An Algorithm for the Proportional Division of Indivisible Items

29 Pages Posted: 11 Jun 2014

See all articles by Steven J. Brams

Steven J. Brams

New York University (NYU) - Wilf Family Department of Politics

D. Marc Kilgour

Wilfrid Laurier University - Department of Mathematics

Christian Klamler

University of Graz

Date Written: May 2014

Abstract

An allocation of indivisible items among n ≥ 2 players is proportional if and only if each player receives a proportional subset — one that it thinks is worth at least 1/n of the total value of all the items. We show that a proportional allocation exists if and only if there is an allocation in which each player receives one of its minimal bundles, from which the subtraction of any item would make the bundle worth less than 1/n.

We give a practicable algorithm, based on players’ rankings of minimal bundles, that finds a proportional allocation if one exists; if not, it gives as many players as possible minimal bundles. The resulting allocation is maximin, but it may be neither envy-free nor Pareto-optimal. However, there always exists a Pareto-optimal maximin allocation which, when n=2, is also envy-free. We compare our algorithm with two other 2-person algorithms, and we discuss its applicability to real-world disputes among two or more players.

Keywords: Fair division, indivisible items, proportionality, envy-freeness, maximinality

JEL Classification: C78, D61, D63, D74, D78

Suggested Citation

Brams, Steven and Kilgour, D. Marc and Klamler, Christian, An Algorithm for the Proportional Division of Indivisible Items (May 2014). Available at SSRN: https://ssrn.com/abstract=2447952 or http://dx.doi.org/10.2139/ssrn.2447952

Steven Brams (Contact Author)

New York University (NYU) - Wilf Family Department of Politics ( email )

Dept. of Politics
19 West 4th St., 2nd Fl.
New York, NY 10012
United States
212-998-8510 (Phone)
212-995-4184 (Fax)

HOME PAGE: http://politics.as.nyu.edu/object/stevenbrams.html

D. Marc Kilgour

Wilfrid Laurier University - Department of Mathematics ( email )

Christian Klamler

University of Graz ( email )

Graz
Austria

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