An Improved Envy-Free Cake Cutting Protocol for Four Agents

TL;DR

This paper presents a simplified and more efficient envy-free cake-cutting protocol for four agents, reducing query complexity significantly compared to previous algorithms, and making the process easier to understand.

Contribution

It introduces an improved algorithm for four-agent envy-free cake cutting that reduces query complexity by a factor of 3.4 and simplifies the existing complex procedures.

Findings

Query complexity reduced by a factor of 3.4

Simpler and more understandable algorithm

Maintains envy-freeness for four agents

Abstract

We consider the classic cake-cutting problem of producing envy-free allocations, restricted to the case of four agents. The problem asks for a partition of the cake to four agents, so that every agent finds her piece at least as valuable as every other agent's piece. The problem has had an interesting history so far. Although the case of three agents is solvable with less than 15 queries, for four agents no bounded procedure was known until the recent breakthroughs of Aziz and Mackenzie (STOC 2016, FOCS 2016). The main drawback of these new algorithms, however, is that they are quite complicated and with a very high query complexity. With four agents, the number of queries required is close to 600. In this work we provide an improved algorithm for four agents, which reduces the current complexity by a factor of 3.4. Our algorithm builds on the approach of Aziz and Mackenzie (STOC 2016) by incorporating new insights and simplifying several steps. Overall, this yields an easier to grasp procedure with lower complexity.