Playing with Dollies

The idea of a Dolly bracket is, I think, a useful one, and so deserves a more precise definition. A Dolly bracket is any bracket designed to return a specific unfair result (in the fairness (C) sense).

In devoting the next post or two to Dolly brackets I don’t mean to suggest that such brackets are to be encouraged. If there were a code of ethics for tournament designers, one of the first rules would be never to create a Dolly bracket. But this is probably reason enough to spend some time playing with them, if only because it’s important to know what does and what doesn’t count as a Dolly bracket.

The cascade bracket that won the fairest of them all pageant is outlandish in that it has almost as many rounds as it has players. But it’s not a Dolly bracket with this configuration of seeds:

cscade

Because it’s the winner of the competition for the best fairness (C), it cannot be a Dolly bracket because it never produces an unfair (C) result. And, silly as it looks, a variant is actually useful in competition.

With five players (and four rounds), this is a bracket that’s often used for the final televised portion of a major bowling tournament. I won’t go into the way professional bowling tourneys are run, but the upshot is that the competition up until the final four games is scrupulously free of bracket effects and the unfairness they can cause. The final games are in knock-out format in order to create the sort of win-or-go-home drama that television demands. Hence, the cascade bracket serves to preserve the results of the qualifying rounds, which are (or at least should be) much fairer than the final. No player can lose more than a single place in the cascade, but television gets its dramatic matches.

Any other use of the cascade bracket is, however, likely to be a Dolly. Here is the same bracket except for the seed lines, from the evil queen scenario:2Dolly

It’s a Dolly bracket because its specifically designed to make it difficult for Snow White, the number one seed, to win against a second-seeded Evil Queen. But the seeds can also be placed to help any other competitor, simply by moving that seed to the top of G1, and arraying the other seeds, in reverse order, below it.

In high-skill games, the Dolly bracket doesn’t work well for lower-ranked teams. Only the Dolly favoring the second seed actually ourperforms the top seed at luck = 1 for a winner-takes-all payout. But with a generous dollop of luck, the Dollies are quite effective, even for the lowest seeds. This table shows the percentage of wins from each position for each of the eight Dollies. To provide a reference for comparison, it also shows the results for strict direct seeding (here called “PBA” in honor of the Professional Bowlers Association, the only group I know that actually uses something similar), and a blind draw.

ranks BD PBA 1 2 3 4 5 6 7 8
1 26.7% 60.1% 73.7% 8.9% 8.8% 8.8% 8.9% 9.0% 9.0% 8.8%
2 18.1% 24.5% 3.5% 65.0% 3.5% 3.5% 3.6% 3.5% 3.5% 3.5%
3 14.1% 9.7% 1.7% 1.8% 60.3% 4.1% 4.1% 4.2% 4.2% 4.1%
4 11.6% 3.6% 2.2% 2.3% 2.4% 56.3% 5.6% 5.7% 5.7% 5.7%
5 9.8% 1.4% 2.8% 3.1% 3.5% 3.6% 52.3% 8.3% 8.5% 8.4%
6 8.1% 0.4% 4.1% 4.4% 4.9% 5.4% 5.8% 47.9% 13.4% 13.4%
7 6.8% 0.1% 5.3% 6.4% 7.2% 7.8% 8.3% 8.9% 42.2% 21.7%
8 4.8% 0.04% 6.7% 8.1% 9.4% 10.5% 11.5% 12.4% 13.5% 34.5%
f(C)      90.74      26.87      43.18       84.91    112.13    132.35    149.35    163.89    177.42    188.90
f(B)    117.51    167.73    198.41     171.01    155.91    143.20    130.50    116.86    100.61      85.47

(The reported averages are from only 100,000 trials each, which make them a little less stable that some I’ve reported.)

The figures in bold show the percentage of wins for the place that the particular Dolly was designed to favor. I’ve also shown the fairness (C) and fairness (B) statistics for each seeding pattern at the bottom of the table. F(C) for the PBA is 26.9, presumably, the fairest-of-them-all result for a WTA payout. This slips to a respectable 43.2 for the one-seed Dolly, and then goes from bad to worse for the other Dollies. The f(B) numbers actually improve a bit as the Dollies favor weaker and weaker teams, but they’re all appalling high – there’s no use of the cascade bracket that’s usable if you really care about fairness (B).

Here’s one additional point. What game should the Evil Queen play? We assume that she’s the second seed, and can use the second-seed Dolly. At luck = zero she never wins – Snow White breezes through all those extra rounds. At luck = 3, she wins 65%. At luck = infinity, she wins 50% – she’s gotten a bye into the finals, but there her fortunes are on a coin toss. So what luck level gives her the very best chance?

Here are some additional runs: Luck = 1, 47.3%. Luck = 2, 63.9%. Luck = 4, 63.7%. Luck = 2.9, 65.1%. If looks like we’re getting close, so now I’ll use million-trial runs instead of 100,000 trial runs. Luck = 2.9, 65.28%. Luck = 3.0, 65.20%. Luck = 2.8, 65.42%. Luck = 2.6, 65.41%. Luck = 2.7, 65.47%.

Enough for now. The maximum winning percentage for a second-ranked team with a Dolly bracket appears to be about 65.5%, in a game with a luck factor of about 2.7. That is  nearly the luck factor for backgammon. This is only appropriate, as backgammon is the preferred game of the Dolly (not her real name) I know.

The rest of the story is a sad one. The computation of the ideal luck level was made somewhat more precise by an itinerant practitioner of tourneygeekery who fell into the clutches of Evil Queen, and was made to work under the lash in her dungeons using only an old IBM mainframe, a simulator written in PL/I, and a rusty abacus. He was eventually rescued when Evil Queen was overthrown, but by then he was a broken man. Out of pity, he was given a sinecure in the court of Handsome Prince and Snow White as Tourneygeek Royale, but never achieved any notable results. And the practice of tourneygeekery fell into disrepute because it was associated with the reviled memory of Evil Queen. This was one of many factors that gradually eroded civic virtue, leading eventually to the fall of the kingdom of Happily Ever After, the NCAA Basketball Tournament, and the Administration of Donald Trump.

 

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