The Ugliest Bottom

Before I start looking into past results to try and root out the baleful influence of the ugly bottom effect, I though I’d try to find a pure example of the effect. And so I’ll try to design a tourney with the ugliest possible bottom.

The feature that earned the original Ugly8 format its name was dropping the A drops and the B drops together into the D round: 8DE: {AB.C.|.X.R}. But if we really want to make the worst possible bracket, we need not just to drop later rounds together, but to drop them before the earlier rounds. And let’s also spread this maximally-ugly bottom into a single cascade. Here’s my candidate for the ugliest possible 8DE: {BC.B.A.A.A.A.R.X}. Here’s what that looks like: medusa8. I dub this the Medusa8 bracket, in honor of the character from Greek mythology who was so ugly that whoever looked at her turned to stone.

Kids, don’t try this at home! This is a format that takes ten, or occasionally even eleven, rounds to play. And I guarantee that everyone who descends into the ugly bottom will complain. But computers don’t mind, so let’s see what the simulator thinks of Medusa8.

OK, a million trials later, Medusa8 earns fairness (C) at 14.36. Pretty good, but it’s still not ugly enough, because it’s still not better than the standard 8DE, at 13.49. It’s doing an impressive job of reducing the skill level of the lower bracket survivor: it’s 0.795 for the standard bracket, and Medusa’s driven it down to 0.652. But there’s still not enough skill in the game. At luck = 1, the average skill of the upper-bracket winner is 1.192, while the mean skill of the best player is 1.422. There are simply too many cases where the someone other than the top dog wins the upper bracket, and in those cases, the perverse ugly bottom effect is inuring to the benefit of the wrong player.

I don’t know how to make Medusa any uglier, so I’ll ramp up the skill level by setting luck = 0.5. Fairness (C) is now 3.91. Close, but not close enough – fairness (C) for the standard 8DE is now down to 3.81.

Well, maybe there’s a skill level at which Medusa8 bests the standard 8DE. But for now, let’s be content with an example of a really ugly bracket that, on the strength of the ugly bottom effect, gets pretty darned close to the standard. The next step is to see what happens when we move to a 65/35 payout scheme.

Here are the results for four 8DE formats, showing the fairness (C) scores with a winner-takes-all payout schedule on the left, and a 65%/35% payout on the right.

winner take all 65%/35%
1 3 1 3
Standard 8DE 13.49 59.15 14.13 54.78
Ugly 8 13.58 59.22 14.81 55.07
Joe’s 8 13.48 57.94 15.75 57.75
Medusa 8 14.36 63.79 19.33 64.78

Joe’s 8 and the Medusa 8 both show pretty clear signs of an ugly bottom effect – they look much better with the winner takes all payout than they do with the 65%/35% split, which restores a clear pecking order among the options. Ironically, the Ugly 8 – the very design in which an ugly bottom effect was first suspected – shows little or no sign of suffering from the effect.

The other striking result is the fall from grace of Joe’s 8DE. It’s not awful, but it’s clearly not as good as the standard bracket, nor even as good as the Ugly 8. It still has a possible advantage because it sometimes saves a round, which the Ugly 8 does not do. But I think that my earlier speculation that it could become a recommended format has been shown to be fanciful. As far as I’m concerned, the state of the art for the 8DE is, in fact, what i though it was at the beginning of this series of posts: the only format that ought to be recommended without reservation.

In a coming post, I’ll revisit the recent ideas about various 16DE brackets to see whether any of those results need to be reconsidered because the earlier tests gave them unwarranted preference as a result of ugly bracket disease.

 

 

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