Connected Equitable Cake Division via Sperner's Lemma

TL;DR

This paper presents a simple Sperner's lemma-based proof for the existence of connected equitable cake divisions among agents, extending results to more general valuation classes including externalities and negative valuations.

Contribution

It introduces a straightforward Sperner's lemma approach to establish connected equitable divisions, broadening applicability to complex valuation scenarios.

Findings

Existence of connected equitable divisions proven using Sperner's lemma

Extension of results to valuations with externalities

Applicability to subclasses of negative valuations

Abstract

We study the problem of fair cake-cutting where each agent receives a connected piece of the cake. A division of the cake is deemed fair if it is equitable, which means that all agents derive the same value from their assigned piece. Prior work has established the existence of a connected equitable division for agents with nonnegative valuations using various techniques. We provide a simple proof of this result using Sperner's lemma. Our proof extends known existence results for connected equitable divisions to significantly more general classes of valuations, including nonnegative valuations with externalities, as well as several interesting subclasses of general (possibly negative) valuations.