I am looking for the best book that contains a mathematically rigorous introduction to game theory. I am a group theorist who has taken a recent interest in game theory, but I'm not sure of the best place to learn about game theory from first principles. Any suggestions? Thanks!

As I note in my comment to the OP, game theory is a big field with several essentially disconnected areas, and one can't really hope for a comprehensive introduction from a single text. I'll recommend two, but this still shouldn't be thought of as a complete introduction. I learned what I know of noncooperative game theory from "Game Theory" by Fudenberg and Tirole. The book is wellwritten if terse, and covers a wide range of topics with a great deal of rigor. I would caution you that the book is written more as a reference than a gentle introduction, but it is certainly selfcontained and I was able to read the book with no previous knowledge of the theory. It is a bit dry, however. As for combinatorial game theory, I'd recommend Berlekamp, Conway, and Guy's "Winning Ways for Your Mathematical Plays," depending on your temperament. The book's style is pretty tongueincheek, and some of the mathematics is nonrigorous (though the details are easy to fill in). But it's an absolutely beautiful book. 


An excellent, rigorous introduction is provided by A Course in Game Theory by Martin Osborne and Ariel Rubinstein. They are very careful in setting up all conceptual machinnery. The book also contains some cooperative game theory at the end. The book can be downloaded freely and legally here. A book that is useful for someone having already someo basic knowledge about game theory and what it is useful for, is Foundations of NonCooperative Game Theory by Klaus Ritzberger. The book is very conceptual and contains a lot of material that is usually not available in textbooks, such as normal form information sets, index theory of Nash components, and the structure theorem of KohlbergMertens. The book is however somewhat short on motivation and is structured along conceptual lines, not pedagogy. 


Daniel Litt's suggestion of Fudenberg and Tirole is a good one. Another standard text that takes a mathematically rigorous approach is Game Theory: Analysis of Conflict by Roger Myerson. 


Game Theory: Mathematical Models of Conflict (Mathematics and Its Applications) [Hardcover] A.J. Jones 


Game Theory by Guillermo Owen. 


For a rigorous introduction to combinatorial game theory, there are two other good references. (1) AN INTRODUCTION TO CONWAY’S GAMES AND NUMBERS by DIERK SCHLEICHER AND MICHAEL STOLL www.mathe2.unibayreuth.de/stoll/papers/games12.pdf which is succinct and might be the best choice given your background. (2) The book `Lessons in Play' by Albert, Nowakowski, Wolfe, is an introduction for undergraduate math majors. 


This is probably not modern enough for you, but I'd suggest von Neumann and Morgenstern's Theory of Games and Economic Behavior. I'd be hard pressed to find anyone more rigorous than von Neumann. Plus, the text is available online. 


An Introductory Course on Mathematical Game Theory by GonzálezDíaz, GarcíaJurado and FiestrasJaneiroReviews can be found here: 1, 2, 3. Taken from the second review:



I would look for introductions to combinatorial game theory by Elwyn Berlekamp alone. Not that the mathematics is any different from other treatments, but it probably stands clearer of the "recreational" background. I'm assuming here that you want to get past the notation for partisan games, Nim as reduction and so on, to issues to do with temperature and/or what you can do with the game tree. (As a go player I do have a bias, since Elwyn is interested in go while John Conway basically isn't beyond stealing the disjunctive game concept; but that's not why I'm commenting in this fashion.) 

