Does the Balanced Bracket Generalize?

We’ve seen some good results from simulation of Joe Czapski’t Balanced 16DE bracket, and a closely related bracket drawn by Sean Garber. But both of these brackets seem to be very specifically drawn for exactly 16 players. If their use is limited to this number, the brackets themselves will not really be very useful.

In this post, I’ll report the results of simulations of the same brackets we’ve considered recently, but with 13 players or teams rather than the full 16.

To begin, however, I should note that at tournamentdesign.org Joe provides a specific bracket for a 13-player double elimination. I’ve redrawn it to conform to the tourneygeek style, but otherwise left it unchanged: jc13de. Unlike Joe’s 16 bracket discussed earlier, this one can, at least, be described in standard notation: A.AB.C.D.|.X.R.

Joe’s 13 is, alas, nothing special. It’s a somewhat eccentric drawing, accommodating three fewer players by means of one bye in the A round, and another in the B round, where it can absorb would more commonly be handled by two more A round byes. This, for reasons explored in Pay Now or Pay Later is a bad idea. As I’ll show below, the bracket performs badly under simulation. So, if Joe’s innovation is to be of use, it needs to work as is, with three round A byes to make the 13 entrants occupy 16 slots.

Here are the entrants in our 13-team bracket bake-off:

A standard, unshifted 16 bracket with three spread byes;
A shifted 16 bracket with three spread byes;
Joe’s specially constructed 13 bracket;
Joe’s 16 bracket with three spread byes;
Sean Garber’s 16 bracket with three spread byes.

Each of these brackets will be tested at luck=1 and luck=3, with a million trials each. A winner-takes-all payout scheme is assumed. Note, please, that these results are not directly comparable to the full-16 brackets discussed in the last couple of posts. The fairness (C) numbers for a bracket with a smaller number of players is bound to be better than one for more players. That’s because the more players there are, the larger the opportunity lesser players to win.

 Joe Czapski 16 15.14 65.86 Sean Garber 16 15.14 65.86 Standard Shifted 15.6 68.37 Joe Czapski 13 15.63 68.42 Standard Unshifted 15.66 68.91

The two balanced brackets, Joe’s and Sean’s, are in a dead heat, both maintaining their advantage over the standard brackets despite the three byes. Joe’s specific 13 bracket has an indifferent result, a little worse than the shifted bracket, a little better than the unshifted one.

It’s hard to generalize about the other brackets on tournamentdesign.org, but none of the half-dozen or so I’ve looked at were as straightforward as the 13 bracket. It may be that some of the other special-size brackets would also perform well. But there’s much to be said for having a single design that can handle a range of entrant numbers. And this experiment suggests that the balanced bracket designs cope just as well with a sprinkling of byes as the standard brackets do.