Mathematics

THE MATHEMATICS OF GAMES AND GAMBLING

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The purpose of this book is to introduce and develop some of the
important and beautiful elementary mathematics needed for rational
analysis of various gambling and game activities. While the only formal
mathematics background assumed is high school algebra, some enthusiasm
for and facility with quantitative reasoning will also serve the
reader well. The book will, I hope, be of interest to:
1) Bright high school students with a good mathematics background
and an (often related) interest in games of chance.
2) Students in elementary probability theory courses who might appreciate
an informal supplementary text focusing on applications to
gambling and games.
3) Individuals with some background in mathematics who are interested
in some common and uncommon elementary game-oriented
applications and their analysis.
4) That subset of the numerate gambling and game-playing public who
would like to examine the mathematics behind games they might
enjoy and who would like to see mathematical justification for what
constitutes '"good" (rational) play in such games.
One guiding principle of the book is that no mathematics is introduced
without specific examples and applications to motivate the theory. The
mathematics developed ranges from the predictable concepts of probability,
expectation, and binomial coefficients to some less well-known ideas
of elementary game theory. A wide variety of standard games is considered
along with a few more recently popular and unusual activities such
as backgammon and state lotteries. Generally it is not the play of the
game. but the reasoning behind the play that provides the emphasis.
Some readers may be temporarily slowed down or frustrated by the
omission of detailed rules for a few of the games analyzed. There are two
important reasons for such omissions. First, I believe that initial exposure
to a new game should be acquired through experience rather than a
formal study of rules. Thus, I would encourage readers unfamiliar with
backgammon to find knowledgeable partners and to learn by playing. In
a classroom setting, the promise of a class tournament is an excellent way
to encourage student~ to master the rules before any mathematical
analysis is engaged in. Secondly, the book is not intended to be a "how
to" document and I would urge readers interested in detailed rules and to consult one of the many specialized books on the particular topic (see
the Bibliography). I give little "advice" that is not justified by prior
mathematical analysis. In most cases it would be unreasonable to claim
that this book will directly improve one's gaming ability, though I would
hope that the insights provided into rational analysis and play will have
some spinoff in that direction. Exceptions to this disclaimer may be
backgammon, which is treated in some detail. and the various gambling
house games, for which it is made quite clear (blackjack being a possible
exception) that the optimal strategy is abstinence.
After an historical and literary initial chapter. subsequent chapters
(except for the last) each include from 10 to 12 carefully selected
exercises. These game-related questions are not primarily intended as
drill problems, but rather for readers genuinely interested in expansion of
ideas treated in the text. Some of the later exercises for each chapter may
be quite challenging and some questions are open-ended, with emphasis
on analysis and explanation rather than numerical answers. Chapter 2
provides the foundation for most of what follows, and should be mastered.
The first two sections of Chapter 4 are needed also for Chapter 5.
Otherwise chapters are. with minor exceptions, independent of one
another. Sections referring to particular games can be included or omitted
according to taste. The Bibliography presents, among other things, full
reference information for books and articles referred to in abbreviated
form in the text.
No serious judgments on the morality of gambling are intended in the
book, though readers will no doubt draw inferences from the tone and
tenor of my remarks at various points. While it should be clear that I am
enthusiastic about games of chance and skill. I hope it is equally clear
that I am even more enthusiastic about the mathematics behind such
games. It is this latter enthusiasm that I hope will be transferred to the
reader.
Numerous institutions and individuals have helped me in shaping this
book. Lake Forest College and three separate groups of students took
their chances with me over the last 3 years in an experimental freshman
level course from which the final text has evolved. The Division of Social
Sciences and Humanities at California Institute of Technology provided
office space and technical assistance. Members of the New Mathematical
Library subcommittee of the Mathematical Association of America have
provided exceedingly thorough and helpful comments on the manuscript.
I would also like to thank Bruce Cain. Charles Maland, and Kathryn
Rindskopf for their valuable suggestions at various stages in the writing
process. Finally. I am indebted to Gertrude Lewin for her typing of the
first draft and Barbara Calli for her lovely artwork and typing of the
final draft.

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Language

English

ISBN/ISSN

0-88385-600-X

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NONE

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Edition

3rd Edition

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