Exploring Fair Division

My latest piece for the New York Times Learning Network is a math lesson exploring basic techniques of fair division.

Fair division is concerned with partitioning a set into fair shares.  “Fair” can take on different meanings in different contexts, but at its most basic level, a share is fair if someone is willing to accept it.

This lesson builds on an excellent article in the NYT about a technique in rent-splitting based on Sperner’s Lemma, an important result in Topology.  The author tells the story of how he and two roommates used the technique to settle on a fair division of rent for three different-sized rooms.

“The problem is that individuals evaluate a room differently. I care a lot about natural light, but not everyone does. Is it worth not having a closet? Or one might care more about the shape of the room, or its proximity to the bathroom.

A division of rent based on square feet or any fixed list of elements can’t take every individual preference into account. And negotiation without a method may lead to conflict and resentment.”

After reflecting on the article, students use the related NYT interactive feature to explore the algorithm and then research basic techniques in fair division like divider-chooser, sealed bids, and the method of markers.  The full lesson is freely available here.