# Cake cutting: Explicit examples for impossibility results

## TL;DR

This paper introduces a new computational model for cake cutting that combines query-based and algebraic operations, demonstrating explicit cases where fair division algorithms cannot achieve equitable or welfare-maximizing solutions.

## Contribution

It proposes a novel model of computation for cake cutting and proves impossibility results using Galois theory, highlighting limitations of existing algorithms.

## Key findings

- Existence of explicit measure pairs with no equitable connected division
- Existence of explicit measures with no welfare-maximizing division
- Impossibility results proven using Galois theory

## Abstract

In this article we suggest a model of computation for the cake cutting problem. In this model the mediator can ask the same queries as in the Robertson-Webb model but he or she can only perform algebraic operations as in the Blum-Shub-Smale model. All existing algorithms described in the Robertson-Webb model can be described in this new model.We show that in this model there exist explicit couples of measures for which no algorithm outputs an equitable fair division with connected parts.We also show that there exist explicit set of measures for which no algorithm in this model outputs a fair division which maximizes the utilitarian social welfare function.The main tool of our approach is Galois theory.