Looking for a mathematically rigorous introduction to game theory
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 non-cooperative game theory from "Game Theory" by Fudenberg and Tirole. The book is well-written 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 self-contained 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 tongue-in-cheek, and some of the mathematics is non-rigorous (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 Non-Cooperative 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 Kohlberg-Mertens. 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.