This is an attractive textbook for an undergraduate course in game theory, aimed at people who, though not necessarily mathematics majors, do have at least some background in the subject. The authors state in the Preface that the book was designed for students who have taken differential calculus and linear algebra, but that even students without this background will be able to understand much of the book. This is true, but I would add the caveat that the reader at least has some background in mathematical proof, to the point where he or she can follow, and occasionally construct, one. For example, although the authors do discuss, as the need arises, the ideas behind such things as mathematical induction and proof by contradiction, these discussions are fairly succinct and for many students may not suffice to serve as a substitute for prior exposure to these ideas.

For students with this modest background, however, the book should prove both interesting and informative. The topics covered here are chosen for a broad and versatile look at the subject, the writing style is clear and enjoyable, examples are plentiful, and there is a good selection of exercises, both computational and proof-oriented. (Solutions are not provided either in the text or on the book’s AMS webpage.)

In terms of level of difficulty, I would describe this text as substantially more student-friendly and accessible than *Games, Theory and Applications* by Thomas, but more mathematically demanding than Straffin’s *Game Theory and Strategy*, a book that has only high school mathematics as a prerequisite.

The first four chapters of the book cover combinatorial game theory. These are games in which the players move sequentially, each with full knowledge of what has happened before, and the outcome depends only on these moves and not on the intervention of random chance. (Typical examples include, for example, chess or tic-tac-toe; games like poker or blackjack obviously do not qualify.) In these four chapters, the authors give basic definitions and examples of both general and special kinds of combinatorial games, analyze the strategy in these examples, discuss game trees, and motivate and prove several “big” theorems, including, quite early in the text, a proof of Zermelo’s theorem, which states that in any combinatorial game there is a strategy that guarantees one player a win, or at worst a draw. Thus, for example, there exists a strategy in chess that guarantees that one of the two players (white or black) will never lose, but because of the complexity of this game, nobody has any idea what that strategy is.

The remainder of the book focuses on “classical” game theory, where players make simultaneous moves, the combination of these moves resulting in an outcome. As is standard in game theory texts, the discussion begins with two-person zero-sum games. The authors begin with examples, discuss the concept of domination, and move fairly quickly to the notion of mixed strategies, providing a proof of Von Neumann’s mini-max theorem. As an example of the authors’ concern for clarity and student accessibility, the proof is given in progressive stages. First, the theorem is proved in the \(2 \times 2\) case, where it can be established by a simple and direct calculation. Then, the proof is given in the \(2\times n\) and \(m\times 2\) cases, and only then is a (geometric, non-constructive) proof given for the general case.

In chapter 7, the authors discuss games that are not necessarily zero-sum. The notion of a Nash equilibrium for such games is introduced in the next chapter and applications to evolutionary biology and economics (Carnot duopoly) are given. The existence of an equilibrium is then actually proved, via Sperner’s Lemma and the Brower fixed point theorem, in chapter 9. The proof given in this chapter is for \(2 \times 2\) games, but the arguments generalize, and the details are provided in an Appendix.

Chapters 10 and 11 discuss, respectively, cooperative games and *n*-person games, including, in the latter chapter, a very brief look at weighted voting and the Shapley-Shubik power index. (The Banzhaf power index, a competing method of assessing voting power, is not discussed.) Finally, chapter 12 discusses, in one section each, several topics in the mathematics of social choice: fair division, stable marriages, and voting theory (specifically, a statement and proof of Arrow’s famous impossibility theorem, which loosely states that there is no such thing as a “perfect” preference voting system; for a more extended discussion of this theorem, see our review of *The Mathematics of Elections and Voting* by Wallis.) The three sections in this chapter are independent of one another, so an instructor can pick and choose.

The last eight chapters of the book are for the most part independent of the first four on combinatorial games (with the exception of chapter 1) but, like the authors, I think that there is some advantage to covering at least parts of both in a single course. This allows a class to see two different “flavors” of the subject, and also to see some some connections between them. The concept of a game tree, for example, is used in both contexts, and an exercise in the text shows that Sperner’s lemma can be used to prove that Hex, a combinatorial game, cannot end in a draw.

In addition to these twelve chapters, there are also three substantial appendices. One of them, as noted above, extends the proof of Nash’s theorem from from \(2 \times 2\) games to more general ones. Another introduces the subject of linear programming and explains its relationship to game theory; this was, I thought, a splendid first look at the subject. Another appendix provides a brief introduction to the work of John Conway connecting numbers with combinatorial games.

In addition to clear and engaging writing, and a good selection of exercises, this book also boasts an excellent bibliography. Based on a quick perusal, I couldn’t identify any text or article that I would recommend to a beginning student that wasn’t already listed there. The only mild defect with the bibliography that I noticed was that some items lacked sufficient identifying information. For example, there is a reference to Ferguson’s *Game Theory*, 2d edition, with no identifying publisher or other details. In fact, this book is available online. There is a reference to William Spaniel’s *Game Theory 101: The Basics*, which presumably refers to a series of youtube videos, but which may also refer to a book of the same name that is based on these videos. There are a couple of other examples as well.

If I have any criticism of the book’s exposition of the mathematical topics, it is that, in order to cover a wide variety of topics, some of them are introduced in what seemed to me to be something of a hurry. I think that Arrow’s theorem, for example, would be more meaningful to a student if some specific examples of voting methods (Borda count, for example) were discussed and specific examples given to show that some of the fairness criteria discussed in that theorem were not satisfied. (If nothing else, this could easily be done in the exercises.) And the concept of a mixed strategy would be better motivated if the notion of a saddle point were discussed in some detail first and some explanation given as to why, if a saddle point exists, that points to the best strategy.

But these are relatively minor objections. By and large this book is a very successful one, and I have no hesitation whatsoever recommending it as a text for an introductory undergraduate course.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.