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Ken Binmore is the broadest thinker working within the classical game theory tradition. Unlike most technicians, he has read widely in philosophy, history, and anthropology, combining a passion for analytical detail with a deep feeling for the broad strokes of human behavior. These characteristics are reflected in this textbook on game theory, which is light-years more sophisticated than the standard fare, yet never sacrifices clarity or expositional elegance on the alter of mathematical or notational rigor. While I would urge anyone who is not math phobic and can recall a bit of high school algebra to tackle this book as an introduction to game theory, I am afraid it will not be widely used in courses because most instructors simply will not have the personal intellectual resources to teach this material. This is because Binmore tackles some of the deepest issues in game theory, whereas most instructors will have had the standard graduate course in which these issues are totally ignored. Moreover, in the interest of clarity, Binmore does not supply the full analytical frameworks in which these deep issues are normally cast, so the instructor will have few resources to deal with the material in a classroom setting. On the other hand, each chapter has plenty of problems that an instructor could use to illuminate the text, say by assigning half to the students and solving some of the remaining problems in class.
Like every textbook writer before him, Binmore treats the Nash equilibrium with great reverence as a solution concept. I consider this a significant error, but at least Binmore tries to explain why (p. 18-19). His answer is sufficiently weak that the critical reader might decide to explore the issue himself. Binmore does not present a set of sufficient conditions under which agents will play a Nash equilibrium (for instance, as presented in the famous paper by Aumann and Brandenburger, 1995). Had he done that, the student might have a better idea of why the Nash equilibrium criterion is of limited value. Binmore's defense of the Nash concept draws on evolutionary game theory, but a notable absence from the book is a treatment of evolutionary game theory. A possible reason for this omission is that the math involved is fairly advanced (dynamical systems theory), but there are versions that avoid these technicalities for beginners (evolutionary stable strategies and stochastic dynamical systems a la Thomas Schelling, Robert Axtell, and Peyton Young).
Among the refreshing positions taken by Binmore in this text is that equilibrium refinements are generally not worth much, except for subgame perfection, and even that is highly suspect except in special situations. Whereas backward induction (a.k.a. finding subgame perfect equilibria) is treated with great reverence in most text books, the technique has been under constant attack theoretically, and it is well known that individuals generally do not use more than a few rounds of backward induction. Binmore actually presents "The Surprise Test" (pp. 45-46) which I believe reveals the deepest contradictions of backward induction, although Binmore believes that the example shows nothing and has a simple non-paradoxical resolution. I believe he is wrong. Binmore's answer is that the teacher makes two statements (you will be tested on day next week, and when the test occurs, you will be surprised). Backward induction shows that the teacher's statement is false, but the student is mistaken by inferring that he will not be tested, since it could be the other half of the teacher's statement that is false. However, in fact, the test is given on Monday, and the student is surprised. So, the teacher was correct, contrary to the backward induction reasoning. Binmore is wrong, because the student was indeed surprised.
Binmore does not particularly care for the concept of rationalizability (it isn't mentioned until p. 424) because it assumes nothing but Bayesian rationality with arbitrary priors. I think this is an error, because it leads him away from an investigation of when even rationalizability is violated. Thus, on p. 153, he confidently asserts "a rational player will never use a strongly dominated strategy." Yet, there are many games of strategic complementarity (e.g., Carlsson and van Damme, 1993), not to mention Basu's Traveler's Dilemma, in which the iterated elimination of strong dominated strategies leads to a unique Nash equilibrium that no collection of reasonable players would ever play. Binmore presents Basu's game in the problems on p. 174, and shows that if players don't care about small amounts of money, there is a plausible Nash equilibrium. This is an interesting idea that is pursued in different ways throughout the book, but is not systematically developed.
One of the most embarrassing questions for classical game theory is why anyone would ever play a mixed strategy in a one-shot game. There are a couple of important attempts at answering this in the literature, and Binmore presents them uncritically. This is uncharacteristic of him. The attempt to define an equilibrium in "conjectures" solves the problem, but says nothing about how people actually play. Binmore presents the usual example of the plausibility of this approach, which uses Throwing Pennies, in which each player "conjectures" the other will use heads or tails with equal probability. But, what if the equilibrium probabilities are 99/100 and 1/100? Why shouldn't the players still play 1/2 and 1/2, in fact? The alternative, Harsanyi's purification theorem (p. 445) deals with this issue better, but it has its own serious limitations, which Binmore does not mention.
Binmore's chapter on game theory and ethics is a gem, and his put-down of Kant in the introductory paragraph is just choice. Since Binmore has written at least three books on this subject, I would have expected more, but this book gives a foundational treatment. Binmore is a noted critique of behavioral economics, which he takes as being an enemy of game theory. However, behavioral economics is bare mentioned in this text, and never in a disparaging way. I think one of the major contributions of game theory is to the methodology of empirical economics, but this aspect of classical game theory is slighted in Binmore's text.
There is much excellent material in this book that I have not had the space to mention, including bargaining and auctions, to which Binmore has personally contributed so much. This book is way beyond virtually all others in exposing the reader to the nitty-gritty issues of classical game theory. Whether that speaks for or against it's being a commercial success remains to be seen.