Combinatorial games: tic-tac-toe theory
Traditional game theory has been successful at developing strategy in games of incomplete information: when one player knows something that the other does not. But it has little to say about games of complete information, for example, tic-tac-toe, solitaire and hex. The main challenge of combinatori...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2008
|
| Schriftenreihe: | Encyclopedia of mathematics and its applications
volume 114 |
| Schlagworte: | |
| Online-Zugang: | BSB01 FHN01 URL des Erstveröffentlichers |
| Zusammenfassung: | Traditional game theory has been successful at developing strategy in games of incomplete information: when one player knows something that the other does not. But it has little to say about games of complete information, for example, tic-tac-toe, solitaire and hex. The main challenge of combinatorial game theory is to handle combinatorial chaos, where brute force study is impractical. In this comprehensive volume, József Beck shows readers how to escape from the combinatorial chaos via the fake probabilistic method, a game-theoretic adaptation of the probabilistic method in combinatorics. Using this, the author is able to determine the exact results about infinite classes of many games, leading to the discovery of some striking new duality principles. Available for the first time in paperback, it includes a new appendix to address the results that have appeared since the book's original publication |
| Beschreibung: | Title from publisher's bibliographic system (viewed on 05 Oct 2015) |
| Beschreibung: | 1 online resource (xiv, 732 pages) |
| ISBN: | 9780511735202 |
| DOI: | 10.1017/CBO9780511735202 |
Internformat
MARC
| LEADER | 00000nmm a2200000zcb4500 | ||
|---|---|---|---|
| 001 | BV043942082 | ||
| 003 | DE-604 | ||
| 005 | 00000000000000.0 | ||
| 007 | cr|uuu---uuuuu | ||
| 008 | 161206s2008 |||| o||u| ||||||eng d | ||
| 020 | |a 9780511735202 |c Online |9 978-0-511-73520-2 | ||
| 024 | 7 | |a 10.1017/CBO9780511735202 |2 doi | |
| 035 | |a (ZDB-20-CBO)CR9780511735202 | ||
| 035 | |a (OCoLC)850520953 | ||
| 035 | |a (DE-599)BVBBV043942082 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-12 |a DE-92 | ||
| 082 | 0 | |a 519.3 |2 22 | |
| 084 | |a QH 431 |0 (DE-625)141582: |2 rvk | ||
| 084 | |a SK 860 |0 (DE-625)143264: |2 rvk | ||
| 100 | 1 | |a Beck, József |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Combinatorial games |b tic-tac-toe theory |c József Beck |
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2008 | |
| 300 | |a 1 online resource (xiv, 732 pages) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b c |2 rdamedia | ||
| 338 | |b cr |2 rdacarrier | ||
| 490 | 0 | |a Encyclopedia of mathematics and its applications |v volume 114 | |
| 500 | |a Title from publisher's bibliographic system (viewed on 05 Oct 2015) | ||
| 505 | 8 | |a pt. A. Weak win and strong draw -- ch. I. Win vs. weak win -- Illustration : every finite point set in the plane is a weak winner -- Analyzing the proof of theorem 1.1 -- Examples : tic-tac-toe games -- More examples : tic-tac-toe like games -- Games on hypergraphs, and the combinatorial chaos -- ch. II. The main result : exact solutions for infinite classes of games -- Ramsey theory and clique games -- Arithmetic progressions -- Two-dimensional arithmetic progressions -- Explaining the exact solutions : a meta-conjecture -- Potentials and the Erdős-Selfridge theorem -- Local vs. global -- Ramsey theory and hypercube tic-tac-toe -- pt. B. Basic potential technique : game-theoretic first and second moments -- ch. III. Simple applications -- Easy building via theorem 1.2 -- Games beyond Ramsey theory -- A generalization of Kaplansky's game -- ch. IV. Games and randomness -- Discrepancy games and the variance -- Biased discrepancy games : when the extension from fair to biased works! -- A simple illustration of "randomness" (I) -- A simple illustration of "randomness" (II) -- Another illustration of "randomness" in games | |
| 505 | 8 | |a pt. C. Advanced weak win : game-theoretic higher moment -- ch. V. Self-improving potentials -- Motivating the probabilistic approach -- Game-theoretic second moment : application to the picker-choose game -- Weak win in the lattice games -- Game-theoretic higher moments -- Exact solution of the clique game (I) -- More applications -- Who-scores-more games -- ch. VI. What is the biased meta-conjecture, and why is it so difficult? -- Discrepancy games (I) -- Discrepancy games (II) -- Biased games (I) : biased meta-conjecture -- Biased games (II) : sacrificing the probabilistic intuition to force negativity -- Biased games (III) : sporadic results -- Biased games (IV) : more sporadic results -- pt. D. Advanced strong draw : game-theoretic independence -- ch. VII. BigGame-SmallGame decomposition -- The Hales-Jewett conjecture -- Reinforcing the Erdős-Selfridge technique (I) -- Reinforcing the Erdős-Selfridge technique (II) -- Almost disjoint hypergraphs -- Exact solution of the clique game (II) | |
| 505 | 8 | |a ch. VIII. Advanced decomposition -- Proof of the second ugly theorem -- Breaking the "square-root barrier" (I) -- Breaking the "square-root barrier" (II) -- Van der Waerden game and the RELARIN technique -- ch. IX. Game-theoretic lattice-numbers -- Winning planes : exact solution -- Winning lattices : exact solution -- I-can-you-can't games -- second player's moral victory -- ch. X. Conclusion -- More exact solutions and more partial results -- Miscellany (I) -- Miscellany (II) -- Concluding remarks -- Appendix A : Ramsey numbers -- Appendix B : Hales-Jewett theorem : Shelah's proof -- Appendix C : A formal treatment of positional games -- Appendix D : An informal introduction to game theory | |
| 520 | |a Traditional game theory has been successful at developing strategy in games of incomplete information: when one player knows something that the other does not. But it has little to say about games of complete information, for example, tic-tac-toe, solitaire and hex. The main challenge of combinatorial game theory is to handle combinatorial chaos, where brute force study is impractical. In this comprehensive volume, József Beck shows readers how to escape from the combinatorial chaos via the fake probabilistic method, a game-theoretic adaptation of the probabilistic method in combinatorics. Using this, the author is able to determine the exact results about infinite classes of many games, leading to the discovery of some striking new duality principles. Available for the first time in paperback, it includes a new appendix to address the results that have appeared since the book's original publication | ||
| 650 | 4 | |a Game theory | |
| 650 | 4 | |a Combinatorial analysis | |
| 650 | 0 | 7 | |a Kombinatorisches Spiel |0 (DE-588)4164753-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Kombinatorisches Spiel |0 (DE-588)4164753-1 |D s |
| 689 | 0 | |8 1\p |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Druckausgabe |z 978-0-521-18475-5 |
| 776 | 0 | 8 | |i Erscheint auch als |n Druckausgabe |z 978-0-521-46100-9 |
| 856 | 4 | 0 | |u https://doi.org/10.1017/CBO9780511735202 |x Verlag |z URL des Erstveröffentlichers |3 Volltext |
| 912 | |a ZDB-20-CBO | ||
| 999 | |a oai:aleph.bib-bvb.de:BVB01-029351052 | ||
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 966 | e | |u https://doi.org/10.1017/CBO9780511735202 |l BSB01 |p ZDB-20-CBO |q BSB_PDA_CBO |x Verlag |3 Volltext | |
| 966 | e | |u https://doi.org/10.1017/CBO9780511735202 |l FHN01 |p ZDB-20-CBO |q FHN_PDA_CBO |x Verlag |3 Volltext | |
Datensatz im Suchindex
| _version_ | 1804176884521500672 |
|---|---|
| any_adam_object | |
| author | Beck, József |
| author_facet | Beck, József |
| author_role | aut |
| author_sort | Beck, József |
| author_variant | j b jb |
| building | Verbundindex |
| bvnumber | BV043942082 |
| classification_rvk | QH 431 SK 860 |
| collection | ZDB-20-CBO |
| contents | pt. A. Weak win and strong draw -- ch. I. Win vs. weak win -- Illustration : every finite point set in the plane is a weak winner -- Analyzing the proof of theorem 1.1 -- Examples : tic-tac-toe games -- More examples : tic-tac-toe like games -- Games on hypergraphs, and the combinatorial chaos -- ch. II. The main result : exact solutions for infinite classes of games -- Ramsey theory and clique games -- Arithmetic progressions -- Two-dimensional arithmetic progressions -- Explaining the exact solutions : a meta-conjecture -- Potentials and the Erdős-Selfridge theorem -- Local vs. global -- Ramsey theory and hypercube tic-tac-toe -- pt. B. Basic potential technique : game-theoretic first and second moments -- ch. III. Simple applications -- Easy building via theorem 1.2 -- Games beyond Ramsey theory -- A generalization of Kaplansky's game -- ch. IV. Games and randomness -- Discrepancy games and the variance -- Biased discrepancy games : when the extension from fair to biased works! -- A simple illustration of "randomness" (I) -- A simple illustration of "randomness" (II) -- Another illustration of "randomness" in games pt. C. Advanced weak win : game-theoretic higher moment -- ch. V. Self-improving potentials -- Motivating the probabilistic approach -- Game-theoretic second moment : application to the picker-choose game -- Weak win in the lattice games -- Game-theoretic higher moments -- Exact solution of the clique game (I) -- More applications -- Who-scores-more games -- ch. VI. What is the biased meta-conjecture, and why is it so difficult? -- Discrepancy games (I) -- Discrepancy games (II) -- Biased games (I) : biased meta-conjecture -- Biased games (II) : sacrificing the probabilistic intuition to force negativity -- Biased games (III) : sporadic results -- Biased games (IV) : more sporadic results -- pt. D. Advanced strong draw : game-theoretic independence -- ch. VII. BigGame-SmallGame decomposition -- The Hales-Jewett conjecture -- Reinforcing the Erdős-Selfridge technique (I) -- Reinforcing the Erdős-Selfridge technique (II) -- Almost disjoint hypergraphs -- Exact solution of the clique game (II) ch. VIII. Advanced decomposition -- Proof of the second ugly theorem -- Breaking the "square-root barrier" (I) -- Breaking the "square-root barrier" (II) -- Van der Waerden game and the RELARIN technique -- ch. IX. Game-theoretic lattice-numbers -- Winning planes : exact solution -- Winning lattices : exact solution -- I-can-you-can't games -- second player's moral victory -- ch. X. Conclusion -- More exact solutions and more partial results -- Miscellany (I) -- Miscellany (II) -- Concluding remarks -- Appendix A : Ramsey numbers -- Appendix B : Hales-Jewett theorem : Shelah's proof -- Appendix C : A formal treatment of positional games -- Appendix D : An informal introduction to game theory |
| ctrlnum | (ZDB-20-CBO)CR9780511735202 (OCoLC)850520953 (DE-599)BVBBV043942082 |
| dewey-full | 519.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.3 |
| dewey-search | 519.3 |
| dewey-sort | 3519.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| doi_str_mv | 10.1017/CBO9780511735202 |
| format | Electronic eBook |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>05749nmm a2200529zcb4500</leader><controlfield tag="001">BV043942082</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">00000000000000.0</controlfield><controlfield tag="007">cr|uuu---uuuuu</controlfield><controlfield tag="008">161206s2008 |||| o||u| ||||||eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780511735202</subfield><subfield code="c">Online</subfield><subfield code="9">978-0-511-73520-2</subfield></datafield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1017/CBO9780511735202</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(ZDB-20-CBO)CR9780511735202</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)850520953</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV043942082</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-12</subfield><subfield code="a">DE-92</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">519.3</subfield><subfield code="2">22</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">QH 431</subfield><subfield code="0">(DE-625)141582:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 860</subfield><subfield code="0">(DE-625)143264:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Beck, József</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Combinatorial games</subfield><subfield code="b">tic-tac-toe theory</subfield><subfield code="c">József Beck</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Cambridge</subfield><subfield code="b">Cambridge University Press</subfield><subfield code="c">2008</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">1 online resource (xiv, 732 pages)</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Encyclopedia of mathematics and its applications</subfield><subfield code="v">volume 114</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Title from publisher's bibliographic system (viewed on 05 Oct 2015)</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">pt. A. Weak win and strong draw -- ch. I. Win vs. weak win -- Illustration : every finite point set in the plane is a weak winner -- Analyzing the proof of theorem 1.1 -- Examples : tic-tac-toe games -- More examples : tic-tac-toe like games -- Games on hypergraphs, and the combinatorial chaos -- ch. II. The main result : exact solutions for infinite classes of games -- Ramsey theory and clique games -- Arithmetic progressions -- Two-dimensional arithmetic progressions -- Explaining the exact solutions : a meta-conjecture -- Potentials and the Erdős-Selfridge theorem -- Local vs. global -- Ramsey theory and hypercube tic-tac-toe -- pt. B. Basic potential technique : game-theoretic first and second moments -- ch. III. Simple applications -- Easy building via theorem 1.2 -- Games beyond Ramsey theory -- A generalization of Kaplansky's game -- ch. IV. Games and randomness -- Discrepancy games and the variance -- Biased discrepancy games : when the extension from fair to biased works! -- A simple illustration of "randomness" (I) -- A simple illustration of "randomness" (II) -- Another illustration of "randomness" in games</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">pt. C. Advanced weak win : game-theoretic higher moment -- ch. V. Self-improving potentials -- Motivating the probabilistic approach -- Game-theoretic second moment : application to the picker-choose game -- Weak win in the lattice games -- Game-theoretic higher moments -- Exact solution of the clique game (I) -- More applications -- Who-scores-more games -- ch. VI. What is the biased meta-conjecture, and why is it so difficult? -- Discrepancy games (I) -- Discrepancy games (II) -- Biased games (I) : biased meta-conjecture -- Biased games (II) : sacrificing the probabilistic intuition to force negativity -- Biased games (III) : sporadic results -- Biased games (IV) : more sporadic results -- pt. D. Advanced strong draw : game-theoretic independence -- ch. VII. BigGame-SmallGame decomposition -- The Hales-Jewett conjecture -- Reinforcing the Erdős-Selfridge technique (I) -- Reinforcing the Erdős-Selfridge technique (II) -- Almost disjoint hypergraphs -- Exact solution of the clique game (II)</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">ch. VIII. Advanced decomposition -- Proof of the second ugly theorem -- Breaking the "square-root barrier" (I) -- Breaking the "square-root barrier" (II) -- Van der Waerden game and the RELARIN technique -- ch. IX. Game-theoretic lattice-numbers -- Winning planes : exact solution -- Winning lattices : exact solution -- I-can-you-can't games -- second player's moral victory -- ch. X. Conclusion -- More exact solutions and more partial results -- Miscellany (I) -- Miscellany (II) -- Concluding remarks -- Appendix A : Ramsey numbers -- Appendix B : Hales-Jewett theorem : Shelah's proof -- Appendix C : A formal treatment of positional games -- Appendix D : An informal introduction to game theory</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Traditional game theory has been successful at developing strategy in games of incomplete information: when one player knows something that the other does not. But it has little to say about games of complete information, for example, tic-tac-toe, solitaire and hex. The main challenge of combinatorial game theory is to handle combinatorial chaos, where brute force study is impractical. In this comprehensive volume, József Beck shows readers how to escape from the combinatorial chaos via the fake probabilistic method, a game-theoretic adaptation of the probabilistic method in combinatorics. Using this, the author is able to determine the exact results about infinite classes of many games, leading to the discovery of some striking new duality principles. Available for the first time in paperback, it includes a new appendix to address the results that have appeared since the book's original publication</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Game theory</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Combinatorial analysis</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Kombinatorisches Spiel</subfield><subfield code="0">(DE-588)4164753-1</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Kombinatorisches Spiel</subfield><subfield code="0">(DE-588)4164753-1</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="8">1\p</subfield><subfield code="5">DE-604</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Druckausgabe</subfield><subfield code="z">978-0-521-18475-5</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Druckausgabe</subfield><subfield code="z">978-0-521-46100-9</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://doi.org/10.1017/CBO9780511735202</subfield><subfield code="x">Verlag</subfield><subfield code="z">URL des Erstveröffentlichers</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-20-CBO</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-029351052</subfield></datafield><datafield tag="883" ind1="1" ind2=" "><subfield code="8">1\p</subfield><subfield code="a">cgwrk</subfield><subfield code="d">20201028</subfield><subfield code="q">DE-101</subfield><subfield code="u">https://d-nb.info/provenance/plan#cgwrk</subfield></datafield><datafield tag="966" ind1="e" ind2=" "><subfield code="u">https://doi.org/10.1017/CBO9780511735202</subfield><subfield code="l">BSB01</subfield><subfield code="p">ZDB-20-CBO</subfield><subfield code="q">BSB_PDA_CBO</subfield><subfield code="x">Verlag</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="966" ind1="e" ind2=" "><subfield code="u">https://doi.org/10.1017/CBO9780511735202</subfield><subfield code="l">FHN01</subfield><subfield code="p">ZDB-20-CBO</subfield><subfield code="q">FHN_PDA_CBO</subfield><subfield code="x">Verlag</subfield><subfield code="3">Volltext</subfield></datafield></record></collection> |
| id | DE-604.BV043942082 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:39:16Z |
| institution | BVB |
| isbn | 9780511735202 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029351052 |
| oclc_num | 850520953 |
| open_access_boolean | |
| owner | DE-12 DE-92 |
| owner_facet | DE-12 DE-92 |
| physical | 1 online resource (xiv, 732 pages) |
| psigel | ZDB-20-CBO ZDB-20-CBO BSB_PDA_CBO ZDB-20-CBO FHN_PDA_CBO |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Cambridge University Press |
| record_format | marc |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Beck, József Verfasser aut Combinatorial games tic-tac-toe theory József Beck Cambridge Cambridge University Press 2008 1 online resource (xiv, 732 pages) txt rdacontent c rdamedia cr rdacarrier Encyclopedia of mathematics and its applications volume 114 Title from publisher's bibliographic system (viewed on 05 Oct 2015) pt. A. Weak win and strong draw -- ch. I. Win vs. weak win -- Illustration : every finite point set in the plane is a weak winner -- Analyzing the proof of theorem 1.1 -- Examples : tic-tac-toe games -- More examples : tic-tac-toe like games -- Games on hypergraphs, and the combinatorial chaos -- ch. II. The main result : exact solutions for infinite classes of games -- Ramsey theory and clique games -- Arithmetic progressions -- Two-dimensional arithmetic progressions -- Explaining the exact solutions : a meta-conjecture -- Potentials and the Erdős-Selfridge theorem -- Local vs. global -- Ramsey theory and hypercube tic-tac-toe -- pt. B. Basic potential technique : game-theoretic first and second moments -- ch. III. Simple applications -- Easy building via theorem 1.2 -- Games beyond Ramsey theory -- A generalization of Kaplansky's game -- ch. IV. Games and randomness -- Discrepancy games and the variance -- Biased discrepancy games : when the extension from fair to biased works! -- A simple illustration of "randomness" (I) -- A simple illustration of "randomness" (II) -- Another illustration of "randomness" in games pt. C. Advanced weak win : game-theoretic higher moment -- ch. V. Self-improving potentials -- Motivating the probabilistic approach -- Game-theoretic second moment : application to the picker-choose game -- Weak win in the lattice games -- Game-theoretic higher moments -- Exact solution of the clique game (I) -- More applications -- Who-scores-more games -- ch. VI. What is the biased meta-conjecture, and why is it so difficult? -- Discrepancy games (I) -- Discrepancy games (II) -- Biased games (I) : biased meta-conjecture -- Biased games (II) : sacrificing the probabilistic intuition to force negativity -- Biased games (III) : sporadic results -- Biased games (IV) : more sporadic results -- pt. D. Advanced strong draw : game-theoretic independence -- ch. VII. BigGame-SmallGame decomposition -- The Hales-Jewett conjecture -- Reinforcing the Erdős-Selfridge technique (I) -- Reinforcing the Erdős-Selfridge technique (II) -- Almost disjoint hypergraphs -- Exact solution of the clique game (II) ch. VIII. Advanced decomposition -- Proof of the second ugly theorem -- Breaking the "square-root barrier" (I) -- Breaking the "square-root barrier" (II) -- Van der Waerden game and the RELARIN technique -- ch. IX. Game-theoretic lattice-numbers -- Winning planes : exact solution -- Winning lattices : exact solution -- I-can-you-can't games -- second player's moral victory -- ch. X. Conclusion -- More exact solutions and more partial results -- Miscellany (I) -- Miscellany (II) -- Concluding remarks -- Appendix A : Ramsey numbers -- Appendix B : Hales-Jewett theorem : Shelah's proof -- Appendix C : A formal treatment of positional games -- Appendix D : An informal introduction to game theory Traditional game theory has been successful at developing strategy in games of incomplete information: when one player knows something that the other does not. But it has little to say about games of complete information, for example, tic-tac-toe, solitaire and hex. The main challenge of combinatorial game theory is to handle combinatorial chaos, where brute force study is impractical. In this comprehensive volume, József Beck shows readers how to escape from the combinatorial chaos via the fake probabilistic method, a game-theoretic adaptation of the probabilistic method in combinatorics. Using this, the author is able to determine the exact results about infinite classes of many games, leading to the discovery of some striking new duality principles. Available for the first time in paperback, it includes a new appendix to address the results that have appeared since the book's original publication Game theory Combinatorial analysis Kombinatorisches Spiel (DE-588)4164753-1 gnd rswk-swf Kombinatorisches Spiel (DE-588)4164753-1 s 1\p DE-604 Erscheint auch als Druckausgabe 978-0-521-18475-5 Erscheint auch als Druckausgabe 978-0-521-46100-9 https://doi.org/10.1017/CBO9780511735202 Verlag URL des Erstveröffentlichers Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Beck, József Combinatorial games tic-tac-toe theory pt. A. Weak win and strong draw -- ch. I. Win vs. weak win -- Illustration : every finite point set in the plane is a weak winner -- Analyzing the proof of theorem 1.1 -- Examples : tic-tac-toe games -- More examples : tic-tac-toe like games -- Games on hypergraphs, and the combinatorial chaos -- ch. II. The main result : exact solutions for infinite classes of games -- Ramsey theory and clique games -- Arithmetic progressions -- Two-dimensional arithmetic progressions -- Explaining the exact solutions : a meta-conjecture -- Potentials and the Erdős-Selfridge theorem -- Local vs. global -- Ramsey theory and hypercube tic-tac-toe -- pt. B. Basic potential technique : game-theoretic first and second moments -- ch. III. Simple applications -- Easy building via theorem 1.2 -- Games beyond Ramsey theory -- A generalization of Kaplansky's game -- ch. IV. Games and randomness -- Discrepancy games and the variance -- Biased discrepancy games : when the extension from fair to biased works! -- A simple illustration of "randomness" (I) -- A simple illustration of "randomness" (II) -- Another illustration of "randomness" in games pt. C. Advanced weak win : game-theoretic higher moment -- ch. V. Self-improving potentials -- Motivating the probabilistic approach -- Game-theoretic second moment : application to the picker-choose game -- Weak win in the lattice games -- Game-theoretic higher moments -- Exact solution of the clique game (I) -- More applications -- Who-scores-more games -- ch. VI. What is the biased meta-conjecture, and why is it so difficult? -- Discrepancy games (I) -- Discrepancy games (II) -- Biased games (I) : biased meta-conjecture -- Biased games (II) : sacrificing the probabilistic intuition to force negativity -- Biased games (III) : sporadic results -- Biased games (IV) : more sporadic results -- pt. D. Advanced strong draw : game-theoretic independence -- ch. VII. BigGame-SmallGame decomposition -- The Hales-Jewett conjecture -- Reinforcing the Erdős-Selfridge technique (I) -- Reinforcing the Erdős-Selfridge technique (II) -- Almost disjoint hypergraphs -- Exact solution of the clique game (II) ch. VIII. Advanced decomposition -- Proof of the second ugly theorem -- Breaking the "square-root barrier" (I) -- Breaking the "square-root barrier" (II) -- Van der Waerden game and the RELARIN technique -- ch. IX. Game-theoretic lattice-numbers -- Winning planes : exact solution -- Winning lattices : exact solution -- I-can-you-can't games -- second player's moral victory -- ch. X. Conclusion -- More exact solutions and more partial results -- Miscellany (I) -- Miscellany (II) -- Concluding remarks -- Appendix A : Ramsey numbers -- Appendix B : Hales-Jewett theorem : Shelah's proof -- Appendix C : A formal treatment of positional games -- Appendix D : An informal introduction to game theory Game theory Combinatorial analysis Kombinatorisches Spiel (DE-588)4164753-1 gnd |
| subject_GND | (DE-588)4164753-1 |
| title | Combinatorial games tic-tac-toe theory |
| title_auth | Combinatorial games tic-tac-toe theory |
| title_exact_search | Combinatorial games tic-tac-toe theory |
| title_full | Combinatorial games tic-tac-toe theory József Beck |
| title_fullStr | Combinatorial games tic-tac-toe theory József Beck |
| title_full_unstemmed | Combinatorial games tic-tac-toe theory József Beck |
| title_short | Combinatorial games |
| title_sort | combinatorial games tic tac toe theory |
| title_sub | tic-tac-toe theory |
| topic | Game theory Combinatorial analysis Kombinatorisches Spiel (DE-588)4164753-1 gnd |
| topic_facet | Game theory Combinatorial analysis Kombinatorisches Spiel |
| url | https://doi.org/10.1017/CBO9780511735202 |
| work_keys_str_mv | AT beckjozsef combinatorialgamestictactoetheory |