{"title":"Computational complexity of necessary envy-freeness","authors":"Haris Aziz , Ildikó Schlotter , Toby Walsh","doi":"10.1016/j.mathsocsci.2023.08.002","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the fundamental problem of fairly allocating indivisible items when agents have strict ordinal preferences over individual items. We focus on the well-studied fairness criterion of necessary envy-freeness. For a constant number of agents, the computational complexity of the deciding whether there exists an allocation that satisfies necessary envy-freeness has been open for several years. We settle this question by showing that the problem is <span><math><mi>NP</mi></math></span>-complete even for three agents. Considering that the problem is polynomial-time solvable for the case of two agents, we provide a clear understanding of the complexity of the problem with respect to the number of agents.</p></div>","PeriodicalId":51118,"journal":{"name":"Mathematical Social Sciences","volume":"127 ","pages":"Pages 86-98"},"PeriodicalIF":0.5000,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0165489623000677/pdfft?md5=bb17a09f7dd443dcd55b7c2ef6b93df3&pid=1-s2.0-S0165489623000677-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Social Sciences","FirstCategoryId":"96","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165489623000677","RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"ECONOMICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the fundamental problem of fairly allocating indivisible items when agents have strict ordinal preferences over individual items. We focus on the well-studied fairness criterion of necessary envy-freeness. For a constant number of agents, the computational complexity of the deciding whether there exists an allocation that satisfies necessary envy-freeness has been open for several years. We settle this question by showing that the problem is -complete even for three agents. Considering that the problem is polynomial-time solvable for the case of two agents, we provide a clear understanding of the complexity of the problem with respect to the number of agents.
期刊介绍：
The international, interdisciplinary journal Mathematical Social Sciences publishes original research articles, survey papers, short notes and book reviews. The journal emphasizes the unity of mathematical modelling in economics, psychology, political sciences, sociology and other social sciences.
Topics of particular interest include the fundamental aspects of choice, information, and preferences (decision science) and of interaction (game theory and economic theory), the measurement of utility, welfare and inequality, the formal theories of justice and implementation, voting rules, cooperative games, fair division, cost allocation, bargaining, matching, social networks, and evolutionary and other dynamics models.
Papers published by the journal are mathematically rigorous but no bounds, from above or from below, limits their technical level. All mathematical techniques may be used. The articles should be self-contained and readable by social scientists trained in mathematics.
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