Use of Linear Programming to Find an Envy-Free Solution Closest to the Brams–Kilgour Gap Solution for the Housemates Problem

Although the Gap Procedure that Brams and Kilgour (2001) proposed for determining the price of each room in the housemates problem has many favorable properties, it also has one drawback: Its solution is not always envy-free. Described herein is an approach that uses linear programming to find an envy-free solution closest (in a certain sense) to the Gap solution when the latter is not envy-free. If negative prices are allowed, such a solution always exists. If not, it sometimes exists, in which case linear programming can find it by disallowing negative prices. Several examples are presented.