[Paper Review] Don't cry to be the first! Symmetric fair division algorithms exist.

Guillaume Chèze|arXiv (Cornell University)|Apr 11, 2018

Game Theory and Voting Systems23 references3 citations

TL;DR

This paper presents symmetric fair division algorithms for cake cutting, ensuring equal treatment regardless of player order. It introduces a symmetric, envy-free algorithm derived from existing envy-free methods and a proportional, symmetric algorithm with O(n³) complexity using Kuhn's algorithm, advancing fairness through a new 'aristotelian fair division' 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.

Motivation & Objective

• To develop symmetric fair division algorithms where player order does not affect outcome fairness.
• To extend existing envy-free algorithms into symmetric variants that preserve fairness regardless of input order.
• To design a proportional and symmetric fair division algorithm with efficient O(n³) complexity in the Robertson-Webb model.
• To formalize and introduce the concept of 'aristotelian fair division' as a fairness principle based on equal shares for equal individuals.
• To critically assess the limitations and assumptions of the Robertson-Webb model in the context of symmetric fair division.

Proposed method

• Adapts existing envy-free cake-cutting algorithms by modifying their execution order to ensure symmetry across players.
• Employs Kuhn's algorithm as a foundation for constructing a proportional and symmetric division with O(n³) query complexity in the Robertson-Webb model.
• Introduces a new fairness criterion—aristotelian fair division—defined as giving equal shares to individuals of equal standing.
• Analyzes symmetry by ensuring that permuting player roles does not alter the expected value each player receives.
• Uses theoretical analysis to prove that the proposed symmetric envy-free algorithm maintains both symmetry and envy-freeness.
• Evaluates the Robertson-Webb model's suitability for symmetric fair division, highlighting potential limitations in modeling symmetric protocols.

Experimental results

Research questions

• RQ1 Can envy-free cake-cutting algorithms be modified to ensure symmetry, so that player order does not affect individual outcomes?
• RQ2 What is the minimal complexity required to achieve both proportionality and symmetry in fair division algorithms?
• RQ3 How can Aristotle’s principle of equal shares for equal people be formalized as a new fairness criterion in cake cutting?
• RQ4 In what ways does the Robertson-Webb model constrain or support the design of symmetric fair division protocols?
• RQ5 Are there inherent trade-offs between symmetry, fairness (envy-freeness/proportionality), and computational efficiency in cake cutting?

Key findings

• A symmetric and envy-free fair division algorithm is constructed by modifying an existing envy-free algorithm to be invariant under player permutation.
• A proportional and symmetric fair division algorithm is achieved with O(n³) query complexity in the Robertson-Webb model, using Kuhn’s algorithm as a basis.
• The paper introduces 'aristotelian fair division' as a new fairness principle, formalizing the idea that equal individuals should receive equal shares.
• The study reveals that symmetry in fair division requires careful protocol design, as standard algorithms often implicitly favor earlier players.
• The analysis suggests limitations in the Robertson-Webb model for expressing symmetric protocols, particularly in capturing the full symmetry of player roles.
• The proposed symmetric algorithms maintain fairness properties (envy-freeness and proportionality) while ensuring that no player gains an advantage from their position in the protocol sequence.