to make sure we have a clear understanding of its virtues and faults.

Here I’ll begin by talking about what would make a shifted bracket more fair or less fair as compared to an unshifted bracket of the same size. In subsequent posts, I’ll use simulation results to show when shifted bracket are better on fairness (C), and when they’re worse.

Though this analysis is applicable to other formats with at least two brackets, for simplicity’s sake we’ll discuss one familiar kind of two-bracket structure: the double-elimination tourney with no recharge. All brackets will be full, with no byes.

The basic pattern of an unshifted lower bracket is to have a consolidation round after each set of drops except the first A drops. This means that for every victory after the first, the structure of the lower bracket rewards the player by skipping a round in the lower bracket.

Whether this is fair or not depends on the degree of skill progression, which in turn depends on skill levels, and on seeding (if there is any). To see the effect of the skill progression, let’s look at the second round of the lower bracket. All of the players in this round have the same record: one win and one loss. But these players are of two kinds.

Half of those players lost the first round or the upper, descended to the lower as an A drop, whereupon they won a match against another A drop. The other half won their first round in the upper, and then lost in the B round, which sends them down to the lower as B drops.

What does this mean about the average skill level of the two kinds of players in this round? Assuming a non-elite tourney, the average player in the A round has a Z score of zero. The winner of that A-round match has a Z score somewhat above zero, and the loser has a score somewhat below zero.

The next round is different in the two brackets. In round B of the upper, both players are, on average above zero. But in the first round of the lower, both players are, again on average, below zero.

Thus, the B-drop player, whose record is W-L, has one win against a player whose average skill is zero, and one loss against a player whose average skill is above zero.

In contrast, the player whose record is L-W has one loss against an opponent averaging zero quality, and one win against an opponent whose average skill is less than zero.

Thus, the second round of the lower bracket is always somewhat unfair. It takes less skill, on average, to be the A drop coming up through the lower than it does to be the B drop. And yet the two are treated equally for all purposes going forward. That’s unfair to the B drop.

But there’s nothing to be done about this. There’s no way (or at least no widely recognized way) to draw the bracket that won’t be unfair to the B drop.

OK, now let’s look at the next round of drops, the C drops, whose record is W-W-L when they drop. Are they fairly treated, or not?

Now we’re in the region where the architecture of the bracket matters. If the C drops are shifted to his the lower early, they’re playing an opponent who’s either W-L-W or one who’s L-W-W. But in either case, by the same logic, the C drop is getting a raw deal. In fact, it’s a bit worse, on average, than the bad deal the B drops got.

If the bracket is unshifted, however, they’ll drop into the fourth round of the lower, where they’ll face an opponent who’s record is either W-L-W-W or L-W-W-W (depending on whether they were an A drop or a B drop). Now, it’s not clear whether the C drops are, on average, better or worse than their opponents. Their two wins and one loss all came, on average, against better opponents, but they do have one fewer win. If there were no skill progression, they’d clearly be making out like bandits, but if the skill gradient were steep enough, they might still be at a disadvantage.

Here, then, is the way to compare the fairness of shifted and unshifted brackets. Look at the relative skill levels of the lower rounds, particularly the rounds that receive drops. If they’re about the same, that’s good, but if they aren’t that’s bad.

Enough theory. Now, let’s dig into some results in the next post.

 

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