The Mathematics of Fairness

We recently played one of my favorite games in my 6th 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?


8 Replies to “The Mathematics of Fairness”

  1. Won’t 45 games be enough? (And you’ll be playing 5 games at a time, if you can figure that schedule out! So it’s only 9 game sessions. Not too bad.)

    1. Yes, of course! Our grid has 100 squares, which is 90 if you remove people playing themselves, and only 45 when you remove the doubles. That’s silly that I missed that when writing this. We even crossed off the reflection across the diagonal (as well as the diagonal) when we made the board. Thanks for the note! Though 45 games is still a lot…

      1. Oh, and also making the schedule was a big challenge. It was tricky for them to figure out who should play who on each round so that everyone was playing at all times. Unfortunately, I was out the next time we had class when they were finishing the tournament, so I’m not sure how they sorted that out. Another math teacher covered the class, though, so I’ll find out how it went soon.

  2. Great discussion. I can imagine the groups discussing differing aspects of fairness and trying to develop a system which allowed for inclusion, but whittling down the competitors in favor of finding a winner.

    I would have loved to have seen what kinds of systems the students came up with.

    Scott Hills

    1. Hi Scott, thanks for the comment. I can try to put up some of their systems. I gave them back to them and they wrote all over them making edits, so they’re a bit of a mess. But I’ll try to find some presentable ones to show.

  3. Lots to “munch” on here. It’s interesting how often fairness comes up in math classes–and yet how many people don’t see how to work social justice issues into the math curriculum. I’m curious, though, about how you might go about “making these connections to broader social issues more apparent to the kids.” Was there something you had in mind? I’d love to hear about it if/when you wind up doing this.

    1. Thanks for the comment. (And nice munching reference… There’s definitely a trend in my blog titles.) Yup, I’m curious about how I’m going to do that, too. It could also be good to make connections to how the students interact with each other in class, because they are very combative and almost militant about complete fairness, yet also quick to grab at personal advantages. I think it’s important to take this opportunity, but I also don’t want to ram social justice down their throats. A big question to discuss is when is it fair to give advantages to people– and when is it fair to make the playing field completely even? This makes me want to look for other mathematical situations which measurements of equality are fuzzy… I’ll certainly keep you posted. I also welcome suggestions!

  4. I love these synchronicities… I had just proposed an arcade game as a hypothetical in my AP Stats class. Ostensibly, wee looking at probability distributions and expected value calculations. Instead, we drifted into a conversation about fairness and balance (recognizing that a fair game for the player is a different question than a fair game for the casino/arcade owner) I am so encouraged to ‘hear’ about these sophisticated and challenging ideas being discussed at such different levels. Thanks for sharing

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