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arXiv · 1807.10906

No bullying! A playful proof of Brouwer's fixed-point theorem

Henrik Petri·Mark Voorneveld
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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 $n$-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 $S$ of children can redistribute their toys among themselves in such a way that all members of $S$ get their favorite toy from $S$, but they cannot bully anyone.

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Henrik Petri, Mark Voorneveld. 2018-07-28. No bullying! A playful proof of Brouwer's fixed-point theorem. https://doi.org/10.1016/j.jmateco.2018.07.001

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