Elementary solution to the fair division problem

TL;DR

This paper introduces an elementary method for solving the fair division problem of continuous resources among participants with charges, using a dynamical system approach with proven exponential convergence.

Contribution

It presents the first analysis of fair division with charges and develops a dynamical system framework for efficient solution convergence.

Findings

Proves exponential convergence to a fair division solution.

Introduces a new approach for fair division with signed measures.

Establishes properties of the dynamical system in resource partitioning.

Abstract

A new and relatively elementary approach is proposed for solving the problem of fair division of a continuous resource (measurable space, pie, etc.) between several participants, the selection criteria of which are described by charges (signed measures). The setting of the problem with charges is considered for the first time. The problem comes down to analyzing the properties of the trajectories of a specially constructed dynamical system acting in the space of finite measurable partitions. Exponentially fast convergence to a limit solution is proved for both the case of true measures and the case of charges.