{"title":"大卫-施迈德勒猜想的解答","authors":"Alain Chateauneuf, Freddy Delbaen, Caroline Ventura","doi":"10.1007/s40505-024-00271-z","DOIUrl":null,"url":null,"abstract":"<p>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 <span>\\(({\\mathbb {N}},\\ 2^{{\\mathbb {N}}})\\)</span> 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.\n</p>","PeriodicalId":40852,"journal":{"name":"Economic Theory Bulletin","volume":"45 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A solution to a conjecture of David Schmeidler\",\"authors\":\"Alain Chateauneuf, Freddy Delbaen, Caroline Ventura\",\"doi\":\"10.1007/s40505-024-00271-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>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 <span>\\\\(({\\\\mathbb {N}},\\\\ 2^{{\\\\mathbb {N}}})\\\\)</span> 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.\\n</p>\",\"PeriodicalId\":40852,\"journal\":{\"name\":\"Economic Theory Bulletin\",\"volume\":\"45 1\",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-07-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Economic Theory Bulletin\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s40505-024-00271-z\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"ECONOMICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Economic Theory Bulletin","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s40505-024-00271-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"ECONOMICS","Score":null,"Total":0}
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.
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.
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