Two posts ago, the fairness (B) statistic was generalized so as to be applicable to any round in a bracketed tourney. And in the last post, I inverted the fairness (B) and fairness (C) statistics to make them easier to interpret. It’s time to put these new tools to work to show how they can provide a clearer picture than the old ones. To do this, I’ll revisit the question discussed by one of the earliest of tourneygeek’s empirical results in Shifting a 32 Bracket.
n.b., this analysis was briefly posted using the old, un-inverted statistics and (much worse) with significant mistakes in the analysis – the current version is, I hope, not only clearer but more correct.
In that post, I analyzed three different double-elimination bracket structures: A.B.|.C.|.D.E.X (the conventional structure); A.B.C.D.|.|.E.X, (the “CD shift”); and A.B.|.C.D.E.|.X, the (“ED shift”).
The drop markers that give the different patterns their standard notations correspond to rounds in the upper bracket (which is not shown). In each case, the A drops go to the first (F) lower-bracket round, and the B drops to the second (G) round. The lower bracket rounds are the F, G, H, J, K, L, and M rounds (and, for the unshifted bracket, an N round).
This table is based on half a million trials of each of the formats, with the prize fund awarded on a 50/30/20 split to the first three places. Luck is set to unity. Where a round receives drops, the round that drops in is noted in parentheses.
|luck = 1.0||A.B.|.C.|.D.|.E.X||A.B.C.D.|.|.E.X||A.B.|.C.D.E.|.X|
|(unshifted)||(CD shift)||(ED shift)|
|f(b:F)||1.65 (A)||0.96 (A)||2.04 (A)|
|f(b:G)||17.12 (B)||16.11 (B)||17.35 (B)|
|f(b:J)||6.97 (C)||20.98 (D)||7.58 (C)|
|f(b:L)||14.33 (D)||0.22||7.51 (E)|
Judged by f(C), the ED shift is the best, followed by the unshifted bracket and the CD shift. But according to f(B), the order is reversed, with CD best followed by the unshifted bracket and the ED shift bringing up the rear. Note that f(B) is the same thing as f(b:A).
I consider f(B) scores of less than one to be quite fair – this statistic is most helpful, I think, in bringing out the problems associated with byes, or with severely unbalanced bracket designs.
The round by round f(b) scores are most interesting, I think, in helping to show how it is that the f(C) scores come to be. The story is told by how each format reacts to the imbalances produced by the B, C, D, and E drops.
In all three formats, the first two rounds of the lower bracket, the F and G rounds, take the A and B drops, respectively. The f(b:F) scores show that taking the A drops is not particularly stressful. This is because the F rounds consists entirely of A drops, and so there’s less scope for imbalance. In contrast, the higher f(b:G) scores show that receiving the B drops does stress the bracket. The B drops are W-L teams, which are (because of skill progression) better than the L-W teams they face.
In the H round, the unshifted format and the ED shift get a breather, as they consolidate teams with like records, but the CD shift is stressed again by the arrival of the C drops. In the J round, all three formats get drops, but the CD shift suffers most.
In the K round, the unshifted bracket and the CD shift consolidate, but the CD shift still has a hangover from absorbing three consecutive rounds of stressful drops. the ED shift gets a relatively benign set of D drops.
By the L round, the CD format has come to rest, but the unshifted format and the ED shift get stressful drops. But the ED format is getting its last drops, which will let it coast to victory in the f(C) competition. CD has its second consecutive consolidation round, and is tranquil, but the unshifted bracket is rocked by D drops.
In the M round, the CD shift gets a relatively mild jolt from the D drops, and the unshifted bracket gets the same in the N round after consolidating in the M round. But the ED shift has received all of its drops, and is coasting. And, as the M round is a money round, the balance in the M round has a disproportionately large influence on f(C). The fact that the ED shift is in balance in the money round is the key to its superiority.
The degree of bracket stress represented by each set of drops is related to the level of skill progression. In the next post, I’ll explore the relationship between these three format further by showing how they react with higher or lower luck parameters.