Fair Division of Multi-layered Cakes

TL;DR

This paper introduces a new model for fairly dividing multi-layered resources among agents, ensuring contiguity and feasibility, with algorithms for exact and proportional allocations for various cases.

Contribution

It presents a novel computational model called 'a pair of knives' and algorithms for exact and proportional multi-layered cake allocations under new constraints.

Findings

Existence of an exact multi-allocation for two agents and two layers.

Feasible and contiguous proportional allocation algorithm for three or more agents and three layers.

A technique for proportional allocations for any number of agents and layers of specific sizes.

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.