This book is designed as a one-semester graduate mathematics course in game theory. In all respects it is a fabulous book that covers all the important topics in game theory at a high but easily understandable level. 

The preface gives a brief introduction to game theory, its definitions, and the differences between cooperative and noncooperative games, along with a brief description of the history of game theory, especially the now famous contributions of John Nash to equilibrium theory.  One learns more about the history in each of the six chapters of the book. This is a second edition of a book published in the same series (Volume 115).  It expands the treatment of cooperative and noncooperative games and adds new allocation rules such as the Myerson value.  In essence, the second edition brings the first edition up to date with many new results and applications.

At the heart of game theory is decision theory, and fundamental to decision theory are preference relations.  One asks when weak preference relations can be represented by an utility function, and the answer is given in the first theorem of the book.  Ordinal and linear utility functions are then described in the remainder of the first chapter and used later in Chapter 3 on extensive games.

Chapter 2 defines and discusses strategic games, where all players make their decisions simultaneously and independently.  They are characterized simply by the players and their payoff functions and their set of possible strategies. Players are assumed to be rational and to maximize their own payoff. The concept of equilibrium is introduced, especially Nash equilibrium.  Since the proofs of the key theorems become increasingly complex, they are reserved for the end of the chapter.  The difficulty of the exercises varies and ranges from easy to hard.

Chapter 3 on extensive games focuses primarily on finite extensive games, though it also touches on repeated games at the end of the chapter.  An extensive game consists of a game tree, a player partition, and an information partition that divides the players according to who has the same information, a choice partition of the alternatives at each decision node, a probability assignment, and a set of utilities representing each player’s preferences.  A nice marketing example is given to illustrate all the components of an extensive game.  The possible strategies available in extensive games (especially behavior strategies) lead to Nash equilibria in extensive games, followed by subgame equilibria, sequential equilibria, perfect equilibria, and proper equilibria.  Many interesting examples are included.  Since this chapter is dense with important definitions, the last section on infinite games could be skipped. Again, the exercises are excellent.

Chapter 4 considers games with incomplete information.  This differs from games with imperfect information. Harsanyi, however, showed that both strategic and extensive games with incomplete information can be represented as extensive games with imperfect information. An assumption is made that the only source of uncertainty comes from the payoffs of the game. Bayesian games and Bayesian equilibria provide the probabilistic structure needed to model these games. An interesting application of Bayesian games is auctions, and this book gives a better than usual introduction.  Exercise 4.4 is a particularly nice example of an auction with only two players, a seller and a buyer, and a Bayesian equilibrium.

Chapter 5 considers cooperative games in detail, with emphasis on bargaining problems, again returning to the work of Nash from 1950. Much of the chapter is devoted to transferrable utility games, where coalitions form and utility is transferred. Numerous examples are given at the outset, and the concept of core is introduced.  It is a subset of the set of imputations, the set of all efficient and individually rational allocations (i.e. no player can get less than he can get by himself).  Allocations can be determined by the Shapley rule to create a Shapley value.  Similarly, other values such as the Banzhaf value exist and are critical to various areas of voting research.  Chapter 5 gets quite dense towards the end, so students may struggle with parts of it.  Chapter 6 provides a set of applications of cooperative games including voting problems, linear programming problems, inventory games, assignment problems, and project planning.

I have never seen a bibliography as extensive as that which appears at the end of this book.  Students will appreciate it.  Similarly, the index is complete and easy to follow.

In summary the organization of this book, the clarity of exposition, the carefully constructed examples and exercises, and the attention to the critical aspects of game theory make this a truly fabulous book for the teacher of a graduate-level game theory class or for the mathematician who just wants to learn game theory.

Margaret (Midge) Cozzens is a Distinguished Research Professor at Rutgers University and Associate Director for Education at the DIMACS Center (Discrete Mathematics and Theoretical Computer Science Center).  She taught graduate game theory at the University of Colorado Denver/Boulder in the early 2000s, and she has run Reconnect workshops for faculty in game theory.  She has had a number of REU and graduate students work on problems and projects related to applications of game theory.  Those who know her know she is not given to exaggeration.  This is a fabulous book.