The Mathematics of Fairness

We recently played one of my favorite games in my 6^th grade class. It’s a fraction game, originally invented by Justin Lanier but altered by me since I’ve started using it. The game is great fun and quite educational—it’s basically war with home-made fraction cards, and I’ll tell you all about how to play it in a future post. This year when we played, though, the most interesting ideas came not from actually playing the game but from trying to make a fair bracket.

In the several years that I’ve played this game with classes of kids, I’ve never had to make the bracket myself. I always have at least one student in the class who plays squash and has way more experience with tournaments and consolation rounds and rankings than I do and is really excited to make a bracket. So, I typically let that kid make the bracket. It’s gone well for the past few years.

But this year, the other kids in the class didn’t want the squash player to be the only one who got to try making a bracket. For better or for worse, I have a class of very competitive kids, and they don’t trust any of their classmates to do anything right. So, as a homework assignment, I asked the kids to design a structure for our tournament. The structure had to be as fair as possible while producing a ranking.

The fair part turned out to be the real challenge. Unfortunately, we had ten different players, both individual kids and pairs playing as a team. Now, ten is very difficult number of players to make into a traditional bracket. During the class discussion leading up to our vote on a tournament structure, I found out that kids dealt with this in various ways. While working on their brackets, they quickly realized that powers of two were best for making brackets. Nothing like designing a bracket for reinforcing the double-ness of a power of two and the lack of double-ness in ten. Lots of byes were going to be necessary. But how many byes? Shouldn’t they be minimized? And how would you determine who gets a bye?

Some kids developed brackets with two byes in the first round and two byes in a middle round. Others had all ten players play in pairs in the first round and worked byes in later. Other kids did way more complicated things. This led to a discussion—is it fairer to have byes in the first round or in later rounds? Which increases the chances that an unworthy deck would get to the final round?

Then there was the question of who gets a bye. Most kids had the byes determined by chance. But a few kids made you “earn” your bye. One girl had byes in middle rounds, with the player who won by the largest amount earning the bye. But in the consolation round, she brought back players who hadn’t lost by very much and let them play again, this time in best of 5 matches to make their second chances “fairer,” to balance the numbers instead. In this way, she emphasized winning in the winners’ tournament and gave second chances in the consolation tournament. This felt most “fair” to her. But does fair mean that everyone has an equal chance of getting a boost or that only those who earn their boost (or were the most unworthily disadvantaged) get one? We talked about this.

Yet another kid decided to abandon a tournament structure altogether. From his perspective, any byes introduced unfairness. No boosts allowed. So, he went for a round-robin, with rankings determined by number of wins and losses. But would a round-robin take so long that its fairness wasn’t worth it? And was the ranking it would produce not quite as ranking-y as the one a tournament would make?

Never had I been in a math class so full of social justice issues. None of the questions we were asking about fairness had correct answers; all are debated frequently in the context of divisive issues ranging from the fairness of the NCAA basketball tournament to affirmative action. I’m considering going back and making these connections to broader social issues more apparent to the kids. Despite being about fairness, though, the work we were doing was still very mathematical.

Eventually, we just voted on it. (Using a first-past-the-post voting mechanism! Not the Coombs method! Or the Borda count! Was it fair??? Aaah fairness!) A narrow plurality chose the round-robin. So, we’re now in the process of playing 90 games of fraction war. Sigh.

(NOTE: It’s actually only 45 games, because I forgot that the board counts each game twice. We took that into consideration on the board, because we crossed out the games mirrored across the diagonal of the grid, but I forgot about that while writing this! Thanks for pointing that out, Sue! It also came up while planning who should play who in each round. That was yet another challenge…)

But, even if the kids get zonked out before finishing the 90 games of fraction war, I feel like the discussions we had were well worth it. We discovered that fairness may depend on context. Equal chances may not be fair; earning your advantage may not be fair either. But sometimes a sacrifice of fairness must be made in the name of accomplishing other goals. Or must it?