Fair distribution of bundles

TL;DR

This paper addresses fair distribution of item bundles among multiple people, providing a method that guarantees fair value sharing and approximate bundle counts using topological techniques, with optimal results for powers of two.

Contribution

It introduces a topological approach to fairly split bundles of items among r persons, breaking at most (r-1)m bundles and ensuring each gets roughly n/r minus a correction term.

Findings

Fair splitting of bundles is possible with minimal breaking.

Guarantees on the number of full bundles each participant receives.

Optimal results achieved when r is a power of two.

Abstract

In this paper, we study the problem of splitting fairly bundles of items. We show that given $n$ bundles with $m$ kinds of items in them, it is possible to distribute the value of each kind of item fairly among $r$ persons by breaking apart at most $(r-1)m$ bundles. Moreover, we can guarantee that each participant will receive roughly $n/r - mr/2$ full bundles. The proof methods are topological and use a modified form of the configuration space/test map scheme. We obtain optimal results when $r$ is a power of two.