# Using the New Tools: the Analysis of the 32 Bracket Continued

In the previous post, the new f(b) measure was used to gain a clearer picture of the way that the unshifted lower bracket for a 32-team double elimination tourney differed from either of the two available shifts. In this post, I’ll continue that discussion, and show how different levels of skill progression, resulting from different luck factors, affect the performance of the three candidates.

For those more interested in the practical result than the details of the method, here’s the result: The ED shift is to be preferred, except in the very low-luck scenario, where it’s a dead head between the unshifted bracket and the CD shift.

Here, again, are the three candidates:

And here are the tables showing f(C), f(B), and f(b) for them, first with luck set to 0.5, as might be appropriate for a tourney for a low-luck competition like go, and then with luck at 3.0, which is approximately correct for a high-luck competition like backgammon. As before, these tables come from runs of half a million trials, with a 50/30/20 payout, and no elite threshold:

 luck = 0.5 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) 2.21 2.21 2.24 f(B) 0.36 0.44 0.88 f(b:F) 2.02  (A) 1.71  (A) 4.24  (A) f(b:G) 26.25  (B) 26.03  (B) 28.32  (B) f(b:H) 0.79 35.33  (C) 3.20 f(b:J) 2.82  (C) 39.35  (D) 3.86  (C) f(b:K) 0.55 11.85 19.44  (D) f(b:L) 9.34  (D) 0.16 19.60  (E) f(b:M) 0.25 3.29  (E) 2.37 f(b:N) 3.35  (E)
 luck = 3.0 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) 38.65 38.22 36.45 f(B) 0.48 0.39 0.62 f(b:F) 1.70  (A) 0.79  (A) 1.28  (A) f(b:G) 2.25  (B) 1.72  (B) 1.91  (B) f(b:H) 1.07 2.00  (C) 0.77 f(b:J) 9.44  (C) 2.26  (D) 8.46  (C) f(b:K) 0.30 0.55 2.32  (D) f(b:L) 10.26  (D) 0.05 0.80  (E) f(b:M) 0.07 3.04  (E) 0.09 f(b:N) 2.96  (E)

As one might expect, f(C) is much better for the low-luck competition – with less of a role for luck in the same number of individual matches, there’s more tendency for the better players to show their advantage.

Overall f(B) is comparable, but the f(b) perturbations in individual rounds tend to be much larger where luck is low, and thus where round progression is high.

The interplay between shift patterns and round progression can be illustrated by considering one set of drops: the D drops. In each case, the D Drop team itself has a record of W-W-W-L.

In the CD shift format, the D’s drop early, into round J. Their opponents there will also have a 3-1 record, though that record might be either  L-W-W-W or W-L-W-W. In a badly drawn bracket, they might also encounter a W-W-L-W opponent, but as generally drawn, the D’s and C’s drop into different quarters of the bracket, and so cannot meet each other as soon as the J round. In a coin-flipping competition, where there is no skill progression, one 3-1 record is as good as any other, and there would be no imbalance. But there is a skill progression – a small one in the high-luck tourney, and a big one in the low luck tourney. So the successive B, C, and D, drops entering the bracket in consecutive rounds show a steady increase in f(b) through the rounds.

In contrast, in the ED bracket, the W-W-W-L D drops can encounter any of these records: L-W-W-W-W, W-L-W-W-W, or W-W-L-W. So some of the opponents will be 4-1 rather than just 3-1. But the fact that the D drop teams earned their three wins against opponents who were, on average, better than the teams coming up through the lower bracket tends to mitigate the imbalances that the D drops introduce. In the low-luck scenario, the C drops cause relatively little imbalance in the ED format – by the next round, the skill advantage of the D drop teams has begun to build again by the time the D drops arrive. In the high-luck scenario, the sweet spot is in round L, which receives the E drops. And, as might be expected, the medium-luck scenario shows the imbalance at its low point in round K, when the D drops actually appear.

The for the unshifted bracket, the situation is also complicated. The D drops L round opponents will always have more wins, but these can be L-W-W-W-W-W, W-L-W-W-W-W, or W-W-L-W-W. Here f(b:L) is a relatively large 10.26, but part of this is caused by the fact that in the L round the D drops are part of every match, whereas in the other two formats, only half of the matches are directly affected by the D drops.

In the low-luck competition, the unshifted bracket comes into its own, coming in in a dead heat with the CD shift with respect to f(C), and beating it no f(B). The ED shift is somewhat worse on both counts, but the differences are not large.

Taken together, the new analysis has the same general result as the old one, all pointing to the superiority of the ED shift.