Envy-free division via configuration spaces

TL;DR

This paper introduces a novel approach to envy-free division problems using configuration spaces and equivariant topology, extending classical theorems to include degenerate preferences.

Contribution

It presents a new topological method for envy-free division, expanding the classical Gale theorem to accommodate degenerate preferences.

Findings

Proves several extensions of Gale's envy-free division theorem.

Demonstrates the effectiveness of configuration space methods in fair division.

Includes applications to preferences with degenerate pieces.

Abstract

The classical approach to envy-free division and equilibrium problems relies on Knaster-Kuratowski-Mazurkiewicz theorem, Sperner's lemma or some extension involving mapping degree. We propose a different and relatively novel approach where the emphasis is on configuration spaces and equivariant topology. We illustrate the method by proving several relatives (extensions) of the classical envy-free division theorem of David Gale, where the emphasis is on preferences allowing the players to choose degenerate pieces of the cake.