A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents

TL;DR

This paper presents the first discrete and bounded protocol for envy-free cake cutting applicable to any number of agents, resolving a long-standing open problem in fair division.

Contribution

It introduces a novel discrete, bounded envy-free cake cutting protocol for any number of agents, with explicit query complexity bounds.

Findings

The protocol guarantees envy-free allocations for any number of agents.

The maximum number of queries for the protocol is extremely high but finite.

Partial allocations achieving proportionality and connectedness can be found with bounded queries.

Abstract

We consider the well-studied cake cutting problem in which the goal is to find an envy-free allocation based on queries from $n$ agents. The problem has received attention in computer science, mathematics, and economics. It has been a major open problem whether there exists a discrete and bounded envy-free protocol. We resolve the problem by proposing a discrete and bounded envy-free protocol for any number of agents. The maximum number of queries required by the protocol is $n^{n^{n^{n^{n^n}}}}$. We additionally show that even if we do not run our protocol to completion, it can find in at most $n^3{(n^2)}^n$ queries a partial allocation of the cake that achieves proportionality (each agent gets at least $1/n$ of the value of the whole cake) and envy-freeness. Finally we show that an envy-free partial allocation can be computed in at most $n^3{(n^2)}^n$ queries such that each agent gets a connected piece that gives the agent at least $1/(3n)$ of the value of the whole cake.