Evolutionary game theory was introduced by the biologist John Maynard Smith almost 40 years ago to model the evolution of individual behaviors under natural selection in biological species. In particular, his concept of an evolutionarily stable strategy (ESS) is used to predict the eventual equilibrium outcome of selection without a detailed analysis of the underlying evolutionary dynamics. More recently, evolutionary game theory methods have spread to many other disciplines, especially those interested in human behavior when individual rational decisions depend on how everyone else is behaving (i.e. on the population state).

*Population Games and Evolutionary Dynamics* by Bill Sandholm has successfully carried out the author’s aim stated at the beginning of the Preface: to “provide a systematic, rigorous, and unified presentation” of these methods. After a brief introductory chapter that motivates evolutionary approaches to population games and puts these in historical context, the remaining 11 chapters are grouped into four parts summarized below.

The book will make an excellent graduate text for an economics course on evolutionary games whose students have taken an introductory course in classical game theory and have a reasonably strong mathematical background to the second year level. It is also suitable for mathematics students interested in evolutionary game theory, but I feel that for students in other disciplines such as biology it will be too difficult except for the highly motivated.

The theory is illustrated by many standard introductory examples as well as others that that will challenge the more advanced students. Especially noteworthy features that enhance its appeal as a textbook are the many exercises (of different levels of difficulty) throughout; the extensive mathematical appendices for most chapters that make the book largely self-contained; the liberal use of geometric intuition and accompanying (color) diagrams; the comprehensive up-to-date bibliography at the end of the book as well as detailed notes on connections to the literature at the end of each chapter. These features make the book quite useful as well for the researcher interested in evolutionary games. In fact, many open problems and directions for future research are discussed.

Part I lays the foundations for population game dynamics, not only through the conventional analysis of Nash equilibrium (NE) but also through a thorough discussion of selection forces when the population is not in equilibrium. No explicit dynamics are given here. The emphasis is on geometric intuition and on three important classes of population games (namely; potential games, stable games, supermodular games) that are used throughout the book.

Part II develops the standard deterministic evolutionary dynamics (replicator dynamics, best response dynamics, etc.) as special cases of the population mean dynamics corresponding to general revision protocols that describe how rational individuals change their strategy choice based on their current information. Part III summarizes known convergence and stability results (both local and global) for NE and ESS of different classes of games and different dynamics using Lyapunov and/or linearization techniques. It also provides an up-to-date discussion of games where standard evolutionary dynamics do not converge (instead exhibiting cyclic or even chaotic behavior).

Part IV investigates the stochastic effects inherent in these revision protocols due to finite population size. It begins by showing that the deterministic dynamics are a good approximation when populations are large and evolution is considered over a fixed finite time interval. Finally, over indefinite time intervals, many of the recent results on the long-term stationary distribution and on stochastic stability are given.

Each of these four parts contains material that is appearing in book form for the first time. The author has done such a good job in connecting and relating the material from the different parts that the reader is left with the impression that the logical organization of these topics is an easy task as he/she enjoys learning about evolutionary games.

Ross Cressman is Professor of Mathematics at Wilfrid Laurier University in Waterloo, Ontario, Canada. He has written two books on evolutionary games: *The Stability Concept of Evolutionary Game Theory *(Springer, 1992); *Evolutionary Dynamics and Extensive Form Games* (MIT Press, 2003).