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.
