Don't cry to be the first!Symmetric fair division exist

TL;DR

This paper develops symmetric, envy-free, and proportional fair division algorithms for cake cutting, introduces aristotelian fair division, and discusses computational complexity within the Robertson-Webb model.

Contribution

It presents new symmetric and envy-free division algorithms, a proportional symmetric algorithm with O(n^3) complexity, and introduces the concept of aristotelian fair division.

Findings

A symmetric and envy-free division algorithm derived from existing envy-free algorithms.

A proportional, symmetric division algorithm with O(n^3) complexity.

Introduction of aristotelian fair division based on Aristotle's principle.

Abstract

In this article we study a cake cutting problem. More precisely, we study symmetric fair division algorithms, that is to say we study algorithms where the order of the players do not influence the value obtained by each player. In the first part of the article, we give a symmetric and envy-free fair division algorithm. More precisely, we show how to get a symmetric and envy-free fair division algorithm from an envy-free division algorithm. In the second part, we give a proportional and symmetric fair division algorithm with a complexity in O(n 3) in the Robertson-Webb model of complexity. This algorithm is based on Kuhn's algorithm. Furthermore, our study has led us to introduce a new notion: aristotelian fair division. This notion is an interpretation of Aristotle's principle: give equal shares to equal people. We conclude this article with a discussion and some questions about the Robertson-Webb model of computation.