Fair division with multiple pieces

TL;DR

This paper investigates envy-free division problems involving multiple pieces and provides bounds on the number of players who can be satisfied with their desired allocations in various settings, using topological and hypergraph theory methods.

Contribution

It introduces new bounds and existence results for envy-free allocations in multi-piece division problems, extending classical fair division theory.

Findings

Existence of allocations where a fraction of players get their desired pieces.

Bounds depend on the number of players, pieces, and divisions.

Use of topological and hypergraph theorems to prove results.

Abstract

Given a set of $p$ players we consider problems concerning envy-free allocation of collections of $k$ pieces from a given set of goods or chores. We show that if $p\le n$ and each player can choose $k$ pieces out of $n$ pieces of a cake, then there exist a division of the cake and an allocation of the pieces where at least $\frac{p}{2(k^2-k+1)}$ players get their desired $k$ pieces each. We further show that if $p\le k(n-1)+1$ and each player can choose $k$ pieces, one from each of $k$ cakes that are divided into $n$ pieces each, then there exist a division of the cakes and allocation of the pieces where at least $\frac{p}{2k(k-1)}$ players get their desired $k$ pieces. Finally we prove that if $p\ge k(n-1)+1$ and each player can choose one shift in each of $k$ days that are partitioned into $n$ shifts each, then, given that the salaries of the players are fixed, there exist $n(1+\ln k)$ players covering all the shifts, and moreover, if $k=2$ then $n$ players suffice. Our proofs combine topological methods and theorems of F\"uredi, Lov\'asz and Gallai from hypergraph theory.