Fair division with multiple pieces

• Published in: Discrete Applied Mathematics Vol. 283; pp. 115 - 122
• Main Authors: Nyman, Kathryn, Su, Francis Edward, Zerbib, Shira
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.09.2020; Elsevier BV
• Subjects: Dual Komiya; Fair division; KKMS; Multiple pieces; Players; Rental harmony in two buildings; Shift assignment; Working hours; Shift assignment; Fair division; Rental harmony in two buildings; Dual Komiya; KKMS; Multiple pieces
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2019.12.018

Given a set of p players, we consider problems concerning envy-free allocation of collections of k pieces from given sets of goods or chores. We show that if p=n and each player prefers k pieces in any division of a cake into n pieces, then there exists a division of the cake where at least pk2−k+1 players get their desired k pieces each. We further show that if p=k(n−1)+1 and each player prefers k pieces, one piece from each of k cakes, in any division of the k cakes into n pieces each, then there exists a division of the cakes where at least pk2−k players get their desired k pieces each. Finally we prove that if p≥k(n−1)+1 and each player prefers one shift in each of k days that are partitioned into n shifts each, then, given that players prefer empty shifts if possible (e.g., if salaries are fixed and do not depend on the number of working hours), there exist n(1+lnk) players covering all the shifts, and moreover, if k=2 then n players suffice. Our proofs combine topological methods and theorems of Füredi, Lovász and Gallai from hypergraph theory.