### 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 us 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.

Original language | English |
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Pages (from-to) | 1-5 |

Number of pages | 5 |

Journal | Journal of Mathematical Economics |

Volume | 78 |

Early online date | 26 Jul 2018 |

DOIs | |

Publication status | Published - 1 Oct 2018 |

### Keywords

- Brouwer
- Fixed point
- Indivisible goods
- KKM lemma
- Top trading cycles

## Profiles

## Cite this

Petri, H., & Voorneveld, M. (2018). No Bullying! A playful proof of Brouwer's fixed-point theorem.

*Journal of Mathematical Economics*,*78*, 1-5. https://doi.org/10.1016/j.jmateco.2018.07.001