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A solution to a conjecture of David Schmeidler

Author

Listed:
  • Alain Chateauneuf

    (University of Paris 1)

  • Freddy Delbaen

    (ETH Zürich
    Universität Zürich)

  • Caroline Ventura

    (University of Paris 1)

Abstract

The purpose of this paper is to disprove an old and interesting conjecture in the theory of cooperative games initially presented as a reasonable conjecture in the seminal work of Schmeidler (J Math Anal Appl 40:214–225, 1972): An exact capacity continuous at the empty set has a countably additive probability in its core. In this paper, we show that this conjecture is not true in general. More precisely, we give a large class of topological spaces for which the conjecture fails, even with the stronger assumption that the capacity is convex. However, on $$({\mathbb {N}},\ 2^{{\mathbb {N}}})$$ ( N , 2 N ) the conjecture holds for convex capacities, as it is easy to show. We prove that in its original form, that is for exact capacities, it fails.

Suggested Citation

  • Alain Chateauneuf & Freddy Delbaen & Caroline Ventura, 2024. "A solution to a conjecture of David Schmeidler," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 12(2), pages 183-189, December.
    • Handle: RePEc:spr:etbull:v:12:y:2024:i:2:d:10.1007_s40505-024-00271-z
    DOI: 10.1007/s40505-024-00271-z
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    Keywords

    A conjecture of David Schmeidler; Core of continuous exact games; Countably additive probabilities; Convex games;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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