Group Decision and Negotiation

, Volume 11, Issue 5, pp 405-414

First online:

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

  • Richard F. PotthoffAffiliated withDepartment of Political Science, Duke University

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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.

bidding envy-freeness fair division housemates problem linear programming