Fair division with divisible and indivisible items

TL;DR

This paper investigates the fair division problem involving both divisible and indivisible items for two participants, establishing conditions for the existence of fair divisions and highlighting their rarity when items are not all divisible.

Contribution

It provides necessary and sufficient conditions for proportional and equitable divisions in mixed divisible and indivisible item scenarios.

Findings

Fair divisions may not always exist with mixed item types.

Necessary and sufficient conditions for proportional division.

Necessary and sufficient conditions for equitable division.

Abstract

In the work the fair division problem for two participants in presence of both divisible and indivisible items is considered. The set of all divisions is formally described; it is demonstrated that fair (in terms of Brams and Taylor) divisions, unlikely the case where all the items are divisible, not always exist. The necessary and sufficient conditions of existence of proportional and equitable division were found.