Fair Division of Multi-layered Cakes
Mohammad Azharuddin Sanpui
TL;DR
The paper addresses fair division of multi-layered cakes under contiguity and feasibility constraints by introducing a pair-of-knives computational model inspired by moving-knife procedures. It proves exact multi-allocation exists for two agents and two layers and shows proportional, feasible, contiguous allocations for three layers and for $m=2^a\cdot 3$ layers with $n\ge m$, providing a constructive algorithm ($\mathcal{A}l$) based on majority switching points. The approach extends prior work (Hosseini 2020; Igarashi & Meunier) and yields practical procedures to compute fair allocations in layered settings. The results advance the theory of fair division in multi-resource settings and open questions on envy-free allocations, arbitrary layer counts, and allocation efficiency.
Abstract
We consider multi-layered cake cutting in order to fairly allocate numerous divisible resources (layers of cake) among a group of agents under two constraints: contiguity and feasibility. We first introduce a new computational model in a multi-layered cake named ``a pair of knives''. Then, we show the existence of an exact multi-allocation for two agents and two layers using the new computational model. We demonstrate the computation procedure of a feasible and contiguous proportional multi-allocation over a three-layered cake for more than three agents. Finally, we develop a technique for computing proportional allocations for any number $n\geq 2^a3$ of agents and $2^a3$ layers, where $a$ is any positive integer.