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The Resource Combinatorial games : tic-tac-toe theory, József Beck

Combinatorial games : tic-tac-toe theory, József Beck

Label
Combinatorial games : tic-tac-toe theory
Title
Combinatorial games
Title remainder
tic-tac-toe theory
Statement of responsibility
József Beck
Creator
Subject
Language
eng
Summary
"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. This is the subject of combinatorial game theory. Most board games are a challenge for mathematics: to analyze a position one has to examine the available options, and then the further options available after selecting any option, and so on. This leads to combinatorial chaos, where brute force study is impractical." "In this comprehensive volume, Jozsef 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."--BOOK JACKET
Cataloging source
UKM
http://library.link/vocab/creatorName
BECK, József
Illustrations
illustrations
Index
no index present
Literary form
non fiction
Nature of contents
bibliography
Series statement
Encyclopedia of mathematics and its applications
Series volume
114
http://library.link/vocab/subjectName
  • Game theory
  • Combinatorial analysis
Label
Combinatorial games : tic-tac-toe theory, József Beck
Instantiates
Publication
Bibliography note
Includes bibliographical references
Contents
  • summary of the book in a nutshell.
  • p. 1
  • Pt. A.
  • Weak Win and Strong Draw.
  • p. 15
  • Ch. I.
  • Win vs. Weak Win.
  • p. 17
  • 1.
  • Illustration: every finite pointset in the plane is a Weak Winner.
  • p. 19
  • 2.
  • Analyzing the proof of Theorem 1.1.
  • p. 32
  • 3.
  • Examples: Tic-Tac-Toe games.
  • p. 42
  • 4.
  • More examples: Tic-Tac-Toe like games.
  • p. 59
  • 5.
  • Games on hypergraphs, and the combinatorial chaos.
  • p. 72
  • Ch. II.
  • main result: exact solutions for infinite classes of games.
  • p. 91
  • 6.
  • Ramsey Theory and Clique Games.
  • p. 92
  • 7.
  • Arithmetic progressions.
  • p. 106
  • 8.
  • Two-dimensional arithmetic progressions.
  • p. 118
  • 9.
  • Explaining the exact solutions: a Meta-Conjecture.
  • p. 131
  • 10.
  • Potentials and the Erdos-Selfridge Theorem.
  • p. 146
  • 11.
  • Local vs. Global.
  • p. 163
  • 12.
  • Ramsey Theory and Hypercube Tic-Tac-Toe.
  • p. 172
  • Pt. B.
  • Basic Potential Technique - Game-Theoretic First and Second Moments.
  • p. 193
  • Ch. III.
  • Simple applications.
  • p. 195
  • 13.
  • Easy building via Theorem 1.2.
  • p. 196
  • 14.
  • Games beyond Ramsey Theory.
  • p. 204
  • 15.
  • generalization of Kaplansky's game.
  • p. 216
  • Ch. IV.
  • Games and randomness.
  • p. 230
  • 16.
  • Discrepancy Games and the variance.
  • p. 231
  • 17.
  • Biased Discrepancy Games: when the extension from fair to biased works!.
  • p. 245
  • 18.
  • simple illustration of "randomness" (I).
  • p. 260
  • 19.
  • simple illustration of "randomness" (II).
  • p. 270
  • 20.
  • Another illustration of "randomness" in games.
  • p. 286
  • Pt. C.
  • Advanced Weak Win - Game-Theoretic Higher Moment.
  • p. 305
  • Ch. V.
  • Self-improving potentials.
  • p. 307
  • 21.
  • Motivating the probabilistic approach.
  • p. 308
  • 22.
  • Game-theoretic second moment: application to the Picker-Chooser game.
  • p. 320
  • 23.
  • Weak Win in the Lattice Games.
  • p. 329
  • 24.
  • Game-theoretic higher moments.
  • p. 340
  • 25.
  • Exact solution of the Clique Game (I).
  • p. 352
  • 26.
  • More applications.
  • p. 362
  • 27.
  • Who-scores-more games.
  • p. 372
  • Ch. VI.
  • What is the Biased Meta-Conjecture, and why is it so difficult?.
  • p. 380
  • 28.
  • Discrepancy games (I).
  • p. 381
  • 29.
  • Discrepancy games (II).
  • p. 392
  • 30.
  • Biased Games (I): Biased Meta-Conjecture.
  • p. 400
  • 31.
  • Biased games (II): Sacrificing the probabilistic intuition to force negativity.
  • p. 418
  • 32.
  • Biased games (III): Sporadic results.
  • p. 430
  • 33.
  • Biased games (IV): More sporadic results.
  • p. 439
  • Pt. D.
  • Advanced Strong Draw - Game-Theoretic Independence.
  • p. 459
  • Ch. VII.
  • BigGame-SmallGame Decomposition.
  • p. 461
  • 34.
  • Hales-Jewett Conjecture.
  • p. 462
  • 35.
  • Reinforcing the Erdos-Selfridge technique (I).
  • p. 470
  • 36.
  • Reinforcing the Erdos-Selfridge technique (II).
  • p. 479
  • 37.
  • Almost Disjoint hypergraphs.
  • p. 485
  • 38.
  • Exact solution of the Clique Game (II).
  • p. 492
  • Ch. VII.
  • I Advanced decomposition.
  • p. 504
  • 39.
  • Proof of the second Ugly Theorem.
  • p. 505
  • 40.
  • Breaking the "square-root barrier" (I).
  • p. 525
  • 41.
  • Brea king the "square-root barrier" (II).
  • p. 536
  • 42.
  • Van der Waerden Game and the RELARIN technique.
  • p. 545
  • Ch. IX.
  • Game-theoretic lattice-numbers.
  • p. 552
  • 43.
  • Winning planes: exact solution.
  • p. 553
  • 44.
  • Winning lattices: exact solution.
  • p. 575
  • 45.
  • I-Can-You-Can't Games - Second Player's Moral Victory.
  • p. 592
  • Ch. X.
  • Conclusion.
  • p. 610
  • 46.
  • More exact solutions and more partial results.
  • p. 611
  • 47.
  • Miscellany (I).
  • p. 620
  • 48.
  • Miscellany (II).
  • p. 634
  • 49.
  • Concluding remarks.
  • p. 644
  • Appendix A.
  • Ramsey Numbers.
  • p. 658
  • Appendix B.
  • Hales-Jewett Theorem: Shelah's proof.
  • p. 669
  • Appendix C.
  • formal treatment of Positional Games.
  • p. 677
  • Appendix D.
  • informal introduction to game theory.
  • p. 705
  • Complete list of the Open Problems.
  • p. 716
  • What kinds of games? A dictionary.
  • p. 724
  • Dictionary of the phrases and concepts.
  • p. 727
  • References.
  • p. 730
Control code
ocn175284055
Dimensions
24 cm.
Extent
xiv, 732 p.
Isbn
9780521461009
Lccn
2008275067
Other physical details
ill.
Label
Combinatorial games : tic-tac-toe theory, József Beck
Publication
Bibliography note
Includes bibliographical references
Contents
  • summary of the book in a nutshell.
  • p. 1
  • Pt. A.
  • Weak Win and Strong Draw.
  • p. 15
  • Ch. I.
  • Win vs. Weak Win.
  • p. 17
  • 1.
  • Illustration: every finite pointset in the plane is a Weak Winner.
  • p. 19
  • 2.
  • Analyzing the proof of Theorem 1.1.
  • p. 32
  • 3.
  • Examples: Tic-Tac-Toe games.
  • p. 42
  • 4.
  • More examples: Tic-Tac-Toe like games.
  • p. 59
  • 5.
  • Games on hypergraphs, and the combinatorial chaos.
  • p. 72
  • Ch. II.
  • main result: exact solutions for infinite classes of games.
  • p. 91
  • 6.
  • Ramsey Theory and Clique Games.
  • p. 92
  • 7.
  • Arithmetic progressions.
  • p. 106
  • 8.
  • Two-dimensional arithmetic progressions.
  • p. 118
  • 9.
  • Explaining the exact solutions: a Meta-Conjecture.
  • p. 131
  • 10.
  • Potentials and the Erdos-Selfridge Theorem.
  • p. 146
  • 11.
  • Local vs. Global.
  • p. 163
  • 12.
  • Ramsey Theory and Hypercube Tic-Tac-Toe.
  • p. 172
  • Pt. B.
  • Basic Potential Technique - Game-Theoretic First and Second Moments.
  • p. 193
  • Ch. III.
  • Simple applications.
  • p. 195
  • 13.
  • Easy building via Theorem 1.2.
  • p. 196
  • 14.
  • Games beyond Ramsey Theory.
  • p. 204
  • 15.
  • generalization of Kaplansky's game.
  • p. 216
  • Ch. IV.
  • Games and randomness.
  • p. 230
  • 16.
  • Discrepancy Games and the variance.
  • p. 231
  • 17.
  • Biased Discrepancy Games: when the extension from fair to biased works!.
  • p. 245
  • 18.
  • simple illustration of "randomness" (I).
  • p. 260
  • 19.
  • simple illustration of "randomness" (II).
  • p. 270
  • 20.
  • Another illustration of "randomness" in games.
  • p. 286
  • Pt. C.
  • Advanced Weak Win - Game-Theoretic Higher Moment.
  • p. 305
  • Ch. V.
  • Self-improving potentials.
  • p. 307
  • 21.
  • Motivating the probabilistic approach.
  • p. 308
  • 22.
  • Game-theoretic second moment: application to the Picker-Chooser game.
  • p. 320
  • 23.
  • Weak Win in the Lattice Games.
  • p. 329
  • 24.
  • Game-theoretic higher moments.
  • p. 340
  • 25.
  • Exact solution of the Clique Game (I).
  • p. 352
  • 26.
  • More applications.
  • p. 362
  • 27.
  • Who-scores-more games.
  • p. 372
  • Ch. VI.
  • What is the Biased Meta-Conjecture, and why is it so difficult?.
  • p. 380
  • 28.
  • Discrepancy games (I).
  • p. 381
  • 29.
  • Discrepancy games (II).
  • p. 392
  • 30.
  • Biased Games (I): Biased Meta-Conjecture.
  • p. 400
  • 31.
  • Biased games (II): Sacrificing the probabilistic intuition to force negativity.
  • p. 418
  • 32.
  • Biased games (III): Sporadic results.
  • p. 430
  • 33.
  • Biased games (IV): More sporadic results.
  • p. 439
  • Pt. D.
  • Advanced Strong Draw - Game-Theoretic Independence.
  • p. 459
  • Ch. VII.
  • BigGame-SmallGame Decomposition.
  • p. 461
  • 34.
  • Hales-Jewett Conjecture.
  • p. 462
  • 35.
  • Reinforcing the Erdos-Selfridge technique (I).
  • p. 470
  • 36.
  • Reinforcing the Erdos-Selfridge technique (II).
  • p. 479
  • 37.
  • Almost Disjoint hypergraphs.
  • p. 485
  • 38.
  • Exact solution of the Clique Game (II).
  • p. 492
  • Ch. VII.
  • I Advanced decomposition.
  • p. 504
  • 39.
  • Proof of the second Ugly Theorem.
  • p. 505
  • 40.
  • Breaking the "square-root barrier" (I).
  • p. 525
  • 41.
  • Brea king the "square-root barrier" (II).
  • p. 536
  • 42.
  • Van der Waerden Game and the RELARIN technique.
  • p. 545
  • Ch. IX.
  • Game-theoretic lattice-numbers.
  • p. 552
  • 43.
  • Winning planes: exact solution.
  • p. 553
  • 44.
  • Winning lattices: exact solution.
  • p. 575
  • 45.
  • I-Can-You-Can't Games - Second Player's Moral Victory.
  • p. 592
  • Ch. X.
  • Conclusion.
  • p. 610
  • 46.
  • More exact solutions and more partial results.
  • p. 611
  • 47.
  • Miscellany (I).
  • p. 620
  • 48.
  • Miscellany (II).
  • p. 634
  • 49.
  • Concluding remarks.
  • p. 644
  • Appendix A.
  • Ramsey Numbers.
  • p. 658
  • Appendix B.
  • Hales-Jewett Theorem: Shelah's proof.
  • p. 669
  • Appendix C.
  • formal treatment of Positional Games.
  • p. 677
  • Appendix D.
  • informal introduction to game theory.
  • p. 705
  • Complete list of the Open Problems.
  • p. 716
  • What kinds of games? A dictionary.
  • p. 724
  • Dictionary of the phrases and concepts.
  • p. 727
  • References.
  • p. 730
Control code
ocn175284055
Dimensions
24 cm.
Extent
xiv, 732 p.
Isbn
9780521461009
Lccn
2008275067
Other physical details
ill.

Library Locations

    • Harold Cohen LibraryBorrow it
      Ashton Street, Liverpool, L69 3DA, GB
      53.418074 -2.967913
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