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.
Unable to display preview. Download preview PDF.
- Abdulkadiroğlu, Atila, Tayfun Sömez, and M. Utku Ünver. (2002). “Room Assignment — Rent Division: A Market Approach”. Typescript, Department of Economics, Columbia University, New York, NY, USA.Google Scholar
- Brams, Steven J., and D. Marc Kilgour. (2001). “Competitive Fair Division,” Journal of Political Economy 109, 418-443.Google Scholar
- Brams, Steven J., and Alan D. Taylor. (1996). Fair Division: From Cake-Cutting to Dispute Resolution. New York: Cambridge University Press.Google Scholar
- Gould, F. J., G. D. Eppen, and C. P. Schmidt. (1993). Introductory Management Science, 4th ed. Englewood Cliffs, NJ: Prentice Hall.Google Scholar
- Haake, Claus-Jochen, Matthias G. Raith, and Francis Edward Su. (2001). “Bidding for Envy-Freeness: A Procedural Approach to n-Player Fair-Division Problems”. Typescript, Institute of Mathematical Economics, University of Bielefeld, Bielefeld, Germany.Google Scholar
- Klijn, Flip. (2000). “An Algorithm for Envy-Free Allocations in an Economy with Indivisible Objects and Money,” Social Choice and Welfare 17, 201-215.Google Scholar