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Publisher:

A K Peters

Publication Date:

2004

Number of Pages:

504

Format:

Paperback

Price:

49.00

ISBN:

1-56881-210-8

Category:

Monograph

[Reviewed by , on ]

Donald L. Vestal

03/5/2005

Mathematics has been used (to varying degrees of success) to analyze games, usually with the intent of finding a “winning strategy.” This book gives a summary of some of the results. The author divides the universe of games into three categories: games of chance (games, such as roulette, whose uncertainty comes from random influences); combinatorial games (games, such as chess and go, whose uncertainty relies on the multiplicity of possible moves); and strategic games (games, such as poker and rock-paper-scissors, whose uncertainty lies in a lack of complete information). The book is divided into three sections, based on these categories.

The section on Games of Chance deals with several games, and uses these to introduce a great deal of probability theory: the basic theory, expectation and variance, the normal distribution, the Poisson distribution, the Monte Carlo method, Markov chains. As expected, it is difficult to derive winning strategies for games of chance — although there is a section which presents a method for “counting cards” in blackjack.

The second section, Combinatorial Games, delves into the notion of game theory, with Zermelo’s Theorem for two-person zero-sum games with perfect information. Several games and their winning strategies are studied, for example, NIM, go, and some of the classic games created by John Conway. This section culminates into a discussion of topics such as artificial intelligence, Turing machines, Gödel’s Incompleteness Theorem, and P-NP-PSPACE-EXPTIME problems. The final section deals with Strategic Games. Here the author considers whether psychology can be more effective than random chance (for example, bluffing in poker). Applying the notion of the minimax value to this picture results in a linear programming problem, and the introduction of the simplex method.

Given the wide variety of (fairly deep) mathematics mentioned here, you can’t expect the book to go into too much detail; even coming in at just under 500 pages, there isn’t enough room to cover these kinds of topics with any depth. However, the book does provide a remarkable summary, replete with numerous footnotes. (Since this book is the English version of a German text, roughly half of the references are German magazines, books, or journals.)

Donald L. Vestal is Associate Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, and a deep admiration for the crime-fighting efforts of the Aqua Teen Hunger Force. He can be reached at vestal@mwsc.edu.

Contents

Preface ix

I Games of Chance 1

1 Dice and Probability 3

2 Waiting for a Double 6 8

3 Tips on Playing the Lottery: More Equal Than Equal? 12

4 A Fair Division: But How? 23

5 The Red and the Black: The Law of Large Numbers 27

6 Asymmetric Dice: Are They Worth Anything? 33

7 Probability and Geometry 37

8 Chance and Mathematical Certainty: Are They Reconcilable? 41

9 In Quest of the Equiprobable 51

10 Winning the Game: Probability and Value 57

11 Which Die Is Best? 67

12 A Die Is Tested 70

13 The Normal Distribution: A Race to the Finish! 77

14 And Not Only at Roulette: The Poisson Distribution 90

15 When Formulas Become Too Complex:

The Monte Carlo Method 94

16 Markov Chains and the Game Monopoly 106

17 Blackjack: A Las Vegas Fairy Tale 121

II Combinatorial Games 135

18 Which Move Is Best? 137

19 Chances of Winning and Symmetry 149

20 A Game for Three 162

21 Nim: The Easy Winner! 169

22 Lasker Nim: Winning Along a Secret Path 174

23 Black-and-White Nim: To Each His (or Her) Own 184

24 A Game with Dominoes: Have We Run Out of Space Yet? 201

25 Go: A Classical Game with a Modern Theory 218

26 Mis`ere Games: Loser Wins! 250

27 The Computer as Game Partner 262

28 Can Winning Prospects Always Be Determined? 286

29 Games and Complexity: When Calculations Take Too Long 301

30 A Good Memory and Luck: And Nothing Else? 318

31 Backgammon: To Double or Not to Double? 326

32 Mastermind: Playing It Safe 344

III Strategic Games 353

33 Rock-Paper-Scissors: The Enemy's Unknown Plan 355

34 Minimax Versus Psychology: Even in Poker? 365

35 Bluffing in Poker: Can It Be Done Without Psychology? 374

36 Symmetric Games: Disadvantages Are Avoidable, but How? 380

37 Minimax and Linear Optimization: As Simple as Can Be 397

38 Play It Again, Sam: Does Experience Make Us Wiser? 406

39 Le Her: Should I Exchange? 412

40 Deciding at Random: But How? 419

41 Optimal Play: Planning Efficiently 429

42 Baccarat: Draw from a Five? 446

43 Three-Person Poker: Is It a Matter of Trust? 450

44 QUAAK! Child's Play? 465

45 Mastermind: Color Codes and Minimax 474

Index 481

Preface ix

I Games of Chance 1

1 Dice and Probability 3

2 Waiting for a Double 6 8

3 Tips on Playing the Lottery: More Equal Than Equal? 12

4 A Fair Division: But How? 23

5 The Red and the Black: The Law of Large Numbers 27

6 Asymmetric Dice: Are They Worth Anything? 33

7 Probability and Geometry 37

8 Chance and Mathematical Certainty: Are They Reconcilable? 41

9 In Quest of the Equiprobable 51

10 Winning the Game: Probability and Value 57

11 Which Die Is Best? 67

12 A Die Is Tested 70

13 The Normal Distribution: A Race to the Finish! 77

14 And Not Only at Roulette: The Poisson Distribution 90

15 When Formulas Become Too Complex:

The Monte Carlo Method 94

16 Markov Chains and the Game Monopoly 106

17 Blackjack: A Las Vegas Fairy Tale 121

II Combinatorial Games 135

18 Which Move Is Best? 137

19 Chances of Winning and Symmetry 149

20 A Game for Three 162

21 Nim: The Easy Winner! 169

22 Lasker Nim: Winning Along a Secret Path 174

23 Black-and-White Nim: To Each His (or Her) Own 184

24 A Game with Dominoes: Have We Run Out of Space Yet? 201

25 Go: A Classical Game with a Modern Theory 218

26 Mis`ere Games: Loser Wins! 250

27 The Computer as Game Partner 262

28 Can Winning Prospects Always Be Determined? 286

29 Games and Complexity: When Calculations Take Too Long 301

30 A Good Memory and Luck: And Nothing Else? 318

31 Backgammon: To Double or Not to Double? 326

32 Mastermind: Playing It Safe 344

III Strategic Games 353

33 Rock-Paper-Scissors: The Enemy's Unknown Plan 355

34 Minimax Versus Psychology: Even in Poker? 365

35 Bluffing in Poker: Can It Be Done Without Psychology? 374

36 Symmetric Games: Disadvantages Are Avoidable, but How? 380

37 Minimax and Linear Optimization: As Simple as Can Be 397

38 Play It Again, Sam: Does Experience Make Us Wiser? 406

39 Le Her: Should I Exchange? 412

40 Deciding at Random: But How? 419

41 Optimal Play: Planning Efficiently 429

42 Baccarat: Draw from a Five? 446

43 Three-Person Poker: Is It a Matter of Trust? 450

44 QUAAK! Child's Play? 465

45 Mastermind: Color Codes and Minimax 474

Index 481

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