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Reviewer: Carlos Linares Lopez

Beyond its obvious attractiveness, game theory is one of the fundamental techniques in decision systems. Its development spans almost a century and is still an open avenue of research. However, since it's crowded with mathematical difficulties that make the methods cumbersome to read and apply, most newcomers to the field find it very difficult to understand. This is not true, though, as this short book nicely shows. As Leyton-Brown and Shoham notice in the preface, it is rather difficult to imagine that anyone who wishes to enter the field of game theory is unaware of many, if not all, of the concepts discussed in the text. Therefore, this is a book with a concise, clear, and organized introduction to the main concepts of game theory. For the ambitious reader, the book often discusses current scientific research. In other words, it is a gentle introduction, covering the basics of game theory to the current state of the art, in easy-to-read and understandable language. The book is very well organized. The chapters follow one another naturally and, surprisingly, the level of difficulty is kept the same throughout the book-compared to other technical books where the difficulty increases as reading progresses. It also does an excellent service of providing a wonderful motivation for the study of game theory, without getting into very involved discussions on rationality and other related concepts that often appear in other books on game theory. The book covers the main topics-representational issues and different sorts of equilibria computing-with clarifying examples. The first chapter is dedicated to the description of games in normal form, the one commonly used, considering all the usual restrictions, such as single played, noncooperative, perfect information, and complete games. These restrictions are dropped one by one in subsequent chapters. The second chapter introduces the intuitive idea of Pareto optimality. It briefly introduces the notion of Nash equilibrium, which is revisited many times in the next chapters in connection with other ideas. The third chapter completes the study of games with simultaneous actions. It discusses a wide variety of equilibria, from the minimax/maximin values to the dominated strategies, and from here to the correlated equilibrium, the trembling hand, the epsilon-Nash equilibrium, and the evolutionary stable strategies. Readers familiar with games as treated in artificial intelligence (AI) will be delighted with the discussion on games with sequential actions, introduced in chapter 4. The last four chapters are devoted to the study of games where the usual restrictions do not apply-that is, those that result from some relaxation. For example, chapter 5 considers the case of imperfect information games, where the player ignores the current state and, therefore, the actions available. Other equilibria exist, and the book explains how to compute them. As in the real world, the strategies used might have to be reconsidered if the game is played repeatedly. The problem is far from easy, and the book introduces the main ideas behind its study. It also discusses and briefly introduces a well-known folk theorem. Chapter 7 addresses the study of games where the agents ignore the games they are playing, provided that some restrictions are met. The book ends with the interesting-but often neglected-case of coalitional games. The book does not end every chapter with a collection of exercises, but the alert reader will find, scattered throughout the book, key questions that better serve this purpose. It also doesn't end each chapter with a proper discussion of historical remarks, but the expositions are often complemented with annotations of who did what and why. Also, although the book does not provide chapter cross-referencing in the beginning, all chapters in the book have plenty of pointers to other sections that make it easy to read them in a different order and still grasp the main concepts. Despite its small size, this book presents a lot of information about game theory. Online Computing Reviews Service