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

Henrik Petri () and Mark Voorneveld
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Henrik Petri: Department of Finance, Postal: Stockholm School of Economics, P.O. Box 6501, SE-113 83 Stockholm, Sweden

No 2016:3, SSE Working Paper Series in Economics from Stockholm School of Economics

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 (search for similar items in EconPapers)
JEL-codes: C62 C63 C69 D51 (search for similar items in EconPapers)
Pages: 9 pages
Date: 2016-04-16, Revised 2017-06-20
New Economics Papers: this item is included in nep-mic
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