Envy-free division in the presence of a dragon
Following a novel approach, where the emphasis is on configuration spaces and equivariant topology, we prove several results addressing the envy-free division problem in the presence of an unpredictable (secretive, non-cooperative) player, called the dragon . There are two basic scenarios. (1) There...
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| Published in | Journal of fixed point theory and applications Vol. 24; no. 4 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.12.2022
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 1661-7738 1661-7746 |
| DOI | 10.1007/s11784-022-00997-y |
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| Abstract | Following a novel approach, where the emphasis is on configuration spaces and equivariant topology, we prove several results addressing the
envy-free division problem
in the presence of an unpredictable (secretive, non-cooperative) player, called
the dragon
. There are two basic scenarios. (1) There are
r
-
1
players and a dragon. Once the “cake” is divided into
r
parts, the dragon makes his choice and grabs one of the pieces. After that, the players should be able to share the remaining pieces in an envy-free fashion. (2) There are
r
+
1
players who divide the cake into
r
pieces. A ferocious dragon comes and swallows one of the players. The players need to cut the cake in advance in such a way that no matter who is the unlucky player swallowed by the dragon, the remaining players can share the tiles in an envy-free manner. We emphasize that in both settings, some players may prefer to choose degenerate pieces of the cake. Moreover, the players construct in advance both a cut of the cake and a
decision tree
, allowing them to minimize the uncertainty of what pieces can be given to each of the players. |
|---|---|
| AbstractList | Following a novel approach, where the emphasis is on configuration spaces and equivariant topology, we prove several results addressing the
envy-free division problem
in the presence of an unpredictable (secretive, non-cooperative) player, called
the dragon
. There are two basic scenarios. (1) There are
r
-
1
players and a dragon. Once the “cake” is divided into
r
parts, the dragon makes his choice and grabs one of the pieces. After that, the players should be able to share the remaining pieces in an envy-free fashion. (2) There are
r
+
1
players who divide the cake into
r
pieces. A ferocious dragon comes and swallows one of the players. The players need to cut the cake in advance in such a way that no matter who is the unlucky player swallowed by the dragon, the remaining players can share the tiles in an envy-free manner. We emphasize that in both settings, some players may prefer to choose degenerate pieces of the cake. Moreover, the players construct in advance both a cut of the cake and a
decision tree
, allowing them to minimize the uncertainty of what pieces can be given to each of the players. |
| ArticleNumber | 81 |
| Author | Panina, Gaiane Živaljević, Rade |
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| Cites_doi | 10.1016/j.jcta.2011.04.007 10.1080/00029890.1980.11995109 10.1016/0097-3165(92)90028-S 10.1016/0022-247X(80)90225-5 10.1093/imrn/rnn153 10.1137/17M1116210 10.1007/BF01769865 10.1112/jlms/49.1.25 10.1007/BF02308547 10.1137/18M1192548 10.1007/s11856-019-1939-6 |
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| Copyright | The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Keywords | 91B32 KKM theorem chessboard complexes 52C99 configuration space/test map scheme dragon marriage theorem Envy-free division 05E45 55M20 |
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| References | CR2 Živaljević, Vrećica (CR18) 1992; 61 CR3 Woodall (CR16) 1980; 78 CR6 Vrećica, Živaljević (CR15) 2011; 118 CR8 CR7 Björner, Lovász, Vrećica, Živaljević (CR4) 1994; 49 CR17 CR9 Asada, Frick, Pisharody, Polevy, Stoner, Tsang, Wellner (CR1) 2018; 32 CR12 CR10 Postnikov (CR11) 2009; 2009 Stromquist (CR13) 1980; 87 Volovikov (CR14) 1996; 59 Gale (CR5) 1984; 13 997_CR2 D Gale (997_CR5) 1984; 13 997_CR12 RT Živaljević (997_CR18) 1992; 61 997_CR6 997_CR3 A Postnikov (997_CR11) 2009; 2009 997_CR17 W Stromquist (997_CR13) 1980; 87 997_CR10 AY Volovikov (997_CR14) 1996; 59 DR Woodall (997_CR16) 1980; 78 S Vrećica (997_CR15) 2011; 118 997_CR9 A Björner (997_CR4) 1994; 49 997_CR7 997_CR8 M Asada (997_CR1) 2018; 32 |
| References_xml | – volume: 118 start-page: 2157 year: 2011 end-page: 2166 ident: CR15 article-title: Chessboard complexes indomitable publication-title: J. Combin. Theory Ser. A doi: 10.1016/j.jcta.2011.04.007 – volume: 87 start-page: 640 issue: 8 year: 1980 end-page: 644 ident: CR13 article-title: How to cut a cake fairly publication-title: Am. Math. Mon. doi: 10.1080/00029890.1980.11995109 – volume: 61 start-page: 309 year: 1992 end-page: 318 ident: CR18 article-title: The colored Tverberg’s problem and complexes of injective functions publication-title: J. Combin. Theory Ser. A doi: 10.1016/0097-3165(92)90028-S – ident: CR3 – ident: CR2 – volume: 78 start-page: 233 issue: 1 year: 1980 end-page: 247 ident: CR16 article-title: Dividing a cake fairly publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(80)90225-5 – ident: CR12 – ident: CR17 – ident: CR10 – volume: 2009 start-page: 1026 issue: 6 year: 2009 end-page: 1106 ident: CR11 article-title: Permutohedra, Associahedra, and Beyond publication-title: Int. Math. Res. Not. doi: 10.1093/imrn/rnn153 – ident: CR9 – volume: 32 start-page: 591 issue: 1 year: 2018 end-page: 610 ident: CR1 article-title: Fair Division and Generalizations of Sperner– and KKM–type Results publication-title: SIAM J. Discrete Math. doi: 10.1137/17M1116210 – volume: 13 start-page: 61 issue: 1 year: 1984 end-page: 64 ident: CR5 article-title: Equilibrium in a discrete exchange economy with money publication-title: Int. J. Game Theory doi: 10.1007/BF01769865 – ident: CR6 – ident: CR7 – ident: CR8 – volume: 49 start-page: 25 year: 1994 end-page: 39 ident: CR4 article-title: Chessboard complexes and matching complexes publication-title: J. Lond. Math. Soc. (2) doi: 10.1112/jlms/49.1.25 – volume: 59 start-page: 324 year: 1996 end-page: 32 ident: CR14 article-title: On a topological generalization of the Tverberg theorem publication-title: Math. Notes doi: 10.1007/BF02308547 – volume: 32 start-page: 591 issue: 1 year: 2018 ident: 997_CR1 publication-title: SIAM J. Discrete Math. doi: 10.1137/17M1116210 – volume: 49 start-page: 25 year: 1994 ident: 997_CR4 publication-title: J. Lond. Math. Soc. (2) doi: 10.1112/jlms/49.1.25 – volume: 87 start-page: 640 issue: 8 year: 1980 ident: 997_CR13 publication-title: Am. Math. Mon. doi: 10.1080/00029890.1980.11995109 – ident: 997_CR3 – ident: 997_CR2 – volume: 118 start-page: 2157 year: 2011 ident: 997_CR15 publication-title: J. Combin. Theory Ser. A doi: 10.1016/j.jcta.2011.04.007 – ident: 997_CR6 – volume: 59 start-page: 324 year: 1996 ident: 997_CR14 publication-title: Math. Notes doi: 10.1007/BF02308547 – ident: 997_CR7 – ident: 997_CR8 doi: 10.1137/18M1192548 – volume: 78 start-page: 233 issue: 1 year: 1980 ident: 997_CR16 publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(80)90225-5 – ident: 997_CR9 doi: 10.1007/s11856-019-1939-6 – volume: 2009 start-page: 1026 issue: 6 year: 2009 ident: 997_CR11 publication-title: Int. Math. Res. Not. doi: 10.1093/imrn/rnn153 – volume: 61 start-page: 309 year: 1992 ident: 997_CR18 publication-title: J. Combin. Theory Ser. A doi: 10.1016/0097-3165(92)90028-S – ident: 997_CR12 – volume: 13 start-page: 61 issue: 1 year: 1984 ident: 997_CR5 publication-title: Int. J. Game Theory doi: 10.1007/BF01769865 – ident: 997_CR10 – ident: 997_CR17 |
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envy-free division... |
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| Title | Envy-free division in the presence of a dragon |
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