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

Author

Listed:
  • Petri, Henrik

    (Department of Finance)

  • Voorneveld, Mark

    (Dept. 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.

Suggested Citation

  • Petri, Henrik & Voorneveld, Mark, 2016. "No bullying! A playful proof of Brouwer's fixed-point theorem," SSE Working Paper Series in Economics 2016:3, Stockholm School of Economics, revised 20 Jun 2017.
    • Handle: RePEc:hhs:hastec:2016_003
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    References listed on IDEAS

    as
    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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    More about this item

    Keywords

    Brouwer; fixed point; indivisible goods; KKM lemma;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies

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