How to cut a cake with a Gram matrix
In this article we study a problem of fair division. In particular we study a notion introduced by J. Barbanel that generalizes super envy-free fair division. We call this notion hyper envy-free. We give a new proof for the existence of such fair divisions. Our approach relies on classical linear al...
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| Veröffentlicht in: | Linear algebra and its applications 2019-01, Vol.560, p.114-132 |
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| container_title | Linear algebra and its applications |
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| creator | Chèze, Guillaume Amodei, Luca |
| description | In this article we study a problem of fair division. In particular we study a notion introduced by J. Barbanel that generalizes super envy-free fair division. We call this notion hyper envy-free. We give a new proof for the existence of such fair divisions. Our approach relies on classical linear algebra tools and allows us to give an explicit bound for this kind of fair division.
Furthermore, we also give a theoretical answer to an open problem posed by Barbanel in 1996. Roughly speaking, this question is: how can we decide if there exists a fair division satisfying some inequality constraints?
Furthermore, when all the measures are given with piecewise constant density functions then we show how to construct effectively such a fair division. |
| doi_str_mv | 10.1016/j.laa.2018.09.028 |
| format | Article |
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Furthermore, we also give a theoretical answer to an open problem posed by Barbanel in 1996. Roughly speaking, this question is: how can we decide if there exists a fair division satisfying some inequality constraints?
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Furthermore, we also give a theoretical answer to an open problem posed by Barbanel in 1996. Roughly speaking, this question is: how can we decide if there exists a fair division satisfying some inequality constraints?
Furthermore, when all the measures are given with piecewise constant density functions then we show how to construct effectively such a fair division.</description><subject>Algorithms</subject><subject>Cake cutting algorithm</subject><subject>Computer Science</subject><subject>Fair division</subject><subject>Gram matrix</subject><subject>Linear algebra</subject><subject>Mathematical problems</subject><subject>Matrix</subject><subject>Multiagent Systems</subject><subject>Proof theory</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9kE9Lw0AQxRdRsFY_gLeAXjwkzuyfJIunItoKBS96XibbDd3YNnWTtvrt3RLx6Gkew-89Zh5j1wgZAub3TbYiyjhgmYHOgJcnbIRlIVIsVX7KRgBcpqLQ6pxddF0DALIAPmK3s_aQ9G1id31CiaUPlxx8v4x6GmidrKkP_uuSndW06tzV7xyz9-ent8dZOn-dvjxO5qkVSvapVqXV3CJUqKmuyUlaCOc4CuGUqqkUwtYWkdvKKiTNsaqcKkjmVOVRijG7G3KXtDLb4NcUvk1L3swmc3PcASpVaJnvMbI3A7sN7efOdb1p2l3YxPMMRyULRCnzSOFA2dB2XXD1XyyCORZnGhOLM8fiDGgTi4ueh8Hj4qt774LprHcb6xY-ONubRev_cf8Az9py9w</recordid><startdate>20190101</startdate><enddate>20190101</enddate><creator>Chèze, Guillaume</creator><creator>Amodei, Luca</creator><general>Elsevier Inc</general><general>American Elsevier Company, Inc</general><general>Elsevier</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0001-9657-4209</orcidid></search><sort><creationdate>20190101</creationdate><title>How to cut a cake with a Gram matrix</title><author>Chèze, Guillaume ; Amodei, Luca</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c354t-958c92c10b19affae4ad3ee2133e55fa833cfc112cbc51a921bbe57a46ab6bbe3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Algorithms</topic><topic>Cake cutting algorithm</topic><topic>Computer Science</topic><topic>Fair division</topic><topic>Gram matrix</topic><topic>Linear algebra</topic><topic>Mathematical problems</topic><topic>Matrix</topic><topic>Multiagent Systems</topic><topic>Proof theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Chèze, Guillaume</creatorcontrib><creatorcontrib>Amodei, Luca</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Linear algebra and its applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Chèze, Guillaume</au><au>Amodei, Luca</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>How to cut a cake with a Gram matrix</atitle><jtitle>Linear algebra and its applications</jtitle><date>2019-01-01</date><risdate>2019</risdate><volume>560</volume><spage>114</spage><epage>132</epage><pages>114-132</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>In this article we study a problem of fair division. In particular we study a notion introduced by J. Barbanel that generalizes super envy-free fair division. We call this notion hyper envy-free. We give a new proof for the existence of such fair divisions. Our approach relies on classical linear algebra tools and allows us to give an explicit bound for this kind of fair division.
Furthermore, we also give a theoretical answer to an open problem posed by Barbanel in 1996. Roughly speaking, this question is: how can we decide if there exists a fair division satisfying some inequality constraints?
Furthermore, when all the measures are given with piecewise constant density functions then we show how to construct effectively such a fair division.</abstract><cop>Amsterdam</cop><pub>Elsevier Inc</pub><doi>10.1016/j.laa.2018.09.028</doi><tpages>19</tpages><orcidid>https://orcid.org/0000-0001-9657-4209</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
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| ispartof | Linear algebra and its applications, 2019-01, Vol.560, p.114-132 |
| issn | 0024-3795 1873-1856 |
| language | eng |
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| source | ScienceDirect Journals (5 years ago - present); EZB-FREE-00999 freely available EZB journals |
| subjects | Algorithms Cake cutting algorithm Computer Science Fair division Gram matrix Linear algebra Mathematical problems Matrix Multiagent Systems Proof theory |
| title | How to cut a cake with a Gram matrix |
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