It has been my experience that when a mathematician hears the phrase "applied mathematics" they are most likely to think of thing such as heat equations and partial differential equations. On the other hand, when a civilian hears the phrase they are more likely to think of questions of probability, statistics, and game theory. T. W. Körner has embraced this latter definition, and written a delightful book entitled Naive Decision Making, which looks at a number of ways in which mathematics can be used to help make a wide variety of decisions in social settings. He constructs a wide variety of models investigating a wider variety of questions and along the way delves into some of the history of the topics as well as giving examples from literature.
Most of the topics covered in his book are things that will be familiar to most MAA members, but many of the individual applications and historical information will not be. The author admits that most of his models will be simplistic and ignore how people actually behave, acknowledging that he may be "like an astronomer who locks herself in a windowless room in order to think undisturbed," but defending the approach by arguing that "it is remarkable how successful [this approach] is in the few cases when it does work."
The book opens, as any good probability book will, with discussions of gambling. In fact, the entire first chapter is entitled "A Day At The Races" and considers questions of placing bets on horses in a variety of situations, depending on how much information you have about the horses, whether you or the bookie bets first, and whether you are allowed to bet negative money, along with other wrinkles. This discussion introduces many key notions in probability theory, which get formalized in the second chapter with discussions of random variables, independence, and many other topics. The discussion may not be at the level one would want from an introductory textbook on probability, but it would be a perfect chapter to assign students as supplementary reading to get a slightly different point of view from the standard treatments.
The third chapter is entitled "The vice of gambling and the virtue of insurance," a title taken from an essay by Bernard Shaw. The chapter opens with a lengthy excerpt from Shaw's essay, and goes on to discuss the issues which arise in this excerpt in more mathematical depth. In particular, Körner discusses annuities and how much we should be willing to pay for them. The book then takes a darker tone (something I never thought I would write in a review of a math book, I have to admit), as there is a section discussing smallpox and its effect on seventeenth century London. There is a lengthy discussion of Daniel Bernoulli's use of life tables to investigate whether the risks of smallpox inoculation outweighed the benefits, and the reader is led through a series of calculations designed to help answer this question. This discussion leads Körner to consider questions of utility functions, and whether people would really choose to maximize their expected lifetime.
Chapter four changes gears and introduces a number of topics from elementary number theory, with an eventual goal of discussing card shuffling and card tricks such as the ones performed by Mr. Jonas in Charles Dickens' Martin Chuzzlewit. Along the way, the book also discusses a bit of cryptography and the Tower of Hanoi problem.
The fifth chapter deals with the question of un-shuffling a deck of cards, and algorithms to efficiently sort the deck. This leads naturally to problems such as The Secretary Problem, in which the goal is to maximize the chance of hiring a good secretary despite not knowing what the full applicant pool will look like, and finding shortest paths in trees.
Other problems considered in the later chapters include designing good voting schemes (for various definitions of "good") and how to match couples for marrying in a way that maximizes the total happiness of the group. The book has a couple of chapters dedicated to game theory, in which the reader will learn how to strategically play Rock-Paper-Scissors and some variants such as Morra, as well as how to behave when one finds oneself in a three-way duel to the death.
With the warning that the early chapters of the book should have convinced the reader to gamble only in rare circumstances, Körner has a chapter entitled "How To Gamble If You Must", looking at a few games of chance and analyzing some strategies for them using simple difference equations and differential equations. The author then returns one more time to probability theory and discusses the binomial distribution in more depth, including a discussion of Hoeffding's inequality and how it can be used to help conduct trials of new drug treatments.
The book closes with a contemplative chapter about the nature of mathematical modelling and its utility in the real world. The author concludes by again acknowledging some of the failings of his models, but pointing out that "mathematicians should not be unduly worried by the limitations of mathematics as a tool for studying the real world. The object of mathematical study is not power or even utility, but pleasure."
Whether or not you agree with this conclusion, Naive Decision Making is a book full of pleasure. The writing is crisp, clear, and full of wit. Körner has managed to strike a nice balance between chatty and technical, between formal proofs and illustrative examples, and between breadth and depth of the topics considered. When trying to decide where on my bookshelf this book belongs, I had a very hard choice to make: There is quite a bit of history, but this is not a history book. There is quite a bit of theory explained from an elementary level, but this is not a textbook. Most of the text is written in the style of one of the many "pop math" books that have come out in recent years, but then you run into a page of equations. Körner certainly assumes some familiarity with calculus and combinatorics and the notions of proof and abstraction that a layperson would be unlikely to have, but he does not assume much more than that.
Luckily, the decision of where to put the book on my shelf has been a theoretical one so far, as I keep lending my copy out to other people. I have already referred a number of my students and colleagues to this book, and I am sure I will refer more in the future. In short, recommending Naive Decision Making is an easy decision to make.
Darren Glass is an Assistant Professor of Mathematics at Gettysburg College whose research interests include Number Theory, Algebraic Geometry, and Cryptography. He can be reached at firstname.lastname@example.org.