11 of 11 people found the following review helpful
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It is perplexing that the explosion in the popularity of game theory has not resulted in a large number of texts on the subject written by and for mathematicians. One can find entire books devoted to the application of game theory to a single subject: economics and biology, of course, but also business management, linguistics, the analysis of voting systems, political science, psychology, the law, etc. However, the classic introduction for mathematicians, first written by Guillermo Owen in the 1960s, remains the standard in that particular niche several decades and three editions later.
Let the prospective buyer be aware of one thing up front. This book was written by a professional mathematician who wrote many papers in game theory. The book is written for a mathematical audience, and it assumes a level of mathematical maturity roughly equivalent to that of the typical senior majoring in mathematics. The book makes unapologetic use of the calculus of several variables, linear algebra, probability theory, statistics, linear programming, convexity in R^n, some results from topology (fixed point theorems), and even Stieltjes integration. Section IV.2, Games on the Square, provides a beautiful application of Stieltjes integration in the analysis of a duel (game of timing) between two combatants; I remember studying Stieltjes integration out of Apostol's advanced calculus text as an undergraduate and wondering what possible use the subject could find. Here is one answer.
Owen last updated the book after the 1994 Nobel prizes in Economics were awarded to John Harsanyi, John Nash, and Reinhard Selten. Many of the topics these men studied, such as subgame-perfect equilibria and games with incomplete information, have been included in the Third Edition. There is even a (very) brief introduction to evolutionary game theory and evolutionary stable strategies, but it occupies only a few pages.
This last remark illustrates the dilemma facing the mathematician who wants to study contemporary game theory. Owen's book remains the unrivaled reference for studying classical von Neumann--Morgenstern game theory from an advanced mathematical point of view. However, much of the activity in game theoretic research during the past three decades has occurred in evolutionary game theory, a subject first created by biologists. To the best of my knowledge, no mathematician who specializes in the field has written the sequel to Owen, providing a mathematically rigorous introduction to evolutionary game theory. It is a reference that is much needed. The handful of books on evolutionary game theory that are most often cited are written by economists (2) and biologists (2); these are wonderful books and serve their intended audiences well, but they are unlikely to be used in advanced mathematics courses where the emphasis is on providing rigorous proofs of all the results that are used. The emphasis and approach is just too different.
If you are looking for a definitive reference on von Neumann--Morgenstern game theory that is written with mathematicians in mind, then Owen remains the first choice. Even as I use other, less demanding books to teach my undergraduates, I find that I continually return to Owen to get the mature view on various topics. For a first course in the subject for undergraduate mathematics majors, I would prefer the book by A. J. Jones; it is far more accessible. But as a reference for the working mathematician, I know of no rival for Owen at the present time.
11 of 15 people found the following review helpful
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I believe this book to be an excellent introduction to game theory, also for self - study purposes. Unfortunately, it seems to contain too many misprints, which are particularly annoying if it is used for self study: You always wonder,whether you are wrong, or the book is. This is all the more amazing as I read the third edition.
4 of 10 people found the following review helpful
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I think this book is definitely aimed at people with at least college level maths background, by which I mean you actually majored in maths, not just took some maths classes. Pretty much the enitre book is filled with theorems, definitions and advanced formuli that require knowledge of advanced calculus, set theory, linear algebra, and graph theory (to name a few) to understand.
It has very little in the way of examples and actual explanations. For example, in chapter 8, when talking about equilibrium pairs, the book doesn't give any methods to actually calculate equilibrium pairs, but just give a whole collection of theorems and definitions that explain what equilibrium pairs are, and then referred readers to other books for the methods to actually calculate them.
If you don't have an advanced maths background, the first problem you'll have reading this book is just understanding all the mathematical notations used in this book. If you can get past that, then you're still faced with trying to understand the implications of all the theories and formuli presented in the book. Suffice it to say that I still couldn't understand 75% of the material in this book, even though this was a textbook for a class I took and I had a professor to explain to me what the book is actually saying.
I understand that this book is supposedly some kind of a seminal work in game theory. But it definitely is not intended for someone seeking an introduction to game theory or someone trying to learn by him/her self.