Game Theory through Examples

Erich Prisner's Game Theory Through Examples is an exemplary contribution to the MAA's series of Classroom Resource Materials. The topics include simultaneous games and sequential games, both zero-sum and non-zero-sum, and with both perfect information and randomness. Mixed strategies for games are discussed relatively late, in chapter 27, but nine example chapters follow, so this topic also is available for exploration. Prisner provides a wealth of example games for students to explore, either individually or in groups, with questions that can take a few minutes to answer to projects that can take a couple of weeks. The electronic format of the text allows easy and timely access to a set of Java applets that allow readers to experience the games being discussed and to a set of Excel spreadsheets that carry out the computations required to analyze the games, so that students need not be bogged down in the mechanical aspects of the subject. Typically, the applets and spreadsheets can be modified by the students so that they can pose questions about variants and explore these on their own.

As Prisner mentions in a helpful Preface, he has included far more material than could be covered in a one-semester course; he provides a helpful guide to the text that displays the relationships among the eight theory chapters (each of which also includes example games), a chapter giving students a brief description of Excel, five history chapters that give the students a sense of the development and relevancy of the ideas, and twenty-four (!) chapters that discuss concrete examples.

Prisner suggests a number of possible audiences for the book, including students in economics, political science, computer science, and mathematics; I believe that the explanations are clear enough, though terse, and the examples rich enough to allow all these groups to enjoy the book. He suggests that his book would be a good secondary text; I agree with this, for if I were teaching from it or using it for independent study, I'd like to have a primary text with more development of the theory and have Prisner's book on hand for concrete examples.

But the principal audience for the text, the one Prisner had in mind, is a general education course in mathematics. The computations in the book are carried out at a pre-calculus level, perhaps even a level of high school algebra, while the level of the ideas and abstractions are appropriate for a college-level audience. It would not fit the particular strictures of a general education mathematics course at my institution, but I can certainly imagine using this text as the basis of an honors seminar with students from a variety of disciplines. And the concrete examples would also assist me in the variety of courses in which I do discuss some topics from game theory (combinatorics, linear programming, the history of mathematics, and graduate courses for middle grades and secondary teachers). My students will particularly enjoy sequential games reminiscent of TV quiz shows, Deal or No Deal, "Waiting for Mr. Perfect," spending money for advertising in elections (especially here in Iowa!), and the simpler variants of poker.

There are a small number of errors in the text, few of which will cause students any difficulty. Exceptions could be on page 201, where the most frequent strategy is said to be scissors rather than paper, and on page 206, where the lines drawn in Figure 27.1 should be described as running from \((0,A_{2,i})\) to \((1, A_{1,i})\) rather than from \((0, A_{1,i})\) to \((1, A_{2,i})\). 

Throughout the book, I am impressed by Prisner's passion for the topics presented. He also provides thoughtful and intelligent commentary on the value of considering psychological and social utility in payoffs (rather than only money), the certainty that not all humans are rational actors in every situation, and the morality of using game theory in settings of wars and duels. It is clear that he has taught this material and listened to his students.

To summarize, this book would be a valuable resource of examples for any instructor teaching topics from game theory. It would also be a good text for a general education course that seeks to engage the students in exploratory projects in this aspect of applied mathematics.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.