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

Henrik Petri, Mark Voorneveld

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2 Citations (SciVal)


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 languageEnglish
Pages (from-to)1-5
Number of pages5
JournalJournal of Mathematical Economics
Early online date26 Jul 2018
Publication statusPublished - 1 Oct 2018


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


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