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Lower bounds on the number of envy-free divisions

Duško Jojić, Gaiane Panina, Rade Živaljević

TL;DR

This work investigates lower bounds on the number of envy-free divisions in two cake-cutting models. It leverages secretive/expelled variants of envy-free results and a configuration-space/test-map framework to quantify multiplicity: a universal lower bound of at least two envy-free divisions in the classical Woodall-Stormquist setting, and a significantly larger bound $\binom{2p-1}{p-1} 2^{2-p}$ in a prime-$p$ measure-equipartition hybrid. The analysis identifies extremal preferences that realize exactly two divisions, showing the underlying incidence graph forms a single $2r$-cycle, and demonstrates how topological methods yield precise counting in the extended model. Together, these results illuminate how additional structure (secretive/expelled participants, measure constraints) enhances the multiplicity of envy-free divisions and showcases the power of topological combinatorics in fair-division problems.

Abstract

We analyze lower bounds for the number of envy-free divisions, in the classical Woodall-Stormquist setting and in a non-classical case, when envy-freeness is combined with the equipartition of a measure. 1. In the first scenario, there are $r$ hungry players, and the cake (that is, the segment $[0,1]$) is cut into $r$ pieces. Then there exist at least two different envy-free divisions. This bound is sharp: for each $r$, we present an example of preferences such that there are exactly two envy-free divisions. 2. In the second (hybrid) scenario, there are $p$ not necessarily hungry players ($p$ is a prime) and a continuous measure $μ$ on $[0,1]$. The cake is cut into $2p-1$ pieces, the pieces are allocated to $p$ boxes (with some restrictions) and the players choose the boxes. Then there exists at least $\binom{2p-1}{p-1} \cdot 2^{2-p}$ envy-free divisions such that the measure $μ$ is equidistributed among the players.

Lower bounds on the number of envy-free divisions

TL;DR

This work investigates lower bounds on the number of envy-free divisions in two cake-cutting models. It leverages secretive/expelled variants of envy-free results and a configuration-space/test-map framework to quantify multiplicity: a universal lower bound of at least two envy-free divisions in the classical Woodall-Stormquist setting, and a significantly larger bound in a prime- measure-equipartition hybrid. The analysis identifies extremal preferences that realize exactly two divisions, showing the underlying incidence graph forms a single -cycle, and demonstrates how topological methods yield precise counting in the extended model. Together, these results illuminate how additional structure (secretive/expelled participants, measure constraints) enhances the multiplicity of envy-free divisions and showcases the power of topological combinatorics in fair-division problems.

Abstract

We analyze lower bounds for the number of envy-free divisions, in the classical Woodall-Stormquist setting and in a non-classical case, when envy-freeness is combined with the equipartition of a measure. 1. In the first scenario, there are hungry players, and the cake (that is, the segment ) is cut into pieces. Then there exist at least two different envy-free divisions. This bound is sharp: for each , we present an example of preferences such that there are exactly two envy-free divisions. 2. In the second (hybrid) scenario, there are not necessarily hungry players ( is a prime) and a continuous measure on . The cake is cut into pieces, the pieces are allocated to boxes (with some restrictions) and the players choose the boxes. Then there exists at least envy-free divisions such that the measure is equidistributed among the players.
Paper Structure (6 sections, 7 theorems, 20 equations, 2 figures)

This paper contains 6 sections, 7 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Envy-free division with either secretive or expelled players
  3. The central new results in the paper
  4. The existence of two different envy-free divisions
  5. Extremal preferences
  6. Simultaneous envy-free division and a measure equipartition

Key Result

Theorem 1.2

(Secretive player) WoodAsadaFrick17MeSuPaZi2022 There are $r-1$ players and the cake is divided into $r$ pieces. If the preferences of players $\{A_i^j\}_{i\in [r]}, j\in [r-1]$, satisfy classical KKM conditions, then there exists a cut $x\in \Delta^{r-1}$ and $r$ bijections $\sigma_i : [r-1] \right In simple words, there exists a cut such that whatever tile is taken by the secretive player $r$, t

Figures (2)

  • Figure 1: We depict the preferences of the first two players, the third ones are analogous.
  • Figure 2: Generating new perfect matchings from old.

Theorems & Definitions (18)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Example 2.2
  • Theorem 2.3
  • proof
  • Definition 3.1
  • Proposition 3.2
  • ...and 8 more