In the last post, I brought the fairness (C) analysis up to date with respect to shifted and unshifted 16 brackets. In this post, I’ll do the same for 32s, with attention to the two different shifts available.
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BBBR:16s
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?
Building a Better Bracket Redux
One of the earliest important results in tourneygeek was that the shifted bracket was, for many purposes, not just a clever way to save a round in a two-bracket tourney, but a way to enhance fairness (C) at the same time. See Building a Better Bracket and subsequent posts.
Recently, however, Slow, Shifty Brackets showed that shifted brackets were, for one particular type of tourney, slower to run than unshifted brackets. This may be added to the finding in Seedy and Shiftless that in some cases unshifted brackets outperform shifted ones with respect to fairness (C).
The initial posts on the virtues of shifted brackets are so old that they preceded the availability of the current simulator, and used a version of the fairness (C) measure that no longer seems adequate. So it’s high time to take another look at the bracket shift to make sure we have a clear understanding of its virtues and faults.
Here I’ll begin by talking about what would make a shifted bracket more fair or less fair as compared to an unshifted bracket of the same size. In subsequent posts, I’ll use simulation results to show when shifted bracket are better on fairness (C), and when they’re worse.
Goodbye Columbus
The design problem that’s been the focus of the last several posts is for the monthly backgammon tourney of the Columbus Backgammon Club, directed by Chris Yep. Chris started by sending a rather innocent-sounding question: should be adopt a shifted bracket for his main-and-consolation event?
Several days and more than sixty simulation runs later, we have an answer: No.
But in the course of finding that answer, we explored a number of issues, and put a finer edge on the bracket he will run. He’s reallocated the prize pool, a bit, and found that he can add to the length of one of the rounds, improving fairness, at a minimal cost in the time needed to run the event. Here are the analyzed brackets for the design with those adjustments: columbus1, and columbus2.
What Chris didn’t realize going into the project is how many moving parts there are to the design of a simple tourney. But he was ready to engage with all of them. Like many directors, I think Chris is used to hearing a lot of objections from his players, and he wants to have an answer for every one of them. I wish him luck.
I learned some things, too, and have a number of things I think I need to look into.
I very much enjoy working through real design problems, and encourage readers to send me queries if you think I can be of help. Or even if you think I can’t – I’d love to get a question or two in the form of, “Here’s my design – don’t you agree that it’s the acme of perfection?” I’m willing to bet that even if I can’t convince you that there’s something you need to change, I can at least raise an issue or two that you haven’t thought through completely.
So, keep those design questions coming!
Revenge of the Pies
This post will summarize the learning from the last couple, which can then be safely ignored.
In my original “pie” post, Dividing the Pie, I briefly discussed the possibility of using the fairness (C) measure to find optimal ways of dividing a prize fund for a particular tourney. But I then said it wouldn’t serve that purpose because the payout scheme would always devolve into a winner-takes-all.
That was incorrect, but incorrect in a really trivial and unuseful way. What I found, after many, many rounds of simulation, was that that happens in some very low-luck scenarios, but not in the more typical case. Well, it turns out that what does happen in a more typical case is that the payout scheme devolves into an equally unhelpful share-the-wealth division.
Nothing to see here. Move along. I had a bad idea, and then spent a couple of days discovering that it was, indeed, a bad idea – just bad in a novel way.
Sorry to have wasted your time. Please come back. I promise to write something better soon.
So why don’t I just delete the posts?
More Pies to Divide
In the last post, Dividing the Pie, Redux, I used an iterative method in an attempt to derive a fairness-(C)-optimal payout schedule for the unshifted, 13-player tourney first introduced in Getting the Most from an Afternoon.
My earlier idea, from Dividing the Pie, was that this wouldn’t work because the was fairness (C) is defined would always prefer a winner-takes-all payout scheme. But this was not the case. I determined that the best payout scheme was 35/30/35.
In this post, I’ll use more or less the same method in an effort to understand, better, just what’s going on.
Dividing the Pie, Redux
Some time ago, in Dividing the Pie I wrote that my new fairness (C) statistic was not going to be useful for testing the relative fairness of various payout schemes. This seems to be another matter on which I was just wrong.
The argument was that the way the statistic worked was to reward a scheme for getting the most money to the best players, so that if the design had any tendency at all to select the best player for the number one prize, it would always tend, on average, to improve fairness (C) if you took money away from the lower places, which would be occupied on average by lesser players, and added it to the top prize.
This seemed an unsatisfying result to me, and it led me to speculate about some other measure that I might develop that would also observe the common-sense notion that it would be fairer, perhaps in a fairness (B) sense, to spread the money around a little.
It turns out that that’s not needful. Fairness (C) alone can be used to guide the division of a prize fund without necessarily recommending a winner-take-all payout scheme.
In this post, I’ll use an iterative process to determine an optimal payout scheme for the 13-player, unshifted bracket that I’ve been discussing for the last few posts.
Finding the Second and Third Best
Getting the Most from an Afternoon looked at the choice of bracket for a time-limited tourney and concluded that moving to the use of a shifted bracket was probably a bad idea because it might make the event run too long.
This post will consider another aspect of the design. How fair is it? And can the fairness be improved by making changes to the distribution of the prize fund?
I’ll begin this analysis by considering the performance of the two brackets in terms of how they tend to distribute the top three prizes. I’ll do this without regard to the way the prize fund is divided. In a subsequent post, I’ll bring the payout schedule back into the analysis.
Slow, Shifty Brackets
In Getting the Most from an Afternoon, I found that shifting the lower bracket resulted in a considerably slower event. This result was explained, in Making a Double Elimination Run Faster, to be caused by the fact that the shift caused the lower bracket matches to have to wait on the results of the upper bracket matches, which were played to a larger number of points.
This is not a general result – playing longer matches in the upper bracket is common in backgammon tourneys, but rare in most other events. But a further experiment shows that, for backgammon, it’s somewhat robust.
Making a Double Elimination Run Faster
In the last post, I showed how I used simulation to address the question of which of two brackets was better for my friend’s monthly backgammon tournaments. Before moving on with that analysis, let’s show how the question arose.
Say you’re running a tourney for nine to sixteen players, and need to achieve a result in a limited time. You can always run a simple four-round knockout tourney, but time is not that limited, and you’d like to offer people more play. So you want to run something along the lines of a double-elimination tourney.
Unfortunately, a full double-elimination tourney with all the bells and whistles can take as many as nine rounds. You don’t have that much time. What are your options?
tourneygeek.com 