Fairly Dividing a Cake after Some Parts Were Burnt in the Oven

TL;DR

This paper addresses the problem of fairly dividing a heterogeneous resource containing both good and bad parts among multiple agents with diverse preferences, ensuring connected pieces and envy-freeness, with proven existence for three agents.

Contribution

It establishes the existence of a fair, connected division for three agents with mixed valuations and provides initial results for more agents, advancing fair division theory.

Findings

Existence of fair division for 3 agents proven.

Connected envy-free division guaranteed for three agents.

Initial positive results for larger groups of agents.

Abstract

There is a heterogeneous resource that contains both good parts and bad parts, for example, a cake with some parts burnt, a land-estate with some parts heavily taxed, or a chore with some parts fun to do. The resource has to be divided fairly among $n$ agents with different preferences, each of whom has a personal value-density function on the resource. The value-density functions can accept any real value --- positive, negative or zero. Each agent should receive a connected piece and no agent should envy another agent. We prove that such a division exists for 3 agents and present preliminary positive results for larger numbers of agents.