Now to begin trying to find Miss FOTA 32. Let’s start simple.
The simplest tournament design that obeys the rules is a straight single elimination. Nothing easier! It plays in five rounds, but we only have to start the E round, not finish it, within the time limit.
Here’s a first run, to see what we’ve got to work with: f(C) = 96.87, T = 98.7. The bad news is that fairness (C) looks pretty awful, but the good news is that the base tourney is using less than 100 of our allotted 500 minutes. How can we best spent the rest of the time to bring down f(C)?
Why is f(C) so bad in the first place? It’s because of the large luck factor. The more luck there is in a match, the more likely it is that the better players will get waylaid before they get the payout they deserve.
The way to spend the extra time is to add games to the matches. Since we’re using about a fifth of our allowed time, let’s start by taking all of the matches and making them best of five games rather than single games. And, while we’re at it, let’s make the final best of nine, because that extra time won’t count against us.
(I’ll show the results for all of these runs in a table below. Each of these runs represents 100,000 trials.)
Good! That brought down f(C) considerably, and it didn’t even use all of our time. In the early rounds, where there are lots of matches to be played, going from a single game to best-of-five multiplies the time required by almost five. But in later rounds, where there aren’t many matches, the fact that some of them won’t go the full five games brings the time requirement down. So, we still have eighty-some more minutes to spend.
Let’s spend it on making the C and D rounds best of seven. Now we’re talking! f(C) has come all the way down below 51. That’s the kind of performance you’d see with single games at luck = 1.5.
We’re using all of the time, now, but perhaps some of it would be better spent in other rounds. How about making the matches 3-5-7-9-9? 3-7-7-7-9? Neither of these seems to help.
Miss SE55779 is our best single-elimination candidate for the coveted Miss FOTA 32 crown.
Time to run our million-trial official simulation to make sure Miss SE55779 meets the rules. We haven’t worried at all about fairness (B) so far – if a perfectly balanced single elimination bracket can’t qualify, then the stage of the Miss FOTA 32 Pageant is going to be a pretty lonely place. But not to worry, f(B) is just 0.367. And the time checks, too – it’s 492 minutes.
Miss SE55779’s fairness (C) score from the long run is 50.92 – on the high side of the confidence interval for the shorter run, but still not bad. Let’s also record the new 95% confidence interval (the judges like to see that): 50.84 to 51.00.
That’s what we’ve got for a single-bracket tourney. Can someone crack the 50 barrier by adding a second bracket? Stay tuned.