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Combinatorial Games: Tic-Tac-Toe Theory Book

Combinatorial Games: Tic-Tac-Toe Theory
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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, solita, Combinatorial Games: Tic-Tac-Toe Theory
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  • Combinatorial Games: Tic-Tac-Toe Theory
  • Written by author Jozsef Beck
  • Published by Cambridge University Press, 4/28/2011
  • "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, solita
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A summary of the book in a nutshell 1

Pt. A Weak Win and Strong Draw 15

Ch. I Win vs. Weak Win 17

1 Illustration: every finite pointset in the plane is a Weak Winner 19

2 Analyzing the proof of Theorem 1.1 32

3 Examples: Tic-Tac-Toe games 42

4 More examples: Tic-Tac-Toe like games 59

5 Games on hypergraphs, and the combinatorial chaos 72

Ch. II The main result: exact solutions for infinite classes of games 91

6 Ramsey Theory and Clique Games 92

7 Arithmetic progressions 106

8 Two-dimensional arithmetic progressions 118

9 Explaining the exact solutions: a Meta-Conjecture 131

10 Potentials and the Erdos-Selfridge Theorem 146

11 Local vs. Global 163

12 Ramsey Theory and Hypercube Tic-Tac-Toe 172

Pt. B Basic Potential Technique - Game-Theoretic First and Second Moments 193

Ch. III Simple applications 195

13 Easy building via Theorem 1.2 196

14 Games beyond Ramsey Theory 204

15 A generalization of Kaplansky's game 216

Ch. IV Games and randomness 230

16 Discrepancy Games and the variance 231

17 Biased Discrepancy Games: when the extension from fair to biased works! 245

18 A simple illustration of "randomness" (I) 260

19 A simple illustration of "randomness" (II) 270

20 Another illustration of "randomness" in games 286

Pt. C Advanced Weak Win - Game-Theoretic Higher Moment 305

Ch. V Self-improving potentials 307

21 Motivating the probabilistic approach 308

22 Game-theoretic second moment: application to the Picker-Chooser game 320

23 Weak Win in the Lattice Games 329

24 Game-theoretic higher moments 340

25 Exact solution of the Clique Game (I) 352

26 More applications 362

27 Who-scores-more games372

Ch. VI What is the Biased Meta-Conjecture, and why is it so difficult? 380

28 Discrepancy games (I) 381

29 Discrepancy games (II) 392

30 Biased Games (I): Biased Meta-Conjecture 400

31 Biased games (II): Sacrificing the probabilistic intuition to force negativity 418

32 Biased games (III): Sporadic results 430

33 Biased games (IV): More sporadic results 439

Pt. D Advanced Strong Draw - Game-Theoretic Independence 459

Ch. VII BigGame-SmallGame Decomposition 461

34 The Hales-Jewett Conjecture 462

35 Reinforcing the Erdos-Selfridge technique (I) 470

36 Reinforcing the Erdos-Selfridge technique (II) 479

37 Almost Disjoint hypergraphs 485

38 Exact solution of the Clique Game (II) 492

Ch. VII I Advanced decomposition 504

39 Proof of the second Ugly Theorem 505

40 Breaking the "square-root barrier" (I) 525

41 Brea king the "square-root barrier" (II) 536

42 Van der Waerden Game and the RELARIN technique 545

Ch. IX Game-theoretic lattice-numbers 552

43 Winning planes: exact solution 553

44 Winning lattices: exact solution 575

45 I-Can-You-Can't Games - Second Player's Moral Victory 592

Ch. X Conclusion 610

46 More exact solutions and more partial results 611

47 Miscellany (I) 620

48 Miscellany (II) 634

49 Concluding remarks 644

Appendix A Ramsey Numbers 658

Appendix B Hales-Jewett Theorem: Shelah's proof 669

Appendix C A formal treatment of Positional Games 677

Appendix D An informal introduction to game theory 705

Complete list of the Open Problems 716

What kinds of games? A dictionary 724

Dictionary of the phrases and concepts 727

References 730


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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, solita, Combinatorial Games: Tic-Tac-Toe Theory

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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, solita, Combinatorial Games: Tic-Tac-Toe Theory

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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, solita, Combinatorial Games: Tic-Tac-Toe Theory

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