MathSciNet has book reviews

Most of the reviews in MathSciNet are for articles, primarily from journals, but also from proceedings volumes and other collections.  Did you know that we also review books?  A book-length treatment of a subject allows the author to stretch out his or her thoughts.  You are expected to explain things in more detail.  Be more expansive.  Make more connections.  Discuss the context and the history of the subject.  As a consequence, a review of a book can offer the reviewer the chance to likewise spread out.  Sometimes, a book review can be fun and still fulfill its purpose of telling the reader about what’s in the book.  Here, Debora Di Caprio and Francisco J. Santos-Arteaga team up to review a book about epistemic game theory.  They start their review by talking about a scene from The Princess Bride.  The movie reference is entirely appropriate, and engages the reader quite nicely.  

MR3155151 
Perea, Andrés
Epistemic game theory.
Reasoning and choice. Cambridge University Press, Cambridge, 2012. xviii+561 pp. ISBN: 978-1-107-40139-6

Many of us in our late thirties came to know about epistemic game theory during our childhood. We did not know that we knew but that did not prevent us from reaching our own set of equilibria. Plus it was a really fun experience to argue with friends and try to outsmart each other. We are of course talking about the film The Princess Bride and its wine on the table scene, where we were introduced to the rational thinking process of Vizzini and listened to him as he formed his beliefs about the beliefs of a masked man regarding his own beliefs, to finally reach, in his case, a suboptimal (though rationally and reasonably chosen) equilibrium. Here is part of that belief formation process (whose copyright belongs to Warner Brothers):

Masked Man: The battle of wits has begun. It ends when you decide and we both drink—and find out who is right, and who is dead.

Vizzini: But it’s so simple. All I have to do is divine it from what I know of you. Are you the sort of man who would put the poison into his own goblet or his enemy’s? Now, a clever man would put the poison into his own goblet because he would know that only a great fool would reach for what he was given. I am not a great fool so I can clearly not choose the wine in front of you. But you must have known I was not a great fool; you would have counted on it, so I can clearly not choose the wine in front of me.

Many years, and several game theory courses, later, we came to terms with the main concepts behind this type of reasoning but we lost part of our childhood wit to formalism. The current book by Andrés Perea brings back the joy of reading about reasoning, beliefs and games and going once again through that first epistemic childhood experience. In other words, this book is for all of you who have rearranged pennies and dimes on a chess board while reading Thomas C. Schelling’s Micromotives and macrobehavior [Norton, New York, 2006] and helped Alice solve the many dilemmas put forward by Ken Binmore.

A bit of formalism is needed to balance the discourse. Epistemic game theory is the study of one’s reasoning process at work in any game-like situation, where the final outcome depends on all players’ choices. It analyses one’s way of reasoning about his own way of thinking and that of all his opponents and what changes in the way of reasoning affect which choices and, hence, the final outcome. It focuses on one’s beliefs about others’ beliefs about others’ beliefs, and so on. It subordinates the concept of optimal choice to that of belief, invalidating the intuitive equivalence between rational and reasonable.

The book is very well organized and very clearly written. Each chapter opens with practical examples based on very simple everyday life situations that put the reader immediately at ease despite the complexity of the argument to be unfolded. Several diagrams and tables support the presentation. In particular, the graphical interpretation of the situations through “belief diagrams” is perfect for understanding the difference between optimal and rational or reasonable choice, and how the equilibria are determined by the subjective beliefs of the player regarding the rationality of the opponents. After the examples and the graphical analysis, an algorithm is introduced to describe how to get to the optimal choice based on the type of beliefs considered in the corresponding chapter. Formal definitions and proofs appear in the second half of the chapters. All the notations and the mathematical tools used are precise and well-explained.

The book is divided into three main blocks, dealing respectively with

• The formation of beliefs in static games (Vizzini forms his beliefs and tells us about them).
• The inclusion of lexicographic beliefs in static games (Vizzini determines his primary beliefs and switches goblets before drinking).
• The revision and formation of conditional beliefs in dynamic games (Vizzini plays the game in a sequel; in this case, with the very same final outcome).

The reader does not need any particular mathematical background to follow the reasoning of the first part of the chapters: an understanding of the notations introduced in the previous chapters is sufficient. Regarding the technical parts, the more expert the reader, the more he will appreciate the style and the coherence of the proofs.

This is a very enjoyable book, from the first to the last page! It is suitable for both undergraduate and Ph.D. students, and will appeal both to the curious reader and to the specialized researcher.

All in all, this is a book that should be read by economists and game theorists, general math enthusiasts and, by all means, anyone who wants to challenge his or her capacity to reason rationally within a strategic environment and enjoy doing so. The unfolding of different dominated equilibria throughout the examples of different chapters should be sufficient for anybody, even without any knowledge of game theory, to enjoy a really fun ride. We know it. You know that we know it and we know that you know that we know it.

Reviewed by Debora Di Caprio and Francisco J. Santos-Arteaga