So far, revisiting the question of the relationship between fairness and bracket shifts has largely confirmed what was found earlier.
The bracket shift’s influence on fairness (C) is benign for both 16s and 32s. This is chiefly because it tends to mitigate the unfairness of dropping the loser of the upper-bracket final directly into the lower-bracket final, dropping it instead to the semifinal of the lower bracket. In 32s, then, the late shift is better than the early shift. And in both 16s and 32s, the drops balance better for high-luck events than they do for low-luck events.
Perhaps it will be worthwhile to look at 64 brackets also, though it’s hard to imagine the results being much different. For this post, however, I’ll look at the one design parameter that has been found to favor unshifted brackets in the past: seeding.
For the purposes of this experiment, I’ll use the tennis-style tiered seeding that I’ve argued elsewhere represents best practice. For 32s I’ll use five tiers: 1, 2, 3-4, 5-8, 9-32; and for 16s I’ll use four: 1, 2, 3-4, 5-16. I’ll assume perfect seeding – that is, that the number one seed really is the best player, and so forth down the table. Other tourney parameters will be the same as they were for recent posts.
Because seeding a tourney intentionally sacrifices fairness (B) to improve fairness (C), the full fairness (b:X) tables are not very helpful. I’ll append one to the end of this post so the you can look at it if you’re curious. But for all other tables, I’ll show only the fairness (b:X) figures for the last three rounds: the final, the lower bracket final, and the lower-bracket semifinal(s). In unshifted brackets (and also in the 32 early shift), these three rounds should be past the region in which the seeding itself has a direct effect on fairness (b:X), but in the late shift, seeding still influences fairness (b:X) for the antepenultimate round (that is, the lower semifinal). In all cases, I’ll report the mean skill levels for the matches in the round, showing the skill of dropped players in bold.
32 A.B.|.C.|.D.|.E.X, luck = 3, f(C) = 72.50:
32 A.B.C.D.|.|.E.X, luck = 3, f(C) = 72.78:
32 A.B.|.C.D.E.|.X, luck = 3, f(C) = 71.02:
With seeding, at this luck level, the late shift is better than the unshifted bracket, which in turn is better than the early shift. The late shift’s gain seems to be coming from better placement of the E drop, which is unfairly advantaged when dropped into the lower final, and somewhat less disadvantaged when it drops into the lower semi.
Here are the results for high-skill events:
32 A.B.|.C.|.D.|.E.X, luck = 1, f(C) = 17.05:
32 A.B.C.D.|.|.E.X, luck = 1, f(C) = 17.56:
32 A.B.|.C.D.E.|.X, luck = 1, f(C) = 18.13:
At the higher skill level, the unshifted bracket is the best of the three, and it’s conspicuously better than the late shift. With the greater level of skill progression, putting the E drop into the lower semi is egrediously unfair to it. In all three designs, the upper winner is significantly better than the lower winner. But in the unshifted bracket, the lower final and semifinal are pretty darned equitable.
Thus, the earlier finding that the unshifted bracket is better for seeded tourneys is partly confirmed. It’s confirmed for high-skill tourneys, but for high-luck events it’s better than one shift, but not as good as the other.
Here, as promised, is a full table of f(b:X) for one design, in this case the high-luck, early shift. The point here is just to show how the seeding distorts the fairness (b:X) calculations for all rounds until near the end:
|ABCDvvEX, luck = 3, f(C) = 72.78|