# Using the new tools: f(C), f(B) and f(b) in the 32 Bracket

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(C) 6.27 6.34 6.13 f(B) 0.55 0.39 0.71 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:H) 0.89 19.82  (C) 1.32 f(b:J) 6.97  (C) 20.98  (D) 7.58  (C) f(b:K) 0.64 5.13 6.33  (D) f(b:L) 14.33  (D) 0.22 7.51  (E) f(b:M) 0.07 4.72  (E) 1.02 f(b:N) 4.70  (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.

## 3 thoughts on “Using the new tools: f(C), f(B) and f(b) in the 32 Bracket”

1. Sean Garber says:

We can call this a backgammoncentric comment. The consolation (one loss) part of the unshifted bracket is very unfair. I’m not even talking about the advantage that the undefeated player has in the grand final, I’m limiting this to the consolation bracket. A 4-1 player may be playing a 7-1 in the final. To me, this is over the top ridiculously unfair. This unfairness doesn’t show up at all in fairness (B) and just barely shows up in fairness (C) because the unfairness could happen to any of the 32 original spots in the bracket. Fairness (B) says that the unfairness is equally spread out over all initial bracket spots, so it is fair. I would personally cringe a bit if someone would put this into the fairness (A) category. This is a real unfairness. It is not about what I am used to or what I am personally comfortable with. Someday Dan, almost certainly not while I am still working, I might just put together a bracket simulator similar to the one that you have here. One thing it would look at would be fairness (E). Fairness (E) would be the standard deviation of the difference of the total number of matches played by the 2 combatants in the consolation final. Examples: If a 5-1 player plays a 5-1 player in the consolation final, the difference is zero. If a 4-1 player is playing a 6-1 player in the consolation final, the difference is 2. Low numbers are better. To me (this is a very backgammoncentric comment) the fairness (E) number is more important than the fairness (B) number or the fairness (C) number, especially when an unshifted type bracket is being considered.

I prefer the ED shift, considering these 3 choices. Probably not a surprise, considering my last paragraph.

The most important fairness (B) number is fairness (B), or fairness (b:A). I have given a quick look at some of your later articles. I bring this up because the important fairness (B) number may now get lost in a sea of other numbers.

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1. Backgammoncentric, perhaps, or at least geared to tournaments with very little skill progression.

I think that the new fairness (b) is also going to be useful for you. Calculating it round by round, it goes haywire with just the sort of things that are going to end up causing the number of matches to be unequal – byes and drops that overlap or split between rounds.

I’m not sure I see much use for your fairness (E) statistic, and I don’t capture the data I need to compute it in the current simulator. But it’s probably related to something I do want to calculate, at some point, so I’ll bear it in mind for a future version. Don’t hold your breath – I have a longish wish list for enhancements I’ve already got uses for. But I’m glad to have suggestions. And some of the more ambitious things on my list have been inspired by the desire to be able to test rather exotic features that appear in some of your brackets (like sending a player to more than one line).

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