# A fair deal for housemates

A new mathematical recipe for fair division allows people to resolve disputes over the splitting up of rent, goods, or even burdensome chores.

Four friends move into a house and find they must choose among four rooms of different size and quality. Instead of sharing the rent equally, they decide to divide the total so that each person ends up satisfied with his or her combination of room and rent. Can they do it?

This is an issue that mathematics can settle, both elegantly and constructively, says mathematician Francis E. Su of Harvey Mudd College in Claremont, Calif. Su has developed a new approach to the question of fair division (SN: 5/4/96, p. 284: https://www.sciencenews.org/sn_arch/5_4_96/bob1.htm) and now offers an interactive Web site (http://www.math.hmc.edu/~su/fairdivision) where people can go to resolve disputes over the splitting up of rent, goods, or even burdensome chores.

In the case of rent, Su’s “Fair Division Calculator” repeatedly queries each housemate on which room he or she would prefer if the total rent was divided in certain ways among the rooms. A mathematical procedure, based on a combinatorial result known as Sperner’s lemma, determines room rents to propose after each housemate’s turn. After many iterations, Su’s algorithm calculates an approximate solution that everyone should be able to live with.

Occasionally, a player can end up with a negative rent—a monetary reward from the other players for putting up with an otherwise undesirable room. The scheme works for any number of participants.

Su continues to refine his methods. He now offers an option in which all the roommates enter bids on all the rooms. His mathematical algorithm then automatically works out a suggested rent allocation without the need for a string of queries.

Steven J. Brams of New York University and D. Marc Kilgour of Wilfrid Laurier University in Waterloo, Ontario, have proposed an alternative approach that works like an auction and avoids negative rents. The players bid competitively on the rooms available, but the player who bids highest for a particular room doesn’t necessarily get it. The procedure assigns rooms to different players, taking into account the sum of the players’ bids. In this method, the price paid for a room depends not just on the winning bid but on lower bids as well.

“Thereby, the market helps to set prices by incorporating the most competitive lower bids into the pricing mechanism,” the researchers say. “It seems only fair that prices should be higher when there is greater demand for items.”