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Connected Equitable Cake Division via Sperner's Lemma

Umang Bhaskar, A. R. Sricharan, Rohit Vaish

TL;DR

This paper investigates fair cake-cutting under the constraint that every agent receives a connected piece and all agents obtain the same utility (equitable division). It introduces the SANN (some agent nonnegative) valuation class, allowing global valuations with externalities and certain negatively valued additive cases, and proves the existence of a connected equitable allocation using Sperner's lemma via vanishing triangulations. The result extends known existence theorems to broader valuation families and connects to several natural subclasses of additive valuations (e.g., identical, split-cake, single-peaked), ensuring connected equitable allocations for appropriate agent orderings. The work also discusses the two-agent case with global valuations, and outlines computational and combinatorial questions, such as finding permutations that satisfy SANN and approximating equitable divisions, highlighting both theoretical and practical significance for fair division with connectivity constraints.

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.

Connected Equitable Cake Division via Sperner's Lemma

TL;DR

This paper investigates fair cake-cutting under the constraint that every agent receives a connected piece and all agents obtain the same utility (equitable division). It introduces the SANN (some agent nonnegative) valuation class, allowing global valuations with externalities and certain negatively valued additive cases, and proves the existence of a connected equitable allocation using Sperner's lemma via vanishing triangulations. The result extends known existence theorems to broader valuation families and connects to several natural subclasses of additive valuations (e.g., identical, split-cake, single-peaked), ensuring connected equitable allocations for appropriate agent orderings. The work also discusses the two-agent case with global valuations, and outlines computational and combinatorial questions, such as finding permutations that satisfy SANN and approximating equitable divisions, highlighting both theoretical and practical significance for fair division with connectivity constraints.

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.

Paper Structure

This paper contains 25 sections, 11 theorems, 11 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Preliminaries
  4. Sperner's Lemma
  5. $n$-simplex.
  6. $k$-face.
  7. Triangulation.
  8. Labeling of a triangulation.
  9. Cake-Cutting
  10. Model.
  11. Allocation.
  12. Valuations.
  13. Global valuations.
  14. Local valuations.
  15. Permuted instances.
  16. ...and 10 more sections

Key Result

Proposition 1

Any Sperner labeled triangulation of an $n$-simplex must contain an odd number of fully labeled elementary $n$-simplices. In particular, there is at least one.

Figures (2)

  • Figure 1: Summary of our results in the form of a Venn diagram. Each rectangle denotes a class of instances (defined in Section \ref{['sec:prelim']}) and each dot-arrow pair denotes the domain on which the corresponding result applies.
  • Figure 2: A three-agent instance with no SANN permutation. For every permutation of the agents, there is an allocation where every agent with a nontrivial piece has negative value.

Theorems & Definitions (21)

  • Proposition 1: Sperner's lemma; S28neuer
  • Example 1: Happy, happier, and happiest agent
  • Remark 1
  • Example 2: Agent ordering matters
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • ...and 11 more