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Luck, Logic, and White Lies: The Mathematics of Games

Jörg Bewersdorff
A K Peters
Publication Date: 
Number of Pages: 
[Reviewed by
Donald L. Vestal
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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

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