A solution to a conjecture of David Schmeidler

IF 0.4 Q4 ECONOMICS

Economic Theory Bulletin Pub Date : 2024-07-26 DOI:10.1007/s40505-024-00271-z

Alain Chateauneuf, Freddy Delbaen, Caroline Ventura

引用次数: 0

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 \(({\mathbb {N}},\ 2^{{\mathbb {N}}})\) 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.

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大卫-施迈德勒猜想的解答

本文旨在推翻合作博弈理论中一个古老而有趣的猜想，这个猜想最初是作为一个合理的猜想出现在施迈德勒的开创性著作中（《数学分析应用》40:214-225，1972 年）：在空集处连续的精确容量在其核心处具有可数可加概率。在本文中，我们证明了这一猜想在一般情况下并不成立。更准确地说，我们给出了一大类拓扑空间，对于这些空间，即使使用容量是凸的这一更强假设，猜想也是不成立的。然而，在 \(({\mathbb {N}},\ 2^{\mathbb {N}})\) 上，凸容量的猜想成立，这一点很容易证明。我们要证明的是，在它的原始形式下，也就是对于精确容量，它是不成立的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Economic Theory Bulletin ECONOMICS-

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期刊介绍： The purpose of Economic Theory Bulletin is to provide an outlet for research in all areas of Economics based on rigorous theoretical reasoning. The Economic Theory Bulletin together with Economic Theory are the official journals of the Society for the Advancement of Economic Theory. The Economic Theory Bulletin is intended to publish: 1. Short papers/notes of substantial interest. Content is subject to the same standards as Economic Theory: research in all areas of economics based on rigorous theoretical reasoning and on topics in mathematics that are supported by the analysis of economic problems. Published articles contribute to the understanding and solution of substantive economic problems. Theory papers with the substance and style for other journals that specialize in short papers are welcomed. Corollaries of already known results in the literature are not appropriate for publication. 2. Survey papers that clearly picture the basic ideas at work in the area, the essential technical apparatus that is used and the central questions that remain open.