Fair Division

Podcast

Fair Division

Transcript

Breaking up fights with math. I'm Bob Hirshon and this is Science Update.

Squabbling roommates fighting over who gets which room and how much they should pay can now get help from an unusual source: mathematics. Mathematician Francis Su at Harvey Mudd college has written a computer program that he says will guarantee what he calls "an envy free" solution to dividing up rooms and rent.

Su:
"And I like to call this result the Rental Harmony Theorem. Because it gives you a way of harmonizing roommates."

Doctor Su actually published a proof for his Rental Harmony Theorem in the December issue of American Mathematical Monthly. And the program that puts the theorem into practice is pretty simple. All you have to do is enter in the number of roommates and the total rent of the house or apartment.

Su:
"And it will ask one of the players/ Hey, which room would you prefer if the rooms were priced like this?/ And based on that response, the interactive algorithm will take that information and decide which player to ask next and how to modify the prices of the rooms."

Doctor Su says that his program and other fairness programs are being used increasingly to settle disputes.

For the American Association for the Advancement of Science, I’m Bob Hirshon.

Making Sense of the Research

Everyone knows you can use math to figure out a paycheck,or divide a pizza. But how about settling arguments? This story talks about a kind of mathematics that looks at disagreements that people have (usually about money or other "number-related" things, but sometimes other things,too), and figures out the most fair way to settle them. One concept you may be familiar with is when two people are dividing a treat, like the last part of a cake, "first person cuts; second person chooses." If the first person knows the second person will get to choose the biggest piece, he or she will be very careful to cut the cake evenly. But things get more difficult, and the math gets more complicated, when there are more than two people involved, and they're dividing up something you can't really cut-- like an apartment, or a roomful of furniture.

Try to answer the following questions based on the transcript:

1. What has mathematician Francis Su come up with to help resolve disputes?
2. What does Dr. Su call his theorem?
3. How can the theorem be put into practice?
4. How does the theorem work?
5. What other kinds of disputes might be resolved using fairness programs?

Going Further

Visit Francis Su's homepage to try out his Fair Division Calculator.

For a good follow-up lesson for grades 6-8, go to Finding Satisfactory Solutions, found under The Nature of Math: Lessons (6-8). In this activity, students decide where to locate ice cream stands in a town so that no one has to travel too far to buy a treat. The problem-solving strategies for this problem give students a chance to grapple with the notion of proof and to decide what makes a solution satisfactory.

For a good follow-up lesson for grades 9-12, go to Evaluating Mathematical Models, found under The Nature of Math: Lessons(9-12). In this lesson, students evaluate a series of mathematical models. Through this activity, students begin to develop an understanding of the fact that mathematics, like science and technology, involves both finding answers to fundamental questions and solving practical problems.