# How To Divide the Rent, a Cake and a Country

by Elaine Schwartz    •    May 1, 2014    •    415 Views

Assume for a moment that you have just rented a 4th floor walk-up apartment for \$3000 with 2 friends. One bedroom is mid-sized and near a fire escape, the second is rather large and has a view of a magnificent tree while the third is tiny, looks out on the alley and is next to the bathroom. How to decide who gets what…and for how much?

Fair resource division is a timeless problem.

Whether dividing an apartment, sharing a cake or resolving a border dispute, the issues are the same. Resource division has always been the economic problem.

In my basic econ course, at the beginning of the year when we focus on how societies deal with producing and distributing limited resources, we always identify the 3 basic economic questions:

• What to produce?
• How to produce?
• Who gets the income?

As always, though, there is so much more.

Investigating the allocation of resources, we could look at fairness. Assuming that fair means envy free, with our roommates, how to treat each one fairly?

An answer takes us to a triangle and a math algorithm that I first saw several days ago in the NY Times. The algorithm developed by Harvey Mudd professor Francis Su requires asking each roommate countless questions about which room he or she might select. You can see just one of many answers from “Chad” below. Dr. Su based his ideas on the work of German mathematician Emanuel Sperner (1905-1980).

In the NY Times Interactive linked below, you can see the other triangles, each displaying a different preference from one roommate.

Dr. Su explained, that “The trick is to design a procedure to have everyone act in their own self-interest and have an outcome that is fair.”

Sounds a bit like Adam Smith. Everyone benefits when people can pursue their self-interest.

Sources and Resources: H/T to the NY Times for their fascinating interactive and article on Francis Su and Sperner’s lemma (a small step toward proving a theorem) and a second much older related commentary. It was especially interesting, though, reading some of Dr. Su’s work because his enthusiasm was so evident.