I offered an undergraduate course in game theory two years ago, and as I performed the usual search for a course text, I was surprised by the shortage of current, up-to-date books on the subject that are written specifically for advanced students of mathematics. There are scores of books on game theory written by and for economists and social scientists; however, most of those books are simply not appropriate for a more rigorous, proof-based course in which the mathematics takes central stage.
After examining dozens of books, I reached the following conclusions.
(1) There is a wonderful old book (1952) written by J. C. C. McKinsey which is mathematically rigorous and written with such clarity that any good student of mathematics could read the book on his own and learn the subject through independent study. Sadly, McKinsey's book (now reprinted by Dover) is too out of date to serve as the primary reference for a modern course. It is still a useful supplement, however. (2) The "classic" mathematical reference by Guillermo Owen is now available in an updated 3rd edition and includes new material on topics of recent interest, such as evolutionary game theory, bargaining, indices of voting power, etc. But while I greatly admire Owen's book as a reference, I find that it is written in that terse, "old school" style that characterized so many classic mathematics texts from the 1960s; the book is heavy on Theorem-Proof-Lemma-Proof, with little in the way of explanatory or motivational text to help the reader gain contextual and historical understanding along with technical proficiency. Even though I am familiar with the material, I read Owen's 9-page Chapter I on games in extensive form and scratch my head; my undergraduate students, who are far less inclined to read critically and repetitively, would never make it through this dense, challenging prose. (3) There are a handful of other game theory texts written by and for mathematicians, but they are either written for a more introductory course (Mendelson, Straffin, Stahl) or are inappropriate for other reasons.
When I finally received a copy of the book "Game Theory" by Antonia Jones, which is difficult to find on library shelves in the U.S., I discovered a text that is mathematically rigorous, extremely well written (very suitable for independent study if you have the background), and full of excellent exercises and applications. A note of caution to university instructors: Dr. Jones has an extensive "Solutions to Problems" section in the back of her text; this is perfect for the student pursuing independent study, but may force the professor to seek additional sources of problems in a traditional course where problems are collected and graded. As a guide to its level: the text assumes a familiarity with multivariable calculus, linear algebra, and elementary probability. There are occasional uses of transfinite ordinals, the fixed point theorems of Brouwer and Kakutani (used to prove the existence of Nash equilibria), and some elementary concepts from point-set topology.
I was told by a colleague that Professor Jones has now moved on to other research interests, but I sincerely wish that she could find the time to return to her wonderful game theory text and write an updated edition. Some exciting applications of game theory have emerged since the 1980 first edition of this book was written; the inclusion of these recent applications would make the text more comprehensive and self-contained, and thus suitable as the sole or primary text for an advanced course. But even without an updated edition, I strongly recommend this book to every mathematician studying game theory on his own and to professors of mathematics who are looking for a text for a first course in game theory at the senior/graduate level. With minor supplements on newer applications, Dr. Jones' text provides (by far) the single best reference that I have been able to find.