The past several posts all amount to another negative result.
I started by revisiting the 16DE “Balanced Bracket” posted by Joe Czapski on his tournamentdesign.org page. The idea was to see if my earlier analysis, which had been done before I’d developed my current fairness (C) metric and tourney simulator, was still valid. Long story short: it was. The Balanced Bracket has some appeal, but it does sacrifice fairness, and that’s a significant disadvantage in my estimation.
This would have been a short single post but for the fact that I made a big mistake. But sometimes you learn more from your mistakes than from your successes, and that happened here. I learned that I need to be more careful about how fairness (C) is used.
The short version of what I learned is that the fairness (C) statistic will consider only what I’ve told it to care about. By using fairness (C) with a winner-takes-all payout scheme, I was telling the system that all I cared about was whether a tourney design was good at identifying a single champion. What I didn’t realize was that a design might look good not so much because it identified the best player, but because it suppressed the chances of other good players who might take the prize away from the best player. This is the “ugly bottom” effect.
The cure for the ugly bottom problem was simple. Instead of calculating fairness (C) based solely on the quality of the champion, I needed to give some weight to the quality of the runner-up.
This highlights, I think, the need to keep in mind more than one aspect of fairness. If you know who the best player is, you can achieve a perfect fairness (C) result by simply giving that player the prize – no need to go to the bother of running a tourney at all. Or, if you cared only about fairness (B), you can achieve perfection by simply drawing lots for the prize – again, no need to bother with running a tourney.
A fair tourney needs to take some account of both fairness (B) and fairness (C)(and perhaps some aspects of fairness (A) as well). By generalizing the way fairness (C) is measured to allow it to be cognizant of more than one place, it became possible to measure fairness (C) without totally disregarding fairness (B). But the tools only work right when you use them right.