The last post discussed, in general terms, the main source of unfairness in the second bracket, and suggested how to tell whether the use of a shift exacerbated or ameliorated that unfairness. It’s time to put the method to the test.
To avoid another source of unfairness that can obscure the fairness problems inherent in the bracket, we’ll discuss only full brackets – that it, brackets with no byes and a number of players that’s an even power of two
We’ll start by looking at the smallest bracket that can be shifted, the 16 bracket. The traditional way to draw this bracket is A.B.|.C.|.DX (16lowerus). It can be shifted, however, by pushing the C and D drops into earlier rounds, eliminating the two consolidation rounds before them, but adding one later: A.B.C.D.|.X (16lowershift).
Does that make the bracket play better, or worse?
As suggested in the last post, the unfairness lurks in the lower bracket where the two players arrive at a match by different paths. To the extent that the players arriving from those different paths have different mean skill levels, the match will be unfair to the better player, who’s getting no credit for traversing the more demanding path.
The round-specific fairness (b:X) measure is admirably suited to show this. Here are the fairness (b:X) numbers for the eleven rounds in an unshifted 16 bracket, player with luck=1:
As we’ve seen in the past, fairness (b:X) values below about one are just showing statistical noise. The full bracket has no structural fairness (B) issues in the upper bracket (rounds A-D), or in any of the lower bracket rounds that do not mix drops with non-drop players (rounds E, G, and I). But there are issues with rounds F, H, J, and K, in which take the B, C, D, and D(winner) drops, respectively.
As discussed in the last post, the second lower bracket round (round F for a 16 bracket) has an unavoidable unfairness. The mean skill level for round A is zero. Because of the relatively low level of luck, the mean skill level in round B, where everyone won the first match is about +0.46, and for round E, where everyone lost the first round, it’s -0.46. Now, the players in round F who are coming in as B drops have an 1-1 record, but that record was earned against better competition, on average, than the players who were originally A drops. Thus, the B-drop lines in round F are +0.07, while the A-drop lines are about -0.07.
In round H, the arriving C drops average +0.51, while the lower bracket survivors are +0.62. That’s because the C drops have a 2-1 record, while the survivors are 3-1. Even though the C drops have faced somewhat tougher competition, they’re still not as good, on average, as players who won another game.
Similarly, in round J the arriving D drop, with a 3-1 record, is +0.89, while the opponent, who’s record will be either 4-1 or 5-1, is +1.05.
For the reconciliation round, round K, the difference is enormous. The undefeated upper-bracket winner is +1.50, while the lower bracket winner, despite having more wins, is only +1.19.
Now, compare the result of the same tourney run with luck=3:
Here, the unfairness of round F is low. The two sort of drops are only +0.013 and -0.013. In round H, the difference is between +0.25 and +0.42. The C drops, who were a bit too generously rewarded in the luck=1 case are even more favored because there’s less skill progression. In round J, the difference is even more stark: +0.48 for the D drop, meeting a 0.73 opponent.
Even with these larger skill differentials, the absolute value of the fairness (b:X) measure is lower, but that’s only because the rewards for skill in the event as a whole are much less certain where luck is to high. For luck=1, fairness (C) is 19.42, while for luck=3 the number is 67.05.
What we see is that in the unshifted bracket, those who stay longer in the upper bracket are rewarded too generously – a bit too generously where luck is low, and way too generously where luck is high.
So what happens with the shift? Here I’ll put in mean skill levels with the problematic rounds:
With the shift, there is no respite from the sort of unfairness that begins in the F round – the C and D drops pile in with no skipped round. And so the severity of the imbalance increases, round by round. But the last two rounds of the lower, I and J, are better than the two corresponding rounds of the unshifted bracket, and round I is much better than round J in the unshifted bracket. And, since the fairness statistics are driven most dramatically by what happens with the in-the-money places, overall the fairness (C) measure improves, 19.07 as compared to 19.42 without the shift.
How about the shift at skill=3?:
Here the results are uncanny – there are no really bad rounds! The D drop lands in round H, but the high luck factor has caused low skill progression. The mean skill levels for round H are: +0.44, +0.44, +0.48, and +0.42 – the last is the D drop, and the next-to-last is a bit higher because it’s in a part of the table that doesn’t get a C drop. Round G, which takes the C drops, is nearly as benign, with skills in the range of +0.22 to +0.25. And the capper is the reconciliation round, round J, where the D(winner) drop arrives at +0.82 to meet a player at +0.89. The little bit of skill progression is very nearly in balance with the extra win needed to make it to the championship match through the lower bracket.
As we might expect, this good fit shows up in fairness (C) measure: 65.22 for the shifted bracket, 67.05 for the unshifted bracket. In both cases, the shifted bracket does better, but in the high-luck scenario it does a whole lot better.
There are, perhaps, some parameters that would make the unshifted bracket the better choice. In earlier experimentation, Seedy and Shiftless, I found that a combination of strict seeding and low luck (I used luck=1 for everything back then) gives the nod to the unshifted bracket. Here I don’t want to do that because the seeding will completely mess up the fairness (b:X) and fairness (C) statistics, so there’s likely to be some level of luck so low that the unshifted bracket would be preferable.
At the extreme, where there’s no luck at all, the unshifted bracket is a slight favorite. But with luck set even as law as 0.5, there’s a slight fairness (C) advantage to the shifted bracket. I’m unaware of any sport or game which would call for such a small luck factor.
For all practical purposes, then, the shifted bracket outperforms the unshifted bracket unless, perhaps, there is seeding involved.
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