Hold everything! I’ve made another boneheaded error, and need to revisit pretty much all of my recent results to correct it.
Here’s the story of how I recognized the problem.
Category: Uncategorized
A Shifted 8DE?
Having discovered a useful alternative to the standard 16-team double-elimination bracket among those available at tournamentdesign.org, I spent some time looking at other brackets posted on Joe Czapski’s web page.
They’re a mixed bag. The 13DE bracket I considered (and essentially rejected) in the last post is not typical, in that it makes no use of “if necessary” matches (other than the recharge round). Most of the other double eliminations make extensive use of those rounds, and that makes them hard for me to evaluate – my simulator wasn’t built with them in mind.
But there is one other candidate I want to consider: Joe’s 8DE. Here is a version adapted to tourneygeek style: jc8. (Here, for contrast, is a standard 8DE: 8destd.)
Is this useful as a shifted 8DE?
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.
Another Balanced Bracket
My friend Sean Garber send me a variation on the one drafted by Joe Czapski, which was discussed in the last post. Here’s an analyzed version, with luck = 1 (to facilitate comparison with Joe’s bracket): sg16bal
A Balanced Bracket Redux
In one of my earliest experiments, I tested one of the “balanced brackets” drawn by Joe Czapski (available from his website tournamentdesign.org). In this post, I’ll revisit that analysis using the better simulator I have now, which ought to give Joe’s creation fairer (in both the fairness (B) and (C) senses) consideration.
Joe has drawn a large number of brackets for double-, triple-, and even quadruple-elimination tourneys. And he’s done so in very creative ways. His guiding objective is to equalize the number of wins needed to win the tourney as a whole. So, for example, the winner of his 16DE will have exactly six wins (and, possibly, one loss), no matter when the one loss happened. In an ordinary bracket, the path of a champion who stays in the winner’s bracket requires only five wins. But the road to redemption for a player who loses in the first round requires eight wins to fight back through the loser’s bracket and a recharge round (or seven if the bracket is shifted).
Balancing the bracket in this way sometimes requires the pairing of players with different won-loss records, and “if necessary” matches that are played or not depending on the result of those unbalanced matches.
BBBR: 128s, results
Here, as promised, are the f(b:X) tables for the five different versions or our hypothetical 128-player double elimination tourney.
(As the formal notation gets a bit unwieldy for a bracket as large as a 128, I’ve named the shifts after the files in the last post.)
The take-aways: the best of the shifted brackets are always better than the unshifted bracket. In the high-luck scenario, all five versions do a remarkably good job of equalizing the final. The “early” shift (which is the only shift that, like the unshifted version, sends the G drop to the lower final) does consistently worse than the other shifts, and for high-skill events it’s even worse than the unshifted bracket.
Somewhat to my surprise, the “super” shift, which squeezes an extra round out of the tourney, does quite well – event to the point of being the best performer for a high-luck event. I’m not ready to anoint it a best practice and post it to the printable brackets page, but it certainly bears further investigation.
BBBR: 128s
It may seem overkill to discuss bracket shifts on a 128 bracket. There are precious few double-elimination tourneys run on a bracket that big, and it’s pretty easy to extrapolate the results from 16s, 32s, and 64s to make a pretty good guess what will happen with 128s. But there are some distinctive things about 128 bracket shifts that make them worth a look.
The main point of interest is that there aren’t just one or two possible shifts of a 128, but a bunch of them.
BBBR: 64s
This is a short post extending the fairness analysis of shifting brackets to a larger example. A 64 bracket can take two shifts, which should be expected to show the same sort of effects as were seen in 16s and 32s. And that’s exactly what happens.
The State of Knowledge
I recently spent some time considering whether I might perform a public service by substantially revising the Wikipedia article on double-elimination tournaments.
It’s pretty dreadful. It’s shallow, and misleading. The one example bracket contained in the article has bad drops. There’s no mention at all of the possibility of shifting brackets to save rounds and (usually) improve fairness. It’s assumed, apparently on the basis of a fairness (A) argument, that there must be a recharge round.
But I decided not to touch the article, even in the comments section. As bad as it is, it probably does represent a consensus of opinion. I couldn’t make it much better without violating Wikipedia’s rules against original research. Wikipedia would not count tourneygeek as a “reliable source”.
BBBR: Seeding
So far, revisiting the question of the relationship between fairness and bracket shifts has largely confirmed what was found earlier.
The bracket shift’s influence on fairness (C) is benign for both 16s and 32s. This is chiefly because it tends to mitigate the unfairness of dropping the loser of the upper-bracket final directly into the lower-bracket final, dropping it instead to the semifinal of the lower bracket. In 32s, then, the late shift is better than the early shift. And in both 16s and 32s, the drops balance better for high-luck events than they do for low-luck events.
Perhaps it will be worthwhile to look at 64 brackets also, though it’s hard to imagine the results being much different. For this post, however, I’ll look at the one design parameter that has been found to favor unshifted brackets in the past: seeding.
tourneygeek.com 