Fairness for Everyone? Do We Need Another Measure?

There is unfinished business from the last couple of posts.

The remaining thing to do in addressing the question of whether a shifted bracket is a good thing or a bad thing is to address the question of whether it is bad because it uses an unbalanced bracket. Is an unbalanced bracket inherently unfair?

Before we do that, however, I want to consider whether it would be a good thing to consider a somewhat different coefficient that looks at the distribution of rewards short of winning the tournament as a whole.

We’ve already seen that the shifted bracket actually increases the fairness score, and least in the context of a 16-team double elimination tournament (16DE), and I have results that show this is also true for a 32DE, which begins to suggest that the technique might be generally beneficial. (I’ll post these results soon.) But in addressing the question of the unbalanced bracket, I became aware of a fairly serious limitation of the fairness score.

The fairness score, you may recall, is intended to measure fairness (C), the extent to which the tournament rewards superior play. For the fairness coefficient that I’ve been using, the only reward that matters is whether or not the team actually wins the tournament. This is a measure that’s worth having, but it does not always tell the whole story. Fairness should pervade the tournament as a whole, not just the play that determines the ultimate winner. In particular, the measure is not always suited to identifying problems with respect to fairness (B), the principle that everyone should be given an equal chance.

In looking at the unbalanced parts of the shifted bracket, I saw that some lines had very slightly lower winning percentages than others. The differences were on the order of a few thousandth’s of one percent winning chances, and I was tempted to dismiss them as irrelevant because they were so small. But were they really so negligible that they could be ignored? There was a difference between two lines in the bracket that ought, according to fairness (B), to have exactly the same chances. Dismissing the problem because it didn’t make a bigger dent in the fairness (C) measure was equivalent to saying, “that’s all right – you weren’t going to win anyway”.

What’s needed, I think, is the ability to look at the distribution of a different kind of reward. Many players enter tournaments with no realistic chance of winning, but still feel they did better or worse according to how many rounds they get to play. In a double elimination tournament, one-quarter of the field is eliminated because they suffer two  straight losses, and another quarter is eliminated with only one win.  A player who wins a couple of matches has every right to say that their result was better than average, and indeed it is. It’s a significant accomplishment, even though the player never got close to the prize money. And for some players, it’s an accomplishment that’s within reach when the trophies and prize money really aren’t.

So, I’m planning to add some logic to my tournament simulator so that it tracks not only the number of tournament wins by player and by line, but also the number of individual match wins by player and by line. This should make it more possible to detect unfairness in parts of the brackets where the overall winning percentage is low.

There will shortly be a post on unbalanced brackets, but that can wait until I have the new tool in place. I doubt that whatever I uncover will be enough to dissuade me from continuing to recommend the bracket shift. But the story is not yet complete.