Luck, Logic, and White Lies

The Mathematics of Games

Jörg Bewersdorff

• Publisher: A K Peters/CRC Press
• Year: 2021
• ISBN:
  9781000372090
• Edition: 2
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

Praise for the First Edition

" Luck, Logic, and White Lies teaches readers of all backgrounds about the insight mathematical knowledge can bring and is highly recommended reading among avid game players, both to better understand the game itself and to improve one's skills."
– Midwest Book Review

"The best book I've found for someone new to game math is Luck, Logic and White Lies by Jörg Bewersdorff. It introduces the reader to a vast mathematical literature, and does so in an enormously clear manner..."
– Alfred Wallace, Musings, Ramblings, and Things Left Unsaid

"The aim is to introduce the mathematics that will allow analysis of the problem or game. This is done in gentle stages, from chapter to chapter, so as to reach as broad an audience as possible... Anyone who likes games and has a taste for analytical thinking will enjoy this book."
– Peter Fillmore, CMS Notes

Luck, Logic, and White Lies: The Mathematics of Games, Second Edition considers a specific problem—generally a game or game fragment and introduces the related mathematical methods. It contains a section on the historical development of the theories of games of chance, and combinatorial and strategic games.

This new edition features new and much refreshed chapters, including an all-new Part IV on the problem of how to measure skill in games. Readers are also introduced to new references and techniques developed since the previous edition.

Features

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• Provides a uniquely historical perspective on the mathematical underpinnings of a comprehensive list of games
• Suitable for a broad audience of differing mathematical levels. Anyone with a passion for games, game theory, and mathematics will enjoy this book, whether they be students, academics, or game enthusiasts
• Covers a wide selection of topics at a level that can be appreciated on a historical, recreational, and mathematical level.

Jörg Bewersdorff (1958) studied mathematics from 1975 to 1982 at the University of Bonn and earned his PhD in 1985. In the same year, he started his career as game developer and mathematician. He served as the general manager of the subsidiaries of Gauselmann AG for more than two decades where he developed electronic gaming machines, automatic payment machines, and coin-operated Internet terminals.

Dr. Bewersdorff has authored several books on Galois theory (translated in English and Korean), mathematical statistics, and object-oriented programming with JavaScript.

*Here is the list of Errata for the second edition of Luck, Logic, and White Lies: The Mathematics of Games: http://bewersdorff-online.de/LLWL-errata.pdf

Part I

Games of Chance

1

Dice and Probability

With a pair of dice, one can throw the sum 10 as the combination either 5 + 5 or 6 + 4. The sum 5 can also be obtained in two ways, namely, by 1 + 4 and 2 + 3. However, in repeated throws, the sum 5 will appear more often than 10. Why?

Although we are exposed in our daily lives to a variety of situations involving chance and probability, it was games of chance that provided the primary impetus for the first mathematical investigations into this domain. Aside from the fact that there is a great attraction in discovering ways of winning at the gaming table, games of chance have the advantage over the rough and tumble of real-life events that chance operates in fixed and precise ways. Thus, the odds, dictated by the laws of probability, of throwing a six, say, are much simpler to calculate than the odds that a bolt of lightning will strike the Eiffel Tower on July 12 of next year. The reason for this is primarily that the situation in a game of chance is reproducible under identical conditions, with the consequence that theoretical results can be checked by experiments that can be repeated as often as one likes, if the results are not already well known as facts of common experience.

The first systematic investigation of games of chance began in the middle of the 17th century. To be sure, there was some sporadic research earlier; indeed, in the 13th century, the probabilities of the various sums that can be obtained with a pair of dice were correctly determined.¹ This fact is particularly noteworthy in that over the following centuries, many incorrect analyses of the same problem appeared. The first to create a systematic, universal approach to the description of problems of chance and probability was Jacob Bernoulli (1654–1705), with his Ars coniectandi (The Art of Conjecture). According to Bernoulli, its object was “to measure the probability of events as precisely as possible, indeed, for the purpose of making it possible for us in our judgments and actions always to choose and follow the path that seems to us better, more worthy, more secure, or more advisable.”² Bernoulli had in mind not only games of chance, but also the problems of everyday life. His belief in the necessity of a mathematical theory of probability is alive and well even today. This was formulated with admirable concision by the renowned physicist Richard Feynman (1918–1988): “The theory of probability is a system for making better guesses.”

¹ R. Ineichen, Das Problem der drei Würfel in der Vorgeschichte der Stochastik, Elemente der Mathematik 42, 1987, pp. 69–75; Ivo Schneider, Die Entwicklung der Wahrscheinlichkeitstheorie von den Anfängen bis 1933, Darmstadt 1988, pp. 1 and 5–8 (annotated references). A historical overview of the development of the calculation of probabilities can also be found in the appendix to the textbook by Boris Vladimirovich Gnedenko, Theory of Probability, Berlin 1998.

Of central importance in Bernoulli’s theory is the notion of probability, which Bernoulli called a “degree of certainty.” This degree of certainty is expressed by a number. As inches and centimeters measure length, so a probability measures something. But what exactly does it measure? That is, what sort of objects are being measured, and what qualities of those objects are the subject of measurement?

Let us first take a single die. We can describe the result of throwing the die by saying, “the result of throwing the die is equal to 5,” or “the result is at most 3.” Depending on what was actually thrown, such a statement may ...