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

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  • Petri, Henrik
  • Voorneveld, Mark

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 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 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.

Suggested Citation

  • Petri, Henrik & Voorneveld, Mark, 2018. "No bullying! A playful proof of Brouwer’s fixed-point theorem," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 1-5.
    • Handle: RePEc:eee:mateco:v:78:y:2018:i:c:p:1-5
    DOI: 10.1016/j.jmateco.2018.07.001
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    References listed on IDEAS

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    1. Shapley, Lloyd & Scarf, Herbert, 1974. "On cores and indivisibility," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 23-37, March.
    2. Voorneveld, Mark, 2017. "An elementary direct proof that the Knaster–Kuratowski–Mazurkiewicz lemma implies Sperner’s lemma," Economics Letters, Elsevier, vol. 158(C), pages 1-2.
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    Cited by:

    1. Данилов В.И., 2019. "Лемма Скарфа И Теорема Брауэра," Журнал Экономика и математические методы (ЭММ), Центральный Экономико-Математический Институт (ЦЭМИ), vol. 55(3), pages 141-143, июль.

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