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Stockholm School of Economics SSE Working Paper Series in Economics, Stockholm School of Economics

No 2016:3:
No bullying! A playful proof of Brouwer's fixed-point theorem

Henrik Petri () and Mark Voorneveld ()

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.

Keywords: Brouwer; fixed point; indivisible goods; KKM lemma; (follow links to similar papers)

JEL-Codes: C62; C63; C69; D51; (follow links to similar papers)

9 pages, April 16, 2016, Revised June 20, 2017

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