No bullying! A playful proof of Brouwer's fixed-point theorem
Abstract
We give an elementary proof of Brouwer's fixed-point theorem. The only mathematical prerequisite is a version of the Bolzano-Weierstrass theorem: a sequence in a compact subset of -dimensional Euclidean space has a convergent subsequence with a limit in that set. Our main tool is a `no-bullying' lemma for agents with preferences over indivisible goods. What does this lemma claim? Consider a finite number of children, each with a single indivisible good (a toy) and preferences over those toys. Let's say that a group of children, possibly after exchanging toys, could bully some poor kid if all group members find their own current toy better than the toy of this victim. The no-bullying lemma asserts that some group of children can redistribute their toys among themselves in such a way that all members of get their favorite toy from , but they cannot bully anyone.
Keywords
Cite
@article{arxiv.1807.10906,
title = {No bullying! A playful proof of Brouwer's fixed-point theorem},
author = {Henrik Petri and Mark Voorneveld},
journal= {arXiv preprint arXiv:1807.10906},
year = {2018}
}