Discrete Envy-free Division of Necklaces and Maps

TL;DR

This paper investigates the problem of dividing discrete necklaces and grids among players in an envy-free manner, highlighting limitations and specific cases where envy-free divisions are achievable, with applications to districting.

Contribution

It introduces the discrete envy-free division problem, analyzes its limitations, and provides new results for small cases and a 2D grid scenario, extending classical cake-cutting theory.

Findings

Usually cannot guarantee envy-free division in discrete settings

Achieves envy-free division in small cases under certain constraints

Provides a 2D grid division model with envy-free properties

Abstract

We study the discrete variation of the classical cake-cutting problem where n players divide a 1-dimensional cake with exactly (n-1) cuts, replacing the continuous, infinitely divisible "cake" with a necklace of discrete, indivisible "beads." We focus specifically on envy-free divisions, exploring different constraints on player-preferences. We show we usually cannot guarantee an envy-free division and consider situations where we can obtain an envy-free division for relatively small. We also prove a 2-dimensional result with a grid of indivisible objects. This may be viewed as a way to divide a state with indivisible districts among a set of constituents, producing somewhat gerrymandered regions that form an envy-free division of the state.